ENCYCLOPEDIA OF PHYSICS EDITED BY S. FLOGGE VOLUME 111/1 PRINCIPLES OF CLASSICAL MECHANICS AND FIELD THEORY WITH 106 FIGURES SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1960 HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLOGGE BAND III/1 PRINZIPIEN DER KLASSISCHEN MECHANIK UNO FELDTHEORIE MIT 106 FIGUREN SPRINGER-VERLAG BERLIN HEIDELBERG GMBH 1960 ISBN 978-3-540-02547-4 ISBN 978-3-642-45943-6 (eBook) DOI 10.1007/978-3-642-45943-6 Alle Rechte, insbesondere das der Obersetzung in fremde Sprachen, vorbehalten. Ohne ausdriickliche Genehmigung des Verlages ist es auch nicht gestattet, dieses Buch oder Teile daraus auf photomechanischem Wege (Photokopie, Mikrokopie) zu vervielfiiltigen. ©by Springer-Verlag Berlin Heidelberg 1960 Urspriinglich erschienen bei Springer-Verlag OHG. Berlin • Giittingen • Heidelberg 1960 Softcover reprint oftbe hardcover 1st edition 1960 Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, daB solche Namen im Sinn der Warenzeichen- und MarkenschutzGesetzgebung als frei zu betrachten waren und daher von jedermann benutzt werden diirften. Contents. Page Classical Dynamics. By Professor Dr. JOHN L. SYNGE, School of Theoretical Physics, Institute for Advanced Studies, Dublin (Ireland). (With 57 Figures) A. Introduction B. Kinematics . 12 I. Displacements of rigid bodies 12 II. Kinematics . 25 III. Mass distributions and force systems . 31 IV. Generalized coordinates 38 C. Dynamics of a particle . 43 I. Equations of motion . 43 II. One-dimensional motions 46 III. Two-dimensional motions . 48 IV. Three-dimensional motions 53 D. Dynamics of systems of particles and of rigid bodies . 56 I. Equations of motion . 56 II. Systems without constraints 69 III. Rigid body with a fixed point . 82 E. General dynamical theory 98 I. Geometrical representations of dynamics 98 II. The space of events(Q T). 105 III. Momentum-energy space (PH) 130 IV. Configuration space (Q) ... 134 v. The space of states and energy ( Q T PH) 143 VI. The space of states ( Q T P) 163 VII. Phase space (PQ) 167 VIII. Small oscillations 180 F. Relativistic dynamics 198 I. Minkowskian space-time and the laws of dynamics . 198 II. Some special dynamical problems 210 III. De Broglie waves 215 IV. Relativistic catastrophes 217 General references. 223 The Classical Field Theories. By Professor Dr. C. TRUESDELL, Bloomington, Indiana and Dr. R.A. TouPIN, U.S. Naval Research Laboratory, Washington, D.C. (USA). (With 47 Figures) . . . . . . . . . . . 226 A. The field viewpoint in classical physics . 226 B. Motion and mass . . . . . . . 240 I. Deformation . . . . . . 241 a) Deformation gradients. 241 b) Strain . . . . . . . 255 c) Rotation. . . . . . 274 d) Special deformations 283 e) Small deformation 303 f) Oriented bodies . . . 309 VI Contents. II. Motion ..... . 325 a) Velocity . . . . 325 b) Material systems 337 c) Stretching and spin 347 d) Acceleration . . . 374 e) Special developments concerning vorticity 385 ei) The vorticity field . . 385 ell) Vorticity averages. . . . . . . . 396 e III) Bernoullian theorems . . . . . . 402 eiV) Convection and diffusion of vorticity. 409 f) Further special motions 430 g) Relative motion 437 III. Mass . . . . . . . . . . 464 a) Definition of mass. . . 464 b) Solution of the equation of continuity 474 c) Momentum. . . . . . . . 481 C. Singular surfaces and waves . . . . 491 I. Geometry of singular surfaces . 492 II. The motion of surfaces . . . . 498 III. Kinematics of singular surfaces 503 IV. Singular surfaces associated with a motion 506 V. Discontinuous equations of balance 525 D. Stress . . . . . . . . . . . . 530 I. The balance of momentum . . 531 II. The stress principle . . . . . 536 III. Applications of CAUCHY's laws 568 IV. General solutions of the equations of motion 582 V. Variational principles 594 E. Energy and entropy . . . . 607 I. The balance of energy 608 II. Entropy . . . . . . 615 a) The caloric equation of state 615 b) The production of entropy 638 III. Equilibrium. . . . 647 F. Charge and magnetic flux 660 I. Introduction . . . 660 II. The conservation of charge and magnetic flux . 666 III. The Maxwell-Lorentz aether relations 677 IV. Conservation of energy and momentum. 689 G. Constitutive equations . . . . . . . . . . . 700 I. Generalities . . . . . . . . . . . . . 700 II. Examples of kinematical constitutive equations 704 III. Example of an energetic constitutive equation. 709 IV. Examples of mechanical constitutive equations 710 V. Examples of thermo-mechanical constitutive equations . 734 VI. Electromagnetic constitutive equations . . 736 VII. Electromechanical constitutive equations . . . . . . . 742 List of works cited . . . . . . . . . . . . . . . . . . . . . 744 Additional Bibliography K: Kinematics of special motions (geometrical theory) 784 Additional Bibliography N: Non-relativistic kinematics and mechanics in generalized spaces ..................... . Additional Bibliography P: Principles of mechanics ... . Additional Bibliography R: Relativistic continuum theories 787 788 790 Contents. VII Appendix. Tensor Fields. By Dr. J. L. ERICKSEN, Associate Professor of Theoretical Mechanics, Mechanical Engineering Department, Johns Hopkins University, Baltimore, Md. (USA). (With 2 Figures) . . . . . . . . . . . . 794 I. Preliminaries . . . . . . . . 794 a) Notation. . . . . . . . . . . . 794 b) Use of complex co-ordinates . . . 796 II. Dimensions and physical components. 797 a) Dimensions of a tensor and its components . 797 b) Physical components . . 802 III. Double tensor fields . . . . 805 a) Definition and examples . 805 b) The total covariant derivative 810 IV. Integrals of tensor fields 813 a) Preliminaries . . . . . . . . 813 b) Circulation, flux, total, and moments 814 c) The transformation of GREEN and KELVIN 815 V. Vector fields a) Vector lines, sheets, and tubes b) Special classes of fields c) Potentials . . . . . . . . VI. Tensors of order two . . . . . a) Proper numbers and vectors b) Powers and matrix polynomials. c) Decompositions . . . . . . . d) Normal and shear components e) Tensor lines and sheets References ......... . Sachverzeichnis (Deutsch-Englisch) Subject Index (English-German) . . 817 817 819 828 829 829 837 840 844 847 850 859 881 Classical Dynamics. By J. L. SYNGE. With 57 Figures. A. Introduction. 1. Classical dynamics defined. Its applicability. For about two centuries (1700 to 1900) physicists recognized only one dynamical theory1• Now three theories exist, of which the third may be subdivided: (i) Newtonian dynamics. (ii) Relativistic dynamics (with quantum theory excluded). (iii) (a) Newtonian quantum dynamics, based on the absolute space and time of NEWTON. (b) Relativistic quantum dynamics, based on the flat space-time of MINKOWSKI or the curved space-time of EINSTEIN. The present article is confined to (i) and (ii), which are separated sharply from (iii) on the philosophical question of determinacy versus indeterminacy. But only parts of (i) and (ii) are included, statics being entirely omitted, and also the dynamics of continua (dealt with in other articles mainly in Vol. VI); in relativity only the special theory is considered, and that briefly, in view of other articles. In fact, for present purposes classical dynamics means the dynamics 2 of particles and rigid bodies, with emphasis on general theory; the essential kinematical preliminaries are included, with finite displacements, mass geometry, force systems, and generalized coordinates. As regards the applicability of classical dynamics, it may be said at once that Newtonian dynamics describes physical phenomena excellently under what may be called "ordinary circumstances", i.e. when applied to problems of engineering or to physical problems involving systems which are neither very large nor very small. Such discrepancies between theory and experiment as do occur may usually be traced to oversimplification in the mathematical model used (see 1 The present article contains only incidental historical references. For the history of dynamics, see R. DUGAS, Histoire de la Mecanique (Neuchatel: Editions du Griffon 1950) and also La Mecanique au XVIIe Siecle (same publisher, 1954). Many detailed historical references will also be found in Voss [27] and WHITTAKER [28] (see List ot General References, p. 223). 2 According to present custom, the word mechanics embraces dynamics and statics, dynamics dealing with systems in motion and statics with systems at rest. This usage disregards the literal meaning of dynamics (t5iivapL>; =force), and was deprecated by Sir WILLIAM THOMSON (Lord KELVIN) and P. G. TAIT in the following words: "Keeping in view the proprieties of language, and following the example of the most logical writers, we employ. the word Dynamics in its true sense as the science which treats of the action of force, whether it maintains relative rest, or produces acceleration of relative motion. The two correspoiioing divisions of Dynamics are thus conveniently entitled Statics and Kinetics". (Preface to Treatise on Natural Philosophy, Vol. 1, Part. 1. Cambridge: University Press 1879-) But the word kinetics did not catch on, perhaps on account of its too great resemblance to kinematics. It was, however, used in the German form (Kinetik) by GRAMMEL [8] p. 305 and WINKEl-· MANN and GRAMMEL [29] p. 373. Handbuch der Physik, Bd. III/1. 2 J. L. SYNGE: Classical Dynamics. Sect. 1. Sect. 2), such as neglect of friction in the model, or the replacement of a (physically) elastic body by a (mathematically) rigid one. Newtonian dynamics may also be used successfully in the kinetic theory of gases and in celestial mechanics (but see below). Defects in prediction appear when (i) relative speeds (u) are not small compared with the speed of light (c), or (ii) masses of atomic size are involved. Since high speeds are attainable in a laboratory only for very light particles, these two conditions are not distinct practically from one another. But we may separate them for purposes of analysis. They represent (i) the boundary where Newtonian dynamics must be replaced by relativistic dynamics, and (ii) the boundary where classical dynamics must be replaced by quantum dynamics. Errors of order (ujc) 2 appear when Newtonian dynamics is applied to bodies in rapid motion. But it is not possible to assess in any such simple way the errors committed when classical dynamics is applied to problems on the atomic scale. Although quantum dynamics uses many of the old words, the mathematical concepts corresponding to them are radically different from those of classical dynamics, and one no longer attempts to formulate atomic problems in a classical way with any feeling of confidence. However, the classical concepts are not wholly abandoned even there, the conservation of momentum and energy, for example, being employed in problems of collision, annihilation or creation of particles on the atomic or subatomic scale (cf. Sect. 120 et seq.). In celestial mechanics, Newtonian dynamics remains the standard basis for computations, and it is remarkably successful. Nevertheless certain small discrepancies between prediction and observation exist!, the most noteworthy being an excess in the rotation of the perihelion of Mercury. This is more simply explained by EINSTEIN's general theory of relativity than by special Newtonian causes introduced to explain it. One concludes that EINSTEIN's theory is the better mathematical model 2, and that Newtonian dynamics must be used with caution in very refined calculations in celestial mechanics. Newtonian dynamics may be used in cosmology3, as an alternative to the general theory of relativity or MILNE's kinematical cosmology. The nature of the subject is such that it is hardly possible to say to what extent any of the proposed theories agree with observation. However, the scientific importance of classical dynamics, Newtonian dynamics in particular, is not to be assessed solely in terms of physical predictions made directly out of it. Newtonian dynamics consists of a body of mathematical conclusions obtained by subjecting certain simple concepts to certain simple laws. In the mathematical development of the subject, general procedures are evolved (Lagrangian and Hamiltonian methods in particular) in which it is convenient to replace the original primitive concepts by more general ones (such as configuration-space and phase-space). It is found that these new mathematical concepts may be taken to represent physical concepts different from those originally envisaged, and so Newtonian dynamics gives birth to new physical conclusions by applying the mathematical ideas inherent in it outside their 1 Cf. J. CHAZY: Tht\orie de la Relativite et la mecanique celeste, Tome 1, Chap. IV, v. Paris: Gauthier-Villars 1928. - G. C. McVITTIE: General Relativity and Cosmology, Chap: V. New York: Wiley 1956. 2 WHITEHEAD's theory of gravitation, based on the special theory of relativity, gives the same rotation of perihelion as EINSTEIN's general theory of relativity [cf. A. S. EDDINGTON: Nature, Lond. 113, 192 ( 1924); J. L. SYNGE: Proc. Roy. Soc. Lond., Ser. A 211, 303 (1952)]. 3 For discussion and references, see H. BoNDI: Cosmology, pp. 7 5, 172. Cambridge: University Press 1952. Sect. 2. Mathematical maps or models. 3 original domain. Examples are the application of Lagrangian methods to electrical circuit theory, and (more striking) the application of Hamiltonian methods in the development of quantum mechan~cs. To pursue this matter a little further, it may be remarked that Newtonian dynamics sets before us for solution sets of ordinary differential equations, and we might therefore classify the subject mathematically as ODE. Hamiltonian methods introduce partial differential equations of the first order, and, when so considered, dynamics might be called PDE1 . The transition to quantum theory via the SCHRODINGER equation involves a passage to partial differential equations of the second order, with consequent classification as PDE2 • Viewed in the light of this process of mathematical development (ODE~PDEc+PDE ), Newtonian dynamics takes on a significance much greater than that indicated by its original scope; it is the parent of new theories in which the original concepts are generalized and rarified, though never entirely lost sight of. 2. Mathematical maps or models. To adapt a famous definition of geometry, physics is what physicists do. Among physicists at large, there is comparatively little inquiry into why or how they do what they are doing, and this is not to be deprecated, because human activities are inhibited by introspection. But there are occasions when a greater danger of intellectual confusion over-shadows the danger of self-analysis. How is it, we ask, that we can tolerate the co-existence of several different dynamical theories, all purporting to describe the behaviour of one single natural world? Is one true, and the others false? Or are all false? There is no doubt that these several theories exist, because men work at them. Nature also exists. The question is: How are these theories related to nature? A satisfying answer is not to be found in discussing whether or not a certain theory gives, or does not give, accurate predictions of the results of certain experiments. The question goes deeper, and it seems possible to approach an answer only by recognizing that, however much they may have been inspired by nature, mathematical theories are no more than maps or models of nature. A "particle" of the natural world (planet, atom or electron) is no more to be confused with the "particle" which represents it in dynamical theory than a city is to be confused with the ink-spot which represents it on a map. But even this analogy does not do justice to the immense gulf separating the natural world from mathematical treatments of it. For an ink-spot on a sheet of paper does at least exist in the natural world (like the city it represents), whereas the essence of the mathematical map or model exists only in the mind, even though the mathematical symbols are written down on paper; mathematic~.! openations involving infinity (differentiation and integration) are purely intellectual concepts, and belong to nature only in so far as the human mind belongs to nature. If it is admitted that mathematical models are to be sharply distinguished from nature, what, then, is their relationship to nature? The relationship seems to be based on certain concepts, the names of which provide a common language for all physicists, experimental and mathematical. These concepts appear as mathematicalconceptsin the model and as physical concepts in the direct discussion of nature. We have, as it were, a dictionary with three columns: N arne of concept Mass Mathematical concept Physical concept A positive number (m) The quantity of matter in a body. A measure of the reluctance of a body to chanpe its velocity. A measure of the capacity of a l:iody to attract another gravitationally. 1* 4 J. L. SYNGE: Classical Dynamics. Sect. 2. A sample row of entries is shown. The entries in the first two columns are complete. But the third entry is only suggestive, for the physical concept demands for its description an account of all the ways in which the idea of mass enters into our understanding of nature, and indeed no description in words can suffice since part of our appreciation of mass arises from muscular sensations and cannot be completely described. This hypothetical dictionary is used as follows. A physical problem is first formulated in terms of physical concepts. It is then translated into mathematical concepts by using the same words, now with their mathematical meanings. Mathematical laws (usually differential equations) are found by a similar translation of physical laws, first stated in terms of physical concepts. The application of these laws to the problem in question then presents a problem in pure mathematics, and, when this problem has been solved, the conclusion is translated into terms of reality by restoring to the words their physical meanings. Such a description of standard procedure in theoretical physics would have appeared ridiculously elaborate a century ago, at which time (even in geometry) there was no clear distinction between physical and mathematical concepts (even in the minds of mathematicians). This distinction is essential for pure mathematics today, since otherwise mathematical reasoning would be confused by contact with the confusions of nature. But physicists today may rightly and honestly dispute the distinction, since it may be their practice and wish to keep mathematical concepts inextricably mixed with physical concepts as a fertile source of new ideas. Clarity and fecundity of thought are not one thing. If the analysis given above is accepted, it clears up the question regardmg the co-existence of several dynamical theories. No one of these theories is true, any more than a map is a true representation of a country. And, as it is convenient to have a variety of maps (on different scales) in order to study the geography of a country, so it is convenient to have a number of maps or models of nature. It is easy to exaggerate the differences between the various models; under ordinary circumstances (cf. Sect. 1) they yield the same information. It is interesting to compare the above ideas with those of BRIDGMAN1. According to his operational method, concepts are defined in terms of physical operations; the concept is synonymous with the corresponding set of operations. Thus, for example, the concept of absolute Newtonian time is to be abandoned because it is impossible to describe experiments by which it ma:y be measured. This would mean that, in the hypothetical dictionary, the first column would read "absolute time" and the second column "a number t", but the third column would be blank, there being no corresponding physical concept. But is this, in fact, the case? There are many physicists, astronomers and engineers who use the word time to refer to a variable t which occurs in certain equations. When they have solved the equations, they pass from their formulae to physical reality, making predictions which are sometimes of very great accuracy, as in celestial mechanics. It is clear that, in their dictionaries, the third column is not blank. If it were, they could not use their results for physical prediction. True, there may be no words in the third column; the entry may lie in the subconscious, in which a great deal of our thought takes place. But an entry of some sort there must be, since otherwise a formula assigning a numerical value to a variable t could not be translated into instructions to direct a telescope in a certain definite manner. 1 P. W. BRIDGMAN: The Logic of Modern Physics, p. s. New York: MacMillan 1951. See alsoP. W. BRIDGMAN: The Nature of some of our Physical Concepts. New York: Philosophical Library 19 52. Sects. 3. 4. Newtonian and relativistic dynamics of a particle. 5 Valuable as the operational method is in clarifying our ideas, it seems that physical concepts are too complex and confused for us to demand that they should always satisfy the operational test. 3. Axiomatics. The word logic has a wide range of meanings, according to context, from the ordinary logic of daily intercourse, through the logic of the expert diagnostician or detective, to the basic logic of twentieth century mathematics1, and beyond that to the more recent developments of mathematical logic. Physical concepts, being by their nature vague, cannot be treated with logical rigour. On the other hand, classical dynamics, if regarded as a purely mathematical theory, admits of an axiomatic basis, as developed by HAMEL [10] and others 2• Therefore it would seem right that any systematic treatment of classical dynamics should start with axioms, carefully laid down, on which the whole structure would rest as a house rests on its foundations. The analogy to a house is, however, a false one. Theories are created in midair, so to speak, and develop both upward and downward. Neither process is ever completed. Upward, the ramifications can extend indefinitely; downward, the axiomatic base must be rebuilt continually as our views change as to what constitutes logical precision. Indeed, there is little promise of finality here, as we seem to be moving towards the idea that logic is a man-made thing, a game played according to rules to some extent arbitrary. To a physicist thoroughly familiar with classical dynamics, as traditionally understood, there is an element of artificiality in the creation of a complete axiomatic base, for he knows that the axioms will be chosen to fit the theory, which he believes he understands already with reasonable clarity, and that the theory will not be changed at all as a result of the axiomatics. But when such a physicist is faced by two different theories and seeks to understand wherein they agree and wherein they differ, he is forced back towards axiomatics in order to understand the agreement and the difference. It might be said that axiomatics spring to life, and promise intellectual excitement outside a restricted circle of axiomatic specialists, only when one seriously considers the creation of new theories by changing the axioms-new theories with physical significance. Therefore, although this article does not offer a treatment of classical dynamics which can be regarded as axiomatic in the modern sense, the relationship between Newtonian dynamics and relativistic dynamics is so interesting that the next two sections are devoted to a comparison between them based on a fairly axiomatic approach. 4. Newtonian and relativistic dynamics of a particle. In this section, and the next, Newtonian dynamics will be denoted by ND and relativistic dynamics by RD. The word event is common to ND and RD. The mathematical concept of an event is a set of four numbers (coordinates) or equivalently a point in a fourdimensional space-time continuum, which is in fact the totality of events. The physical concept is a sharply localized occurrence of very brief duration. 1 Any logical system, if it is to avoid vicious circles, must start with undefined terms and unproved propositions. Cf. 0. VEBLEN and J. W. YouNG: Projective Geometry, Vol. 1, p. 1. Boston: Ginn 1910. 2 For some recent work on this, and references to older work, see J. C. C. McKINSEY, A. C. SuGAR and P. SuPPES: Axiomatic foundations of classical particle mechanics. J. Rational Mech. Anal. 2, 253-272 (1953); J. C. C. McKINSEY and P. SuPPES: Transformations of systems of classical particle mechanics. J. Rational Mech. Anal. 2, 273-289 (1953); J. C. C. McKINSEY and P. SuPPES: Philosophy and the axiomatic foundations of physics. Proc. Xlth Internat. Congr. of Philosophy, Vol. VI, p. 49-54, 1953; H. RuBIN and P. SUPPES: Transformations of systems of relativistic particle mechanics. Pacif. J. Math. 4, 563-601 (1954). 6 J. L. SYNGE: Classical Dynamics. Sect. 4. Throughout the rest of this section, and indeed throughout the whole article, concepts are considered only as mathematical concepts. As explained in Sect. 2, the corresponding physical concepts are sometimes extremely complicated; we cannot reason about them with a degree of precision suited to the present occasion. For these physical concepts, the reader must consult his own private threecolumn dictionary (cf. Sect. 2); if the required entry in the third column should happen to be non-existent, he must fill it from some other source of information. In ND an event has an absolute position and an absolute time (t). The totality of all possible positions form an absolute space of three dimensions. Two absolute positions define a distance, and absolute space is Euclidean when this distance is used as metric. This implies the existence of coordinates (x, y, z) such that the element of distance d a is da = (dx 2 + dy 2 + dz2)!. (4.1) There is a 1 : 1 correspondence between all possible events and the totality of tetrads (x, y, z, t), with variables in the range - oo to + oo. In RD (only the special theory of relativity is considered here) two events define a separation. There exist coordinates (x, y, z, t), ranging from- oo to+ oo, such that the separation ds between adjacent events is1 ds = 1 dx2 + dy2 + dz2 - dt2 Jl. (4.2) There is a 1:1 correspondence between all possible events and the tetrads (x, y, z, t). The word particle is common to ND and RD. The history of a particle is a curve in space-time (world line), and may be described by equations of the form x=x(t), y=y(t), z=z(t). (4.3) In ND the derivatives of the functions in (4.3) (the components of velocity) may have any values. In RD these derivatives are bounded by ( dx )2 ( dy )2 ( dz )2 dt +Tt +Tt < 1 • (4.4) so that along the world line of a particle we have ds = (dt2 - dx2 - dy 2 - dz2)l; (4.5) this is called the element of proper time. We may use proper time as parameter on a world line, writing its equations in the form x = x(s), y = y(s), z = z(s), t = t(s), (4.6) instead of as in (4.3). These four functions satisfy the equation (4.7) The word mass is common to ND and RD (also called proper mass in RD, but the single word mass will be used here). It is a number m associated with a particle 2 ; it may be constant, or it may vary along the world line. In ND a force with components (X, Y, Z) may act on a particle. In that case the world line satisfies the equations of motion :t (m ~;)=X, !_(m~) = y dt dt ' -~-(m!:.!___) =Z dt dt . (4.8) ------ 1 A factor c2 is usually inserted before dt2 in (4.2} (cf. Sect. 107}, but we can make c = 1 by changing the unit in which t is measured. 2 In RD proper mass is often denoted by m0 , the symbol m being used forrelative mass (cf. Sect. 108). Note that, throughout this article, m means proper mass. Sect. 4. Newtonian and relativistic dynamics of a particle. 7 If (X, Y, Z) are given functions of the quantities dx dy dz m, x, y, z, t, dt' lit' dt, (4.9) we say that the particle moves in a given field of force. In that case the equations of motion, together with an equation m = m (t) (usually m = const), determine a unique world line corresponding to assigned initial values of the quantities (4.9). In RD a four-force with components (X, Y, Z, T) may act on a particle. Then th.e equations of motion are d~(m~~)=X, :s(m~~)=Y, :s(m~;)=z, :s(m::)=T.(4.10) It follows from (4.7) that these equations of motion imply dm = T !!__-X !_x__- Y !_y- Z !_!__ (4.11) ds ds ds ds ds · If (X, Y, Z, T) are given functions of the quantities listed in (4.9) we say that the particle moves in a given field of force. Then the Eqs. (4.10) determine a unique world line, and also m along it, corresponding to assigned initial values of the quantities listed in (4.9). Let us now take m = const in ND and in RD. In ND the equations of motion read (4.12) In RD, we have by (4.11) T = X!:.!__ + Y !-!__ + Z _dz_ dt dt dt ' (4.13) so that only X, Y, Z, and not T, are to be regarded as arbitrarily assigned. The last equation in (4.1 0) is implied by the first three, and, if we take t for parameter, the equations of motion (4.10) may be written where p = X _ _ n:z_ '!_y_ d_x y2 y dt dt ' Q - 2'_ - !'!_ !1'_ dY_ - y2 y dt dt ' R _ Z m dy dz -?-rTt-a:t· 1 y = v1 - v2' d2 z m---=R dt 2 ' v2 = ( ~: r + ( -t:-r + ( ~; r (4.14) (4.15) If we compare the Eqs. (4.12) for ND with (4.14) for RD, we observe only a formal change from (X, Y, Z) to (P, Q, R). However there is a difference. Suppose (X, Y, Z) independent of velocity (dxjdt, dyjdt, dzjdt), as is often the case in ND. Then (P, Q, R) depend on velocity, tending to zero as v tends to unity, i.e. as y tends to infinity. This fact, and the inequality (4.4) connected with it, distinguishes RD from ND, as far as the motion, in a given field of force, of a particle of constant mass is concerned. 8 J. L. SYNGE: Classical Dynamics. Sect. 5. A much more important difference between ND and RD emerges when we consider, not a single particle in a given field of force, but a system of particles. moving under forces which are due to the particles alone. 5. Newtonian and relativistic dynamics of a system. To fix the ideas, consider two physical concepts: (i) the solar system, (ii) a free rigid1 body. For the solar system, Newtonian dynamics (ND) sets up as mathematical model a system of P particles with constant masses m; (i = 1, 2, ... P). For the several particles we have equations of motion of the form d2 z· m.--!-=Z· • dt2 .. (i = 1, 2, ... P), (5.1) the force (X;, Y;, Z;) being given, in accordance with NEWTON's law of gravitation, by X-= G m. ~ m;(x;- x;) ) s s.L.J a J r~; = (x;- ~;)2 + ~~- Y;)2 + (z;- z;)2, (5.2) with similar expressions for Y; and Z;. The summation for f is from f = 1 to f=P, with f=f=i; G is the gravitational constant. We have, then, in (5.1), (5.2) a set of equations adequate to determine (x;, Y;, z;) (i = 1, 2, ... P) as functions of t and of the values of (5.3) for t=O. For a free rigid body, we again take a system of P particles and the equations of motion (5.1). With these we associate conditions of rigidity: (5.4) where a1; are constants, the distances between the particles. As for the forces, they are taken to be of the form X;=~X;;. lj=~Y;,., Z;='f.Z;1, 1 1 i X;;=- X1; = A;;(X;- X;). (5.6) (5.5) where Here Au(= A;;) are unknown; on eliminating them, we have in (5.1), (5.4), (5.5) (5.6) a set of equations adequate to determine (x;, Y;, z;) (i = 1, 2, ... P) as functions oft and the initial data (5.3). which must be chosen to satisfy (5.4) and the equations obtained by applying djdt to (5.4). These mathematical models of the solar system and of a rigid body are mathematically clear and physically satisfactory, in that a vast number of satisfactory physical predictions have been made by their use. But we may ask: What is the most general model of a system of particles of constant masses permissible in Newtonian dynamics? 1 This word provides a good example of that confusion regarding physical concepts (Sect. 2) which makes it difficult to treat them logically. On one occasion, a physicist might say: "I mounted the interferometer on a rigid base" (meaning, perhaps, a block of stone). On another occasion he might say: "There are no rigid bodies" (meaning that any body is deformed by su~ficiently great stress). These statements are meaningful, when taken separately; taken together, they make nonsense, and would ruin any attempt at a logical argument. In a mathematical model this sort of confusion should not be allowed to occur. Sect. s. Newtonian and relativistic dynamics of a system. 9 In attempting to answer this question, let us confine our attention to a system of P particles which is closed or isolated in the sense that all forces are due to these particles alone (and not to any external agency), and in which the particles are free in the sense that there are no rigid bonds between them. We write down 3 P equations of motion of the form (5 .1), with the understanding that the forces (Xi,¥;, Zi) depend only on the instantaneous state of the system. For simplicity, let us suppose that they depend only on positions and velocities, so that they are functions of the 6 P quantities We ask: What functions are permissible? NEWTON's Third Law1 gives a partial answer, stating essentially that the forces exerted on one another by two particles, A and B, act on the line A B, in opposite senses and with a common magnitude. This is equivalent to saying that (Xi, Y;, Z;) are of the form shown in (5.5) and (5.6), but it gives no information as to the nature of the coefficients Aii except for the symmetry condition Aii = Ai;; they might be any functions of the quantities (5.7). A more complete answer to our question is given by the following Axiom of homogeneity (i = 1, 2 , ... P) . (5 .7) v Fig. 1. Interaction in Newtonian dynamics. and isotropy of space: The set of equations determining the motion of the system has the same form for all coordinate systems (x, y, z) obtained from one such coordinate system by a translation and rotation of axes. To enlarge on this, we note that (4.1) determines the coordinate system (rectangular Cartesian coordinates) only to within an orthogonal transformation (cf. Sect. 9). The above axiom demands the invariance of the equations of motion under such orthogonal transformations, provided they be proper (no reflections included). In variance under translation implies the homogeneity of space and invariance under rotation implies isotropy. In variance under reflection in a plane (improper transformation) would imply equivalence of the two screw-senses. To explore the most general type of force system satisfying the axiom, we note that the transformation considered corresponds precisely to a rigid-body displacement. Thus the axiom is satisfied if the force system is" rigidly attached" to the instantaneous configuration of the particles. To see what this means, consider a system of four particles as in Fig.1. The positions at timet are A, B, C, D, and the velocities are the four vectors marked v. We have to assign the four vectors marked F, the forces on the particles. The axiom demands that these forces can be described in terms of the tetrahedron A BCD and the four vectors v, rigidly attached to the tetrahedron, in such a way that if these defining elements are moved rigidly together in space, the forces F are carried rigidly along with them. The idea is extremely simple, concepts of elementary Euclidean geometry replacing the formal equations. If we abandon NEWTON's Third Law, but accept l Quoted in a footnote to Sect. 26. 10 J. L. SYNGE: Classical Dynamics. Sect. 5 the axiom of homogeneity and isotropy, then any force system so constructed is permissible. If we demand invariance under reflections also, then a reflection of the defining elements in a plane must reflect the forces in that plane. NEWTON's Third Law is consistent with the axiom of homogeneity and isotropy, but it restricts the interactions to the lines joining the particles, and thus fails to cope with electrodynamic interactions other than simple CoULOMB attractions and repulsions. However, electrodynamic interactions should be treated relativistically, and, in the systematic development of Newtonian dynamics, we shall accept NEWTON's Third Law, since otherwise we could not establish the fundamental principles of linear and angular momentum (cf. Sect. 44). Fig. 2. Interaction in relativistic dynamics. We now pass to the relativistic dynamics (RD) of a system. The requirement that the separation between adjacent events should have the form (4.2) limits the class of permissible coordinates (x, y, z, t) to those obtained from one such set by a LoRENTZ transformation (cf. Sect. 106). Just as in ND we imposed an axiom of homogeneity and isotropy of space, so in RD we impose a similar axiom in space-time. For a closed or isolated system of P particles, the Axiom of homogeneity and isotropy of space-time is as follows: The equations determining the motion of the system must be invariant under a proper1 LoRENTZ transformation. A LORENTZ transf<,>rmation may be regarded as a translation and rotation of space-time, a rigid-body transformation with "rigidity" understood in terms of separation (4.2). Therefore, just as permissible Newtonian force systems can be discussed in terms of rigid constructions in space, so permissible relativistic force systems can be discussed in terms of rigid constructions in space-time. But there the analogy between ND and RD ends. In ND we have, at any (absolute) time t, a configuration of particles with attached velocity vectors as in Fig. 1, but in RD we have only a set of world lines, and there is no obvious way to set up a unique correspondence between the events on the several world lines To correlate events with the same value of t would not be a LORENTZinvariant procedure. The most natural way to correlate events on the world lines is by means of null lines. Consider a system composed of two particles with constant proper masses m1 , m2 • Let J.Ji, JVa be their world lines (Fig. 2). It is convenient to use Minkowskian coordinates x. (small Latin suffixes take the values 1, 2, 3, 4 with summation understood for a repeated suffix), writing (5.8) where i= V-1. Let A, with coordinates x,, be an event on J.Ji. Draw the null cone 2 into the past from A as vertex; let it cut JVa at B, with coordinates x;, so that BA is a null line, and we have (x,- x;) (x,- x;) = 0. (5.9) 1 A LORENTZ transformation is a linear transformation with determinant + 1 or - 1. In the former case, it is proper; in the latter case, improper (cf. Sect. 106). 2 Cf. Sect. 107. Sect. 5. Newtonian and relativistic dynamics of a system. 11 Let 1 A dx, t A A'=~_! atE (5.10) r = ds a ' r ds ' where ds, dsl are elements of proper time on Jli, ~respectively. The pair of world lines and the events A, B on them provide us with the following vectors: d ;., d ;.; x, - x; , A,, A; , -ris , H, .. .. (5 .11) Equations of the form (5 .12) for Jli, together with similar equations for ~, will constitute a suitable formulation of the two-body problem in RD, provided that X, is a vector constructed out of the vectors (5 .11) and invariants formed out of them. Thus we might take, as a fairly simple choice, satisfying this requirement, X ( I) R , , I + jl d).. d).~ r = IX X, - X, + p Ar + Y Ar U ds + e (IS' 1 where the coefficients are assigned functions of the invariants W = (xn- x~) An, W ( 1) d).n = x,.-xn dS' W 1 = (x~- xn) A~, AnA~, ) - Xn Xn ds~- d ).I An ds~' W'- ( I - ) d).~ J ~' d).n d).n d).~ "n ds ' dS ds 1 • (5 .13) (5.14) In writing down the equations for~. we use events C, D as in Fig. 2 instead of A, B and make appropriate changes. The preservation of the mass m1 imposes the condition [d. (4.11)] X,A,= o, or ex w - p + y A, A; + d, ~:; = o; the preservation of m2 imposes a similar condition. (5.15) (5 .16) Although the equations as in (5 .12) satisfy the condition of LoRENT-z-invariance, they present a mathematical problem far more complicated than what we meet in ND. They appear to be differential equations, but they involve the two events A, B, and have the nature of difference equations on account of the "retarded" effects due to B. The commonly accepted equations of motion for a pair of particles carrying electric charges e1 , e2 are as in (5.12); the coefficients in (5.13) are given by1 (5 .17) 4 2 - el e2 !!!_ nc e- w12 , the other coefficients being zero. 1 The charges are measured in HEAVISIDE rational units; to change to Gaussian electrostatic units, delete the factor 4n. The factor c is the speed of light. For the derivation of these formulae, see W. PAULI, Relativitatstheorie, Encykl. d. Math. Wiss. V 19, p. 645. Leipzig und Berlin: Teubner 1920, where, however, there is a misprint in the last line-for (X,u') 3 read (X,u') 2. Cf. al~o J. L. SYNGE: Relativity: The Special Theory, pp. 394, 423. Amsterdam: North-Holland 1956. 12 J. L. SYNGE: Classical Dynamics. Sect. 6. To summarize, for a single particle in a given field of force, we have a set of three differential equations to solve, whether the dynamics be Newtonian or relativistic. But, for a system of interacting particles, the differential equations of Newtonian dynamics are replaced in relativity by differential-difference equations; these present such great mathematical difficulties that only certain limiting cases can be handled by approximate methods 1. B. Kinematics. I. Displacements of rigid bodies. 6. Displacements parallel to a plane. The position of a rigid body is determined by the position of any plane section of it, and so the displacements of a rigid body which are parallel to a fixed plane may be discussed in terms of the displacements of a lamina in a plane. Such a displacement may be described by stating that two points of the lamina, initially at A and B, are moved to positions A' and B'. This displacement may be broken down into a sequence of two displacements: (i) a translation carrying A to A', and (ii) a rotation about A'. Or we may start with a rotation and follow it by a translation. It is convenient to use complex numbers (z = x + i y) to describe displacements in a plane. A translation, represented by a complex number t, causes a transformation z' =z+t. For a rotation through an angle{) about a point c, we have z'- c = (z - c) eiif. If we use T and R as symbols for the operations of translation and rotation respectively, and denote by R T and T R (read from right to left) the combined operations, then we have the following transformations for the two orders of the operations: R T: z' = c + (z + t- c) eiD, } T R: z' = t + c + (z - c) eiD. (6.1) In general these transformations are different (R T =1= T R), and this is perhaps the simplest example of two operations in mechanics which do not commute. We have RT= TR only in the exceptional cases where t=O or {)=O. Consider the first of (6.1). To find a fixed point for the displacement R T, we are to put z' =z; this gives for z the equation z (1 - eiD) = c + (t- c) eiD. (6.2) Unless{)= 0 (or a multiple of 2n) this equation has a unique solution. Therefore every rigid plane displacement (except a pure translation) has a unique fixed point. Equivalently, any rigid plane displacement (except a translation) can be produced by a rotation about a suitably chosen centre. This centre can be found (and indeed the above theorem proved) by means of a simple construction. Let the displacement send A to B and B to C. Then unless the displacement is a pure translation, the right bisectors of A B and B C meet at some point D; D is the required centre. The resultant of two plane displacements {D2D1) is therefore the resultant of two rotations {R2R1), and this resultant is itself a rotation (R3); we may write (6.3) 1 Cf. C. G. DARWIN: Phil. Mag. (6) 39, 537 (1920). - J. L. SYNGE: Proc. Roy. Soc. Lond., Ser. A 177, 118 (1940). Sect. 7. EuLER's theorem. 13 Two rotations about different centres do not commute (R2R1 -{=-R1R2). It is easy to construct the two resultants. Let A1 , A 2 be the centres and {)1 , {)2 the angles of rotation. Then the angles of R2R1 and R1R2 are the same, viz. 2cp=ff1 +ff2 , and their centres are C21 and C12 as in Fig. 3; these centres are reflections of one another in the line A1 A 2 • A lamina moving in a plane (or a rigid body moving parallel to a plane) has three degrees of freedom, since the position of the lamina is fixed when we know the two coordinates of any point of it and the inclination to a fixed direction of any line in it. Thus the configuration-space is of three dimensions (Sect. 62). It has the same type of connectivity as an infinite cylinder, there being one independent irreducible circuit, corresponding to a complete rotation of the lamina (Sect. 63); and it is flat with respect to the kinematical line element (Sect. 84). 7. EuLER's theorem. Consider a rigid body with a l?t ( fixed point 0. Draw a 4~ '7----t--------+----4> 4 spherical surface S with unit radius and centre 0. The position of the body is fixed by the positions of those points of it which lie on S, and any displacement of the body which leaves 0 fixed is a rigid transformation of S into itself. Fig. 3. Resultant of two rotations in a plane. C12 is the centre for R1 R2 and C21 the centre for R2 R1 • EULER's theorem: Any rigid displacement of a spherical surface into itself leaves two points of the surface fixed, these points being diametrically opposed to one another. This theorem can be proved (and the fixed points found) by the construction given after (6.2), the straight right bisectors in the plane being replaced by great circles on the sphere. The exceptional case (pure translation in the plane) cannot arise in the case of a sphere, for two great circles must intersect!. EuLER's theorem may be expressed by saying that a rotation about a point is a rotation about some line through that point. This property (fixed point implies fixed line) is due to the oddness of the dimensionality of space. The composition of two rotations about a point can be effected as in Fig. 3, with the straight lines replaced by great circles on the sphere. The points A1 and A2 are now points where the sphere is cut by the two axes of rotation with A1 on one axis and A 2 on the other. The only essential difference is that although the angle of the resultant rotation is still 2 cp, where cp is the angle so marked in Fig. 3, we now have (7.1) E being the spherical excess of the triangle A1A2C12 . It is clear that finite rotations about a point do not commute (R2R1 -{=-R1R2), unless they are about a common line 2• :For a proof of EuLER's theorem based on the fact that a real orthogonal matrix has one eigenvalue equal to ± 1, see GoLDSTEIN [7], pp. 118-123. 2 For a number of interesting special results on finite rotations and fuller demonstrations of the matters dealt with here, see LAMB [14], Chap. I. 14 J. L. SYNGE: Classical Dynamics. Sect. 8. A rotation is specified by three parameters, e.g. the angle of the rotation and two of the direction cosines of its axis. Therefore a rigid body with a fixed point has three degrees of freedom, the same number as for a lamina moving in a plane. But in spite of the analogies between these two systems, there is a great algebraic difference. In the plane case we can use complex numbers very simply as in (6.1), but much more complicated methods are required to deal with finite rotations about a point, as will be seen in the following sections. The configuration-space (Sect. 62) has three dimensions in both cases and its connectivity is similar (one independent irreducible circuit, Sect. 63); but, whereas the configuration-space of a lamina is flat with respect to the kinematical line element (Sect. 84), that of a body with a fixed point is curved. 8. General rigid body displacements. Let D denote any displacement ·of a rigid body. Suppose that D carries a point of the body from a position A to a position A'. Then D can be accomplished in two steps: (i) a translation A A', and (ii) a rotation about A'. This is the standard way of describing a general displacement, the point A of the body being called the base-point. A set of orthogonal triads originally parallel to one another is changed by D into a set of orthogonal triads parallel to one another. Hence it is clear that the rotation (i.e. the direction of its axis and the angle of rotation) is independent of the choice of base-point. But the translation depends on the choice of base point. CHASLES' theorem: Any rigid body displacement is equivalent to a screw. A screw is defined as the resultant of a rotation and a translation parallel to the axis of rotation; it is easy to see that for a screw the translation and the rotation commute. To prove CHASLES' theorem, we start with any base-point and break down the translation into two translations, one (say ~) parallel to the axis of rotation and the other (say T2) perpendicular to that axis. Then T2 and the rotation are plane displacements with a common plane, and may be compounded into a single rotation with a new axis parallel to the old one (cf. Sect. 6). This rotation and ~ together form a screw. This proves CHASLES' theorem. A general rigid body displacement can be reproduced by two half-turns, a half-turn being a rotation about a line through two right angles (equivalently, reflection in a line). The axes of the two half-turns cut perpendicularly the axis of the equivalent screw; their distance apart is half the translation of the screw; and the angle between them is half the angle of rotation of the screw1 . A free rigid body has six degrees of freedom, three for the translation and three for the rotation. Its 6-dimensional configuration-space has one independent irreducible circuit, and is curved with respect to the kinematical line element (Sect. 84). We have observed the analogy between a plane displacement (translation + rotation) and a displacement of a 3-dimensional rigid body with a fixed point (rotation). There is a similar analogy between the general displacement of a rigid body (translation + rotation) and a displacement of a 4-dimensional rigid body with a fixed point (rotation). We do not meet 4-dimensional rigid bodies in Newtonian physics, but, in the special theory of relativity, LoRENTZ transformations with fixed origin [put B, = 0 in (107.5)] may be regard as 4-dimensional rotations, modified on account of the indefinite metric of space-time. 1 For further details and other properties of finite displacements, see LAMB [14], Chap. I. Sect. 9. Orthogonal matrices. 15 9. Orthogonal matrices. The following notation wiU be used for matrices: M is the transpose of M. Mt is the complex conjugate of M. 1 is the unit matrix. M is orthogonal if MM = 1 (equivalently, M-1 = 1lf), and proper or improper according as det M = 1 or - 1. M is unitary if MMt = 1 (equivalently, M-1 =Mt). M is symmetric if M = M. M is Hermitian if Mt = M. r is the column matrix (x, y, z). Let (I, J, K) be an orthogonal triad of unit vectors (orthonormal triad) with origin at 0. Let a rigid body be given a rotation about 0, and, as a result, let (I, J, K), regarded as fixed in the body, be carried into the orthonormal triad (i,j, k). There exists, then, a matrix of scalar products (or relative direction cosines): M = (;:: ; :~ ; ::) . (9.1) K·i K-j K-k Using orthogonal projections and introducing the column matrices of vectors (9.2) we may exhibit the connection between the two triads in the following forms: (9-3) Let i have direction cosines (l1, m1, n1) relative to (I, J, K), with a similar notation for j and k. Then ( l1 l 2 l3 ) M= m1 m2 m3 , n1 n2 na (9.4) From the orthonormality of the triads we have six conditions, of which the first two are [2 [2 [2 l l l ( ) 1+ 2+ a=1, 1m1+ 2m2+ ama=O, 9.5 and it follows that M is an orthogonal matrix: (9.6) Hence det M = ± 1. If the triad is not moved at all, we have M =1, det M =1; therefore, by continuity, M is proper for all rotations. An improper orthogonal matrix corresponds to a rotation combined with a reflection in the origin. We shall confine the argument to rotations, although some of the formulae are applicable to improper orthogonal transformations. So far we have avoided the use of coordinates. But now let OXY Z be axes fixed in space, and let (I, J, K) coincide with them, so that/= ( 1, 0, 0), J = (0, 1, 0), K= (0, 0, 1). Consider any point of the body. Let its coordinates be (x, y, z) before the rotation and (x', y', z') after it. Then the initial and final position 16 J. L. SYNGE: Classical Dynamics. vectors of the point are respectively r=xi+yJ+zK,} r'=xi+yj +zk, so that x' = r' · I= l1 x + l 2 y + l3 z, etc. , and the coordinate transformation may be expressed in the matrix forms r'=Mr, Sect.<). (9.7) (9.8) (9.9) the second following from the first by (9.6). Note the linearity of the transformation, a most important fact. Further, note the formal interchange of M and Min the comparison of (9.3) and (9.9). For a given rotation, the matrix M depends on the choice of the triad (I, J, K), or, equivalently, on the choice of the axes OXY Z. By choosing K along the axis of rotation, we can simplify M to the form ( cosx -sinx M= sinx cosx 0 0 where X is the angle of the rotation. (9.10) When we compound a sequence of rotations, we do not, as a rule, keep the axes OXYZ fixed once for all. We may follow the scheme T--o-~--->- T2 --• · ·---Tn 1 ---+ t, (9.11) M, M, M, Mn-1 - Mn which shows, under each passage from triad to triad, the corresponding matrix as in (9.1) or (9.4). The resultant rotation is then, as in (9.3), (9.12) To get the corresponding coordinate transformation, using fixed axes coincident with T, we must, as in the transition from (9.3) to (9.9), replace M by its transpose, obtaining r'=Mr, (9.13) Note that the matrices are written down here in their natural order, so that the operations are actually applied in the reverse of that order. The difference between (9.12) and (9.13) (interchange of M and M) can be a source of such petty confusion. And there is yet a third way of looking at the rotation. We may hold a point fixed in space, and consider its coordinates (x, y, z) relative to the old triad T and its coordinates (x', y', z') relative to the new triad t. Then the transformation is r'=Mr, (9.14) which is similar to (9.12). To clarify the situation at this stage, we may sum up by saying that a proper orthogonal 3 X 3 matrix M can be interpreted in the following four consistent ways: (i) A table of scalar products (9.1) of the old triad (I, J, K) and the new triad (i,j, k). (ii) Rotation of an orthonormal triad, T~t with f=MT. (iii) Coordinate transformation with the axes fixed in space and the point carried with the body, r~r' with r' =Mr. (iv) Coordinate transformation with the point fixed in space and the axes carried with the body, r~r' with r' =Mr. Sect. 10. Rotation in terms o£ its axis and angle (Eulerian parameters). 17 10. Rotation in terms of its axis and angle (Eulerian parameters). An ordered orthogonal triad of vectors (I, J, K) has two possible orientations, right-handed and left-handed. At a point on the earth's surface, we get a right-handed triad by taking I horizontal and pointing to the East, J horizontal and pointing to the North, and K directed vertically upward. In this article, all triads, including triads of coordinate axes, will be chosen right-handed, in accordance with current practice, unless specially noted otherwise. The positive sense of rotation about K is the sense of the 90°-rotation which carries I into J. Any rotation about an axis lying on a unit vector U can be described by a symbol [U, x], where x is an angle of rotation, counted positive in the positive sense just defined. The correspondence between rotations V and such representations is, however, multiple valued. For, if R denotes any rotation about U, we may write symbolically R = [ U, X + 2 n :n:] = [- U, - X + 2 n :n:] ( 10.1) n being any integer. We reduce the multiplicity of the correspondence by introducing a vector V and a scalar e as follows: V=Usinix. e=costx. (10.2) Let the components of V on any axes be (A, p,, v); then (10.2) imply (10.3) It is easy to see that any set of values (A, p,, v, e) satisfying (10.3) determine a unique rotation, but that to a given rotation there correspond two sets of values of these quantities. We may write symbolically o p R ={A, p,, v, e} ={-A,- p,,- v,- e}. (10.4) Fig. 4. Axis . V and angle of rotation X· The quantities (A, p,, v, e) are EULER's parameters1. They serve to describe the configurations of a rigid body with a fixed point. For we can pass from ·any given initial configuration C0 to a final configuration C by a definite rotation R, and R is determined by (A, p,, v, e). In view of (10.3) and (10.4), we may make the following statements, if we regard (A, ft, V, e) as rectangular Cartesian coordinates of a point in Euclidean 4-space: (i) Any point on the hypersphere (10.3) determines a final configuration of the body. (ii) Any final configuration of the body determines a pair of diametrically opposed points on the hypersphere ( 10.3). (iii) There is a continuous 1 : 1 correspondence between the final configurations of the body and the straight lines drawn through the origin in a space of four dimensions. Since (A, p,, v, e) describe a rotation about a fixed point, the matrix M of Sect. 9 is expressible in terms of them: this is done as follows. Let P(r) and P'(r') (Fig. 4) be the initial and final positions of a point of a body which experiences the rotation ( 1 0.2). Let N be the common foot of the 1 The notation follows F. D. MuRNAGHAN: The Theory of Group Representations, p. 328. Baltimore: Johns Hopkins Press 1938. - WHITTAKER [28], p. 8, uses (~. TJ. C. z). Handbuch der Physik, Bd. III/1. 2 18 J. L. SYNGE: Classical Dynamics. Sect 11. perpendiculars dropped from P and P' on V; let NP - = p, and let s be a unit vector such that (p, s, V) form a righthanded orthogonal triad. Then But --~ ------)- ~ r' = 0 N + N P' = 0 N + p cos X + s p sin X. p=r-ON, - and so by ( 1 0.2) Now r' = r cos X+ ON (1-~ x) + sinx (Vxr)fV l = r(e2 - V2) + 2 V20N + 2e(Vxr). V20N=V(V·r), - and so the transformation r-+r' is r' = r(e2 - V2) + 2V(V. r) + 2e(Vxr), or in matrix form r' = Mr, where ( A.2-,u2-v2+e2 2(A.,u-ve) 2(vA.+,ue)) M= 2(A.,u+ve) ,u2-v2-A.2+e2 2(,uv-A.e) , 2(vA.-,ue) 2(,uv+A.e) v2-A.2-,u2+e2 A., fl, v being the components of V, so that ( 1 0.5) (1 0.6) (10.7) ( 10.8) (10.9) (10.10) Note that M is unchanged if we replace (A., ,u, v, e) by (-A., - ,u, - v, -e), as of course must be the case. 11. Eulerian angles 1• Let a rotation about 0 carry the orthonormal triad (I, J, K) into (i,j, k). We break this rotation into three rotations (Fig. 5). First, rotate about K so as to make the new position of the plane (I, K) contain k, say through an angle ([!; this gives a transformation l I 1 = I cos ([! + J sin ([!, ) (I, J, K) -+ (I1 , J 1 , K1) ~ = - I sin ([! + J cos ([!, K1 =K. (11.1) Secondly, rotate about J 1 to bring K1 to k, say through an angle{}; this gives a transformation l I 2 = I 1 cos{} - K1 sin{}, ) (Il, Jl, K1)-+ (I2, J2, k) J2 =J1, k = I 1 sin {} + K1 cos {}. (11.2) 1 The notation used here follows WHITTAKER [28], p. 9, and has the advantage that (D, q;) are the usual polar angles of the vector k. For alternative notations, cf. H. TIETZ, this Encyclopedia, Vol. II, p. 135; APPELL [2] II, p. 151 (he interchanges rp and 1p); GOLDSTEIN [7], p. 107 (he discusses various usages). Sect. 11. Eulerian angles. 19 Finally, rotate about k to bring I 2 to i and J 2 toj, say through an angle tp; this gives the transformation ~ i = I 2 cos "P + J 2 sin "P,) (I2, J 2, k)- (i,j, k) j = -I2 sin tp + J 2 costp, {11.3) k=k. The angles ({}, q;, tp) are the Eulerian angles. Their values determine the position of the triad (i,j, k) relative to (I, J, K). They may be given any values whatever, but all positions of (i,j, k) are obtained by letting them cover the following ranges: 0 :;;;. {} :;;;:. 17:, ) 0:;;;. q; < 21t, Os;;tp<21t. ( 11.4) From the above equations of transformation we can express (i,j, k) as linear functions of (I, J, K), and hence obtain a matrix M of scalar products as I~---+--..,.-~kj in (9.1) or direction cosines as in (9.4). This matrix M is exhibited compactly as follows in an abridged notation in which c =cos, s =sin, and the subscripts 1, 2, 3 refer to {}, q;, tp Fig. s. Eulerian angles 8, p 0 ') I 0 e'''~' sin j{} cos j{} 0 e~i>p =(cos j{} e- ~i('l'+'l'l -sin j{} e- ~i(tp->p)) sinj{}eAi(tp->p) cosi{}e!i(q;+>p) · J (15.2) In terms of the stereographic parameters, we have, by (13.9) T=( p q), -q p (15.3) and so the transformation ( 15.1) may be written ( z' x' +iy' x' -, i y') = ( p q) ( z . x - i y) (P - ~,) ' z ,- q p x + zy z q p (15.4) PP+qq=1. The unitary character of T may be verified immediately. In terms of the Eulerian parameters, we have, by (13.9), = ( (! - iv - iA - fl). T . , . ' - z 11 + p, (! + zv (15.5) or ( 15 .6) 1 The matrix T differs from the corresponding matrix given by GoLDSTEIN [7], p. 116, which has a factor i in two elements. This is because our rotation ( 11.2) is about the y-axis, whereas in GoLDSTEIN's definition of the Eulerian angles his second rotation is abont the x-axis. As a result 111 appears in his work where 112 appears here. Sects. 16, 17. Frames of reference. Velocity of a particle. 25 so that the transformation (15.4) may be written ~'a +y'a +z'tJ = (-iA.tJ1 -if1-tJ2-iva3 +e1) X } x (xa1 + ya2+za3) (iA.a1 +ifta2+iva3 +e1), A.2+!-l2+v2+ e2=1. (15-7) This is equivalent to the quatemionic formula (12.6): x'i + y'i +z' k = (A.i +!-li +vk+ e) (xi+ Yi +zk) (- Ai -!-li -vk+ e),} ( 8) A.2+!-l2+v2+e2=1. 15. 16. Infinitesimal displacements. An infinitesimal displacement of a rigid body is reducible to an infinitesimal translation and an infinitesimal rotation. On account of their infinitesimal character (all differentials of order higher than the first being neglected), the translation and the rotation commute. Indeed, any two infinitesimal displacements commute, and this makes the discussion of infinitesimal displacements comparatively simple. To discuss an infinitesimal rotation, we take the rotation-angle X in (10.2) to be infinitesimal, so that V = -!x and e = 1. Let X be an infinitesimal vector with magnitude z, lying on the directed axis of rotation. Then X= 2 V, and ( 1 0.8) gives, for the infinitesimal displacement resulting from the rotation x. 1'1 -1'= xxr. ( 16.1) The vector character of this equation tells us that we can compound infinitesimal rotations by adding the corresponding vectors X according to the parallelogram law. In matrix form (16.1) reads r'-r=Mr, (16.2) where z1 , z2 , Xa are the components of X· The matrix is skew-symmetric; it is easy to prove directly that any orthogonal matrix which differs infinitesimally from the unit matrix, differs from it by a skew-symmetric matrix. II. Kinematics. 17. Frames of reference. Velocity of a particle. Let S0 be absolute fixed space (cf. Sect. 4), and let S be any rigid body, fixed or moving. S constitutes a frame of reference; if 0 x y z are any rectangular axes fixed in S, then to any event there corresponds a tetrad of numbers (x, y, z, t), t being the absolute Newtonian time (cf. Sect. 4). Kinematics deals only with relative motions, and for kinematical purposes S is as good as S0 ; it is only in dynamics proper (i.e. when forces producing motion are considered) that the distinction between S and S0 appears. Let P be a moving point or particle. Its velocity relative to S is the vector . (. . ") V='f'= x,y,z, ( 17.1) the dot indicating ·differentiation with respect to t. The velocity is tangent to the trajectory of P and of magnitudes, where sis arc-length on the trajectory. The magnitude of the velocity is called speed. Let xe be curvilinear coordinates in S; then the line-element of S is of the form ds2 =gea dxl1 dxa, suffixes taking the values 1, 2, 3, with summation understood for a repeated suffix. The contravariant and covariant velocity vectors 26 J. L. SYNGE: Classical Dynamics. Sect. 18 are respectively VII= · · ( ) XII, V11 = g110 X 6 • 17.2 The physical component of velocity in the direction of a unit vector with contravariant components All (and covariant components A11 ) is the orthogonal proj.ection of (17.1) on that direction, i.e. the invariant! v(A)=v11 AII=VIIA11 • (17.3) In cylindrical coordinates (r, f(J, z), the physical components of velocity along the coordinate lines are (r, r9-;, z). (17.4) In spherical polar coordinates (r, {}, f{J), the physical components of velocity are Fig. 7. (p, r) coordinates. (r, rD, r sin {}9-;). (17.5) In (p, r) coordinates in a plane (Fig. 7), the components v, along the radius vector and v .L perpendicular to it are p;. v,=r, V_L=- . (17.6) Vr2- p2 18. Acceleration of a particle. Hodograph. The acceleration a of a point or particle is the rate of change of velocity: a= v = r = (x,y,z). (18.1) Resolution along i, the unit tangent vector to the trajectory, and j, the unit vector along its principal normal, gives • • v2 • d v • v2 • a=vt+-J=v-t+-J, e ds e (18.2) where v is the speed and e is the radius of curvature of the trajectory. For curvilinear coordinates xe with ds2 = g110 dx11 dx0 as in Sect. 17, the contravariant acceleration vector is 2 all= XII+{:.} xPx•, where the CHRISTOFFEL symbol is defined by ( 18.3) { e} =gear .. 'JI a] 2r"'JI a]= ogpa + og.a- ogl'. ge"ga,=f5ae· (18.4) pv lP' ' ' lP' ' ox• oxl' ox0 ' r Here 6~ is the KRONECKER delta (=1 if (!=a, =0 if e=J=a). It is usually easier to calculate the covariant acceleration vector: a ar ar a=-~---- e dt oXe axe ' (18.5) The physical component 3 of acceleration in the direction of a unit vector All is (18.6) 1 If the coordinates are orthogonal (g110=0 for e=t= 11), this definition is satisfactory. But when the coordinates are oblique, it is sometimes convenient to define physical components by i:>blique resolution, and care is needed to avoid confusion of ideas; cf. C. TRUESDELL: Z. angew. Math. Mech. 33, 345 (1953); 34, 69 (1954). 2 Cf. A. J. McCoNNELL: Applications of the Absolute Differential Calculus, Chap. 1 7. London and Glasgow: Blackie 1931; I. S. SoKOLNIKOFF: Tensor Analysis, Chap. 4. New York: Wiley 1951; J. L. SYNGE and A. ScHILD: Tensor Calculus, Chap. 5. Toronto: University of Toronto Press 1952. 3 See footnote in Sect. 17 with reference to oblique coordinates. Sect. 19. Angular velocity of a rigid body. 27 For cylindrical coordinates (r, rp, z) we have d s2 = dr2 + r 2 d rp2 + d z2 , ( 18. 7) and the physical components of acceleration along the coordinate lines are a,= r- rcp2 , a = _!_ _!__ (r2m) 'P r dt r a,=z. ( 18.8) For spherical polar coordinates (r, {), rp) we have ds2 = dr2 + r2 d{)2 + r2 sin2 {) drp2 , (18.9) and the physical components of acceleration along the coordinate lines are1 a,= r- r0.2 - r sin2 f)cp2 ' l a{}= : :t (r20.) - r sin{) cos{) g;2, (18.10) - 1 d ( 2 . 2{) ') a'P- rsin{} dt r sm rp · The curve with equation r' = v (t), where v (t) is the velocity of a moving point, is called the hodograph of the motion. For constant velocity, the hodograph is a single point; for constant acceleration, it is a straight line; for motion with a=;= krjr3 (inverse square law), it is a circle 2• 19. Angular velocity of a rigid body. Let S0 be a rigid body relative to which a second rigid body S moves. For simplicity of description we may think of S0 as fixed in absolute space. Having selected a particle P0 of S as base-point, we can describe any infinitesimal displacement of S by giving the displacement of P0 and the rotation about P0 • The latter is an infinitesimal vector x. as in (16.1). We define the angular velocity w of S by the equation x= wdt, ( 19.1) dt being the infinitesimal time-interval during which the displacement takes place. The velocity v of any particle P of S is then v =v0 + w X r, ( 19.2) where v 0 is the velocity of P0 and r the position vector of P relative to P0 • The vectors v 0 , wand r may be described by giving their components at the instant t along any orthonormal triad T, which may be fixed in 50 , or in S, or may be moving relative to both S and S0 • It is usually most convenient to fix Tin S, but in cases of symmetry it may be better to fix one vector of T in S and confine another to a plane fixed in S0 , as we shall see later (Sect. 56). Since v 0 is separated from w in ( 19.2), we can discuss the angular velocity of S as if P0 were fixed. To express w in terms of the Eulerian angles (Sect. 11), we may take T to be the triad (i,j, k) of Fig. 5, fixed in S. The displacement in time dt can be produced by infinitesimal rotations d{), drp, d1p about J 1 , K, k, respectively, and so wdt = ~d{) + Kdrp + kd1p. ( 19.3) 1 These and other accelerations may also be calculated by means of moving axes; cf. WHITTAKER [28], p. 18, where a number of special coordinate systems are discussed. 2 Cf. MAcMILLAN [17] I, p. 285. 28 J. L. SYNGE: Classical Dynamics. On resolving~' K, k along (i,j, k), we obtain w = w1 i + w2 j + w3 k, I w1 = {). sin tp - cp sin {} cos tp, w2 = {). cos tp + cp sin {} sin tp , W3 = ti; + cp cos {}. Similarly we can resolve on (I, J, K), fixed in S0 (Fig. 5), obtaining w = Q1J + Q2J + !l3K, !11 = -{J.sinrp+ti;sin{}cosrp, !12 ={).cos rp + ti; sin{} sin rp, Q3 = cp + tiJ cos{}. Sect. 19. ( 19.4) ( 19. 5) To discuss angular velocity in terms of quaternions and the Eulerian parameters, we let the quaternionic units (i, f, k) correspond to axes fixed in S0 • The position r of a particle at time t is given, as in (12.6), by r=qroq-1, where r 0 = x0 i + y0 1. + z0 k, the initial position of the particle, and From (19.6) we have We note that and that q=l.i+!-lf+vk+e, l q-1=-J..i-f-lf-vk+e, ).2 + #2 + v2 + 122 = 1 . so that these products are vectors. If ( 19.6) ( 19.7) (19.8) (19.10) (19.11) is the angular velocity, resolved along the fixed axes, then, by (19.2) with v0 =0, ; = f(!lr- r!l). (19.12) Comparing this with (19.8), we have ar- ra = 0, where (19.13) (19.14) Since a is a vector and ran arbitrary vector, (19.13) implies a=O, and therefore the angular velocity is (19.15) Sect. 20. Moving axes. Absolute and relative rates of change of a vector. 29 evaluating this product by (19.9), we find the components of angular velocity, relative to the fixed axes (i, f, k): By (19.15} we have D1 = 2(~e- A(! -~iv + 1-li?, l D2 = 2(/J,e- 1-le -VA+ vA), !la = 2 (v e -vi!- il-l + A,U). q-1Qq = 2q-1q = w1 i + w2 f + w3 k, w1 , w2 , w3 being defined by this equation; hence Q = w1 qiq-1 + w2 qf q-1 + w3 qkq-1 , (19.16) (19.17) (19.18) and we recognize w1 , w2 , w3 as the components of angular velocity on that triad which initially coincided with (i, f, k). Carrying out the calculation in (19.17), we have the components of angular velocity on the moving axes1 : wl=2(~e-Ae+f1.v-f-tv), l w2 = 2 {it e - 1-l e + _v A - v ~) ' Wa= 2(ve- vi!+ Af-t- A/J,) 0 (19.19) 20. Moving axes. Absolute and relative rates of change of a vector. The theory of moving axes is often found difficult and confusing on account of the demands it makes on one's power to visualize bodies in motion. If one gets confused, the best plan is to think in terms of infinitesimal displacements, resolving the actual displacement which occurs in time dt into a set of elementary displacements, each due to a different cause. There is no question of the order in which these causes are applied, because infinitesimal displacements commute with one another. For brevity, such an analysis into infinitesimal causes is omitted in the derivation of the formulae which follow. Let (i,j, k) be an orthonormal triad, rotating with angular velocity w. Let V be a vector, with components (l';,, ~. Va) on the rotating triad, so that V= l';,i + ~j + "Vak. (20.1) The rates of change of the vectors (i,j, k) are the same whether the origin of the triad is fixed or moving; they are determined solely by the angular velocity ro. If the origin is fixed, these rates of change are the velocities of points with position vectors (i,j, k), and so, by (19.2) we have di • dt=WXt, dj • dk k dt = ro XJ, -(j( = ro X . Differentiating (20.1}, we see that the absolute rate of change 2 of Vis dV dV Iii-= Tt + w XV, where ~ = ~1_ • + _cl__lJ_ • + ~1._ k. dt dt 1. dt J dt ' this is the relative rate of change of V. 1 See WHITTAKER [28], p. 16, for an alternative derivation 2 WHITTAKER [28], p. 17, calls it the time-flux. (20.2} (20.3} (20.4} 30 J. L. SYNGE: Classical Dynamics. Sect. 20. If, in particular, V = w, we have dw /Jw dt=dt' (20.5) the absolute and relative rates agreeing. For the absolute second derivative we have d2V IJ2V IJV /Jw ~=~+ 2WX M+dt X V+w X (w XV). (20.6) We shall apply these formulae to the calculation of velocity and acceleration. Let S0 be fixed absolutely. Let S be a rigid body in general motion, ~arrying an orthonormal triad (i,j, k), the origin 0 of this triad having a position vector T 0 (t) relative to some origin in S0 • Let P_ be any moving point or particle, not necessarily attached to S; its absolute position vector '1', relative to the origin in S0 , may be written 'l'='l'o+'l'', where r' =xi+ yj + zk, - (20.7) (20.8) this last being in fact the vector OP; (x, y, z) are the coordinates of P as judged by an observer attached to S, which is, in fact, a moving frame of reference. By (20.3) the absolute velocity of P is df' df'0 + df'' + , , V=dt=dt dt=V0 V +wX'I', (20.9) where v 0 =absolute velocity of 0, v' = ~:' =relative velocity of P, as observed by an observer carried by S; its components are (%, y, z), w =angular velocity of S. If P is attached to S, then v' = 0; for this reason the remaining part (v0 + w X r') is called the velocity of transport. Differentiation of (20.9) gives the absolute acceleration of P in the form where a = a0 + a' + a,+ a,, a0 = dd~o = absolute acceleration of 0, /j2 , a'= IJ; =xi+ y j + z k = relative acceleration, a, = 2 w x v' = CoRIO LIS (or complementary) acceleration, a, = w x .,., + w X (w X '1'') = w X .,., + w (w · r') - r' w2 • Here w=dwfdt=!JwjM. If P is attached to S, then a' = ac = 0 and a= a0 + a1; this part of (20.10) is called the acceleration of transport. (20.10) (20.11) (20.12) (20.13) Sect. 21. Mass-centres Moments and products of inertia. 31 If w is constant, then (20.14) where R is the vector drawn perpendicularly from the axis of w to P; this may be called centripetal acceleration. III. Mass distributions and force systems. 21. Mass-centres. Moments and products of inertia. The mass-centre of a system of particles with masses m, and position vectors r; is the point 2.: m;T; i r =-.,..,--. J:.,m; If the system is rigidly displaced, its mass-centre is carried rigidly with it. (21.1) In a uniform gravitational field, the gravitational forces acting on a system of particles are statically equivalent (or equipollent) to a single force acting at the mass-centre. For that reason, the mass-centre is commonly called the centre of gravity. The word barycentre is also used. In this article the term mass-centre will be used throughout. For a continuum of density e the mass-centre is defined as in (21.1), with summations replaced by integrations: r = f/:d~._ (21.2) This formula applies to distributions of mass over volumes, surfaces, or curves, d r being respectively an element of volume, surface, or length, and e being the appropriate density. If a system S consists of n parts 5; (i = 1, 2, ... , n) with masses m; and masscentres r,, then the mass-centre of S may be found by replacing each part 5; by a particle of mass m; at r,, and using the formula (21.1). The linear moment of a particle with respect to the origin is the vector mr, and its quadratic moment with respect to the axes of coordinates is the matrix or tensor ( mx2 myx mzx mxy my2 mzy mxz) myz . mz2 (21. 3) The linear and quadratic moments of a system are given by summation or integration. Moments and products of inertia are closely connected with the quadratic mome:1t. The moment of inertia of a particle P of mass m about a line L is mp2, where p is the perpendicular distance of P from L. Its product of inertia with respect to a pair of perpendicular planes is mpq, where p, q are the distances of P from the planes, taken with appropriate signs. The moments and products of inertia of systems are found by summation or integration1. Thus, for a discrete system of particles, the moments of inertia with respect to the coordinate axes Oxyz are A = l: m; (Y7 + z~), B = ~ m, (z~ + x~) , } c = 2: m; (xi + y~), (21.4) 1 If a system of total mass m has a moment of inertia I about a line L, then the radius of gyration k about L is defined by mk2 =I. )2 J. L. SYNGE: Classical Dynamics. Sect. 22. and the products of inertia with respect to the coordinate planes are F=L,m,y,z,, G=L,m;Z;X;, H=L,m,x,y,. (21.5) j i i If V = (l, m, n) is a unit vector through the origin, then the moment of inertia about V is, by the definition, I= 1:m,p: = 1:m,!T;XV! 2 ) i i = A l2 + B m2 + C n2- 2F m n - 2 G n l - 2H l m. (21.6) Since I is a quantity independent of the choice of directions of coordinate axes, the elements of the symmetric inertia matrix · ( A -H -G) 1= -H B -F -G -F C (21.7) are the components of a tensor of the second order. For a continuous distribution A=fe(Y2+z2)d-r, F=feyzd-r, B=Je(z2+x2)d-r, G=fezxd-r, c = J e(x2+ y2) d-r,} (21.8) H=fexyd-r. 22. Theorem of parallel axes. Principal axes of inertia. Let L be any line and L0 a parallel line through the mass-centre of a system. The theorem of parallel axes states that (22.1) where I, I 0 are the moments of inertia about L, L0 , respectively, m is the total mass of the system, and p the perpendicular distance between L and L0 • This is easy to prove, as is also a similar theorem for products of inertia. From (22.1) it follows that the moment of inertia about a line L0 passing through the mass-centre is less than that about any parallel line L. Principal axes are defined in terms of stationary values of the moment of inertia. To each unit vector V drawn through the origin (which may be any point) there corresponds a moment of inertia I, as in (21.6). The direction cosines (l, m, n) of a vector V for which I has a stationary value satisfy Al-Hm-Gn=Il,) -Hl+Bm-Fn=Im, -Gl-Fm+Cn=In, (22.2) where I is the stationary value. The three principal moments of inertia at the origin are these stationary values, which are the roots of the cubic determinantal equation A-I -H -G -H B-I -F =0. (22.)) -G -F C-I The three roots are real and positive: real because the matrix (21.7) is symmetric, and positive because the form (21.6) is positive-definite1. 1 Except in the case when all the mass lies on a single line through the origin; then it is semi-positive-definite, and one root is zero, while the other two are equal and positive. Sect. 23. Linear momentum. 33 The directions defined by (22.2), in which I has any one of the three values determined by (22.3), are the principal axes of inertia at the origin; these axes form an orthogonal triad 1, and the three planes determined by it are the principal planes. Products of inertia with respect to principal planes vanish, and for that reason (since it greatly simplifies the work) one uses, almost always, axes which coincide with the principal axes of inertia. Then the inertial properties are described by the three principal moments of inertia, A, B, C, and the moment of inertia about any line through the origin with direction cosines (l, m, n) is given by (22.4) For the geometrical representation of inertial properties one uses the momenta! ellipsoid 2 with equation A x2 + B y2 + C z2 - 2F y z - 2 G z x - 2 H x y = 1. (22. 5) The moment of inertia about any line L drawn through the origin is 1/r2, where r is the radius vector of this ellipsoid, drawn in the direction of L. Two mass distributions are equimomental if they have the same moment of inertia about any arbitrary line. It follows that two equimomental systems have the same mass-centre, the same total mass, the same principal axes of inertia at the mass-centre, and the same principal moments of inertia there. The converse is also true. A uniform triangular plate of mass m is equimomental to a set of three particles, each of mass m/3. placed at the middle points of the sides. A uniform solid tetrahedron of mass m is equimomental to a set of five particles, one of mass 4m/5 placed at the mass-centre, and the other four, each of mass m/20, placed at the vertices3. 23. Linear momentum. The linear momentum of a particle is mv, where m is the mass and v the velocity. The linear momentum M of a system is the sum of the linear momenta of its parts: M=Lmivi, M=fevd-r:. i (23.1) All the theory with which we are here concerned is applicable both to discrete systems of particles and to continuous systems, so that we have two types of formulae, one involving summation and the other integration. As the passage from the one to the other is obvious, only the summation will be shown in general. The velocity v, of any particle of a system may be written Vi=V +v;, (23.2) where v is the velocity of the mass-centre and v~ the velocity relative to the masscentre. By (21.1) we have (23-3) 1 If two of the principal moments of inertia are equal, only one member of the triad is determined, and the triad may be completed by taking any orthogonal pair orthogonal to the determined member; this is the case of axial symmetry. If all three principal moments of inertia are equal, any orthogonal triad is principal; this is the case of spherical symmetry. 2 For further details about the momental ellipsoid and the ellipsoid of gyration (the polar reciprocal of the momental ellipsoid with respect to its centre), and for the theory of moments of inertia generally, see RouTH [22] I, pp.16-22; see also AMES and MURNAGHAN [1], pp. 191-196, APPELL [2] II, pp. 1-17, and }UNG [12]. 3 Cf. ROUTH [22] I, pp. 22-27. Handbuch der Physik, Bd. 111/1. 3 34 J. L. SYNGE: Classical Dynamics. Sect. 24 where r; is the position vector relative to the mass-centre; hence (23.4) and so, by (23.1) and (23.2), the linear momentum of the system is M=mv, (23.5) where m is the total mass of the system. The linear momentum of any system is the same as that of a single fictitious particle, with mass equal to the total mass of the system, moving with its mass-centre. 24. Angular momentum. The angular momentum1 of a particle about a point 0 is h = rxM = rxmv = m (rxv), (24.1) where m is the mass, v the absolute velocity 2, and r the position vector relative to 0. It is, in fact, the moment of the linear momentum. The three components read v 0 Fig. 8. Angular momentum. h,. _ m(yv.- zv1), ) h1 - m(zv,.- xv.), h,=m(xv1 - yv,.). (24.2) The (scalar) angular momentum about a directed line through 0 is the orthogonal projection of h on that line. Thus, if V is a unit vector on the line, the scalar moment is v. (rxM) =1'· (Mx V) =M· (Vxr). (24.3) The angular momentum of a system is the sum of the angular momenta of its parts: h = L 1';Xm;V; = L m,(r,xv,). (24.4) i i The point 0 with respect to which angular momentum is calculated may be fixed or moving. To investigate the effect of changing from one such point to another, consider two points 0, 0* with absolute velocities v, v*, and let - 00* =1' (Fig. 8). Let there be any system of particles, a typical particle Po having position vectors r,, r; relative to 0, 0*, respectively, so that r, = r + r~. Then the angular momenta about 0, 0* respectively, are h = ~ m;r,xv,, i h* = ~ m,r;xv;, i (24.5) (24.6) 1 The old term moment of momentum is obsolete. Following APPELL [2] II, p. 157. one might call it kinetic moment (moment cinetique). However, the term angular momentum is now standard in English practice. But it is hard to find a suitable symbol for it, avoiding all possible confusion with symbols having accepted meanings. Following WHITTAKER'S notation [28], p. 60, the symbol his used in this article; since quantum mechanics is excluded, there can be no confusion with PLANCK's constant h. But in other contexts, a different symbol is advisable. GOLDSTEIN [7], p. 379. uses L, in spite of the standard use of L for the Lagrangian function. APPELL (loc. cit.) and P:ERi:s [20], p. 14, use a. 2 We are still dealing with kinematics and the absolute space of NEWTON is not involved. Absolute velocity here means velocity relative to some frame S0 which is used throughout the argument, and is spoken of as fixed. Sect. 25. Kinetic energy. 35 where V; is the absolute velocity of P;. Let v; be the velocity of P; relative to 0*, so that V; = v* + v;; then h = 1.: m; (r + r;) X (v* + v;), h* = 1.: m; r; X (v* + v;), i and hence h = mrxv* + rxl: m;v; + h*, where m is the total mass of the system. (24.7) (24.8) (24.9) This formula is particularly useful when 0* is the mass-centre. For then the middle term disappears, and we have h = h0 + h*, where h0 = rxmv*, h* = 1.: r;x (m;v;); {24.10) (24.11) h0 may be called the orbital angular momentum and h* the spin angular momentum, to borrow the terms of quantum mechanics. We note that h0 is the angular momentum about 0 of a fictitious particle of mass m, moving with the masscentre, and h* is the angular momentum of the system about the mass-centre; in computing h* it is actually a matter of indifference whether we use the velocities relative to the mass-centre, as in (24.11), or the absolute velocities. It is well to emphasise that, in any computation of angular momentum, we need to specify (i) a frame of reference relative to which velocities are measured, and (ii) a point about which moments are taken. For a rigid body turning about a fixed point 0 with angular velocity w, the angular momentum about 0 is h = 1.: m;r;XV; = 1.: m;r;x (wxr,) =w 1.: m.,rz- 1.: m;r;(w · r;). (24.12) Resolution of this vector equation on any orthogonal triad gives the components h1 = A w1 - H w2 - G w3 , ) h2 : -Hw1+ Bw2 -Fw3 , h3 -- Gw1 - Fw2 + Cw3 , (24.13) where (w1 , w2 , w3) are the components of wand A, B, C,F, G, Hare the moments and products of inertia relative to the triad. If the triad is principal, then these equations simplify to (24.14) To secure this simplicity for all time, it is, in general, necessary to fix the triad in the body. But in a case of symmetry (A= B), a triad with one member along the axis of symmetry suffices. Then the triad is fixed neither in space nor in the body; note that in (24.14) the components of angular velocity are those of the body, not the triad. 25. Kinetic energy. The kinetic energy of a particle is T = !mv2, where m is the mass and v the absolute velocity; for a system of particles the kinetic energy is T = 1.: -! m; vz = ! 1.: m; V; • V;. (25.1) i 3* J. L. SYNGE: Classical Dynamics. Sect. 25. Let v be the absolute velocity of the mass-centre of a system, v, the absolute velocity of a particle of the system, and v~ its velocity relative to the mass-centre. Then v,=v+v~. and (25.1) gives T -.! 2+ ~ 1 1 2 - 2 m v L.. 2 m, v, , (25.2) i since L m,v~ = 0; here m is the total mass. This is the theorem of KoNIG: the i kinetic energy of any system is the sum of two parts: ( i) the absolute kinetic energy of a fictitious particle of mass m moving with the mass-centre, and ( ii) the kinetic energy of the motion relative to the mass-centre. For a rigid body turning about a fixed point with angular velocity w, the kinetic energy is T=!L:m,(wxr,) 2 l i (25.3) =! (Aw~ + Bw~ + Cw~- 2Fw2 w3 - 2Gw3 w1 - 2H w1 w2), where A, B, C, F, G, Hare the moments and products of inertia. When principal axes are used, this reduces to T = ! (A w~ + B w~ + C w~) . (25 .4) When the positions of the principal axes are described by the Eulerian angles (Sect. 11), by (19.4) the kinetic energy is T = ! A (0 sin VJ- cP sin{} COSVJ)2 +! B (iJ. cos VJ + cP sin{} sin VJ) 2 + } (25.5) + !C('IjJ +ci;cos{))2. For rotation about a fixed axis, whether principal or not, we have (25.6) where I is the moment of inertia about the axis. In terms of the symmetric inertia matrix I,. of (21.7), the kinetic energy is T= T(w) = !I,.w,w., (25.7) with summation over 1, 2, 3 understood for repeated subscripts here and below, and by (24.13) we have h, = I,.w., T =! h,w,. In terms of the reciprocal matrix],., satisfying I,.],,=tJ.,, we have w,=],.h., T=T(h) =i],.h,h •. Therefore 8T(w) = h 8w, '' For principal axes we have the formulae T= ~ (Aw~ + Bw: + Cw~) I = _!__ (hf + n: + hi) 2 A B C = + (hlwl + h2w2 + h3w3). J (25.8) (25.9) (25.10) (25.11) Sect. 26. Force systems. 37 26. Force systems. Consider forces F; (i = 1, 2, ... , P) acting on P particles with position vectors r; relative to a base-point 0. The force F; is regarded as made up of two parts: an external force F;' and an internal force F;", the latter being the resultant of reactions exerted on the particle i by the other particles of the system. We accept, as an hypothesis or axiom, NEWTON's Third Law1 : this states that the force exerted by particle i on particle f is equal and opposite to the force exerted by particle f on particle i, and that the two forces lie on the line joining the particles. This law, commonly called the law of Action and Reaction, is equivalently expressed by writing p F;"= 2: A;i(r1 - r;) (i = 1, 2, ... , P), i~l where Aii are scalar factors satisfying A;i=Ai;· (26.1) Now, for any system of forces (.F; acting at r;), the total force F and the total moment (or torque) G about the base-point 0 are respectively p G = l:r;XF;. (26.2) The given system of forces is said to be equipollent 2 to a single force F acting at 0 and a couple G. Note that F is a bound vector and G a free vector. If we change the basepoint 0, F remains unchanged but G changes; by suitable choice of 0 we can reduce the force system to a wrench, i.e. we can make G = pF, where p is a scalar factor, called the pitch of the wrench. The vector pair (F, G) is called a motor 3 or a torsor 4 . It does not change if we slide the vectors F; along their lines of action, and hence it is easily seen that the internal forces F;" contribute nothing to it. Therefore the total force and the total moment are given by P P F=l:F;', G=l:r;x.F;', (26.3) i~l i~l in which formulae only the external forces occur. This elimination of internal forces is of fundamental importance in Newtonian dynamics. The work done by5 a force Fin a displacement br of its point of application is F · br, and for a system of forces the work is bW = 2: F; · br;. (26.4) ; If the forces act on a rigid body, and this body is given an infinitesimal displacement consisting of an infinitesimal translation br0 and an infinitesimal rotation bx about a base-point, then .ll .ll .ll , (26.S) ur; = uro + uxxr;, 1 "To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts". Sir IsAAC NEWTON's Mathematical Principles of Natural Philisophy and his System of the World, MoTTE's translation revised by F. CAJORI, p. 13. Berkeley: University of California Press 1946. Cf. Sect. 5 of the present article for a more general law, consistent with the homogeneity and isotropy of space. 2 The word equivalent is often used, but it is misleading, since there is equivalence only if the particles on which the forces act form a rigid body. 3 For references to motor symbolism, see Sect. 49. 4 Cf. PERES [20], p. 9. 5 The work done against the force is - F · (j '1'. J. L. SYNGE: Classical Dynamics. Sect. 27· where r~ is the position vector relative to the base-point. The work done in this displacement is dW = F · dr0 + i~~~ • (dx_xr~) } = F · dr0 + G · dx_, in the notation of (26.2). (26.6) Given a plane II passing through a point 0, any infinitesimal rotation about 0 may be resolved into two infinitesimal rotations, one (d1 x_} about an axis perpendicular to II and the other (!52 x.) about an axis in II. If a couple is represented by a pair of equal opposite forces (magnitude P, distance apart p) in the plane II, then the work done by it in an infinitesimal rotation is ± p P d1x, the sign depending on whether the senses of the couple and the rotation are the same or opposite. If two equal and opposite forces F, -F, act at points r, r', then the work done in arbitrary infinitesimal displacements is dW = F · (dr- dr'). (26.7) If, further, the forces act along the line joining the points, we may write F = {} (r-r'), where {} is some scalar factor, and (26. 7) becomes (26.8) This vanishes if the displacements are such as to keep the distance I r-r'l unchanged. Since by NEWTON's Third Law the reactions between any pair of particles of a system satisfy the above conditions on F, it follows that no work is done by the reactions in a rigid body. Other cases of workless reactions are (a) reactions at smooth contacts, (b) reactions at rolling contacts (with no sliding). All such reactions disappear from those general equations of dynamics which are based on energy and work. IV. Generalized coordinates. 27. Holonomic systems. Moving constraints. A configuration of a system composed of P particles may always be described by giving the 3 P coordinates of the particles, but a smaller number suffices if these 3 P coordinates are connected by equations of constraint. Thus, if the system is rigid and has a fixed point, three coordinates suffice (e.g. the Eulerian angles of Sect. 11). Any set of parameters whic!l completely determine the configuration of a system are called generalized coordinates; their rates of change are generalized velocities. Suppose that N generalized coordinates qe (and no smaller number) describe the configurations of system, and that it is possible to vary qe arbitrarily and independently without violating the constraints (such as rigidity); then we say that the system is holonomic, with N degrees of freedom. The following are examples of holonomic systems, with the numbers of degrees of freedom indicated: rigid body with fixed point (3). free rigid body (6), rigid body moving parallel to a plane (3), rigid body in contact with a fixed plane (5). A system may be subject to moving constraints (e.g. a particle may be constrained to move on a surface which itself moves in some prescribed manner). Then, to describe a configuration, we need the time t as well as the generalized Sect. 28. Non-holonomic systems. 39 coordinates q11 , and the system is holonomic if arbitrary independent variations of q11 and t do not violate the constraints. A system with fixed constraints is scleronomic; with moving constraints, rheonomic. If (x;, y,, z1) are the coordinates of the i-th particle of a system, relative to fixed axes 0 xyz, the selection of generalized coordinates sets up equations of the form X0 = x.(q1 , q2 , ... qN, t), Yi = y.(q1 , q2 , ... qN, t). Z; = Z; (q1 , q2 , .. • qN, t), (27.1) t being absent for a scleronomic system1• The kinetic energy of a rheonomic system is T = 2 1 L; ~ m; ( X;+ '2 y, '2 + Z; '2) ) i=1 N N = i 2;1aeaq/Ja + L aeqe +a; 11 ,a=1 e=1 (27.2) here a11 a, a11 and a are functions of q1 , ... qN, t. For a scleronomic system this reduces to N T = 1"' . . ( ) 2 £.., aoaqpqa, 27.3 p,a=l a positive-definite 2 quadratic form by virtue of the definition of T, the coefficients a9 a being function of q1 , ... qN. 28. Non-holonomic systems. Consider a rigid lamina which can slide freely over a fixed plane; this is a scleronomic holonomic system with three degrees of freedom. But now suppose that a small sharp blade is fixed in the lamina, the blade being capable of motion only in the direction of its length. If (x, y) are the Cartesian coordinates of the blade and {} its inclination to the x-axis, then (x, y, {}) form a system of generalized coordinates for the lamina, but they are subject to the non-integrable relation {~=tan-D. (28.1} The number of generalized coordinates cannot be reduced below three, but these three coordinates are not freely variable. We call a system non-holonomic when it is impossible to describe the configurations by generalized coordinates q11 (e = 1, 2, ... N) and the time t, with q11 and t freely and independently variable. In such cases there are present certain non-integrable 3 equations of constraint of the form N L Ac 11 dq0 + Acdt = 0, (c = 1, 2, ... M); e=l (28.2) 1 But even for a system without moving constraints it may be convenient to use equations of the form (27.1). For example, to study the motion of a free rigid body (e.g. a rocket) relative to the earth (the motion of the latter being known), we might let q1 , q2 , •.. q6 describe the configuration of the body relative to axes fixed in the earth; then the equations giving the coordinates of the particles of the body relative to fixed axes will be of the form (27.1), the time t entering through _the earth's motion. From an analytic standpoint, it is sometimes convenient to use the word scleronomic when t is absent from (27.1) and rheonomic when it is present, without regard to the physical system under consideration. 2 T > 0 unless q1 = · · · = qN = 0. 3 PERES [20], p. 218, calls a system semi-holonomic if the equations of constraint are integrable, with constants of integration dependent on initial conditions. Mathematically, a semi-holonomic system is not very different from a holonomic system, since the integrated equations of constraint may be used to reduce the number of generalized coordinates. 40 J. L. SYNGE: Classical Dynamics. Sect. 28. here Ace and Ac are functions of the variables (q, t). The system is said to have N- M degrees of freedom. The formulae (27.2) and (27.3) for kinetic energy are valid for non-holonomic systems. Non-holonomicity usually occurs in systems with rolling contacts, the condition of rolling being the equality of the instantaneous velocities of the two particles at the point of contact, one particle belonging to each body1 • The following examples illustrate non-holonomic constraints due to rolling. oc) Sphere rolling on a horizontal plane. Let (I, J, K) be a fixed orthonormal triad with origin 0 in the given plane and K vertical (Fig. 9). As generalized coordinates we take (x, y, {}, cp, VJ) where (x, y) are the coordinates of the point of contact relative to axes 0 xy drawn along I and J, and ({}, cp, VJ) are the Eulerian angles (X,y) Fig. 9. Sphere rolling on plane. (Sect. 11) which describe the position of an orthonormal triad (i,j, k), fixed in the sphere, relative to (I, J, K). Relative to the centre C of the sphere, the position vector of the point of contact is- bK, where b is the radius of the sphere. Therefore, by ( 19.2), the velocity of that particle of the sphere which is instantaneously in contact with the plane is xi+yJ+wx(-bK), (28.3) where w is the angular velocity of the sphere. Resolving the angular velocity along (I, J, K), we have w =!J1I +D2J -T-!J3K, (28.4) where the coefficients are expressed in terms of the Eulerian angles and their rates of change as in (19.5). Substituting {28.4) in (28.3), and equating the result to zero (condition of rolling), we get the following two non-integrable equations of constraint: x- b (~cos cp + ,P sin{} sin q;) = 0, } y- b (D sin cp - ,P sin{} cos q;) = 0. The sphere has 5 - 2 = 3 degrees of freedom. (28.5) {3) Circular disc rolling on a horizontal plane 2• Let (I, J, K) be a fixed orthonormal triad with origin 0 in the given plane and K vertical (Fig. 10). With these as axes, the position vector of the centre C of the disc is r=xi +yJ +zK. (28.6) Let (i,j, k) by a second orthonormal triad centred at C, i being directed to the point of contact, j being horizontal, and k perpendicular to the plane of the disc. 1 It is by no means always desirable to use generalized coordinates in the discussion of non-holonomic systems; RouTH [22] II, pp. 165-205, applies direct methods with great elegance. For an extensive treatment of non-holonomic systems, with problems worked out in detail and consideration of non-linear constraints, see HAMEL [11], pp. 464-507. See also WINKELMANN and GRAMMEL [29], pp. 434-440. For the dynamics of non-holonomic systems in the present article, see Sects. 46, 48, 85. 2 For the dynamics of a rolling disc, see APPELL [2] II, pp. 253-258. Sect. 28. Non-holonomic systems. 41 Let {} be the inclination of - i to K, cp the inclination to I of the tangent to the disc at the point ofcontact, and 1p the inclination to i of a radius fixed in the disc. Then (if b = radius of disc) z = b cos{}, (28.7) and (x, y, {}, cp, 1p) form a system of generalized coordinates. z K The velocity of C is v=xi+yJ-bsin{}0-K, (28.8) and the angular velocity of the disc is w =-Jj + rj;K + ipk, (28.9) as we see on increasing each of the angles in turn. Hence, by ( 19.2), the velocity of the particle of the disc instantaneously .x I in contact with the plane is Fig. 10. Circular disc rolling on plane. v + w X b i =xI+ y J- b sin{} J K + b J k + b ( ip + rp sin{}) j, (28.1 0) smce We have K = - cos{} i + sin {} k. j =cos cpi +sin cpJ, k =cos{} sin cp I- cos{} cos cpJ +sin{} K, (28.11) } (28.12) and when these are substituted in (28.10), the condition of rolling gives the following two non-integrable equations of constraint: x + b cos {} sin cp J. + b cos cp ( ip + sin {} rp) = 0 , } y- b cos{} cos cp J. + b sin cp (ip +sin{} rp) = 0. The disc has 5 - 2 = 3 degrees of freedom. y) Axle with pair of wheels which roll on a plane. We suppose that each wheel is free to turn about the axle. (If they were both fixed to the axle we would have a holonomic .x system with one degree of freedom.) Let b be the radius of each wheel and 2 c the length of the axle. We take generalized coordinates (x, y, cp, 1p, 1p') (28.13) as shown in Fig. 11. Then, computing the components of velocity, parallel and perpendicular to the axle, of the two Fig. II. Axle with pair of wheels which roll on a plane. particles instantaneously in contact with the plane, and then equating these components to zero to satisfy the condition of rolling, we get the following three 42 J. L. SYNGE: Classical Dynamics. equations of constraint: x cos q; + y sin q; = 0 , ) - x sin q; + y cos q; + c rp + b ip = 0 , - x sin q; + y cos q; - c rp + b ip' = 0 . The last two yield the integrable combination 2 c rp + b ( ip - ip') = 0. If we are given initial values of the three angles, we have 2 c q; = A - b ( 1p - 1p'), Sect. 29· (28.14) (28.15) (28.16) where A is a given constant; we may then take (x, y, 1p, 1p') as generalized coordinates, with the two non-integrable equations of constraint: x cos q; + y sin q; = 0 , } - x sin q; + y cos q; + t b ( ip + ip') = 0. (28.17) The system has 4- 2 = 2 degrees of freedom. 29. Generalized forces. Work. Potential function. Consider a system of P particles, in general rheonomic and non-holonomic. Let forces with components (X;*,¥;*, Zt) act on the particles with coordinates (x;, Y;. z;). Then, in any completely arbitrary set of infinitesimal displacements of the particles, all constraints being disregarded, the work done by these forces is p t5 W = L (X£ h; + Y;* t5y; + Z;* t5z;). (29.1) i=l Now if (qe, t) describe the configurations of the system, we have equations of the form (27.1), and so the work done in the displacement corresponding to arbitrary variations (t5qg, t5t) is N t5 w = L Q: t5qe + Q* t5t, (29.2) (>=1 where p Q* = L (x~ ox; + Y* oy; + Z* ~z;_) e i=l • oqg • oqe • oqe ' (29.3) p Q* = '\' (x* ox; + Y* oy• + Z* ~zi_) LJ • ot • ot • ot . i=l (29.4) The quantities Q: are generalized forces; they are usually easier to calculate from (29.2) than from (29.3). The total force (X;*, Y;*, Z;*) acting on a particle of the system can in general be split into two parts: (X;, Y;, Z;) = given force 1, such as gravity, } (X;, Y/, z;) =force of constraint. Then the total generalized force Q: can be split into two parts: Q:= Qe+ Q~. 1 Also called applied force. (29.5) (29.6) Sect. 30. Basic equations. 43 so that Qe is the given generalized force and Q~ the generalized force of constraint; we suppose Qe to be known functions of q1 , ... qN, t, ql> ... qN, and possibly of ij1 , ... ij N also. The importance of this splitting is due to the fact that in many dynamical sys~ems the forces of constraint are workless, by which we mean that these forces do no work in a displacement ~qe which satisfies the instantaneous constraint (with ~t=O). This implies N L: Q;~qe = 0, e=l so that the work done by all forces in this displacement is N ~W= L Qe~qe. e=l If there exists a function V(q1 , •.• qN, t) such that oV Qe=- oqe' so that by (29.8) ~W=-~V, (29.7) (29.8) (29.9) (29.10) then Vis the potential function 1, or potential energy, of the system. More generally, if there exists a function V(q1 , ••• qN, t, q1 , ... qN) such that (29.11) then V is an extended potential function. C. Dynamics of a particle. I. Equations of motion. 30. Basic equations. NEWTON's first two laws are embodied in the equation d dt (mv) = F, (30.1) where m is the mass of a particle, v its absolute velocity, and F the force acting on it. If m is constant (as we shall assume it to be), then (30.1) is equivalent to ma=F, (30.2) where a is the absolute acceleration. Let xe be curvilinear coordinates in absolute space (cf. Sects. 17, 18). Then, by (18.3), the equation of motion (30.2) may be written in the contravariant form · (30.3) Here Ffl is the contravariant forcevector; the covariant force vector~ may be calculated from the invariant formula (30.4) 1 French writers prefer to speak of a force function U, where U = - V. 44 J. L. SYNGE: Classical Dynamics. Sect. 31. where (J W is the work done by the force in an arbitrary displacement (Jxe, and P is obtained from ~ by the formula pe = geP F,_.. (30.5) By (18.5) the covariant form of the equation of motion is1 d aT aT ma =-------=F e dt axe axe P' (30.6) where T = fmge"xvx", (30.7) the kinetic energy of the particle. If a potential function 2 V(x, t) exists [cf. {29.9)], such that (30.8) or, more generally, an extended potential function 2 V(x, x, t) such that [ cf. {29.11)] {30.9) then {30.6) may be written {30.10) where L is the Lagrangian function L= T-V. (30.11) For motion in a plane, the most convenient coordinates are usually either rectangular Cartesians (x, y), for which the equations of motion are mx=X, my=Y, (30.12) or polar coordinates (r, if), for which they read m(r-r0.2)=R, m+-ft(r2 0.)=8, (30.13) where R is the radial component of force and 8 the transverse component. In space, we may conveniently use cylindrical coordinates (18.8) or spherical polars (18.10). 31. Energy. Angular momentum. From (30.2) we deduce dT aT= F. v, (31.1) so that the ra,te of increase of kinetic energy is equal to the rate of working (or activity or power) of the force F. If there exists a potential function V(x) (independent oft), then {31.1) gives the energy integral E being the total energy. Also, by (30.2) we have or T + V = E = const, rxma=rx F, dh ---= G dt ' 1 These are Lagrangian equations, as in ( 46.1 7). (31.2) {31-3) (31.4) 2 Here and later, X stands for the three quantities xe, and i for the three quantities xe. Sect. 32. Moving frames of reference. 45 where h is the angular momentum about the origin 0 and G the moment of F about 0. If the line of action ofF passes through 0, we have G=O and h = const. (31.5) This is the integral of angular momentum. In the case of a particle moving in a plane under the action of a force directed to, or from, the origin, it gives r2 J. = const , (31.6) from which it follows that the radius vector drawn from the origin to the particle traces out equal areas in equal times. It is useful to remember that (31.7) where v is the speed and p the perpendicular dropped from the origin on the tangent to the orbit. 32. Moving frames of reference. Let S be a rigid body which has a prescribed motion relative to absolute space. This motion is described by selecting some base point 0 in S, giving the absolute velocity v 0 (t) of 0, and also giving the angular velocity w of S. We take S as a moving frame of reference. Then the absolute acceleration a of a particle, acted on by a force F, may be broken down as in (20.10}, and the equation of motion (30.2) may be written m a' = F + F0 + F, + Fe, where m is the mass of the particle and F0 =-ma0 F, = - mac= , -2m w X v', l Fe=- ma1= -mwXT' -mwx (<.oXT') =-mwXT'- mw (w · 1'') + mw21''. (32.1) (32.2) In (32.1) a' is the relative acceleration, i.e. the acceleration as observed by an observer carried along with S, and we may say that, relative to a moving frame of reference, NEWTON's law of motion holds in the form ma'=F', (32.3) where F' is made up of the real force F and the three fictitious forces Fo, F,, Fe; here F0 arises solely from the acceleration of the base point (and is present even if S is not rotating) ; F, is the CoRIOLIS force, a force of great importance in meteorology and i:ndeed in all phenomena involving the rotation of the earth; Fe is of the nature of centrifugal force, although that term is properly applied only when w is constant, in which case we have [ cf. (20.14) J (32.4) If the motion of Sis a uniform velocity, we have a0 =w=0, and (32.1) gives ma'=F, (32.5) which means that such a frame is a Newtonian frame. This result is known as Newtonian relativity; st3.!1:ing with an absolute space S0 and NEWTON's law of motion relative to S0 , we find that this law holds also for any frame S in uniform motion. 46 J. L. SYNGE: Classical Dynamics. Sect. 33. II. One-dimensional motions. 33. Simple harmonic oscillator. Damping. A simple harmonic oscillator is a particle which moves on a line under the influence of a restoring force directed to a point 0 on the line and proportional to the distance from 0. If a frictional damping force, proportional to the velocity and opposing it, is added, and also a disturbing force, the equation of motion reads x + 2px + P2 x =X (33.1) where restoring force = - m p2 x, l damping force= - 2m px, disturbing force = m X, (33.2) m being the mass of the particle. For the undamped undisturbed oscillator, the motion is given by where x+p2 x=O, } x = a cos (p t + e), a = amplitude\ p 2 :n; • d' t' = peno 1c une, __p_ = v = frequency, 2:n: p = 2nv = w =circular frequency, p t + e = fP = phase angle, e =phase constant. (33.3) The oscillator is said to be "in the same phase" for two values of fP differing by a multiple of 2n. The solution (33.3) may also be written in complex form x=AeiPt, (33.4) where A is a complex amplitude, which includes the phase constant; the physical displacement xis the real part of (33.4). If X= 0, the general solution of the damped Eq. (33.1) is, in complex form, x = A e"'' + B e"•', where n1 , n2 are the roots of (33.5) (33.6) These roots may real or complex, giving non-oscillatory and oscillatory motions respectivelys. When damping and disturbing forces are both present, the latter being a given function oft, the general solution of (33.1) is I x =A e"•'+Be"•' + - 1-jX(-r){e"•(t-Tl- e"•(t-Tl}d-r. (33.7) "t- nz 0 1 Sometimes 2a is called the amplitude, and a the semi-amplitude. 1 The damping constant p. being positive, the real part of each root is necessarily negative; for details, see SYNGE and GRIFFITH [26], pp. 163-166. Sect. 34. Circular and cycloidal pendulums. 47 For a sinusoidal disturbing force, we write X(t) = X 0 eiqt; (3 J.8) then (33-7) becomes (33-9) where A' and B' are new arbitrary constants. As t--+ oo, the first two terms die away, and we are left with the forced oscillation X 0 eiqt X 0 eiqt X=(iq-n1)(iq-n2) p2-q2+2ip.q" (33.10) The amplitude is large (resonance) if (p- q) and I' are small, that is, when the frequency of the disturbing force is nearly equal to that of the undamped free oscillator and the damping is smalF. 34. Circular and cycloidal pendulums. Consider a particle of mass m moving under gravity on a smooth vertical circle of radius l (circular pendulum). If {} is the angular displacement from the downward vertical, the total energy is t m l2 .0.2 + m g l ( 1 - cos{}) = E = const, (34.1) and the equation of motion is (34.2) For small amplitudes, this reduces to the Eq. (33-3) for the harmonic oscillator and the periodic time is In general, we get an oscillatory motion if Wo = l~lo=o < 2P, and the motion is given by sin if}= sin foc sn P (t- t0), where oc is the maximum value of {}, so that . 1 1 w0 sm -- oc = - -- - 2 2 p ' (34.3) (34.4) (34.5) (34.6) t0 is an arbitrary constant, and the Jacobian elliptic function sn has sin ioc for modulus 2• The period is l (34.7) = 2 pn [ 1 + ( ~ r k2 + ( ~:~ r k4 + .. ·] J where k =sin t oc; this gives, to order oc2, r=2n V-}[1 +-~i]. (34.8) 1 Cf. (104.19) for the generalization of (33.10) to an oscillating system with more degrees of freedom. 2 Cf. WHITTAKER [28], p. 72; SYNGE and GRIFFITH [26], p. 371. 48 J. L. SYNGE: Classical Dynamics. Sect. 35. If w0 > 2p, the motion is no longer oscillatory, the particle going round and round the circle. In this case the solution is1 sin ! D- = sn ~ (t - t0), (34-9) where k = 2Pfw0 < 1, and the modulus of sn is k. If w0 = 2p, the particle reaches the highest point of the circle in an infinite time, the motion being given by 2 sin !-D=Tanp(t-t0). {34.10) For motion under gravity on any smooth curve in a vertical plane, the equation of motion is .. dz s + gds = 0' (34.11) where s is arc length and z height above some fixed level. If z = k s2, we get s + 2gks =0, (34.12) (34.13) the same form of equation as for a harmonic oscillator. The period is then independent of amplitude, and the curve (34.12), which is a cycloid, is called a tautochrone for the force of gravity3• III. Two-dimensional motions. 35. Projectiles. For a particle of mass m which moves in a uniform gravitational field, in a resisting medium with density depending only on height, the equations of motion are · x=-l(v,y) :. y=-g-l(v,y)~. (35.1) Here the x-axis is horizontal, the y-a~is is directed vertically upwards, and ml(v, y) is the force of resistance, opposing the motion; vis the speed. If I= 0, we get the elementary parabolic trajectory, x=a+bt, y=c+et-igt 2 , (35.2) where a, b, c, e are constants determined by the initial conditions'. If l=kv, the Eqs. (35.1) have simple exponential solutions 5• Integration can also be carried out explicitly6 for l=k0 +kv", which contains, as a special case 7, resistance varying as the square of the speed (I= k v2). The integration of equations of the form (35.1) is a central problem in exterior ballistics, the function l(v, y) being given numerically or by some empirical formula 8. Numerical integration is used. 1 Cf. WHITTAKER [28], P· 73- 2 The hyperbolic functions sinh, cosh, tanh are printed Sin, Cos, Tan throughout this Encyclopedia. 3 For tautochrones and brachistochrones (curves of quickest descent), see APPELL [2] I, pp. 478-489; MACMILLAN [17] I, pp: 322-329. 4 For geometrical constructions connected with parabolic trajectories, see APPELL [2] I, p. 374; LAMB [13], p. 72; SYNGE and GRIFFITH [26], p.151. 5 Cf. SYNGE and GRIFFITH [26], p. 159. 8 Cf. APPELL [2] I, p. 383; MAcMILLAN [17] I, p. 256. This is the integrable case of LEGENDRE. 7 Cf. SYNGE and GRIFFITH [26], p. 157. 8 For the older approach to exterior ballistics, seeP. CHARBONNIER: Traite de Balistique exterieure (2vols.) (Paris: Doin&Gauthier-Villars 1921-1927); and for the modern approach, see E. J. McSHANE, J. L. KELLEY and F. RENo: Exterior Ballistics (Denver: University Press 1953). Sect. 36. KEPLER problem. 49 36. KEPLER problemr. In the KEPLER problem, a particle of mass m is attracted to (or repelled from) a fixed point 0 by a force which varies as the inverse square of its distance r from 0. Let the inward component of this force be m11fr2 (ft positive for attraction, negative for repulsion). By symmetry, the orbit is plane, and, in terms of polar coordinates (r, {}) in the plane of the orbit, the equations of motion are [cf. (30.13)] (36.1) where h is a constant, the angular momentum per unit mass [ cf. (31.6) J. The kinetic and potential energies per unit mass are V= _ _!!_ r (36.2) and we have the equation of energy T+ V=E, (36.3) where E is a constant, the total energy per unit mass. Putting u=1/r, and eliminating t from (36.1), we get d2 u p, d{}2- +u = h2 (36.4) and hence u = }!_ + c cos ({} - {} ) h2 0 ' (36.5) where C, {}0 are constants of integration. From (36.3), C is determined in terms of E and h, and we can make {}0 = 0 by choice of the line {} = 0; then the equation of the orbit may be written 1 1 u =--; = T (1 + ecos{}), (36.6) where h2 l = ---, 1-' V--2-Eh2 e = 1 + -1-'2·- • (36.7) The orbit is a conic section of eccentricity e; it is an ellipse, parabola or hyperbola according as E 0, respectively, the centre of force being a focus of the conic. If the force is repulsive, then !-' < 0, E > 0, and only the hyperbolic orbit occurs; it is that branch of the hyperbola which is convex towards 0. For an elliptical orbit, the semi-axis-major a and the eccentricity e determine the orbit; likewise E and h determine it. The relations between these constants are a=-_!!_ 2E' E =- __!!__ 2a' The speed v at distance r is given by v2=t-t (~ - :). (36.8) (36.9) 1 For the solution of the KEPLER problem by means of the HAMILTON-JACOBI equation, see Sect. 78, or APPELL [2] I, 592- 596. For the relativistic KEPLER problem, see Sect. 11 5. For a force of the form r-2 <1>({}), see APPELL [2] I, p. 408. Handbuch der Physik, Bd. III/I. 4 50 J. L. SYNGE: Clas~ical Dynamics. Sect. 37. and the period is T = -- 1 - e2 = 2n - = • 2na2 v-- vaa np. h , v- 2E3 (36.10) We cannot enter here into the details of elliptical (planetary) orbits, fundamental in celestial mechanics1. 37. Central forces in general. For a particle of mass m repelled from a fixed point 0 by a force mF(u), where u= 1jr, the Eq. (30.13) give (37.1) and hence (37.2) The case of attraction is here included, F being then negative. A potential V exists, given by r V(u) =-f Fdr, . (37.3) •• where r 0 is any constant, and the energy equation T + V =E leads to ( du)2+ 2 =2(E-V) d{} u h2 ' (37.4) which is, of course, a first integral of (37.2). This equation can be solved by a quadrature, being of the form 2 ~)2 = I ( ) = 2 (E - V) - 2 d{} u h2 u . To relate the time t to the variables (u, D), we have, by (37.1), dt = r2d{} = ± du h hu2Vf(u)' the sign being chosen to make dt positive. (37.5) (37.6) The apsides of an orbit are those points at which r is a maximum or a minimum; thus apsides occur for u=u1 , u=u2 , where (37.7) The apsidal angle is (37.8) The whole orbit may be obtained from the part between two adjacent apsides, since the orbit is symmetric with respect to any apsidal radius. The whole orbit is contained between (and touches) two concentric circles, but in exceptional cases the radius of the inner circle may be zero and the radius of the outer circle infinite. · 1 For the anomalies, KEPLER's equation and LAMBERT's theorem, see APPELL [2] I, pp. 434-448; WHITTAKER [28], pp. 89-92; WINTNER [30], Chap. 4. Also MACMILLAN [17] I, pp. 278-292, where a repulsive force is included. 2 An equation of this type is of frequent occurrence in dynamics, leading in general to periodic solutions. Elliptic functions may be discussed in terms of such an equation; cf. PERES [20], pp. 107-122, SYNGE and GRIFFITH [26], pp. 364-370. Sect. 38. Stability of circular orbits. 51 Borrowing the words from astronomy, we may call any one of the apsides on the inner circle perihelion, and any one of the outer apsides aphelion. When the force varies as r(F=ek2 r, e= ±1), the motion is more simply discussed by means of rectangular Cartesians; we have X = f k2 X' ji = f k2 y. If e = - 1, the orbit IS a central ellipse, with equations x = A cos k t + B sin k t, } y = C cos kt + D sin kt. (37-9) (37.10) If e = + 1, the solution is similarly expressed in hyperbolic functions; the orbit is a central hyperbola. In special cases (vanishing angular momentum) the orbit is a straight line through the origin; then we have a simple harmonic oscillator in the case e= -1. 38. Stability of circular orbits. For a circular orbit of radius r = 1fu, we require f(u)=O, f'(u)=O. (38.1) The first condition is a consequence of (37.5), and the second follows from (37.2), which is equivalent to (38.2) Putting u = u0 for the circular orbit, we write u = u0 +; (38.3) for a disturbed orbit (;is small), and obtain from (38.2) and (38.1), to the first order in ;, d2~ 1 i:f"( ) df)2 = 2 s- Uo . This has sinusoidal solutions (giving stability) if and only if f"(u0 ) < 0. (38.4) (38.5) Thus we have the following criterion for stability of a circular orbit of radius r = 1/u, described under a central attractive force: -1 I" ( ) U=-1----<0. 1 d ( F ) 2 h2 du u 2 (38.6) Here F is negative, the magnitude of the attractive force being - mF. But, by (37.2), applied to the circular orbit, we have and so the criterion for stability may be written dF 3Faq-qe+ Tt (t = 1, ... P), (46.3) e~l e so that the components of velocity are functions of the 2N + 1 variables (q, q, t). For the partial derivatives of these functions, we have oi-; or; otie aq;;, or; d or; (. P. N) -8-=dt-8- t=1, ... ,(2=1, ... ' q(! q(! as is easily verified. The kinetic energy is p T= i L m;r;· r;, i~l a function of (q, q, t), and its partial derivatives are (46.4) (46.5) (46.6) Sect. 46. LAGRANGE's equations. Ignorable coordinates. 59 Hence, by (46.4), p !_ oT -~ = "1\'m.r·· .. oT; ( ) d "' " L...J ,, " e=1, ... N. t uq e uqe i=l uqe (46.7) Let ~q~ (e = 1, ... , N) be an arbitrary set of infinitesimals, and let ~ri be corresponding displacements of the particles of the system, obtained by differentiating (46.1) with t held fixed, so that (46.8) (46.9) Note that this is a purely kinematical result, no forces or equations of motion having been used; nor have the equations of constraint (46.2) been involved. We now introduce the generalized force Q:. split into two parts as in (29.6): Q:=Qe+Q~ (e=1, ... N), (46.10) where Q~ is the given (or applied) force, and Q~ the force of constraint. We assume the constraint workless, in the sense that for all values of ~qe satisfying N LAce ~qe = 0 (c = 1, ... M). e=l (46.11) (46.12) We tum now to n'ALEMBERT's principle (45.1). Choose any ~qe satisfying (46.12), and let ~r; be the corresponding displacements as in (46.8). We use (46.9) to change the first term in (45.1), and for the last term we have (46.13) in view of (46.11). Thus (45.1) gives ~ ( d oT oT ) L...J dt aq-aq- Qe ~qe = O, e=l e e (46.14) for all ~qQ satisfying (46.12). From this last equation we obtain at once LAGRANGE's equations of motion for non-holonomic systems: d oT oT ~ diaq-aq=Qe+L..J#cAc~ (e=1, ... N) e e c=l (46.15) where De are undetermined multipliers. These equations are to be supplemented by the equations of constraint (46.2) in the form N LAce qe + Ac = 0 (c = 1, ... M), (46.16) e=l so that we have in all N +M equations for theN +M quantities (q1 l' #c). 60 ]. L. SYNGE: Classical Dynamics. Sect. 46. We can write down the Eqs. (46.15) explicitly for any given system as soon as we given the form of the function T and the forms of the functions Q11 ; these last are most easily obtained in practice by calculating the work {J W and using (46.13). {3) Holonomic systems. For a holonomic system we can takeN to be the smallest possible number of generalized coordinates. The #-terms disappear from (46.15), and we have LAGRANGE's equations of motion for a holonomic system: (46.17) But even though the system be holonomic, it is sometimes convenient to use more than the minimum number of coordinates; then the equations of motion are as in (46.15), supplemented with (integrable) equations of the form (46.16). If a holonomic system possesses a potential function V as in (29.9) (in other words, a potential energy), or an extended potential function as in (29.11}, the equations of motion (46.17) may be written d oL oL di a--- a-= o (e = 1, ... N), qe qe (46.18} where L=T-V. (46.19) Here L is a function of the 2 N + 1 variables (q, q, t) ; it is called the Lagrangian function, or the kinetic potential. The peculiar virtue of (46.18) is that the equations of motion of a system can be written down once a single function is given. Also, if two different physical systems have Lagrangian functions of the same form, then they behave in the same way, If we multiply (46.18} by q11 and sum with respect to (J, the result may be rearranged to read d ( ~. oL ) oL di L..Jqlly--L +ae=O. 11=1 qe (46.20} If L does not depend explicitly on t (and this may happen even for a rheonomic system), we have 8Lf8t=O; in that case (46.20) gives a constant. N • oL Lqey--L=K, e=l qe (46.21) The kinetic energy is quadratic in the generalized velocities (q11}, and we may write it T= I;+ 7;. +To. (46.22) where the subscripts indicate degrees of homogeneity in the generalized velocities. If V = V(q), an ordinary potential function, application of EULER's theorem for homogeneous functions to (46.21} gives (46.23) If, further, the system is scleronomic, we have T= T2 , and (46.23) becomes T+V=K, (46.24) the equation of energy or integral of energy, as in (45.4). Sect. 46. LAGRANGE's equations. Ignorable coordinates. 61 y) Ignorable coordinates. Consider a holonomic system with Lagrangian L. If some one coordinate qe is absent from L, that coordinate is said to be ignorable1, and the process described below is called the ignoration of coordinates. If qe is ignorable, then the corresponding Lagrangian equation of motion (46.18) gives (46.25) a constant. This is a first integral of the equations of motion. If q1 , ..• , qM are ignorable coordinates, there are M integrals like (46.25). Solving these equations, we obtain the velocities corresponding to the ignorable coordinates (that is, q1 , ... qM) as functions of the other coordinates and velocities, of the time t, and of the constants c1 , ... eM. The Routhian function R, defined as M M R=L- Lqe :~ =L- 'L,qPcP, e=l qe p=l (46.26) can be expressed in the form R = R(qM+l• ... qN, t, qM+l• ... qN, c1 , ••• eM). (46.27) This function, as we shall now show, may be used to replace Lin the equations of motion. Since the dynamical system may be thought of in any configuration at any time with any generalized velocities, the 2N -M + 1 quantities (46.28) may be regarded as independently variable; equivalently, the 2N -M + 1 quantities (46.29) are independently variable. On applying such a variation, we get from (46.27) and (46.26), with (46.25), (46.30) Therefore, treating the variations of the quantities (46.29) as independent, we have i:JR i:JL i:JR i:JL (e =M + 1, ... N) (46.31) i:Jqe i:Jqp ' i:Jqe i:Jqe and i:JR i:JL i:JR Tt=Tt· -a~;-= -q~~ (e = 1, ... M). (46.32) (/ 1 The words kinosthenic and cyclic are also used, particularly the latter, which is a pity, because cyclic may be needed in a topological sense; cf. Sect. 63. The word ignorable has been used in different senses: (i} absent from T, and (ii} absent from L; cf. GoLDSTEIN [7) p. 48; LANCZOS [15], p. 125. 62 J. L. SYNGE: Classical Dynamics. Sect. 47. Substituting from (46.31) in LAGRANGE's equations (46.18), we get equations of motion in the form d oR oR a~- 8 - = o (e = M + 1 .... N). t q(/ qll (46.33) The only unknowns in these equations are theN-M non-ignorable coordinates q11 • The equations contain the constants c1 , •.. , eM. The original equations (46.18), being N equations of the second order, formed a system of differential equations of order 2N; in (46.J3) we have a system of order 2N -2M, the Lagrangian form being preserved, with R in place of L. The passage from (46.18) to (46.33) is the process of ignoration of coordinates. If (46.33) have been solved for the non-ignorable coordinates, the ignorable coordinates are given by f oR q11 =- ac;;dt (e=1, ... M). (46.34) 47. HAMILTON's equations. Consider any system with N degrees of freedom for which the motion is given by the Lagrangian equations (46.18), L being any function of the generalized coordinates q11 , their derivatives q11 , and the time t. Define the generalized momenta Pe by oL Pe =---,-.---- (e = 1, ... N). r..·q2 (47.1) If (47.2) as is in general the case, (47.1) can be solved for the generalized velocities, so that we have (47.3) Then L can be expressed as a function of the 2N + 1 variables (q, t, p), and so also can the Hamiltonian function H defined by N H(q, t, p) = ~ P11 q11 - L. (47.4) e=l We now regard the 2N + 1 quantities (q, t, p) as independently variable, the generalized velocities being expressed in terms of them by (47.3). For an arbitrary variation we have all summations running 1, ... , N. By (47.1} the second and third summations on the right cancel, and we obtain the equations :~ =qe, .:~ =- :~. ~ =- ~~ (e = 1, ... N). (47.6) On making use of (47.1), we can now write the Lagrangian equations (46.18) in the form • oH Pe = - oqll (e = 1, ... N). (47.7) Sect. 48. APPELL's equations. 63 These are HAMILTON's equations of motion; they are also called canonical equations. The passage from LAGRANGE's equations to HAMILTON's is a purely mathematical process, without reference to the original dynamical system; thus HAMILTON's equations hold for any system provided LAGRANGE's equations hold in the form (46.18); in particular, (47.7) are the equations of motion of any holonomic system (rheonomic or scleronomic), provided a potential function V, or an extended potential, exists. For the rate of change of H we have, using (47.7). • N ( oH • oH • ) oH oH H= L aqqe+ -apPe + Tt = ae· e=l e e Thus, if H does not involve t explicitly, so that oHjot = 0, we have H = const, (47.8) (47.9) which may be called an integral of energy. If T = T2 +I;.+ T0 as in (46.22), and V = V(q), then N H = Ltle :L - L = T2 - T0 + V, (47.10) e=l qQ and if T = T2 (the case most commonly encountered in dynamics), then H=T+V. (47.11) so that in this case H is the total energy of the system, kinetic and potential. If, in addition to the forces taken care of by the potential function V (or the extended potential), forces Qe act on the system, the Lagrangian equations (46.18) are modified to read a oL oL --.-- - = Qe (n = 1, ... N). dt eqe oqe 0: (47.12) To convert these into Hamiltonian form, we note that (47.6) were obtained by purely mathematical manipulations, without reference to the equations of motion, and they are valid in the present case. It follows that the Hamiltonian equations (47.7) are modified to read . oH . oH Q ( N) qe = ape , Pe =- 8qe + e e = 1,... . (47.13) 48. APPELL's equations 1. For a system of P particles, the energy of acceleration is defined as P S 1'\' .. .. (8) = 2 L. m r, · r,, 4 .1 i=l accelerations replacing velocities in the definition of kinetic energy. If qe ((} = 1, ... N) are generalized coordinates so that r,=r; (q, t), then (48.2) 1 Cf. P. APPELL: Sur une forme generate des equations de la dynamique. Paris: GauthierVillars 1925; APPELL [2] II, pp.388, 412, 498; NoRDHEIM [18], p.69; PERES [20], p.219. 64 J. L. SYNGE: Classical Dynamics. Sect. 48. We can therefore write S = S(q,q,ij, t). (48.3) This function of 3 N + 1 quantities is APPELL's function; it is quadratic in the second derivatives iju, and its partial derivative with respect to one of them is [ cf. ( 46.7) J (48.4) Thus, if the system is holonomic and the q's form a system of coordinates of minimum number, LAGRANGE's equations (46.18) lead at once to APPELL's equations of motion (48. 5) Suppose now that constraints, in general non-holonomic, are imposed by the equations N LAced%+ Acdt = 0 (c = 1, ... M) (48.6) e=l as in (46.2), so that LAGRANGE's equations take the form (46.15). Using the purely kinematical result (48.4), we can express (46.14) as follows: for all <5qu satisfying By (48.6) we have N 85 N L F·- <5qe = L Qe 15 %, e=l qe e=1 N 2.: Ace <5qe = 0 (c = 1, ... M). e=l N LAcufJe+Ac=O (c=1, ... M), e=l and hence, differentiating with respect to t, N L Aceiie + Bc(q, q, t) = 0 (c = 1, ... M), e=l (48.7) (48.8) (48.9) (48.10) Be being a function of the 2N + 1 variables indicated. These last equations may be used to express ijl> ... ijM in terms of the 3 N- M + 1 quantities and so we can write S(ql, ... qN, ql, ... fJN, ijl, ... ijN, t) } = S(ql, ... qN, ql, ... qN, ijM+l' ... ijN, t). If we give variations bij1 , ••• , bijN, arbitrary except for the conditions N 2.: Ace bije = 0 (c = 1, ... M), e=l it follows from (48.10) and (48.11) that N N - "' 85 (j •• "' 85 (j •• L. Y qe= L. -~ q,. e=l qe r=M+l q, (48.11) (48.12) (48.13) Sect. 49. Equations of motion of a rigid body. This is equivalent to saying that N 85 N 8S 2.: a··- IJqQ = 2.: 8"' IJq, e~l qQ r~M+l q, for all variations /Jq1 , ... , 1Jq1 .. ,- which satisfy N LAce IJqe = 0 (c = 1, ... M). e~l We now define Q M +1, ... Q N by the condition that N N l: Qe 1Jqe = l: Q,bq, e~l r~M+l 65 (48.14) (48.15) (48.16) for all variations satisfying (48.15). Then, by (48.14) and (48.16), we may write (48.7) in the form and since these variations are arbitrary, we have as - ~-- = Q r(r = M + 1 , ... N) , vq, (48.17) (48.18) which are APPELL's equations of motion in a form valid for non-holonomic systems1. 49. Equations of motion of a rigid body. Consider a rigid body of mass m with mass-centre 0 and principal moments of inertia A, B, Cat 0. The four numbers m, A, B, C specify the body dynamically. Let q1 , q2 , q3 be generalized coordinates describing the position of 0 in absolute space 50 , and let q~, q;, q~ be generalized coordinates describing the position of the body relative to 0, i.e. specifying the directions of principal axes fixed in the body relative to axes fixed in space. By the theorem of KoNIG (Sect. 2;), the kinetic energy of the body may be written (49.1) where T0 is the kinetic energy of a particle of mass m moving with 0, and T' is the kinetic energy of the motion relative to 0; these functions are of the forms 3 3 T: ,_, ( ) . q' T' " ' ( ') . ' . ' 0 = LJ aQ" q qe "' = LJ aea q qeqa. p,a=l p,a=l (49.2) If the coordinates q are rectangular Cartesian coordinates (x, y, z), we have T0 = fm(x 2 + y2 + z2 ), (49-3) and if the coordinates q' are the Eulerian angles ({}, rp, 'If), we have, as in (25.5), T' = fA ( J sin 'If - cp sin {} cos 'If) 2 + f B ( {} cos 'If + cp sin {} sin 'If) 2 + + i- c (ip + cp cos {}) 2• } (49.4) In the case of axial symmetry (A= B), this simplifies to T' = fA (g,2 + cp 2 sin2 {}) + f C (ip + cp cos {})2. (49.5) I For the application of these equations to servo-mechanisms (asservissements), see APPELL [2] II, pp. 412-416. Handbuch der Physik, Bd. lll/1. 66 J. L. SYNGE: Classical Dynamics. Sect. 49. Let Q2 , Q~ (e = 1, 2, 3) be generalized forces such that the work done in an arbitrary displacement is 3 3 6W= 2; Qe6qe + 2: Q~6q~. (49.6) e ~I e~I We have then the six Lagrangian equations of motion, as in (46.17), ~~ :~!-~~~ = Qe, l !:_oT'- oT'- , (e=1,2,3). dt"''' ,,-Qe oq e uqe (49.7) The coordinates q are separated from the coordinates q' on the left hand sides, but this separation does not, in general, extend to the right hand sides; in other words, the problem of the motion of a rigid body does not, in general, split into two problems. These Lagrangian equations hold for a rigid body without constraint. They hold also in the case of constraints, provided we include in Qe and Q~ contributions from the forces of constraint. In discussing the motion of a rigid body, it is often more convenient to use the principles of linear and angular momentum instead of LAGRANGE's equations. By (44.4) and (44.7) (dropping the asterisks), we have the two vector equations ma=F, h=G, where a = acceleration of the mass-centre 0, h =angular momentum about 0 of the motion relative to 0, F = total external force, G =total moment of external force about 0. (49.8) (49.9) In order to use these equations for the determination of the motion, we have to resolve them into components along some orthonormal triad; we can choose a triad fixed in absolute space, a triad fixed in the body, or neither of these. We shall later consider a triad fixed in absolute space; for practical purposes it is best to choose a moving triad which is a principal triad of inertia for the body. But let us start by taking any arbitrary orthonormal triad (i,j, k), rotating with angular velocity .2. Resolving along this triad, we have h= ~+h j. + h3 k,) G = G1 t + G21 + G3 k, .Q =!2ri +!2d +!23k. Then, by (20.3), we can write (49.9) in the form (jh Tt+.2xh=G, or. explicitly, h1-h2!23+h3!22=G1, l h2- h3!21 + hrD3= G2, h3- h1!22 + h2!21 = G3. (49.10) (49.11) (49.12) Sect. 49. Equations of motion of a rigid body. 67 Let us now choose (i,j, k) be a principal triad of inertia at 0, the corresponding moments of inertia being A, B, C. If w is the angular velocity of the body, we have then, by (24.14), W = Oh i + W 2 j + Wa k, } (49.13) h =A w1 i + B w2 j + C w3 k. There are three cases to consider. rx) Unsymmetrical body (A, B, and C all different). In this case the triad (i,j, k), if it is to be principal, must be fixed in the body. Therefore .2= w, and (49.12) with (49.13) give us EuLER's equations of motion for a rigid body A 1 - ( B - C) w2 wa: G1 , l B w2 - (C- A) Waw1 - G2 , (49.14) C wa- (A - B) w1 w2 = Ga. {J) Body with axis of symmetry(A=B=f=C). Now (49.14) simplify to A (A - C) w2 wa: G1 , l A w2 - (C- A) w3 w1 - G2 , Cwa= Ga. (49.15) But we are no longer compelled to fix (i,j, k) in the body in order to have a principal triad; it is sufficient to fix k in the body. Doing this, we have Da remaining arbitrary, and (49.12) becomes Aw1 -Aw2 Da + Cwaw2 = G1 , l Aw2 +Aw1 Da- Cwaw1 = G2 , Cw3 =Ga. The first two equations may be exhibited in complex form 1 : Aw+(AQa-Cwa)iw=F,} w = w1 + i w2 , F = G1 + i G2 . (49.16) (49.17) (49.18) y) Body with spherical symmetry (A =B=C). Now (49.14) simplify to (49.19) for axes fixed in the body. In this case we can choose .2 arbitrarily; if we choose .2=0, (49.12) gives (49.19), but with the components taken on axes with directions fixed in space. Returning to the motion of the mass-centre as given by (49.8), we may use a moving triad (i,j, k) here also. Let v be the absolute velocity of 0, and let it and F be resolved along the triad, so that we have v = v1 ~ + ~ + v3 k, } F = 1\ t + F;J +Fa k. (49.20) 1 Useful in ballistics and other problems of stability; cf. K. L. NIELSEN and J. L. SYNGE: Quart. Appl. Math. 4, 201 ( 1946); E. J. McSHANE, J. L. KELLEY and F. V. RENO: Exterior Ballistics, p. 176 (Denver: University Press 1953); S. O'BRIEN and J. L. SYNGE: Proc. Roy. Irish Acad. A 56, 23 (1954). The complex notation is also useful in the theory of KOWALEWSKI's top (Sect. 56). 5* 68 J. L. SYNGE: Classical Dynamics. Then {49.8) may be written or, explicitly, m ( ~: +~X v) = F, m(v1 - D3 v2 + D2 v3) = }\, l m(v2 - D1 v3 + D3 v1) = ~. m(v3 -D2 v1 +D1 v2) =Fa. Sect. SO. (49.21) (49.22) In Case IX, we put ~=w. Then we have in {49.14) and (49.22) six equations for the six components of v and w. When these have been found as functions of t, the complete determination of the motion demands a further step. Assigning six generalized coordinates (q) to the body, we express the six components of v and w as functions of (q, q); then, v and w being known, we have six differential equations of the first order to determine (q) as functions of t, and so complete the description of the motion. Cases {J andy are treated similarly. In Casey we may put ~=0, and then (49.22) read (49.23) The important case of a rigid body turning about a fixed point (see Sect. 55-57) is covered by the preceding theory. The body has three degrees of freedom, and its motion is determined by (44.5), which is formally identical with (49.9). But now h is the angular momentum about the fixed point, and G the total moment of forces about that point. With this change of interpretation, all the work from (49.10) to (49.19), inclusive, is applicable; but now A, B, Care prin-cipal moments of inertia at the fixed point, and not at the mass-centre. In the case of a freely moving rigid body, explicit reference to the mass-centre is avoided in the motor symbolism (Motorrechnung) of STUDY and VON MISEs1. 50. Moving frames of reference 2• Let S be a rigid body in some quite general prescribed motion. We take S as a frame of reference, and seek the equations of motion of a dynamical system relative to this frame. In Sect. 32 we treated this question for a single particle; now we take a dynamical system consisting of P particles, with scleronomic holonomic constraints, so that there exist generalized coordinates qe (e = 1, ... N) determining the configuration of the system relative to S, these coordinates being freely variable without violation of the constraints. For example, S might be the earth, having an orbital motion around the sun and also a rotation about its axis; the system might be a rigid body having one point attached to the earth; qe might be the three Eulerian angles relative to a triad fixed on the earth. The plan is to reduce S to a Newtonian frame by the introduction of fictitious forces. As in (32.1), the equations of motion of the particles may be written (50.1) here F; is the total real force on the particle (applied force+ reaction of constraint), and the other three vectors on the right hand side are as in (32.2), but with a suffix i attached tom, r', v'. Let IJqe be an arbitrary variation of the generalized coordinates, and let IJr~ be the corresponding displacements of the particles 1 R. VON MISES: Z. angew. Math. Mech. 4, 155, 193 (1924). -FRANK [5] pp. 98-102. - C. B. BIEZENO: Handbuch der Physik, Vol. 5. pp. 247-250. Springer: Berlin 1927. - WINKELMANN and GRAMMEL [29] pp. 373-378. - L. B:RAND, Chap. II of op. cit. in Sect. 12. - W. RAHER: Ost. Ing.-Arch. 9, 55 (1954). 2 For alternative methods of treating relativ~ motion, see APPELL [2] II, pp. 360-376. Sect. 51. Two-body problem. 69 relative to S. Taking the scalar product of (50.1) by or; and summing fori from toP, we get p p p p p l: m; a:. or;= l: Iii. or;+ l: F,;. or;+ l: F,;. or;+ l: F;;. or;. (50.1 a) Since a; is the acceleration of the particle relative to S, by the purely kinematical formula (46.9) we have P I I N ( d oT1 oT') L m;a;. Or;= L 7iJ oq - Bq- oqQ, i=1 e=l (! e (50.2) where T'(q, q) is the kinetic energy of the motion relative to S, i.e. p p T l ( ") _ 1 "' " I " I _ 1 "' 12 q,q -2L.Jm;r;·r;- 2 L.Jm;V;. (50. 3) i=l i=l To deal with the other summations in (50.1 a), we first define generalized forces Q0 by P N l:lii ·or:= 2: Qe oqe, (50.4) i=l e=l this being the work done by the applied forces, the frame S being held fixed during the virtual displacement; the reactions of constraint do not appear in Qv. Next we define A0 , B0 , C2 (e = 1, .. . N) by the equations. [cf. (32.2)] p p N l:F,;· or;=- l: m;ao. or;= L Aeo%, i=l i=1 e=l p P N l:F;,;. or;=- 2 L m,(wxv;). or;= L Be oqe, i=l i=l e=l p p p (50.5) L 11;; · b r; = - L m; ( c.O X r;) · or; - w · L m; ( w · r;) or; + i=l i=l i=l P N + w2 2: m;r;. or; = 2: ca oqe, i=l e=l the variations bq0 being arbitrary. In these expressions a 0 and ware respectively the acceleration of the base-point 0 (fixed in S), and the angular velocity of S; they are given functions of t (since the motion of S is prescribed), and hence A0 and C0 are functions of (q, t), while Be is a function of (q, q, t). After these preparations, we can use (50.1 a) to obtain at once LAGRANGE's equations for the motion relative to S in the form (50.6) the last three terms being fictitious generalized forces due to the motion of S. II. Systems without constraints. 51. Two-body problem. Consider two particles with masses m1 , m2 , attracting or repelling one another with equal and opposite forces acting along the line joining the particles and depending only on the distance between them. Fig. 13 shows the case of repulsion. 70 J. L. SYNGE: Classical Dynamics. Sect. 51. Let r 1 , r 2 be the position vectors of the particles relative to some fixed origin; let P be the force exerted by the first particle on the second. Then the equations of motion are (51.1) The position vector of the second particle relative to the first is (51.2) and from (51.1) we obtain Mr=P, (51. 3) Thus, by abandoning the absolute frame of reference, and using an accelerated frame attached to the first particle, we reduce the two-body problem to the oneP body problem, the mass of the second particle being fictitiously changed 1 in this process, but the force being unchanged. We can now apply to (51-3) the theory of central forces as developed in Sect. 37. But note that in (3 7.1) F is force per unit mass; in using -P (37.1) we are to put F=P/M, where !PI is 0 Fig. 13. Two-body problem. the magnitude of P and P is positive for repulsion and negative for attraction; for the potential V of (37-3) we have , V= -M-1 f Pdr. (51.4) ,, We can also simplify the two-body problem by using a suitable Newtonian frame of reference. By (51.1) we have (51.5) and this tells us that the mass-centre is unaccelerated; a frame in which it is at rest is Newtonian. Using this frame, and taking the origin at the mass-centre, we have (51.6) It is then sufficient to work with only one of the Eq. (51.1). If the scalar law of repulsion or attraction is P=P(r), the second of (51.1) may be written (51.7) Again we have a one-body problem. Now the mass is unchanged, but the law of force is altered. In the case of the inverse square law, we have P = kfr2, and the motion relative to the mass-centre is given by (51.8) Viewed in a fixed frame of reference, the orbits of the two particles are twisted curves in space. It is much easier to study the motion in a moving frame, attached either to one particle as in (51.3), or to the mass-centre as in (51.7), for then the orbits are plane. Note that the assumption regarding the interaction between the particles rules out magnetic interaction and electromagnetic retarded effects. 1 The quantity Min (51.3) is called the reduced mass. Sect. 52. Capture and scattering. 71 52. Capture and scattering1• Consider two particles which interact as in Sect. 51, the scalar force between them at distance r being P (r), positive for repulsion and negative for attraction. We suppose that for large r this force goes to zero at least as fast as r 1-'(e > 0), so that the potential V exists as in (51.4) with r0 = oo. Fort=- oo the particles are infinitely distant from one another. They approach and interact. We are interested in the result of the encounter, i.e. the state of affairs at t = + oo. If r--+0 as t-+ oo, or more generally if r is bounded as t-+ oo, we have capture. If r-+ oo as t-+ oo, we have scattering. We wish to determine whether capture or scattering occurs in any encounter for which the initial conditions are prescribed, and, in the case of scattering, to find the directions in which the particles are scattered. It is advisable to view any particular encounter in three frames of reference: SM, the mass-centre frame, in which the masscentre is fixed. SR, the relative frame, in which the particle m1 is fixed. SL, the laboratory frame, in which the particle m1 is at rest for t = - oo. W=WK 0 0 Fig. 14. Diagram showing initia ~state before an encounter .. We use parallel axes in all three frames. The frames SM and SL are Newtonian, but SR is accelerated; however SR is Newtonian for t=- oo, and fort=+ oo also if scattering takes place. We shall denote initial velocities by v1 , v2 , indicating the frame of reference with M, R, or L, as in (v1)M, (v2)R· Then mr (vr)M + m2 (v2)M = 0, (vr)R = (vr)L = 0, l (v2)R = (v2)L = (v2)u- (vrhr = ( 1 + ::) (v2Lu, (52· 1) so that in all three frames v2 has the same direction. In SM the initial data provide a pair of infinite parallel lines (the asymptotes of the initial trajectories); in SR and SL we have a point (position of m1) and an infinite line (the asymptote of the initial trajectory of m2}. A single diagram as in Fig.14 serves to display the initial data of a given encounter, no matter which frame is used. Here k is a unit vector drawn on the line of v2 . In S R or SL the point 0 is the initial position of m1 , and in SM it is any point on the asymptote of the initial trajectory of m1 . The vector b is drawn from 0 to meet the line of k at right angles. It is the impact vector and its magnitude b is the impact parameter or collision parameter; it is in fact the shortest distance between the two asymptotes of the initial trajectories, viewed in any unaccelerated frame. The relative velocity appears in Fig. 14 as w = v2 - v1 = w k; it is of course the same for all the frames. (52.2) 1 Cf. CoRBEN and STEHLE [3] pp. 86-90; GoLDSTEIN [7] pp. 81-89; D. BoHM: Quantum Theory, Chap. 21. New York: Prentice-Hall 1951. For details of encounters between particles obeying various particular laws of force, between smooth spheres and between rough spheres, see S. CHAPMAN and T. G. CowLING: The Mathematical Theory of NonUniform Gases, Chaps. 10, 11. Cambridge: University Press 1952. Cf. also H. GRAD: Comm. Pure Appl. Math. 2, 331 (1949) and his article on the kinetic theory of gases in Vol. XII of this Encyclopedia. For relativistic capture and scattering by a fixed centre, see J. L. SYNGE: Relativity: The Special Theory, p. 426. Amsterdam: North-Holland Publishing Co. 1956. 72 J. L. SYNGE: Classical Dynamics. Sect. 52. We deal with the encounter primarily in 5 R, passing immediately to results in 5 M, and with a little more complication to results in 5L. In 5R the particle m1 remains permanently at 0 in Fig. 14 and m2 describes an orbit in the plane (b, k). By (37. 5), (51. 3) and (51.4), this orbit is determined by (~;f=l(u), l(u)= z(Eh-;_!1_-u2 , (s2.3) where 00 1 U=-r ' V(u) = m1_±mz J P(Y) dY, mlm2 (52.4) (Y, {}) being polar coordinates in the plane of the motion. Here, in terms of the initial data, E = iw2 , h = bw. (52.5) The time is given by [cf. (37.6)] u t=-1 f ~--du h u2Vf(u) . (52.6) ffu_) Fig.15. Graphs of f(u): capture for C1 and C,, scattering for 1:. Fig. 16. Capture: r~o as t~ oo. The outcome of the encounter depends entirely on the function f (u), i.e. on the form of the function V(u), on the masses of the particles, and on the two constants (b, w). We have (52.7) so that the graph of I (u) starts above the u-axis (Fig. 15). If it does not meet that axis at all (curve C1), then l(u) >0 for all u, and the orbit spirals in to 0, with capture resulting (Fig. 16). If the graph touches the u-axis at u=u0 (C2 in Fig. 15), there is an apse with apsidal distance Y = Y0 = 1/u0 ; but this apse is never attained, because I (u) contains (u- u0) 2 as a factor and the integral in (52.6) diverges; the result of the encounter is capture as shown in Fig. 17. If, finally, the graph cuts the u-axis at u=u0 (curve .E in Fig. 15), an apse occurs in finite time, and we have scattering as shown in Fig. 18. Omitting head-on collisions; for which b = 0, it is clear that capture is impossible in the case of a repulsive force, since such a force bends the trajectory away from 0. Assuming now that scattering takes place (the force being either repulsive or attractive), we proceed to calculate the scattering angle XR as shown in Fig. 18, which shows also the base line {} = 0, the two asymptotes of the orbit, and the apse A at which dufd{} = 0. The apsidal distance is OA = Y0 = 1/u0 and the apsidal angle, as shown, is u, .,_ =I v~iu) . (52.8) 0 Sect. 52. Capture and scattering. 73 Since the orbit is symmetrical about the apsidalline OA, the scattering angle is XR = n- zoc = n- 2j'l;~~u) . 0 (52.9) With the convention as to sign adopted here, we have 0~ XR~ :n; for repulsive scattering and - oo 0). Zero-scattering can occur if, and only if, there is a cut-off of interaction, with P(r) =0 for r>r1 , say, so that if b>r1 the two particles pass one another with straight trajectories. We define the total cross section for scattering to be the area lis of II corresponding to significant scattering; thus lis= f adD, (52.29) an improper integral over the unit sphere with the point e = 0 excluded. If Ilo is the area of II corresponding to zero-scattering, then Ilc+Ils+Ilo is the total infinite area of II; consequently at least one of lie, II., Il0 must be infinite. The determination of the density a is merely a question of finding the mapping ratio as in (52.27). The scattering vector is s = i sin e cos cp + j sin e sin cp + k cos e' and it traces out the solid angle dD = !sin@d@dcp!. Hence a---- dll I b db dtp0 ! b I db I ----- -- - dD - lsin@d@dtpl - sin@ d@ · since dcp0 =dcpR = dcpL. (52.30) {52.31) (52.32) In the frame SR we know XR as a function of (b, w) [cf. (52.11)]. Let us solve for b as a function of (XR' w) and then substitute XR = eR for repulsive scattering, as in (52.22}, and XR= ±8R modulo 2n for attractive scattering, as in (52.23). Thus we getl b = b(@R, w). (52-33) Then (52.32) gives for the density aM= aR = siri~-; I dtR I· (52.34) the right hand side being a function of @R and W; in this, as in the other differentiations, w is held fixed. For the laboratory frame SL, we get from (52.32) (52.35) and by (52.26) this may be written (52.36) (1 + m: + 2 mz cos@R)~ ml ml =aR--· !1 + :: coseRI We would like to express aL explicitly as a function of @Land w, but we cannot. The best plan is, for given w, to regard (52.25) and (52.36) as expressions for eLand aL in terms of the parameter eR. We can however use approximations when the mass-ratio m2/m1 is small, for then @L and aL differ little from @R and aR. 1 This can be a multiple valued function in the case of attractive scattering. Sect. 52. Capture and scattering. 77 In view of the symmetry of the mapping about the axis k in Fig. 19, it is often convenient to use the dilferential cross section for scattering in the ring B, B +dB. This is the area of II which is mapped on to the ring, and its value is 2na sin B I dB! = 2nb I db!. (52.37) Here follow details for some special cases of scattering. a.) Smooth elastic spheres. A collision between two smooth elastic spheres may be regarded as an encounter between two particles with a cut-off at r=D, whereD is the sum of radii of spheres; we take V(u)=O foru< 1/D and V(u)-+ oo as u-1/D from below. The solution of (52.10) is u0 =1fD, and by (52.9) the scattering angle is u, f du . b XR = 7l- 2 v--~-- = 7l- 2 arcsin -D • b-2- u2 0 Thus b=D cos! XR =D cos! BR, and by (52-34) the density is (52.38) {52.39) which is independent of BR and w. To get av we would use (52.25) and (52.36). {3) CoULOMB scattering. Take P(r) = .!!_ r2 ' V(u) = ku, {52.40) (k >0 for repulsion, k <0 for attraction). Then by (52.10) 1 2k u 2 f(u) = -1)2 - 1)2U,2" - u = (u0 - u) (u + u1), {52.41) where Uo= b2~2 (-k+ Vk2+b2w4), ul=b2~2 (k+ Vk2+b2w4), (52.42) By (52.9) the scattering angle is (52.43) so that fork >Owe have O RuTHERFORD's scattering formula; it holds for both repulsive and attractive CouLOMB fields. For the frame SR the energy of the incident particle at infinity is jm2w2• 78 J. L. SYNGE: Classical Dynamics. Sect. 52. For the laboratory frame we have, by (52.25) and (52.36), (1 + m} + 2 m2 cos@R)1 l m 1 m 1 ; J 11 + :: cos@RI (52.47) thus eLand (JL are expressed in terms of the parameter eR. For two particles of the same mass, we have m1 =m2 and 1 fJL= - 2 fJR, y) Inverse cube law. Take P(r) = ~-, (k >0 for repulsion, k <0 for attraction). By (52.10) we have t(u) = .L - u2 (-k __ + 1) b2 b2w2 • (52.48) (52.49) (52.50) Capture occurs if k <- b2w2• If k exceeds this value, there is scattering at the angle In the case of repulsion we have eR = XR and hence ~) Inverse fifth power1• Take 1 P(r) = ~, V(u) = 4 ku4 , r We have by (52.10) where I( ) _ 1 k u' 2 _ k ( 2 2) ( 2 + 2) U-bZ-2b2w2-U-2b2w2Uo-U u ul u~= ~ (- b2w+ Vb4w2+ 2k), ) u~ = ~ (b2w + Vb4w2 + 2k). (52.51) (52.52) (52.53) (52.54) (52.55) (52.56) 1 Repulsion varying inversely as the fifth power of the distance is of importance in the kinetic theory of gases; cf. CHAPMAN and CowLING (op. cit. in preceding footnote) pp. 170 to 174, where a force varying as any power of r is discussed. Sect. 53. n-body problem. 79 The scattering angle is given by the elliptic integral (52.57) 53. n-body problem. The n-body problem is concerned with the motion of n particles which attract one another gravitationally according to the law of the inverse square. If m; (i = 1, ... n) are the masses of the particles, r; their position vectors, and r;i= -ri;=r;-r;, then the equations of motion are (i = 1, ... n), (53.1) where G is the gravitational constant. The system has three integrals of linear momentum and three integrals of angular momentum, contained in the vector formulae £ m,r, = M = const, l £ ~~:i X r; = h = const. i~l There is also the integral of energy: where T + V = E = const, 1 n • • T = 2 L mi r; . r;, i=l n V = _ ~ Gm;m;. .L.J r .. i,i=t '1 j>i (53.2) (53.3) (53.4) The velocity of the mass-centre is constant, and we can, if we wish, use a frame of reference in which the mass-centre is permanently at rest. Seven integrals of linear momentum, angular momentum, and energy exist also if the forces are of a more general type, provided they act in equal opposite pairs along the lines joining the particles and depend only on the mutual distances. The inverse square law is, however, definitely involved in jACOBI's equation, which reads where d2 ,p --=2T+V dt2 ' n if>= i L: m,r;. i~l (53.5) (53 .6) This rather striking result is most easily proved1 by applying HAMILTON's equations of motion (Sect. 47) to any system with N degrees of freedom having a Hamiltonian of the form H(q, p) = T(p) + V(q), (53.7) where T is homogeneous of degree 2 in the generalized momenta (p) and V is homogeneous of degree -1 in the generalized coordinates (q); the Hamiltonian of the n-body problem is of this form. By virtue of the homogeneities, we have NoH L-a-qe =- V. e=l qe (53.8) 1 Cf. WHITTAKER [28], p. 342, for a different proof. 80 J. L. SYNGE: Classical Dynamics. Sect. 53- If we define lJI by (53.9) (53.10) This is the general form of jACOBI's equation; in the n-body problem we have df/J n • N dt= Lm;r;·r;= LPeqe=lJI. and (53.10) gives {53-5). The quantity i~l e~l n t/>' = _!_ " m; mi r~- 2 L. M '1' i,j~l j>i (53.11) (53.12) where M is the total mass of the system, is independent of the frame of reference, and it is easy to see that t/>' = t/> when the origin is taken at the mass-centre. Hence jACOBI's equation may also be written in the form d2 f/J' = 2 T' + V dt2 ' (53.13) where T is the kinetic energy relative to the mass-centre. If n=2, we have the two-body problem (Sect. 51), which is easily solved. But for n > 2 the problem is of great mathematical difficulty. The case n = 3 (three-body problem) has been of particular interest to mathematicians and possesses a vast literature 1. In the three-body problem there are 9 coordinates and 9 momenta, and the Hamiltonian equations of motion form a system of order 18. By means of the integrals (53.2) and (53-3) it is possible, by application of canonical transformations2, to reduce the order from 18 to 63 ; if the particles move in a plane, the reduction is from order 12 to order 4. Although no general formal solution of the three-body problem is known, there exist special solutions known as LAGRANGE's particles 4, in which the configuration is a rigid line or triangle; these motions are as follows: (a) The particles remain always on a straight line rotating with an arbitrary constant angular velocity, which determines the mutual distances of the particles. (b) The triangle formed by the particles remains equilateral and of constant size, rotating in its plane with an arbitrary constant angular velocity, which determines the size of the triangle. 1 For modern accounts, see WINTNER [30], Chap. 5, and C. L. SIEGEL: Vorlesungen iiber Himmelsmechanik. Berlin: Springer 1956. For current literature on the n-body problem, see the Subject Index of Mathematical Reviews under the heading "Astronomy: 3 and n-body problem"; about fourteen papers appear each year on the average. 2 For canonical transformations, see Sects. 87, 91, 95 of the present Article. 3 See WHITTAKER [28] Chap. 13; FRANK [5], p. 171; GRAMMEL [8], p. 346; G. D. BIRKHOFF: Dynamical Systems, Chap. 9. New York: American Mathematical Society 1927. 4 Cf. WHITTAKER [28], p. 406; RoUTH [22] I, p. 232; C. CARATHEODORY: Sitzgsber. Bayer. Akad. Wiss., Math.-nat. Abt. 1933, 257 (Gesammelte mathematische Schriften, Bd. 2, p. 387. Miinchen: Beck 1955). For elementary solutions of the n-body problem, see HAMEL [11], pp. 449-464. For the stability of LAGRANGE's particles, see WHITTAKER [28], pp. 409-412, and GRAMMEL [8], pp. 370-372. Sect. 54. Periodic structures. 81 54. Periodic structures. Let particles, each of mass m, be attached at equal intervals along an infinite straight string, which is massless. If the particles execute small transverse oscillations, the displacements Yp (t) satisfy the equations (54.1) where a2 =Sf(md), S=tension, d=separation of particles. Equations of the same form occur for longitudinal oscillations if there are elastic connections between the particles. If the initial conditions are (54.2) the solution of (54.1) is oo I Yp= 2: [otp+lhl(2at)+f1p+lf h 1 (2aT)dT], (54.3) 1=-00 0 where ] 21 is the BESSEL function of order 2l. Using the recurrence formulae for BESSEL functions!, it is easy to verify this solution. The above is the simplest example of a vibrating lattice, which may more generally consist of particles of several masses, and may be two-dimensional or three-dimensional, as in the crystal lattice of a solid body. The spatial periodicity of the system is an essential feature 2• For a finite string with fixed ends, carrying n equal particles equally spaced, we have equations of motion as in (54.1), but now with end conditions: Yp= a2 (Yp+l - 2Yp + Yp-1) Yo= Yn+l = 0. (P=1, ... n)} To solve these equations, we substitute Yp = 'Y/p cos (cot+ e) (p = 0, 1, ... n + 1), where 'Y/p, co and e are constants; then (54.4) become a2 "'pH+ (co2 - 2a2) 7Jp + a2 'Y/p-1 = 0 7Jo = "'n+l = 0. This set of ·equations is satisfied by (p = 1, ... n).} . 7Jp=Rezp (P=0,1, ... n+1). (54.4) (54.5) (54.6) (54.7) 1 A very convenient list of formulae for BESSEL functions is given inN. W. McLACHLAN: BESSEL Functions for Engineers. Oxford: Clarendon Press 1934. 2 For the vibrations of lattices, with an historical.introduction and a discussion of electrical systems mathematically equivalent to the mechanical structures, see L. BRILLOUIN: Wave Propagation in Periodic Structures. New York and London: McGraw-Hill 1946. To supplement the history given by BRILLOUIN, it may be noted that HAMILTON worked intensively on this subject under the title "Dynamics of Light", but published only a brief account of his work; see W. R. HAMILTON: Mathematical Papers, Vol. 2, pp. 413-607. Cambridge: University Press 1940, HAMILTON obtained the formula (54.3) above by operational methods, the BESSEL functions appearing as integrals (op. cit., pp. 451, 576). For loaded strings, chains of rods or gyrostats, and networks, see RouTH [22] II, Chap. 9; for loaded strings and molecules, see CoRBEN and STEHLE [3], Chap. 8. For the BoRN-V. KARMAN theory of the specific heat of solids, see M. BLACKMAN, this Encyclopedia Vol. VII part 1, p. 330. Handbuch der Physik, Bd. III/I. 6 82 J. L. SYNGE: Classical Dynamics. Sect. 55. provided the complex z's satisfy a 2 Zp-1-r + (w2 - 2a2) Zp + a 2 Zp_ 1 = 0 Re Zo = Re zn-1-1 = 0. Choose z0 = - i{J, a pure imaginary, and write (p = 1, ... n),} zp=-i(JeiP'P (P=0,1, ... n+1). (54.8) (54.9) Then all of (54.8) are satisfied provided only two equations are satisfied, viz. a 2 ei"' + (w2 -2a2) +a2 e-i"' = 0 '} . . (54.10) Re z fJ e •(n-1-l)_!_!!___ cos (2 at sin ~( r :n:_) + s,) r~l n + 1 2 n + 1 (p = 1, ... n) .J The complex amplitudes Zp (54.9) may be displayed on a circle in the complex plane as in Fig. 20. III. Rigid body with a fixed point. 55. Rigid body under no forces 1• Consider a rigid body on which no external forces act. By (44.4) its mass-centre has a constant velocity, and by (44.7) the motion relative to the mass-centre satisfies h*=O, (55.1) where h* is the angular momentum about the mass-centre. Relative to the mass-centre, the body has three degrees of freedom, and the three scalar equations contained in (55.1) suffice to determine the motion. 1 For analytical details and diagrams, see APPELL [2] II, pp. 164-195; MAcMILLAN [17] II, pp. 192-216; RouTH [22] II, Chap. 4; SYNGE and GRIFFITH [26], J?P· 418-429; WHITTAKER [28], pp. 144-155; WINKELMANN and GRAMMEL [29], pp. 390-404; R. GRAMMEL: Der Kreisel, Bd.1, pp.121-164. 2nd. Edn.: Berlin: Springer 1950. Sect. 55. Rigid body under no forces. If external forces act but have no resultant moment about the mass-centre, the motion relative to the mass-centre is again given by (55.1). This situation arises when a rigid body moves in a uniform gravitational field; then the masscentre moves in a parabola, but the motion relative to the mass-centre is uninfluenced by gravity. If the rigid body is not free, but has a fixed point 0 about which it can turn freely, and if there act on the body no external forces except the reaction maintaining this constraint, then, as in (44.5), we have h = 0, (55.2) h being the angular momentum about the fixed point. The mathematical problems presented by (55.1) and (55.2) are identical, except for the fact that in (55 .1) moments of inertia are to be taken relative to the mass-centre and in (55 .2) they are to be taken relative to the fixed point. In the following discussion we shall deal with (55 .2), with the body turning about a fixed point 0; but the argument applies also to motion about the mass-centre in free motion. Let (i,j, k) be a principal orthonormal triad fixed in the body, and let w be the angular velocity of body and triad. Then (55.3) where A, B, C are the principal moments of inertia at the fixed point 0. By (55.2) the vector his fixed in space, and its magnitude his a constant. We have then a constant. By (25.4) the kinetic energy T is given by A w~ + B w~ + C wi = 2 T, and this is constant since the reaction of constraint does no work. (55.4) (55.5) The motion may be given a vivid and simple description due to POINSOT1 . The POINSOT ellipsoid, with equation (55.6) is fixed in the body, and the motion is described by saying that this ellipsoid rolls on the invariable plane, which is the plane (fixed in space) drawn perpendicular to the fixed vector h at a distance 2 Tfh from 0. The vector drawn from 0 to the point of contact is the angular velocity vector w; the curves traced by this point of contact on the ellipsoid and the plane are called respectively the polhode and the herpolhode. According to EuLER's equations (49.14), the components of angular velocity satisfy A 1 - ( B - C) w2 w3 = 0, l Bw2 -(C-A)w3w1 =0, (55.7) Cw3 - (A -B)w1w2=0. The Eqs. (55.4) and (55.5) are integrals of these equations. Assuming the body unsymmetric, so that A, B and Care distinct, and choosing the triad (i,j, k) so that A > B > C, we obtain an analytic solution of the problem as follows. 1 L. PorNSOT: Theorie nouvelle de la Rotation des corps. Paris: Bachelier 1851. This is interesting historically, because PorNSOT revolted against the purely analytical approach to dynamics advocat:d by LAGRANGE. 6* 84 J. L. SYNGE: Classical Dynamics. Sect. 55. The Eqs. (55.4) and (55.5) are solved for w1 and w3 , and the solutions are.substituted in the second of (55.7). This gives a differential equation for w2 , of which the solution is an elliptic function. Two cases have to distinguished, according as h2 is greater than or less than 2 BT. The solutions are as follows1, expressed in terms of Jacobian elliptic functions of modulus k. h2 > 2BT: m1 =IX dn P (t- t0), m2 = {3 sn P (t- t0), {J = v 2AT -h2 p = v-(h2 -2C T) (A -B) B(A-B) ' ABC ' w3 = y en P (t - t0) , ) k=vB-C.2AT-h2. (55.8) A-B h2 -2C T h2 <2BT: w1 = IX en P (t - t0), m2 = {3 sn P (t - t0), P = V'J2~! -h2) (B-C) ABC ' w3 = y dn P (t - t0) , ) k =VA -B . n2 -2cT . B-C 2AT-h2 {3= In both cases we have Vh2 - 2CT IX= A(A-c)' 2A T- h2 'Y = - C(A - C) • (55 .9) (55.10) Once these components of angular velocity have been found, the description of the motion is completed by introducing the Eulerian angles{}, q;, 'P (Sect. 11) to describe the position of the triad (i,j, k) relative to a fixed triad (I, J, K). Choosing Kin the direction of h, one obtains {} and 'P from {} Cw3 cos =-h-, and q; by a quadrature from Bw tantp= --A 2 , wl (55.11) sin '0 ip = w2 sin 1p- w1 cos 1p. (5 5.12) In the above procedure we make use of the last row of (11.5) and (19.4). The Eqs. (55 .7) have special solutions in which any one of the three components of angular velocity is a constant and the other two components vanish. These correspond to steady rotations about the three principal axes. To· discuss the stability of these steady motions, we note that (55.4) and (-55.5) may also be expressed as follows in terms of the components of h on (i,j, k): ~ + h~ + h: = h2 ' l ~i+~~+~= T. (55.13) Taking (~, h2 , h3) as rectangular Cartesian coordinates in a representative space, we see that the steady rotations correspond to the points (h, 0, O). (0, k, o), (0, 0, h). The Eqs. (55.13) restrict the representative point t-o a curve which is the intersection of a sphere and an ellipsoid, and by examining the forms of these curves it is easy to see that steady rotations about the axes of greatest and least moments of inertia are stable, while a steady rotation about the axis of intermediate moment of inertia is unstable 2• 1 If }!2 = 2BT, the solution is exponential; cf. RouTH [22] II, p. 120. 1 The convenience of this representation is due to the fact that we have to deal with a sphere and an ellipsoid, whereas, if we stick to angular velocity, (55.4) and (55.5) provide two ellipsoids. For a discussion of the two approaches, with diagrams, see ScHAEFER [23], pp. 434-447. Sect. 56. Spinning top. If the body has an axis of inertial symmetry, so that A = B =f: C, the motion is greatly simplified. The PorNSOT ellipsoid is now an ellipsoid of revolution, and the motion is described by saying that a right-circular polhode cone, fixed in the body, rolls on a right-circular herpolhode cone, fixed in space. The cases A >C and A< C have to be distinguished; in the former case the cones are outside one another, but in the latter case the polhode cone (or body cone) contains the herpolhode cone (or space cone)l. 56. Spinning top. The toy spinning top is a solid of revolution which· is set spinning about its axis of sym111etry and placed in· contact with a horizontal plane. The essential feature of this system is that we have a rigid body moving in contact with a fixed horizontal plane under the action of two forces, namely, the force of gravity acting at the mass-centre and the reaction at the point of contact. The contact may be regarded as smooth, in which case we have a bolonomic system. Or it may be rough enough to prevent sliding; then the system is non-holonomic. Or it may be imperfectly rough, in which case the body slides or rolls according to circumstances 2• a.) Unsymmetrical top. In the usual mathematical idealization, the top is regarded as a rigid body with a fixed point 0 (the apex or vertex of the top); it moves under the influence of two forces, viz. the force of gravity acting at the mass-centre D and the reaction at 0 required to hold 0 fixed. The dynamical specification consists of seven numbers: the mass m, the principal moments of inertia A, B, C at 0, and the coordinates ~. 'YJ, C of D relative to the principal axes at 0. The theory of such a top applies also to the motion about its masscentre of a free rigid body acted on by forces equipollent to a single force with fixed magnitude and direction, acting at a point fixed in the body; with this interpretation, the theory has some significance in ballistics, the force being due to the resistance of the air. A top is said to be symmetrical if A=B, ~=rJ=O, (56.1) so that OD is an axis of inertial symmetry. Otherwise the top is unsymmetrical. To discuss the motion of an unsymmetrical top, we may use LAGRANGE's equations with the kinetic energy expressed in terms of the Eulerian angles ({}, fll, 'I') as in (49.4). But we keep the physics of the problem better in mind by using EULER'S equations (49.14), which may be written where A (B- C) w~ =- mg (rJK3 - l;K2), l Bw2 - (C- A) w3 w1 =- mg (C K1 - ~K ), C w3 - (A - B) w1 w2 = - mg (~ K2 - 'YJ K1), 1 For description and diagrams,' see SYNGE and GRIFFITH [26], p. 427. (56,.:;2) (56.3) 2 The best general reference for the treatment of these problems is RouTH [22] II, Chap. 5. For the effects of friction, see also J. H. }ELLETT: A Treatise on the Theory of Friction, Chaps. 5 and 8 (London: Macmillan 1872); and GRAMMEL, pp. 107-121 of op. cit. in Sect. 55. In a recent toy, the tippe-top, the body has a spherical base, and it turns over when set spinning with a sufficiently high angular velocity; for theory see C. M. BRAAMS: Physica, Haag 18, 497 (1952); k D. FoKKER: Physica, Haag 18, 503 (1952); F. A. HARINGX: De Ingenieur 4, Technisch Wetenschappelijk Onderzoek 2 (1952); N. M. HuGENHOLTZ: Physica, Haag 18, 515 (1952); S. O'BRIEN and J. L. SYNGE: Proc. Roy. Irish Acad. A 56, 23 (1954); D. G. PARKYN: Math. Gaz. 40, 260 (1956). 86 J. L. SYNGE: Classical Dynamics. Sect. 56. a unit vector directed vertically upward (Fig. 21). Since K is fixed in direction, we have • aK o = K = Tt + wxK, (56.4) or, explicitly, K. K K ) • 1 + W2 a - Wa 2 = 0, K2 +w3 K1 -w1 K3 =0, (56.5) K3 + w1 K2 - w2 K1 = 0. In (56.2) and (56.5) we have six differential equations of the first order for w1 , w2 , w3 , K1 , K 2 , K 3 , which quantities are expressible in terms of the three Eulerian angles and their first derivatives. 0 K These equations possess the following integrals, resulting from the constancy of total energy, the vanishing of the moment of gravity about the vertical through 0, and the fact that K is a unit vector: f(Aw~ + Bwi + Cw~) + + m g (~ K1 + 1J K2 + C K3) = const, l Aw1 K1 + Bw2 K2 + Cw3 K3 = const, K~+K~+K~=1. (56.6) In certain special cases additional integrals exist. Of these, the most famous is KoWALEWSKI's integral, which occurs in the case where A=B=2C, C=O. (56.7) We make 1]=0 by changing to new principal axes (i,j, k). Then (56.2) and (56.5) give Fig. 21. The unsymmetrical top. The force of gravity is - mgK. 2~+~ww =~{3K ,} K + tK w3 = twK3 , (56.8) where w =w1 +iw2 , K =K1 +iK2 , {3=mg~fC. On eliminating K3 , we get ·:t (wZ- {3 K) + i w3 (wz- {3 K) = 0, (56.9) and hence :t log ( w2 - {3 K) = - i w3 . (56.1 0) On adding the complex conjugate we obtain the required integral (w2 - {3 K) (w2 - {3 K) = const, or (w~ - w~ - fJ K1) 2 + (2w1 w2 - fJ K2) 2 = const. (56.11) (56.12) This integral, together with the first two of (56.6), gives us three equations in the Eulerian angles and their first derivatives; these equations can be solved in terms of hyperelliptic functions 1 . 1 The above argument follows RoUTH [22] II, pp. 159-161, or APPELL [2] II, pp. 209 to 211. For a treatment of KOWALEWSKI'S top by LAGRANGE'S equations, see WHITTAKER [28], pp. 164-t67. These references may be consulted for other integrable cases of the unsymmetrical top. For much detailed work on the unsymmetrical top, including the use of HAMILTON's equations, see HAMEL [11], pp. 407-449. See also GRAMMEL, pp. 164-214 of op. cit. in Sect. 55. Sect. 56. Spinning top. 87 (3) Symmetrical top: general motion. To deal with a symmetrical top, satisfying (56.1), it is convenient to use the two orthonormal triads (i,j, k), (1, J, K) shown in Fig. 22. (1, J, K) is fixed in space, with K pointing vertically upward; k lies along the axis of symmetry 0 D and j is horizontal. The position of the triad (i,j, k) is described by the co-latitude{} and the azimuthal angle rp shown in Fig. 22. The angular velocities w and .2 of the body and of the triad (i,j, k) respectively are where W = WI~ + ~ + W 3 k, } .2 =Dit +il2J +D3 k, wi = QI =sin {}tjl, w2 = D2 = - J, Q~ = cos{}tjl; the angular momentum about 0 is h=Awii+Aw2 j+Cw3 k; (56.15) the moment of gravity about 0 is G=akx ( -mgl()=-mgasin{}j, (56.16) where OD=a. The equation of motion is (56.17) (56.13) (56.14) and this leads to three differential equations for{}, rp and w3 • But it is convenient to proceed indirectly!. By (49.17) we have W 3 =S, (56.18) Fig. 22. Fixed triad (I, J, K) and moving triad (i,j, k) for symmetrical top. a constant (the spin of the top). We have also { 1 (h·K) = h·K= G ·K= 0,) h·K=a, (56.19) a constant (the angular momentum about the axis K), and we have the integral of energy (56.20) a constant. Substituting from (56.14), (56.15), and (56.18) in (56.19) and (56.20), we get the following two equations for {} and q;: Asin2 {}tjl=a-{Jcos{}, ) A(/}2 + sin2 {}tjl2) + ~ = 2(E- m g a cos{}), (56.21) where fJ = C s. Eliminating tjl and putting cos{}= x, we get the differential equation (56.22) 1 For a direct treatment of the symmetrical top by LAGRANGE's equations, see WHIT TAKER [28], pp. 155-163, where both Eulerian angles and CAYLEY-KLEIN parameters are discussed. For a symmetrical top on a smooth plane, see WHITTAKER [28], pp. 163-164. 88 J. L. SYNGE: Classical Dynamics. Sect. 56. Since I (x) is positive in the motion (except when x = 0), and I ( -1) < 0, I (1) < 0, this cubic in x has three real zeros x1 , x2 , x3 , such that - 1 < X1 < X2< 1 < X3 , (56.23) special cases of equality being disregarded here. The variable x oscillates on the range x1 .:;;;; x .:;;;; x2 , and the solution is cos{} = x = x1 + (x2 - x1) sn2 n (t - t0), (56.24) where (56.25) k being the modulus of the Jacobian elliptic function sn. The azimuthal angle cp is given by . oc-flx cp = A (1 _ x2) • (56.26) It is clear that cp has one sign throughout the motion if, and only if, a./{3lies outside the range (x1 , x2). The motion is most clearly followed by tracing the path of the point k on the unit sphere; its polar coordinates are (fJ, cp). This path is bounded by the two circles x = x1 (above) and x = x2 (below), and the path crosses itself if, and only if, cp changes sign during the motion1 . y) Symmetrical top: steady precession. Any motion of the top may be maintained by applying to it a torque G = h, according to (56.17). Consider, with the notation of Fig. 22, the steady motion given by {} = const, cp = p t, w3 = s, (56.27) where p and s are any constants. By (56.14) and (56.15) we have then w1 =!21 = p sin{}, w2 =!22 = 0, !23 = p cos{},) h=ApsinfJi+Csk, h = Qxh = psinfJ(Ap cos{}- Cs)j. (56.28) The torque required to maintain this motion is precisely the gravitational torque (56.16) provided that p and s satisfy the equation Csp-Ap2 cosfJ=mga. (56.29) This is the equation defining the steady motions of the symmetric top with its axis inclined at an angle{} to the vertical; s is the spin and p is the precessional angular velocity with which the axis of the top rotates about the vertical drawn through the vertex of the top. Given any values of p and fJ, a spins can be found to satisfy (56.29). Conversely, given s and{}, (56.29) is satisfied by two real values of p, viz. (56.30) 1 For further details on the motion of a symmetrical top, see APPELL [2] II, pp. 197-209; MAcMILLAN [17] II, pp. 216-249; RouTH [22] II, Chap. S; SYNGE and GRIFFITH [26], pp. 432-440; WINKELMANN and GRAMMEL [29], pp. 406-422. Reference may also be made to the classical treatise of F. KLEIN and A. SoMMERFELD: Ober die Theorie des Kreisels. Leipzig: Teubner 1897-1910. For much detailed information about the theory of the top and gyroscopic applications, see R. GRAM MEL: Der Kreisel, 2 vols. 2nd. Edn.: Berlin: Springer 1950. See also A. GRAY: A Treatise on Gyrostatics and Rotational Motion. London: Macmillan 1918. Sect. 56. Spinning top. 89 provided that 2 4A m g a cos{} s >--c2--· (56.31) If s is large, one of these precessional angular velocities is small and the other is large; the small value is approximately mga P = -cs · (56-32) which is a very useful simple formula from which the spin can be computed from measurement of the slow precession. c5) Stability of a sleeping top. A symmetrical top is said to be sleeping when it spins about its axis of symmetry with that axis vertical. In this motion, with f(x) f(x) I Xo>l Fig. 23. Graph of the fundamental cubic for a stable Fig. 24. Graph of the fundamental cubic for an unstable sleeping top. sleeping top. spin s, the constants IX, {J, E which occur in the cubic f(x) of (56.22) have the values IX = fJ = C s, E = t C s2 + m g a, (56.33) and the cubic is f(x) = ----- (1 - x) 2 1 + x- -- - . 2mga ( C2s2 ) A 2A mga (56. 34) This has a double zero at x = 1 and a single zero at C2s2 X = X = ---- -- - 1 0 2Amga ' (56.35) unless it happens that x0 = 1, in which case there is a triple zero. Let us suppose that x0 9=1. Then we have two cases: x0 >1 as shown in Fig. 23, and x0 <1 as shown in Fig. 24. Any disturbed motion for which the constants IX, {J, E differ little from the values (56.33) will have its range of oscillation (x1 ,x2) controlled by a cubic function, as in (56.22), with a graph which differs little from the graph shown in Fig. 23 or 24, whichever applies to the undisturbed motion. The disturbed graph (indicated by the broken lines in the figures) will have zeros at (x1 , x2), where x1 0 and (ii) a period of restitution with c <0. In the period of compression there act impulsive forces just sufficient to reduce c to zero. Denoting them by (-In, In), and writing v~, v~', ... for quantities at the end of the period of compression, we have (v~'-v )=-In, m2 (v;'-v 2)=In, ) h~'- h1 =-r1 xin, h~- h2 = r 2 xin, c" = n · (u~- u~) = 0. (59.8) The 13 scalar equations contained in (59.8) suffice to determine I and the state of motion at the end of the period of compression. As for the period of restitution, it is assumed that the impulsive forces for this period are proportional to the impulsive forces during compression; the factor of proportionality, denoted by e, is called the coefficient of restitution. Its value ranges from e = 0 (inelastic collision) to e = 1 (elastic collision); collisions with intermediate values of e are called semi-elastic. The final result of the collision is given by substituting F=(1+e)In (59.9) in (59.1), the value of I having been found from (59.8). It can be shown that e = 1 implies T' = T as in (59.5) As an illustration of the coefficient of restitution, consider the collision of two smooth homogeneous spheres. In this case the impulsive forces have no moments about the mass-centres, and (59.8) reduce to I= _mtmz_n. mt + mz (vl- v2) l m1 m2 c -~~ m--;' where c is the initial speed of compression. By (59.1) and (59.9) we have (59.10) (59.11) (v~-v )=-(1+e)In, m2 (v;-v 2)=(1+e)In, (59.12) 96 J. L. SYNGE: Classical Dynamics. and therefore the velocities after collision are v' = v - __ »Z~c- (1 +e) n ) 1 1 ml + m2 ' v~=t, + -~L-(1+e)n. ml + m2 The loss of kinetic energy is Sect. 60. (59.13) (59.14) 60. Minimal theorems in impulsive motion 1• For a system of P particles, acted on by impulsive forces F;, we have, by (58.2), (60.1) where W; are arbitrary vectors and V;, v~ the velocities of the particles before and after the application of the impulsive forces. Here and below, summations are fori= 1, . .. P. The Eq. (60.1) may be regarded as a form of n'ALEMBERT's principle (Sect.45), valid for impulsive motion. The system may be subject to constraints, due to which certain particles are fixed or confined to smooth fixed curves or surfaces, or the distances between certain particles are kept constant (rigidity). Such constraints may persist through the application of the impulsive forces, or they may suddenly come into existence, or they may be suddenly abolished. In any case we can break down the impulsive forces into (60.2) where P; are given or applied impulsive forces, and R; are impulsive forces of constraint. The latter satisfy ~ ~R,·w,=O, (60.3) if w, are velocities satisfying the constraint. oc) CARNOT's theorem (first part). Theorem: If there are no applied impulsive forces, the sudden introduction of a constraint reduces the kinetic energy 2• Toprovethistheorem, we choose W 0=vi in (60.1); the right hand side vanishes, and we have ~ m, (v; - v;) · v; = 0. (60.4) The loss of kinetic energy can therefore be expressed as a positive-definite expression: T- T' = l ~ m, v, · v, -l ~ m, v; · v; } = l ~ m, (v;- v;) · (v,- v:J > 0. (60.5) fJ) CARNOT 's theorem (second part). Theorem: Kinetic energy is increased when rigid bonds are broken by an explosion. By an "explosion" we understand that impulsive forces operate between the particles of the system, in equal opposite pairs acting along the joining line, like action and reaction in NEWTON's Third Law. Although they occur in these balanced pairs, these are applied impulsive forces P;, not impulsive forces of constraint R;; the latter will be present in those bonds of rigidity which remain unbroken. 1 Cf. RoUTH [22] I, pp. 298-304; APPELL [2] II, pp. 527-539. 2 The kinetic energy is unchanged in the exceptional case where the introduction of the constraint changes no velocity. Exceptional cases like this are omitted in the enunciations and proofs. Sect. 60. Minimal theorems in impulsive motion. 97 For the same geometrical reason as that by which ordinary reactions in rigid bonds do no work, we have V; being the velocities before the explosion; we have also L,R;·t';=O, and hence, on putting W;=V; in (60.1), L, m; (v; - v;) · V; = 0. (60.6) (60.7) (60.8) Consequently the gain in kinetic energy can be expressed as a positive-definite expression : T' - T = t L, m; v; · v; - t L, m; V; · V; } = t L, m; (v; - v;) · (v; - v;) > 0. (60.9) y) KELVIN's theorem. Theorem: If a system, initially at rest, is set in motion by applied impulsive forces acting on named particles of the system, these impulsive forces being such that the velocities of the named particles have prescribed values, then the kinetic energy is less than that of any hypothetical motion in which the constraints of the system are satisfied and the named particles have the prescribed velocities. Let v~ be the actual velocities and V~1 the velocities in the hypothetical motion, so that v~ = V~1 for the named particles. Then (60.10) if W; = v~- v;1 , because ~ = 0 except for the named particles and W; = 0 for them. Further ~ L R;. W; = 0, (60.11) because both v: and v; 1 satisfy the constraints. Putting V; = 0 m (60.1), since the system is initially at rest, we have then '\' I ( I ") LJ m;V; · V;- V; = 0; (60.12) hence T"- T1 can be expressed as a positive-definite expression: T il Tl 1 '\' " " 1 '\' ' I } - = 2 LJ m,.· V; · V; - 2 L. m; V; · V; 1 '\' ( " I) ( , I) = 2 LJ m; V; - V; · V; - V; > 0. (60.13) (J) BERTRAND's theorem. Theorem: A system in motion is acted on by given applied impulsive forces, and as a result its kinetic energy becomes T'. Then T' > T", where T" is the kinetic energy resulting from the application of the same impulsive forces to the same initial motion, but now subject to constraints consistent with that motion. Let V; be the initial velocities, v; the final velocities in the absence of the additional constraints, and v;l the velocities in the presence of those constraints. If ii: and il;l are the impulsive forces of constraint in the two cases, we have L ~~ " "'R~" " R; . V; = 0' LJ i . t'; = 0' (60.14) since it is a question of additional constraints. Then, by (60.1), P; being the given applied impulsive forces, we have "'m- (v~- v.) · v" = P. · t{ 1 L...J } ~ ~ -~ , 1 l' '\' (" ) " p~ " LJ m1 V; - t'; · V; = ; · V; , (60.15) Handh11rh drr Physik, Brl. III/I. 7 98 J. L. SYNGE: Classical Dynamics. Sect. 61. and so, by subtraction, {60.16) Therefore T' - T" may be expressed as a positive-definite expression: (60.17) E. General dynamical theory. I. Geometrical representations of dynamics. 61. The role of ~eneral dynamical theory. The most obvious goal of all dynamical theory is to solve dynamical problems which arise in physics or astronomy. Starting from a physical concept (Sect. 2), such as the solar system, we set up a corresponding mathematical concept, or mathematical model, and try to solve the differential equations belonging to that model. But it is not altogether clear what we mean when we speak of solving a set of differential equations. True, a problem is regarded as solved when the coordinates of the particles of the model at time t have been expressed as simple functions of t and of those parameters which define the initial positions and velocities. But what are simple functions? We no longer regard a function f (t) as a formal expression in t, but as a quantity determined by t, and it is impossible to draw a sharp line between functions which are simple and those which are not. If we drop the word simple and speak merely of functions, then every dynamical problem is solved as soon as it has been well stated, because the differential equations, with the initial conditions and the value of t, determine the coordinates at time t. This is not mathematical hair-splitting, but hard fact, for modern methods of electronic computation provide arithmetical solutions to dynamical problems to any desired degree of accuracy, the differential equations being replaced by difference equations. In ballistics, for example, this modern method has largely replaced the search for formulae representing the solution1 . But, though precise definitions may elude us, there can be no doubt that the two-body problem has a simple solution, whereas the three-body problem has not. In the case of the two-body problem we have formulae involving parameters; we can change the values of those parameters, and so study what we may call the mathematical structure of the class of all solutions, with intellectual satisfaction and understanding. We can, moreover, form accurate vivid mental pictures of the behaviour of the two bodies, so that their motion becomes almost as real to us as the motion of a piece of machinery working before our eyes. In the case of the three-body problem, a numerical solution, based on assigned values of the parameters involved, tells us how the bodies move under those given circumstances. But a single numerical solution doe:> not reveal the mathematical structure of the problem, nor does a collection of such solutions. In this case, as in many others, we must seek an understanding of mathematical structure by examining the differential equations themselves. But we can be more ambitious. We can aim, not at understanding the mathematical structure of some selected dynamical problem, but at understanding the mathematical structure of a class of problems so wide that we may regard the whole of dynamics as our objective. We shall concentrate our attention 1 The historical development of the idea of an "unsolved" dynamical problem is discussed by WrNTNER [30], p. 143. Sect. 61. The role of general dynamical theory. 99 on those systems to which LAGRANGE's equations of motion, or HAMILTON's, are applicable, but that includes a very wide range of problems indeed. We recognize two purposes in the study of general methods in dynamics. First, the practical purpose, to increase our power in solving specific problems by developing standard techniques with a wide range of applicability. Secondly, the intellectual purpose, to understand the mathematical structure of dynamics. But it is not as simple as that. The development of quantum mechanics out of classical dynamics shows that, in the long run, an understanding of the mathematical structure of dynamics may have more practical results (i.e. results which increase our knowledge of the physical world) than a concentration on specific problems of the type for which dynamical methods were originally designed. With this in mind, the emphasis in the following account of general dynamical theory will be laid on mathematical structure, specific dynamical problems being regarded more in the nature of illustrations than as objectives in themselves. We seek then to understand the mathematical structure of dynamics. But here we encounter a formidable difficulty, because understanding is a very personal matter. It is not a question of accepting this theorem or that, but of gaining an overall picture in which the details are subordinated to a central idea, and opinions are bound to differ widely as to the best central idea to choose. What is psychologically satisfying to one man may not suit another. In elementary dynamics we all meet on common ground, for we share a. vivicl geometrical intuition of the motion of a particle, and to this intuition the formulae we use are subordinated. There exists, in fact, a triangle of mental links of this nature: physical concept geometrical picture formulae But when we pass on to more complicated dynamical systems, the geometrical picture becomes harder and harder to follow, and it is thrust into the background, so that dynamics becomes, to a great extent, a matter of formulae only. This is satisfactory for those who take pleasure in purely formal arguments, but for most of us the loss of geometrical intuition is a serious handicap. In the present article an attempt is made to give geometrical intuition its just place in general dynamical theory by the systematic use of representative spaces in which the motion of a representative point corresponds to the motion of a dynamical system1. 1 The geometrical approach to dynamics seems to have originated with H. HERTZ: Die Prinzipien der Mechanik in neuem Zusammenhang dargestellt (Leipzig 1894). An English translation by D. E. JONES and J. T. WALLEY ( 1899) has been republished recently: H. HERTZ: The Principles of Mechanics (New York: Dover Publications Inc. 1956). This contains an Introductory Essay by R. S. CoHEN (with a bibliography) and a Preface by H. VON HELMHOLTZ. See also G. RicCI and T. LEvi-CIVITA: Methodes de calcul absolu et leurs applications. Math. Ann. 54, 125-201 ( 1901) and Paris: Blanchard 1923; J. W. GIBBS: Elementary Principles in Statistical Mechanics (Yale: Scribner 1902; see The Collected Works of J. W. GIBBS, Vol. 2. New York-London-Toronto: Longmans & Green 1928); J. L. SYNGE: Phil. Trans. Roy. Soc. Lon d., Ser. A 226, 31 -106 (1926); L. BRILLOUIN: Les Tenseurs en mecanique et en elasticite (Paris: Masson 1938); LANCZOS [15]; Pll.ANGE [21]. 7* 100 J. L. SYNGE: Classical Dynamics. Sect. 62. General dynamical theory occupies a curious position in physics. Historically, it has been suggested by, and developed in terms of, the Newtonian dynamics of particles and rigid bodies. But we feel an urgent need to give it a wider scope, presenting it as a consistent mathematical theory applicable to any physical system the behaviour of which can be expressed in Lagrangian or Hamiltonian form. There is a temptation to present it as pure mathematics, and the exposition which follows is a compromise. The argument is not of sufficient precision to satisfy a modern pure mathematician, but at the same time it does attempt to exhibit a mathematical structure independent (except for suggestion and explanation) of the preceding part of this article. Everything is based on a Lagrangian or a Hamiltonian, or an equivalent concept. Kinetic energy, so important in the direct physical applications of Newtonian dynamics, plays a minor part (in illustrative examples and in Sects. 84 and 85), and the Hamiltonian is not restricted to be a quadratic function of the generalized momenta, as it always is in Newtonian dynamics. 62. Representative spaces. The following representative spaces will be used to elucidate dynamical theory1 : Name Configuration space . Space of events . . . Momentum-energy space Phase space . . . . . . Space of states . . . . Space of states and energy Symbol Q QT PH QP QTP QTPH I Dimensionality I N N+1 N+1 2N 2N+ 1 2N+2 Coordinates Here and throughout PartE, except where noted to the contrary, small Greek suffixes take the values 1, 2, ... N, and small Latin suffixes the values 1, 2, ... N + 1, with the summation convention for a repeated suffix in each case. The representative spaces are listed above in order of increasing dimensionality; they will be treated in a different, more convenient, order. Our aim is to present dynamical theory in a fairly abstract way, so that the results may be applicable outside the range of traditional Newtonian dynamics. But, to keep our feet on the ground, let us briefly consider the above representative spaces relative to a Newtonian system (holonomic, and either scleronomic or rheonomic) with N degrees of freedom and generalized coordinates qP. The system possesses a Lagrangian L(q, t, q) and moves in accordance with LAGRANGE's equations of motion (Sect. 46); equivalently, it possesses a Hamiltonian H(q, t, p) and moves in accordance with HAMILTON's equations (Sect. 47). We shall call the system conservative if t is absent from H (or, equivalently, absent from L), so that we have as in (47.9), this implies aH --- = o· at ' H=E, (62.1) (62.2) a constant of the motion. All motions for which E has a common value constitute an isoenergetic dynamics 2• 1 There are other representative spaces which might be considered, such as the space of 3N dimensions, with coordinates qQ, qp, Pp· used by P. A.M. DIRAC, Canad. J. Math. 2, 129-148 (1950). 2 A family of orbits which have the same constant energy is also called a natural family; cf. \VHITTAKER [28], p. 387. Sect. 62. Representative spaces. 101 Of all the representative spaces, Q is the simplest. If the system consists of a single particle moving in ordinary space, then Q is ordinary space; and if the particle is constrained to move on a surface or curve, then Q is that surface or curve. However, the picture of the totality of trajectories is somewhat complicated, for a trajectory is not determined by a point in Q and a direction in Q (i.e. the ratios dq1 : dq2 : ••• dqN)· For a conservative system, a single infinity of tra~ jectories correspond to a given direction at a point (consider a particle in a gravitational field); for a non-conservative system, there is a double infinity of trajectories. The totality of trajectories is easier to visualize in Q T, in which a trajectory is determined by a point and a direction (i.e. the ratios dq1 :dq2 : ••• dqN:dt, which ratios are the generalized velocities). This applies whether the system is conservative or not. Moreover, the treatment oft on a parity with the coordinates qu makes Q T a suitable background for relativistic dynamics. - The space PH is of rather secondary importance. It is useful when we deal with an encounter, in which a number of particles, initially in free independent motion, come under the influence of one another, and then separate with final motions free and independent. When the particles are moving freely and independently (i.e. before and after the encounter), the representative point maintains a fixed position in PH, and the effect of the encounter is to move this point from one such position to another. The space Q P is, thanks largely to the work of J. W. GIBBS on statistical mechanics, probably the best known of the spaces listed above. If the system is conservative, then the totality of trajectories appears as a congruence of curves in Q P, one curve passing through each point. This is a satisfyingly simple picture, but it is complicated in the non-conservative case, for then there is a single infinity of trajectories through each point. Moreover, it is not well suited to relativity, for which t should be treated on a parity with qQ. In the space Q T P the time t is treated on a parity with the coordinates qQ and the momenta Pe; the Hamiltonian H(q, t, p) is a function of position in the space. The picture of the trajectories is simpler than in Q P for a non-conservative system, for now we have a congruence of curves, one through each point. Q T P differs from Q P and Q T PH in having an odd dimensionality-an important difference from a mathematical standpoint. The space Q T PH provides the most general approach to dynamics. In it t and H are treated on a parity with qQ and pQ, so that there is complete forinal symmetry. The 2N +2 coordinates fall into two groups, (q, t) and (p, H), the two groups being almost interchangeable in the dynamical theory. To preserve symmetry, a dynamics is best defined, not by giving H as a function of (q, t, p), but by writing down an energy equation involving, in general, all the 2N + 2 coordinates of Q T PH. This equation defines a surface of 2N + 1 dimensions in Q T PH, and the representative point is confined to this surface; but it is sometimes convenient to use an energy function instead of an energy equation, in order to deal with the whole space instead of merely with this surface. The use of Q T PH is immediately suggested by HAMILTON's optical method1 in which all the coordinates in space are treated on a parity. The symmetrical approach to dynamics is indicated in HAMILTC1N's calculus of principal 1 W. R. HAMILTON: Mathematical Papers I. Cambridge: "L'niversity Press 1931. See footnote in Sect. 67. 102 J. L. SYNGE: Classical Dynamics. Sect. 63. relations1, and in modern times it has been revived 2• All theory developed in Q T PH can be immediately transferred to isoenergetic dynamics in Q P by a mere reduction of dimensionality. 63. Topological remarks. As a simple illustration of a topological situation often encountered in a more complicated form, consider a particle moving on a circle with equation (63 .1) where (~. 'Y}) are rectangular Cartesian coordinates and a a constant. The configuration space Q is the circle itself, and we may assign a generalized coordinate q by writing ~=a cosq, 'YJ = asinq, - oo 0), (64.4) and, by EuLER's theorem for homogeneous functions, (64.5) As regards smoothness, we shall assume the existence and continuity of whatever derivatives we may require. Should piecewise discontinuities arise for consideration, they can be dealt with on the occasion. In a region of overlapping coordinate systems, A transforms as an invariant in the sense of tensor calculus. If the two coordinate systems are x: and x, and the two Lagrangians are A* and A, then A* (x*, x*') =A (x, x'). (64.6) The Lagrangian action 2 along any directed curve r drawn from a point B* (where u = u1) to a point B (where u = u2 > u1) is defined to be u, AL(T) = J A(x, x')du, (64.7) 1 The present discussion is intentionally kept on a somewhat abstract level, in order not to limit the scope of ultimate applications of the theory. The word "Lagrangian" forms a link with the more concrete theory of Sects. 46, and hence with physical concepts (Sect. 2). f "'L 2 Unfortunately the simple word action is commonly used for ··~. dqQ (cf. GoLDSTEIN [7], . oqP p. 228; WHITTAKER [28], p. 248), and there 1s no commonly accepted name for the more fundamental integral (64. 7); it will be called Lagrangian action in this article. 106 J. L. SYNGE: Classical Dynamics. Sect. 64. so that AL(T) is a functional of the curve r. By reason of the homogeneity of A, AL(F) is independent of the parametrization. The element of Lagrangian action lS dAL =A (x, X 1 ) du =A (x, dx). (64.8) There is, in general, no connection between the action from B* to B and the action from B to B*, even though the curve is the same in both cases. By imposing an element of action A (x, dx) on Q T, we make it a FINSLER space in the language of geometry. If A (x, dx) is the square root of a homogeneous quadratic form in the differentials, then Q T is a Riemannian space. The element of Lagrangian action {64.8) may be written more explicitly as {64.9) By reason of the homogeneity of A, dAL is independent of the choice of the parameter u. Choose u=t; then, by (64.1), dAL = A(ql, ... qN, t, ql, ... qN, 1) dt. (64.10) Define the function L(q, t, q) by L(ql, ... qN, t, ql, ... qN) =A (ql, ... qN, t, ql, ... qN, 1). (64.11) Then the element of Lagrangian action is dAL = L(q, t, q) dt. (64.12) This function L (q, t, q) we call the ordinary Lagrangian function (cf. Sect. 46). The two functions, A (a function of 2N + 2 quantities, positive homogeneous of degree unity in the last N + 1 of them) and L (a function of 2N + 1 quantities, with no such condition imposed), are equivalent to one another, in the sense that one determines the other. Given A, we get L by (64.11); given L, we get A by equating (64.9) and (64.12) and dividing by du. Thus =L(ql, ... qN,t,ql,···qN)t 1 (64.13) A(x1 , ••. xN, xN+I• x~, ... x.~. x;,+-1) l ( X~ XN ) 1 = L xl, ... xN, xN+l• --, --' ... --,--~ xN+l' XN+l XN+l which has the required homogeneity. To find the relationships between the partial derivatives of A (x, X1 ) and L(q, t,q), we vary x, and x; in (64.13). This gives But aA ..: aA ..: I I ( aL ..: aL ..: aL ..: . ) L ..: I ~a~ ux, +-a-, ux, = t -a- uqQ +--at ut + -;;•- uqQ + ut . x, x, qe oqe (64.14) (64.15) Substituting this in (64.14), writing he for c5qe, h~ for c5q~, c5x;_,+ 1 forM, and equating the coefficients of the 2N + 2 independent differentials c5 x,, c5 x;, we get ~=tl_aL_ axe aqe ' aA aL (64.16) Sect. 65. First form of HAMILTON's principle. LAGRANGE's equations of motion. 107 65. First form of HAMILTON'S principle. LAGRANGE's equations of motion. Let r (Fig. 32) be any curve joining B* to B. We vary it to a neighbouring curve~ with end points D*, D. Whatever parameter u is used on T, we choose the parameter u on ~ with the same end values, u1 , u2 • Then the variation of the Lagrangian action is t5AL = AL(~)- AdT) l = j't5Adu= j'(:~·bx,+ :~ax;)au. u1 U 1 Integration by parts gives [ OA ]u=u, j"'( d oA oA) t5AL= o;-,-t5x, - -a·-~-,--~- t5x,du. UXy U=Ul U UXy uX1 Let us now restrict the varied curve ~ by demanding that its end points coincide with B* and B. The first term on the right hand side of (65.2) disappears. If t5AL=O for all variations of r to ~' arbitrary except for the condition of fixed end points, then r must satisfy the EULER-LAGRANGE equations d oA oA ~---,--~-=0. (65-3) du ox, ox, We may refer to these equations as LAG RANGE's equations of motion, and to the curves satisfying them as rays or trajectories 1 . (65.1) (65.2) /} The variational equation Fig. 32. Variation of Lagrangian action. t5 fA (x, x') du = 0, (65.4) for fixed end points, is equivalent to the set of differential equations (65.3). We call (65.4) the first form of HAMILTON's principle 2• This principle can also be written t5JL(q,t,q)dt=O, (65.5) for fixed end values of qe and t, and this leads at once to LAGRANGE's equations of motion in the form [cf. (46.18)] d oL 8L it a,z;- oqQ = o, (65.6) which are equivalent to (65.3). The expression occurring here is called the Lagrangian derivative of L. 1 The word trajectory links the mathematical concept with the physical concepts of dynamics. The word ray links it with optics, and that may seem out of place in the present connection. But the wave mechanics of DE BROGLIE and ScHRODINGER has weakened the barrier between dynamics and optics. If we need the word wave in dynamics, the word ray enters naturally with it. Moreover, the present exposition of general dynamical theory owes as much to HAMILTON's method in optics as it owes to his method in dynamics, for in his optics all the coordinates were put on a parity, whereas in his dynamics the time was privileged. 2 From a mathematical standpoint, (65.3) and (65.4) are different ways of saying the same thing. We seem to be indulging in a plethora of verbiage. But words are important, as indicated in Sect. 2. The differential equations (65.3) cannot be put into words which are more illuminating than the bare formulae. HAMILTON's principle can: it says that the Lagrangian action has a stationary value for a trajectory. These words are easier to link up with physical concepts than mathematical formulae are. 108 ] . L. SYNGE: Classical Dynamics. Sect. 66. There are N + 1 equations in (65.3) and only N in (65.6). However, (65.3) are identically related, for we have, remembering the homogeneity of A, x; (d~t ~:,- --~1-) = d~ (x;~1--)- x;'-::.-- x;~~ = !~- -~~ = 0. (65.7) , r ' , ' Provided the Eqs. (65.6) can be solved for q, (and we shall suppose that they can), these equations determine a ray in Q T corresponding to assigned initial values of q,, t and q,. Then, through each point x, of Q T and -in each direction (defined by the ratios dx1 :dx2 : ••• : dxN+I) there passes a unique ray or trajectory. We may use the term A-dynamics to refer to theory based on the variational Eq. (65.4) and its extremals (65.3); likewise the term L-dynamics for the variational Eq. (65.5) and the extremals (65.6). They are essentially equivalent to one another, L-dynamics being a form of A-dynamics in which tis treated in two capacities: a coordinate in Q T and a parameter on a curve in QT. We include them both under the title Lagrangian dynamics. 66. Two illustrative examples. LAGRANGE's equations (65.6) form the link between the dynamical systems most commonly encountered and the present more abstract theory. This theory applies to all physical systems which behave in accordance with (65.6), whether those systems are truly dynamical or not. The system may consist of electrical circuits, with the generalized velocities corresponding to currents. In the truly dynamical domain, the present theory applies, by virtue of (46.18), to all holonomic systems for which the generalized forces are derivable from a potential function or an extended potential function. In such systems the kinetic energy is always quadratic in the generalized velocities, and so also is the Lagrangian L ( = T- V) when V is an ordinary potential. In the present general theory no such restriction is placed on the function L(q, t, q), which is to be regarded as an arbitrary function of its 2N + 1 arguments. To illustrate the general procedures, we shall keep before us the two following examples: rx) R S (relativistic system): We take for homogeneous Lagrangian A(x, x') = Vb-. x; x; +A, x;, (66.1) where b,,(=b,,) and A, are functions of the x's. This is a generalization of the relativistic Lagrangian for a charged particle moving in a given electromagnetic field 1 . The homogeneous Lagrangian is simpler than the ordinary one, which, by (64.11), reads L(q, t, q) = (beaiJ/la + 2be,N+I tle + bN+l,N+l)i + Aeqe +AN +I· (66.2) {3) ODS (ordinary dynamical system): This is a holonomic scleronomic conservative system with an ordinary potential function, so that L(q, q) = T(q, q)- V(q) = i aeaqeqa- V, (66.3) the coefficients aea(= aael and V being functions of the q's. Thus tis absent from L. In this case the homogeneous Lagrangian is, by (64.13), (66.4) a little clumsier than L(q, q). 1 Cf. Sect. 114. Sect. 67. The energy equation !1 (x, y) = 0 and the Hamiltonian H (q, t, p). 109 It is clear that the theory for RS will be simpler if we use A(x, x'), and the theory for ODS simpler if we use L(q, q). HAMILTON's principle reads RS: 0 J [(b, dx,dx,)~ + A,dx,] = 0, ODS: bf(faeariila- V)dt=O. For RS LAGRANGE's equations of motion read and if we choose for u that parameter on the ray which makes they simplify to _d_(b x')+(o~,-~As)x'_i}/)st x;x; = 0, du rs s ox5 ox, s ox, or b "+(oA_,__!As_) '+(~s_obts). ''=O T s xs ~ ~ xs " ~ xs XI . ux5 uXr , uXt uXr For ODS LAGRANGE's equations of motion read d oT oT or or •• { } • • a oV qQ + f"QV q,,qV =- ae EJqa' where the CHRISTOFFEL symbols are [,u v a]= 1_ ( oa'"a +!Java- ~a'"v). ) ' 2 oqv oq'" oqa ' {J'.}= aea[,uv,a], aeaa'"a=b~. (66.5) (66.6) (66.7) (66.8) (66.9) (66.1 0) (66.11) (66.12) (66.13) (66.14) 67. The energy equation .Q(x, y) = 0 and the Hamiltonian H(q, t, p). We now make a fresh start. We have before us a space Q T of N + 1 dimensions with coordinates x,, expressed in alternative notation as (67.1) Instead of imposing a Lagrangian A(x, x') or L(q. t, q) on Q T, we impose an energy equation 1 Q(x, y) = 0, (67.2) connecting the coordinates x, of a point in Q T with a vector y,. This equation may be regarded geometrically as attaching to each point of Q T a surface (of N dimensions) in an (N + 1)-dimensional space tangent to Q T, y, being coordinates in that tangent space. In a domain where two coordinate systems (x, x*) overlap, y, is to transform as a covariant vector: * OX5 y, = Ys ox* . r 1 The reason for this name will appear at (67.8). (67. 3) 110 J. L. SYNGE: Classical Dynamics. Sect. 68. This makes (67.4) so that this Pfaffian form is invariant. For purposes of general theory it is often best to treat all the coordinates symmetrically, and then we leave (67.2) in this general implicit form. But we sometimes find it convenient to solve for YN+t• so that the energy equation reads 1 YN+t + w(xl, ... xN, xN+l• YJ, ... YN) = 0. (67.5) We define Pv and H by the equations Yg = Pe• YN+l =-H (67.6) (note the minus sign). Then (67.2) expresses a relationship between the 2N + 2 quantities qe,t,pe,H, and (67.5) expresses Has a function of the others: or, if we like, H = (JJ (ql, ... qN' t, pl, ... PN), H = H(q, t, p). {67.7) (67.8) We make a suggestive link with physical concepts (to be strengthened later) by calling H the Hamiltonian and y, the momentum-energy vector; for brevity we may refer to y, simply as momenta, if there is no danger of confusion. Since, for the commonest systems, the Hamiltonian is equal to the total energy, it seems appropriate to call (67.2) the energy equation, since it is equivalent to (67.8) 2• 68. Second form of HAMILTON'S principle. HAMILTON's canonical equations of motion. LetT be any curve in Q T joining a point B* to a point B. We define the Hamiltonian action along T to be the integral (68.1) the vector field y, along r being assigned in any way consistent with the energy Eq. (67.2). Thus the Hamiltonian action is not a functional ofF alone, but also of the assignment of y, along r. We now vary r as in Fig. 32 of Sect. 65, at the same time varying y, consistently with (67.2), and obtain for the variation of Hamiltonian action t5AH = J (t5y, dx, + y, t5 dx,) } = [y, t5x,] + J (t5y, dx,- t5x, dy,). (68.2) 1 The equation may have several roots, so that w is multiple valued; in that case we concentrate on one of the values. We might of course solve for any of the y's, but for present purposes YN+l is best. 2 In this account of general dynamical theory, I follow the pattern of HAMILTON's method in geometrical optics, which is essentially the same as his dynamical method, but more compact, because more symmetrical. To gain this compactness, one treats the coordinates q~ on a parity with the time t, and the momenta Pe on a parity with the negative of the Hamiltonian, -H. This symmetry is present, in thought if not in notation, in E. GARTAN, Le~ons sur les invariants integraux (Paris: Hermann 1922); one meets it also in LANCzos [15], pp. 185-192, and in GoLDSTEIN [7], p. 243. LANczos uses the name auxiliary condition for (67.2) above. In HAMILTON's optics, this equation is the equation of the surface of components and the quantities y, are the components of normal slowness; cf. W. R. HAMILTON: Mathematical Papers, Vol. 1, pp. 291, 303 (Cambridge: University Press 1931). For modern general treatments of geometrical optics, see C. CARATHEODORY: Geometrische Optik (Berlin: Springer 1937), or, following HAMILToN's ideas more closely, J. L. SYNGE: J. Opt. Soc. Amer. 27, 75-82 (1937). Sect. 68. Second form of HAMILTON's principle. Since Q = o for F and for the varied curve, we have (JQ = .fl!! (Jx + _B_!J_ (Jy = 0. ox, r oy, r If we hold the end points fixed, then (68.2) becomes bAH= J (by,dx,- bx, dy,). 111 (68.)) (68.4) In view of the side condition (68.3), a curve of stationary Hamiltonian action, i.e. a curve satisfying (J Jy,dx,= 0, Q(x, y) = 0, with fixed end points, satisfies the equations oQ dx, = dw a ' Yr 8Q dy =- dw ··- - r ox, ' (68.5) (68.6) where dw is an infinitesimal L_.._GRANGE multiplier. Hence the extremal satisfies the differential equations dx, oQ dw oy, ' dy, dw C.Q (68.7) -ax:· We call (68.5) the second form of HAMILTON's principle and {68.7) HAMILTON's equations of motion. Equations of this form are called canonical. A curve, with attached vectors y,, satisfying this principle (or, equivale11:tly, these differential equations), we call a ray or trajectory; the curve in Q T and the associated vector field on it are described by equations of the form x, = x, (w), Yr = y, (w) . (68.8) These functions are determined by (68.7) if initial values of the x's and the y's are assigned. The parameter w in (68.7) is a special parameter in the sense that it cannot be changed once the function Q has been assigned; for the element of Hamiltonian action is dx, oQ dAH= y,dx,= y,-d dw = Y,-;;;--dw, W uy, (68.9) and this determines dw. But, of course, a relationship may be expressed by different equations, and if we change from an equation Q = 0 to an equation Q* = 0, both expressing the same relationship, then the corresponding parameters <>atisfy dw* dQ dw dQ* (68.10) We note that Q = const is a direct formal consequence of the differential equations (68.7); for we have dQ oQ dx, oQ dy, -·- = ... -·- ---+ ·- ...... -- = 0. dw ox, dw oy, dw (68.11) The theory presented here is fundamental and we shall express it in the unsymmetrical notation also. HAMILTON's principle 1 ( 68. 5) reads (J f (Pe dqe - H dt) = o, H = H(q, t, p), (68.12) l This general form of HAMILTON'S principle is due to HELMHOLTZ; cf. LEVI-CIVITA and AMALDI [16] II2, p. 559- 112 J. L. SYNGE: Classical Dynamics. Sect. 68. with fixed end points in QT. The general variation of this integral is But (68.14) and so bA8 = [Pe bqe- H bt] + J {aPe (dqe- ~~ dt)- ) - bqe (dPe + -~~ dt) + bt (dH- ~~ dt)}. (68.15) The first term on the right hand side vanishes for fixed end points, the variations bq , ape, bt remaining arbitrary on the rest of r. Hence the variational equation (6S.12) leads to HAMILTON's equations of motion for the rays or trajectories in the form . oH . oH (68 6) qe = --p~' PQ=- - 8-q;, .1 agreeing with (47.7). Like (68.7), these equations are canonical. We get also dH dt oH ot · (68.17) This tells us that H is constant along a ray or trajectory if t is absent from the function H(q, t, p) (conservative system, cf. Sect. 62). We may use the term Q-dynamics to indicate theory based on (68.5) and H-dynamics for theory based on (68.12). They are different ways of looking at the same thing, and we include them both under the title Hamiltonian dynamics. The relative advantages and disadvantages of these two aspects of Hamiltonian dynamics are closely analogous to the relative advantages and disadvantages of expressing the equation of a surface in the two forms f (x, y, z) = 0 and z = f (x, y); although, to improve the analogy, one should think of an even number of variables. Q-dynamics seems preferable for general arguments in which it is desirable to put all the 2N +2 quantities on the same footing, whereas Hdynamics is in many ways preferable from an analytical standpoint Thus, the equations of motion (68.16) present themselves clearly as a system of order 2N (2N equations, each of the first order), whereas in (68.7) we see a system apparently of order 2N + 2. This latter order is reduced to 2N + 1 by dividing all the equations by dxN+1fdw, so that xN+l (the time) becomes the independent variable, and the energy equation Q (x, y) = 0 implies the further reduction of order to 2N. We shall return to the question of order in Sect. 91. Comparing (68.7) with (68.16), we see that in H-dynamics the special parameter w is the timet. In Q-dynamics, w has in general no simple physical meaning, but if we use the equation Q (x, y) = 0 to make Q + 1 homogeneous of degree unity in the y's, so that we have (JQ 0 Y -- = y -- - (Q + 1) = Q + 1 = 1 ' oy, ' oy, ' (68.18) then it follows from (68.9) that w is the Hamiltonian action A8 . In his optical work HAMILTON adopted this homogenization as standard procedure, but it will not be used in this article because it is more convenient to leave the form of the function Q (x, y) unrestricted. Sect. 69. Equivalence of Langrangian dynamics and Hamiltonian dynamics. 113 69. Equivalence of Lagrangian dynamics and Hamiltonian dynamics. We understand Lagrangian dynamics to be the theory of Sects. 64 and 65, based on a homogeneous Lagrangian A (x, x') or an ordinary Lagrangian L(q, t, q); and Hamiltonian dynamics to be that of Sects. 67 and 68, based on an energy equation Q(x, y) =0 or a Hamiltonian H(q, t, p). We shall show that these two dynamics are essentially equivalent, with some additional generality in Hamiltonian dynamics in the matter of the definition of the momentum-energy vector. We shall establish the essential equivalence by setting up a correspondence or equivalently A(x, x') ~Q(x, y) = 0, L(q, t, q) ~ H(q, t, p). (69.1) (69.2) When this has been done, dynamics may be treated indifferently in terms of A or L or Q = 0 or H. The correspondence is established by demanding the equality of Lagrangian action and Hamiltonian action for an arbitrary curve in QT. Let us start with an assigned homogeneous Lagrangian A (x, x'), and define Yr by . (69.3) These partial derivatives are homogeneous of degree zero in the derivatives x;, and therefore involve only theN ratios x~:x~: ... x;,+I• in addition of course to the coordinates x,. Elimination of these ratios from theN+ 1 equations (69-3) gives an equation which we write 1 Q(x, y) = 0. (69.4) Then along any curve with parameter u and with x; =dx,jdu, the element of Lagrangian action is, by (64.8), dAL=A(x, x') du = x; ~~-du = y,dx,. ux, (69.5) By (68.1), this is the element of Hamiltonian action dAH, corresponding to the energy equation (69.4). Actually, dAH is more general than dAL, because in it the momentum-energy vector y, is restricted only by the energy equation (69.4), whereas in dAL this momentum-energy vector is precisely specified for any curve by (69.3). However, if we vary a Hamiltonian ray or trajectory, keeping the ends fixed, then bAH=O for all variations by, consistent with il=O, and therefore, in particular, for by, consistent with (69.3). Thus bAH=O implies bAL = 0, and this tells us that the Lagrangian rays coincide with the Hamiltonian rays. The above procedure establishes the one-way correspondence A(x, x') ~Q(x, y) = 0; (69.6) given a homogeneous Lagrangian, we obtain the energy equation by elimination from (69.3), as described. The equivalent one-way correspondence L(q, t, q) ~ H(q, t, p) (69.7) 1 We assume that only one equation results from this elimination; there might be more Cf. P.A.M.DmAc: Canad. J. Math. 2, 129-148 (1950). Handbuch der Physik, Bd. III/I. 8 114 J. L. SYNGE: Classical Dynamics. Sect. 69. may be obtained directly from theN+ 1 equations (69.8) by eliminating the N quantities qe and expressing H as a function H(q, t, p). Let us now start with Hamiltonian dynamics, writing down an energy equation Q(x, y) = 0. (69.9) On an arbitrary curve in Q T, with equations x, = x, (u), the momentum-energy vector y, is to be regarded as arbitrary, except for this energy equation. But let us restrict the choice of y,. to what we shall call the natural momentum-energy by imposing the equations dx, = {} ap_ du oy,' (69.10) where {} is an undetermined factor. We recognize here part of the equations of motion (68.7). After solving theN +2 equations in (69.9) and (69.10) for y, and{} as functions of the x's and their derivatives x; = dx,Jdu, we define the function A by A(x, x') = y,x;. (69.11) Then the element of Hamiltonian action may be written dAu = y,dx, = y, x;du =A(x, x') du, (69.12) which is the element of Lagrangian action for the homogeneous Lagrangian A (x, x'), as in (64.8) (for the homogeneity of A see below). Collecting the equations, we may say that the one-way correspondence Q(x, y) =0--+A(x, x') (69.13) is obtained from the equations x; = {}~Q, uy, A=y,x~. Q(x,y)=O, (69.14) by eliminating{} and y, and solving for A in terms of the remaining quantities. As for homogeneity, if these equations are satisfied by certain values (A, x;, {}.), they are also satisfied by (kA, k x;, k{}) for any k, and this indicates that A (x, x') is homogeneous of degree unity, and not merely positive homogeneous. However, the function A may have several branches, and multiplication by a negative k may involve a change of branch, as illustrated by an example in Sect. 70; only positive homogeneity is demanded in Sect. 64. Similarly, the one-way correspondence H(q, t, p) --+ L(q, t, q), (69.15) by which we pass from a given Hamiltonian to an equivalent Lagrangian, is obtained from the equations . aH . qe = fipe • L =qePe- H; (69.16) we are to eliminate Pe and express L in the form L(q, t, q). We have now established the essential equivalence of Lagrangian and Hamiltonian dynamics. The correspondences are illustrated in Sect. 70, and their geometrical significance is explained in Sect. 71. Sect. 70. Examples of LAGRANGE-HAMILTON correspondences. 115 70. Examples of LAGRANGE-HAMILTON correspondences. We consider the systems RS and ODS of Sect. 66. For RS, we start with Then (69-3) gives A (x, x') = VE;:·.x;x~ +A, x;. 8A y,= aX'= v and elimination gives the energy equation Q(x, y) = t[brs(y,- A,) (y,-A5 ) -1] = 0, where brs b,m = r5:,.; the factor i is a mere notational convenience. (70.1) (70.2) (70.3) If we start from the energy equation (70-3), and seek to recover A, we write down (69.14), which read X~=-&brs(y -A.), A=y,x;, D(x,y)=O. (70.4) Hence y,- A,= -&-l b, 5 X~, 2D=-&-2 b, 5 x;x;-1 =0, -& = e vb::-x; x;' where e = ± 1 (all square roots are taken positive). Therefore (70.5) c brs x; y, =A,+ -Vb::~,;,~~ , (70.6) and we get the two Lagrangians A+ (x, x?: A, x: ~ v,b,s x: x:,} (70_7) A_ (x, X)- A, x, Vbrs x, X5 • Fork >0, we have then A+ (x, kx') = kA+ (x, x'), A_ (x, kx') = kA_ (x, x'), (70.8) and for k to B by a curve C and write B U(B) =Jy,dx,, (74. 5) the integral being taken along C. We suppress the dependence on B, because we keep this point fixed once for all. Note that, in (74.5), y, is not the natural momentum-energy vector associ a ted with C [ cf. ( 69.1 0) J , co/Jerenl syslem of' but the vector field given by (74-3) for the coherent rllfs or lrajeclories system. By (74.4) the integral (74.5) has the same Fig. 34. The one-point characteristic value for all reconcilable circuits in R. This means function V(B) ~ jy,dx,. that, if R is simply connected, then U is a singlevalued function of those coordinates which fix the position of B. If R is multiply connected, then U is a multiple-valued function. Let C1 , C2 , ... Cm be a complete set of independent irreducible circuits. Then any two curves joining B differ by a set of such circuits, and we have U(B) =U0(B) + nd1 + nd2+ ··· + nmfm, (74.6) where U0(B) is one determination of U, ] 1 =I} y,dx,, ... fm =I} y,dx,, (74.7) C, Cm and n1 , n2 , ••• nm are integers, positive, negative or zero 2. If the rays or trajectories fill QT (forming a congruence of curves), then in (74.5) we can give arbitrary variations to the coordinates of B (say x,). Then oU(x) y, = --8--x-' 1 (74.8) P = oU(q, t) _ H = oU(q, t) e oqe ' ot ' or, equivalently, (74.9) and therefore, by (74.1), U satisfies the HAMILTON-JACOBI equation D(x, ~~) = 0, (74.1 0) 1 Or family_· Cf. P. A.M. DIRAC: Canad. J. Math. 3, 1-23 (1951), who remarks that "presumably the family has some deep significance in nature, not yet properly understood". 2 The fact that U(B) is multiple-valued when R is multiply connected is intimately connected with rules of quantization; cf. A. EINSTEIN: Verh. dtsch. phys. Ges. 19, 82-92 (1917); J. L. SYNGE: Phys. Rev. 89, 467-471 (1953). See also Sects. 98, 99; where the ]'s are action variables. Sect. 75. Waves of constant action. 123 or, in the unsymmetrical notation, ~ + H(q. t, ~¥) = o. (74.11) We call U(B) the one-point characteristic function of the coherent system of rays or trajectories. True, it depends on the point B<0>, but only trivially, for a change in B(oJ merely adds a constant to U(B). 75. Waves of constant action (Lagrangian or Hamiltonian)1. HUYGENs' construction. For a coherent system of rays or trajectories, as in Sect. 74, we define the waves of constant action (Lagrangian or Hamiltonian-they are the same) to be the loci 2 U(B) = const. (75.1) The rays cut across the waves as in Fig. 35, which shows the waves W*, W with points B*, Bon them, the curve joining these points being a ray F. The action along this ray is the same as the action along any other ray drawn from W* to W, and is in fact equal to the integral J y, dx, taken along any curve in R drawn from W* toW, y, being the vector field defined by the coherent w* system as in (74-J). Fig. 35. Rays or trajectories in In terms of the theory as here presented, it would be QT cutting across waves w•, w. meaningless to ask whether the rays cut the waves orthogonally; we have no Riemannian metric in Q T, and the concept of the orthogonality of a curve and a subspace is not invariant under coordinate transformations. This objection does not, however, apply to the momentum-energy vector y,, since it is a covariant vector (to make y, dx, invariant, dx. being contravariant). This vector y, is in fact orthogonal to the waves, in the sense that y,bx,=O (75.2) for every infinitesimal displacement bx, in a wave; this follows from (74.5), bU being zero for such a displacement. The wave W may be generated from the wave W* by HuYGENs' construction as follows (Fig. 36). Let B* be any point on W*. From B* draw rays in all directions in Q T and measure off on them an action A = U(W) - U(W*), (75.3) where these are the values of U on the two waves. This gives us an N-space, say VN, with equation w* Fig. 36. HuYGENs' construction S(x*, x) =A, {75.4) inQT. where S is the two-point characteristic function. TAr is thus itself a wave, with B* as source. In (75.4) the quantities x* are fixed (coordinates of B*), and the 1 For waves in configuration space Q, see Sect. 81. 2 In the language of the calculus of variations, they are transversals, the rays or trajectories being extremals. The condition of coherency (74.4) is called the condition of MAYER; it may be regarded in a sense as a condition of irrotationality as in hydrodynamics, cf. A. EINSTEIN: Sitzgsber. preuss. Akad. Wiss., phys.-math. Kl. 46, 606~ 608 ( 191 7). CARATHEODORY, p. 249 of op. cit. in Sect. 71, calls the set of extremals and waves a vollstandige Figur. 124 J. L. SYNGE: Classical Dynamics. Sect. 76. quantities x are current coordinates on lj,. We shall show that Tj, touches W at that point B at which the ray T from B* cuts W. First, B must lie on VN, because it is contained in the class of all points at action-distance A from B*. Further, if we give an infinitesimal displacement bx, to B, transferring it to a neighbouring position B' on W, the action for the ray joining B* to B' exceeds A by y, bx, [cf. (72.5)], and this vanishes by (75.2). Thus, to the first order, B' lies on Tj,, and this establishes the tangency of V!v and W at the point B. It is clear, then, that W is an envelope, in the spaceR of the coherent system, of the family of N -spaces (75 .4), the value A being held fixed and B* being allowed to range over the initial wave W*. Since these N-spaces are themselves waves from sources on W*, we have HUYGENS' construction. We can, of course, regard the generation of one wave from another in this way as taking place in infinitesimal steps [make A infinitesimal in (75.4)]. Viewed in finite terms or infinitesimally, we have a contact transformation which establishes a correspondence between the points of the two waves, with tangent elements associated with the points; a tangent element here means the totality of infinitesimal vectors bx, satisfying (75.2) for given y,. 76. Determination of waves from initial data. Method of characteristic curves 1 • In the preceding work, the domain filled by the coherent system of rays was a subspace R of QT, or possibly QT itself. Let us now suppose that the rays fill QT, or a portion of it, so that they form a congruence. We seek to determine the waves from suitable initial data; we wish to solve for U the partial differential equation Q(x, y) = 0, au y,= ox-· r (76.1) subject to initial conditions which assign to U the value U0 (in general not constant) over a subspace l:M of QT, the dimensionality M being anything from zero (a point) 2 toN. The waves will then have the equations U = const, and U will be the one-point characteristic function of the coherent system rays or trajectories appropriate to the initial conditions. The method is the method of characteristic curves, which curves in the present instance are the rays or trajectories. We write down the ordinary differential equations or equivalently, in terms of a parameter w, dx, dw (!!} ay, , dy, dw These equations determine a unique solution (!!} ox, . x, = x,(w), y,=y,(w) if the values of x, and y, are assigned for w = 0. (76.2) (76. 3) (76.4) 1 Cf. CARATHEODORY, Chap. 3 of op. cit. in Sect. 71, also T. LEvr-CrvrTA: Caratteristiche dei sistemi differenziali e propagazione ondosa (Bologna: Zanichelli 1931), or in French: Caracteristiques des systemes differentiels et propagation des on des (Paris: Alcan 1932). 2 The case M = o is covered by Sect. 72, and may be disregarded in this work. The reader is reminded that QT is of (N + 1) dimensions and that Latin suffixes range from 1 to (N + 1); cf. Sect. 62. Sect. 77. jACOBI'S complete integral o£ the HAMILTON-jACOBI equation. 125 Let B*, with coordinates xi, be any point on .EM (Fig. 3 7) and let yi be chosen to satisfy Q(x*, y*) = 0, (76.5) and also (76.6) for every infinitesimal displacement in .EM. It is impossible to enter here into all possibilities. The conditions (76.5) and (76.6) may be inconsistent, in which case no solution U of (76.1) and the initial conditions exists; and even if the conditions are consistent, certain degeneracies may occur. We shall merely follow a general argument, asserting that (76.5) and (76.6) contain M + 1 conditions, and hence leave Yi with N- M degrees of freedom. We use xi and yi as initial values for (76. 3). From the first of (76. 3), these values determine a direction in Q T; there are ooN-M such directions at each point of .EM and there are ooM points on .EM, with the result that we get a congruence of curves (rays or trajectories) filling QT. Let B, with coordinates x,, be any point of QT. Through B draw the curve Fbelonging to the above Congruence; let F CUt J:M at B*, with coordinates xi. Then define U(x) by B U(x) = U(x*) + J y,dx,, (76.7) B• y,. wOYe: ll ~ const the term U(x*) being the assigned value U0 and the integration being along r. On varying B, and consequently B*, we get Fig. 37. Waves in Q T obtained from initial data by the method of characteristic curves. CJU(x) = CJU(x*) + y, Clx,- Yi Clxi + J (Cly, dx,- Clx, dy,). (76.8) Now (76.3) imply dQjdw=O, and hence Q(x, y) = 0, (76.9) on account of (76.5). Hence CJD=O, and the integral in (76.8) vanishes; further, by (76.6), the equation reduces to Therefore CJU(x) = y,Clx,. au y,= ax, , (76.10) (76.11) and so, by (76.9), U satisfies the partial differential equation (76.1); by (76.7). it satisfies the initial condition. Thus the required solution has been found. To revert to a point raised in Sect. 72, we are not so much interested here in finding a formula for U(x) as in setting up a scheme which determines it, and indicates the conditions under which it may not exist. 77. jACOBI's complete integral of the HAMILTON-jACOBI equation. Suppose that we seek all the rays or trajectories, with their associated momentum-energy vectors, for a dynamical system with the energy equation or equivalently Q(x, y) = 0, H = H (q, t, p). (77.1) (77.2) 126 J. L. SYNGE: Classical Dynamics. Sect. 77. The plan of jACOBI was not to attempt to integrate directly the ordinary differential equations of motion, but to work with the HAMILTON-JACOBI equation, which, corresponding to (77.2), reads au ( au) ----+Hqt-=0 at ' ' aq · (77-3) He reduced the problem of motion to that of finding a complete integral of this equation of the form U = ](q1, ... qN, t, a1 , ... aN)+ aN+I· (77.4) Here the a's are arbitrary constants; a complete integral must contain N + 1 of them, but one can be additive, since only derivatives of U occur in (77,3). jACOBI's theorem asserts that, if be are any constants, then the equations b = ai (77.5) Q aae determine the totality of all the rays or trajectories, and the equations a I PQ = aqe determine the associated momenta; or equivalently that the equations . aH qe = ape and . aH Pe =- aqe are satisfied by virtue of (77.5) and (77.6), and that (77.5), and (77.6) all the solutions of (77.7) and (77.8). (77.6) (77.7) (77.8) contain To prove this, we note first that (77.5) are N equations involving qo and t, and so determine functions qe (t) for any values of the constants; further, that (77.6) then give associated Pe. The derivatives of the q's and p's are obtained by differentiating (77. 5) and (77.6) with respect to t; this gives (77.9) and • a2 I . a2I Pe= 8qeaqa qa+ Fq~at · (77.10) Since (77.4) satisfies (77-3), we have a I + H (q t ~I) = o at ' ' aq (77.11) for arbitrary values of the a's, and so a2 I oH 82 I -~ -+ --------- = 0 oae at ap(} aq(} aae 0 (77.12) Comparing this with (77.9), we see that (77.7) is satisfied. Now differentiate (77.11) with respect to qe, obtaining --~ I- + _;;_~ + ql-l___~:J__- 0 (77.13) aqe at oqe apa aqa aqe - · Use (77.7) in this and compare with (77.10); we see that (77.8) is satisfied. Sect. 77. JACOBI'S complete integral of the HAMILTON-JACOBI equation. 127 Since the number of constants (ae, be) is 2N, the same as the number of initial values (qe, Pe) required to determine a solution of (77.7), (77.8), JACOBI's theorem is established. We may say that any dynamical problem is essentially solved if we can find a complete integral as in (77.4). If a complete integral of the HAMILTON-JACOBI equation is given, as in (77.4), then the two-point characteristic function S(qt . ... q'/.i, t*, q!, ... qN, t) is obtained1 by eliminating the N constants ae from theN+ 1 equations S = ](q .. t, a .. ) -. ](q*, t*, a), } oj(q, t, a) oj(q*, t*, a) ---a;;-= ---i~Q---·, (77.14) The connection between the complete integral and the two-point characteristic function is closely related to the fact that a complete integral of the partial differential equation D(x _OJ_)= 0 ' ox ' (77.15) say S(x*,x), in which x; are arbitrary constants, may also be regarded as any solution of the same equation if we take as independent variables the 2N + 2 quantities x, and x;, of which the second set do not appear explicitly in the equation. Whichever point of view we take, we are led to the fundamental determinantal equation [;2 s det ----- = 0 ox,ox~ ' (77.16) as in (73 .1). jACOBI's theorem may be treated symmetrically as follows. Let S(x*, x) be a complete integral of the quantities x; being arbitrary constants. Then, defining y, by as we have and Define y; by Yr=7fX···, r * os y, =- ox*. r (77.17) (77.18) (77.19) (77.20) (77.21) Now (77.19) implies the determinantal equation (77.16), and hence (77.21) implies a relationship between the quantities x* and y*, say D*(x*, y*) = 0. (77.22) 1 Cf. A. W. CoNWAY and A. J. McCoNNELL: Proc. Roy. Irish Acad. A 61, 18-25 (1932); w. R. HAMILTON: Mathematical Papers, Vol. 2, pp. o13-621 (Cambridge: University Press 1940); also LANCZOS (1.5], p. 262. 128 J. L. SYNGE: Classical Dynamics. Sect. 78. If we give to these quantities any constant values consistent with this equation, then the equations oS(x*, x) y,=··ax--' r * oS(x*, x) y, =- ox* r (77.23) define a curve x,(w), with associated y,(w), satisfying (for some parameter w) dx, oD(x, y) dy, oD(x, y) dw oy, dw ox, (77.24) To prove this, we differentiate (77.23) with respect to w, obtaining dy, o2 S dx5 dw ox,ox5 -dw ' o2 S dx5 0 = ·-·----- ax: OXs dw • (77.25) and the result follows on comparing these equations with (77.19) and (77.20). It is important to note that we do not actually need the Eq. (77.22); for we require only 2N + 1 equations to define a curve with associated vector y,, and we can get these equations from (77.23) by omitting one of equations on the right, so that only N of they* are actually involved; they, and the constants x*, may be given arbitrary values. 78. The practical use of jACOBI's theorem. Separation of variables. If, as often occurs in practice, the Hamiltonian does not contain the time explicitly (conservative system, cf. Sect. 62), we apply the procedure described in the preceding section by seeking a function ] as in (77.4) of the form ] =-Et + K(q1, ... qN, a1 , ... aN_1, E), (78.1) where a1 , ... aN_ 1 , E are arbitrary constants. Then, by (77.3), K is to satisfy ( aK aK) H q1, ... qN,-a-, ... -,- =E. ql uqN (78.2) When such a complete integral has been found, the Eqs. (77.5). (77.6) give, for the trajectories, the equations BK BK BK b1 = -,-, ... bN-1 = -,--' bN = - t + "'E ' ua1 uaN-l u (78.3) and P1 = :~ . · · · PN-1 = a::_1 , PN = :~ · (78.4) The first N -1 equations in (78.3) determine a curve in the space Q, and the last gives the time t. H has a constant value in the motion (a consequence of oHfot=O), and this constant value is E. The HAMILTON-JACOBI equation (78.2) may sometimes be solved by separation of variables. Let H be of the form (78.5) where H1 , A1 are functions of q1 , P1 only, H 2 , A2 functions of q2 , P2 only, and so on. Then (78.2) may be written in the form Dl + D2 + ... + DN = 0' (78.6) where (78.7) Sect. 78. The practical use of JACOBI's theorem. Separation of variables. 129 with similar expressions for D2 , 00. DN. We satisfy (78.6) by taking K = K1(q1, a1, E)+ K 2(q2, a2, E)+ 00 • + KN(qN, aN, E), (78.8) and making K1 , K2 , ..• KN satisfy {78.9) and similar equations, the quantities a1 , a 2 , ••• aN being constants which are arbitrary except for the condition (78.10) so that there are N -1 of them independent. Now (78.9) is an ordinary differential equation. When solved for dK1fdq1, it gives K1 by a quadrature; similarly K 2 , 00. KN are obtained by quadratures, and we have a complete integral of the HAMILTON-jACOBI equation (78.2) in the form (78.8), containing N arbitrary constants [we recall that an additive constant was thrown away in (77.4) in passing from U to ]] . Systems of the LIOUVILLE type1 are particular cases of (78.5); for them the kinetic and potential energies are of the forms T = i (Al + 0 0 0 +AN) (Bl qi + 0 0 0 + BNq~)' l V = T-i + ... + VN (78.11) Al + ... +AN' where A1 , B1 , v;_ are functions of q1 only, A2, B2, ~ functions of q2 only, and so on. The corresponding Hamiltonian is H = t(pi/Bl + ... + p'j.,(BN) + J'i + ... + VN (78.12) A1 + ... +AN ' and the ordinary differential equations as in {78.9), by which the problem is reduced to quadratures, now read (~~: r = 2B1(EA1- v;_ + al), ... ( ~~; r = 2BN(EAN- ~+aN). (78.13) As a simple example, consider the KEPLER problem 2 (Sect. 36). For polar coordinates r, {}, we have the Lagrangian and the momenta are oL . p, = ---af = r' {78.14) (78.15) (We have taken the mass of the particle to be unity). The Hamiltonian is H = r _i~ or + J. (){} ~ - L = T + V l (78.16) = _1_ (p2 + _1_p2)- fl 2 ' r 2 fJ r ' 1 For further details, and for the more general systems of STAECKEL, which also yield to the method of separation of variables, see APPELL [2] II, pp. 437-440; LEVI-CIVITA and AMALDI [16] II2, pp. 415-424. 2 Cf. CoRBEN and STEHLE [3], pp. 251-257 for a detailed three-dimensional treatment, with consideration of the BoHR-SOMMERFELD quantum conditions. See also APPELL [2] I, pp. 592-596. Handbuch der Physik, Bd. III/I. 9 130 J. L. SYNGE: Classical Dynamics. Sect. 79. which is of the form (78.12). The HAMILTON-JACOBI equation (78.2) reads _1_ [( oK )2 + _1_ ( oK )2]- 1!... = E (78.17) 2 or r 2 (Jf} r ' and we obtain a complete integral in the form K = F(r, a, E)+ a{}, (78.18) where a and E are arbitrary constants, and F is given by a quadrature from ~)2 = 2£ + 3:!!_ __ a2 . dr . r r2 (78.19) By (78.3) the trajectories have the equations dF oF bl = aa + {}, b2 = - t + a E- . (78.20) It is obvious that the above method can be used when the potential is any function of r. Separation of variables demands a special choice of coordinates. Thus, in the KEPLER problem, rectangular Cartesians would not do. In the problem of two centres of attraction, the variables may be separated (the system being reduced to LIOUVILLE type) by transforming from rectangular Cartesians (x, y) to elliptic coordinates (q1 , q2) by the formulae (78.21) the centres being at x = ±c. If one attracting centre is removed to infinity, and its strength infinitely increased, we get in the limit the problem of a charged particle moving in a field which is the superposition of a uniform electric field on a CouLOMB field (STARK effect) 1 . The relativistic KEPLER problem is treated in Sect. 116. III. Momentum-energy space (PH). 79. The space PH and the momentum-energy characteristic function. In PartE II we took the (N + 1)-dimensional space of events Q T as the background for dynamical theory, using the coordinates x, where 2 (79.1) Let us now view dynamics in the (N + 1)-dimensional momentum-energy space PH in which the coordinates are y, where Ye=Pe' YN+I= -H. We start from the beginning, assuming an energy equation .Q(x, y) = 0, or equivalently, on solving for YtV+l• H =H(q, t, p). (79.2) (79. 3) (79.4) For any curve r in PH with equations y,= y,(u), we define a new type of action by the integral (79.5) 1 Fora detailed treatment, see CORBEN and STEHLE [3], pp. 258-264; see also APPELL [2] I. pp. 602-607; GRAMMEL [8], p. 321; PERES [20], pp. 243, 244. For a Lagrangian treatment of the problem of two centres, see WHITTAKER [28], pp. 97-99. 2 For notation, see Sect. 62. Sect. 79. The space PH and the momentum-energy characteristic function. 131 where x,(u) are arbitrary except for (79.3); equivalently, A=J(qedPe-tdH). (79.6) On varying F, we get (79.7) If we hold the end points of r fixed (i.e. the end values of y,), the variational equation <5A = <5 J x, dy' = 0, (79.8) with the side condition (79-3), leads at once to the canonical equations dx, oQ dy, oQ ( ) dw oy, ' dw - ox, ' 79.9 for some parameter w. These are the same equations as in (68.7), and we may call the curves satisfying them rays or trajectories as before. It is clear then that dynamics in PH based on the energy equation (79.3) and the variational equation (79.8) is the same as the dynamics in Q T based on the same energy equation and HAMILTON's principle (68.5), i.e. <5A = <5fy,dx,= 0, (79.10) for fixed end points in QT. The two actions are connected by A +A= J y,dx.+ f x,dy, } = x,y,- xiyi, (79.11) where (x*, y*) refer to the initial point and (x, y) to the final point. In Q T we think of x, as current coordinates on a curve and y, as an associated vector field; in PH, we reverse the roles. There is a formal duality between the two representations; we could, for example, use the technique of Sect. 69 to define a ''homogeneous Lagrangian" in PH as a function of the quantities y and y' (where y;=dy,fdu), and pac;s to an "ordinary Lagrangian", a function of Pe' H and dpefdH. We now introduce in PH the momentum-energy characteristic function 1 W(C*, C) defined by W(C*, C)= f x, dy,, (79.12) the integral being taken along the ray or trajectory joining the points C* and C of PH. On varying these end points, we get from (79.7) (79.13) If the variations <5y, and <5yi can be taken arbitrarily (i.e. if there are rays joining arbitrarily varied positions of C* and C), then ow ow * -oy,- = x,, &;,; =- x,, (79.14) and hence, by (79.3), W(y*, y) satisfies the partial differential equations n(-~~-. y) = o, n(- :; , y*) = o. (79.15) The values of xi and x, define a ray or trajectory, and hence they determine Yi and y,; conversely, the values of Yi and y, in general define a ray or trajectory, 1 In optics, this is HAMILTON's T-function, also known as angle-characteristic and angleeikonal; it is the basis of the theory of the aberrations of optical instruments. It is denoted by Where to avoid confusion with kinetic energy. 9* 132 J. L. SYNGE: Classical Dynamics. Sect. 80. and hence they determine x: and x,. The relationship between the two-point characteristic function S ( x*, x) and the momentum-energy characteristic function W(y*, y) is W(y*, y) + S(x*, x) = x,y,- x;y;. (79.16) In addition to these two characteristic functions, there are two other mixed characteristic functions: and F(x*, y) = S(x*, x)- x, y,, G(x, y*) = S(x*, x) + x;y;. (79.17) (79.18) If arbitrary variations are permissible, we have and oF_ * ox* --Y,' T oG ·a-x= y,, , oF -0- =- x,, y, oG * ~=x,. y, (79.19) (79.20) 80. Encounters. We assumed, following (79.13), that C* and C can be given arbitrary displacements tJy; and tly, in PH, but there are important cases where this cannot be done. Let us consider a ray or trajectory F in Q T, joining a point B* to a point B (Fig. 38). Suppose that in a domain M* of Q T, containing B*, the function D(x, y) is independent of its first set of arguments, the x's, and that the same is true in a domain M containing B. We then write the energy equation D*(y*) = o in M*, .Q(y) = 0 in M. (80.1) Fig.38.Arayortrajectoryinevent-space By (79.9), the y's are constant along a ray in M* Q T with straight initial and final portions. or in M, and in fact the ray is a straight line in each of these domains of QT. It is clear from (80.1) that y; and y, cannot be given independent variations. Solving (80.1), we get y;+l =-w* (Yt, ... y;), or equivalently H*=H*(p*), H=H(p). (80.2) (80.3) We are in fact dealing with a system for which, initially and finally, the Hamiltonian depends only on the momenta, as is the case for a free particle or a set of free particles without interactions between them. Now (79.13) gives Jtw-( 0(}))1: ( * * ow·)Jt * u - X~- XN+l oy~ uy~- X~ - XN+l oy3 uVQ, which shows that W is a function only of the 2N quantities y~, y:. notation, W is a function only of p~, p;, and (80.4) reads tlW=(qQ-t !~)t1Pe-(q:-t* ~;;)tlp:. (80.4) In the other (80.5) Under these circumstances W is to be regarded as an arbitrary function of its 2N arguments, and is not required to satisfy any partial differential equation Sect. 80. Encounters. 133 as in (79.15). Since Pe• P: can be given arbitrary independent variations, (80.5) gives ;:: = - q: + t* ~~; ' ) ow oH (80.6) ope = qe - t ope . Regarding the function W as assigned, these are the equations of the initial and final rays or trajectories, the momenta Pe, P: having constant values by virtue of the canonical equations (79.9). When viewed in Q T, these initial and final rays are straight lines; when viewed in PH, they are mere points, each lying on a certain surface of N dimensions, as given in (80.1) or (80.3) (Fig. 39). Consider now an encounter, as in the kinetic theory of gases, of a system of n particles which interact with one another, the generalized forces being derivable from a potential function, or an extended potential, so that the dynamics is Hamiltonian. Write ~ = m2 = m3 for the c mass of the first particle and q1 , q2 , q3 for its rectangular Cartesian coordinates; write m4 = m5 = m6 for the mass of the second particle and q4 , q5 , q6 for its coordinates; and so on. Then the Lagrangian is N L = i L mA q~ - V(q, q), (80.7) A=l where N = 3 n. We suppose that V is a func- SJ*(y *) -o tion of the positions and, perhaps, the Velo- Fig. 39. Encounter viewed in momentum-energy cities of the particles, vanishing when the dis- space PH. ~~~~:((~ :'~i~:~';!!~~~ point c• and tances between the particles tend to infinity. Let the particles start at infinite distances from one another, approach, interact, and finally withdraw to infinite distances again. To avoid the awkward limits, t-+ ± oo, let us suppose that the interactions are completely absent for t < t't and for t > t0 , being switched on and off as smoothly as we like. This means that we modify (80.7) by writing V(q, t, q) for V(q, q), with the understanding that V = 0 except for t't < t < t0 • Then for the initial and final states we have the energy equations N A;l (80.8) H*=i1:m;;t 1 P1 2 for t*t0 . A=l We are now in a position to describe the effect of any possible Hamiltonian encounter in terms of a single function W of the arguments (80.9) in the sense that, if this characteristic function is assigned, then the initial and final trajectories are given by (80.6), the quantities (80.9) having arbitrary constant values, and the functions H*, H being as in (80.8). In view of the axiom of homogeneity and isotropy in Newtonian dynamics (Sect. 5), the function W is not completely arbitrary. If p*<1>, ... p*<"> are the individual momentum vectors (in ordinary space) of the individual particles before the encounter, and p<1>, •.. p the individual momentum vectors after the encounter, then W can involve the 6n components of these vectors only in the form of invariants under rigid body displacements. Thus, if there are only 134 J. L. SYNGE: Classical Dynamics. Sect. 81. two particles, W must be a function of the nine scalar products (the last made a sum for symmetry) p(1).p(1), p(2) . p(2)' 1 p*(2). p*(2), p(1) . p* (1), p(2). p* (2)' (80.10) p(1) . p* (2) + p(2) . p* (1)' or an equivalent set. IV. Configuration space 1 (Q ). 81. Reinterpretation of dynamics in the space Q. Rays and waves in a coherent system 2. The space of configurations Q, in which the coordinates of a point are W(t} r the N generalized coordinates qe of. the dynamical system, gives the most natural geometrical representation; if the system consists of a single particle, the representative point in Q is identified with the position of the particle in ordinary space. All that has been said in Part E II about dynamics in Q T can be reinterpreted in Q. A ray or trajectory, which was a curve in Q T, now appears in Q as a moving point, the time t being definitely a parameter and not a coordinate; the coordinates qe and the associated momentum Pe satisfy the Cai_lonical equations Fig. 40. A ray or trajectory r in Q and a moving wave of constant Lagrangian or Hamiltonian action. In time dt the ray displacement is D D' and the wave . oH p eH (B1.1) qe = ope ' e = - aq;. But while Q may seem easier to think about than displacement DE'. Q T, the waves of constant action for a coherent system discussed in Sect. 75 present a rather complicated moving picture when viewed in Q. In Q T the rays or trajectories are fixed curves and the waves fixed surfaces of N dimensions, as shown in Fig. 35 (p.123); in Q the former are moving points and the latter moving surfaces of N- 1 dimensions, with equations U(q, t) = const, (81.2) where U is the one-point characteristic function. In Fig. 40, F is a ray or trajectory, and D, D' the positions on it of the representative point at times t, t + dt respectively. W is the wave which passes through the point D at time t, and W' is the same wave at time t+dt. It is important to note that in general D' does not lie on W'; in other words, the representative point does not ride on a wave. To explore the relation between the velocities of the representative point (the ray velocity) and the wave (the wave velocity), we note first that the ray velocity is tie, but that, for a general Hamiltonian dynamical system, there is no way of converting this contravariant vector into an invariant speed. To investigate wave velocity, we take a pointE on W', adjacent to D, and denote the displacement DE by {Jqe; then by (81.2), since it is a question of one moving wave, we have aU aU oqe {Jqe + fitdt = 0, (81.3) 1 Often called q-space. 2 Cf. LEVI-CIVITA and AMALDI [16] II2 , pp. 456-469. For the general theory of waves and characteristics, see T. LEVI-CIVITA, op. cit. in Sect. 76. For some interesting general remarks relative to wave mechanics, see T. LEVI-CIVITA: Bull. Amer. Math. Soc. 39, 535-563 (1933). . Sect. 81. Reinterpretation of dynamics in the space Q. 135 or, by (74.9), (81.4) To follow the usual practice in finding wave velocity, we should take the displacement bqe along the normal to W at D. There is a normal, defined by oUfoqe ( = Pe), but this is a covariant vector, whereas CJ% is contravariant, and we cannot (in any invariant sense) speak of them as having the same direction, without limiting the generality of the dynamical theory. The best we can do is to take CJ% along the ray (taking E at E' in Fig. 40), so that R bqe = dqe = qe dt, (81.5) the proportionality factor R being describable as R = ray velocity __ _ wave velocity measured along ray Multiplying (81.5) by Pe and using (81.4), we get R = Pe qe_ = f'_e_ oH H H ope' (81.6) (81.7) So far the argument has been of maximum generality. Consider now the ordinary dynamical system1 (ODS) of Sects. 66 and 70; for it we have T = iae,ijeqa, L = T- V(q),} (81.8) H = faea Pe Pa + V. The tensor aea provides us with an invariant kinematical line element ds in Q, defined by 2 d 2 _ d d (81. 9) s -aea qe qa. It is natural to define the ray speed v by v2 = ( '!_~) 2 = a q' q' = 2 T = 2 (E - V) dt e a e a ' (81.10) where E is the constant 3 total energy (E = H = T + V). In terms of the metric (81.9), the wave W has a contravariant unit normal ne given by aea ~l!_ ne= oq" Vat-'" au au_ oq~" aq. aea Pa ---- lfa~-<•pl-'p: aea Pa Vi(Y- V) 0 (81.11) To find the normal velocity of wave propagation, we take E on the normal to W at D, so that (81.12) where d{} is an infinitesimal multiplier, and (81.4) gives d{} H dt (81.13) 1 The essential requirement here is the tensor aQa• and this is available for rheonomic systems also [cf. (27.2)]; some slight modification of the statements is necessary. 2 This kinematical line element is discussed more fully in Sect. 84. 3 Constant, that is, for the ray or trajectory F; it is not necessary to take E constant throughout the coherent system. For the derivation of the formula (81.14) for the wave velocity from the HAMILTON-JACOBI equation, see E. SCHRODINGER: Ann. Physik (4) 79, 489-527 (1926); Abhandlungen zur Wellenmechanik p. 494 (Leipzig: Johann Ambrosius Barth 1927). See also L. BRILLOUIN: Les Tenseurs en mecanique et en elasticite, Chap. VIII (Paris: Masson & Cie 1938), and GOLDSTEIN [7], p. 307. 136 ] . L. SYNGE: Classical Dynamics. Sect. 82. Hence the normal velocity u of the wave is given by 2= aeabqebqa = eap p (!_f!_) u dt2 a e a dt 2 l H2 H2 £2 = aeap;p; = 2(H- V) = 2(E- V). (81.14) 82. Isoenergetic dynamics in Q and its relation to general dynamics in QT. Consider a conservative dynamical system; the time t is absent from the Hamiltonian, so that we have H =H(q, p). Then along any trajectory H(q, p) =E. (82.1) (82.2) We shall call E the total energy, since it is precisely that for ordinary dynamical systems. We now study isoenergetic dynamics (cf. Sect. 62) in the space Q, by which we understand that Eisa constant, not merely along each trajectory, but for the whole system of trajectories considered, and indeed for those varied curves which we have occasion to use. In fact, Eisa constant built into the isoenergetic dynamics. Let E be an assigned number, and let the system start from a point D* of Q at timet* with any initial momenta satisfying (82.2). Let Fbe the curve described in Q in accordance with the equations of motion. Then, if D is the position of the representative 'point on F at time t, the Hamiltonian action from D* to D is, by (68.1), AH(F) = J (Pedqe- H dt) = J Pedqe- (t -t*) E. (82. 3) r r Consider any adjacent curve I;_ joining D* and D, described in the same time interval (t*, t) and having associated with it a vector field Pe satisfying (82.2) and approximately the same as on F. Then AH(I;_) =fPedqe- (t-t*)E, r, and hence But by HAMILTON's principle (68.12) we have bAH=O, and therefore the trajectory in Q satisfies (without reference to time) the variational equation and side condition H(q, p) - E = 0, (82.4) the end points in Q being fixed. If we now compare this isoenergetic dynamics in Q with the general dynamics in Q T, based, as in (68.5), on the variational equation bfy,dx1 =0, .Q(x,y)=O, (82.5) we recognize at once the complete identity of the two dynamics, save for the trivial difference in dimensionality, N for (82.4) and N + 1 for (82.5). The transition x,----">-qe, Y,----">-Pe' is effected as follows: } .Q(x, y) = 0-----">-H(q, p)- E = o. ( 82" 6) All the theory developed in E II for general dynamics in Q Tis available for isoenergetic dynamics in Q, and we shall discuss some aspects of this in Sect. 83. Sect. 83. MAUPERTUIS action. 137 We note that the timet hasdisappeared from (82.4), so that, if we use nothing but (82.4), we can hardly expect the time to reappear. But it does. For if we apply to (82.4) the argument used at (68.5) to find the extremals, we get the equations dqe _ oH ape _ oH dw - ape ' ·a--w· - oqe ' (82.7) where w is some special parameter. But we know that the trajectories satisfy the canonical equations (68.16), that is dq~ _ oH ape _ oH """"dt - ope ' """"dt - - oqe ' (82.8) and, comparing these with (82.7), we see that, though we chased tout, it comes back as the special parameter in the canonical differential equations derived from (82.4). If we have solved a dynamical problem in Q, obtaining a curve and a momentum field along it, we can use any one of (82.8) to find the time t; or we can use some derived equation, such as dt=_t~ Pa8Hf8Pa . (82.9) There is another way of keeping the time in isoenergetic dynamics which can be the cause of no little confusion. This method puts a parameter t on each varied curve according to the following plan. Given any arbitrary motion qe = qe (t), not in general satisfying the canonical equations (82.8), there is a natural momentum [cf. (71.11)] Pe(t) given by (82.10) where Lis the Lagrangian. By adjusting the parameter twe can satisfy H(q,p) =E; then we can state the variational equation (82.4) in a more restricted form as f oL . ( oL) t5 oq;qedt=O, H q,aq -E=O, (82.11) the variation being for fixed end points in Q, but not for fixed end times, the timet on each curve being obtained from the second equation in (82.11). This variational equation is more restricted than (82.4) because (82.1 0) is more restrictive than H(q, p) =E. It seems less confusing to work with the more general equation (82.4) and restore the time only on trajectories, as in (82.9) or in an equivalent way. 83. MAUPERTUIS action, the two-point characteristic function for isoenergetic systems, the homogeneous Lagrangian, and JACOBI's principle of least action. In isoenergetic dynamics in Q, we define the MAUPERTUIS1 action as A =fPedqe, where the p's and q's satisfy the energy equation H(q, p) =E. (83 .1) (83 .2) 1 Although it is the custom to use the single word action for A as in (83.1), (83.3) or (83.4) (cf. WHITTAKER [28], p. 248, GoLDSTEIN [7], p. 228), it seems desirable to have an adjective to distinguish this integral from the Lagrangian or Hamiltonian action of Sect. 64, 68. The name of MAUPERTUIS appears to be most commonly associated with the integral A, particularly in the form (83.4) or m J v ds for a single particle, and will be used here as a distinguishing adjective, although historically it might be juster to call this the Eulerian action; cf. DUGAS, pp. 250-264, of op. cit. in Sect. 1. 138 J. L. SYNGE; Classical Dynamics. Sect. 83. If we further restrict prJ to be natural momenta as in (82.10), we have - J oL . A = 8 .-qedt. qe (83-3) For the system ODS of (81.8), we have A=2fTdt. (83.4) Pursuing the analogy with general Hamiltonian dynamics in Q T, we define, in isoenergetic dynamics in Q, a two-point characteristic function by (83. 5) the integral being taken along the ray or trajectory joining the point D* to the point D. Variation of the end points gives (83.6) and hence, if arbitrary variations are permissible, as is in general the case, es es * a-q; = Pe• oq: = - P(}. (83.7) It follows from (83.2) that 5 satisfies the two partial differential equations H(q, ~:)=E, H(q*,-,-:~-)=E. (83.8) We recognize the HAMILTON-jACOBI equation in the form (78.2). In the timeless theory of isoenergetic dynamics in Q, a coherent system of rays or trajectories appears as a set of fixed curves and the associated waves as a set of fixed surfaces, two waves being separated by a constant amount of MAUPERTUIS action. Just as in Q T we can pass from an energy equation .Q(x, y) =0 to a homogeneous Lagrangian A (x, x') by the Eqs. (69.14), so in isoenergetic dynamics we can find a homogeneous Lagrangian A (q, q') (where q~ = dqefdu, u being any parameter) by eliminating D- and Pe from the N + 2 equatwns I - .o. oH(q!..t1_ A- p I H( p) E o q(}-·v op(} ' = (}q(!, q, - = . (83.9) The rays or trajectories in Q then satisfy the variational equation tJ J A(q, ql) du = 0, for fixed end points in Q. Let us carry out this process for a system with the Lagrangian L( ') 1 • • • V q, q = 2aeaqeqa + aeqe- • (83.11) where ava(=aae), arJ and V are functions of the q's only. (This is a little more general than the ODS of Sect. 66 and (81.8), for which av = 0.) We first find the Hamiltonian as follows : (83.12) Sect. 84. The kinematical line element. 139 Then we proceed by (83.9): (83.13) and so we have the homogeneous Lagrangian for isoenergetic dynamics in Q: A (q, q') = v2 (E- V) Vaea q~ q; + aeq;. (83.14) The variational equation (83.10) may be written 15 J(V2(E- V) Vaeadq07ii, + aedq0) = 0. (83.15) If we put a0 = 0, so that the system becomes ODS, and use the kinematical line element (81.9}, this becomes <5JV(E- V)ds=O, (83.16) which is known as jACOBI's principle of least action1• This may be interpreted geometrically by saying that the trajectories are geodesics in the space Q if it is endowed with the Riemannian line element (83.17) which may be called the action line element. This metric is singular on the locus V =E, which corresponds to a state of instantaneous rest for the system, since V=E implies T=O. 84. The 'kinematical line element. In this section and the next, we abandon Hamiltonian dynamics temporarily. Consider a dynamical system with kinetic energy T = lg0 aqeqa. (84.1} Since this work is guided by the techniques of tensor calculus, we denote the coordinates by qe rather than qe, since dqe is a contravariant vector. We represent the system by a point in the space Q, which we endow with the kinematical line element2 (84.2) already used in (81.9} (we change a to g to avoid confusion with acceleration). Before introducing forces, we consider kinematics. Any motion qe = qe (t) defines a curve in Q, and at each point of the curve a contravariant velocity vector which may also be written (8q) (84.4} 1 For other derivations of this principle, see APPELL [2] II, p. 4 54; GoLDSTEIN [7], p. 232; PERES [20], pp. 229, 248. Written in the equivalent form d fT dt = o, with the side condition dE= 0, this is sometimes called HoLDER's principle. This principle and HAMILTON's principle are both contained in the principle d J (2 T...:.. .A. E) dt = 0, with the side condition d(EA-ldtA)= o, where A is any constant; cf. E. STORCH!: Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 14,771-778 (1953). 2 This is almost the line element of HERTZ (pp. 55. 62 of op. cit. in Sect. 61); he divides by the total mass of the system, so as to give ds the dimensions of a length. 140 ] . L. SYNGE: Classical Dynamics. Sect. 84. where (84.5) the unit tangent to the curve. There is also a contravariant acceleration vector IJve ae=--IJt ' (84.6) the absolute derivative of ve, the absolute derivative of a vector field Ve(u) along a curve qQ(u) being1 (84.7) Substitution from (84.4) in (84.6) gives ae = v A,e + "v2ye = v .!-11__ A,e + "v2ye ds ' (84.8) where ye is the unit first normal to the curve of motion and" the first curvature 2, defined by IJ ;.e = "' .. e e u 1 ---.. 0 IJs "'• ' geu Y Y = ' ",;;;;, . (84.9) Thus the acceleration of the representative point can be resolved along the tangent and first normal in just the same way as in (18.2) for a moving particle. Since we have a fundamental tensor geu in Q, and its contravariant conjugate geu, we can pass from contravariant components to covariant, and vice versa. The covariant acceleration is _ u _ IJve _ d oT oT ae - ge (J a - Tt- dt afi! - aqe. (84.10) Passing from kinematics to dynamics, we introduce the generalized force Qe, a covariant vector defined by (84.11) !5W being the work done in a displacement oqe (OW is an invariant). LAGRANGE's equations of motion are a oT aT Tt oqe- aqe = Qe, (84.12) and these may be written (84.13) where Qe is the contravariant force vector. In words: acceleration=force 3. It should be noted that, while the physical dimensions of the several components ot a vector depend on the choice of coordinates, the magnitude v of the velocity vector has the dimensions [M~LT- ] and the magnitude a of the acceleration vector has the dimensions [ Mi LT- 2]. 1 For further details about absolute derivatives, see ] . L. SYNGE and A. ScHILD: Tensor Calculus, pp. 47-51. Toronto: University Press 1952. For the CHRISTOFFEL symbols, see (18.4). 2 We might call this geometrical curvature, to distinguish it from the dynamical curvature of Sect. 85. To within a constant factor," is the curvature of HERTZ (cf. pp. 74-77 of op. cit. in Sect. 61). 3 If Qe = 0, then ae = o, and hence, by (84.8), "= o, so that the trajectory is a geodesic if no forces act. Sect. 85. Least curvature. 141 We can form a geometrical picture of the problem of stability as in Fig. 41, rand F' being two adjacent trajectories. There are two simple ways of correlating points on them: (i) isochronous correspondence, in which we correlate positions with the same value oft, the infinitesimal deviation vector being ~e in Fig. 41; (ii) normal correspondence, in which the infinitesimal deviation vector 'YJe is orthogonal to r, so that 'YJeve = 0. {84.14) Between the two types of deviation vector we have the relation ~e = 'YJe + {}ve, {84.15) in which, by virtue of (84.14), {} = t;ava {84.16) 2 • v The isochronous vector ~e satisfies the equation of deviation1 {84.17) r Fig. 41. Deviation of trajectories in the space Q, with the isochronous deviation vector ~ and the normal deviation vector "'ll. where R~a,... is the curvature tensor of the space Q with metric {84.2), R~ap• = a:" {:.}- a~• {a~}+{;.} {T~}- {aT,..} U.} • {84.18) and QTa is the covariant derivative of the contravariant force, {84.19) The equation of deviation for normal correspondence is a little more complicated. Substitution from (84.15) in {84.17) gives, with the aid of (84.13), (84.20) with {84.14), we have then N + 1 equations for{} and 'YJe. 85. Least curvature. With the notation of Sect. 84, we define the dynamical curvature K of an arbitrary kinematical motion with acceleration ae, in the presence of given forces Qe, as the positive square root of (85.1) From the positive-definite character of kinetic energy, this expression is nonnegative; K =0 if, and only if, the equations of motion (84.13) are satisfied. We impose constraints, in general non-holonomic, given by [cf. (46.2)] (85.2) 1 Cf. J. L. SYNGE: Phil. Trans. Roy. Soc. Lond. A 226, 31-106 (1926), and Tensorial Methods in Dynamics (University of Toronto Studies, Applied Mathematics Series No.2; Toronto, University Press 1936). The latter c- Xs -<>- Ys• uXs <~Ys (88.2) so that the integral is actually of the form I(A ,B; C)= f (Xsdxs + Y_;dys), (88.3} 1 Remember that we have imposed the condition w' = w; if we relax this, a necessary and sufficient condition for CT is ./ rJ = p.r, where p. is a scalar multiplier. A linear transformation~= J ~·is called symplectic (or the matrixJ is called symplectic) if J satisfies (87.16); cf. H. WEYL: The Classical Groups, Chap. 6 (2nd Edn. Princeton: University Press 1946). WINTNER [30]. pp. 17, 29, 45, uses l to denote the matrix r; SIEGEL (p. 9 of op. cit. in Sect. 53) uses ~- Sect. 88. where Generating functions. )( 1 OX; s= Ys- Yr OXs, Y 1 OX; .=- y,--,-. uYs 147 (88.4) If we agree to keep A fixed once for all, we may denote the integral by I(B;C). And if we make B coincide with A, so that C is a circuit, we may denote the integral by I(C). Other similar obvious notations are used below. Giving arbitrary variations to A, Band C, we get from (88.1), on integration by parts, tH (A, B; C) = [y, !5x,- y; !5x;]!J } (88.5) + f [(dx, !5y,- !5x,dy,)- (dx; !5y;- !5x; dy;)]. Suppose that (x, y)-+ (x 1 , Y1) is CT. Then, on account of the bilinear invariant (87.19), the integral on the right hand side of (88.5) vanishes, and we have tH (A, B; C) = [y, !5x,- y; !5x;]!]. (88.6) A number of consequences follow: (i) Since the variation of I vanishes when A and Bare held fixed, I(A, B; C) has the same value for all reconcilable curves joining A and B; symbolically, I(A, B; C) =I(A, B). (ii) If A is held fixed, then I(B; C) =I(B), a function of B only, multiple valued in the case of multiple connectivity; we have (88.7) for any arbitrary variation 1 of B. (iii) If A and B coincide, so that C is a circuit and we can write I (C) for the integral, then tH (C)= 0 for an arbitrary variation of the circuit. This implies that I (C) has the same value for all reconcilable circuits, and I (C)= 0 for a reducible circuit. Equivalently, (88.8) for every reducible circuit; in words, the circulation of action in a reducible circuit is invariant under CT. In the case of an irreducible circuit, the effect of a CT is to increase or decrease the circulation by an amount which is the same for all reconcilable circuits. By (88.7) we have this: if (x, y)-+(X1 , Y1) is CT, then the Pfaffian (88.9) is an exact differential. Let us prove the converse. Given a transformation (x, y)-+(X1 , Y1) such that y,!5x,- y;~5x; = tH(B), (88.10) where I(B) is some function of (x, y), let us take a 2-space x,= x,(u, v), y,= y,(u, v), so that (x, y, X1 , y 1 , I) are all functions of u and v. Then OX, - 1 OX; _ _ Of y, ov y, ov - ov . (88.11) Differentiating with respect to u, interchanging u and v, and subtracting, we get {u, v} = {u, v}' (88.12) 1 This means arbitrary variations of (x, y) or equivalently of (x', y'); the CT may be such that arbitrary independent variations cannot be given to (x, x'). This occurs in the case of MATHIEU transformations (see later). 10* 148 J. L. SYNGE: Classical Dynamics. Sect. 88. in the notation of (87.21). This establishes the existence of the bilinear invariant, and hence the canonical character of (x, y)-+(x', y'). We have now three tests for CT: (i) the symplectic test (87.16), based on the matrix r, (ii) the bilinear test (87.19), and (iii) the exact differential test(88.10). Canonical transformations may be generated as follows. Let G1(x, x') be an arbitrary function. If we define (y, y') by then 8G1 (x, x') y,=~-, r (88.13) (88.14) an exact differential in the space (x, x'). But this procedure does not necessarily give us a transformation (x, y)-+(x', y'), because we have no assurance that the Eq. (88.13) can be solved for (x, y) in terms of (x', y') or vice versa. To examine this, we differentiate (88.13), obtaining uy,=-~-uxs .i EJ2Gl .~: +~ux,, EJ2Gl .~: ' ) ux, u.X5 uX1 uX5 .~: ' EJ2Gl .~: EJ2Gl .~: ' uy,= -~uxs-~-ux,. ux, uX5 ux, uX5 (88.15) Let us impose on G1(x, x') the condition EJ2G det~ =I= 0. (88.16) uX1 uX5 Then (88.15) can be solved for (bx, by) in terms of (bx', by') and vice versa. These solutions can be integrated, because, if we pass round a small circuit in the space of (x, x'), we are led round small circuits in the spaces of (x, y) and (x', y'). Hence we get a reversible transformation (x, y)-+(x', y'); by virtue of (88.14), it is a CT. Thus, starting from a generating function G1(x, x'), arbitrary except for the inequality (88.16), we obtain aCT from (88.13) or (88.14). This powerful method of generating CT does not, however, yield all CT. It does not yield those CT for which there exist one or more relationships1 between the variables (x, x') or in particular the CT of MATHIEU 2 for which Y, bx,- y; c5x; = 0. (88.17) Similarly, the alternative generating functions described below fail to yield certain special CT. But to understand the general theory of CT it seems advisable to neglect such special cases, and to suppose the CT under consideration such that any of the following sets of (2N + 2) variables forms a coordinate system in the space Q T PH, in the sense that the variables in any set may be varied arbitrarily and independently: (x,y), (x',y'), (x,x'), (x,y'), (y,x'), (y,y'). The formula (88.14) may be written in the following equivalent forms: y, bx,- y; bx; = bG1 , x, by,- x; by;= r5 G2 , Yr bx, + x; by;= r5G3 , x, dy, + y; dx; = c5G4 , 1 See WHITTAKER [28], p. 294. 2 See WHITTAKER [28], p. 301. (88.18 a) (88.18b) (88.18c) (88.18d) Sect. 88. where Generating functions. G2 = x,y,- x;y;- G1 ,) G3 = x;y; + G1 , G4 =x,y,-G1 • We have then four different ways of generating CT: , i:JG1 (x, x') Yr =- i:Jx' • r ' i:JG2(y, y') x, = - - oy~-- ' , oG3 (x, y') x, = -ay·~ -- ' i:JG4 (y, x') X=---- r i:Jy, ' , oG4 (y, x') y,=-w-· r 149 (88.19) (88.20a) (88.20b) (88.20c) (88.20d) Any one of these formulae gives a CT, the generating function involved being arbitrary except for an inequality of the type (88.16). Here are some particularly simple examples of CT. First, using (88.20a): G(x, x') = x,x;, y,= x;, i:J2G det~=1,} ux.,uxs y; =- x,, (88.21) so that the variables are interchanged, with a reversal of sign as indicated. Secondly, using (88.20c): G(x, y') = x,y;, ()2G det- 0 0 , =1,} x, Ys x, ' = x,, (88.22) so that this is the identical transformation. Thirdly, using (88.20c) again, take ()2G i:lfs G (x, y') = /,(x) y;, det -<>-- 0 1 = det- =f= 0, (88.23) ux, Ys ox, the functions fs (x) being arbitrary except for this last condition. We get y,=~~s y;, x;=/,(x), r (88.24) so that we have an arbitrary transformation (x)-+ (x') and y, transforms like a covariant vector. This is an extended point transformation1• In (q, t, p, H) notation [cf. (86.1)], the CT (88.20) read as follows: G1 = G1 (q, t, q', t'): t =- ac2 i:JH' _ H' = _ i:JG1 } (88.25 a) i:Jt' • t' = i:JG2 i:JH'' } (88.25 b) 1 WHITTAKER [28], p. 293. GoLDSTEIN [7], p. 244, calls it simply a point transformation. These references may be consulted for further details about CT, with examples. See also WINTNER [30], p. 34 and CARATHEODORY, pp. 78-102 of op. cit. in Sect. 71. 150 J. L. SYNGE: Classical Dynamics. Sect. 88. G3 = G3 (q, t, p', H'): 8G3 - H = 8G3 Pe = oq~ ' ot ' t' = _ oG3 ) (88.25 c) oH' · G4 = G4 (p, H, q', t'): t =- 8G4 aH' p' = ac-~._ e oq~ ' The following CT leave the time unchanged: G3 (q, t, p', H') =-tH' + g(q, t, p'): P' og H=H'-~ e = eq;; · at • G4 (p, H, q', t') =- H t' + g(p, q', t'): t = t', p~ = ::~-' - H' = 8G4_ i:Jt' • t' = t. H'=H-}J_ ot' · The following CT leave the Hamiltonian unchanged: ) (88.25 d) ) (88.26c) ) (88.26d) G3 (q, t, p', H') =-tH' + g(q, p', H'): ag H=H', ' i:Jg Pe = aqe • q~ = ap~ • I i:Jg ) t =t- oH'' (88.27c) G4 (p,H,q',t') = -Ht'+g(p,H,q'): qn = og t = t'- ~ p' og H' =H. " ope ' i:JH ' e = i:Jq~ ' The following CT leave both time and Hamiltonian unchanged: G3 (q,t,p',H') = -tH'+g(q,p'): P og H = H', , og e = oqe ' qe = ap~ ' G4 (p, H, q', t') =- Ht' + g(p, q'): og , P' og q - t=t, - e - ap-; • e - i:Jq~ ' t' = t. ) H'=H.) We shall now show1 that aCT is unimodular in the sense that detJ = 1. ) (88.27d) (88.28c) (88.28d) (88.29) Take the CT to be generated as in (88.20c) and differentiate. In matrix notation we have by = A bx + B by', bx' = B bx + C by', (88.30) where A=(~) ox, ox, ' ( i:J2G ) C=~i:J3'. Yr Ys (88-31) These equations may be written by -Abx=Bby', } - ii c5 x = - bx' + C by', (88.32) or (-A 1)· (bx) ( 0 B) (bx') - iJ o by = -1 c ,by' · (88.33) 1 Cf. CARATHEODORY, p. 92 of op. cit. in Sect. 71. Sect. 89. POISSON brackets and LAGRANGE brackets in QTPH. 151 Comparing this with (87.6), we have ( ~ 1 ~) - C= ! ~) J · (88.34) and (88.29) follows on taking the determinants of the two sides. 89. POISSON brackets and LAGRANGE brackets in QTPH 1• Let u, V be two functions of the (2N + 2) independent variables (x, y); the PoissoN bracket [ u, v] is defined by ou ov ov ou [u,v] =------=-[v,u]. (89.1) · ox, oy, ox, oy, Let (x, y) be functions of the two independent variables u, v; the LAGRANGE bracket {u, v} is defined as {u v} = ox, oy,- i3x, oy, =- {v u}. (89.2) ' ou ov ov ou ' In Sect. 97 we shall use the same notation with qe, Pe substituted for x,, y,; the following results can be translated immediately from (x, y) to (q, p). If u, v, ware any three functions of (x, y), then [[u, v], w] + [[v, w], u] + [[w, u], v] = 0. This POISSON-JACOBI identity is easy to prove by direct calculation 2 • In terms of the matrix r of (87.11), we have ou ou 75% ( ov) [ u, v J = (a;-,-oy) r :; , ox oy ov ( ox) {u, v} =(au-, -a;-) r :~ . (89.3) (89.4) Under an arbitrary transformation (x, y)-+(x', y'), with u and v treated as invariants, we have the formulae of transformation [ cf. (87.6) and (87.1 0) J ( ox) (ox') :: =J E . ~:) =J-1(:;) ou ou ' 8Y -8? (89. 5) and the same equations with u replaced by v. Hence (89.4) gives (89.6) 1 It used to be the general custom to denote the PoissoN bracket by (u, v) and the LAGRANGE bracket by [u, v], cf. WHITTAKER [28), pp. 298, 299. That no~tion is used by H. TIETZ, this Encyclopedia, Vol. II, pp. 194, 195. But in the application of classical dynamics to quantum theory, it has been found more convenient to denote the PoiSSON bracket by [u, v]; cf. P. A.M. DIRAC: Quantum Mechanics, p. 94 (Oxford: Clarendon Press 1930). Following GOLDSTEIN [7), pp. 250, 252, we shall denote the PoiSSON bracket by [u, v] and the LAGRANGE bracket by {u, v}. 2 For an indirect proof, see APPELL [2] II, p. 445; NoRDHEIM and FuEs [19], p. 107; CARATHEODORY, p. 55 op. cit. in Sect. 71. 152 J. L. SYNGE: Classical Dynamics. Sect. 89. If the transformation is CT, then, by (87.15) and (87.16), these equations become [u, v] = [u, v]', {u, v} = {u, v}'; (89.7) the POissoN and LAGRANGE brackets are invariant under CT. Returning to an arbitrary transformation (x, y)---* (x', y'), let uA represent the variables x~, ... xN+l• y~, ... YN+l• capital Latin suffixes taking the values 1, 2, ... 2N +2, with the summation convention. We have then two skewsymmetric (2 N + 2) x (2N + 2) matrices, a PoissoN matrix P with elements =[uA,u ] and a LAGRANGE matrix L with elements LA8 ={uA,u8 }. The element A C of the product LP is Now ox, ouB _ oy, ouB _ 0 ouB oy5 - -8ua ox5 - ' (89.9) and hence (89.10) In fact, we have LP= -1, L= -P-1, P= -L-1 . (89.11) This relation between the PoiSSON and LAGRANGE matrices is true for an arbitrary transformation (x, y)---* (x', y'). Further we have, for arbitrary independent variations, (<51 ;r', t51y') L (~:::) = <51 UA { UA, UB} <52uB = ( ox, _OY_,_ _ ox, oy, ) 15 u 15 u OUA OUB OUB OUA 1 ~ 2 B = <51x,t52yr- <52xrb1Yr (b2;r) = (b1 a:, <51y) r b2 Y = (b1 a:', b1y') J rJ ( :.~::). , 2Y (89.12) Therefore, for an arbitrary transformation (x, y)---* (x', y'), the LAGRANGE matrix L is related to the Jacobian matrix J by L=JrJ. (89.13) For aCT, we have then by (87.16) L = r, P = - L-1 = - r-1 = r. (89.14) In the above theory of POISSON and LAGRANGE brackets there has been no reference to an energy function Q. We now introduce this, and consider a ray or trajectory satisfying the canonical equations (86.6). Let F(x, y) be any function. Then as the representative point moves along the ray or trajectory, we have (89.15) Sect. 89. POISSON brackets and LAGRANGE brackets in QTPH. 153 In particular, the canonical equations themselves niay be written ~~ = [x,,.Q], ~~ = [y,,.Q]. (89.16) Thus the POISSON brackets are intimately connected with the motion of a dynamical system. It follows from (89.15) and the PoiSSON-jACOBI identity (89.3) that, for any two functions l(x, y) and F(x, y), d d-w [!, F] = [[I, F], .Q] =- [[F, .Q], I] - [[.Q, I], F]. (89.17) If I and F are constants of the motion, so that df dF dW = [I,.Q] = 0, dW = [F,.Q] = 0, (89.18) then it follows from (89.17) that the POISSON bracket [I, F] is also a constant of the motion (POissoN's theorem1). Let us now use the (q, t, p, H) notation, related to the (x, y) notation by (86.1), taking the energy function in the form .Q(x, y) = YN+I + ro(xl, ··· xN+I• Yt, ··· YN) as in (86.3), or, equivalently, .Q(x, y) =- H + w(q, t, p). The canonical equations (86.6) read Thus t=w+const, and we have dH ow -lit = -----at . (89.19) (89.20) (89.21) (89.22) On the energy surface .Q = 0, we may substitute H for ro, and the equations take the usual form dH oH dt at · (89.23) For any function F(q, t, p, H) we have by (89.15) dF _ fJF f)Q +~~-- fJQ fJF -~~~ dt - oxe oye oxN+I fJYN+I oxe oye oxN+t OYN+t ( 89. 24) _ fJF ow + oF ow oF + ow oF - oqe ope Tt - oqe Tie- Tt aii · On the energy surface .Q = 9 we may write H instead of w, and obtain dF oF fJH oF dt = Tt + Tt oH + [F, H]qp• (89.25) 1 Cf. APPELL [2] II, p. 447. 154 J. L. SYNGE: Classical Dynamics. Sect. 90. where oF oH oH oF [F, H]qp = oqe oPe - oqe oPe · (89.26) If F=F(q, t, p), this becomes dF oF Tt = Tt + [F, H]qp• (89.27) and if F = H it becomes simply dH oH dt 0 t . (89.28) 90. Canenical transformations generated by the canonical equations. The basic relative integral invariant. The CT we have been considering are finite transformations. To get an infinitesimal CT, i.e. one near the identical transformation, we recall that the identical transformation was given in (88.22); accordingly, following the plan (88.20c), we introduce the generating function G3 (x, y') = x,y; +F(x, y') du, (90.1) the function F being arbitrary and du being an infinitesimal constant. This gives the CT _ , + d . oF(x, >'}_ , _ + d . oFhil_ Yr- Yr U "x ' x, - X, U " ' ' U T Uh (90.2) or, to the first order, dx = x' - x = du · oF(x.yJ ) T T I OYr > d , d oF(x, y) y=y-y=- U·--- r ' 1 Ox, ' (90. 3) in which we have replaced y' by y in the partial derivatives. If we write the canonical equations (86.6) in the form d -d oQ(x,y) x,- W· " , uy, dy = - dw · oQ (x, 2:'2_ r ox, ' (90.4) and compare these with (90.3), we are led to say that the canonical equations generate an infinitesimal CT, the increment dw in the special parameter playing the part of the infinitesimal constant du, and the energy function Q the part of the function F. However, there is a change in point of view which may be a source of considerable confusion. Hitherto we have regarded a CT as a change of the labels attached to fixed points in the space QTPH, but we have regarded the canonical equations as the description of the motion of a representative point in QTPH for some fixed coordinate system. This duality of interpretation is present in all transformation theory, and we face it by recognizing two alternative interpretations of a CT: (i) We have an assembly of geometrical objects (points in QTPH) to which different sets of labels (x, y), (x', y') may be attached. (ii) We have a Euclidean space E2N+2 with one fixed set of rectangular coordinate axes, and (x, y), (x', y') are the coordinates of two different points of E2N+ 2 relative to that set of coordinate axes. According to the first point of view, a transformation (x, y)-+ (x', y') changes the labels on fixed points; according to the second, it moves the points, the space Sect. 90. Canonical transformations generated by the canonical equations. 155 E2N+2 being transformed into itself as a whole. If we putF=Q and du=dw, there is complete formal agreement between (90.3) and (90.4); this common form can be interpreted geometrically in either of these two ways. So far we have considered only infinitesimal CT generated by the canonical equations. In the interpretation (ii) above, we see all the points of E2N+2 given infinitesimal displacements, corresponding to some fixed infinitesimal value of dw. However, from the group property of CT it follows that a succession of infinitesimal CT is itself aCT, and we are led to the conclusion that, if we follow the points of E2N+2 along the rays or trajectories, with a common value of the finite increment L1 w for them all, then the resulting transformation of E2N+2 into itself is a finite CT. We shall now show how a generating function for this finite "CT may be constructed, the integration of the canonical equations of motion being assumed. For any curve C in QTPH along which a monotone parameter u is assigned, the integral G = J {y,dx,-Q(x, y) du} is meaningful. A general variation gives (90.5) 15G=[y,l5x,-QI5u]+ } (90.6) + J (15y, dx,- 15x, dy,- 15Q du + dQ 15u). 8 (x,yJ (x*,y*) Let us seek those curves C for which 15 G = 0 when the Fig. 42. Construction of a generating variation is arbitrary exceptfor the following conditions: function G (x•,x, L1 w) in Q r PH. (i) The end values x:, x, are fixed. (ii) The increment L1 u in the parameter u along the curve is fixed. We can put 15 u = 0 in (90.6), and we get Therefore the required curves satisfy dx, ()[) dy, ()[) du oy, ' du ax, . (90.8) This tells us that these curves are rays or trajectories, and also that the parameter u on any one of them is the special parameter (u = w). Moreover, from the nature of the variational principle used here, it follows that the 2N + 3 quantities (x*, x, Llw), chosen arbitrarily, determine the value of an integral G(x*, x, Llw) = J {y,dx,- Q(x, y) dw}, (90.9) calculated along a ray or trajectory (Fig. 42). On giving arbitrary variations to the 2N +3 quantities (x*, x, Llw), we get from (90.6) ac ac * fu = y,, ox*-= - y, ' (90.10) , , and also o~Gw = -Q(x, y) = -Q(x*. y*), (90.11) Q being constant along the ray or trajectory by (90.8). We recognize in (90.10) the CT (88.20a), and so we conclude that for any assigned value of L1 w, the function G (x*, x, L1 w), thus obtained by integrating along rays or trajectories, is the generating function of a finite CT which transforms the space QTPH into itself. We have, in fact, a one-parameter family of CT, with parameter L1 w. 156 J. L. SYNGE: Classical Dynamics. Sect. 90. Assuming the integration of the canonical equations (90.8), the generating function is constructed in the following steps: (i) Take any point B* in QTPH with coordinates (x*, y*). (ii) Through B* draw the ray or trajectory C satisfying (90.8) and proceed along it until the special parameter w is increased by an assigned amount L1w. Let B (x, y) be the point so obtained. Then we have functional relationships x,= x,(x*, y*, L1w), y,= y,(x*, y*, L1w). (90.12) (iii) Solve the first set of these equations, obtaining y: = y: (x*, x, L1 w). (90.13) (iv) Calculate the integral (90.9) from B* to B along C, (x, y) being functions of (x*, y*, w) of the form (90.12); thus G appears as a function of (x*, y*, L1 w). (v) Substitute from (90.13) to obtain G(x*, x, L1w). Fig. 43. A circuit C and a tube of tbe natural congruence in Q T PH. In (88.8) we established the invariance of the circulation of action under CT, a result capable of a dual interpretation according to the way in which we regard the CT. In (88.8) it was a question of changing the labels on fixed points. To get the other point of view, it is simplest to start all over again. Fig. 43 shows a circuit C in QTPH and a tube containing C, this tube consisting of rays or trajectories (part of the natural congruence). It is convenient to reserve d for a displacement along the natural congruence, so that ()!} dx,= dw · - 8-, Yr ()!} dy,=-dw·-~-· ux, (90.14) We shall use b for a displacement along C, so that the circulation inC is u(C) = ~=f=O. ux,uy5 Then, as in (88.20c), the equations ac , ac y r = ex, ' x, = oy; (91.3) (91.4) 158 J. L. SYNGE: Classical Dynamics. Sect. 92. define a CT (x, y)-+(x', y'). For the energy function (always treated as an invariant in the sense of tensor calculus) we have Q'(x', y') =!J(x, y) =D(x, ~~) = y;_,+l· (91.5) Accordingly the new equations of the natural congruence read (91.6) and integration gives x~ =au, x;.+l = w, } y~ = bu, y~+ 1 = k, (91.7) where afl, bu, and k are (2N+1) constants, the values of which depend on the particular ray or trajectory under consideration. We note that the special parameter w is equal to the coordinate x;.+l, a trivial constant of integration having been dropped. Since, for arbitrary values of the constants, (91.7) represents a congruence of straight lines all parallel to the axis of x;.+l, we have succeeded in transforming the natural congruence into a congruence of parallel straight lines. The energy surface Q = 0 has been transformed into the plane y;_,+l = 0. By working in QTPH instead of in QT, we are able to present the theory of Sect. 77 in a more general way. The partial differential equation (91.2) is in fact the HAMILTON-JACOBI equation in a general form. To establish the connection, let us pass over into (q, t, p, H) notation, taking the energy function in the form D(x, y) = YN+l + w(xl, ··· xN+l• Y1• ··· YN). (91.8) Then by (86.1) the partial differential equation (91.2) becomes ~~ +H(q,t,~f) = -H', (91.9) if we write H instead of was a functional symbol; we need a solution G(q, t, p', H') such that (91.10) at ap~ at oH' We may regard (91.9) as a partial differential equation in the independent variables (q, t), the quantities (p', H') being constants. The first step towards integration is to put G= -H't+ U(q,t); (91.11) then U has to satisfy au ( au) ae+H q,t,Bi =o, (91.12) which is in fact the HAMILTON-JACOBI equation (77.3). We need a complete integral in order to carry out the transformation (91.4). 92. The order of the canonical equations, and its reduction by means of a first integral. We return to the question of the order of the equations of motion, raised near the end of Sect. 68. Given an energy function D(x, y), the canonical equations dx, oQ _dy, oQ dW oy, ' dw ox, (92.1) Sect. 92. The order of the canonical equations. 159 form a system of order 2N + 2. If we multiply across by dwjdxN+l' we get dxe _ o!Jfoye dye - - o!Jfoxe (92.2) dxN+r- oilfoYN+r ' dxN+r- o!JfoYN+I' and dyN+r _ o!JjoxN+r (92. 3) dxN+r - o!JfoYN+~ This is a system of order 2N + 1; the independent variable xN+l is contained in Q. We know that D(x,y)=c, (92.4) a constant along each trajectory. Let this equation be solved for YN+I' so that we have (92.5) Substituting this in (92.2), we have a system of order 2N, containing the constant c. If these equations are solved for x1 , ... xN, y1 , ... YN, then YNH is given by (92. 5). Thus the canonical equations (92.1) are reducible to order 2 N; but the reduced Eqs. (92.2) are not in canonical form. Suppose now that, instead of being given an energy function (which leads to a natural congruence filling QTPH), we are given an energy surface with equation Q(x, y) = 0. (92.6) The trajectories are now confined to this surface. We have still the Eqs. (92.1) to (92.3) and we also have (92.4) and (92.5) with c = 0. But now we are concerned with a surface, and the equation for that surface may be put into different forms. In fact, the functional form of Q is not prescribed, and we are entitled to change it, so that the equation of the energy surface reads We have then D(x, y) = YII'+l + w(xr, ... xN+I' Yr, ... YN) = 0. ~!!___ = 1 OYNtr ' and the Eqs. (92.2) become dxe - oQ dxN+l- oye' (92.7) (92.8) (92.9) This system is of order 2N. When we base dynamics on an energy surface in QT PH, the equations of motion can be reduced to order 2N, with preservation of the canonical form, if we ( i) write the equation of the energy surface in the form (92.7), and (ii) take xN+r as parameter. Note that the parameter xN+I is now contained in Q, whereas w was not contained in Q in (92.1). The standard translation from (x, y) to (q, t, p, H) is given as in (86.1) by Xe=qe, XN+l:t, } (92.10) Ye=Pe' YN+1--H. But in view of the symmetry in the (x, y) notation, there is no necessity to stick to this translation; we are at liberty to permute the suffixes on x, (making the same permutation for y,). Thus, in (92.7), YN+I need not necessarily stand for - H; it might stand for p1 , in which case the parameter in (92.9) would not be t, but q1 . This versatility of the ( x, y) notation should never be forgotten. 160 J. L. SYNGE: Classical Dynamics. Sect. 92. In what follows we shall assume that an energy function is given. Consider a system which possesses a first integral F(x, y), by which we mean that dF = [F .Q] = 0 dw ' [cf. (89.15)], so that F(x, y) = const (92.11) (92.12) along each trajectory. We shall discuss the reduction of the order of the canonical equations (92.1) from 2N + 2 to 2N by means of this first integrall, with preservation of the canonical form and of the special parameter w. Let G (x, y1 ) be a solution of the partial differential equation (92.13) satisfying ()2G det ~--, =F 0. ux,uy5 (92.14) This procedure is, in fact similar to that of Sect. 91, the HAMILTON-jACOBI equation (91.2) being replaced by (92.13), and one might ask why we should spend our time on (92.13) when a solution of (91.2) solves the problem of motion. The answer is that the practical feasibility of getting an explicit solution depends very much on the complexity of the function involved (.Q and F respectively). It may well happen that F is much simpler than .Q. Having solved (92.13), we apply the CT ac y,=i}X· , I ac x, = ay_;_, and the equations of motion transform to dx; 8!1 1 dy~ 8!1 1 dw oy; ' dw ax; ' the new energy function being .Q1 (X1 , y1 ) = .Q(x, y). We have then - 1-=- YN+l =--F x,- =--F(x,y)=O, a!11 d I d ( ac) d oxN+l dw dw ox dw so that the variable x~+l is absent from .Q': If, now, we select from (92.16) the 2N equations dx~ dw oil' dy~ oil' oy~ • aw - ax~ • (92.15) (92.16) (92.17) (92.18) (92.19) (92.20) we have a set of canonical equations of order 2N, as required; the new energy function contains y~+l as a constant. If we confine our attention to trajectories on the energy surface .Q = O, we can effect a further reduction of 2 in the order. In the new coordinates, the energy surface has the equation .Q (X~, ... X~, y~, ... Y~+l) = 0; (92.21) 1 The argument given here, being set in QT PH, has an appearance of greater generality than other treatments; cf. NoRDHEIM and FuEs [19], p. 115. See PRANGE [21], pp. 713-726, for a discussion, with considerable detail, of the simplification of a canonical system by knowledge of a first integral. Sect. 92. The order of the canonical equations. 161 we solve this equation for one of the y's, say yi.r, and proceed as at (92.7), obtaining 2N- 2 canonical equations analogous to (92.9). Thus, using a first integral and an energy surface, the order is reducible to 2N- 2. Some illustrative examples follow. a.) Reduction of order by ignoration of a coordinate or by the integral of energy in a conservative system. As will be seen, the work here is actually trivial, but it will serve to explain the method. Suppose that one of the coordinates, say xN+l• is absent from Q(x, y). In view of what has been said above about the symmetry of the notation, this may mean either that ·(i) the system has an ignorable coordinate (Sect. 46), or (ii) that the system is conservative [tis absent from H(q, t, p)]. Both cases are covered by the following argument. Our assumption is oQ ------0 OXN+l - ' and consequently we have the first integral F(x, y) = YNH = const. The partial differential equations (92.13) has the very simple form and a suitable solution is oG , ~=YN+l• u~N+l G(x, y') = x,y;. By (92.15) this gives the identical transformation y, = y;, x,= ' x,, (92.22) (92.23) (92.24) (92.25) (92.26) and the problem of reduction in order to 2N is solved by merely picking out from the canonical equations the 2N equations (92.27) To complete the reduction to order 2N- 2 on the energy surface, we write the equation of that surface in the form Q(x, y) = YN + w(x1 , 00 • xN, y1 , 00 • YN-l• YNH) = 0. (92.28) Then (92.27) give, as in (92.9), dx1 ow -dxN = oy;' dyl ow (92.29) dxN ox1'_' Let us translate this into (q, t, p, H) notation, using (92.10). We start with an energy function Q(q, p, -H), t being absent by hypothesis. We have the first integral H=E, and, as in (92.27), the equations of motion Handbuch der Physik, Bd. III/l. (92.30) (92.31) (92-32) 11 162 J. L. SYNGE: Classical Dynamics. Sect. 92. We now take an energy surface .Q=O, and write its equation in the new form (92.33) the constant E having been substituted for H. Then, as in (92.29), we have the equations of motion dql ow -dq;.- = op----; • dpl ow dqN = -aq;, (92.34) Here w contains the independent variable qN. Thus, in a conservative system, we reduce1 the order of the canonical equations to 2 N- 2 by means of the integral of energy H =E. {J) Reduction of order by means of an integral linear in the momenta. Suppose that a system has a first integral Y1 + Y2 + y3 = const. (92.35) It is convenient to modify the above plan, and use a generating function G (x 1 , y) and the CT oG 1 eG x, = ~' y, = ----,---,-. (92.36) uy, ux, We seek G to satisfy 1 aG Y1 + Y2 + Ya = Y1 = w · 1 A suitable solution is G = x~ (YI + Y2 + Ya) + x~ Y2 + · · · + x~+l YN+I; it satisfies fJ2G det~-~-=f=O, ux,uy5 (92.37) (92.38) (92.39) so that (92.36) gives aCT (x, y)->-(X1 , y'). Since y~ is a constant of the motion, .Q'(x1 , y1 ) lacks x~. and the new equations of motion read dx~ f)[J' dxN+I f)[J' l (JW= ey; • =-,-, dw OYN+I dy; B!J' dYN+I f)[J' (92.40) dw- - ox~' dw - OXN+I' a system of order 2N. Integrals of the above type occur in the 3-body problem (Sect. 53); in that case we have three integrals of linear momentum, Yt + Y" + Y1 = Y~ , l Y2 + Ys + Ys = Y2, Ya + Y& + Ye = Y~, (92.41) the right hand sides being constants of the motion. The system has 9 degrees of freedom (N =9). By taking the generating function G(x', y) = x~(Yt + Y4 + Y7) + x~(Y2 + Ys+ Ys) + } + x~(Ya+ Y&+ Ye) + X~Y4+ ··· + X~oYto• (92.42) ------ 1 Cf. WHITTAKER [28], p. 313, for a different description of this reduction. Sect. 93. Circulation theorem. 163 we eliminate x~, x;, x~ from the transformed Q, in which y~, y~, y~ appear as constants. The canonical equations of the form (92.1) are thereby reduced from order 20 to 20-6=14. But if we work in the space QP (Sect. 96), using an energy function H(q, p) instead of Q(x, y), the reduction in order is from 18 to 121 . y) Reduction of order by means of an integral of angular momentum. Suppose that F(x, y) = X1Y2- X2Y1 = y~, (92.43) a constant of the motion. According to (92.13) we are to solve the partial differential equation (92.44) A suitable solution is G (x, y') = y~ [y; (xi+ x~) +arc tan ::] + x3 y~ + · · · + xN+I Y~+I; (92.45) it satisfies the determinantal condition (92.14), and gives the CT (92.46) The variable x~ is absent from the new energy function. The generating function (92.45) is reached by the use of polar coordinates, x1 = rcos&, x2 = rsin{}. VI. The space of states (QTP). 93. Circulation theorem. In the (2N + 1)-dimensional space of states Q T P the coordinates 2 of the representative point are qe, t, Pe. The Hamiltonian H is not a coordirtate, but a function of position in Q T P: H = H(q, t, p). (93 .1) The canonical equations of motion are . oH qQ = ope' (93.2) 1 For the reduction of the order of the equations of motion in the 3-body problem from 18 to 6, see WHITTAKER [28], pp. 340-351. 2 We shall work in the small, and avoid reference to overlapping coordinate systems (cf. Sect. 63). For notation, see Sect. 62. 11* 164 J. L. SYNGE: Classical Dynamics. Sect. 94- Written in the form dq1 dqN dp1 dPN oH =···=-oH =~=···= oH dt 1 ' (93-3) op1 opN oq1 - oqN in order to put all the coordinates on a parity, these equations exhibit the natural congruence of trajectories in Q T P, one curve passing through each point. We note that (93-2) imply (93.4) To discuss circulation in Q T P independently of the discussion in Sect. 90 for Q T PH, we define the circulation in any circuit C to be "(C) = ~ (Pe {Jqe- H {Jt). (93-5) c Giving to C any infinitesimal displacement d (not necessarily along the natural congruence), and integrating the varied expression by parts, we get d" (C)= q'l ... p~-1 =---. u oqf,;_1 (96.16) Having done this, we can effect a further reduction of 2 in the order by means of the integral of energy (96.13). Note that solving the HAMILTON-jACOBI equation is equivalent to determining the motion. The solution of (96.15) may be a much simpler matter, for F(q, p) may be a simple function like P1 +P2+P3 [cf. (92.35)]. 97. Non-conservative systems. Canonical transformations in QP. POISSON brackets and LAGRANGE brackets 2• We now turn to the general non-conservative system with Hamiltonian H(q, t, p) involving the time, so that BH at =t= 0. (97.1) 1 This formal argument is valid only in the small; cf. Sects. 63, 100. 2 Cf. WHITTAKER [28], Chap. 11. Sect. 97. Non-conservative systems. Canonical transformations in QP. 171 Since the energy function in QTPH was Q(x, y) and not Q(x, y, w), we cannot apply the QT PH-theory by a simple reduction in dimensionality, as we did for a conservative system in Sect. 96. It is true that QTPH-theory is valid in all generality, but it is set in a space in which t is a coordinate, and we have now to demote t to the position of a mere parameter. Consider a function G(q, q', t) and the transformation (q, p)-+ (q', p') given by p' = _ ac (q, q'_._!l_ e aq~ . (97.2) Then (97.3) or Pe bqe- H(q, t, p) bt = p~ bq~- K(q', t, p') bt + bG, (97.4) where K(q',t,p') =H(q,t,p) + aG(~l·t). (97.5) Now G is a function of position in the space QTP [since we can solve (97.2) for q~ in terms of (q, t, p)], and we can apply the Pfaffian argument as in (94.13); this tells us that the canonical equations (97.6) transform into ., aK qe=-ap~' (97.7) the Hamiltonian being changed as in (97.5). Thus (97.2) is a canonical transformation (CT) in Q P, the generating function G containing t as a parameter. This treatment in QP is less general than the treatment given in Sect. 94 for QT P, because we have not in the present instance transformed the time. For time-preserving CT in QTPH, see (88.26). Let u and v be any two functions of the 2N + 1 quantities (q, p, t), i.e. functions of position in QP and of the parameter t; their POISSON bracket is defined as au av av au [u v] = ------~- (97.8) ' aqe ape aqe ape · As the representative point moves along a trajectory, the rate of change of any function F(q, p, t) is (97.9) In particular we have, as an alternative expression of the canonical equations, (97.10) We shall now show that if u (q, p, t) and v (q, p, t) are two constants of the motion, then so also is their PmssoN bracket [ u, v J. We are given _1!_11,_ = ~ + [u H] = 0 t!!' = ~ + [v HJ = 0. dt at ' • dt at ' (97.11) 172 J. L. SYNGE: Classical Dynamics. Sect. 98. Now by (97.9) we have d a Iii [u, v] =at [u, v] + [[u, v], HJ, (97.12) and in this equation we can change the last term by the PorssoN-JACOBI identity [cf. (89-3)] [[u,v],w] + [[v,wJ,u] + [[w,u],v] =0, so that, remembering the skew-symmetry of PoiSSON brackets, :t [u,v]= :t [u,v]-[[v,H],u]+[[u,H],v]. Applying (97.11}. we get :t [ u, v] = :t [ u, v] + [ ~~ ' u] - [ ~; ' v] = 0' which establishes the result. Consider now an oo2 family of trajectories with equations q" = qe(u, v, t), Pe = Pe(u, v, t), (97.13) (97.14) (97.15) where u and v are constant along each trajectory. Then the LAGRANGE bracket (97.16) is a function of u, v and t: we shall prove by direct calculation that this LAGRANGE bracket is a constant of the motion. We have oqe _ . _ oH ope _ · _ iJH 7ft- qe- ape • at- Pe- - aq(! • and these are functions of u, v and t. Then !:_(oq(! ape)=~(aq~ oPe)=~(aH)f!?_e __ oqe~(oH) 1 dt au ov ot au ov au ape ov au ov oqe _ o2H oqa oPe+ o2H oPa oPe_ - apeoqa au-av- ap(!oPa ali"Tv . oqe o2H oqa oqe o2H opa -au oqe oqa Tv- au oq(! oPa av _ o2H opa oPe_ o2H oqP oqa - app opa au Tv oq'l oqa au Tv. Interchanging u and v and subtracting, we get d lii{u,v}=O, establishing the result. (97.17) (97.18} (97.19) 98. Non-conservative systems. Absolute integral invariants in Q P. LIOUVILLE's theorem. We continue to consider a general system for which H = H (q, t, p). Let capital suffixes A, A1 , ••• take the values 1, 2, ... 2M where M :;;,N, N being, as always, the number of degrees of freedom of the system. Consider an oo2M family of trajectories with equations qf! = qe(u, t), P(! = Pe(u, t), (98.1} where u stands for a set of 2M quantities uA which are constant along each trajectory. Sect. 98. Non-conservative systems. Absolute integral invariants in QP. 173 For any fixed value oft, the Eqs. (98.1) define a surface of 2M dimensions immersed in QP. Let D be a domain in this surface, limited by bounds on the ranges of uA. Introduce the following 2M x 2M determinant, a function of the u's and oft: oqe. oqe. oqe. ou1- au; ... ou2M oqe. oqe. oqe. au; au; ... OU2M oqeM oqeM oqeM au; ou2 ... au;~ opa, opa, opa, au; ou2-- ... ou2M (98.2) Here e1 , ••• (IM, 0'1 , ... aM are any numbers in the range 1, 2, ... N. Then the integral J LIM(el, ... eM, 0'1, ... aM) dul ... du2M D (98.3) is invariant in the sense that it has the same value no matter what parameters 1 uA are used in D. Define rpM by rpM= LIM(e1 .... eM, e1 .... eM), with the summation convention operating; and define IM = ~! J rpMdui ... du2M· D Its value is independent of the choice of parameters uA in D. (98.4) (98.5) Using the permutation symbol BA, ... A,M, which is skew-symmetric in all its suffixes and equal to unity when they read 1, 2, ... 2M, we may write out the determinant (98.2) explicitly. But we need only rpM as in (98.4); it reads (98.6) There is summation here for each e on the range 1, ... N and for each A on the range 1, ... 2M. In terms of LAGRANGE brackets, we have rpM= (-!)M BA 1 ···A 2M {UA1 , UAM+l} {UA 2, UAM+>} ... {UAM' UA 2M}. (98.7) Each LAGRANGE bracket is independent oft by (97.19). Therefore rpM is independent of t, and we conclude that the integrals I M, as given in (98.5) for M = 1, 2, .. . N, are absolute 2 integral invariants. The case M = N is of particular interest. The absolute integral invariant IN=;, J rpNdui ... du2N D (98.8) is now an integral extended over a 2N-dimensional portion D of the 2N-dimensional phase space QP. This domain changes with t, the representative points 1 Provided that their orientation is not changed; cf. H. D. BLOCK: Quart. Appl. Math. 12, 201-203 {1954). 2 Called absolute {not relative) because D need not be a closed domain (e.g. a circuit). 174 J. L. SYNGE: Classical Dynamics. Sect. 98. being carried along according to the canonical equations (Fig. 45). For any chosen value of t, we may use (q, p) as coordinates in D, so that U1 = ql, ·· · uN = qN, uN+I = Pt• · · · U2N = PN· (98.9) Then (98.2) and (98.4) give LlN(!?t• · · · !?N • 0'1, ···aN) = ee•···eN ea, ... aN'} tPN= N!, (98.10) and the integral invariant IN becomes IN= I dql··· dqNdPt··· dPN= I dqdp, (98.11) D D to use an abridged notation. Calling this integral the volume 1 of D, we have LIOUVILLE's theorem: the volume of any portion of QP is conserved when thereFig. 4 5. Conservation of volume in Q P (LIOUVILLE'S theorem). presentative points whick compose it move in accordance with the canonical equations. This result is so fundamental .in statistical mechanics that we shall look at it in two other ways. First, LIOUVILLE's theorem, as proved by him 2, is aCtually more general, neither the evenness of dimensionality of the space nor the canonical form of the equations of motion being required. Consider the equations where the right hand sides satisfy oXA =0 OXA ' (98.12) (98.13) the suffix A ranging 1, 2, ... M, with the summation convention. Instead of following LIOUVILLE, we may use a hydrodynamical argument. The Eqs. (98.12) define a velocity-field vA =XA in an M-space in which xA are taken as rectangular Cartesian coordinates, and, just as in ordinary hydrodynamics ~+~+~ ox oy oz is the expansion (rate of increase of volume per unit volume), so in this M-space ovAfoxA is the expansion, volume being defined as I dx1 • •. dxM. Then, by virtue of (98.13), volume is conserved. The details of proof can be filled in by expressing the rate of increase of a volume, moving according to (98.12), as an integral over the (M -1)-space bounding it, and applying GREEN's theorem.·· It is evident that (98.13) is true in particular if M is even and (98.12) are canonical. The second alternative approach is through canonical transformations (CT). The essential point here is that the jACOBIAN of aCT is unity [cf: (88.29)], this being true even if aCT (q, p)-+ (q', p') contains t as a parameter. Two conclusions follow from this. First, without referring to motion at all, we recognize the integral IN of (98.11) as a suitable definition of volume, since volume so defined has the same value for all coordinates (q, p) in QP obtained from one set 1 GrBBS called it extension-in-phase; cf. footnote to Sect. 96. 2 J. LIOUVILLE: J. de Math. 3, 342 ( 1838); the famous theorem is a secondary result in his paper. Sect. 99. Action-angle variables. 175 of coordinates by CT1 . Secondly, volume is conserved in the motion because motion in accordance with the canonical equations consists of infinitesimal CT [cf. (90.4)]. In statistical mechanics 2 we consider a vast number n of identical Hamiltonian systems, differing only in their initial conditions. The superposition of these systems in the space QP gives an ensemble, a "fine dust" of representative points with a probability density l(q, p, t) such that n I dq dp is the number of representative points in the volume element dq dp at timet. As the element dqdp moves with the dust according to the canonical equations, its volume is conserved and also the number of representative points in it. Hence d 1/d t = 0, or, equivalently, Of 8t + [!, H] = 0. (98.14) This is the fundamental partial differential equation to be satisfied by the density 1. determining I for any t when I is given for t = 0. 99. Action-angle variables 3. Action-angle variables were introduced by C. DELAUNAY for the discussion of astronomical perturbations 4 • Later, they were found to be admirably suited to the older form of quantum mechanics, for the BOHRSoMMERFELD quantization consisted in making each action variable an integral multiple of PLANCK's constant h. As treated below, the theory of action-angle variables depends on the separation of variables in the HAMILTON-JACOBI equation. Even though the space QP should have Euclidean topology, one or more of the separating variables may be cyclic (e.g. an azimuthal angle) 5• However, the presence of cyclic coordinates is not an essential feature of the theory and their inclusion makes the discussion a little more complicated. Therefore it will be assumed that they are absent, essential modifications due to their presence being noted where necessary. Let the Hamiltonian 6 H(q, p) be such that the HAMILTON-JACOBI equation is of the separable type (Sect. 78). By this we mean that, in the 2N-dimensional space of the variables (q, P'), the partial differential equation has a solution of the form H (q, ~~) = p~ G(q, p') = Gl(ql, P') + G2(q2, p') + ... + GN(qN, P')' p' standing for the N quantities p~, and the determinantal condition CJ2G det oqe opa =f= 0' being satisfied; in other words (99.2) is a complete integral. (99.1) (99.2) (99.3) 1 There is no definition of volume invariant under arbitrary transformations of (q, p). 2 SeeR. H. FowLER: Statistical Mechanics (Cambridge: University Press 1936); A. I. KHINCHIN: Mathematical Foundations of Statistical Mechanics (New York: Dover Publications, Inc. 1949); A. MuNSTER: Statistische Thermodynamik (Berlin: Springer 19 56); also the article by E. A. GuGGENHEIM in Vol. III, part 2 of this Encyclopedia. 3 For various treatments of action-angle variables, with illustrative examples and reference to quantum conditions and adiabatic in variance, see M. BoRN: The Mechanics of the Atom, Chap. 2 (London: Bell1927); CoRBEN and STEHLE [3], pp. 239-264; FuEs [6]; GOLDSTEIN [7], pp. 288-307; LANCZOS [15], pp. 243-254; A. SoMMERFELD: Atomic Structure and Spectral Lines, Vol. 1, pp. 615-623 (translated from Sth. German Edn. by H. L. BROSE, London: Methuen 1934; 3rd Edn.). 4 Cf. WHITTAKER [28], pp. 426, 431. 5 Note that the word cyclic is used in a topological sense, and is not to be confused with ignorable (cf. Sect. 63). 6 The system is assumed to be conservative, i.e. oHjot = 0. 176 J. L. SYNGE: Classical Dynamics. Sect. 99. A canonical transformation (CT) is defined by P _ OG(q.~ 1_ oG(q,p1) Q - 0qQ I qQ- op~ (99.4) We observe the effect of separation here; when written more explicitly, the first set of these equations read P _ oG1(q1, p'J p _ ac2(q2,p1 ) 1 - oql , . 2 - oq2 , (99.5) each equation contains only one p and the corresponding q, but of course all the quantities p~ are, in general, involved in each equation. The new Hamiltonian is H1 (q 1 ,p1 )=H(q,p)=H(q, ~~)=p;_,, (99.6) and the new equations of motion are • I oH1 • 1 aH, ( ) qe = ap~ , Pe =- oq~ . 99.7 Therefore all the quantities (q1 , p1) are constant along each trajectory, except q;_,; for it we have q;_,= 1, so that q;_,=t+const. We can write p;_, = E, (99.8) E being the constant value of H on the traFig. 46. Circuit I',(p') or .r,(j) in the plane: jectory. By the CT (99.4) we have transformed the canonical variables q, p,. the trajectories into parallel straight lines, as in Sect. 96. Consider a representative plane Il1 in which q1 , P1 are taken as rectangular Cartesian coordinates (Fig. 46). If the N quantities p~ are held fixed, the first equation in (99.5) defines a curve in Il1 ; denote this curve by F1(p1). Similarly the other equations in (99.5) define curves ~(P'), ... JN(P') in representative planes Il2 , ... liN. We now assume that these curves are all closed1 ; in each of the representative planes we have ooN circuits FQ(p1 ), a set of circuits, one in each plane, being determined by the values of the N quantities p~. Define quantities fe by the formulae h = <} pldql, ... lN= <} PNdqN, (99.9) I't(P') I'N(P') these being in fact the "areas" contained within 2 the several circuits determined by the values p~. Assuming that ofe det -, =F 0, (99.10) BPa we have a two-way functional relationship {]) ~ (p1), and we may express p~ as function of the ]'s: p~ = p~(]). (99.11) Substituting these functions in (99.1) and (99.2), we have a solution of ( oG*) 1 H q,aq = PN(]) (99.12) 1 In the case of a cyclic coordinate, the circuit may appear like T 1 in Fig. 31, p. 104. 2 Or under, in the case considered in preceding footnote. Sect. 100. The periodic property of angle variables. 177 in the 2N-dimensional space of the variables (q, ]), this solution being of the form (still separated) (99.13) This function we now use as generating function for aCT (q, p)-+ (w, ]), expressed by the formulae P = _CJ__G* (q_J)_ oG* (q, J) e oqe , we=-~· (99.14) The quantities fe are called action variables and the quantities we angle variables. When action-angle variables are used, the Hamiltonian involves the action variables only; we write it H*(J). The canonical equations of motion read • oH* • oH* le=- owe =0, we=aie-, (99.15) so that the ]'s are constant along each trajectory and the w's are given by w11 = v11 t + <511 , where vf! and <511 are constants, the former being oH* "'a= ofe . The constant value of H along the trajectory is H=E=H*(J), the constant E being the total energy in ordinary dynamical systems. (99.16) (99.17) (99.18) 100. The periodic property of angle variables. Let F(J) be any circuit in the space QP such that all the action variables fe are constant along it. A position of the representative point Bin QP determines positions of representative points B11 in the planes lie (Fig. 46), and when B is carried once round F(J), these points Be are carried round the respective circuits !;(]), perhaps several times over. Symbolically, we may write F(J) = nil I; (]) ' (100.1) where the coefficients ne are integers (positive, negative or zero). In (100.1), and below, the summation convention operates as usual over the range 1, ... N. On carrying B round F, the generating function G*(q,]) is increased by L1rG* = <} ~~: dqe = <}Pedqe. (100.2) T(J) 1 (J) To examine this final expression, we write out the first set of the transformation Eqs. (99.14): (100.3) The first of these equations sets up a connection between p1 and q1 which is the same for the point Bon F(J) and the corresponding point B1 on I;_(]); therefore, by the definition of h. in (99-9), we have c}P1 dq1 = n1 c}p1 dq1 = n1 k (100.4) ru> r. Thus (100.2) gives (100.5) Handbuch der Physik, Bd. III/1. 12 178 ] . L. SYNGE: Classical Dynamics. Sect. 100. We see that, on the basis of assumptions explicit and implicit, the functions Gt (q1 , ]) , ... Gt;(qN, ]) are necessarily multiple valued functions of their q-arguments. Now vary the action variables ~ infinitesimally, the circuit F(J) varying in consequence. The n's in (100.5), being integers, cannot change under this infinitesimal variation, and we get 15LlrG* = nu15k On the other hand, from (100.2) we have MrG* = 15 ~ Pu dqu = ~ (15Pu dqQ- d 15e dpe). T(J) T(J) (100.6) (100.7) This last expression is the bilinear form invariant under CT (cf. Sect. 96), and therefore 15LirG* = ~ (15 ~ dwe- 15we d fe). (100.8) T(J) But the ]'s are constants on both the varied and unvaried circuits F; therefore d~=O and 15~=const, and we have 15LlrG* = 15~Llrwp, (100.9) where Llrwe is the increment in wP on passing once round F(J). Comparing this last equation with ( 1 00.6), we may state the result: On taking the representative point B once round any circuit F(]) in QP for which the action variables ~ all have fixed values, the increment in the angle variable we is (100.10) where nP is the number of times the point Be goes round the circuit J;(]) in the plane ne. In particular, if we hold all the quantities (q, ]) fixed except q1 , then the point B1 moves on the curve Jl (]) in Il1 , and the points B2 , •.• BN remain fixed. This causes the representative point Bin QP to move on some curve, and when B1 has completed a circuit of Jl(]), B has completed some circuit Jl*(J) for which the numbers nu of (100.1) are n1 = 1, n2 = · · · = nN = 0. Substituting in (100.10) and writing Ft for F, we have Lfr,•W1=1, Lfr,•W2 =0, ... Lfr,•WN=O. Hence more generally, with the notation interpreted as above, Now let the second set of equations in (99.14) be solved for qP: qe = qu (w, ]) . (100.11) (100.12) (100.13) (100.14) If we fix all the quantities (w,]) except w1 , we leave one degree of freedom in the representative point B in QP, the P's being given to within that one degree of freedom by (100.3). Then B moves on some curve in QP, and the "projected" points Be move on the curves J;(J). Let w1 be increased continuously from 0 to 1, the other w's being held fixed as aforesaid. From inspection of (100.12) we conclude that, when this operation has been completed, the point B1 will Sect. 100. The periodic property of angle variables. 179 have gone once round !;,{]) and the points B2 , 00. BN will have been restored to their original positions without having gone round their respective circuits. The same argument can be applied to increases of unity in each of the angle variables individually, and we conclude that the functions (100.14) are periodic in each of the w's with period unity1• We can therefore expand these functions in FouRIER series of the form q =~A 0 e2ni(n,w,+ .. ·+nNU'N) (100.15) {] L...J e, n11 ••• n_.y , (n) the summation running over all integer values of the n's (positive, negative and zero), and the A's being complex functions of the action variables J, such that a reversal in the signs of all then's turns an A into its complex conjugate. Then, by (99.16), the motion of the system is given by q = ~ B 0 e2ni(n,v,+ ... + "N"N)t (100.16) Q ~ Q,nlr···"..v ' (n) the B's being functions of the ]'s. In this sense, the quantities v~ are "frequencies". If we know the function H*(]), they can be calculated at once from (99.17) by differentiating this function. It has been assumed that the curves in the planes II~ defined by the Eqs. {100.3) (for fixed values of the ]'s) are closed, these closed curves being in fact the circuits f'e (]). This assumption does not at all imply that the motion of the system is periodic: we see from (100.16) that it is periodic if, and only if, the ratios of the frequencies v~ are rational numbers. A system is called degenerate if the frequencies satisfy a relation of the form (100.17) where s1 , •.. sN are integers or zero, with at least two non-zero. Degeneracy occurs when the Hamiltonian H*(]) involves the action variables in certain ways which the following example illustrates. Suppose the Hamiltonian is of the form H*(]) = /(K, h_, lr,, 00 ·lv), } K = m1.h + m2h +rna fa, (100.18) where the m's are integers. Then BH* of '~~I= 8]1 = ml BK' (100.19) and we have a double degeneracy: (100.20) As indicated in Sect. 63, the exposition of general dynamical theory in this article is, from the standpoint of modern pure mathematics, on a rather low level of precision. As far as theory in the small is concerned, it would not be hard to make those additions which would make it precise, but action-angle variables take us out of the small into the large through the introduction of the circuits f'e{]). This raises topological questions of considerable complexity, not touched on here. 1 A cyclic coordinate will not be periodic. It is increased by its cyclic constant, and (100.15) and (100.16) are modified by the addition of other terms; cf. FuEs [6], p. 140, GOLDSTEIN [7], p. 295. . 12* 180 J. L. SYNGE: Classical Dynamics. Sect. 101. VIII. Small oscillations. 101. Reduction of energies to normal form. Normal modes and frequencies. Degeneracy. Consider a dynamical system with N generalized coordinates 1 q~ and Lagrangian L=T- V, T 1 ()"o"a = 2aea q q-q ' V = V(q), the system moving in accordance with LAGRANGE's equations a oT oT dt oqe oqe av oqe 0 This is the ordinary dynamical system (ODS) of Sect. 66. (101.1) ( 1 01.2) We geometrize in configuration space. Q. The representative point describes a trajectory in accordance with (101.2), the trajectory being determined by an initial point qe and an initial velocity qe. If, at a certain point in Q we have ~=0 oqe ' (101.3) then no motion results if the velocity vanishes; these N equations define equilibrium configurations, and we may expect in general to find a discrete set of such points in Q, since the number of equations is equal to the number of coordinates. We shall now discuss small oscillations about equilibrium. By a change of coordinates, we can make the equilibrium point the origin 0 (qe=o there), and also make V=O at] 0, since potential energy is always undetermined to within an additive constant. Expanding aea(q) and V in power series about 0, the principal parts of T and V become T -.! ·e·a V lb e,.,a - 2 ae a q q ' = 2 e a q '1 • ( 1 01.4) Here the coefficients are constants, and ae a= aae, bQ a= bae. The equations of motion (101.2) now read aeaqa+beaqa=O. (101.5) For the sake of mathematical clarity, it is wise to forget that we are dealing with an approximation, and regard the Eqs. (101.4) and (101.5) as defining our problem, with qe finite; the homogeneity of the system permits this. The straightforward practical method of solving (101.5) is to substitute (101.6) where oce are constant complex amplitude factors and w a circular frequency. On eliminating the oc's from (101.5), we get the secular equation (101.7) from which to determine the values of w. Any positive root w2 gives a real w; this is a normal circular frequency, and the corresponding normal mode of oscillation is given by the real part of ( 1 01.6), the amplitude factors being solutions of ( 101.8) The ratios of the oc's are real, but they have an arbitrary common complex factor. The above method is difficult to follow if the secular equation ( 101. 7) has repeated roots, and a much deeper insight into the mathematical structure of 1 To agree with tensor notation, we use superscripts here. See Sect. 62 for summation convention. Sect. 101. Reduction of energies to normal form. 181 the problem presented by (101.4) and (101.5) is gained by starting afresh, using the geometry of the space Q. We assume (as is the case for all natural systems) that the kinetic energy is positive-definite. Then the quadratic form A= aeaqeqa (101.9) is also positive-definite, and there exists a linear homogeneous transformation (q) ~ (q') which makes 1 If we denote finite increments by L1, the formula D2 = ae a L1 qe L1 go = L1 q? + ... + L1 q;J defines a finite Euclidean distance D between any two points of Q. (This is the integrated form of the kinematical line element of Sect. 84.) The kinetic energy is T=taeaqeqa=t(q~ +q~ + .. · +q;J). (101.12) We may now treat Q as a Euclidean Nspace, qe being oblique Cartesian coordinates and q~ rectangular Cartesian coordinates. We are concerned with the geometrical form of the equipotential surfaces, which have the equations B = beoqeqa= b~aq~q~= const, (101.13) ( 101.11) Fig. 47. Reduction to normal coordinates by b~a being the new coefficients after application maxima (the case where N ~ 3). of the transformation. To investigate the principal axes of the equipotential surfaces, and to discover whether they are ellipsoidal or hyperboloidal in character, we proceed as follows. It is convenient to have before us at the same time expressions in both the coordinate systems (q) and (q') ; the former will be written on the left, and the latter on the right. On the sphere SN-l with equation (101.14) B is a function of position, and it attains a maximum value (say 21) at two or more points of SN_1 ; let U(Ile (or U~(Il) be the coordinates of such a point (Fig. 47). NOW intersect the sphere s_, -1 by the plane orthogonal to this last vector, with equation a u(I)e qa = 0 or U'(I) q' = 0 ea e e ' (101.15) thus obtaining a sphere of N- 2 dimensions (say SN_2). On SN_ 2 B attains a maximum (say ).2) at two or more points; let u<2le (or u~< >) be the coordinates of such a point. We have the orthogonality condition a U(l)e U<2>a = 0 or U'e(I) U'e<2> = 0. ea ' (101.16) Next we cut SN_ 2 by a plane orthogonal to U<2>e, obtaining a sphere SN_3 , and proceed as before. Carrying on this process, we arrive at a circle 51 and finally at a pair of points 50 • I It is convenient to denote the new coordinates by q~ (not q' e); for transformations conserving the form (101.10), there is no distinction between contravariant and covariant quantities. 182 J. L. SYNGE: Classical Dynamics. Sect. 101. In this way we get a set of N mutually orthogonal unit vectors, U(a) e or U~(a), with numbers A.a associated with them, these being the maxima of B under the conditions stated above. Then, by an orthogonal transformation, (101.17) we change to new rectangular Cartesian coordinates q~ with axes in the directions of the aforesaid orthogonal vectors, and it is easy to show, by reason of the maximal properties, that the form B lacks all product terms when expressed in the coordinates q~. Dropping the double primes on the final coordinates, we may state this result: Given two quadratic forms, A and B, with A positive-definite, there exists a linear homogeneous transformation which turns A into a sum of squares and B into a form lacking product terms; equivalently, the kinetic and potential energies (101.4) can be transformed into T = t (q~ + q~ + · · · + q~), } V= i(A.lq~ + A.2q~ + ... + A.Nq~). (101.18) These final coordinates are normal coordinates. There is no implication in the above argument that the A.'s are positive or that they are distinct; they are of course real, since the argument did not go outside the real domain 1. Once the energies have been put into the normal form (101.18), the discussion of the motion is extremely simple, for the equations of motion (101.2) become (101.19) in which the variables are separated. Depending on the sign of the A. contained in it, the solution of any one of these equations is as follows: q = a cos VI t + b sin VI t if A. > o , ) q = a t + b if A. = 0, q = a Cos V- A. t + b Sin V- A. t if A. < 0, (101.20) where a and b are constants. The equilibrium is stable under any one of the following equivalent conditions: (i) All the A.'s are positive. (ii) be a qe qa is a positive-definite form. (iii) The potential energy Vis a true minimum at the equilibrium configuration. If any one of the A.'s is zero or negative, the equilibrium is unstable. In the case of stability, the equipotential surfaces are ellipsoidal; in the case of instability, they are ellipsoidal (with V a maximum at the centre) or hyperboloidal or cylindrical. Suppose the equilibrium stable, so that each normal coordinate varies sinusoidally as in the first of ( 1 01.20). Then a normal mode of oscillation is one in which only one normal coordinate oscillates, the others being zero, and the normal frequencies ve and normal circular frequencies we are (101.21) 1 The transformation of the energies to normal form, as in (101.18), whether by means of maximal properties or otherwise, is basic in all thorough treatments of the theory of small oscillations. Cf. CORBEN and STEHLE [3], Chap. 8 (where there are a number of examples of systems with few and with many degrees of freedom); GoLDSTEIN [7], Chap. 10; WHITTAKER [28], Chap. 7. Sect. 102 The effect of constraints. 183 If two or more of these frequencies coincide, the system is degenerate 1 . In a normal mode, the representative point in Q performs a harmonic oscillation on a straight line. These straight lines are the principal axes of the equipotential surfaces ( 101.13) when these surfaces are referred to the coordinate system (q'). In a non-degenerate system, these lines are fixed in direction; for a degenerate system, they are partially indeterminate, lying in a plane of two or more dimensions (according to the degree of degeneracy). and the normal mode can be performed on any one of these lines, the normal coordinates in that case being partially indeterminate. In a completely degenerate system, the direction of a normal mode of oscillation is completely arbitrary; in this case the equipotential surfaces are spheres in the coordinate system (q'). Under arbitrary initial conditions, the system performs a motion which is a superposition of all the normal modes; in general, the orbit in Q is a very complicated curve, and the motion is periodic only if the ratios of the normal frequencies are rational. The roots of the equation det (aea A- bea) = 0 (101.22) are invariant under linear transformations of the q's. For normal coordinates, as in ( 1 0 1.18), this equation becomes (101.23) Therefore A.1 , A.2 , ••• )..N are the roots of ( 1 01.22) ; they are in fact the eigenvalues of the matrix bea relative to the matrix aea· If A is any one of these eigenvalues, the equations (101.24) define the corresponding eigenvectors for any system of coordinates. From the invariance of these equations it is easy to see, by using normal coordinates, that these eigenvectors point in the directions of the normal modes of oscillation. Since (101.22) is the same as (101.7). and (101.24) is the same as (101.8) (except for trivial change in notation), we recognize the mathematical significance of the simple substitution (101.6), which remains, when all is said, the most practical technique for dealing with vibration problems. A degeneracy is indicated by a multiple root of the secular equation (101.7). 102. The effect of constraints. To a system with kinetic and potential energies as in (101.4), let a constraint (102.1) be applied, the A's being constants. If we regard the energies in ( 1 01.4) as approximations, valid for small values of the velocities and coordinates, then (102.1) may be thought of as arising from any constraint which is independent of the time; it may even be nonholonomic, there being no distinction in a linear approximation between holonomic and non-holonomic. As in (46.15). the equations of motion of the constrained system are ( 102.2) where{} is an undetermined multiplier. To investigate the motion, we substitute (102.3) 1 This word is overworked; cf. Sect. 100 for a different meaning. 184 J. L. S"XNGE: Classical Dynamics. Sect. 102. in (102.1) and (102.2); on eliminating the ex's and{} we get the following determinantal equation for the circular frequency w: I agaw2- bQa AC?I = 0. (102.4) Aa 0 That is the practical plan. But to find the relationship between the frequencies of the unconstrained and the constrained systems, it is best to use normal coordinates of the unconstrained system, as in {101.18). Then, with A. written for w2, (102.4) becomes LI(A.) = o, (102.5) where A. - A.l 0 AI 0 A.-A.2··· A2 Ll (A.) =- (102.6) 0 0 ···A.-A.N AN AI A2 AN 0 the numbers A0 being now the coefficients in the equation of constraint (102.1) when expressed in the normal coordinates. On expansion, we have Ll (A.) = AHA. - A2) (A. - Aa) . ·. (A.- AN) + ) + A: (A.- A.I) (A. - Aa) · · · (A.- A.N) + + A: (A- AI) (A. - ).2) ... (A.- AN) + • 0 • 0 •• 0 0 •••• 0 + A~(A- AI) (A- A2) ... (A.- AN-I). (102.7) Here A1 , A.2 , ••• A.N are the squares of the circular frequencies of the unconstrained system. Suppose that the unconstrained system is non-degenerate; then, by a mere interchange of coordinates, we can arrange that At< A2< ... O, LI(AN-1)<0, LI(A.N-2)>0, ... , (102.9) and therefore Ll (A.) has N -1 real zeros, separating the numbers A.1, ... A.N. Under these circumstances (i.e. in what we may call the general case) the constrained frequencies separate the unconstrained frequencies. To allow for the possibility of one or more ofthe A's vanishing, this statement must be weakened to (102.10) where 'II indicates an unconstrained frequency and p' a constrained frequency. Degeneracy may be produced by constraint; in geometrical language, an ellipsoid possesses circular sections. The effect of applying a constraint to a degenerate system is best illustrated by an example. Take N = 5 and suppose At< A2 = Aa = A4 < A5 , (102.11) Sect. 103. Dissipative systems. Gyroscopic stability. 185 so that there is a triple degeneracy in the unconstrained system. Then (102.7) becomes Ll (A) =A~ (A- A2) 3 (A- A5) + ) + (A~ +A~ +A~) (A- A1) (A - A2) 2 (A- As) + +A~ (A- A1) (A- A2) 3 • (102.12) Supposing that none of the A's vanish, the graph of Ll (A) is now as in Fig. 48. We have and, near A= A2 , Ll (A) "'(A~+ A:+ A~) (A2 - A1) (A- A2) 2 (A2 - As)< 0. The constrained system has nor- L1f).J mal frequenciesv~O. (103-14) Then, on expanding the determinant, we have an algebraic equation of degree 2N, the coefficient of s2N being positive. We seek necessary and sufficient conditions that all its roots should have negative real parts (a slightly stronger condition than the requirement of stability, for which a zero real part would suffice). In the argument which follows 1 , the evenness of the degree of the equation plays no part, and it is convenient to write the equation A ~ C l(s)=a0 s"+a1 s"-1+ .. ·=0, a0>0. (103.15) Fig. 49. Interlocked polynomials. If, in the complex plane of s, we lead s along the imaginary axis from - co to + oo, then the increase in arg I (s) is precisely n times the number of roots of I (s) = 0 with negative real parts. To use this fact, we write s = i y and l(s) = l(iy) = i"(P .. - iP .. _1) (1 03 .16) P,. = ao y" - a2 yn-2 + ... ' } pn-1 = a1 yn-1- aa y"-3 + ... . where (103.17) Thus, for son the imaginary axis, we have ( 103 .18) and, if all the zeros off (s) have negative real parts, then arc tan (P .. _1(P,.) decreases by nn as y goes from - oo to + oo. This occurs if, and only if, the polynomials (Pn, P,._1) are interlocked in the following sense: (i) All the zeros of P,. (y) are real. (ii) All the zeros of P .. _1 (y) are real and separate those of P .. (y). (iii) The relation between P .. and P .. _1 is as shown in Fig. 49; P,._1 is positive at A and negative at B, these being successive zeros of P,., with P,. positive between them. This relation is equivalent to the condition that P,. and P .. _1 have the same sign in the limit y---* + oo. (In Fig. 49 arc tan P,._1fP .. decreases by n as we go from A to B.) Accordingly the question of the negative character of the real parts of the zeros of l(s) is equivalent to the question of the interlocking of (P,., P .. _1). To discuss this, we change the notation, writing a0=A,., a1 =A,._1 and Pn= A,.y"+ B .. y"-2+ ... , } pn-1 = An-1 yn-1 + Bn-1 yn- 3 + ... {103-19) 1 E. J. RouTH; Stability of a Given State of Motion, Chap. 3 (London; Macmillan 1877); PERES [20], p. 265; RouTH [22] II, Chap. 6; WINKELMANN and GRAMMEL [29], p. 480. 188 ] . L. SYNGE: Classical Dynamics. Sect. 103. By the following formulae we define a sequence of polynomials P,._2 , P,._3 , ••• P0 and a sequence of numbers A,._2 , A,._3 , ••• A0 (the coefficients of their highest powers): P,._2 = A,._zy"-2 + B,._zy"-4 + ... = A,.yP,._l -A,._lP,., Pn-3 = A,._3yn-a + B,._ay"-o + · · · = A,._l YPn-2- A,._zPn-1• Pz =A2yz + Bz P1 =A1y Po=Ao = A4YPs- AaP4, = AaYPz- AzP3, =AzyPl-AlPz. (103.20) As in (103.15), we take A,.= a0 > 0. Suppose {P,., P,._1) interlocked, which implies A,._1 >0. We shall prove that (P,._1 , P,._2) are interlocked. To do this, we see from the first of (103.20) that, at any zero of P,._1 , P,._2 and P,. have opposite signs. Therefore P,._2 has all its zeros real, and they separate those of P,._1 • In the limit y-+ + oo, P,._2 has the same sign as it had at the greatest zero of P,._1 ; that sign is the sign of -P,. at that zero, and is positive (cf. Fig. 49). Therefore A,._2 > 0; (P,._1 , P,._2) are interlocked. Proceeding step by step in this way, we see that all the pairs (P,._2 , P,._3), (P,._3 , P,._4), ••• {P1 , P0) are interlocked1, and (given A,.> 0) we conclude that the interlocking of (P,., P,._1) implies A,._1 > 0, A,._2 > 0, ... , A1 > 0, A0 > 0. (103.21) To prove the converse, we assume (103.21) with A,.> 0. For large y, the terms on the right in (103.20) are separately of higher order than the terms on the left; hence all the P's have the same sign in the limit y-+ + oo, and that sign is positive, the sign of A0 • Thus, by the last of (103.20), P2 is positive at infinity and negative at y = 0, where P1 = 0. Hence (P2 , P1) are interlocked, and, working up the table step by step, we conclude that {P,., P,._1) are interlocked. Thus (103.21) are necessary and sufficient conditions for the interlocking of (P,., P,._1), or, equivalently, for the negative character of the real parts of all the zeros of f(s). The quantities occurring in (103.21) may be expressed in determinantal form as follows, in terms of the coefficients in the formula (103.15) for f(s) 2: a1 a0 0 A,._a= as a2 al , all a4 aa with the understanding that a,= 0 for r > n. 0 0 (103.22) 1 In the case of (.ll. P0 ). interlocking means merely that P1 has the sign of Po as y-+ + oo. 2 CH. HERMITE: Crelle's J. 52, 39 (1850). - A. HURWITz: Math. Ann. 46, 273 (1895); cf. R. GRAMMEL: Der Kreisel, Bd. 1, p. 259. Berlin: Springer 1950. Sect. 104. Forced oscillations. Resonance. Operational methods. 189 104. Forced oscillations. Resonance. Operational methods. If, for a system moving in accordance with (103.1), all the roots of (103.8) have negative real parts, then, no matter what the initial conditions may be, the system ultimately tends to rest at the origin. In addition to the forces already represented in ( 103.1) we now supply a disturbing force Qe (t), regarded as a given function of t, and so modify the equations of motion to read (104.1) If the applied force is simple harmonic with circular frequency w, we write Qe = ~ eiwt. (104.2} After a long time, no matter what the initial conditions are, the system will tend to a forced oscillation given by (104-3) the complex amplitude factors a.e being found by substituting in ( 104.1) ; they must satisfy the equations (104.4) This is no eigenvalue problem; it is merely a question of solving a set of linear equations. But the eigenvalue problem represented by (103.8) is closely connected with the solution of (104.4), because the amplitude factors become large when the disturbing frequency is close to a natural frequency, or, more accurately, when iw is close to one of the roots of (103.8). Then we have resonance. Such problems are most compactly discussed by operational methods, and we shall show how to obtain the solution of (104.1) for a general disturbing force, not necessarily of the form (104.2}. For initial conditions we take qe = ne, qe = ve for t = 0. Let I denote the operation 1 of integrating with respect to t: t If(t)=Jf(r:}dr:. 0 Apply the operator I to (104.1} and use (104.5); this gives aea (ir- v") + ce.,(q"- n") +be., I q" =I Qe. On repeating this operation, we get Aeaqa =Be+ I2 Qe, where Aea = aea + Ceai + beai2' } Be= ae.,na + aeai va + ce.,I na. (104.5) (104.6) (104.7) (104.8) (104.9) The Eq. (104.8} is equivalent to (104.1) and (104.5}; if it is satisfied, they are, as we may see on differentiating it. 1 For details of the operational method, see H. JEFFREYS and B. S. JEFFREYs: Mathematical Physics, Chap. 7 (Cambridge: University Press 1956; 3rd. Edn.). They use Q for the integration operator, here changed to I to avoid confusion with the generalized force Much difficulty is caused in the operational method by the use of the HEAVISIDE operators p and p-1, which are not commutative, and, although they lend some formal simplicity to the work, they will not be used here. For operational methods based on the LAPLACE transform, see R. V. CHURCHILL: Modern Operational Mathematics in Engineering (New York and London: McGraw-Hill 1944); N. W. McLACHLAN: Modern Operational Calculus (London: Macmillan 1948); K. W. WAGNER: Operatorenrechnung (Leipzig: Johann Ambrosius Barth 1940) (lithoprinted Ann Arbor: Edwards 1944); I. N. SNEDDON: Fourier Transforms (New York: McGraw-Hill 1951). and this Encyclopedia, Vol. II, p. 251. 190 J. L. SYNGE: Classical Dynamics. Sect. 104. The essence of the operational method consists in treating the operator I as if it were a number, the validity of results obtained in this way being checked afterwards. We treat Apa as a matrix of numbers and define DQa as the cofactor of the element Apa• so that (104.10) where D is the determinant D = det (apa +Cpa!+ bpal2). (104.11) Multiplying (104.8) by DP~' and then dividing across by D, we get q~'= ~ DP~'(BP+I QP). (104.12) The general rule of operational calculus is to expand any rational fraction in I as a power series in I. This ensures the commutativity of two such fractional operators. Now D is a polynomial of degree 2N in I; it is related to the polynomial L1 of ( 103 .8) by D (I) = J2N L1 (;). The fraction 1/D admits a unique power series expansion, ~ = C0 + C1 l + C2 12 + ···, ( 104.13) (104.14) and if this is substituted in (104.12) we get on the right hand side a mathematically meaningful expression, provided that the infinite process converges. If it does, then ( 1 04.12) is the solution of the differential equations ( 104.1) with the initial conditions (104.5). ".13ut although 1/D is basically interpreted as an infinite series of operations, it not actually necessary to use an infinite series in order to calculate the solution (104.12). For, by (104.13), (104.15) where a 0 = det aP a and s1 , .•• s2N are the eigenvalues, satisfying ( 103 .8). We can then make resolutions into partial fractions as follows: DP~' Be = ___!1_ + __!!'{_____ + ... + K~N ) D 1 - s1 I 1 - s2 I 1 - s2 N I ' DPI' I L~~' Li" L PI' ___ D ___ = -1---s1 -J + -1---s-2 -J + ... + 1- ~:NI ' (104.16) the numerators of the fractions on the left being polynomials of degree 2N -1 in I; here the K's are constants depending on the initial data, and the L's are constants independent of the initial data, depending in fact only on the three matrices apa• bpa• cpa· The expressions (104.16) are now substituted in (104.12), and the following formulae 1 applied: -~1 1=erxt ) 1- a. I ' ~I~f(t) = Jt f(-r) erx(t-r)d-r, 1- rx.I 0 ( 104.17) 1 These formulae are easily established; cf. JEFFREYS and JEFFREYS, p. 233 of op. cit. in preceding footnote. Sect. 105. Oscillations about steady motion. 191 rx being any constant. The solution (104.12) takes the form q~-'=Ki_e +K~e'· + ... +K~Ne •Nt+ ) + j[Li~-' es,(l-r) + L~~-' es,(t-r) + ... + L~~ e"•N(t-r)] QQ(r) dr. (104.18) 0 The above result is very general, except for the assumption made implicitly at ( 1 04.16), that the eigenvalues are distinct; in cases of degeneracy the partial fraction expansions must be modified by the inclusion of fractions with higher powers in the denominators. Let us now assume the disturbing force simple harmonic, as in ( 1 04.2). Further, let us assume that all the eigenvalues are distinct and have negative real parts. Then, omitting those terms which tend to zero as t--+ oo, we get from (104.18) the following expression for the forced oscillation due to the applied force ( 1 04.2): (104.19) Since the frequency w is here involved only where shown explicitly, the formula exhibits clearly the phenomenon of resonance 1 . 105. Oscillations about steady motion or about a singular point in phase space ( Q P). Transformation of H to normal form. Consider a dynamical system with N degrees of freedom and Hamiltonian H(q, p) from which some of the coordinates (q) are absent (ignorable coordinates). A steady motion is defined to be one in which the non-ignorable coordinates and the corresponding momenta are constant. Let there be M ignorable coordinates qA (A= 1, 2, ... M), the non-ignorable coordinates being written Q r(F = 1, 2, ... N- M), with a similar notation for the momenta. Then the Hamiltonian may be written H =H(Q, p. P), (105.1) and the equations of motion are • 8H qA = opA ' (105.2) · oH Qr= 8Pr' · oH Pr=- 8Qr' (105.3) The conditions for steady motion are Qr= 0, Pr= 0. (105.4) Combining these last conditions with (105.2), we get, for a steady motion, (105.5) where cxA are constants, so that the ignorable coordinates increase at constant rates and the corresponding momenta are constant. Combining (105.4) with (105.3), we get, for a steady motion, 8H oQr = 0 • ( 105 .6) 1 Cf. (33.10) for the harmonic oscillator. 192 J. L. SYNGE: Classical Dynamics. Sect. 105. To obtain a steady motion, we have to satisfy these 2 (N- M) equations by assigning appropriate values to the 2N -M constants (Q, p, P). Thus we may in general expect to find ooM steady motions, M being the number of ignorable coordinates. If a system in steady motion is disturbed in such a way that the· constants P.A are not changed, we may investigate the oscillations about steady motion by using the equations of motion (105.3), linearised by assuming (Q, P) to have values adjacent to the constant values which they have in the undisturbed steady motion. Having thus formulated the problem of the oscillations about steady motion of a system possessing ignorable coordinates, we now present the same problem in a different way without reference to steady motions or ignorable coordinates. Given a Hamiltonian H(q, p), the canonical equations • aH • aH qe=ape' Pe=-aqe (105.7) define a direction of motion at every point in phase space ( Q P), except at those singular points where the 2N equations 1!~ = o aH = o (105.8) oqe ' ape are satisfied. Since there are 2N quantities (q,p), we may expect in general to find a finite number of singular points in Q P. Each such point represents a complete history of the system, because the equations of motion (105.7) are satisfied by constant values of (q, p), provided these constant values satisfy (105.8). We shall now investigate trajectories in Q P adjacent to a singular point. On comparing (105.6) and (105.8), we recognize that such an investigation is at the same time an investigation of oscillations about a state of steady motion for a system with ignorable coordinates. The present approach has the advantage that we come at once to the heart of the matter. The discussion which follows is closely connected with the matters treated in Sect.103. But here we shall use Hamiltonian methods rather than Lagrangian. In the Lagrangian method we are restricted to transformations of the coordinates (q), the transformation of momenta (p) (if we bring in momenta at all) being a derived transformation; in the Hamiltonian method we can use canonical transformations (CT). Singular points in QP are invariant under CT. For (105.8) are equivalent to ~H = 0 for an arbitrary variation of position in Q P, and this is invariant since His invariant under CT. Let qfJ = ae, p(J =be be a singular point. The generating function (105.9) gives the CT Pe= ;~ =P~+be, q~= :~~ =qe-ae, (105.10) so that the singular point becomes q; = 0, p; = 0. We shall now use these new canonical variables, dropping the primes. We assume H(q, p) expansible in a power series about the singular point. The constant term in the expansion is ineffective in (105.7), and we drop it. Then, in view of (105.8), we have, to the second order inclusive, H(q, P) = !Aeaqeqa + B~aqePa + !CeaPePa• (105.11) where the coefficients are constants. Sect. 105. Oscillations about steady motion. 193 For compactness, we introduce the notation of Sect. 87, with a slight modification of dimensions, since we are now in Q P (of 2N dimensions) instead of in Q T PH (of 2N + 2 dimensions). We write (105.12) then (105.11) may be written 2H=zHz, (105.13) where His a symmetric 2N X 2N matrix (H =H). As in (87.11), we write where 1 is now the unit N X N matrix, and note the properties r=-r, r-1 =-r, J'2 = - 1(2 N), det r = 1 . Then, as in (87.13), the equations of motion read z=rHz. (105.14) (105.15) (105.16) To find the nature of the motion, we seek to separate the variables in these equations, and this we do by applying a CT which reduces the matrix H to a simple (normal) form1. Consider the linear equations r Hz = A. z, or Hz = - A. r z, (105.17) and the associated determinantal equation det (H + A. r) = o, (105.18) which has 2N roots, real or complex, and not necessarily distinct. For any root A we have (since det r= 1) det [r(H + A.r)r] = o. (105.19) Transposing this matrix and using (105.15), we obtain det (H - A r) = o, (105.20) and comparison with (105.18) shows that, if A. is a root, then so also is -A.. We can write the whole set of roots as ± A.1 , ... ± A.N, and exhibit them in the form of a diagonal matrix L=(Lo 0 ) 0 -L0 ' (105.21) where L0 is the diagonal NxN matrix with elements ().1 , ... A.N). We shall assume that A1 , ••• AN are distinct 2• Then a set of solutions of (105.17) may be exhibited as a 2Nx2N matrix Z such that z-rrHZ=L. (105.22) 1 Cf. WHITTAKER [28], pp. 427-429; C. LANczos, Ann. d. Phys. (5) 20, 653-688; C. L. SIEGEL, pp. 76-80 of op. cit. in Sect. 53. 2 The present argument applies only to this non-degenerate case. The case of repeated roots is covered by the WEIERSTRASS contour-integration treatment of stability of motion, as given by WHITTAKER [28], pp. 197-202. Handbuch der Physik, Bd. III/1. 13 194 J. L. SYNGE: Classical Dynamics. Sect. 105. However (and this is important) this equation does not determine Z uniquely. If Z is any solution-matrix, then so is ZP, where Pis any diagonal matrix; and if Y and Z are two solution-matrices, then Z=YP, where P is some diagonal matrix. Let Z be any solution-matrix. Define Y by y = - (rz r)-1 = - r z-1 r. Then y-1 r H Y = r Z r · r H. r z-1 r = - r Z H r z-1 r. But, transposing (105.22), we have znrz-1 = -L, and so y-1 rHY= rLr=L. (105.23) (105.24) (105.25) (105.26) (105.27) Thus Y is also a solution-matrix, and so is connected with Z by (105.23), where P is some diagonal matrix. Hence P= y-1Z=- rzrz, rP=ZrZ, and, transposing, we get Pr=rP, from which it follows that P is of the form where P0 is a diagonal NxN matrix. (105.28) (105.29) (105.30) In the class of solution-matrices satisfying (105.22) we now seek one (say J) satisfying the symplectic condition [ cf. (87.16) J JrJ=r, (105.31) and this we do by taking any solution matrix Z, diagonal P by (105.28), defining D by forming the corresponding D= ( o p-1 0) 0 1 I and writing J=ZD. It is easy to verify that nrD = rP-1 , and hence, using (105.28), J rJ = D Z r Z D =Dr P D =Dr D P = r, establishing ( 10 5 .J 1). With J defined by (105.33), we apply the CT z=Jz', and the Hamiltonian of (105.13) becomes 2H = z' H'z', H'=JHJ. (105.32) (105.33) (105.34) (105.35) (105.36) (105.37) Sect. 105. Oscillations about steady motion. 195 Since J is a solution-matrix, (105.26) gives JHrJ-1 = -L, JH= LJr, (105.38) and so (105.39) Thus by a CT we are able to transform a quadratic Hamiltonian to the normal form (105.40) where the A.'s are the roots of (105.18). The new coordinates (q', p') are in general complex. If we apply the further CT with generating function N "2 G (q' p") = "" (q' p" - }._ ~ _ }. _ ;. ' 2) ' L..J I! II 2 ). 4 Q q(! ' (!=I (! (105.41) so that (without the summation convention) P' oG p" 1 A. , , _ oG _ , p;; 11 = oq~ = 11 -2 (!q(!, q(!- ap~'- qe- --;;-· (105.42) the Hamiltonian changes into an alternative normal form, H" = t (Pi' 2 + ... + p;; 2 _ ;.~ q~' 2 _ .•. _ ;.~ q;J 2) • (105.43) If we use the form (105.40), the equations of motion read . , a H' ~ , p", oH' 1 , qt= ap~ =,"J.qt····· ~=--aq~-=-AtPt····· (105.44) in which the variables are separated, and the motion is given by (105.45) the constant coefficients depending on the initial conditions. It is clear that the motion given by these equations is stable if, and only if, all the A's are pure imaginaries. It is easy to show that if H(q, p) is positive-definite, then all the roots of (105.18) are pure imaginaries, even in the degenerate case of repeated roots. Let A., z (with z=j=O) be any solution of (105.17). Let z be the complex conjugate of z, and zt the transpose of z (i.e. the row matrix). Then zt Hz=- A.zt r z. (105.46) From the positive-definite character of H, the left hand side is real and positive. Therefore A. =l= 0 (no zero roots) and zt rz=J=O. (105.47) Thisquantityiseasily seen to be a pure imaginary, and hence A. is a pure imaginary also. We have the following result: the motion of a system with homogeneous quadratic Hamiltonian His stable if His positive-definite1• 1 We have proved it here only in the non-degenerate case of distinct eigenvalues; for the degenerate case, see preceding footnote 13* 196 J. L. SYNGE: Classical Dynamics. Sect. 106. 106. Perturbations. Consider two dynamical systems, Sand 5', with canonical variables (q, p), (q', p') and Hamiltonians H(q, t, p), H'(q', t, p') respectively. They are completely independent.· Suppressing suffixes, we may write their equations of motion in the form 5: oH q=e;;· P= • -iii; 8H l ., oH' ·, oH' (106.1) 5': q =·ap', p =- oq'. Now think of S and S' as forming a single system S + S' with a Hamiltonian H(q, t, p) + H'(q', t, p') + K(q, q', t, p, P'), (106.2) K being an interaction Hamiltonian. (As a very simple example, we might take S and S' to be two free particles and K to be the potential due to their mutual gravitational attraction.) The equations of motion of S + S' are l . oH aK q-- -+ - s S': - ap ap, + ., oH' oK q = iFf/+ op'-, p = _ oH oq _ !~ oq ' l ., oH' oK p = - ·aq'- - aq'. (106.3) The effect of K is to perturb the original motions ( 106.1) of S and S'. If the derivatives of K are small, the values of q, p, q ', p' at a given point of the space QT P of S + S' are nearly the same in the perturbed and unperturbed motions. But in a very long time significant changes in the motion may result; in that case we speak of secular changes. For the system formed by the sun and the planets, the Hamiltonian may be written in the form H = T(S) + 1: T(P) + 1: V(S P) + 1: V(P P'), (106.4) where T(S) is the kinetic energy of the sun, T(P) the kinetic energy of a planet, V(S P) the mutual potential energy of the sun and a planet, and V(P P') the mutual potential energy of two planets. To put this into perturbation form, we denote by 50 a fictitious sun fixed at the origin and define V' ( S P) by V'(S P) = V(S P)- V(S0 P). Then the Hamiltonian (106.4) may be written H =H(S) + L,H(P) + K, where H(S) = T(S), H(P) = T(P) + V(S0 P), (106.5) (1o6.6) (106.7) and K stands for the terms not otherwise accounted for. K is the perturbing Hamiltonian. The unperturbed motion is known, for H(S) corresponds to the free motion of a particle and H(P) to the KEPLER problem. The practical significance of ( 1 06.6) lies in the fact that K is small; the actual motion of the solar system is a perturbation of a state of motion in which the sun is at rest, and the planets describe fixed elliptical orbits with the sun as focus, without mutual interaction. The basic idea behind perturbation theory is this: starting at t=t1 , the motion up to t = t2 of the complete system (including the perturbation) differs little from the unperturbed motion, provided we start the two motions from the same Sect. 106. Perturbations. 197 point in Q T P-space and do not make the interval t2 - t1 long. Assuming the unperturbed motion known, the effect of the perturbation during such a finite interval can be found by approximate methods 1. Without making any approximations based in smallness of the perturbing Hamiltonian, the general perturbation problem may be stated as follows: Given the general solution of the canonical equations (106.8) of unperturbed motion, it is required to set up a technique to find the motion for the Hamiltonian H = H0(q, t. p) + H1(q, t, p), (106.9) H1 being the perturbing Hamiltonian. Let the solution of (106.8) be qe = qe (c, t), Pe = Pe (c, t), (106.10) where c stands for 2N arbitrary constants cA, capital suffixes having the range 1, ... 2N; these quantities are constant along each unperturbed trajectory. Solving (106.10), we have (106.11) these 2 N functions being determined by the form of the function H0(q,t,p). Let us view the situatl' on 1. n Q T p Fig. so. Perturbation viewed in Q T P. r, = unperturbed trajectories, r = perturbed trajectory. (Fig. SO) 2 • .E2N is the surface t = 0, B is any point, and To is the unperturbed trajectory through B, cutting .E2N at B*, say. Since cA are constant along T 0, we have at B* (106.12) Thus cA form a system of coordinates on .E2N; (c, t) form a system of coordinates in Q T P, but not a canonical system in general. The unperturbed trajectories To form a system of projection-lines by which a point B is projected into B*, the corresponding values of cA being given by (106.11). Consider now a perturbed trajectory T. At B its direction differs from the direction of T 0 , and as the representative point B traverses T, its projection B* moves on .E2N. For that reason, the method given here is called the method of variation of constants, since cA are constants for T0 but not for r. The perturbation problem is reduced to the study of the way in which cA vary with t as the representative point traverses T; if we knew this, then we would know T, its equations in the form CA = fA(t) (106.13) determining a curve in Q T Pin the coordinate system (c, t). 1 For a detailed treatment of perturbations, with use of action-angle variables, see FRANK [5], pp. 127-156, and FUES [6]. 2 Remember that any set of canonical equations defines a congruence of curves in QTP, one curve through each point; cf. Sect. 93. 198 J. L. SYNGE: Classical Dynamics. On Fwe have, by (97.9), cA being the function (106.11), cA= o;:-+[cA,Ho+Hl], and on Fo. since cA is constant, O=ii/+ OCA [ J CA,Ho. Hence, by subtraction, we have on F cA = [cA, H1]. Sect. 107. (106.14) (106.15) ( 106.16) By virtue of (106.10), or equivalently (106.11), the right hand side is a function of (c, t), and so we have here a set of 2N equations to determine the functions ( 106.13) and hence the perturbed motion. The Eqs. (106.16) may be put in a different form. By (106.10) we can express H1 as a function of (c, t): H1(q, t, p) = K(c, t). ( 106.17) Then oK ocB (106.18) and [ H J ocA oHl oH1 ocA [ J oK CA, 1 = ~q Bp - ~q ap = CA,CB fie-· Q e Q Q B (106.19) Thus (106.16) may be written (106.20) So far everything is exact. But if the derivatives of H1 lire small, or equivalently the derivatives of K are small, then the right hand sides of (106.16) and (106.20) are small. The projected point B* moves slowly over l:2N, and we may approximate to its motion in the finite interval (tv t2) by substituting in these right hand sides the values of cA for t = t1 and integrating by quadratures1. F. Relativistic dynamics 2• I. Minkowskian space-time and the laws of dynamics.~ 107. LoRENTZ transformations. Small Latin suffixes will now have the values 1, 2, 3, 4, and small Greek suffixes the values 1, 2, 3, with summation understood in either case for a repeated suffix. Let x' be real coordinates of an event in the 4-dimensional manifold of spacetime; and let the separation ds between adjacent events be given by (107.1) 1 For perturbation theory from the standpoints of Lagrangian equations and contact transformations, see CoRBEN and STEHLE [3], pp. 306-312. 2 The basic laws of Newtonian and relativistic dynamics were contrasted in Sects. 4 and s. Essential formulae are repeated here, but no more will be said about the peculiar difficulties surrounding relativistic systems. As general references for the special theory of relativity, the reader may consult 0. CosTA DE BEAUREGARD: La Theorie de la Relativite restreinte (Paris: Masson & Cie. 1949); P. G. BERGMANN: Introduction to the Theory of Relativity (New York: Prentice-Hall 1942), and this Encyclopedia, Vol. IV; A. EINSTEIN: The Meaning of Relativity (5th Ed.; Princeton: University Press 1955); HALPERN [9}; M. VON LAUE: Die Relativitatstheorie, Bd. 1 (5th. Ed.; Braunschweig: Vieweg 1952); C. MoLLER: The Theory of Relativity (Oxford: Clarendon Press 1952); A. PAPAPETROU: Spezielle Relativitatstheorie (Berlin: VEB Deutscher Verlag der Wissenschaften 19 55); J. L. SYNGE: Relativity: The Special Theory (Amsterdam: North-Holland Publishing Co 1956). Sect. 107. LORENTZ transformations. 199 where the coefficients are functions of the coordinates, and e 1s an indicator chosen equal to + 1 or -1 so as to make ds real. In the special theory of relativity, with which alone we are concerned here, space-time is flat, which means that there exist real coordinates (x, y, z, t) such that ds 2 = e (dx 2 + dy2 + dz2 - c2 dt2), where c is a fundamental constant (the speed of light). (107.2) It is convenient to introduce Minkowskian coordinates, with "imaginary time ", defined by X1 = X, X3=z, x2 = y, } (107.3) x4 = i c t; then (107.2) may be written compactly as ds2 = e dx, dx,. ( 107.4) Minkowskian coordinates are used throughout this article; they have the great convenience that, for them, covariant components of vectors and tensors are the same as contravariant components, and we can write all vectors and tensors with subscripts, thus avoiding a notational complication. If the imaginary time x4 should ever be a source of confusion, we can at once pass from Minkowskian past Fig. 51. Null·cone. 'Past and future. Timelike, spacelike and null vectors. coordinates x, to real Cartesian coordinates x' by writing x(/ = xC!, x4 = ix4• We shall have occasion to pass to real coordinates in Sect. 111 in order to discuss a matter of sign. The group of LORENTZ transformations consists of those transformations (necessarily linear) which conserve the quadratic form dx, dx,. Any such transformation is of the form X~= A,s X5 + B,, where the coefficients satisfy (107.5) (107.6) Comparison with (9.6) shows that A is, formally, an orthogonal matrix, and this suggests that a LORENTZ transformation is a "rigid" displacement of spacetime into itself. This is true, in a sense, and very important for the understanding of the LORENTZ transformation, but the presence of imaginary elements in A (due to the imaginary time) makes the geometry of the LoRENTZ transformation essentially different from the geometry of orthogonal transformations of a 4-space with four real coordinates. The null-cone drawn from any event a, as vertex (Fig. 51) has the equation (x,- a,) (x,- a,) = 0. (107.7) Events exterior to the null-cone satisfy (x - a,) (x,- a,) > 0, ( 107.8) 200 J. L. SYNGE: Classical Dynamics. Sect. 108. and those interior to it satisfy (x,- a,) (x,- a,)< 0. (107.9) The interior is divided into two parts, according as (x4 - a4)ji is positive or negative; these two parts are respectively the future and the past with respect to the event a,. A displacement dx, from a, into the future or into the past is timelike, and for it e = -1; a displacement on the null-cone is null, and one pointing outside the null-cone is spacelike (e = + 1). The null-cone remains invariant under a LORENTZ transformation, and so do its interior and exterior regions. But future and past may be jnterchanged. However, for our purposes we reject LORENTZ transformations which cause this interchange. By (107.6) the Jacobian J of a LoRENTZ transformation is]=± 1; the transformation is proper if J = + 1 and improper if J = -1. It is a basic postulate of relativity that all the laws of dynamics should be invariant under proper future-preserving LORENTZ transformations. This is equivalent to saying that the laws are capable of geometrical construction in terms of the geometry of Minkowskian space-time (cf. Sect. 5). It is further assumed that any displacement dx, along the world line of a material particle is timelike. This is equivalent to saying that no particle can travel as fast as light (see Sect. 108 below). The abstract concept of separation ds is given physical content by the assertion that, along the world line of a material particle, ds is a measure of proper time, i.e. the time recorded by a standard clock carried with the particle. An observer is said to be Galileian (or to use a Galileian frame of reference) if the separation ds between any two events is expressible as in (107.2) or (107.4) in terms of his coordinates. When two Galileian observers, 5 and 5', observe the same event, their observations are connected by a LORENTZ transformation. By cooperation between the two observers in the choice of space-axes, the LoRENTZ transformation connecting the two observations may be put into the simple form x' = y (x - v t), here y'= y, z'=z, Y = v v2 ' 1--c2 t' = y (t - ~-;} (107.10) (107.11) and v is the relative velocity of 5 and 5'; more precisely, the velocity of 5' relative to 5 is (v, 0, 0), and the velocity of 5 relative to 5' is (-v, 0, 0). 108. Kinematics in space-time. 4-momentum. a.) Velocity of a point. Consider a moving point (not necessarily a particle); the equations of its world line may be written x, = x, (x), ( 108.1) where X is a monotonic parameter. The 3-velocity of the point is (108.2) v = VvQ vQ, (108-3) is greater than, equal to, or less than c, according as the 4-vector dx,Jdx is spacelike, null, or timelike. Sect. 108. Kinematics in space-time. 4-momentum. 201 For a material particle, this 4-vector is timelike, and we define the 4-velocity to be the timelike unit vector A= dx, r ds ' satisfying J.,J.,= -1. By (108.2) the relations between 3-velocity and 4-velocity are . Ao ve = zc--;:--, 4 1 y= -~--- v /-v2 · 1--c2 (108.4) (108.5) ( 1 08.6) For a point moving faster than light, 4-velocity may be defined as in (108.4); we change -1 to + 1 in (108.5), and the relations with 3-velocity are as in (108.6) except that y is changed toy* where *- 1 y - vv2 -. ---1 c2 (108.7) For a photon we assume that dx,fdx is a null vector. Thus the speed of a photon is c. It has no definable 4-velocity; we cannot use ( 1 08.4) because d s = 0. fJ) Velocity of a wave. An equation (1 08.8) defines a 3-space in space-time. We may call it a 3-wave. It is the history of a 2-wave, the instantaneous 2-wave being given by putting x4 = const in ( 1 08.8). The normal 3-velocity of the 2-wave is easily seen to be1 (108.9) where F,. = oFjox,. Hence the speed u of wave propagation satisfies (108.10) Since F, 4 is pure imaginary, u is greater than, equal to, or less than c, according as the 4-vector F,, (which is the space-time normal to the 3-wave) is timelike, null, or spacelike. y) 4-acceleration. For a material particle the 4-acceleration is the 4-vector d)., d2 x, ds ds2-. (108.11) It is orthogonal (in the space-time sense) to the 4-velocity, because (108.5) gives , d)., = 0 A, ds · (108.12) 1 Cf. SY:- r -x,.x,. (111.4) and elimination gives the energy equation 2!J(x, y) = y,y, + m2 e2 = 0, (111.5) from which the space-time coordinates are absent; the canonical equations ( 11 0.7) give dx, dy, = O. dW=y,, dw (111.6) We can write (111.4) in terms of 4-velocity: y,=meA.,. (111.7) Thus the Hamiltonian 4-vector is tangent to the world line in the case of a free particle; this is not generally true for a particle in a field. From ( 1 08.6), ( 11 0.19) and (111.7), we see that the Hamiltonian 3-momentum is (111.8) and we have also . iE y,=~mye=c' (111.9) where E is the relative energy, as in Sect. 108. By (110.19) we have then H=E. (111.10) To find the form of the Hamiltonian function, we are to solve (111.5) for y4 , obtaining (111.11) in which the+ sign is to be chosen on account of (111.9). Then, as in (110.20), the· Hamiltonian is H=eVPoPo+mzez. (111.12) 208 J. L. SYNGE: Classical Dynamics. Sect. 111. It remains to consider the ordinary Lagrangian for a free particle. Applying {110.15) to {111.2), we get so that mc2 L dt = mcds = -dt, y mc2 R2 1 L = -~ = mc2 1 --= mc2 - - mv2 + · · · y c2 2 the unwritten terms involving the square and higher powers of vjc. This differs in three ways from the Lagrangian L=T=imv2 (111.13) (111.14) (111.15) of a free particle in Newtonian dynamics. First, through the presence of the constant m c2 , representing proper energy; secondly, through the minus sign in -jmv2 ; and thirdly, through the terms not written explicitly. If the Lagrangian is merely an integrand to be used in a variational equation for the determination of equations of motion, then the presence of mc2 is trivial, since it gives the same contribution to JL dt for all the varied motions; nor is the sign in - imv2 significant, since - L yields the same extremals as L. As for the unwritten terms, they represent the sort of relativistic correction one expects to find, tending to zero with vfc. But if action itself is physically significant, then the difference between ( 111.14) and (111.15) is significant. For a particle at rest, (111.14) gives a positive action mc2 Jdt, whereas (111.15) gives zero. If we compare two particles, one at rest and the other moving, (111.14) ascribes the greater action (in a given time interval) to the particle at rest, whereas ( 111.15) ascribes the greater action to the moving particle. Let us return to the change of sign introduced in (110.4) and consequentially in ( 11 0.8), and consider the definition of the Hamiltonian 4-vector when the coordinates are real. If we use real coordinates x' in space-time, we have to distinguish between covariant and contravariant vectors, the transition from the one to the other being effected by means of the fundamental tensor gmn of (107.1). However, the geometry of space-time is not changed if we reverse the signs of all the quantities gmn· Hence, when we diagonalize gmn by using real Cartesian coordinates, there are two choices: we can take the diagonal form to be or we can take it to be (gmn) = ( + 1, + 1, + 1, - 1), (gmn) = (- 1, - 1, - 1, + 1). ( 111.16) (111.17) Some writers prefer the first, some the second. There is no physical difference between them, but ( 111.16) has the advantage that we can pass to the unit matrix (Minkowskian coordinates) by introducing one imaginary coordinate, whereas ( 111.17) requires three imaginary coordinates. Now, with real coordinates and gmn and gmn respectively as above, we would write (111.1) as (111.18) and partial differentiation gives in each case the same covariant vector: (111.19) Sect. 112. The two-event characteristic function and the HAMILTON-JACOBI equation. 209 This single covariant vector yields two (opposed) contravariant vectors, according as we use (111.16) or (111.17); they are, respectively, ( 111.20) The first of these points into the past, the second into the future. Now there are two possible definitions of the covariant Hamiltonian 4-vector, viz. y,= ± oAjox'', and it is convenient to choose that sign which makes the corresponding contravariant 4-vector point into the future, at least in the case of a free particle. Thus we need two different definitions of y, according as we use ( 111.16) or ( 111.17) : they are OA y, = - ox'' for (111.16}' (111.21) OA y,= ox'' for (111.17). (111.22) In Minkowskian coordinates we do not have to distinguish between covariant and contravariant vectors; we have already seen that ( 111.21) is the appropriate definition, since (111.7} shows that y,, so defined, points into the future. 112. The two-event characteristic function and the HAMILTON-JACOBI equation. The two-event characteristic function S(x*, x) is, as in (72.1) with a change of sign, S(x*, x) =- J y,dx,, integrated along the trajectory joining the two events. Hence as y,=-~· , * as y, = a-x•-. , (112.1) ( 112.2) Substitution m the energy equation equation .Q (x, y) = 0 gives the HAMIL TON-JACOBI .Q(x,- ~~)=o. (112.3) If, as in Sect. 77, we have a complete integral S(x*, x) of (112.3), then the Eqs. (112.2) determine the trajectories and associated Hamiltonian 4-vectors; in these equations we are to regard the quantities (x*, y*) as constants. Actually we need only six constants, since there are oo6 trajectories. If one of the (x*) is taken additive to S, and the other three (x*! and the first three of the (y*) chosen arbitrarily, then the first three equations in the second set in (112.2) give a trajectory, and indeed all the trajectories, since six arbitrary constants are involved. For a free particle as in Sect. 11"1, we have S(x*, x) = - y. (x,- x:), y, being constant along the trajectory. Putting s = V- (x,- x:) (x,- x:), we have, as in (111.7) _ A _ me (x,- xi) y,- me ,- 5 , and hence S(x*, x) =me V- (x,- x:) (x,- x:) = mes. Handbuch der Physik, Bd. III/1. 14 (112.4) (112.5) (112.6) (112.7) 1/ 210 J. L. SYNGE: Classical Dynamics. Sects. 113. 114. II. Some special dynamical problems. 113. Hyperbolic motion. In Newtonian dynamics we can prescribe the motion of a particle, and calculate the force that gives this motion. Similarly, in relativity we can prescribe a world line and calculate the corresponding 4-force consistent with given constant proper mass of the particle, using the formula (113.1) One of the simplest world lines is the pseudocircle with equations (113.2) where a is a constant. Since the first equation reads, in real coordinates, (113.3) this is called hyperbolic motion (Fig. 52). The parametric equations of the world line are x1 = aCosq;, x4 = ia Sin q;, and hence d s2 = - dx~ - dx~ = a2 d q;2, (113.4) (113.5) so that the surviving components of the 4-velocity are ~ dxl s· ,., =~= 1nm ds T' ~ dx4 • C A4 = d5 = t osq;. (113.6) By (113.1) the required 4-force is X = _m_ Cos m = _!I'IX 1 a T a2 1 • 1 X im s· mx, 4=---a mq;=~, X2 =X3 =0. j (113.7) Fig. 52. Hyperbolic motion Thus the 4-force is directed from the origin of space-time, with a magnitude proportional to the Minkowskian distance. 114. Particle in a potential field. Harmonic oscillator. Let V(x1 , x2 , x3) be a potential function, and let , _ v--, iV x~ A(x , x) -me -x , x-., ----- c (114.1) be the homogeneous Lagrangian for a particle of constant proper mass m moving in this field. This expression is not LORENTZ-invariant; we are using a special frame of reference. We can write ·--- iVl A (x .A) =me V-A A _,_ _ ___!_ I 1 t C I and the corresponding ordinary Lagrangian L is given by Ldt=A(x,dx} -:-mcds+ Vdt,) L =me + V. I' (114.2) (114.3) Sect. 114. Particle in a potential field. Harmonic oscillator. 211 By (110.13) the equations of motion are ds d (- md) + i c u~ ~v ;.4 = 0' l d ( i v) -ds - me }.4 - c = 0, ( 114.4) so that Jt (myve) =- ·~~, l myc2 +V = K, (114.5) where K is a constant (constant of energy). To discuss a harmonic oscillator, we consider motion along the xcaxis with V = i k2 x~. Let the particle start from the origin with v = v0 • Then, by the last of (114.5), we have (114.6) The particle comes to rest at x =a (we put x1 = x for simplicity) where a is given by (114.7) and the constants v0 and a are related by 1 mc2 l K = mc2 + 2 k2 a2 = v1 ---~~ , 1 1 ~ ·2-- k2 a2 = mc2(Yo- 1)' y j 0 = v~c.~~~! . (114.8) By (114.6) we have ~ = v~(K~~~~~~ -, (114.9) and so the period -c of the harmonic oscillator is given by x=a a T = 4 J d t = 4 J d;- x=O o a = -:- I v(Il~ ~)~~: ~ (114.10) 0 where (114.11) The formula (114.10) is exact; expanding in powers of x, we get ym ( 3 k2a2 ) -c = 2 n -k- 1 + -16 -mc2 + · · · ' (114.12) in which the first term is the Newtonian period. 14* 212 J. L. SYNGE: Classical Dynamics. Sect. 115. 115. Charged particle in electromagnetic field. An electromagnetic field with electric vector Ee and magnetic vector HP can be described by a skew-symmetric tensor F.s where E1 = iF14 , H1 = F;a, the corresponding 4-potential rp, is such that F.s= rrs,r- rrr,s> the commas denoting partial differentiation. (115.1) (115.2) For a particle of constant proper mass m and charge e, moving in a given electromagnetic field 1, we take the homogeneous Lagrangian A (x, x') =me v=-.x;.x;- ~ rp,x;, or, in terms of 4-velocity, A (x, A) =me V-A,~~-~ rp,Ar The corresponding ordinary Lagrangian is mc2 e • L =-~-·· rp, X+ v l' c Q Q ' where v = I!Cf!4 i ' the potential energy of the particle. The equations of motion (110.13) give (115.3) (115.4) (115.5) ( 115.6) (115.7) The term on the right is the LORENTZ ponderomotive force. This equation is of the form (109.1), and the condition (109.3) for constancy of proper mass is satisfied on account of the skew-symmetry of the electromagnetic tensor. These equations may also be expressed in the vector form [cf. (40.1), (40.2)]: :de (myv) =e(E+: xH), l (115.8) lit(mye2) =ev·E. To get the energy equation Q(x, y) =0, we write the equation for the Hamiltonian 4-vector {115.9) and so obtain (since A,A,=-1) 2f2(x, y) = (y,- ~ rp,) (y,- d rp,) + m2e2 = o. (115.10) The canonical equations are dx, 8Q e l ;~ ~ ~·;;, y, ·; (::~ : ~.) ~ •.• (115.11) 1 The units are electrostatic. Sect. 116. Relativistic KEPLER problem. 213 By (110.20) the Hamiltonian is [we solve (115.10) for y4] H = c~, = V ± c V(Pe -~- IPe)(Pe- ~-q;e) + m2c2. (115.12) The Hamiltonian 3-momentum is e Pe =my vii+ r;IP11· (115.13) By (112.3) and (115.10), the HAMILTON-jACOBI equation is (os e )(os e ) -+-IP --+--IP + m2c2= 0. ox, c , ox, c , (115.14) In the case of an electrostatic field, we put !p11 =0, eqJ4=iV; then we have --;(as- v)2 +~~ + m2c2= o. c ot axil OXe (115.15) In terms of a complete integral, the motion is given by (112.2). 116. Relativistic KEPLER problem. Consider a particle, of constant proper mass m and chargee, moving in the field of a charge e' of opposite sign, fixed at the origin. If e, e' are measured in Gaussian electrostatic units, the field and 4-potential are (116.1) The equations of motion ( 11 5 .8) give d k'l' Tt(yv) =--;a· d k dt (y c2) = - rs v. 'I'. (116.2) where r is the position vector of the moving charge and ee' k=-->0. m (116.3) It follows that the motion is plane, and if we use polar coordinates (r, {)) in the plane of the orbit, we get an integral of angular momentum and an integral of energy k "c2 --=W r r ' (116.4) (116.5) where A and W are constants. Putting f! = 1/r and eliminating the time, the differential equation of the orbit comes out to be1 (116.6) Assuming the coefficient of e to be positive, and putting (116.7) 1 Cf. BERGMANN, p. 218 of op. cit. in Sect. 107; SYNGE, p. 398 of op. cit. in Sect. 107. 214 J. L. SYNGE: Classical Dynamics. Sect. 116. we obtain the equation of the orbit in the form - = 1 n = -~-c [(W--- p · cos (p {} + C) + - 2 )~ kW] r o: Ap2 c4 Ac3 ' (116.8) where C is a constant. The condition for a finite orbit is - c2< W 1 moving with ( 121.6) (121.7) ( 121.8) (equality occurs only if ve = v~, in which case there is no collision), and hence m'>m+m. (121.9) Thus total proper mass is always increased in a completely inelastic collision. The increase in total proper energy is m' c2 - m c2 - m c2 ; (121.10) if the speeds of the incident particles are small compared with c, it is easy to show that this increase is approximately equal to the heat generated in such a collision according to Newtonian dynamics (cf. Sects. 58, 59). A completely inelastic collision of a material particle and a photon is similarly treated. This represents absorption of a photon by a material particle. Here also the total proper mass is increased; since the proper mass of the photon is zero, this means that we get a material particle with a greater proper mass than that of the incident material particle. 220 J. L. SYNGE: Classical Dynamics. Sect. 122. An elastic collision is characterized by two conditions: (i) The number of particles is unchanged. (ii) The proper mass of each particle is unchanged. The result of an elastic collision between two material particles is very easy to describe in the mass-centre reference system: the speeds of the particles are unchanged, and they recede from the collision in opposite directions, the line of these directions being undetermined by the conservation law. The vector diagram in the space PH is as in Fig. 55. In the elastic collision of a material particle iniliol 0 final and a photon (the CoMPTON effect, see Sect. 122), the frequency of the photon is unchanged when the mass-centre reference system is used. 122. CoMPTON effect. An elastic collision between a material particle and a photon is called the COMPTON effect. As indicated in Sect. 120, there is a twofold indeterminary in the result of the collision. Fig. 55. Vector diagram of elastic collision. Fig. 56. CoMPTON effect in the laboratory frame of reference. To discuss the collision in a general Galileian frame of reference, we write M,, M; for the 4-momenta of the material particle before and after the collision, and P,., P; for the 4-momenta of the photon. We have the following equations: M; ~P~=M,+ ~·- 2 ) M,M,- M,M,- m, P; P; = P,P,= 0. ( 122.1) These six equations contain all the information available for the determination of the eight quantities M;, P;. The description of the outcome of the collision depends on the frame of reference employed. The description is simplest for the mass-centre reference system (see Sect. 121}, but in the usual description, as given below, one uses the laboratory frame, i.e. the frame in which the particle is initially at rest!. Fig. 56 shows the photon (with frequency v) approaching from the left, and being scattered (with frequency V1 ) at an angle {}; the material particle recoils with speed V1 at an angle cp as shown. From (120.4) and (120.5) it follows that the three lines of motion are coplanar and that m Y1 V1 cos cp + !!i_ cos{} = hv , ) c c I I • h111 • {} my v smq;--c-s1n =0, my' c2 + h V1 = m c2 + h v. ( 122.2) 1 See J. L. SYNGE, pp. 193-199 of op. cit. in Sect. 107 for further details and descriptions in other frames of reference. Sect. 123. Angular momentum and mass-centre. 221 Assuming{} to have any value (this is part of the indeterminacy of the problem), we find, after a little calculation, where t _ cot!D anqJ-~, v' _ 2ksinHV1+k(2+k)sin2 !D "7:- 1 + 2k (1 + k) sin2 !D , 1 1+2k(1+k)sin2 !D y = v v;2 = t+2ksin2 !D 1--C2 v' = v t+2ksin2 lD' k - ..!!!'.._ (122.4) - mc2 • 123. Angular momentum and mass-centre 1• Let x, be any event in the history of a particle, and let M, be its 4-momentum at that event. Then the angular momentum of the particle at that event, relative to the origin of the spacetime coordinates, is defined to be the skewsymmetric tensor H,.= x,M.- x.M,. (123.1} More generally, the angular momentum relative to an event a, is defined to be I (122.3) H,. (a) = (x,- a,) M.- (x.- a5 ) M, } (123.2) = H,.- a,M. + a.M,. Fig. 57. World lines of a system of particles interacting at point catastrophes. If the particle is free, then its world line is straight and M, lies along it; in that case H,. (a) is independent of the particular event x, chosen on the world line. Consider now a system of particles, interacting with one another only by catastrophes in which the world lines intersect. Collisions may be elastic or inelastic; particles may combine and new particles may be formed; the particles may be material particles or photons. The essential condition is that 4-momentum should be conserved at each catastrophe, and that each catastrophe should take place at a single event. Fig. 57 illustrates the world lines of such a system. Let II be any spacelike 3-space. Let x, be the event where a typical world line cuts II, and let M, be the corresponding 4-momentum. Then each world line cutting II gives an angular momentum relative to the origin as in (123.1), and we get a total angular momentum for the system by adding the contributions of the several particles. If we move II in space-time, this total angular momentum certainly remains constant until II crosses a catastrophe, because each particle is free between catastrophes. Further, there is no change in the total angular momentum when II does cross a catastrophe, because only a single event is involved and the total 4-momentum of the particles involved is conserved. In fact, both the total 4-momentum and the total angular momentum of the world lines cutting II are independent of the choice of II; they are constants of the system. 1 See the books by Mt11LLER and SYNGE cited in Sect. 107; also C. M0LLER: Ann. Inst. H. Poincare 11, 251-278 (1949) and Comm. Dublin Inst. Adv. Studies, Ser. A 1949, No. s. 222 J. L. SYNGE: Classical Dynamics. Sect. 124. Let us now use for the system the symbols previously employed for a single particle, writing H,s = total angular momentum relative to the origin, H, 5 (a) = total angular momentum relative to a,, M, = total 4-momentum. We have as in (123.2), but now with an extended meaning, H, 5 (a) = H, 5 - a,A[. + a5 M,. Consider the four equations H, 5 (a)M5 =0, (123.3) (123.4) of which only three are independent on account of the skew-symmetry of H, 5 (a). Substitution from (123.3) gives (123.5) These equations locate a, on a straight line in space-time, with equations HrsMs + .o.M a,= M M 'If r> n " (123.6) where {} is a variable parameter. This is the history of the mass-centre of the system, mass-centre being so defined relativistically. This history is parallel toM,. If we use a frame of reference with time-axis parallel to M, (this is the masscentre reference system of Sect. 120), we haveM;,=O; (123.6) gives a mass-centre fixed at the point with coordinates - He4 ( ) ae=M.-. 123.7 4 If, on the other hand, we leave the directions of the space-time axes arbitrary, but move the origin to a position on the history of the mass-centre, then (123.5) is satisfied by a,= 0, and therefore H, 5 A{.= 0. (123.8) 124. Particles with spin. The Eq. (123.8) suggests that the spin or intrinsic angular momentum of a particle with 4-momentum M, should be represented by a skew-symmetric tensor H, s satisfying the condition H, 5 A{.= 0. {124.1) Then the angular momentum of the particle about any event a, will consist of two parts: an orbital angular momentum (x,- a,) M.- (xs- as) M, (124.2) and a spin angular momentum H, s, satisfying ( 124.1) and independent of the choice of origin and of the event a,. Any skew-symmetric tensor can be represented by two 3-vectors, and we may describe the spin by the two 3-vectors, ~ and He*, where H1 =iH14 , =i~ , H1* = J4.a, H2* = Hal• {124.3) The factor i is inserted here to yield a real 3-vector, 4 being a pure imaginary in Minkowskian coordinates. General references. The four equations contained in (124.1) give the vector equation H=H*x_!_ c , where v is the velocity of the particle, and also the scalar equation H·v=O, 223 (124.4) (124.5) which is of course a consequence of (124.4). By (124.4) the two spin vectors are perpendicular to one another. If we use the rest frame of the particle, thus making v = 0, we have H = 0 by (124.4), and hence only one spin vector, H*. The spin tensor yields a LoRENTZ-invariant: !)2 = jH,sHrs= H*2- H2. (124.6) This expression is always positive, since we can make H = 0 by choice of frame of reference, and hence Q is real. For a particle moving under the influence of a 4-force X, and a torque Y,s (=-Y.,), equations of motion have been proposed1, which read as follows in the notation of the present article: H, 5 A5 =0, A,A,=-1, } M;=X,, H;5 =M,A5 -M.A,+ Y,s; (124.7) here A, is the 4-velocity and the prime means dfds along the world line. The mass of the particle being defined by m = - M, A,, the above equations imply M,= m A,+ H, 5 A; + Y,sAs· (124.8) If X,=O and ¥, 5 =0, then M, is a constant 4-vector, and the orbit is a circle in that frame of reference for which M1 = M2 = 1\fa = 0. I wish to thank Professors C. LANczos and C. TRUESDELL for discussions and advice during the preparation of this Article, and Dr. L. BAss for assistance in proof-reading. General references. This is a selection of textbooks and articles on classical dynamics, with brief notes, each of which is intended to give some general indication of the nature of the contents. [1]. AMES, J. S., and F. D. MuRNAGHAN: Theoretical Mechanics. Boston: Ginn 1929. - Vector analysis, including theory of screws. Kinematics. Dynamics of a particle and of a rigid body. Lagrangian and Hamiltonian equations. Variational principles. HAMILTON-J ACOiU equation. POISSON brackets. Relativity. [2] APPELL, P.: Traite de Mecanique rationnelle. Paris: Gauthier-Villars, Tome I, 1941 (6th. Edn.); Tome II, 1953 (6th Edn.). - A classical treatise, presenting the subject in detail and with great clarity. Sparing use of vector notation. Tome I deals with kinematics, statics and the dynamics of a particle. Tome II deals with systems, halonomic and non-holonomic, Lagrangian and Hamiltonian equations with associated general theory, shocks and percussions. Three further volumes deal with continuous media, rotating fluid masses and tensor calculus. [3] CoRBEN, H. C., and P. STEHLE: Classical Mechanics. New York: Wiley, and London: Chapman & Hall 1950. - A modern textbook, with emphasis placed on those parts of the subject most pertinent to quantum mechanics. Vector and matrix notation used. Hamiltonian theory, PoiSSON brackets, and contact transformations. Introduction to special theory of relativity. 1 J. FRENKEL: Lehrbuch der Elektrodynamik, Bd. 1, p. 353. Berlin: Springer 1926. - M. MATHISSON: Acta Phys. Polan. 6, 163, 218 (1937). - J. WEYSSENHOFF and A. RAABE: Acta Phys. Polan. 9, 7, 19 (1947). - J. WEYSSENHOFF: Acta Phys. Polan. 9, 26, 46 (1947) See also 0. C. DE BEAUREGARD, p. 122 of op. cit. in Sect. 107. 224 J. L. SYNGE: Classical Dynamics. [4] FINZI, B.: Meccanica Razionale. 2 volumes. Bologna: Zanichelli 1948. - A general textbook on mechanics, with some attention given to Lagrangian and Hamiltonian methods. Relativistic mechanics. Statistical mechanics. [.5] FRANK, PH.: Analytische Mechanik. Die Differential- und Integralgleichungen der Mechanik und Physik, Teil 2, pp. 1-176. Braunschweig: F. Vieweg & Sohn 1927. - Lagrangian and Hamiltonian equations, transformation theory, HAMILTON-JACOBI equation, action-angle variables, stability, rigid motions, perturbations. [6] FuEs, E.: Storungsrechnung. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 131 to 177. Berlin: Springer 1927. - Multiply periodic motions, action-angle variables, degeneracy, adiabatic invariants, development in powers of parameter, secular disturbances, DELAUNAY's method, time-dependent disturbances. [7] GoLDSTEIN, H.: Classical Mechanics. Cambridge, Mass.: Addison-Wesley 1950. - Emphasis on techniques required in quantum mechanics. Matrix and vector notations used. Special relativity. Hamiltonian equations, canonical transformations, small oscillations, introduction to Lagrangian and Hamiltonian formulations for continuous systems and fields. [8] GRAMMEL, R.: Kinetik der Massenpunkte. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 305-372. Berlin: Springer 1927. - Dynamics of a particle, free or constrained. Motion relative to rotating earth. Two-body problem. Three-body problem. Stability. [9] HALPERN, 0.: Relativitii.tsmechanik. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 578-616. Berlin:Springer 1927. - pynamics of a particle and of a continuum in special relativity. Light quantum. General relativity. [10] HAMEL, G.: Die Axiome der Mechanik. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 1-42. Berlin: Springer 1927. -Newtonian laws. Construction of mechanics from continuity hypothesis, from rigid body, from particles. Construction from Lagrangian and energetic principles. Non-classical forms of dynamics. Absence of contradictions. [11] HAMEL, G.: Theoretische Mechanik. Berlin: Springer 1949. - A comprehensive textbook, with detailed treatments of rigid body, n-body problem and non-holonomic systems; 263 pages devoted to problems and their solutions. [12] JuNG, G.: Geometrie der Massen. Encyklopii.die der mathematischen Wissenschaften, Vol. IV.1, pp. 279-344. Leipzig: Teubner 1901-1908. - A specialized article on linear, quadratic and higher moments, with bibliography up to 1903. [13] LAMB, H.: Dynamics. Cambridge: University Press 1929 (2nd Edn.).- Elementary textbook, without vector notation, noteworthy for direct simple treatment of plane problems. [14] LAMB, H.: Higher Mechanics. Cambridge: University Press 1929 (2nd Edn.).- Sequel to [13]. Geometry of finite rotations, screws and wrenches. Motion of rigid body in space. Lagrangian and Hamiltonian equations. Vibrations. [1.5] LANczos, C.: The Variational Principles of Mechanics. Toronto: University of Toronto Press 1949. - D' ALEMBERT's principle. Lagrangian and Hamiltonian equations. Canonical transformations. HAMILTON-JACOBI theory. Emphasis on the geometry of phase-space. [16] LEVI-CIVITA, T., and U. AMALDI: Lezioni di Meccanica Razionale. Bologna: Zanichelli; Vol. I, 1923; Vol. Il1, 1926; Vol. liz, 1927. - A comprehensive treatise, comparable to APPELL [2]. Vol. I deals with kinematics, geometry of masses, and statics. Vol. II1 , deals with dynamics of a particle, LAGRANGE's equations, stability of vibrations. Vol. liz deals with dynamics of a rigid body, Hamiltonian theory, variational principles, and impulsive motion. [17] MACMILLAN, W. D.: Theoretical Mechanics. New York and London: McGraw-Hill; Vol. I, 1927; Vol. II, 1936. - Comprehensive textbook with much detail. Vol. I deals with orbits, ballistic trajectories, Lagrangian and Hamiltonian equations and variational principles for a particle. Vol. II deals with a rigid body, with fixed point or rolling, impulsive forces, Lagrangian and Hamiltonian methods in general, method of periodic solutions. [18] NoRDHEIM, L.: Die Prinzipe der Dynamik. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 43-90. Springer: Berlin 1927. - Differential and integral principles, virtual work, D'ALEMBERT'S principle, principles of GAUSS, HERTZ, HAMILTON, JACOBI. [19] NoRDHEIM, L., and E. FuEs: Die HAMILTON-JACOBische Theorie der Dynamik. GEIGERSCHEEL, Handbuch der Physik, Vol. V, pp. 91-130. Berlin: Springer 1927. - Canonical transformations, POISSON and LAGRANGE brackets, HAMILTON-JACOBI equation, eikonal. [20] PERES, J.: Mecanique generale. Paris: Masson & Cie. 1953. - Compact textbook dealing with moments of inertia, non-holonomic constraints, virtual work, dynamics of a particle and of a rigid body, equations of LAGRANGE, APPELL and HAMILTON, HAMILTON-JACOBI equation, stability about equilibrium or steady motion. Shocks and percussions. General references. 225 [21] PRANGE, G.: Die allgemeinen Integrationsmethoden der analytischen Mechanik. Encyklopadie der mathematischen Wissenschaften, Vol. IV.2, pp. 505-804. Leipzig: Teubner 1904-1935. - Detailed treatment of n'ALEMBERT's principle, LAGRANGE's equations, variational principles, variation of constants, HAMILTON's optics, characteristic function, HAMILTON-JACOBI equation, separation of variables, integral invariants, systematic integration of canonical systems, canonical transformations, substitution or generating functions, equivalent systems. [22] RouTH, E. J.: A Treatise on the Dynamics of a System of Rigid Bodies. London: Macmillan; Vol. I, 1897 (6th Edn.); Vol. II, 1905 (6th Edn.). German translation by A. ScHEPP, Leipzig 1898. Vol. II reprinted by Dover Publications, New York, 1955. - Although old-fashioned in notation, this book remains the most useful reference for miscellaneous special information on the dynamics of rigid bodies. Strings and membranes are also treated. The arrangement is unsystematic, but there is a good index. [23] SCHAEFER, CL.: Einfiihrung in die theoretische Physik, Bd. 1. Berlin: W. de Gruyter 1950 (5th Edn.). - The first 469 pp. of this book treat the dynamics of particles and rigid bodies in considerable detail, Lagrangian and Hamiltonian methods being included. The rest of the book (507 pp.) deals with the mechanics of continua. [24] ScHOENFLIES, A., and M. GRDBLER: Kinematik. Encyklopadie der mathematischen Wissenschaften, Vol. IV.1, pp. 190-278. Leipzig: Teubner 1901-1908. - Finite displacements, velocity, acceleration, linkages, mechanisms. [2.5] STACKEL, P.: Elementare Dynamik der Punktsysteme und starren Korper. Encyklopadie der mathematischen Wissenschaften, Vol. IV.1, pp. 436-684. Leipzig: Teubner 1901 -1908. - Direct treatment of the dynamics of particles and rigid bodies, with bibliography and detailed historical references. Gyroscopic motion handled rather fully, with diagrams. [26] SYNGE, J.L., and B.A. GRIFFITH: Principles of Mechanics. New York-Toronto-London: McGraw-Hill1959 (3rd Edn.). -Textbook of statics and dynamics up to gyroscopic theory. Motion of charged particles in axially symmetric electromagnetic field. Methods of LAGRANGE and HAMILTON. Vibrations. Introduction to relativity. [27] Voss, A.: Die Prinzipien der ratione lien Mechanik. Encyklopadie der mathematischen Wissenschaften, Vol. IV.1, pp. 3-121. Leipzig: Teubner 1901-1908. - History and philosophy of mechanics, from GALILEO and NEWTON. Eliminati0 p. of force by KELVIN and HERTZ. Principles of n'ALEMBERT, FOURIER, GAUSS, HAMILTON. Principle of energy. [28] WHITTAKER, E. T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge: University Press 1937 (4th Edn.). Reprinted by Dover Publications, New York 1944. The 2nd Edition (1917) translated into German by F. and K. MrTTELSTEN-ScHEID (Berlin: Springer 1924). - Standard treatise, with systematic arrangement of material. More compact than APPELL [2] or LEvr-CrVITA and AMALDI [16], the usual problems of the dynamics of particles and rigid bodies being treated by Lagrangian methods, without vector notation or diagrams. The second half of tl:e book deals with Hamiltonian systems, integral invariants, transformation theory, first integrals, three-body problem, theory of orbits. [29] WINKELMANN, M., and R. GRAMMEL: Kinetik der starren Korper. GEIGER-SCHEEL, Handbuch der Physik, Vol. V, pp. 373-483. Berlin: Springer 1927. - The top, symmetric and unsymmetric, with many diagrams. Relative motion of a rigid body on the rotating earth. Systems of rigid bodies. Gyroscopic stability. [30] WrNTNER, A.: The Analytic Foundations of Celestial Mechanics. Princeton: University Press 1947. -The main theme is then-body problem, but the book contains a compact critical treatment of Hamiltonian methods and canonical transformations, with interesting historical notes and references at the end. Handbuch der Physik, Bd. III/!. 15 The Classical Field Theories. By C. TRUESDELL and R. ToUPIN 1• With 4 7 Figures. With an Appendix on Invariants by j. L. ERICKSEN. A. The field viewpoint in classical physics. 1. Corpuseies and fields. Today matter is universally regarded as composed of molecules. Though molecules cannot be discerned by human senses, they may be defined precisely as the smallest portions of a material to exhibit certain of its distinguishing properties, and much of the behavior of individual molecules is predicted satisfactorily by known physical laws. Molecules in their turn are regarded as composed of atoms; these, of nuclei and electrons; and nuclei themselves as composed of certain elementary particles. The behavior of the elementary particles has been reduced, so far, but to a partial subservience to theory. Whether these elementary particles await analysis into still smaller corpusdes remains for the future. Thus in the physics of today, corpusdes are supreme. I t might seem mandatory, when we are to deal with extended matter and electricity, that we begin with thelaws governing the elementary particles and derive from them, as mere corollaries, the laws governing apparently continuous bodies. Such a program is triply impractical: A. The laws of the elementary particles are not yet fully established. Even such senior disciplines as quantum mechanics and general relativity remain open to possible basic revision and not yet satisfactorily interconnected. B. The mathematical difficulties are at present insuperable. (Even on a lower level they remain: As is well known, the "proof" that a quantum-mechanical system may be replaced by a classical system in first approximation is defective.) C. In such special cases as have actually been treated, the mathematical "approximations" committed in order to get to an answer are so drastic that the results obtained are not fair trials of what the basic laws may imply. When such a result appears in disaccord with experience, we are at a loss whether to assign the blame to the basic laws themselves or to the mathematical process used in the subsequent derivations. 1 Acknowledgment. The authors are deeply indebted to Professor Dr. K. ZOLLER for thorough criticism of most of the manuscript and proofs. They are grateful also to Professors J. L. ERICKSEN and W. NoLL and to Dr. B. CoLEMAN for help in certain passages. During portions of the period of preparation of this treatise, TRUESDELL's work was supported by an ONR contract {1955 to 1956) the U. S. National Science Foundation {1956), the John Sirnon Guggenheim Memorial Foundation (1957), the Mathematics Research Center, U.S. Army, University of Wisconsin (1958). and the National Bureau of Standards (1959). During 1957 he was on sabbaticalleave from Indiana University. While all parts of this work have been discussed and revised jointly, Chaps. A to E and the firsthalf of Chap. G were written by TRUESDELL; Chap. Fand the second half of Chap. G, by TouPIN. Sect. 1. Corpuseies and fields. 227 But more than this, such a program even if successful would be illusory: (a) The future discovery of new entities within the present "elementary" partides would nullify any daim for such results as predictions from "basic" laws of physics. Indeed, within any corpuscular view the possibility of an infinite regress is logically inevitable. (b) The details of the behavior of the corpuscles are extraneous to most mechanical and electromagnetic problems. Materials whose corpuscular structure is quite different may exhibit no perceptible difference of response to stress. A voiding illusory complications, it is possible to construct a direct theory of the continuous field, indefinitely divisible without losing any of its defining properties. The field may be the seat of motion, matter, force, energy, and electromagnetism. Theories expressed in terms of the field concept are called phenomenological, because they represent the immediate phenomena of experience, not attempting to explain them in terms of corpusdes or other inferred quantities. The corpuscular theories and the field theories are mutually Contradietory as direct models of nature 1• The field is indefinitely divisible; the corpusde is not. To mingle the terms and concepts appropriate to these two distinct representations of nature, while unfortunately a common practice, leads to confusion if not to error. For example, to speak of an element of volume in a gas as "a region large enough to contain many molecules but small enough to be used as an element of integration" is not only loose but also needless and bootless. In a deeper sense, the continuous field and the assembly of corpusdes may be set into entire agreement. Adopting the viewpoint of statistical mechanics, consider a dassical system of mass-points of any kind whatever, and assign a probability to its initial conditions. Extending a notable success by IRVING and KIRKWOOD 2, NoLL3 has defined certain phase averages which he proved to satisfy exactly the laws of balance for a continuous field. This result, not a limit formula or approximation, is an exact theorem on distributions in phase space. Thus those who prefer to regard dassical statistical mechanics as fundamental may nevertheless employ the field concept as exact in terms of expected values. While sometimes the phenomenological approach is regarded as only approximate, the result just described shows that in representing matter as continuous rather than discrete we can in fact make no statement that is inconsistent with the statistical view of matter as composed of dassical molecules, so long as we confine attention to the exact and generat theory of continuous media 4• This treatise presents the exact and general theory of the continuous field. 1 The formal "derivations" of the field equations from the mass-point equations of mechanics given in many textbooks are illusory, such a derivation being impossible without added assumptions which are rendered superfluous by a direct approach to the continuum. The difficulty can be avoided by a formulation of the fundamental equations as Stieltjes integrals (cf. Sect. 201); in essence, this was done by EuLER [1752, 2, §§ 20-22]. 2 [1950, 12]. Square brackets refer to the bibliography. 3 [1955, 19]. In NoLL's paper precise conditions of regularity for the density in phase are stated. The molecules, not restricted in variety, are supposed free of constraints but otherwise subject to arbitrary mutual and extrinsic forces. The expression for the resultant extrinsic force in general does not depend only on the extrinsic forces to which the molecules are subject; otherwise the agreement stated in the text above is unqualified. An extension to quantum-mechanical systems is given by IRVING and ZwANZIG [1951, 12]. 4 That is, only the generat equations expressing the balance of mass, momentum, and energy in the continuous field have been derived. There is no indication that any special theory of continuous bodies, such as the theory of perfect fluids, is consistent with statistical mechanics. In fact, a simple field theory seems to emerge only in approximation, and from a simple molecular picture an extremely complicated field theory results. Also, the exact agreement does not extend to thermodynamics, which from the statistical standpoint appears tobe only an approximate theory. 15* 228 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 2, 3. 2. Classical mass-points and classical fields. From the time of NEWTON until relatively recently, many natural scientists considered the mass-point the fundamental quantity of nature, or at least of mechanics. They believed that matter was composed of many very small particles obeying the laws of classical mechanics, and that, consequently, the behavior of gross matter could be predicted, in principle, to any desired accuracy, from a knowledge of the intermolecular forces. Thus continuum mechanics appears as an approximate or at best secondary theory within classical mechanics. While this tradition clings on in physics teaching today, it is not realistic. Aside from the as yet unconquered mathematical difficulties in putting this ideal program into practice, the program itself is out of keeping with modern views on matter. The smallest units of matter are no Ionger believed to obey the laws of Newtonian mechanics, except approximately and in circumstances rarely occurring in dense matter. Nevertheless, conditions in which the classicallaws of momentum and energy fail perceptibly for tangible portians of matter are extremely rare if not altogether unknown. To cite an example, no corpuscular theory based on Newtonian mechanics has produced formulae for the specific heats of solids which agree with experimental values. N evertheless, there is not the slightest indication that a solid body when heated and set in motion fails, as a body, to obey the classical laws of balance of mass, momentum, and energy. In fact it is almost the rule that Newtonian mechanics, while not appropriate to the corpuscles making up a body, agrees with experience when applied to the body as a whole, except for certain phenomena of astronomical scale. Only paedagogical custom has bindered general realization that as a physical theory, continuum mechanics is better than masspoint mechanics 1• Indeed, in physics it is inappropriate to lay down the laws of classical mechanics for small bodies, to which in general they do not apply, and thence to derive or state by analogy the corresponding laws for extended bodies, to which they do apply. Rather, the process should be reversed: Classical mechanics is the mechanics of extended bodies 2• There remain certain special problems, particularly problems in celestial mechanics, ballistics, and mechanisms, where the mechanics of mass-points is accurate. As is shown in Sect. 167, problems of this kind are easily and reasonably regarded as special cases within continuum mechanics. 3. Experiments and axioms. It has become fashionable to present the foundations of theoretical physics in terms of experiments. Indeed, since physics is intended to predict numerous phenomena of nature from knowledge of a few, the preconception that a given physical discipline should be derivable from the results of certain basic experiments is most appealing. In fact, however, an experimental approach to mechanics and electromagnetism is not practical. The field, infinite in extent and indefinitely divisible, is by its very nature not measurable directly. The "experiments" sometimes used as the starting point for paedagogical treatments of field theories are a posteriori verifications at best; always unperformed and often unperformable, too often they are mere hoaxes. Moreover, they belie the true course by which the field theories have developed. Experience has been the guide, thought has been the creator 3• Not only does any theory reduce and abstract expericnce, but also it overreaches it by extra 1 Cf. TRUESDELL [1952, 22, PP· 79-80]. 2 Cf. HAMEL [1908, 4, p. 351]. Note also that from a theory of phase averages over systems governed by quantum mechanics IRVING and ZwANZIG [1951,12] infer the classical equations of balance of mass, momentum, and energy. 3 Cf. e.g. DUGAS [1954, 6], TRUESDELL [1956, 23]. Sect. 3. Experiments and axioms. 229 assumptions made for definiteness. Theory, in its turn, predicts the results of certain specific experiments. The body of theory furnishes the concepts and formulae by means of which experiment can be interpreted as in accord or disaccord with it. To overturn a theory by the results of experiment, we seek the aid of the theory itself; in terms of the theory, from experiment we may find agreement which develops cÖnfidence in the theory, but establish a theory by experiment we never can. Experiment, indeed, is a necessary adjunct to a physical theory; but it is an adjunct, not the master. While most theoretical physicists seem to act in accord with the above views, they rarely admit to holding them. Therefore a fuller explanation, largely a paraphrase of a work by SouTHWELL1, is appended. The "operational" system accepts as basic only quantities susceptible of direct measurement and, connecting them, laws which are to be tested by experiment. From these laws, logical inference is to derive a system shown by actual trial to keep contact with physical experience at every stage. Apart from the practicallimitations in checking any theoretical "law", there is a deeper objection against this view of physics, in that it rests on a circularity: No experiment can be interpreted without recourse to ideas in themselves part of the theory under examination. Similarly, no quantity can be measured in the absence of a theory explaining the experiment. Consider the measurement of "mass" by weighing or by impact; in the former case the law of falling bodies andin the latter case the law of conservation of momentum, both employing the concept of mass, are used to complete the measurement. If we seek to verify NEWTON's "law" that "Every body continues in its state of rest, or of uniform motion in a right line, unless compelled to change that state by forces impressed upon it," 2 we require a free body, unavailable because all bodies in the Iabaratory are subject to the earth's attraction. Indeed, we try to neutralize that attraction, as in "ATwoon's machine": The body is connected by a light string passing over a freely running pulley with a second body of equal weight, and it is found that, started with any initial velocity, the test body retains its velocity almost unchanged. Casting aside the small observed retardation, doubtless arising from friction, we still cannot accept this result as a proof of the "law" in question. The body found to move with substantially uniform speed and direction is not a free body, and without the principles of mechanics, themselves dependent upon the law we are supposedly establishing by experiment, we cannot justly assert that the forces present do in fact neutralize each other. More elaborate application of the principles of mechanics is required if we are to reason that the inertia of the pulley has no effect on the ideal experiment. Further, to estimate the "experimental error" in the real experiment, we require a hypothesis of friction and an application of the laws of mechanics both for the effect of this friction and for the partially counteracting effect of the inertia of the pulley. Such difficulties are avoided by the postulational standpoint, according to which physics, as an abstract discipline, may employ any variables and any consistent initial assumptions or "laws" which are convenient. In construction of this mathematical system it is not necessary to maintain contact with experiment at every stage. The system is an abstract model, designed to represent some of the observed phenomena of the physical universe, but directly concerned only with ideal bodies. Same few of the properties of theseideal bodies are postulated; the numerous remainder is tobe derived mathematically. Whether these derived properties correspond with physical observationisaseparate question, tobe decided by subsequent comparison with experiment. But the available tests apply only to the system as a whole: We cannot devise an experimentsuch as to verify any one of its assumptions apart from the rest. Naturally it is possible to construct an ideal system without relevance to physics. However, since experience is the guide, entirely wrong physical theories have been rare. Rather, a weil thought theory usually turnsout to have relevance for certain physical phenomena but to be in error for others. Such is the case with the classical field theories. Their failures areweil known and have provided the impetus for "modern" physics. Often their successes are forgotten. It is classical physics by which we grasp the world about us: the heavenly 1 [1929, 9]. 2 [1687, 1, Lex 1]. 230 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 4. motions, the winds and the tides, the terrestrial spin and the subterraneous tremors, prime movers and mechanisms, sound and flying, heat and light!. Thus the classical field theories have won indisputable permanence in the language by which we speak of nature. Whatever the future revisions of theories of the structure of matter, the place of the class~cal field theories will remain unchanged. This permanence, along with the difficulties mentioned in Sect. 1, makes necessary a complete and independent presentation of the foundations of the classical field theories. Being mathematical disciplines, they should be derived from axioms. Indeed, as his sixth problern HILBERT2 set the construction of a set of axioms, on the model of the axioms of geometry, for "those branches of physics where mathematics now plays a preponderant part; first among them are probability theory and mechanics." Like all of his problems conceming physical applications of mathematics, his proposal for mechanics has received little attention. The possibility that the future may revise the physics of small corpusdes does not reduce the need for axiomatic treatment of the field theories. Physics, like mathematics, may be constructed precisely at several different levels. The interconnection of the different levels, either exactly or by approximation or by addition of new axioms, then fumishes definite mathematical problems 3• Having reached agreement that we should base the classical field theories on a set of axioms, we must now admit, ruefully, our inability to do so. In our opinion, none of the attempts to form such a system has been successful. Only in very recent years has an adequate set of axioms for pure mechanics, at last, been constructed by Non 4• To present his development of the subject here would be premature, since a correspondingly clear and precise formulation of irreversible thermodynamics is not yet available. We regard the fully invariant formalism for electromagnetic theory given in our Chap. F as being essentially an axiomatization of the subject. Non and CoLEMAN have disclosed to us the outline of what appears to be a satisfactory basis of general thermodynamics in deformable media. Thus there are grounds for expecting that HILBERT's program will shortly be actualized. Despite the lack of complete axiomatic formulation, the generat equations governing the classical fields are known and universally accepted. The present article is devoted to a formally precise study of these general equations. Any future axiomatization, if successful, will necessarily lead to these same equations. 4. Mathematics and its physical interpretation. That a branch of theoretical physics is a mathematical science by no means implies its aim or interests to be those of pure mathematics. Rather, the Problems are set by the subject. The developments must illumine the physical aspects of the theory, not necessarily in the narrower sense of prediction of numerical results for comparison with experimental measurement, but rather for the grasp and picture of the theory in relation to experience. In this spirit do we pursue our subject, neither seeking nor avoiding mathematical complexity5• 1 Cf. V. BJERKNES et al. [1933. 3, Vorwort]. 2 [1900, 5]. 3 In mathematics the economy of such independent constructions has lang been realized. E.g., to construct the complex numbers we presume the properties of the real numbers given; for the real numbers, those of the integers; and for the integers, mathematical logic. To approach fluids in terms of nuclear physics is like treating functions of a complex variable with the apparatus of formal logic. 4 [1957. 11] [1958, 8] [1959 • .9]. 5 The expression of such a program has been attributed to KELVIN and TAIT, but we are unable to trace the reference. Sect. s. Exact and approximate theories. 231 Some will reproach us with too much abstract and useless formalism. Not forgetting that such deprecation was bestowed upon WHITTAKER's Analytical Dynamics half a century ago, we are confident that the reader of half a century hence will regard our compromise of the moment as erring rather toward insufficient use of the mathematical tools available. Any mathematical theory of physics must idealize nature. That much of nature is left unrepresented in any one theory is obvious; less so, that theory may err in adding extra features not dictated by experience. For example, the infinity of space is itself a purely mathematical concept 1 , and all theories erected within this space must share in the geometrical idealization already implied. Indeed, it is difficult to find any theory that does not contain infinities, and infinities, by definition, are immeasurable. While at one time certain theoretical statements were regarded as "laws" of physics, nowadays many theorists prefer to regard each theory as a mathematical model 2 of some aspect of nature. In a sense, then, every theory is only "approximate" in respect to nature itself. This unavoidable defect in theory is often taken as a patent for "approximate" mathematics in the deductions from it. Indeed, while mathematics is generally understood to proceed by entirely logical processes, were the "derivations" in some of the accepted physical papers of today translated into common reasoning, they would fail to meet the logical standards of a competent historian or bibliographer. All too often is heard the alibi that since the theory itself is only approximate, the mathematics need be no better. In truth the opposite follows. Granted that the model represents but a part of nature, we are to find what such an ideal picture implies. A result strictly derived serves as a test of the model; a false result proves nothing but the failure of the theorist. To call an error by a sweeter name does not correct it. The oversimplification or extension afforded by the model is not error: The model, if well made, shows at least how the universe might behave, but logical errors bring us no closer to the reality of any universe. In physical theory, mathematical rigor is of the essence. In this ti-eatise we attempt to keep the argument rigorous. However, nothing is gained by laboring elementary details. We presume that the reader knows infinitesimal calculus, simple algebra, and tensor analysis; that he can supply for himself, without repetition on our part, conditions sufficient for interchanging differentiations, inversion of functions, expansions in power series, etc. Roughly speaking, our proportion of what is said to what is left unsaid is that which is customary in works on differential geometry. 5. Exact and approximate theories. While every theory is a model of nature, and thus not "exact" in relation to it, nevertheless there is a government among theories. A theory is tested by experiment, and a range of confidence in it is established. In this sense, a given theory is "good"; if the range of application is greater than another's, it is the "better" of the two. For example, the theory of the flow of viscous compressible fluids should suffice to predict definite results, fit for experimental test, concerning the propagation, absorption, and dispersion of sound in fluids. That such results have never been obtained is only from our lack of sufficient mathematics. Instead, a perturbation scheme has been used to infer equations governing "small" motions. The resulting acoustical theory is presumed to yield an" approximation" to the better but intractable theory of fluids. Any given theory may be laid down as "exact ". It is then a definite mathematical problern to discover the relation of its results to those derived from 1 EULER [1736, 1, § 8]. 2 HELMHOL TZ [ 1902, 4, § 1]. 232 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 6. 7. other theories, considered as "approximate" in respect to it. Problems of this kind are important and difficult, indeed in most cases too difficult for the mathematics available today. We do not attempt to study them in this treatise. Neither co we presentl the unjustified linearizations or formal schemes of perturbation which occupy much of the literature. Our scope is restricted to exact treatment. 6. Closed systems and armatures. In the nineteenth century there was a search for sets of physical laws which should include the maximum range of physical phenomena yet remain sufficiently specific to predict definite results in particular cases. The culmination of this trend came in the systems of ]AUMANN and LoHR 2, in which an all-embracing set of equations goveming mechanics, electrodynamics, che~ical reactions, diffusion, heat transfer, electromechanical effects, etc. are postulated. Current knowledge of the structure of matter (cf. Sect. 1) has destroyed the raison d' etre of such closed systems as weil as rendering them impractical. Rather, the classical field theories offer us armatures on which particular models of cxtendcd matter and electricity may be built. In this spirit, it is inclusiveness ratl:.er than particular problems that we seek here. For example, it is often claimed tlmt in nature, if we Iook closely enough, only conservative forces occur; that such dfects as friction are gross appearances resulting only from Iack of knowledge of the underlying conservative process. But natural problems are not confined to those on the smallest or largest scale. The world about us, as we see it, must be mastered and controlled. Situations incompletely described are the rule, not the exception, and we must formulate good theories for these limited aspccts of nature. Our object is a generat framework 3 for such theories. The most general motions, the most general stresses, the most general flows of energy, and the most general electromagnetic fields, fumish the subject of this treatise. 7. Field equations and constitutive equations. Motion, stress, energy, entropy, and electromagnetism are the concepts upon which field theories are constructed. Certain laws of conservation or balance are laid down as relating these quantities in all cases. These basic principles, which are in integral form 4, in regions where 1 An exception is Parte of Subchapter BI, which is included only so as to aid the reader in connecting the general theory with the results assumed in the ordinary theory of elasticity. B [1911, 7], [1918, 3]; [1917, 5]. 3 The field viewpoint is excellently apt to secure such generality, while to overcome the complications of corpuscular theories it is usual to make simplifying hypotheses which sharply !essen their scope. Cf. HELMHOLTZ [1902, 4, § 2]. As was remarked by LAGRANGE [1788, 1, Part 1, Sect. IV, § II, '1[9] the field view has precisely the same mathematical advantage over the corpuscular view as the differential theory of curves over polygonal approximations. 4 The view that all naturallaws should be expressed by integrals is generally attributed to the Göttingen lectures of HILBERT. That jump conditions arenottobe derived from smooth solutions was clearly understood by STOKES [1848, 4, p. 353]: " ... I wish the two subjects to be considered as quite distinct." In recent years mathematicians have created various kinds of "generalized solutions ", whereby, granted certain purely analytic presumptions as to the intended meaning of a problern formulated in terms of differential equations, discontinuous solutions may be inferred from continuous ones. We regard these approaches not only as demanding unnecessary mathematical apparatus but also as concealing the simple and immediate nature of physical laws. Neither in respect torigor nor in any other regard do they offer advantages over HILBERT's program of stating physicallaws in integral form. While the work of ZEMPLEN [1905, 7, p. 438] and HELLINGER [1914, 4], which reflects HILBERT's influence, rests upon variational principles, the program of postulating integral conservation laws, which we follow here, seems first to have been laid down by KoTTLER [1922, 3 and 4]. Cf. also CARTAN [1923. 1, introd., §§ 8, 81], KoTCHINE [1926, 3, § 3]. As remarked by VAN DANTZIG [1937. 11], it is obvious that notions like differentiability can have no empirical basis at all. Since, on the contrary, Sect. 8. The nature and plan of this treatise. 233 the variables change sufficiently smoothly are equivalent to differential field equations; at surfaces of discontinuity, to fump conditions. The field equations and jump conditions form an underdetermined system, insufficient to yield specific answers unless further equations are supplied. Within the embracing concept of the balanced fields, it is possible to define ideal materials 1 by certain further conditions. These defining conditions are called constitutive equations. The most familiar constitutive equations, here expressed in words, are: The distances between particles do not change (Rigid body). The stress is hydrostatic ( Perfeet fluid). The stress may be determined from the stretching alone (Viscous fluid, perfectly plastic body). The stress may be determined from the strain alone (Perfectly elastic body). The flux of energy is a linear function of the temperature gradient (Classical linear heat conduction). The thermodynamic affinities are linear functions of the thermodynamic fluxes ("Irreversible thermodynamics "). The diffusion velocity of a constituent of a binary mixture is proportional to the gradient of its peculiar density (Classicallinear mass diffusion). The electric displacement is proportional to the electric field; the magnetic induction is proportional to the magnetic intensity (Classical linear electromagnetism). The constitutive equations and field equations together, along with the jump conditions and boundary conditions, should lead to a definite theory, predicting specific answers to particular problems. For some of the special materials listed above, this definiteness has been proved through theorems of existence and uniqueness. The present treatise is devoted to the generat principles of balance alone. Thus we deal only with the field equations and fump conditions. Our last chapter mentions guiding principles by which rational constitutive equations may be formulated. The theories of certain particular ideal materials fill several later volumes of the Encyclopedia. 8. The nature and plan of this treatise. We present the common foundation of the field viewpoint 2 • We aim to provide the reader with a full panoply of touls of research, whereby he himself, put into possession not only of the latest discoveries but also of the profound but all too often forgotten achievements of previous generations, may set to work as a theorist. This treatise is intended for the specialist, not the beginner. Necessarily it presents the foundations of the field theories, not as they appeared in the last century and linger on in the textbooks, nor as the experts in some other domains such simple theories as the dynamics of perfect fluids and linear electromagnetism are known to furnish inadequate models of experience when the fields appearing are required tobe everywhere continuous, no physical principle should be stated in differential form. The older approach, still followed in many textbooks, either sets up more special postulates for discontinuities or employs an unrigorous Iimit process from the differentiable case. 1 The program of mechanics was laid out by EuLER [17 52, 2, § 19] (cf. also his remarks on rigid bodies and ideal fluids [1769, 1, § 12]); of continuum mechanics in particular, by CAUCHY [1823, 1], but the later emphasis on linear problems in very special theories caused it tobe largely forgotten until it was stated anew by v. MrsEs [1930, 3]. 2 The only single work attempting even a major part of our subject is the book of BRILL [1909, 2]. 234 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 9. may think they ought tobe presented 1, but as they are cultivated by the specialists of today. This treatise is organized as follows: 1. Kinematics, including conservation of mass, in Chaps. B and C. 2. Balance of momentum, Chap. D. 3. Balance of energy, including the thermodynamics of irreversible deformations, Chap. E. 4. Balance of electromagnetism, Chap. F. 5. Guiding principles for constitutive equations and examples of them, Chap. G. Foramore detailed plan, see the table of contents. We interpret "classical" in the narrower sense, as confined to phenomena in Euclidean three-dimensional space 2 and governed by N ewtonian mechanical principles. However, it would be crippling to maintain this restriction in dealing with electromagnetism. The four-dimensional viewpoint adopted in Chap. F necessitates further kinematical developments there and results in some duplication of subject. On the other band, to have begun with a world-invariant formalism in the earlier chapters would have greatly lessened their direct usefulness to specialists in mechanics. That over one half of the work is devoted to kinematics, the mathematical description of motion, is not malapropos. As the need for more and more general field theories has grown, the preliminary light which kinematics unencumbered by physical restrictions can provide, always appreciated by virtuosi of mechanics3, has become a necessity. In presenting here as our Chaps. Band C the first general treatise on the kinematics of continua, we believe that we look toward the future course of the field theories. 9. Tradition. We have tried to supply full and correct attributions, not only for historical perspective but also in plain justice. If the name attached to many a proposition is but a small one, that is all the less reason that its owner should be pilled of what little he wrought by a no greater name of today, whose slight capacities are scarcely increased by wilful or heedless ignorance of what others have done. However, the multitude of detailed citations should not prevent the great names from emerging. Our subject is largely the creation of EULER and CAUCHY. If we present their results in forms often very different from the original, in return we have included many of their discoveries that have not previously found a place in expositions. Not only will these names be the most frequently encountered, but also their appearances are at the crucial theorems and definitions. Next come STOKES, HELMHOLTZ, KrRCHHOFF, KELVIN, MAXWELL and HUGONIOT. In the twentieth century, HADAMARD and HILBERT 4 continued and deepened the tradition. That no one later name is frequently cited does not indicate that the subject is dead. Rather, after a generation of 1 Cf. KELVIN and TAIT [1867, 3, Preface]: " ... where we may appear to have rashly and needlessly interfered with methods and systems of proof in the present day generally accepted, we take the position of Restorers, not of Innovators." 2 At the end of the treatise, references to relativistic and to non-Euclidean or higherdimensional but non-relativistic theories are given in Special Bibliographies R and N, respectively. 3 We follow the tradition of EuLER, CAUCHY, and KELVIN. Cf. the remarks of HELMHOATZ [1858, 1, lntrod.J, ZHUKOVSKI [1876, 7], ST. VENANT [1880, 10] and jAUMANN [1905, 2, Introd.]. 4 While HILBERT published nothing relevant to our subject, his personal influence was widespread and continuing (cf. e.g. ZEMPLEN [1905, 7], HELLINGER [1914, 4, footnote 6]), and the organization used in his Göttingen lectures [1907, 3 and 4] has influenced ours. Cf. also footnote 4, p. 232. Sect. 10. General scheme of notation. 235 quiescence, in very recent years it has experienced a revival in a form more compact and general, and, we believe, closer to nature. 10. General scheme of notation. We make frequent and essential use of notations and results given in the appendix, "Invariants ", by J. L. ERICKSEN. Citations of that article are indicated by the prefix App. Thus "(App. 7.1)" refers to Eq. (7.1) ofthat appendix, and "Sect. App. 7" refers to its Sect. 7. The following partial table explains the basis for selection of notations. It has not been possible to maintain this scheme without exception, nor to avoid use of the same symbols in different senses in widely separate passages. Full-sized characters: I talic letters A, a, ... , etc.: Sealars and the kernel indices of vectors and tensors. For distinctions, see Sects. App. 15, 14. Bald-face letters A, a, ~ ... : Vectors, tensors, and matrices. For explanation, see Sect. App. 3. Roman numerals I, II, III, II, III, etc.: Principal invariants and moments of second order tensors (Sect. App. 38). H ebrew letters ~. ~ •... : Scalar coefficients in expressions for isotropic functions (Sect. 299). Russian capitals .IJ., U .... : Other special scalar invariants. Script letters d, .a, ... : Regions, surfaces and curves. German letters ~. a, ... , W, a, ... : Quantities defined by integrals. Black-letter capitals ~. W, ... : Similarity parameters. Sans-serif capitals M, U, ... , M, U, ... : Dimensional units and dimensions (Sect. App. 7). Creek letters e, {}, tP, ff!, ... : Angles, thermodynamic variables, world tensors. Indices: Italic and Creekindices K, k, T, y, ... : Tensorial indices. Roman indices s, t, ... , S, T, ... : Descriptive marks. Bald-face indices: Tensors or matrices from which an invariant is constructed. For example, Ia, IIa, and IIIa are the principal invariants of a. German indices a, ~ •... : Enumerative indices, not indicating tensor character. The corresponding numbers are written in italics. Thu" b! and c~ stand for the sets of contravariant and covariant components of the vectors b 1 , b 2 , ... , b1 and c1, c2, .•. , c1. A A,Ak A,Ak Symbol a, ayß• A, ArA b, by 6 • B, BrA B,Bk c c, Ckm C,CKM List of frequently used symbols Name Abnormality of a vector field Axis of finite rotation Magnetic potential First fundamental form of a surface Second fundamental form of a surface Magnetic flux density Position vector of the center of mass CAUCHY's deformation tensor GREEN's deformation tensor Velocity of light in the aether Absolute concentration of the constituent ~ Mass supply of the constituent ~ Place of definition or first occurrence (App. 30.1) (37.17) Sect. 276 (App. 19.7) (App. 21.5) Sect. 274 (161.5) (26.1) (26.2) Sect. 280 (158.4) (159.1) 236 C. TRUESDELL and R. TouPIN: The Classical Field Theories. d(n) da d,d,.". Symbol da, d~, da, Da, D~, na, etc. D,Dk ek,k, ... k,.• ek,k, ... k,. e, e11 "., E, EKM e, 6,."., E, 'EKM E,E11 f, 111 Fk".p, pabp, etc. g g, Ckm g", gf g,grA h, hk H,H11 i, ik, i[W] f, 1 J,]k l, l,., lkm m(n) m, mkqp M,M11 n p, pk,P, PK .P(=oi:) ii(=~l p p P, [cp] 2{ ~.~" ~ (t, (tlGl, ~km (t, ~k tl. g:,. .(), _()[Gl, .\),., .\),. m .() . .\),. l,Jlfll,~km 3.3" ~ ß, ßlGl, 2,., 2km W1 '· \13,. Special notations. Name Speed of propagation of a surface Local speed of propagation of a surface Velocity potential Electric potential Acceleration potential CAUCHY's spin tensor Vorticity vector Spatial diffusion vector or tensor Material diffusion tensor } Wryness of an oriented body General co-ordinates of places or events General symbols for particles } Deformation gradients Velocity field Aceeieration field Partide of the constituent 2l Velocity of the constituent 2l Reetangular Cartesian co-ordinate of places Reetangular Cartesian initial co-ordinates of particles Physical components of stress Christofiel symbols Jacobians Elements of arc } Elements of area Elements of volume Unit and dimension of action Unit and dimension of length Unit and dimension of mass Unit and dimension of charge U nit and dimension of time Unit and dimension of temperature Unit and dimension of magnetic flux Virtual work Potential of free charge Total internal energy EuLER's tensor Electromotive intensity Totalforce Total moment of momentum Potential of free current Tensor of inertia Conduction current Kinetic energy Total torque Mass Totallinear momentum 237 Place of definition or first occurrence (183.1) (183.4} (88.2) Sect. 276 (109.1) (86.1) (86.2) (101.5) (101.12) (61.7) (14.1). (152.5) (14.1) (17.1) (67-2) (98.1) (158.1) (158.2} (13.1) (13.1) (204.1) (16.1) (20.3) (20.5). (20.8) (20.9) Sect. 288 Sect. App. 8 Sect. 155 Sect. 270 Sect. 65 (246.2) Sect. 270 (232.1) Sect. 283 (240.1) (168.4) Sect. 274 (196.1) (166.2) Sect. 283 (168.4} Sect. 274 (94.1), (166.4) (196.1) (155.1) (166.1) 238 C. TRUESDELL and R. TouPIN: The Classical Field Theories. 0 ~ ~(K) 'WK 9k d.9",. d#,; "· rxr rxa Pa Symbol 'i' 'i'(N,M)•'i'(n,.n) r. 'i'ro r. r!:,p. rJ.jp. ~s ß(N)• ß(n) ßa ß:O LI Lf,LJFab q;, (/JrtJ 'P X W,Wkm n,ntJs 1 w [W] Hl- ~ dr dt dc dt ~d Tt curl div dual rot Guide through this treatise. Name Specific volume Absolute world velocity vector Thermodynamic substate World potential of free charge-current Thermodynamic coefficient Electromagnetic field Specific free energy Specific enthalpy Angular velocity of a frame or rigid motion "\Vorld vorticity tensor Unit tensor or matrix "Trident", i.e., arbitrary scalar, vector, or tensor J ump of W across a surface Impulse of W at an instant Material derivative of W Co-rotational time flux Convected time flux Displacement derivative Curl of a k-vector Natural divergence (In earlier sections, "div" and "curl" are applied only to three-dimensional vectors) Dual of a k-vector Naturalrotation of a k-vector 239 Place of definition or first occurrence (156.3) (153.3) (246.1) Sect. 283 (247.9) Sect. 270 (251.1) (251.1) (143.2), (143.14) (154.5) Sect. App. 3 (173.1) ( 194.13) (72.2) (148.7) (150.5) (179.5) Sect. 268 Sect. 268 Sect. 267 Sect. 268 11. Guide through this treatise. The chapters, and often even major subdivisions of the chapters, are largely independent of each other. We have tried to organize the ideas in such a way as to minimize cross-referencing. In most cases a reader with some experience in the subject will be able to start at any point he pleases. Chap. B collects and organizes all the researches we have been able to find on the kinematics of continuous media. Because of its completeness, it contains much dassie material found also in other works, but" some parts of it deserve particular notice. Subchapter I of Chap. B concerns the theory of a single deformation; that is, of finite strain and local rotation. Part f, concerning oriented bodies, presents apparatus for a type of physical theory as yet little studied but likely to be of future value. Subchapter II, which can be followed without reading its predecessor, is a treatise on the kinematics of continuous motions, i.e., deformations changing with time. Part e of this subchapter is directed especially toward interpretations in the flow of fluids and should be omitted by readers interested primarily in new theories; the essentials are given in the preceding parts. Part g, concerning relative motion, is conceptually the most important in the chapter and most apt to be enlightening in new studies of material behavior. Subchapter III concerns mass and momentum and includes general formulae of transformation under change of frame and also general solutions of the equation of continuity by means of stream functions of various kinds. 240 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 12. Chap. C contains the kinematical theory of slip surfaces, shock waves, and other kinds of surfaces of discontinuity. A general equation of balance, or "conservation law", was stated in Sect. 157 and was shown tobe equivalent to a certain differential equation in regions of sufficient smoothness; · at the end of Chap. C, this same equation of balance is applied to a surface of discontinuity and shown to be equivalent to a certain jump condition. Thus all the laws of classical physics may be derived by a uniform process from appropriate integral equations of balance. Chap. D presents the laws of classical mechanics and the general theory of contact forces or stress, in terms of which mechanical theories of continuous media are formulated. Subchapter III gives many theorems on mean values of the stress as determined from boundary conditions when the stresses themselves are not uniquely determined. Subchapter IV is intended to be an exhaustive treatise on solutions of the equations of motion or equilibrium of a general continuum by means of stress functions. Subchapter V derives all the variational principles of mechanics that partake of any considerable generality. The first part of Chap. E presents the general theory of energy in unexceptionable terms. The rest of the chapter concerns the more dubious subject of thermodynamics, set within field concepts. Entropy is taken as the primitive idea here, and emphasis is put upon exact formal properties of equations of state involving arbitrarily many variables. The rate of production of entropy is calculated. A general theory of the motion of heterogeneaus media may be collected from Sects. 158, 159, 215, 254, 255, 259, and 261 (cf. also Sect. 295). Chap. F, on electromagnetism, begins by a study of a tour-dimensional spacetime in which no geometrical structure is presumed. In such a space-time the basic laws of conservation of charge and conservation of magnetic flux are then stated. These laws turn out to have an entirely general form, applicable alike in classical electromagnetism and in relativistic theories. A generallaw of conservation of energy-momentum, including as a special case the classical momentum principle developed in Chap. D, is formulated. Additional assumptions that characterize classical or relativistic electromagnetism are then analyzed. Chap. G concerns constitutive equations defining particular materials. In contrast to the exhaustive earlier chapters, this one is selective. After a list of the principles used in forming constitutive equations, there follow brief sections on several classical or recent theories. These sections illustrate both the theoretical principles governing choit;e of constitutive equations and the immediate applicability of some of the general theory given in the preceding chapters. B. Motion and mass. 12. Scope and plan of the chapter. This chapter presents the kinematics of continuous media in a Euclidean space of three dimensions. We treat those aspects which are fundamental, either for intuitive grasp or for solution of problems in the various field theories. We omit the older type of kinematical research, where motions preserving the similarity of certain classes of geometrical figures are analyzed; this work is the subject of special Bibliography K at the end of the treatise. As to be expected of a general treatise on the kinematics of continua, this one is divided into three parts: Subchapter I presents the analysis of a single deformation, or strain; Subchapter II, of motion, which consists in a family of deformations continuously varying in time; Subchapter III, of mass. Sects. 13, 14. General co-ordinates. Invariant description. Duality. 241 The general theory is due primarily to EuLER (1745-1766) and CAUCHY (1823-1841); important special concepts and results were added by D'ALEMBERT (1749), GREEN (1839), STOKES (1845), HELMHOLTZ (1858), KELVIN (1849- 1869), E. and F. CosSERAT (1909), ZoRAWSKI (1911), and many others. While many expositians of the subject have been published, no other single work presents even a major part of the topics in this chapter1. I. Deformation. a) Deformation gradients. 13. Geometrical axiom. The proper content of this subchapter begins at Sect. 15, the function of the first two sections being only to present some geometrical preliminaries. Henceforth, except when the contrary is explicitly stated, we refer to Euclidean three-dimensional space, and we employ only real co-ordinates. Many of the results that follow hold in fact in Riemannian or even affine n-space and in complex co-ordinate systems; these generalizations will be obvious to those expert in geometry. An exact and straighttorward embodiment of the assertion that space is Euclidean is the existence of a reetangular Cartesian co-ordinate system2, to which all points may be referred. One such system, called the common frame, is laid down at the beginning, other systems of reference being defined later in terms of it. When co-ordinates in the common frame are to be written out, instead of using ZK and zk we generally write Z=iX+jY+kZ, z=ix+jy+kz, (13.1) where i, j, k are unit co-ordinate vectors in the common frame. While any one of the infinitely many possible reetangular Cartesian systems may be selected as "the" common frame, the choice is made once and for all. Thus, for example, we shall not use the notation ( 13.1) unless both points are referred to the same reetangular Cartesian system. 14. General co-ordinates. Invariant description. Duality. In Euclidean space it is permissible to restriet all considerations to reetangular Cartesian co-ordinates, to use absolute notations which eschew Co-ordinate systems, or to employ preferred curvilinear nets. Any of these three styles suffices for proving general theorems; each shows peculiar advantages in certain special problems; and each has its passionate and exclusive devotees. We prefer to use two independently selected generat curvilinear co-ordinate systems 3, one at Z and the other at z. To help 1 Surveys of certain parts are included in the following works, of which those distinguished by an asterisk possess the merit of originality, at least in part: KELVIN and TAIT* [1867, 3], ZHUKOVSKI* [1876, 7], E. and F.CossERAT* [1896, I] [1909, 5], JAUMANN [1905, 2], LovE* [1906, 5, Appendix to Chap. I], CAFIERO [1906, I], HEuN [1913, 4], ARIANO [1924, I] [1925, I] [1928, I], L. BRILLOUIN [1925, 2] [1938, 2, Chap. X], PLATRIER [1936, 8], SJGNORINI [1943, 6], ToNOLO [1943, 8], NovozHrLov* [1948, 18], GREEN and ZERNA [1950, IO] [1954, 7, Chap. II], MURNAGHAN [1951, 18], TRUESDELL* [1952, 2IJ [1953, 32] [1954, 24], Mrsrcu [1953. I9], DEFRISE* [1953, 8], NoLL [1955. I8, Chap. 1], DovLE and ERICKSEN [1956, 5]. . 2 We remind the reader of the notations explained in Sects. App. 2 and App. 3. 3 This scheme was introduced by E. and F. CossERAT [1896, I, Chap. IV], put into tensorial form by MURNAGHAN [1937, 7, § 1], and deve!oped by TRUESDELL [1952, 2I, §§ 12-22] [1953, 32], DoYLE and ERICKSEN [1956, 5, § 111], and TOUPIN [1956, 20, § 3]. GmBs [1875, I, p. 185] in using two reetangular Cartesian systems noted that "It is not necessary, nor always convenient, to regard these systems of axes as identical ... ". Handbuch der Physik, Bd. III/1. 16 242 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 14. the reader find his way through the literature, we summarize some other schemes in Sect. 66B. Our formalism of course includes the special case when all points are referred to the common frame. Thus any reader unfamiliar wi th tensor analysis may follow most of the development by interpreting covariant derivatives simply as partial derivatives in a single reetangular system 1 . Let Z and z be points, and in the neighborhood of each let general curvilinear co-ordinates X and x be given by their equations of transformation from the common frame, as follows (Fig. 1): X =X(Z), Z =Z(X); x = x (z) , z = z (x) . ( 14.1) Since the choices of co-ordinates employed at X and x should be entirely free and independent, the theory should be invariant under general changes of co-ordinates, X*= X* (X) and x* = x* (x). (14.2) This suggests use of the double tensors 2 defined in Sect. App. 15, with the interpretation y given there. Throughout this subchapter, we shall describe conditions at x and X in symmetrical formalism, invariant with respect to choice of co-ordinates. Majuscule letters, in general, will refer to X; minuscules, to x. Often weshall give a particular procedure for constructFig.L Independentgeneral curvilinear co-ordinates. ing quantities T~:::lJf~:::P' in terms of information at two arbitrary points X and x. Now suppose we interchange the roles of X and x systematically but otherwise follow the same procedure. We thus obtain dual quantities t";;::.f,~:::lrJ,. Our symmetrical notation achieves the economy expressed in the following, purely formal, principle of duality 3 : In any given equation, mafuscules and minuscules may be interchanged. In applying this rule, the interchange must be effected both in the kernel letters and in the indices. Henceforth, except in the most important cases, we shall not write down the duals. Also, for special reasons we shalllater introduce special notations departing from the principle of duality, the most important of these being u, E, and R. For an easy example, consider the following prescription: Let P be the position vector from a fixed point to X. The dual prescription is: Let p be the position vector from a fixed point to :r. The resulting components pk obviously have no general relation to the components pk= gkPK of P shifted to :r. Similarly, if certain fields T, a, and A have been shown to satisfy the identity ykK= ak A K, then dual definitions willlead to the identity tK k= AK ak· The components tK k of the double field t, in general, will bear no relation to the corresponding converted components ofT; that is, yKk=g~gfJ{TmM=l= tKk· The line elements at X and x will be written ( 14.3) 1 Once and for all, however, we warn such a reader that deformation is a general point Iransformation (Sect. 15). Thus, willy nilly, he is employing the idea of general Co-ordinates, for this idea is inherent in the theory of deformation. 2 MICHAL [1947, 9, Chap. XIV] was the first author to apply them to the theory of deformation. No earlier work was expressed in fully invariant form. 3 In essence this principle was certainly known to CAUCHY and was used by FINGER [1892, 4]. The nearest we have found to a formal statement is a remark of LE Roux [1911, 9]. ects. 15, 16. Continuity. 243 where G and gare given in terms of the transformations (14.1) from the common frame to the arbitrarily selected co-ordinate systems by the usual formulae: G _ _ {) azP azo KM- gKM- 'PQ oXK oXM ' (14.4) The Riemann tensors RKMPQ and rkmpq based upon the components GKM and gkm vanish identically. Since X and ~ are points in the same space, it is obvious that the components gkm and GKM are components of the same metric tensor. That is, if G is the dual of g, then GKM = ~M = gi. gli{ gkm and Gkm = gkm• as is proved formally in Sect. App. 15. We are thus justified in avoiding the kernel index G entirely, as usually we will, but in some cases it helps to avoid ambiguity if we write G11 , G12 , etc. for the covariant components of the metric tensor in the co-ordinates at X; 15. Deformation. This subchapter constructs the mathematical apparatus describing the deformation of a portion of matter from one configuration into another (Fig. 2). Let a typical point Z be carried into z: Z=z(Z}, Z=Z(z). (15.1) In explicit notation, ( 15.1) reads 1 x = f(X, Y, Z), etc.,} (15.2) X= F(x, y, z), etc. This transformationwill be called the deformation. Our object in this subchapter is to analyse the major properties of the deformation. 0 ------ . )(' Fig. 2. Defonnation. With quantities associated with Z we shall by precise definitions set into correspondence certain quantities associated with z, saying then that these quantities are deformed by (15.1). As Z runs over a set of points fJt, its image z under the mapping (15.1) runs over a set of points, say i. In physical contexts the deformation of 91 into ~ will be produced by applying forces to the material. To promote physical interpretation we shall sometimes speak of fJt and i as the undeformed and the deformed material, respectively. More often, however, weshall prefer a symmetrical nomenclature for a mathematically symmetrical situation and shall speak accordingly of the material about Z and the material about z. 16. Continuity. We now lay down the a~iom of continuity: Throughout fJt and i, the deformation (15.1h and its inverse (15.1) 2 are single-valued and as many times continuously dilferentiable as required. Not only does this axiom cast aside deformations so irregular as to be useless in physics, but also it implies as a special case the permanence of matter: No region of positive finite volume is deformed into one of zero or infinite volume. For this permanence it is necessary that the Jacobians of (15.1) do not vanish; weshall not lose essential generality by assuming oo > izJZI > o. (16.1) 1 This is the notation of EuLER [1762, 1] [1770, 1, § 100]. The now more common notation a, b, c for X, Y, Z in this connection was introduced by LAGRANGE [1788, 1, Part II, Sect. II, ~ 4]. 16* 244 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 16. It follows also that ( 15.1) carries every region into a region, every surface into a surface, every curve into a curve. Another corollary of the axiom of continuity is the principle of impenetrability 1 : One portion of matter never penetrates within another. But in addition, the assumption of continuity excludes many singularities of physical interest. Such exclusion is legitimate because in the physical problems of the classical field theories singularities are confined to points, lines, or surfaces over an interval of time, or to instants: In a word, they are isolated. Thus they may be given special attention. Generally they fall under one of two types: boundary surfaces or wave surfaces whose specification is included at least in part in the definition of the problem, and singular points engendered by the differential equations of the particular material. The theory of the latter is outside the scope of the present treatise, which describes only general features common to all materials; the form er are analysed in Chap. C. The assumption of continuity restricts the results of the present chapter to regions where phenomena are occurring smoothly. In any particular application, this assumption will generally be valid only in portions of the entire space under consideration. For most of the theorems which follow, it is enough that the deformation possess two continuous derivatives; the reader whose taste favors weakening hypotheses of smoothness may easily satisfy it, though not sufficiently to include in the results of this chapter the essentially different discontinuous motions analysed in Chap. C. We now generalize the terms and assumptions used in Sect. 15 and hitherto in this section so as to allow use of the general co-ordinate systems introduced in Sect. 14. We refer to X and x as co-ordinates in the undeformed and deformed material, respectively. From (15.1) 1 and (14.1) 2 we have x =x(z(Z(Xl)), or, for short, in co-ordinate form, x = x(X), X =X(x); xk = jk (XI, X2, xa), XK = FK (Xl, X2, X3)' k = 1, 2, 3; } K=1,2,3. ( 16.2) ( 16-3) We employ only co-ordinate systems such that in the axiom of continuity we may replace z by x and Z by X. In particular, oo> Jx/XJ >0. ( 16.4) Of frequent use is the absolute scalar I whose numerical value is the Jacobian I z/ZI: By ( 16.4) follows I-- jdet gk~ I xfXI = Jz/ZJ. Vdet gKM (16.5) ü a2 > a 3 • Let so that bl = al-t (al + aa) = t (al- aa)' l b2 = a2 - t (a1 + a3 ), b3 = a3 - t (a1 + a3) = t (a3 - a1) = - b1 • Clearly b1>b2>b3 , b1>0. From (23.1) and (23.2) we have u 0 = u + t (a1 + a3) v, u ~ b ·V; (23 .2) (23-3) ( 23.4) thus from given u and v one can construct u 0 . We thus concentrate on constructing u for given v. Refer b to a reetangular Cartesian principal co-ordinate system zk. In this system, and (v1)2 + (v2)2 + (va)2 = 1 u2 ""' u · u = v · b 2 • v = (b1 ) 2 [(v1) 2 + (v3) 2] + (b2) 2 (v?) 2 ') = (b1)2 + [(bz)2- (b1)2J (v2)2' = (b1)2 + [(bz)2- (bl)2] cos2 , (23. 5) where is the angle determined by v and the proper vector of b corresponding to b2 , cos ""' v2 • The angle 1p determined by u and this proper vector is given by cos1p = u2 ju = b2vd(b1)2 + [(b2)2- (b1)2J cos2}-~. } = b2 cos {(b1) 2 + [(b2) 2 - (b1) 2] cos2 } -§. (23 .6) Projecting u and v on a plane z2 = const yields the components (b1 v1 , 0, - b1 v3 ) and (v1, 0, v3) respectively, so the projected vectors make equal angles with the z1-axis, supplementary angles with the z3-axis. The above facts imply that u is given by the following construction: In the plane of the proper vectors corresponding to the two larger proper numbers b1 and b2 , draw a unit vector making the same angle with the proper vector corresponding to b2 as does v. In the same plane, draw a vector of magnitude {(b1)2- [(b2) 2- (b1) 2 ] cos2 }~ which makes the angle 1p, given by (23.6), with the proper vector corresponding to b2 , and which lies on the same side of this proper vector as does the first vector. Rotate the two vectors thus constructed ab out the proper vector corresponding to b2 through equal but opposite angles until the first vector coincides with v. The second vector will then coincide with u. 24. MOHR'S mapping. For any symmetric tensor a in three dimensions, one has (24.1) where X(v) and Y(v) are, respectively, the normal component and the maximum shear component of a corresponding to the unit vector v. Eqs. (24.1) define a mapping of the unit 1 [1877, 1]. An alternative procedure is given by GuEST [1939, 9]. 254 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 24 sphere v · v = 1 into the X- Y plane. Since (24.1) 2 is unaltered if Y is replaced by - Y, the image of the unit sphere is a region t symmetric with respect to the X-axis. In reetangular Cartesian principal Co-ordinates, (24.1) reads X = a1 (v1)2 + aa (va)a + aa (va)2 • Y2 = (a1)2 (vl)2 + (aa)2 (v2)2 + (aa)2 (va)2- X2 • } (v1)2 + (v2)2 + (va)2 = 1 • (24.2) whence (24.)) The image of a curve on the sphere whose radius vector makes a con~tant angle with the proper vector of a corresponding to ab• i.e., a curve vb = const, is thus a circle with center on the X-axis at X= i(ac +ab). (b, c, b =1= ), whose squared radius is given by the right member of (24.3) 2 • From (24.2)3 and (24.3). the region t consists of the points interior to the largest and exterior to the two smallest of the three circles given by (X _ ab ~ a, r + y 2 = (ab ~ a, r b =I= c (24.4) points on the boundary being included in t. If a1 =a2 =a3 , t reduces to the point (a1 , 0). If two, but not three, proper values of a coincide, t reduces to a circle. In all other cases, its area is non-zero. The circles (24.4) -more precisely, their centers-uniquely determine the proper numbers of a. For two tensors a and b suchthat b = o · a · o-1, where o is orthogonal, the corresponding regions t coincide; conversely the. boundary of t determines a to within an orthogonal transformation. Adding to a a tensorproportional to 1 does not alter Y but adds a constant to X. The effect is thus to translate the region t parallel to the X-axis. Since X and Y arehomogeneaus of degree one in a, multiplying a by any non-zero scalar factor effects a uniform expansion 1 of 1. It is easy to show, conversely that, if the regions 1 corresponding to two tensors a and b differ only by a translationparallel to the X-axis and a uniform expansion, then, assuming a =1= 0, there exist scalars A and B and an orthogonal tensor o such that 2 b = A o · a · o-1 + B1. In any number of dimensions (24.1) defines a mapping of the unit sphere into the X- Y plane. In two dimensions, the unit circle v · v = 1 is mapped onto the circle (24. 5) In this case (24.6) so the center and radius of the circle (24.5) are easily obtained from the components of a given in any co-ordinate system3 . There seems tobe no equally simple method of constructing the region t in three dimensions from components given in an arbitrary co-ordinate system. There is an extensive Iiterature concerned with applications of MoHR's mapping, much of it to theories of yield or rupture of solids 4• By making transformations of the formX*=X* (X, Y, a1 , a2 , a3). Y*= Y* (X, Y, a1 , a2 , a3). one can obtain other plane representations of a which are essentially equivalent to MoHR's, but which may be more convenient for some purposes. CHARREAU 5 has considered the 1 The facts thus far noted are contained in papers by MoHR [1882, 3] [1900, 8] [1914, 8, pp. 192-235]. JUNG [1947, 7] gives an earlier (1866) reference to CuLMANN, which may contain some of the material given above. We have been unable to see CuLMANN's work. 2 KLOTTER [1933, 7] discusses the relation between the maps of a, its deviator 0a, and the tensor b defined by b""'Aa + B1. 3 Further discussion of the plane case is given, e.g., by NADAI [1950, 20, Chap. 10], FRAGER and HODGE [1951, 20, Chap. 5], DEWULF [1947, 4], FADLE [1940, 10], and WISE [1940, 18]. 4 Cf. NADAI [1950, 20, Chap. 15], ToRRE [1946, 8] [1951, 26], BöKER [1915,1], v. KARMAN [1911, 8] [1912, 5] for discussion and further references. 5 [1945, J]. Sect. 25. Stretch, extension, elongation, and shear. 255 case X*= X, Y*= k (X2+ Y2), where k is a positive constant, Y* then being proportional to the squared magnitude of the vector a · v, CHARREAU and DuPONTl discuss the case it being assumed that a1 > a2 > a3 . The latter transformation maps the unit sphere v · v = 1 onto a triangular region in the X*, Y* plane. b) Strain. 25. Stretch, extension, elongation, and shear. The change in length and relative direction occasioned by deformation is called, loosely, strain2• By (20.3), an element of arc dX at Xis deformed into an element of arc d~ at ~- Since (20.3) is linear and homogeneous, the ratio of lengths dxfdX is independent of the original length d X and hence for given displacement gradients is a function only of the direction of dX. Let N be a unit vector along dX at X. Then the stretch A.(N) in the direction of N is defined by dx A(N) = dX' while the extension 15(N) is defined by 15(N) = A(N) - 1 0 (2S.1) (25.2) For the range of possible values of A. and 15, the axiom of continuity (Sect. 16) yields 0 < A(N) < oo, - 1 < 15(N)< tX>. (25.3) In view of the remarks at the beginning of Sect. 20 we may interpret (25.1) by means of the length L of a finite arc~ emanating from X and tangent there to N, and the length L +L1 L of the arc c at ~ into which ~ is deformed. Wehave , L" L + LlL ( ) A(NJ= tm- -- 25.4 L-->0 L We agree not to dualize the concept of stretch. That is, letting l+L1l be the length of an arc c' emanating from ~ and tangent there to n, while l is the length of the arc ~ at X from which c' was deformed, we set If we choose c' as c, then l = L and L1 L = L1l, and we get A(N) = A(n) • (25 0 5) (25.6) This identification is not necessary, however, and it is convenient to specify stretch sometimes in terms of directions at X, sometimes in terms of directions at ~. The important fact just established is that the totality of stretches is the same, whether the undeformed or the deformed material is used for reference. Elements parallel to N are lengthened or shortened according as A(NJ > 1 or A.(NJ < 1, or according as 15(NJ is positive or negative. Doubling the length corresponds to A. = 2, 15 = 1 ; halving the length, to A. = i, d = - i. Thus A. is 1 [1945, 1]; [1944, 5]. 2 This term is due to RANKINE [1851, 1, Sect. I, § 5], bothin generaland for the tensor E satisfying the relations (57.10). 256 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 25. multiplicatively symmetric with respect to lengthening and shortening but not additively so, while ~ is not symmetric at all. An additively symmetrical measure of strain I (Ä.) must satisfy 1(1/Ä.) =-I(Ä). (25.7) Among the infinitely many smooth functions that conform to this requirement arel I(Ä) =log Ä., 4 I (Ä.) = - arc tan Ä. - 1 , n - oo < 1 (A) < + oo, l - 1 < I (A) < + t . (25.8) For doubling the length these measures have the respective values log 2 and 4/n arctan 2- 1. The difficulties which stand in the way of using such measures as thesewill be mentioned in Sect. 33. The stretch itself furnishes the most immediate measure of strain and is the basic concept in all serious studies of the subject. Fora given pair of points in a given deformation, there are in general infinitely many stretches, but they are not independent; as we shall see in Sect. 27, a properly selected set of three suffices to determine them all. Returning to the definition of extension, if before calculating the ratio of lengths we project d~ upon the direction of dX, the result is called the elongation e(N) in the direction of dX: _ dxKdXK B(N)=(d~-1. (25 .9) Thus for elements which are turned through an obtuse angle the elongation is always negative. Elongation is not necessarily a measure of change of shape. For example, in a rigid rotation through a positive angle, every extension is zero, but every element not parallel to the axis suffers a negative elongation. More generally, from the definitions (25.1), (25.2), and (25.9) follows (25.10) it is possible for e(N) to assume any finite value. In the case of an element whose direction is unaltered by the deformation, we have e(N) = ~(N). The condition e(N) = -1 is necessary and sufficient that the element be turned through a right angle, irrespective of its stretch. An identity connecting elongation and stretch is given as (35.3) below. Let dX1 and dX2 be two elements at X, and let them be deformed into the elements d~l and d~2 at ~. Let unit tangents be Nl' N2' nl' n2 0 If e(N,, N,) is the angle between dX1 and dX2 , while f9(N,,N,) -y(N,,N,) is the angle between d~1 and d~ • then Y(N,,N,)• the decrease in angle, is the shear of the directions N 1 , N2 . The concept of shear is not dualized. Thus with n1 , n2 as defined we have (25.11) The axiom of continuity forbids a shear equal to a right angle: I y I<-! n for all pairs of directions. From the linearity of (20.3) follow certain symmetries. First, the stretches of oppositely directed elements are equal: (25.12) 1 According to MEHMKE [1897, 5, §I], (25.8h was used by IMBERT in 1880 to describethe extension of rubber. Cf. LuDWIK [1909, 7, Pt. 1, §I]. Sect. 26. The deformation tensors of CAUCHY and GREEN. 257 Second, the shears of oppositely directed pairs of elements are the same, and reversal of one of a pair of elements changes the sign of the shear: (25.13) In most cases there is thus no loss in generality if we take shear as an acute angle: 0 ~ y < {- n. 26. The deformation tensors of CAUCHY and GREEN. The theory of finite strain is the creation of CAUCHY1. He observed that (20.2) put into the formula d52=gKMdXKdXM for the squared element of length at X gives (26.1) c is Cauchy's deformation tensor 2• As we shall see, aU changes of length and angle are easily calculated from the values of the components ckm· The formulae dual to (26.1) are (26.2) C is Green's deformation tensor 3• Since c and C are metric tensors, their matrices are non-singular. In the common frame CAUCHY's and GREEN's tensors have the familiar forms 4 c x x = ( 1 + :; r + ( t: r + ( ~; r. CX'" = (1 + Ou). ou_ + jv_ (1 + 01!_) + OW OW • ax oY ax aY ax aY · (26.3) ( ou )2 ( ov )2 ( ow )2 Czz = 1 - -ox + Bx + fiX- ' c =-(1- ~)~-- _av(1- 81!_) + j_!l!_!_UJ_ zy OX oy OX oy ox oy ' ... ' where u, v, w are the components of the displacement vector (19.1). The tensors c and C are different tensors; i.e., ckm=I=Ckm in general. A formula relating c to Cis given as (37.6) below. Since C and c serve to measure all lengths, it is obvious that the stretches and shears can be expressed in terms of them. In fact, for the stretches in the directions of the unit vectors N and n, by (25.4) and (25.5) we get (26.4) (the dissymetry in the definition of stretch accounts for the failure of duality here). Thus the normal components (Sect. App. 45) of C and c in the directions N and n, respectively, are the squares and reciprocal squares of the stretches in those directions. If we let A1 and ./.1 be the stretches and Ll 1 and <51 the extensions in 1 [1823, J] [1827, 2] [1841. J]. 2 [1827, 2, Eqs. (10), {11)]. Cf. also LE Roux [1911, 9]. 3 While the tensor C appears formally in work by PIOLA [1836, I, Eq. (139)] [1848, 2' ~ 34], its components were first interpreted by GREEN [1841, 2, pp. 295-296]. Cf. also CAUCHY [1841, J, §I, Eq. (15)]. 4 Corresponding explicit forms for cylindrical and revolution co-ordinates are written out by MAZZARELLA [1954, 15]. Handbuch der Physik, Bd. III/1. 17 258 C. TRUESDELL and R. TOUPIN: The classical Field Theories. Sect. 26. the directions of the X1 and x1 co-ordinate curves, then1 (26.5) Since there need be no connection between the choices of ;E co-ordinates and of X co-ordinates, there is no simple relation between A1 and .A.1 . When it is necessary to observe this distinction, we use the symbols A and L1 for stretch and extension of elements at X. The shear J'(N1, N2) of the directions NI, N 2 may be calculated from cos e(N1,N2 ) = gKM Nlli~M, (€) ) _ ... CKMNIK ~M cos (NJ, N2) - J'(NJ, Nz) - -V-- p V =R 5 , CpQ ~ ~Q CRs Nz Nz (26.6) = 1 c NK N.M Ä(NI)Ä(N2) KM I II . Thus the shear component (Sect. App.45) of C for the directions NI andN2 is insufficient to determine the shear of those directions. In fact, if we write S for the right-hand side of (26.6)a, we obtain siny(N1,N2 ) = Ssin8(N1,N2 )-V1-S2 cos8(N1,N2 ). (26.7) From (26.6), a necessary and sufficient condition that the shear of the directions NI , N 2 be zero is that the ratio of the corresponding shear components of C and g equal the product of the stretches in the directions N 1 and N 2 • For orthogonal shears (26.7) reduces to (26.8) so that the vanishing of an orthogonal shear component of C is necessary and sufficient for the vanishing of the corresponding orthogonal shear. When applied to the directions of the X1 and X2 Co-ordinate curves, (26.6) and its dual become (26.9) where I;, 2 is the shear of the directions tangent to the X1 and X2 co-ordinate curves at X, while y12 is the shear of the directions which are deformed into the tangents to the x1 and x2 curves at ;E. Since the two systems of co-ordinates may be chosen independently at will, there is no simple relation between I;, 2 and y12 • In the orthogonal case, i.e., 812 = !n and 012 -y12 =in, (26.9) reduces to . ." 1 cl2 0 ,, cl2 sm _L 12 = A A . v-v- ' sm ?'12 =- JL1 JL2 V V 0 1 2 Gn G22 gn g22 (26.10) From (26.1 Oh we see that in the orthogonal case the shear component I C 121/V G11 V G 2 2 constitutes an upper or a lower bound for sin I;, 2 according as surface area in the X 3 co-ordinate surface be increased or decreased in deformation. 1 Theseinterpretationsand (26.10) are due to GREEN [1841, 2, pp. 295-296]. Cf. E. and F. CossERAT [1896, 1, § 3]. While they may now be read off from familiar results in differential geometry, GREEN obtained them long before general co-ordinates were introduced in three dimensions, and in fact much of the theory of curvilinear co-ordinates grew out of continuum mechanics. Sect. 27. The strain ellipsoids and the principal stretches. I: Geometrical treatment. 259 Shear is a property of a pair of directions. When these directions are selected as co-ordinate directions, it is obvious that the results obtained are not independent of the choice of co-ordinates. Shear is not an invariant concept unless the directions to which it refers are kept fixed. As we shall see in the next section, in any deformation at any points X and ~ it is always possible to find a system of orthogonal co-ordinates in which the Co-ordinate shears all vanish. The maximum shears will be determined in Sect. 28. A deformation is rigid if the distance between every pair of points is left unchanged. Necessary and sufficient that a deformation be rigidisthat at each point C=C=l. (26.11) At a single point where (26.11) is satisfied weshall say the deformation is locally rigid; when no confusion is likely, the qualification "locally" may be omitted. In order for every portion of a curve '?/ to be carried into a corresponding portion of a curve c of the same length, it is necessary and sufficient that the stretch in the direction of the tangent to '(]' be 1 at each point. A differential equation for inextended curves has been derived and discussed by CASTOLDI1. 27. The strain ellipsoids and the principal stretches. I: Geometrical treatment. CAUCHY created his representation of symmetric tensors by quadric surfaces, which has been explained in Sect. 21, as a means of visualizing strain, inertia, and stress. The quadrics of C and c weshall call the strain ellipsoids at X and ~. respectively2• That these quadrics are indeed ellipsoids may be proved in many ways. The simplest proof consists in noting that by the axiom of continuity, there exists a sphere about ~ which contains the image of every sufficiently small sphere about X, and the only quadric contained in a sphere is an ellipsoid. This proof 3 involves the undesirable complications mentioned at the end of Sect. 20. An algebraic proof was given in Sect. App.43. For a formal yet geometric approach, consider the differential elements dX which sweep out a sphere of radius K at X, and hold K fixed. By (26.1) these dX are deformed into differential elements d~ which sweep out at ~ the quadric surface (27.1) Since g is a Euclidean metric tensor, the firstform is positive definite, and hence the quadric of the second form is an ellipsoid. Select a vector dX1 at X, and let dX2 be any vector in the plane perpendicular to dX1 , so that gKM dXf dX!f =0. By (20.2) and (26.1h follows (27.2) Now the gradient vector of the strain ellipsoid (27.1) at the terminus of d~1 is 2ckm dx'J'. Since (27.2) asserts that d~2 is perpendicular to this gradient, we have derived Cauchy's firstfundamental theorem4 : An element of arc and its normal plane in an infinitesimal sphere at X are deformed into an element of arc and its confugate plane in the strain ellipsoid at ~. This theorem is often phrased 1 [1950, 3]. 2 The quadric of C was called the strain ellipsoid by KELVIN and TAIT [1867, 3, § 160]; the quadric of c, the reciprocal strain ellipsoid by LovE [1906, Ii, § 6]. 3 This and other arguments based on geometrical considerations regarding infinitesimals were given by ST. VENANT [1864, 4] [1880, 10]. 4 [1828, 3, Ths. I and II]. Let .'JI'be the planeelementnormal to dX, and Iet it be deformed into p; according to GALLI [1933, Ii], for small deformation the normaltop subtends equal angles with d;n and dX, but this relation does not hold for arbitrary strains. 17* 260 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 2i. as follows: Perpendicular diameters of an infinitesimal sphere at X are deformed into conjugate diameters of the strain ellipsoid at ~. and conversely (Fig. 4). Cf. also the presentation in Sect. 21. Now a quadric has three diameters that are perpendicular to their conjugate planes, these diameters being its axes. The axes of the strain ellipsoid at ~ and X are called the principal axes of strain at ~ and X, respectively. Thus it follows from CAUCHY's theorem that there exists an orthogonal triad at X which Fig. 4. Perpendicular diameters of a sphere at X are deformed into conjugate diameters of an ellipsoid at a:. is deformed into an orthogonal triad1 at ~. It is but a change of words to say that there exists at ~ an orthogonal triad which results from the deformation of a certain orthogonal triad at X. Therefore, by the property dual to the foregoing, the original orthogonal triad at X is a set of principal axes of strain at X. Thus follows another corollary of CAUCHY's theorem: The deformation rotates the principal axes of strain at X into the principal axes of strain at ~ (Fig. 5). For derivation of the last corollary, we have tacitly assumed that the lengths of the axes of the strain ellipsoid at ~ are all unequal, so that the principal axes Fig. 5. The strain ellipsoids at X and a:, showing the rotation of the principal axes of strain. "the" axes at ~ those directions into deformed. of strain are uniquely determined. If the ellipsoid degenerates into a spheroid or a sphere, an infinite number of orthogonal triads are deformed into orthogonal triads. Any one of these may be selected and called "the" principal axes of strain. In this case, so that the last corollary will remain correct as stated we take for which "the" axes at X are The stretch in a given direction was defined in Sect. 25. Since the diameters of an infinitesimal sphere at X are deformed into diameters of the strain ellipsoid at ~. the ratios of corresponding diameters are exactly the stretches. That is, the stretches in the several directions at a point vary as the distance from the point to the surface of the ellipsoid. Thus, as CAUCHY 2 remarked, the stretches are distributed symmetrically about the principal axes: in particular, any stretch that is not a principal stretch is experienced either in no direction or in infinitely many directions. Cf. the remarks at the end of Sect. 21. Similarly, the ratios of diameters of a sphere at ~ to corresponding diameters of the strain ellipsoid at X are the reciprocals of the stretches. Thus the Iongest axis of the ellipsoid at X is rotated into the shortest axis of the ellipsoid at ~. and if for one of the ellipsoids there is an axis of intermediate length, it is deformed into one of intermediate length for the other. Hence the strain ellipsoid at X degenerates to a spheroid or a sphere if and only if the strain ellipsoid at ~ does so. The interpretation of stretches as proportional to diameters of an ellip1 By applying WEINGARTEN'S conditions (App. 48.3), TONOLO [1949, 32] has obtained necessary and sufficient conditions that the orthogonal triple of plane elements so determined envelop a triply orthogonal system of surfaces. 8 [1823, 1] [1827, 2]. Sect. 28. The strain ellipsoids and the principal stretches. Il: Algebraic treatrnent. 261 soid yields at once Cauchy's second fundamental theorem1 : At any point X there exists a direction in which the stretch is not less than in any other direction; a second, perpendicular to it, in which the stretch is not greater than in any other direction. The stretches in these two directions and in a mutually perpendicular third direction are uniquely defined pure numbers, the principal stretches A1 , A2 , A3 , satisfying (27.3) If A1 > A2 > A3 , the stretch A2 is a minimax. The lengths of the axes of the strain ellipsoid at x stand in the ratio A1 : A2 : A3 to one another; the lengths of the corresponding axes of the strain ellipsoid at X, in the reciprocal ratios. The directions in which A 1 , A2 , A3 are the stretches are the principal directions of strain at X; alternatively, they are the principal directions of strain at x. The principal stretches are the most important quantities connected with the strain. Their magnitudes determine the range of stretches, since every stretch A must satisfy ~ A~ A3 • If we know the attitudes of the principal axes of strain, either at x or at X, the values of the three principal stretches enable us to construct the strain ellipsoids and hence to determine the stretch in general as a function of direction. The extensions ~a corresponding to the principal stretches Aa are called the principal extensions. For a deformation to be locally rigid it is necessary and sufficient that a=1,2,3. (27.4) In a rigid displacernent, the strain quadrics are spheres, but the converse is not true. Indeed, suppose the quadric at X be a sphere. Then all stretches are equal, and C KM= Ä2gK M, so that frorn (26.6) follows 2 Y(NJ, N2) = 0. Thus the deforrnation is conforrnal. Again the converse is not true, since an inversion is conforrnal but need not produce equal stretches in all elernents3. 28. The strain ellipsoids and the principal stretches, 11: Algebraic treatment. While the geometric proofs given in the foregoing section are rigorous, some readers will prefer algebraic demonstrations, which in any case bring with them formal apparatus useful later as well as some additional results. We preserve unity of treatment by making no reference to any of the theorems already derived by geometric means. However, we leave to the reader the geometrical interpretation of the formulae we now derive. Since by its definition (26.1) 2 the tensor c is real and symmetric, we may apply to it a now celebrated theorem of CAUCHY, first proved in this very connection 4 (cf. Sect. App. 37): The proper numbers c0 arereal; proper vectors corresponding to distinct proper numbers are orthogonal; and there exists a reetangular C artesian frame such that at x llc!ll = C1 0 0 c2 0 . c3 (28.1) Since c is a Euclidean metric tensor, we have ckk > 0 in all co-ordinate systems, and hence by the theorem just stated it follows in particular that Ca > 0. The 1 [1841, 1, §I] (in part, [1823, 1] [1827, 2]). z CrsoTTI [1944, 3, § 3]. 3 Ün the basis of theorerns of CAUCHY and LIOUVILLE, SIGNORINI [1943, 6, ~~ 15-16] develops the properties of conforrnal strains. 4 Andin connection with the tensor of inertia, see Sect. 168 [1828, 1, pp. 15-17] [1841. 1, Th. VIII]. 262 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 28. directions of the co-ordinate axes in a system for which (2801) holds are principal directions of strain at ~- Dualstatements hold for C, whose proper numbers are C0 o Letl the vectors n0 be an orthogonal unit triad in the directions of the principal axes of strain at ~. so that while gkm n~ ng' = c5ab o Set N K-XK k/V- a = ;kna Ca 0 Then by (2602) 2 , (1702), and (2802) follows and gKMN{- Nr =gKMX~Xt!,n~nl:'fVcac{,, l = ckm n~ nl:'fVc;i:-1,= cagkm n~ ngjVca cb, = c5nb o (28.2) (28.3) (28.4) (28o5) The results (28.4) and (2805) assert that the transformation (28-3) carries the orthogonal triad of unit proper vectors na of c with proper numbers Ca into an orthogonal triad of unit proper vectors Na of C with proper numbers Ca given by 1 C0 = --0 Ca (2806) The formula (28o3) sets the principal directions of strain at ~ into one-to-one correspondence with the principal directions of strain at X, the dual formula being (2807) Since the principal stretches An are defined as the stretches in the principal directions of strain, by (2801) and (26.4) we get (2808) incidentally verifying (2806)0 Hence by (2503) follows O (29.1) where we are using the notation of associated components. To interpret e-r, we calculate the magnitude of the element of area da at x in terms of the components dAK at X. By using (20.8), (30.5), and (29.1) 2 we obtain (da) =J2gkmX~mx;-;,.dAKdAM, 1 = IIIc CK M dAK dAM. J (29.2) Comparing this result with (26.2) 1 shows that the tensor IIIc e-r measures the ratios of areas in the same sense that the tensor e measures the ratios of lengths 2• The principle of duality enables us to assert an analogaus interpretation for IIIc e-r. If, in analogy to the theory of deformation of lengths, we were to construct a theory of deformation of areas, the formula (29.2) and its dual would enable us to proceed in steps parallel to those we have based upon (26.1) and (26.2). This fact enables us to assert a second principle oj duality: In any proposition concerning change of lengths and of angles between elements of length expressed in terms of the tensors e and c, a valid Proposition results if we replace "length ", e, and c by "area ", IIIc e-r, and IIIc e-r, and conversely. In particular, since e-r and e are co-axial, the extremal changes of area occur in elements normal 1 The tensor c-1 was introduced by PIOLA [1833. 3, § 5] [1836, 1, Eqs. (142), {143), (152)]. FINGER [1894. 4, Eqs. (12), (31)] introduced IIIc c-1 and c-1. Cf. also [1892, 4]. 2 TONOLO [1943, 8, § V.4] derived a formula which is essentially (29.2h; by substituting in it the dual of (37 .8), he observed that the rotation tensor cancels out, and hence the ratio dafdA can be calculated from the components C KM, as is immediate from our result (29.2) 2 . In this paragraph and the next two we follow TRUESDELL [1958. 10]. 264 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 30. to the principal axes of strain, and the greatest (least) change of area occurs in the element normal to the axis of least (greatest) stretch. In fact, the extremal changes of area are AaAb, a=f=b, satisfying .?. .1. ~.?. .?. ~.?. .?. • In the notation of associated components, (26.1) 2 and (26.2) 2 read (29.3) From this result and from (29.1) we have~ ~"' - (jP X + (jP X a//k"'_- (jm k;K + (jk m;K axK - k K;m m K;k, 0 p - p X pX , ;p X;K Ckm;p = XK;k Xfmp + XK;m X[\p, (29.4) (}km= xk;M xm + xm;M xk ;K ;MK ;MK· From (29.1h and (29.3) 2 we get (29.5) But by (App. 38.10) we have (29.6) Combining (29.5) and (29.6) yields an identity due to FINGER2 : (29.7) TRUESDELL3 noted that each of the deformation tensors may be expressed as the trace of a matrix of derivatives of one set of deformation gradients with respect to one another. The formulae areimmediate consequences of (17.5): k axK;k Cm=- axm;K' -lk oxk;K cm=-axK;m' (29.8) and their duals. By the theorem of SPOTTISWOODE given in Sect. App. 40, we may order the -1 -1 proper numbers Ca and Ca of c-1 and C-1 in such a way that (29.9) The tensor c-1 is the deformation measure most commonly used in modern work on finite deformation, since ( 1) it is an ordinary symmetric tensor whose principal axes are the principal axes of strain in the deformed material (this distinguishes it from x~K, which is a non-trivially double field with 9 independent components rather than 6, and from C, whose principal axes are the principal axes of strain in the undeformed material), (2) it is a quadratic polynomial in derivatives with respect to points in the undeformed material (this distinguishes it from c), and (3) its proper numbers are the squares of principal stretches (this also distinguishes it from c). 30. The strain invariants. The principal invariants Ia, Ila, lila of a tensor a have been discussed in Sect. App. 38. By (28.8) and (29.9) we have the following 1 Cf. BONVICINI [1932, 3] [1935, 2]. 2 In the form given by FINGER [1894, 4, Eq. (34)], the invariants of c are replaced by those of C through (30.2). 3 [1952, 20, § 14]. Sect. 30. The strain invariants. 265 expressions for the principal invariants of C, c-1, c, and c-1 in terms of the principal stretches 11.0 and principal extensions /)0 : Ic = Ic-1 = ll.j + A§ + II.~ = (1 + b1) 2 + (1 + b2) 2 + (1 + b3) 2 , Ilc = IIc-1 = II.] A§ + A~ II.~ + II.~ 11.], =(1 +b1) 2 (1 +b2) 2 +(1 +bJ2(1 +153) 2 +(1 +1.53) 2 (1 +151) 2 , IIIc= IIIc-1 = ;tpp~ = (1 + 15 1) 2 (1 + 152) 2 (1 + r5J2, 1 1 1 Ic = Ic-1 = -,2 + -1T + -"2 ' '-I Az 1.3 1 1 1 Ilc = IIc-1 = _12 02- + ·-12 12 + 12 "2- ' 11.1 Az Az A3 ''JI~J 1 Irre = IIIc-1 = 02 02 12 • /,JAzAJ Hence follow the fundamental identities 1 : IIc Ic = IIIc' (30.1) (30.2) As a corollary of these formulae and of the representation theorem for isotropic scalar functions 2 we have the fundamentallemma: An absolute scalar function of any one of the tensors c, c-1, C, C-I equals an absolute scalar function of any other. From (30.1) follows oo>I>O, oo>Il>O, oo > III > o, (30. 3) where the invariants are calculated from any of the measures C, c, c-1, c-1, or from C" or c", for any n. Moreover, since the principal invariants determine unique values of the proper numbers (Sect. App.38), corresponding to any assigned values of I 0 , II0 , and III0 satisfying (30. 3) and to any assigned principal directions at X, there exists a unique deformation tensor C. In particular the conditions l=Il=3, III = 1, (30.4) for any of the tensors cn or cn, are necessary and sufficient forarigid deformation. When the stretches are great, I0 , II0 , and III0 are also very large; when the stretches are small, I0 , II0 , and III0 are much smaller. The invariants of c show a reciprocal behavior. E.g., if the deformation doubles alllengths, then I0 , II0 , and III0 assume the values 12, 48, 64; Ic, Ilc, and IIIc, t. 1\. and lh· The axiom of continuity is violated at a point where any of the principal invariants equals 0 or oo. If I 0 = 0, all lengths are annulled; if I 0 = oo, at least one stretch is infinite. 1 Doubtless these identities were known to CAUCHY. FINGER [1894, 4, Eq. (28)s] noted (30.2)1 but apparently did not realize its importance. (30.2) 2 follows from (30.2h by the principle of duality. Cf. also [1892, 4]. (30.2}a is really obvious, but apparently was first given, in the moreelaborateform (31.5) 6 7 , by ALMANSI [1911, 2, § 2] and HAMEL [1912, 4, §§ 364 to 365]. In the form (31.5), all three identies are given an elaborate derivation by MuRNAGHAN [1937. 7, Appendix]. Apparently RIVLIN [1948, 27, § 3] was the first to state them in the simple and immediate form (30.2). As has been observed by SIGNORINI [1943, 6, '1!13], since C and c-1 have the sameproper numbers, (30.2) is no more than a rewriting of the weil known relation between the invariants of a matrix and of its inverse, immediate from SPOTTISWOODE's theorem. 2 Since /(0 c o-1) =f(c) for any orthogonal transformation 0, f(c) =g(cl, Cz, cJ), where the function g is a symmetric function of its three arguments. Hence g (c 1 , c2 , c3) = h (Ic. Ilc, Illc). 266 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 31. For any deformation, we derive by (26.2), the rule for the determinant of a product of matrices, and (16.5) the identity IIIc = det c~ = det gpq det gP Q (det x~K) 2 = p. (30.5} Hence by (20.9) follow CAUCHY's formulae 1 dv = VI:IIcdV, dV = vnrcdv. (30.6} From these results and the inequalities (App. 39.10}a 4 we derive bonnds for the change of volume in terms of the first invariants of c' and C: (1 )-! dv (1 )! I. :::::; --- :::::; -- Ic 3 c - dV - 3 ' (30.7} with equality holding if and only if all three principal stretches are equal. It is of some use to express the principal strain invariants in terms of the elementary symmctric function of the principal extensions, which we shall denote by I6 , II6 , and III6 . Then 2 (30.8) As a measure of strain magnitude we suggest (30.9) This measure vanishes if and only if the deformation is rigid, and it increases if the absolute value of any one extension increases while the others are held constant; moreover, lengthening and shortening are measured symmetrically. A simple measure of shear intensity is given by 3= Va~ sin Ya· (30.10) By (28.12}, it is easy to express this measure in terms of the principal invariants of C, but the result is complicated. The octahedral invariant Uc, defined by (App. 38.11), is a measure of the intensity of the shear components of C, but for large strain this bears no simple relation to 3. 31. The classical strain tensors and elongation tensors. Much of the older work on finite strain employsone orother of the strain tensors 3 E and e, defined by 2E = C - 1, 2 e = 1 - c. (31.1) So as to reflect the dissymmetry in the definition of stretch (Sect. 25}, the defini tion (31.1) 2 takes - e as the dual of E. 1 [1827, 2, Eq. (28)]. 2 The formula for the ratio of volumes was noted by BoussiNESQ [1912, 1, § 15]. 3 E was introduced by GREEN [1841, 2, p. 29] and Sr. VENANT [1844, 3] [1847, 3, § 2] and is probably the commonest strain measure even today. ALMANSI [1911, 1, § 2] and HAMEL [1912, 4, § 363] introduced e, which has figured largely in the Italian Iiterature and more recently has become popular with British authors, who often attribute it to CoKER and FILON [1931, 3, § 3.06]. Cf. also L. BRILLOUIN [1925, 2, § 4] [1938, 2, Chap. X, §VII] and MURNAGHAN [1937, 7, § 1]. Sect. 31. The classical strain tensors and elongation tensors. To interpret E, note that we may put (26.6) into the form 1 Ä(N,) Ä(N2) cos (E)(NJoN2)- l'(N"Nz))- cos E)(N,,N2)) = CKMNfNzM- cos(oi)(N"N~), = 2EKMNfNzM. 267 (31.2) In terms of the gradients of the displacement vector, by (19.4) we have 2 E K _ (K + 1 P;K M- U ;M) 2 U UP;M• k - (k -.! p;k em- u ;m) 2 u ftp;m• (31.3) the latter being the dual of the former since -u is the dual of u. The vanishing of either E or e is necessary and sufficient for a rigid displacement. The normal components of E and e are called normal strains; the shear components, shear strains. The principal strains Ea and ea are related to the principal stretches Äa and principal extensions da by 2Ea= Ca-1 = A~-1 = (1 + da) 2 -1 = 2da+ 6~. } 2ea= 1- Ca= 1- Ä.-; 2 = 1- (1 + 60)-2 = 2Ea(1 + c5a) 2 ; (31.4) hence by (25.3) follows -l < Ea < oo, - oo < ea < t. Formulae expressing the principal invariants of E and e in terms of those of c, C, etc., follow by (30.1). For examplea, lc = lllc Ilc-' = 3 - 2 le, ) Ilc= Illclc-'= 3- 41e+ 41Ie, (dvfdV) 2 = 1 + 21E + 41IE +SillE= IIIc, = (1 - 21e + 41Ie- 8IIIet1 = 11r1, (31.5) where in the last sequence (30.6) has been used. From these formulae and (30.3) may be read off4 inequalities satisfied by invariants of E and e. We may split the displacement gradients uK;M and uk;m into symmetric and skew-symmetric parts: uK;M=EKM+RKM• EKM~u the axes at X may be determined but not those at x. From yet others, such as the six ckm> we may find the axes at x but not those at X. Quantities of the last two types are called strain measures at X and at x, respectively, while quantities of the first type are measures of both strain and rotation. From the foregoing remarks and the representation theorem for isotropic tensor functions 3 follows the theorem of equivalence: A ny uniquely invertible isotropic second order tensor function of c is a strain measure at x; of C, at X. Also from the general theory of isotropic functions it follows that the two sets of strain measures obtained by the foregoing theorem consist in symmetric tensor fields, those of the former set having as their principal directions the principal axes of strain in the deformed material, while those of the latter set have as their 1 [1892, 8, §§ 8-9]. 2 [1949, 36]; [1948, 23, § 3]; [1949, 26, § 3J. 3 While the assertion has a lang history, the first general proof is due to RrvuN and ERICKSEN [1955, 21, § 29]. Fora simpler proof, see §59 of Mathematical principles of classical fluid mechanics by J. SERRIN, this Encyclopedia, Vol. VIII/1. Sect. 33. Certain particular strain measures. 269 principal directions the principal axes of strain in the undeformed material. Examples for the deformed material are c, c-1, and e; for the undeformed material, C, c-1, and E. It is possible to describe strain correctly by a measure which is not a tensor, but there can hardly be any advantage, and attempts of this kind have usually led to confusion if not disaster. I t is obvious that a description of strain in terms of a strain measure at ~ is not in generat equivalent to one in terms of a strain measure at X. In certain special situations the two descriptions may become equivalent, e.g. for phenomena in certain isotropic materials (the much abused term isotropic will be explained in Sect. 293y), or when the two sets of principal directions happen to coincide. 33. Certain particular strain measures. Many students prefer to use a tensor whose proper numbers reduce for small extensions to the principal extensions !5a themselves. By (31.4), E and e are such tensors, but so also is ___! (1 - cK) K -+- 0 2K ' 1 ' (33.1) where we employ the K-th power of a tensor as defined by (App.41.2). Thus this requirement does not define a unique measure. The measures C} and c- ~ are attractive in that their proper numbers are exactly the principal stretches, but fractional powers are difficult to use in practice, since the components of such a tensor referred to co-ordinates other than principal are in general complicated infinite series in the displacement gradients. For example _ 1 _ c • = (1- 2e) • 1 = L., ~ - (2n-1)!! _nT ___ e n , l n=O 00 (2n-1)!! n = L --2n--;!- (1 - c) . n=O (33-2) By the theorem at the beginning of Sect. App. 42, the series converges if .A.3 >2-~. diverges if .A- <2-~. Similar objections apply to transeendental measures such as H- t log C = t log (1 + 2E), } h = - t log c = - t log (1- 2e), (33-3) where the definitions, which may be achieved by analytic continuation of the series for the logarithms, have a sense1 for all deformations since c and C are positive definite (Sect. 28). Since ha =Ha= log Aa, we have V-- dv Ih = In= log III0 = log -iV, (33.4) and accordingly the deviators of Hand h are zero for a uniform dilation ( Sect. 4 3) and thus may serve as distorsion tensors measuring change of shape apart from change of volume 2• While logarithmic measures of strain are a favorite in 1 Rather than using the concepts of Sect. App. 42, it is preferable, in analogy to (App. 41.2), to define H as the unique symmetric tensor whose principal directions are the principal axes of strain at X and whose proper numbers are log Äa. 2 REINER [1948, 23, § 7]. Cf. also the next footnote. 270 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 33A. one-dimensional or semi-qualitative treatments 1, they have never been successfully applied in general. Such simplicity for certain problems as may result from a particular strain measure is bought at the cost of complexity for other problems. In a Euclidean space, distances are measured by a quadratic form, and attempt to elude this fact is unlikely to succeed. Most modern work uses the metric tensors c and C or their reciprocals c-1 and c-1. . 33A. Appendix. History of the theory of strain. 1. Early work. The idea of strain, that is, of relative rather than absolute change of configuration, seems to begin with the definition of what we here call extension, viz . changeinlength extens10n == . . , ongmal length which was introduced by BEECKMAN (1630) and ]AMES BERNOULLI (1705); the latter understood that a law relating stress to extension characterizes a material, while formulae relating force to change in length, favored by empiricists such as HooKE (1675), can refer only to a particular specimen. A detailed history has been written by TRUESDELL [1960, 4, §§ 3, 8, 13, 20]. The history of one-dimensional strain measures given by MEHMKE [1897, 5] is misleadingly incomplete; the account of the early work by TonHUNTERand PEARSON [1886, 4, Chap. I] is inaccurate as weil. 2. Infinitesimal strain. The theory of small strain in a three-dimension~l medium was created by EuLER ( 17 50-1770), who dealt with time rates. For displacement, the theory of small strain was fully elaborated by CAUCHY (1822-1840), who obtained it by specialization from his general theory of finite strain. References to this work will be given in the course of our exposition of it in Part e of this subchapter. While CAUCHY was certainly influenced by the researches of FRESNEL, it is CAUCHY's special achievement to have disengaged and developed individually the concepts of strain, stress, and elasticity, which in FRESNEL's writing appear to be for the most part taken for granted. The work on special problems concerning deformable solids from the time of GALILEO (1638) through the researches of HoOKE (1675), LEIBNIZ (1685), }AMES BERNOULLI (1691-1705), PARENT (1713), EULER (1720-1776), DANIEL BERNOULLI (1734-1766), and COULOMB (1773-1784), even including NAVIER's derivation of the general equations of linear elasticity (1821), suffers from lack of a definite and explicit concept of strain as distinct from displacement but comprising extension, dilation, bending, shearing, and all other special deformations. A history of this work to 1788 is given by TRUESDELL [1960, 4]. 3. Finite strain. References to all important work on three-dimensional finite strain have been given already in connection with the theory itself. Nothing was done prior to CAUCHY's time, and little has been added by the extensive subsequent literature. CAUCHY hirnself acknowledged, if somewhat vaguely [1841, 1, second sentence], a debt to earlier work in differential geometry. The mostrelevant part is provided by EuLER's and GAuss's work on the line element on a surface and in particular the theory of applicable surfaces; however, these developments, the history of which has been written by SPEISER [1955, 23] [1956, 19], by no means sufficed to make the theory of strain evident, especially since the now familiar formulae analogous to (26.3) for a curved surface embedded in Euclidean space, while discovered by EuLER, were not published until much later [1862, 3]. 3a. Non-tensorial measures. GREEN in his first formulation [1839, 1, p. 249] laid down a reetangular system at X and took the extensions and changes of mutual angle suffered by the co-ordinate lines as measures of strain. KELVIN remarked that the usual strain measures are unsymmetrical in this sense, that each normal component is a function of the extension in but a single direction, while each shear component, being essentially a change of angle, depends upon the extensions in a pair of directions; he constructed a symmetrical specifica1 Cf. LUDWIK [1909, 7, Pt. ,, § 1], HENCKY [1928, 4, § 1] [1929, 2] [1929, 3, § 2], WEISSENBERG [1935, 10, pp. 59-60], BIEZENO and GRAMMEL [1939, 2, Chap. 1, '\115], REINER [1948, 23]. Noneofthis work is unequivocal as regards generalfinite strain, for which the definitions (33.3) were given by MURNAGHAN [1941, 3, p. 127) and RICHTER [1948, 24, § 2). Later RICHTER [1949, 26, § 3] worked out various special properties of h and H. Noticing that the condition of vanishing in uniform dilation does not determine a unique strain measure, RICHTER proposed a set of axioms, including a superposition principle for coaxial stretches, and showed that there are at aJ and X unique distorsion tensors [1949, 26, § 4] which satisfy them. This corrects an earlier attempt by MouFANG [1947, 10]; cf. [1948, 25]. RICHTER's distorsion tensors are complicated algebraic functions of e and E, respectively. Sect. 34. Conditions of compatibility. 271 tion of strain in terms of the extensions of the edges of a tetrahedron [1877, 5, Chap. VIII, Ex. 1] [1902, 5]. DoRN and LATTER [1948, 8] used the logarithms of the principal stretches and the cosines of the angles. SwAINGER [1947, 15 and 16] [1948, 29 and 30] [1949, 30] [ 19 50, 29 and 30] [ 19 54, 23] claims to define a new linear strain measure, embellished with polemies against the classical strain measures and the authors who use them; his views have been criticized by GoRDON [1950, 8], TRUESDELL [1952, 21, § 151 and § 171] [1955, 29], and RicHTER [1955. 20], among others, andin his numerous publications we have been unable to find anywhere a prescription for calculating his strain measure from a given displacement (but see the table below). The authors favoring logarithmic measures such as (25.8h and (33.3) usually call them "natural"; this term is applied by YosHIMURA [1953, .36] to -logtan}(}:n-y), where y is the shear. 3b. Table of tensor measures. Author SIGNORINI [1930, 6, § 9]. (Most of SIGNORINI's work employs E or e) BIOT [1939, 3, § 1] [1939, 4, p. 118] [1939, 5, p. 108] [1940, 3, § 1] MuRNAGHAN [1941, 3, pp. 127-128] (in reference to BIOT), RICHTER [1948, 24, §2] SWAINGER according to HERSHEY [1952, lJ] and REINER [1954, 20, § 4] MooNEY [1948, 15, pp. 435-436] ÜLDROYD [1950, 23, § 6] Definition }(c-1 -1) C~-1 Special Property The case K= -1 in (33.1) The unique strain measure at X whose proper numbers are ba , The unique strain measure at x I whose proper numbers are Aa I The unique strain measure at x whose proper numbers are - ba (IIJc)lc-~~ III=1 (IIIc)-kc dV=Illdv See also footnote 1, p. 256 and the formulae for logarithmic measures in Sect. 33. REINER [1948, 23, §§ 3, 5. 7] [1954, 20] and HERSHEY [1952, 11] present reviews of various definitions of strain and give comparative numerical values in simple cases. HANIN and REINER [1956, 11] work out the forms of the coefficients in formulae expressing one of the above measures as an isotropic function of another (cf. Sect. 33). 3 c. Other discussions of strain. KILCHEVSKI [ 1938, 5, § § 3-7] regards the components of the metric tensors at X and at X as generalized anholonomic components of the same tensor. Setting Bf!;f ~ tgK M gkm• we have gkm = Bf!;f gKM, etc. He introduces a very generalmeasure of deformation which includes as special cases both the deformation tensor c and the stretching tensor d ( Sect. 82). HENCKY [ 1949, 13] on the basis of a discussion of a special kind of finite defdrmation of a finite element criticizes all strain measures which depend only on the deformation gradients and obtains formulae of the "projective" type L1 xk = A k LfXK, in which Ak depends on the shape of the original element and on a certain arbitrary vector; in passing to the Iimit of infinitesimal volume he gets a formula different from (20.3), but LoDE [1954, 13, § 3.3] pointsout an error in his work. LODE hirnself prefers to use x~K and rediscovers such formulae as (20.8). ' A correct theory of affine type is mentioned at the end of Sect. 20; cf. also Sect. 61. 34. Conditions of compatibility. Given the deformation (16.2), it is a straighttorward matter to calculate the six components CKM by (26.2) 2 or to calculate any other measure of strain. Conversely, we may ask how to calculate the deformation when the components CKM are given as functioris of X. To solve this problern we have to integrate a system of six partial differential equations in three unknowns: k m -I (Xl x 2 xa) ( 34.1) gkmX;KX;M- KM • • · Such a system, being overdetermined, in general admits no solution unless the assigned functions fKM satisfy a condition of integrability. In the present case, by (26.2) 1 CKM is a metric tensor of Euclidean threedimensional space. According to a well known theorem asserted by RrEMANN, a symmetric tensor akm is a metric tensor for Euclidean space if and only if it is a non-singular positive definite tensor such that the Riemann-Christoffel 272 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 34 tensor Rka~ps forrned from it vanish identically1. Various forrns of this tensor in the co-ordinates yk are R(a) _ a [-o { r } _ _ o { r } { s } { r } _ { s } { r }] kmpq- kr oyP mq oyq mp + mq sp mp sq ' where Hence 1 ( iJ2akq iJ2akp iJ2amp iJ2amq ) = 2 aymoyP- -aymoyq + oyk oyq - ayk oyP + + a,s ({;J {msp}- {krp} {~q})' =-+ °:~~-~u bk~ bp~ + a,s ({krq} {msp}- {/p} {~q}), = O[~m. q] - ~JI:m,,_p_l + { r }Cr mp]- f r} [r mq] oyP oyq kq ' \kp ' ' [p, km J == __1__ ( oakp + oapm - oakm)' 2 aym oyk oyP R(c) -o kmpq- ' R(cJ -o KMPQ- , (34.2) (34-3) (34.4) where in the former equations y =X, in the latter, y =X. There are six algebraically independent non-identically vanishing components of Rkmpq in a space of three dimensions. Either of the two sets of Eqs. (34.4) is referred to as the conditions of compatibility. Many equivalent forms have been found, of which we record only that given by GRAIFF 2 : EKM,PQ + -1 EPQ,KM- EKP,MQ- EMQ,KP + I +CRs [(EMR,K + EK R,M- EKM,R) (EPs,Q + EQs,P- EFQ,s)- - (EMR,Q + EQR,M- EMQ,R) (Eps,K + EKs,P- EPK,s)J = 0, (34.4a) where the covariant differentiation is based upon gJL· 1 This theorem was asserted rather vaguely in RIEMANN's second Habilitationsschrift (1854) [1868, 13, §§ II 2, II 4, III 1]; in his Paris prize essay (1861) [1876, 5, Pars secunda], RrEMANN proved necessity and asserted that sufficiency is not difficult to prove. Priority in publication belongs to CHRISTOFFEL [1869, 1, §§ 1-6] [1869, 2] and L!PSCHITZ [1869, 3, § 7] [1870, 2, Part I, § 8]. Fora modern proof, cf. e.g. VEBLEN [1927, 8, § V4]. Although geometry and mechanics were closer tagether in the last century than today, it was lang before students of elasticity theory (even including some great geometers) recognized the connection between the conditions of compatibility and the equivalence problern of Riemannian geometry. 2 [1958, 3, Eq. (10)]. References for the linearized case will be given in Sect. 57. The general case has been treated by MANVILLE [1904, 5], MARCOLONGO [1905, 4], RIQUIER [1905, 5], CAFIERO [1906, 1, Chap. I, § 3], CRUDELI [1911, 3], BURGATTI [1914, 1, § 3], SIGNORINI [1930, 4, § 2] [1942, Jl, p. 60] [1943, 6, Chap. 1, ~ 20], KILCHEVSKI [1938, 5, § 18] (including a fourdimensiona] generalization), ToNOLO [1943, 8, §IV], SETH [1944, 11], ZELMANOV [1948, 39, Eqs. (6),(7)], NovozHrLov [1948, 18, § 39], ÜLDROYD [1950, 23, § 5], GREEN and ZERNA [1950, 10, § 4], ZERNA [1950, 36, § 4], SEUGLING [1950, 26], LODGE [1951, 15], GALIMOV [1951, 9], PLATRIER [1953, 21, 22], KOPPE [1956, 14]. FADOVA [1889, 7] was the first to recognize the geometrical nature of the problem, for the linearized case (cf. also the works of CAFIERO and CRUDELI just cited). Sect. 43. Conditions of compatibility. 273 In some problems, it is useful to measure the strain with regard to different configurations for different particles 1. The duals of Eqs. (34.4a) are satisfied, but (34.4a) themselves need not be, and the tensor RifkPQ may be taken as a measure of the incompatibility of the reference configurations for different particles. The conditions (34.4) arenot independent since R(al, in virtue of its definition (34.2), satisfies the identities of BrANCHI 2, R (a)h + R(a)h + R(a)h _ 0 mpk, q mkq, p mqp, k- • (34.5) The implications of these identities in respect to solution of (34.4) arenot known; a simpler analogous problern which has been solved will be rnentioned later on in this section. lf a:1 (X) and a:2 (X) are any pair of solutions of (34.4), the difference p 1 -p2 of the corresponding position vectors is a rigid displacement. In a problern formulated entirely in terms of the deforrnation, the conditions of compatibility need not be regarded, since they are satisfied automatically in virtue of the definitions of c and c: lt is only in a problern where the components of c or of C are thernselves taken as basic unknowns that the conditions of cornpatibility becorne additional equations to be solved. For such a problern any additional conditions imposed on c or on C must be compatible with (34.4). Consider the quantities Rkpmn defint>d by Rkpmn == t (akn, mp + apm, k••- akm, np - apn, km) +) -1 + ars (Apn.,Akns- Ap,.,Akms), Akpm == t (akm, p + apm, k- akp, ml • (34.6) where the comma denotes the covariant derivative based upon the metric tensor gkm· From this definition, the quantities Rkpmn form components of a tensor field; in a Euclidean space, we may choose a reetangular Cartesian co-ordinate system and by inspection of (34.2) 2 conclude that R* = R in this co-ordinate system and hence in all co-ordinate systems. Thus in writing the conditions of compatibility ß = 0, we may if we please replace partial derivatives by covariant derivatives 3 . We may pose an analogous question in regard to the elongation tensor E. In this case, the problern is a linear one: To find necessary and sufficient conditions that given a symmetric tensor E there exist a vector u such that (34.7) 1 The intended application is to bodies initially stressed though free from applied Ioads. The stress may then be thought of as arising in response to deformation from natural states which are different for different particles. There is no natural state for the body as a whole. This idea was first put forward, it seems, by EcKART [1948, 10, §§ 1-2], though his discussion is obscured by remarks about cutting a body apart and by failure to take account of the fact that in a deformation in Euclidean space the conditions of compatibility need not be satisfied at points where the axiom of continuity (Sect. 16) is not satisfied. Cf. the criticism by TRUESDELL [1952, 21, § 18']. While the idea that yield may be represented by a deformation from a hypothetical non-Euclidean space was expressed by KoNDO [1949, 15] [1950, 14-16] [1954, 10] and by BrLBY, BuLLOUGH and SMITH [1955, 2, §§ 2-3], the first clear statement and formalism isthat of KRÖNERand SEEGER [1959, 8, §§ 1-2]. 2 E.g., EISENHART [1926, 1, § 26]. For generalizations, cf. ScHOUTEN [1954, 21, Chap. III, § 5]. 3 ERICKSEN [1955, 6]. Handbuch der Physik, Bd. III/1. 18 274 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 35· The desired condition is easily shown to have the following equivalent forms 1 : M~1 b~gffRT;SU = 0, eKMP eRST EMs; PT= 0, l EKN;MP + EPM;KN- EKM;NP- EPN;KM = 0, (34.8) the second of which is valid only in three-dimensional space, while the others remain true in n-space. There are six linearly independent, not identically vanishing conditions (34.8). While they resemble in some ways the conditions (34.4), in principle they are of a different kind, for they are conditions of integrability for the differential system (34.7) when (34.4) holds, i.e., when the space is assumed to be Euclidean. Conditions of this kind are discussed from a more general standpoint in Sect. 84. If we set (34.9) then we have identically ~ +J~ +J~ - KPMN;R KPNR;M KPRM; N- 0, (34.10) a relation formally analogaus to the Bianchi identities (34.5) but presupposing that the space is flat. The conditions of compatibility (34.8) may be written in the form Jtf}.MN = 0. W AsHizu 2 has shown that in virtue of the identities (34.10) and GREEN's transformation, the conditions (34.8) may be divided into two sets of three, viz., l(El = JCE) = J(E) = 0 and l(E) = llU ~1 3~ llll l (E) = J(E) = 0, such that if both sets are satisfied upon the boundary of a region then 2323 3131 ' the vanishing of either set in the interior implies the vanishing of both sets. Thus the degree of redundancy that the identities (34.10) imply in the conditions of compatibility (34.8) is rendered definite. If E satisfies (34.8), and if 0uK is any solution of EKM = u(K; M), then the most general solution is (34.11) where P is any vector satisfying P;~ = blf, ii is any skew-symmetric tensor satisfying RKM;Q=O, and Bis any vector satisfying BK;M=O. That is, the displacement corresponding to a given elongation tensor is indeterminate to within a vector having the same form as an infinitesimal rigid displacement. c) Rotation. 35. Fundamental theorem. The results of CAUCHY presented in Sects. 25 to 27 show that the stretch of every element is determined by either of the strain ellipsoids, providing the orientations of its axes in the common frame be specified. Both the stretch and the change in direction of every element are known if either ellipsoid and both sets of principal axes of strain are specified. Hence follows the fundamental theorem : The deformation at any point may be regarded as resulting from a translation, a rigid rotation of the principal axes of strain, and stretches along these axes. The translation, rotation, and stretch may be applied in any order, but their tensorial measures arenot independent of this order. This theorem, while obvious from CAUCHY's work, was not stated by hima; not only did he wait fourteen years after completing the theory of strain in all 1 References to proofs are given in Sect. 57. 2 [1958, 12]. 3 It is given rather vaguely by KELVIN and TArT [1867, 3, § 182], and apparently LovE [1892, 8, § 10] was the first to assert it explicitly. Sect. 36. Thc mean rotation tensor of CAUCHY and NovozHrLov. 275 detail before giving a theory of rotation (Sect. 36 below), but also his measure of rotation is not one of those that come at once to mind on stating the fundamental theorem. The theory of finite rotation has always presented singular difficulty, although the essential idea is simple. Perhaps the trouble lies in its being a truly Euclidean concept: Often those least desirous of generality are also least successful in grasping the special peculiarities afforded by a special case. The angle {} through which the element dX is rotated by the deformation follows at once from (20.3): (3 5 .1) with the convention 0 ~ {} ~ n. In particular, we may choose dX as a unit vector N, obtaining cos{} = ,..!- gKMgr XkpNK NP') 11(1\) ' 1 mxK k p , gkmgK ·pn n · ll(n) ' Stretch, extension, rotation, and elongation (Sect. 25) are related thus 1 : s = }.. cos {} - 1 = (j cos {} - 2 sin 2 t {}, this being but another form of (3 5 .2). (3 5 .2) (3 5. 3) If we let na be a proper vector of c, we get for the angle {}a through which the deformation has turned it (3 5 .4) These three angles specify the rotation, since the a-th axis of strain at ~ lies on a cone of vertex angle {}a about the a-th axis of strain at X shifted to ~- If the three angles {}a vanish, or, what is the same thing, if any two of them vanish, the rotation itself is said to vanish, and the deformation is called a pure strain. It is important to realize that while by definition the principal axes of strain are not rotated in a pure strain, it by no means follows that no linear elements suffer rotation. This is grasped most easily through an example. Suppose two tri-axial ellipsoids have axes mutually parallel hut of lengths in pairwise different ratios. Then a linear transformation of one into the other changes the orientation of every radius vector not lying along one of the axes. An algebraic criterion for invariant directions will be given in Sect. 38. 36. The mean rotation tensor of CAUCHY and NovoZHILOV. CAUCHY 2 took as a measure of rotation the mean values of the angles through which all elements in each of three perpendicular planes are turned. Refer all quantities to the common frame. Let Nx be a unit vector perpendicular to the X-axis atZ, so that in terms of a real angle (/) we may write N x = j cos ifJ + k sin ifJ . (36.1) 1 BoussrNESQ [1877, 1, § 3] described this formula and gave further discussion of the rotation by means of the apparatus of Sect. 23. 2 [1841, 1, Th. IV]. 18* 276 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 36. CAUCHY's departure from the ideas of Sect. 3 5 was to consider not {} as given by (35.2) but rather the angle {}x between Nx and the projection of nx upon the Y-Z plane. The slope of this projection is given by and t z,ycosiP+z,zsiniP an ({! = -'-.:...._---:,........,.__:.:=--::- y,y cosiP + y,z sin IP ' Therefore (36.2) asserts that whence follows tan fJx + tan IP 1 - tan fJx tan (/) z, y cos IP + z,z sin IP y, }" cos IP + y,z sin IP ' X y, y cos2 IP + (y,z + z, y) sin IP cos IP- z,z sin2 IP ' (36.2) (36.4) tan {} = z,ycos2fP+ (z,z- y,y) sin IPcos IP- y,z:sin2__l -Ryz+Eyzcos21P+ i(Ezz-Eyy) sin21P (36-5) ,..,., - ,..,.,- - --.....,- ---, 1 + Eyy cos2 IP + Ezz sin2 IP + Ezy sin 21P where RandEare defined by (31.6). Forthedetermination of {}x, the x-component of the deformation is not used. The expression (36.5) is periodic of period :n; in C/J, reflecting the fact that oppositely directed elements suffer equal rotations. From (36.5), {}x as an angle satisfying 0 ;o;;,{}x-;;;, :n; is uniquely determined with an important exception: {}x = 0 and {}x = :n; are indistinguishable. CAUCHY's measure of rotation about the X-axis is 2n " Xx = - 1 -J{}x ( C/J) difJ = _!_ J {}x (C/J) d(/J · 2n n (36.6) 0 0 However, CAUCHY failed to notice that because of its equivocal treatment of {}x =0 and {}x= :n; his formula (36.5) does not always determine {}x uniquely and hence is not sufficient to calculate Xx· To remedy this difficulty, we may easily replace (36.5) by a formula for cos {}x. However, despite the elegance of CAUCHY's concept, his measure of rotation has never been used. First, the angles Xx, XY, Xz do not form a vector field1. Second, the values of the mean rotations about three perpendicular axes do not suffice to determine the rotations of the individual elements at Z. NovozHILOV2 has most happily modified CAUCHY's definition by putting 2n tan Tx = _.! __ Jtan {}x (C/J) difJ 2n 0 (36.7) in place of (36.6). The integration can now be performed explicitly; from (36.5) 2 it is easy to see that in fact tan Tx = - Ryz ~ -- 2n j. 2n ~ ~ diP ~ , l 0 1-:Eyycos2fP+. Ezzsin2fP+Ezysin21P -Ryz V(1 +Eyy){1 +Ezz) -.E.h . 1 Their law of transformationwas calculated by CAUCHY [1841, 1, §I, Eq. (37)]. 2 [1948, 18, § 7]. (36.8) Sect. 37. Algebraic proof of the fundamental theorem. 277 This elegant formula of NovozHILOV shows that RKM is a measure of the mean rotation suffered by elements in the K-M plane 1 . While we have used a special Co-ordinate system, our results are easily put in invariant form. Let X1 = const be a surface at X, let X2 and X3 be Co-ordinates on that surface, and consider only transformations of these surface COordinates. Then, with dummy indices restricted to the range 2, 3, (36.5) reads -R23 +eKMNK E: NP tan{}x, = ~ 1 +EgsNQ N 5 (36.9) while (36.8) reads (36.10) where Ji is the two-dimensional tensor obtained from E by suppressing all components having the index 1. The formula (36.10) justifies our calling R the mean rotation tensor. Since R is a tensor, if it vanishes in some one co-ordinate system, it vanishes in all. In view of (36.10), therefore, if the mean rotations of the elements in three perpendicular planes at a point be 0 or n radians, the mean rotation of the elements in every plane at that point is 0 or n radians. As is indicated by the two possibilities in the foregoing statement, R, while indeed a measure of mean rotation, is not a measure of rotation alone. The easiest way to see this is to consider a rigid rotation through n radians: x =-X, y = - Y, z =Z. In such a rotation, R = 0. Since R = 0 if and only if there exists a displacement potential 2 U: (36.11) we may call deformations for which R =0 potential deformations. These will be characterized in Sect. 38. If we replace NovozHILov's measure of rotation by CAUCHY's or by the mean of cos {}x, we get a measure of rotation: only, but there is no simple expression for these measures in terms of ii and E. Since ii is a skew-symmetric tensor of;second order, we may always replace it by an axial vector, which we choose to define as follows: R-i curl u. (36.12) The context will make clear whether the vector or the tensor is intended by the symbol R. 37. Algebraic proof of the fundamental theorem.r. The fundamental theorem of Sect. 3 5 is but an interpretation of the polar decomposition theorem in Sect. App. 43. Historically, however, the former was much the earlier and in fact gave rise to the latter. It is illuminating to give a different proof of the fundamental theorem, 1 As was observed by KELVIN and TAIT [1867, 3, § 190], KELVIN's transformation (Sect. App. 28) yields an obvious but not very illuminating connection between Rand the mean rotation around a closed circuit: ~uKdXK = fJ?KMdAKM. 't' Y' The interpretation of R given by M. BRILLOUIN [1891, 1, § 1] is faulty, being based on an error in calculation. 2 LOVE [1892, 8, § 12]. Handbuch der Physik, Bd. Ill/1. 18a 278 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 37. based indeed on the same ideas as the proof of FINGER, the CossERATS, and AUTONNE given in Sect. App. 43, but using the explicit formulae1 of Sect. 28. Let Na be an orthogonal triad at X, na an orthogonal triad at ~. If we shift the Nato ~. we can then exhibit a unique orthogonal tensor (cf. Sect. App. 43) R which rotates the shifted triad Na into the triad na: -1 -1 k Rk mNK Rk NK NK ,..KRm k RK k na= mgK a = K a' a =15m kna= kna, (3 7.1) where the intermediate components RkK = Rk m g'f{ represent the translation from X to ~ followed by the rotation R. In terms of the reciprocal triads Na and na, we easily see that Rand R-1 assume the forms -1 RkK = n~ N]i, RKk = NaK n~, (3 7.2) where we employ the summation convention for diagonally repeated German indices. The duals of (3 7.1 )1 are also formulae for the Na in terms of the na. Comparison of them with (37.1) 2 shows that R-1 is the dual of R, as is geometrically obvious, and yields the formulae -1 Nf =RKMgttn~, n~ =g'J.tRMKNf. (37.3) For the rotation of the reciprocal triads we have (37.4) Thus far the triads na and Na are arbitrary. Henceforth we restriet them to be directed along the principal axes of strain at ~ and at X. There follows the necessary and sufficient condition for a pure strain: (37. 5) Co-ordinate forms of this condition are Rkm = b~, Rkm = gkm• RkK = g1c, etc. There is an important formula which with some justice may be called FINGER's theorem 2 : (37.6) whereby the n-th powers of c-1 and C are related. To prove it, we need only observe that by (28.6) and the theorems on matrices given in Sect. App. 37, the tensors on the two sides of the equation are symmetric tensors having the same proper numbers, while by the definition of R the principal axes of the two tensors coincide. A moreformal proof can be constructed from (App. 37.12)a, (37.1), and (37.4); also, the result may be read off from (App. 41.2). We turn now to a proof of the fundamental theorem. By the dual of (28.3) and (37.1) follows · X;K = X;M UK = X;M a K = a na K• ( 37. 7) k k S!.M k NMNa v-c kNa } = RkM VCaNaM N]i. 1 We follow TouPIN [1953, 32, pp. 595- 597] [1956, 20, § 4]. 2 FINGER [1892, 4] studied the properties of a · a' and a' · a, where a is any non-singular matrix. Most of his results have been stated in Sect. 22. If we identify a with 1\ :r7 Kil, then one of his formulae is (37.6) with n = 1. The theoremwas given by RICHTER [1952, 17, § 2) for n = t; by NoLL [1955. 18, § 2b] for n = 1; and by TouPIN [1956, 20, § 4] for the general case. Sect. 37. Algebraic proof of the fundamental theorem. By (App. 37.12)3, this is equivalent to ! • ! X~K = RkM Ctf = gtRPM Ctf = Rkmg'~Ctf. If in this we substitute the dual of (3 7.6) with n = - t, we get k -lkRm -lk Rm -.k -lk m RM X;K = Cm K = Cm k 5K = Cm gM K· Now in a pure strain, by (37.5) we get from (37.8) and (37.9) the formulae .. k _ k Ct M _ -i k m • x-;·K- g M K- Cm gK, in a translation, which isarigid pure strain, by (26.11) follows ~K =rl-; 279 (37.8} (37-9) (37.10) (37.11} while in a more general rigid displacement, by (26.11) we get from (37.8) and (37-9) the formulae ~K = gkMRMK = Rkmg'ß. (37.12} In view of these special cases and (17.11), we see that the four formulae (37.8) 2 , (37.8)3 , (37-9) 2 , and (37-9) 3 constitute statements of thefundamental theorem. For example, (37.8) 2 asserts that to obtain the set of nine x~x we may begin with stretches -"a along the principal axes of strain at X, then rotate those axes into the principal directions of strain at ~. then translate from X to ~- The remaining three formulae express similar decompositions in different orders. The dual formulae give corresponding expressions for X{).. From (37.8h follows (37.13) which may be regarded as an expression for the rotation tensor as an infinite series in the displacement gradients. The formulae (37.2) are much simpler, but they cannot be used until both sets of principal axes of strain have been calculated. By (19.3), equivalent to (37.8) is !. uK;M = Rx° CQM - gKM · (37.14) Hence we may connect the elongation tensor E and the mean rotation tensor ii with the rotation tensor R and the deformation tensor C: (37.16) No such simple formula of composition holds when ii =l= 0. For the general expression of a spatial rotation in terms of angles .or other parameters, the reader is referred to standard treatises on kinematics. Here we mention only EuLER's theorem that every rotation about a point may be regarded as a rotation about a line through the point and hence may be character- 280 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 38. ized by a unit vector and an angle. In the langnage of rnatrices, this theorern asserts that II RKMII if not the identity rnatrix possesses a real unit vector invariant, its axis A, uniquely deterrnined up to sign, and a real scalar invariant, its angle{), uniquely determined up to a convention of sign and quadrant. An expression for R in terrns of these two invariants is 1 RK.u = cosf} b~ + (1 - cosf}) AK AM+ sin {) eKMPAP. (37.17) If we write R = R (A, fJ), then for the inverse rotation we have R-1 = R(A,- fJ): (37.18) In other words, the duals of A and {) are A and - {}, Also RKMAM=AK, RKK=R\=1 +2cosf}, R[KM]=sinf}eKMPAP. (37.19) This last forrnula asserts that the axial vector ! Rx points along the axis of rotation and has rnagnitude sin {). Alternative necessary and sufficient conditions for pure strain are RKK = 3 ' R[K M] = 0' {) = 0. Perhaps preferable to {) or RKK as a scalar rneasure of rotation is I\= sin2 ! {) = t (3 - RKK), (37.20) (37.21) since I\ vanishes if and only if the deforrnation isapure strain, and for deforrnations which are not pure strains it satisfies O< I\~ 1, with I\= 1 corresponding to a rotation through angle n. In surnrnary, every deformation carrying a neighborhood of X into a neighborhood of :r is specified locally by the translation from X to :r, the axis A, the angle of rotation {), either of the two sets of principal axes of strain, and the three principal stretches Aa . A study of further rnatters concerned with the separation of strain frorn rotation has been rnade by SIGNORINI 2. 38, Invariant directions. A different question, first considered by KELVIN and TAIT 3, is to find the particular elernents dX which suffer no rotation. It is easiest to start with (20.3), which yields as the condition for such an invariant elernent gff dx~ = gff ~M dXM = b dXK, (38.1) where b is a real factor of proportionality. Thus it is necessary and sufficient that b satisfy det [(b- 1) b~- u~M] = 0, (38.2) where we have used (19.4). This is a real cubic. Since it has at least one real root, we obtain the theorem of Kelvin and Tait: In any deformation, at least one direction is left unaltered. By (16.4), b -:fo 0. Let D be the discrirninant of (38.2). Then if D> 0, there are three distinct invariant directions 4 ; if D 0 the three invariant directions need not be mutually orthogonal, but if they are, then a fortiori they are principal axes of strain at X and at 3J. Cf. the theorem of KELVIN and TAIT presented in Sect. App. 37. By (31.6h, the tensor whose proper numbers satisfy (38.2) is symmetric if and only if ii = 0. If ii = 0, (38.1) becomes identical with the equation for the principal elongations. Therefore R =0 is a necessary and sufficient condition that the principal axes of strain be invariant as lines. Alternatively, a necessary and sulficient condition that the principal axes of strain be deformed into themselves is that they coincide with the principal axes of elongation; equivalently, that there exist an orthogonal triple of invariant directions 1• Thus, in particular, the condition (38-3) is necessary for pure strain. But it is not sulficient 2• For in (38.1) it is possible that b< 0; i.e., what we have called an invariant direction includes the possibility of a reversal of sense. Suppose now (38.3) holds. Then, as already established, the principal axes of strain are invariant and coincide with the principal axes of elongation. By (16.4), it is impossible that just one or all three principal axes may be reversed in sense, but the case of two reversals is possible and corresponds to a rotation through angle :n; about one principal axis 3• Since when ii =0 the roots ba of (38.2) are related to the principal elongations through ba -1 =Ba, we see that if R =0 then either Ba> -1 for a = 1, 2, 3 or eise B1 > -1, B2 < -1, B3< -1, where B1 ~ Bz ~ B3 • Hence we have the following theorem: In apotential deformation which is not a pure strain, the axis of rotation is the axis of greatest elongation and the angle of rotation is a straight angle. The axis of greatest elongation may be either the axis of least stretch or the axis of greatest stretch. In terms of the measure l\ defined by (37.21), we have the following criterion: if R=O, then either l\=0 or l\=1. The former case isapure strain; the latter, a non-pure potential deformation in which the axis A points in the direction of greatest elongation. Conversely, l\ = 0 implies ii = 0, but in generat l\ = 1 does not imply R = 0. If we refer Rk m and RKM to the principal axes of strain, for a pure strain they both reduce to the unit matrix, while for a non-pure potential deformation they both reduce to one of the forms diag ( 1, - 1, - 1), diag (- 1, 1, -1),diag (-1, -1, 1). Our considerations here are all local. By the axiom of continuity Sect. 16 it follows that if R =0 along a curve, upon a surface, or throughout a region, then in the entirety of the said manifold the deformation is a pure strain or a non-pure potential deformation according as it is one or the other at some one point of the manifold. 1 DARBoux [1901, 3] [1910, 5, §§ 319-330] found necessary and sufficient conditions that E have a triply orthogonal set of isostatic surfaces when R = 0. Cf. Sect. App. 48. He proved also that corresponding to any triply orthogonal family of surfaces there exist infinitely many potential deformations. ToNOLO [ 19 52, 20] has replaced DARBoux's complicated analysis by a short and elegant demonstration. 2 Almost the entire literature on finite strain (e.g. LovE [1927, 6, § 33]) follows KELVIN and TAIT [1867, 3, § 183] in asserting erroneously that (38.3) is a sufficient condition for pure strain. Apparently the error and its correction were first noted by SIGNORINI [1943, 6, ~ 14]. 3 Our convention regarding the definition of principal axes when there are two or more equal principal stretches (Sect. 27) makes this case no exception to our statement here. 282 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 39. In a potential deformation, (31.10) reduces to K ~K 1 ~KP ~ EM=EM+aE EpM, (38.4) Thus in a potential deformation Eis an isotropic function 1 of E. Moreover, from (38.4) we get Ea = !5a or - (2 + t'5a), (38.5) where for a pure strain the former alternative holds in all three cases, while for a non-pure potential deformation the former alternative holds only for the elongation along the axis. Fora potential deformation, from (37.15h we get i -1 ~ CKM=RKPCpQRMQ· (38.6) The rotation R therefore leaves the quadric of cl, and hence also the strain ellipsoid, invariant; this gives a second proof 2 of the theorem above. A different condition for invariant directions follows from the theorem at the beginning of Sect. App. 37: If a deformation with di~tin'Ct principal stretches is the succession of two pure strains, then there are three invariant directions. 39. Composition of strains. For two successive defonnations, the first from X to X, the second from X to ~. we have the simple fonnula (17.10) for composition of the defonnation gradients. No such simple result is valid for the corresponding strains. We now seek to relate the Cauchy tensors for these two defonnations to the Cauchy tensor for tfie defonnation from X to ~- Equivalently, we compare the two measures of strain at ~ from the two different unstrained states X and X. By (26.1} 2 we get 3 ckm =gKMX~Xf'!o =gKMX~KXftX~kX~m•} =hMX~kX~m• (39.1) where X~= oXKJoXK, x~k == oXKJox", and lc is the Cauchy tensor measuring the strain from X to X. By the dual of (37.8) follows -1 -1 i i ckm = lcK MRKpRMq2c~2c'f.., (39.2) where 2c is the Cauchy tensor measuring the strain from X to ~- This is the general law of composition of strains, or of change of strain reference. As would be expected, to calculate the strain from X to ~ a knowledge of the strains from X to X and from X to ~ is insufficient. In addition, knowledge of the rotation from X to ~ is necessary. In case the displacement from X to ~ is a pure strain, (39.2) becomes (39-3) which is not the ordinary matrix law of composition, being of the form c = 2cl · 1c · 2cl. In the special case when ZK = Ä.b~-ZK, so that the strain from Z to 1 In a work written in 1956 but not published, TouPIN has shown that in the case when the principal stretches are all distinct or all equal, R = 0 is necessary and sufficient that every principal direction of E be a principal direction of E. In the case when two principal stretches are equal, the strain ellipsoid is a spheroid, and rotating it about its axis of symmetry produces an equal elongation in all elements in its equatorial plane. Therefore the elongation quadric is also a quadric of revolution with the same axis. If the angle of rotation is not 0 or n, we have .ii =1= 0, yet every principal axis of E is also a principal axis of E. This theorem was stated by ToUPIN without proof. 2 Given in the work of TouPIN mentioned in the preceding footnote. 3 The dual is given by LovE [1892, 8, § 10]. Sect. 40. Isochoric deformations. 283 Z isauniform dilation (cf. Sect. 43), we get the obvious relation (39.4) hence (39-5) The multiplicative resolution (39.2) gives in general the simplest expression for a given strain in terms of composite strains and rotations. For some problems, however, a more complicated additive resolution is preferable. To obtain it, we begin with the dual of (39.1) 3 : cKM = 2cKMx~Kx~M = (gKM+22EKM) x~Kx~M· } = 1CKM+22EKMX~KX~M· (39.6) In this we substitute the formula that results from replacing ~ by X in (19.4), so obtaining1 EKM=1EKM+2EKM+21UP;(K2EJ:I) } + 2EPQ lUP;K JUQ;M· (39-7) The apparent simplicity of this formula is somewhat deceiving, since all tensors relating to either of the deformations are shifted to the point X. As would be expected of a result dual to (39.2), this one calculates the total strain in terms of the second strain, the first strain, and the first rotation. A more complicated result follows from (39-7) if we replace 1EKM and 1uK;M by their expressions in terms of ifKM and iiKM· By setting 1E =0 in (39.7) w~ get an expression for the change in the components of strain induced by a superposed rigid deformation. From (39.6) it follows that c = 1c if and only if 2E = 0; this is another way of saying that the components of c characterize the deformation to within a rigid deformation. d) Special deformations. 40. Isochoric deformations. There are four important special types of deformation defined by conditions referring neither to a particular direction nor to a special geometrical configuration. The first three, rigid deformation, pure strain, and potential deformation, were defined in Sects. 26, 35, and 36. The fourth, isochoric deformation, is defined by the condition that volumes be unaltered. By (20.9), (16.5), (30.6), and (30.8) 4 , we get as alternative necessary and sufficient local conditions for an isochoric deformation and many other forms are easily found. In view of (40.1)a, by specialization from the fundamental theorem on isotropic scalar functions we conclude that in an isochoric deformation, an isotropic scalar function of c is equal to a function of Ic and IIc only. These two invariants satisfy a number of simple relations. First, (30.2) becomes (40.2) 1 Cf. SIGNORINI [1943, 6, ~ 8]. 284 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 41. Second, since Illc=Ä.~.:l~Ä.~ = 1, one of the principal stretches, say Ä.3 , can always be eliminated in terms of the others. For example, from (30.1) follows I0 =Ic-,= Ä.~ +Ä~+ ;.,\_~. (40.3) From this formula and (40.2) we conclude Ic, Ilc, Ic-'• Ilc-' ~ 3. (40.4) where the lower bound 3 is assumed in a rigid deformation. This is to be contrasted with the general case, where all that can be inferred is that the principal invariants are positive. In an isochoric deformation, from putting (40.1) 5 into (30.8)1 and (30.8) 2 we derive Ilc- 3 = Ic-,- 3 =- 4 II.,- 2 III., + (II.,+III.,) 2 , } (40.5) Ic- 3 = IIc-•- 3 =- 4 II., -10 III6 + II~ + 4 II6 III6 + 2III~. On the one hand, these identities put into (40.4) yield inequalities connecting 3 the principal extensions. On the other hand, if we set c52= L c5~, then as c5-+0 we have from (40.5) «=1 (40.6) This, as RIVLIN1 has remarked, is tobe contrasted with the general case, where all that can be concluded isthat Ic-3, Ilc-3, and IIIc-1 are O(c5). 41. Plane strain2• A deformation is said tobe a plane strain if there exists a family of parallel planes which are individually preserved, while the family of lines normal to these planes is preserved as a family. If we choose the preserved planes as reetangular co-ordinate planes and the remaining co-ordinate surfaces as cylinders normal to these planes, the deformation assumes the form x"=x"(Xl.,X2), k=1,2, z=Z=x3 =X3 • (41.1) To visualize the strain and discuss it, it suffices to restriet attention to points in the z = 0 plane, to replace the strain ellipsoids there by their elliptical sections by that plane, etc. The Cauchy and Greentensorsand the rotation tensor of a plane strain have matrices of the type 0 0 0 0 1 (41.2) Hence the z-direction at a point is both a principal axis of strain and the axis of rotation, so that the angle of rotation is the common angle through which the two principal axes lying in the z = 0 plane are tumed. A necessary and sufficient condition that one principal stretch be 1, and hence a necessary but not sufficient condition for a plane strain, is that the principal invariants of any one of the tensors c, C, c-1 and C-1 satisfy (App. 38.8). Since III =Ä.P.~. by (40.1) 3 it follows that a plane strain is isochoric if and only if the principal stretches are reciprocal to one another. By (App. 38.8), a necessary but not sufficient condition for plane isochoric strain is I0 = II0 . By (40.2h follows then Ic= Ic: In a plane isochoric strain, each principal invariant of any one of 1 [1951. 22, § 21]. 2 More detailed analysis is given by SIGNORINI [1943. 6, ~ 31]. Sect. 42. Homogeneaus strain. 285 the tensors c, C, c-1, and C-1 equals the corresponding principal invariant of any other. Adeformation is said tobe a generalized plane strain 1 when (41.1) is replaced by k=1, 2, z=f(Z). (41.3) The planes Z = const are preserved as a family, but not necessarily individually. For example, there may be a uniform stretch Ä normal to these planes, so that z = ÄZ. A number of properties of plane strain carry over to the generalized case. For example, the z-direction is again both a principal axis of strain and the axis of rotation. The principal stretch in this direction is Ä = f'(Z). Since Ä1 and Ä2 are independent of Z, and since IIIc = J.7 Ä~ Ä 2, it follows that a generalized plane strain is isochoric if and only if Ä = const and Ä1 Ä2 = }.-1 • 42. Homogeneous strain. A deformation is called a homogeneaus strain when every straight line is deformed into a straight line. Equivalently, every ellipse (including the circular special case) is deformed into an ellipse, or every ellipsoid is deformed into an ellipsoid. A formal and invariant local condition for homogeneous strain is ( 42.1) of the many equivalent forms of this condition we note only 1\;K=O, A~M=O, CKM;P=O. (42.2) The most convenient condition is that in the common frame the deformation ( 1 5 .1) has the form Z=D·Z+B. z = n-1 . (z- B) ' ' (42.3) where matrix notation is used and where D is a real matrix with positive determinant and Bis a real constant vector. When referred to the common frame, all the measures of deformation and rotation (such as c, C, R, !\, A, R, E, etc.) are constants. There is no loss in generality in supposing the origin of the common frame so chosen that R = 0. Homogeneous strain was introduced and studied exhaustively by KELVIN and TAIT2 • For homogeneaus strain, the reservations expressed in Sect. 20 become unnecessary, for finite line segments are deformed according to the same law as are differential elements in a general deformation. By uniformly extending initially circular or reetangular frames across which marked cloth or rubber sheets had previously been stretched, WEISSENBERG 3 has obtained the beautiful illustrations of homogeneaus strains which are shown here as Fig. 6. By (20.10), we may choose to regard any deformation satisfying the axiom of continuity as a homogeneaus strain, with error 0 (dX2). Thus for general deformations all properties which depend only on x7K may be developed by application of appropriate theorems on homogeneous strain, and many authors attack the subject in this way. We follow the reverse procedure to this extent, that we leave to the reader the specialization of general results already obtained to the case of homogeneous strain. The following sections develop the properties of certain particularly important types of homogeneaus strain and certain of their generalizations. The co-ordinate system used is always reetangular Cartesian. 1 The case z = ÄZ was introduced by LovE [1906, 5, § 94]. 2 [1867, 3, §§ 155-189]. Further geometrical properties were discussed by IsE: [1890, 5, §§ II-III]; algebraic treatments were given by METZLER [1894, 6] and CAFIERO [1906, 1, Chap. Il]. 3 [1935, 10, pp. 85-87] [1949, 37]. 286 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 42. Sect. 42. Homogeneaus strain. 287 Fig. 6 b. Various defonnation ellipsoids (after \\rEISSENBERG). 288 C. TRUESDELL and R. TOUPIN: The Ciassical Field Theories. Sect. 42. A necessary and sufficient condition for a homogeneaus isochoric strain to be plane is that the principal invariants of any one of the tensors c, C, c-1, and c-1 satisfy I= II. (42.4) Fig. 6c. Various defonnation ellipsoids (alter WEISSENBERG). This is immediate from (App. 38.8), since by hypothesis the deformation is a homogeneous strain. Since J) in (42.3) is the matrix of deformation gradients, by (37.8) and (37-9) we have D = R · cl = c - L R, (42.5) where R is the rotation tensor defined by (37.1). GRroLI 1 has obtained a theorem of approximation which characterizes R. Let z = z (Z) and z* = z* (Z) be any two homogeneaus defor1 [1940, 12]. Sect. 42. Homogeneaus strain. mations, and define their deviation d(z*, z) by d(z*, z) ==o f Jz*- zJ2 dV, r. 289 (42.6) where j';, is a sphere of fixed radius a about Z. Given z, we now seek to determine z* as a rigid deformation minimizing d (z*, z). Since z* = R* · Z + B*, where R* is a rotation Fig. 6d. Various deformation ellipsoids (after WEISSENBERG). matrix, we have d(z*, z) = JJ (B*- B) + (R*- R · C~) · ZJ r. 2 dV, l = JJR· {R-1 · (B*- B) + (R-t·R*- c') ·Z}J2dV, r. = JJB' + (R' - C~) ·ZJ 2 dV, r . where R'~ R-1 · (B*- B), Handbuch der Physik, Bd. lll/1. R' = n-1 ·R*, (42.7) (42.8) 1') 290 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 43. and where the factor R, since it does not affect the length being integrated, was dropped in passing from (42.7) 2 to (42.7)a. Hence d(z*,z) = ~na B' + 2B' · (R'- C~) · fZdV + l -r-. + (R'- C~) · ( fZZdV) · (R'- cl)'. -r-. Fig. 6e. Various defonnation ellipsoids (alter WEISSENBERG). Since 1';, is a sphere, by choosing the origin of Z at its center we get d(z*, z) = !na3 B' 2 + 1 4 5 na5 (RJ.m-lkm) (~km.- tkml.l = fna3 [B' 2+ }a2 (3 + Ic- 2RkmCkm)], (42.9) (42.10) where we have used BoRCHARDT's theorem in Sect. App. 40 and the fact that (R')-1 = (R')'. k Since R' is a matrix of direction cosines, we have I Rl.ml ;:;;: 1, and h ence J Ri,mCkml ~ ~. - 3 with equality holdingifandonlyif R' = l. Since3+Ic-2I ;=11 =o ~6~. from (42.10) we conclude that a~l (42.11) where equality holds if and only if B' = 0, R' = 1. These conditions, by (42.8). are equivalent toB*= B, R* = R. We have proved Grioli's theorem: Let a given homogeneaus strain be decomposed into a lranslation, rolation, and a pure strain; then the Iranslaiion and the rotation are precisely those defining the rigid deformation whose deviation from the given strain is the least possible. 43. Uniform dilation. A pure homogeneaus deformation in which the principal stretches have a common value, A., is called a uniform dilation. For such a defor- Sect. 44. Simple extension. 291 mation we have D =Al, c = C-1 = - 1 1 C = c-1 = A.2 1 1 A2 ' ' Ic = 3 A2, Ilc = 3 A4, IIIc = A6, J (43.1) where 0 < A. < oo. An invariant necessary and sufficient condition that a pure homogeneaus deformation be a uniform dilation is 1 (+rr=(~ IIY=III, (43.2) where the invariants are calculated from any one strain measure. This is so because (43.2) is necessary and sufficient that the strain measure have equal proper numbers. 44. Simple extension. A pure homogeneaus deformation in which two but not three principal stretches are equal is called a simple extension. In a reetangular co-ordinate system whose axes are the principal axes of strain, we have BA. o o D = o BA. o o o A. ß2 A2 c = c-1 = 0 0 O< B 0 or v < 0, the cross sections normal to the axis wane or wax as the length along the axis increases, while if v = 0, lengths normal to the axis are unchanged. The extension is isochoric if and only if B =A.-~, in which case (44.2) 2 becomes v = 1 ;:~~ = ~ + 0 (az) (44.3) as Oz-+0. Some authors 2 restriet the term simple extension to the case when v =0. Such a pure homogeneaus strain may be characterized by the necessary and sufficient conditions 3 IIE= IIIE= 0, the stretch being determined by IE= ,12-1. A corresponding invariant condition for the case when v does not necessarily vanish is more elaborate 4 : 18 I II III- 4P III + J2 112- 4IP- 27IIJ2 = o, (44.4) 1 A more elaborate but equivalent condition was given by SrGNORINI [1930, 6, § 10]. 2 E.g., LovE [1892, 8, § 3]. 3 CAFIERO [1906, 1, Chap. li, § 16]. 4 A more complicated but equivalent condition is given by SrGNORINI [1930, 6, § 10]. 19* 292 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 45. where the invariants may be calculated from any strain measure. To see that this condition is necessary and sufficient, recall that the matrix of any strain measure may be reduced to diagonal form by a suitable choice of co-ordinates (Sect. 28). From the definition of a strain measure (Sect. 32), forapure homogeneous deformation to be a simple extension it is necessary and sufficient that exactly two proper numbers of the strain measure be equal. But (44.4) 1 asserts that the discriminant of the cubic equation for the proper numbers vanishes, while (44.4) 2 asserts that the case of Sect. 43 is excluded. Q.E.D. lt is easy to consider a more general homogeneaus extension in which all three stretches are unequal. By the fundamental theorem of Sect. 27, however, any pure homogeneaus strain if referred to principal axes is of this form, and hence this case, being special only in choice of Co-ordinate system but not intrinsically, is of no interest. A deformation, not necessarily homogeneous, is a pure extension 1 if a certain triply orthogonal system of planes is deformed into itself. Referred to principal axes, such a deformation assumes the form x=f(X), y=g(Y), z=h(Z), and hence Ax=f', Ay=g', Az=h', whence follow formulae similar to (44.1). In particular, Illc= (f'g'h') 2 , and hence by obtaining the general solution of f' g' h' = 1 we conclude that an isochoric pure extension is necessarily homogeneous. Y,y / ~~----~--~~-------+--f--+~~ Z,z ,-i---KY ' I I L :r I I I I typical plane of shear Fig. 7. Simple shear. --typical axis of shear and rotation 45. Simple shear. Consider a homogeneaus strain such that 1. Two orthogonal families of parallel planes are individually preserved. 2. The lines normal to one family remain normal to that family. 3. The lines common to both families are not stretched. Such a homogeneaus strain is called a simple shear; the planes whose normals remain normals are called the planes of shear, while those of the second family are called the shearing planes (Fig. 7). It is obvious that the lines common to both families are individually preserved. The direction normal to a plane of shear is the axis of the shear. Since simple shear is the most important and instructive of all special deformations 2, we shall analyse it in detail. 1 TRUESDELL [1952, 21, § 42B]. 2 The concept of shear received currency relatively late in the history of continuum mechanics. While in 1770 EDLER had briefly and clearly explained and calculated not only the time rate of extension but also the time rate of shearing (for the results and reference, see Sects. 82-82A below). his work on this subject escaped notice until 1955. According to ToDHUNTERand PEARSON [1886, 4, § 726 and Appendix, Note A (6)], the first clear discussion of shear in reference to solids was given by VrcAT (1831). earlier remarks being due to YouNG (1807). The first analysis of finite shear was given by KELVIN and TAIT [1867, 3, §§ 169 to 176]. SrGNORINI [1943, 6, ~ 12] approaches simple shear through a preliminary characterization of all deformations that leave a single plane point-by-point invariant. Sect. 45. Simple shear. 293 Ignoring, as usual, a possible translation, we let the planes of shear be the planes Z =const; the shearing planes, Y = const. Hence z =Z, y = Y, and x=MX +KY +LZ. The condition that the Z-direction be preserved yields L = 0, while the condition that the stretch in the x-direction be 1 yields M = 1. Thus X=X +KY, y=Y, z=Z, 1 K 0 D= 0 1 0. 0 0 1 -=O henceforth. To visualize the shear, imagine that each of the planes Y = const be slid in the X-direction by an amount proportional to its distance from the Y = 0 plane. Such a displacement is easily approximated with a pack of cards. The stretch suffered by lines in the Y-direction is V 1 + K2 • Either from the definition of simple shear or from the fact that det D = 1 it is obvious that simple shear is an isochoric deformation. The point (- t K, 1, 0) is carried into the point ( + t K, 1, 0). Hence the length of the line whose slope angle is t n + arc tan t K in the Z = 0 plane in the undeformed material is unaltered. It follows that in any given simple shear there are two families of parallel planes that are transported rigidly, i.e., undeformed; both these are normal to the planes of shear, one set being the shearing planes themselves and the other the planes subtending the angle t n + arc tan t K from these planes. The Z-direction is of course a principal direction of strain. The other principal directions of strain are easily found, since by the symmetrical distribution of stretches and the extremal property of the principal directions (Sect. 27) it follows that the principal directions bisect the angles between the undeformed planes. In the undeformed body, one of these families of planes is inclined at the angle in+ t Are tan t K to the shearing planes, and in theseplanes lie the elements suffering greatest stretch. Hence and from (45.1) the magnitudes of the two principal stretches in the plane of shearing are easily calculated: .41=1 +tK2 +KV1 +tK2, l /.~=1/1.1=1 +tK2 -KV1+iK2. (45.3) For small shears we have /.1 =1 +tK +O(K2), /.2=1-tK +O(K2), while the principal axes approach the bisectors of the co-ordinate axes. The shear experienced by the pair of elements ( 1, 0, 0) and (0, 1, 0) is T, the angle of shear, but this is not the greatest shear. As follows from the general theory in Sect. 28, the greatest orthogonal shear is experienced by the elements bisecting the principal axes of strain in the plane of shear. One of these elements is inclined at angle tarctantK to the shearing planes. By applying (45.3) to (28.12) we conclude that the magnitude of the greatest shear, y, is given by tany = K V1 + tK2 • (45.4) Hence for small shears y =K +0 (K3) = r +0 (r3). 1 Some authors (e.g. LovE [1892, 8, § 3]) use the termangle of shear for the angle through which an unstretched fibre in a plane of shear but not in a shearing plane is turned. As shown below, this angle is 2Arc tan t K = 2ff. Handbuch der Physik, Bd. III/1 . 294 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 45. The principal axes of strain in the planes of shear in the deformed material are the lines into which the two previously determined axes are deformed. They may be calculated by the same construction, starting from the deformed rather than the undeformed positioris of the lines whose length is unaltered. Hence the principal axis of greatest stretch in the deformed material subtends the angle ± n-i arc tan !K from the shearing planes. The axis of rotation is parallel to the Z axis, and the angle of rotation is given by Y,y {) = arc tan i K = arc tan (i tan r). (45.5) For small shear, {)=ir+O(r:i)= iK +O(K3). In terms of the rotation, we may write (45.3) in the form A.1 =sec{) + tan {), } (45.6) · A.2 = sec{) - tan {). Hence 1 The amount of shear is the ditference of the principal extensions. Fig. 8. Simple shear. A homogeneaus isochoric pure x,x strain is sometimes called a pure shear 2• The results just established are equivalent to resolution of a given simple shear into a pure shear followed by or preceded by a rotation about an axis normal to its plane. The features of shear just discussed are illustrated in Fig. 8, which is accurately drawn for a shear angle in. The portion of material shown is, beforedeformation, the intersection of a unit cube with a typical plane of shear; the particle Z selected for illustration is at the center of this area. The foregoing geometric treatment, which follows KELVIN and TAIT, is elegantly simple. For solution of mechanical problems, however, it is more convenient to have at hand the explicit formulae for various tensors introduced earlier in the chapter. The Cauchy and Green tensors and their reciprocals have the values C= 1 -K o 1+K2 o , 1 1 K o 1+K2 o , c-1 = 1+K2 K o 1 0 ' . 1 1+K2 -K o 1 0 1 1 Ic = II0 = 3 + K2, IIIc = 1. (45.8) The proper numbers of all four deformation tensors are 1 and .l.i and ;.~ as given by (45.3). The unit proper vectors of C corresponding to these proper numbers, in the reverse order, are NJ,N2= vi+i(tK±V1+tK2)' N3=k, (45.9) 2 + tK2 ± KV1 + !K2 1 CAFIERO [1906, 1, Chap. II, § 16]. 2 LOVE [1906, 5, §2]. Sect. 45. Simple shear. 295 while the corresponding unit proper vectors of c are i+i(-tK±V1+tK2 ) nJ,n2= V , n3=k. z+tK2=t=KV1 +tK2 (45.10) Since we are employing the common frame, there is no need to distinguish the different types of components of R, and by substituting (45.9) and (45.10) into (37.2) we get tK --- 0 V1+tK2 v1 + tK2 R = IIRMmll = -tK 1 (45.11) v1 +tK2 v1 +tK2 0 0 0 1 where the row index is M and the column index is m. Hence the axis of rotation is given by A = ± k, while by (37.21) the angle and measure of rotation are given by 1 cos{} =V , 1 +tK2 1\ - _1__ (1 - 1 ) - 2 v1 +tK2 . This result for {} agrees with (45.5) and also follows directly (45.10) and the fact that cos {} =nf N1M =nf N2M. By (45.8), for the strain tensors E and e we have 0 e= -!K 0 --!K2 0 , 0 E = . tK2 ~ , 0 -!K 0 l le =-IE = 2 Ile = 2 IIE =--! K2 , while for the tensors of elongation we have o !K o 0 0 0 (45.12) from (45.9) and (45.13) (45.14) Hence, as is obvious, the elongations of elements in the co-ordinate directions are a11 zero. The directions of elements suffering greatest and least elongation lie in the plane of shear and are inclined at angle i-n to the shearing planes, the extremal elongations being ±-! K. The axial vectors of mean rotation are given by r = ii = -! curl u = - -! K k = - tan {} k. (45.15) Of course this vector and the axis A are parallel, but only for small rotation does the magnitude of this vector equal the angle of rotation (45.5). This result is to be Contrasted with the formula for the cross of R: -!Rx =-sin{}k, (45.16) which follows from the general formula (37.19) 4 and the fact that we may choose - k as the axis. The concept of simple shear is not invariant, since it singles out a particular rotation. We may approach an invariant characterization as follows: A homogeneous isochoric strain may be regarded as simple shear followed by or preceded 296 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. by a rotation about the axis of shear if and only ifl ! 0 = II0 ; Sect. 46. (45.17) equivalently, if and only if it is plane. The amount of shear is Vro- 3. The equivalence of the two conditions follows from (42.4). The theorem itself isgeometrically obvious. For an algebraic proof, we need only notice that in any plane iso~ choric strain, as was remarked in Sect. 41, we have Ä1 = 1/.1.2 , and hence by (45 .6) and (45.5) follow uniquely determined values of {} and K. The foregoing theorem is particularly useful when applied to deformations which arenot homogeneous, as follows: At a point where III0 =1 and I0 =II0 , the measures of strain and rotation have the same values as for a simple shear of amount K = VIo __:::-3, in the plane normal to the principal direction whose stretch is 1. 46. Composition of shears and other special homogeneous strains. By ( 17.11), a succession of homogeneaus strains may be regarded as a single homogeneaus strain, but the result depends on the order in which the strains are taken, as follows from (17.12). However, no such rule as (17.11) holds when the matrices D are restricted to special subclasses such as pure strains, simple extensions, or simple shears. That is, while homogeneaus strains constitute a group, none of the previously defined special types, apart from uniform dilation, form subgroups. This fact suggests that these special deformations may be used to build up a general homogeneaus deformation. Two resolutions of this kind are stated now without proof2: 1. Any homogeneaus strain may be produced by the succession of a simple shear, a simple extension (without transverse contraction) normal to the plane of shear, a uniform dilation, and a rotation. 2. Any homogeneaus strain may be produced by the succession of three simple shears in mutually perpendicular planes, a uniform dilation, and a rotation. As is indicated by No. 2, the result of two successive shears will in general be complicated 3 • However, in the special cases when the planes of the two shears are perpendicular and one or two families of planes are individually preserved in each deformation, the formulae aresimple enough tobe worth noting. First 4, if both the Y = const and the Z = const planes are preserved, we get D= 0 1 K 1 1 K 0 2 = 0 1 K 1 1 0 0 · 010, 1 0 K2 l 001 001 001 =Dl·Dz =Dz·Dl. (46.1) Second 5, if only the Y = const planes are individually preserved, it follows that 1 Kl 0 1 Kl 0 D= 0 1 0 0 1 0 0 K2 1 0 0 1 (46.2) =Dl·Dz =Dz ·Dl. I LovE [1892, 8, § 7]. 2 The proofs are easy from geometrical considerations (cf. KELVIN and TAIT [1867, 3, §§ 178-185]) or by theorems on the factorization of matrices. An analysis of shears into products of reflections has been given by WILSON [1907, 8]. 3 Cf. LovE [1892, 8, Note AJ. • E. and F. CosSERAT [1896, 1, § 7]. 6 TRUESDELL [1952, 21, § 42G]. Sect. 47. Generalized shear. For these two cases we get, respectively, 1 K 1 K 2 1 C= 1 +K~ K 1 K 2 , 1 +K~ Hence in both cases K 1 0 1 +K~+Kj K2 1 I0 = II 0 = 3 + K~ + K~, III0 = 1 . 297 (46.3) (46.4) By the criterion given at the end of Sect. 45, each of these composite shears may be regarded as a simple shear of amountl K = V K~ + K~. The principal axes of the resultant shear are of course different in the two cases. The planes of shearing are the planes normal to the proper vectors corresponding to the proper number 1. For the former, this gives K2Y=K1 Z; for the latter, K2 X=K1 Z. 47. Generalized shear. In the deformations 2 X= X f(Y) + g(Y), y = h(Y), z=k(Z)+l(Y), (47.1) the planes Y =const areslidparallel to each other, but the direction and amount of the slide varies arbitrarily from plane to plane. At the same time, there are stretches in the X, Y, and Z directions; in particular, Äx=f(Y) and Äz=k'(Z) = k'(k-1(z-l(YJ)), so that in a given plane Y = const the stretch Äz depends only on z, and the stretch Äx is constant. This dass of deformations reduces to a special kind of generalized plane strain if l (Y) = 0; it includes simple extension, simple shear, and pure shear as special cases. We readily find that (Xf' + g') I 0 C= (Xf'+g')2+h' 2+l' 2 k'l' k'2 f2 + (X I g I'+ g't (X I g I'+ g') h' (X I g I'+ g') l' h'2 Ic = f2 + h' 2 + k'2 + l'2 + (X f' + g')2' IIc = f2(h'2 + k'2 + ['2) + k'2 [(X f' + g')2 + h'2J, IIIc = f2h'2k'2. h' l' l'2 + k'2 (47.2) Hence such a deformation is isochoric if and only if k' =const =Ä, say, while f(Y)h'(Y) =1/J... Another generalization of simple shear, given by x=X+f(Y), y=Y+g(Z), z=Z+h(X), (47.3) 1 For the former case, this was observed by LovE [1892, 8, § 10(iv)]. 2 This class of deformations combines the features of (46.2) and of a class considered by ADKINS [1955, J, § 3]. 298 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 48. may be regarded as the succession of three variable shears in mutually perpendicular planes1 . Forthis case we get 1 + h' 2 !' h' 1 + /' 2 !' h' c = 1 + f' 2 g' c-1 = 1 + g' 2 g' 1 +g'2, 1 + h' 2 Ic = 3 + !' 2 + g' 2 + h' 2' IIc = 3 + !' 2 + g' 2 + h' 2 + !' 2 g' 2 + g' 2 h' 2 + h' 2 !' 2, IIIc = (1 + /' g' h') 2 • (47.4) Thus the deformation (47.3) is isochoric if and only if at least one of the functions f, g, h reduces to a constant; that is, if and only if at least one of the three constituent shears is a simple shear. By (45.17), it is locally a simple shear if and only if at least two of the three constituent shears are simple shears. SrGNORINI 2 has determined the most general deformation in which all components of E except a single shear component vanish. 48. Simple torsion. When a cylinder is twisted uniformly along its length, we get the isochoric deformation r=R, 0=8+KZ, z=Z, (48.1) where R, 8, Z and r, (), z are cylindrical polar co-ordinates referred to the same system. Torsion deserves detailed analysis because it is the most important simple deformation which is neither a homogeneaus strain nor a plane strain. The constant K is the twist per unit length. Since G11 = 1, G2 2 = R2; G33 = 1, g11 =1, g22 =r2, g33 =1, while all other components GMP and gmp vanish, from (26.2h and (29.1) 2 we get IIC1!11 = ~ ~ ~ ll~ tll = ~ 1 + 2 ~ • l 0 K R2 1 + K2 R2 0 K r2 1 (48.2) 10 = Ilc = 3 + K2 R2 , III0 = 1 . The physical components of C and c are precisely equal to their counterparts for simple shear as given by (45.8), provided we take the plane of shear as the tangent plane to the cylinder R = const and the shearing planes as the planes Z = const, at the same time replacing K by KR. In other words, simple torsion may be regarded as effecting on each cylinder R = const, when cut along a generator and developed upon a plane, a simple shear of magnitude KR. Thus a complete analysis of torsion may be read off from the results obtained in Sect. 45. The formulae so gotten for R and other tensors are expressed in terms of physical components rather than tensor components. The angle of rotation is given by {} = arc tan i KR, (48.3) the principal stretches are 1, sec{} + tan {}, sec{} - tan {}, etc. 1 This is not the succession of the three shears X 1 =X+f(Y) X 2 =X1 x=X2 Y, = Y Y2 = Y1 + g (Z1 ) y = Y2 Z 1 =Z Z2 =Z1 z=Z2 +h(X2), their composition being the isochoric deformation x = X+ f ( Y), y = Y + g (Z), z = Z + h(X + /(Y)). 2 [1943, 6, ~ 30]. Sect. 49. Generalized torsion, shear, and inflation of a circular cylinder. 299 49. Generalized torsion, shear, and inflation of a circular cylinder. A family of deformations of torsional type is given by1 r =f(R), () = g(@) + h(R) k(Z), z =l(Z) +m(@) +n(R), (49.1) where the co-ordinates are chosen as in Sect. 48. If this deformation is considered throughout an entire cylinder, rather than merely a sector, g and m must be periodic of period 2 n. The function f represents a radial expansion of the cylinders R = const; g, an annular expansion; k, a torsion of magnitude varying from cylinder to cylinder, as moderated by h; l, an extension along the axis; m, a shearing of the generators of the cylinders; and n, a shearing of the cylinders along their generators. We readily calculate JJCEJI = I' 2 + f2 h' 2 k2 + n' 2 f2 g' h' k + m' n' (12 g' h' k + m' n')jR2 (12 g' 2 + m' 2)jR2 / 2 h' k' h k + l' n' / 2 g' h k' + l' m' f'2 y2 I' h' k f2 h' k' h k + l' n' (12 g' h k' + l' m')jR2 , f2h2k'2 + l'2 f'n' f'h'k r2(h'2k2+g'2jR2+h2k'2) f' n' r 2 (g' m'jR2 + h k' l' + h' k n') g' m'jR2 + h k' l' + h' k n' , l'2 + m'2jR2 + n'2 Ic = f2(h'2k2 + g'2jR2 + h2k'2) + f'2 + l'2 + m'2jR2 + n'2, II0 = f2 [(h k' n'- h' k l')2 + {(h k' m'- g' l') 2 + (g' n'- m' k h') 2}jR2] + +I' 2 [f2 h2 k' 2 + l' 2 + (12 g' 2 + m' 2)jR2, III0 = / 2 /' 2 (g' l'- m' h k') 2jR2 • (49.2) To find the isochoric subdass of these deformations, we set IIIc= 1 and solve the functional differential equation so obtained. If h = const, the result is r= VAR2 + B, () = ce + DZ, z = Ef) + FZ + n(R), A (CF- DE) = ± 1; (49.3) if h' g' k' l' m' oj= 0, the result is ' ~V I ii~) ':~- + B ()=Cf)+ K + h(R) (DZ + L). z = E f) + F Z + n (R), ACF=±G, if h' =I= 0, g' = 0, the result is ADE= 'f1; (49.4) , ~V 1 2~:,~:-: ~. \ (49.5) O=K+h(R)(DZ+~). z=l(Z)+EfJ+n(R); if h' =I= 0, l' =I= 0, the result is r= JR -~+B, 2Avdv l () = g(fJ) + h(R) (D~ + L), z = Ef) + n(R); (49.6) 1 This family includes one introduced by ERICKSEN and RrvLIN [1954, 6, §§ 8-10]. 300 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 50. if h' =!= 0, m' = 0, the result is r = YAR2+ B, if h' =!= 0, k' = 0, the result is (J = C @ + h (R) k (Z), z=FZ + n(R); o = ce + h(R), z = FZ + m(@) + n(R), (49.7) (49.8) where in all cases insignificant additive constants have been dropped. Numerous interesting cases are included, of which we mention but a few. If A < o, r is a decreasing function of R in (49.3h; such deformations represent the eversion of a hollow cylinder. For a: cylinder which is subjected first to longitudinal stretch Ä and then to twist T, put A =Ä-1, B=O, C= 1, D=rfÄ, E=O, F=Ä, n=O. If A =00 f(2nK2 ), C = 2nf00 , F = K2, B = D = E = n = 0, the deformation represents the closing of a circular cylindrical shell of wedge angle @0 so as to form a complete cylinder, furnishing an example of the distorsioni introduced by WEINGARTEN and VoLTERRA1• lf A = C =F = 1, B = D = E = 0, then (49.2) reduces to the forms included in (45.8), up to a permutation of co-ordinates, if K is replaced by n'. Thus the result of a shearing along the generators is locally the same as a simple shear of amount n' in the planes e = const. A similar remark holds for the case f' = g' = l' = 1, k = 1, m = n = 0, except that physical components must be used, the amount of shear is rh', and the planes of shearing are the Z = const planes. y 50. Bending of a block into a cyL __ I I I lindrical wedge. As our first example of the advantage of the use of different co-ordinate systems at X and ~. we consider the deformation 2 r =f(X), z = h(Z), O = g(Y),} (50.1) where for the most applications it is L-___ ......... _ _,_ ___ ..J....I __ ..J...._ ____ % desirable, but not necessary, to reFig. 9. Bending of a block. gard the reetangular Cartesian X, Y, Z system and the cylindrical polar r, 0, z system as having a common origin and a common z-Z axis. The family of planes Z = const is preserved; the Y = const planes are deformed into planes intersecting along the z axis; the planes X= const are deformed into concentric circular cylinders whose axis is the Z-z axis. The deformation is most easily visualized by restricting attention to a block whose faces are X, Y, Z planes and which does not include the origin (Fig. 9). Since GKM = (jKM, while g11 = g33 = 1, g22 = r 2, gkm = 0 if k=l=m, by straighttorward calculation from (26.2} 2 and (29.1h we get IJc~ll =IIC~II = /' 2 0 0 r2g'2 0 h'2 Ic = f'2 + r2g'2 + h'2, Ilc = f'2h'2 + r2g'2(f'2 + h'2), Illc = r2 I' 2 g' 2 h' 2. (50.2) To interpret the formulae just obtained, note that corresponding components of c-1 and C have the same numerical values for a given particle; of course, if we regard c-1 as a function of ~ and C as a function of X, the functions are different. For example, i: = C~, but if we take c-1 as a function of ~ and C as a function of X, we have c1/ = [ r g' (g-1 (0)) ]2 = C~ = [/(X) g' (Y) ]2. 1 [1901, 15]; [1905, 6]. 2 We follow TRUESDELL [1952, 21, § 421]. SIGNORINI [1943, 6, § 29] approaches the case z = Z through conditions of integrability for the pure strain. Sect. 51. Bending of a block into a wedge. 301 Since (50.2) is in diagonal form, the directions of the co-ordinate curves bothat X and ~ are principal directions of strain, and the principal stretches are f', rg', h'. The contravariant components of the unit proper vectors, expressed in the appropriate co-ordinates, are given by N 1 = (1, 0, 0), n 1 =(1,0,0), whence follows by (37.2h N 2 = (0, 1, o), n2 = (o, : o), 1 0 0 r-1 0 1 N 3 = (0, 0, 1), l n3 = (0, 0, 1), (50. 3) (50.4) To calculate Rk,,. from this result by the formula Rkm = g~ RkK• we need the values of the shifter gf, given by (App. 17.2) for the case when the co-ordinate systems at X and at ~ have a common origin and a common Z-z axis. Hence follows cos () - r sin () 0 IIRk mll = + sin () cos () 0 , 0 0 1 so that the axis of rotation is the z axis, and by (37.21) we get cos f} = cos () , or f} = ± () : (50.5) (50.6) Apart from a possible reflection, the angle of rotation is the azimuth angle. The result is obvious, but we have given the foregoing formulae in illustration of the general apparatus of Sect. 37. The isochoric subdass of (50.1) is r=V2AX+B, O=CY, z=DZ, ACD=±1, (50.7) where two constants of integration have been adjusted so as to center the deformed block with respect to the co-ordinates at X. If we wish to allow shearing of the cylindrical surfaces, we may generalize (50.1) by resulting in 1n~11= r = f(X), 0 = g(Y) + k(X). z = h(Z) + l(X). f'2 y2 I' k' f'l' f'2 + f2 k'2 + 1'2 f2g' k' I' k' y2 (g'2 + k'2) k'l' IIC~II= f2g'2 f'l' r2k' l' h'2 + 1'2 Ic= f'2 + r2 (g'2 + k'2) + h'2 + 1'2, Ilc= f'2h'2+ r2[g'2(f'2+ h'2+ 1'2) + h'2k'2], IIIc = r2 f'2 g'2 h'2. (50.8) h'l' 0 h'2 (50.9) The isochoric subdass of these deformations is obtained by adding to the second members of (50.7) the functions 0, k(X), l(X). 51. Bending of a block into a spherical wedge. If we let the co-ordinates at ~ be spherical polar, rp being the azimuth angle, we get in analogy to (50.8) r=f(X), O=g(Y)+k(X), rp=h(Z)+l(X). (51.1) 302 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 52. Analogaus calculations yield !' 2 y2 !' k' y2 sin2 () !' l' llc~ll = I' k' Y2 (g' 2 + k' 2) r2 sin2 () k' l' f' l' r2 k' l' r2 sin2 () (h' 2 + l' 2) f' 2 + r2k' 2 + r2sin2 () l' 2 r2g' k' r2sin2 () h' l' IIC~il = r2g'2 (51.2) Ic = f'2 + r2 (g' 2 + k' 2) + y2 sin2 () (h' 2 + l' 2), IIc = r4 sin2 () (g' 2 h' 2 + h' 2 k' 2 + g' 2l' 2) + y2 sin2 () h' 2 !' 2 + y2 !' 2 g' 2, IIIc = r4sin2() j'2g'2h'2. When the shearing functions k and l vanish, the co-ordinate axes are principal axes of strain, and Counterparts of the results at the beginning of Sect. 50 are easily obtained. For the isochoric case, it is easy to show that h' and k must be constants. Without loss of generality we can then set k = 0, h' = H, and obtain r=yAX+B, 3 cos () = ± A H Y + C, cp = HZ + l (X). (51.3) 52. Inflation, torsion, and shear of a sphere. Let the co-ordinates at x and at X be spherical polar, and suppose Then r = f(R), () = g(@) + h(R) k((/J), cp = l((/J). j'2 r 2 I' h' k I' 2 + y2 h' 2 k2 r2g' h' k r2 g' h' k y2 g'2 r2hh' k k' r2g' hk' 0 2 . 2 () _! sm -- h k' l' R 2sin2EJ 2 . 2 () ~m-~l'2 R2 sin2EJ IIC~il= R2 R2 R2 r2 h h' k k' r2 g' hk' r2 h2 k'2 + r2 sin2 () 1'2 ---- R 2sin2EJ R2 sin2 EJ R2 sin2EJ 2 '2 2(h2k'2+ . 2 01'2) I = /'2 + r2h'2k2 + r _!_ . + _!'_ -~srll __ C R 2 R 2sin2EJ ' r2 j'2 g'2 r2 [1'2 sin2 () (1'2 + r2 {h'2 k2 + g'2fR2}) + j'2 h2 k'2] IIc = R2 + R2sin2EJ ' 4 . 2 () III = __"_s~ /'2 '2l'2. 0 R4sin2EJ g (52.1) (52.2) The isochoric subdass in general does not appear to be easy to determine unless h and k are constants. Taking them as zero yields for the isochoric subdass ( cos e) () = ± arc cos - A L , (52.3) Sects. 53, 54. Small principal extensions. 303 where insignificant constants have been dropped. Forthis subclass, as also more generally whenever h and k are constants, the co-ordinate directions are principal axes of strain. Even though the two sets of co-ordinate surfaces coincide, it does not then follow that the deformation is a pure strain. In fact, it is evident that the principal axes of strain experience non-zero rotation unless Since f may be a decreasing function of R, eversion of a spherical shell is included as a special case in (52.1) and (52.3). e) Small deformation 1• 53. The meaning of "small ". The looseness with which the term "small" is often used has led to much confusion and some error. In this work, unless the contrary is stated explicitly, by "A is small when B is small" we mean "B--+0 implies A--+0". By "A(yl,y2, ... ,y")~B(y ,y , ... ,y") when Y1 , ••• ,y" aresmall" (53.1) we mean "A and B are analytic functions of y1 , ••• ,y" at y1 = y2 = · · · = y" = 0, and the expansions for A and Bin powers of the Yk have the same non-identically zero leading terms ". (As is weil known, successive applications of this definition may lead to seemingly inconsistent results 2.) While a more general definition could easily be formulated, this one suffices for unequivocal and correct statements of results of the type used by authors who neglect terms because they are "small ". We must warn the reader however, that it is unjustified to assume that the sequences of deformations standing behind the Iimit processes used in the following purely kinematical statements are compatible with any particular theory of continuous bodies. For example, we sometimes consider extension and rotation as independently variable, but in any particular theory of bodies a sequence of deformations is produced by a sequence of Ioads, and whether it is possible to choose the Iatter in such a way as to Ieave the Iocal rotation fixed but vary the extensions is du bious. It does not seem possible to give a precise meaning to the concept of "numerically small" often encountered in physical works. For example, if by "x <1" Wl" agree to mean "x ~ w- 6 ", then certainly 106 • w- 6 is not small. However, when this concept of smallness is used, the users invariably manipulate it in a fashion justified only if the sum of any finite number of small quantities is small, and a constant multiple of any small quantity is small. 54. Small principal extensions 3• When the three principal extensions Oa are small, by (31.4) we have (54.1) From (30.8) and (App. 39.11) it follows that every one of the components Ex M and ekm is small. By (26.4) and (31.1), the extension in every direction is small, 1 We do not have space to include the !arge body of results on infinitesimal deformation of surfaces. 2 For example, if I= y3 and g = y, I and g are small when y is small, and also y is small if either 1 or g is small. However, if we set we obtain V ~ 13 + I g2 + I g3 = y5 + y6 + y9' v ""' Ia = y9 when g is small ""' 1 g2 + 1 ga = y5 + y6 when 1 is small ""' Ia + 1 g2 = y5 + y9 when both I and g are small ""' y5 when y is small. Thus to assume both I and g small is not the same thing as to assume y small. 3 While the results of this section are due to CAUCHY [1827, 2], he did not distinguish the case considered here from the more restricitve one considered in Sect. 56. 304 C. TRUESDELL and R. TOUPIN: The Ciassica! Fie!d Theories. Sect. 54- and (54.2) From (26.9) and (31.1) we see that the shears are small, and after some manipulation we derive the formula r. R; 2EKM __ . cosece --(EKK + EMM)cote K=j=M. (54-3) KM VgK KWM M KM gKK gMM KM• As we have said, from (54.1) it follows that the shears are small. It would not be sufficient to replace (54.1) by the assumption that the extensions in some non-principal orthogonal triad of directions are small. To see this, note that by (26.5) and (31.1) the vanishing of the extensions in three orthogonal directions yields lE= 0 but imposes no bounds on IIE and IIIE. Hence the t5a may be arbitrarily !arge. To describe the case considered in this section without reference to the principal directions we should have to speak of small extensions and shears. For an orthogonal system of co-ordinates at X it follows that E~ R! the extension in the XI direction, 2 (yg11 /yg;-2 ) E~ R! the shear of the XI and X2 directions. In other words, for the physical components EIIR; Ll22 tr23 (54.4) L133 In an orthogonal co-ordinate system, the physical components of E for small extensions equal the corresponding extensions and halves of the shears. The dual formula holds for e, but there is in general no simple relation between the two sets of extensions and shears. From (54.1), (31.S)s and (31.5h we getl (54. 5) For small extensions, the first invariants of E and e equal the relative increment of volume. The fundamental identities (30.2) now assume the forms (54.6) while the fundamental lemma of Sect. 30 may be expressed thus: Any scalar function of E equals the same scalar function of e. Also from (30.8) follows 2 (54.7) It is only in the case of small extensions that the tensors E and e are convenient measures of strain. From (31.1) follows a formula which in itself includes much of what we have just derived: Ca R! 1 + E. (54.8) By (3 7.15) we now get simple expressions for the measures of rotation and elongation: RKM R; R[KM] + R[Kp EMJP• l EKM R; R(KM) + R(Kp EM)P- gKM• RKM R; b~ + RKM + E~- E~- Efr UKp, (54.9) 1 CAUCHY [1827, 2, Eq. (33)]. 2 Conditions of compatibility (Sect. 34) appropriate to the case of small extension have been discussed by CASTOLDI [1954, 2]. Sect. 56. Small extensions and small rotation. 305 where we ha ve retained terms of order 1 in the extensions. If we revert to our usual usage of "small ", we get simply -1 RKM R:; b~ + u~M• RKM R:; b~- u~M· l (54.10) Note that for thesesimple formulae tobe true, the rotations need not be small. E.g., in any rigid rotation (54.10) holds, with R:; replaced by =. 55. Small rotation1. The term "small rotation" shall refer to the case when the rotation-& of every element is small. By (35-3) follows 2 e=b-!(1 +b)-&2 +(1 +b)O(-ß-4). (55 .1) Also, from (36.7) and (36.8) it follows that Tx issmalland is given by -RYZ Tx R:; (55.2) V(1 +Eyy) (1 +Ezz) -Elz . Consequently R is small when the rotation is small. By the results of Sect. 38, then, a small potential deformation is a pure strain; alternatively: For the case of small rotation, a necessary and sulficient condition for pure strain is ii = 0. From (37.17) follows from which is plain the vectorial character of small rotations: 1RKP2RP M R:; b~ + eKMP (-ß-1 Af + ß-2 A:). (55.3) (55.4) Also, comparison of (55.3) 1 with (55.3) 2 shows that the rotation tensor R is completely characterized by its cross; writing R==!Rx, we have (55.5) Extending the convention of Sect. 36, we now use the same symbol R to denote at will either the rotation tensor or its cross; the context will show which one is intended. 56. Small extensions and small rotation. When both the rotations and the extensions are small, we may combine the results of the last two sections. From (55.1) it follows that the elongations are small, and hence the components ifKM are small. By (55.2) and (36.12) we get (56.1) For small extension and rotation, the length of the profection of the vector R upon a given direction is the mean rotation about that direction3• While ii is a small 1 That it is possible in interesting cases of deformation for the rotation to be numerically !arge when the extension is numerically small was noticed by ST. VENANT [1844, 3] [1847, 3], who mentioned the following examples: a thin sheet may be bent back so that its two ends touch, a long slender shaft may be twisted through several diameters. Cf. PoiNCARE [ 1892, 11, § 2]. ALMANSI [1917, 1, § 3], KAPPUS [1939, 11, § 4, footnote]. It is possible to exhibit families of such deformations in which E--*0 while E remains non-zero. See [1944, 4], where CISOTTI, using a previously derived result [1932, 2], has calculated E for the case of a rigid displacement, when E = 0. Cf. also CAsToLm [1954, 3]. 2 NovozHILOV [1948, 18, Eq. (1.110)]. 3 CAUCHY [1841, 1, Ths. 5. 6, 7]. Handbuch der Physik, Bd. III/1. 20 )06 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 57. vector representing mean rotations and R as given by (55.5) is a small vector representing rotation of the axes of strain, the two vectors are distinct unless further assumptions are made. Recalling our definition of "small" in Sect. 53, we see that in the present case {31.10) reduces to 1 Efr ~ Efr +t (- EKP RMP- RKP EMP + RKP RMp), (56.2) . while (55.1) becomes (56.)) 57. Small displacement gradients. If the extensions and rotations are small, it follows that the gradients uK;M are small. However, according to the definition in Sect. 53, to suppose conversely that the uK;M aresmall is not the same thing. In fact, if we retain only terms of lowest order in expansions in powers of the uK;M we get not only all the results of Sects. 54 to 56 but also in place of (56.2) simply 2 E~E. e~e: (57.1) When the displacement gradients are small, the strain and elongation tensors are equal, their physical components being related to the extensions and shears by (54.4). By observing (57.1) and (56.1), we see that theinvariant identity (57.2) now expresses an additive decomposition of any displacement into a pure strain and a mean rotation. This is Cauchy's 'l"esolution of a displacement with small gradients 3 . For an alternative interpretation based on the rotation of the principal axes, we may calculate the rotation R from {54.9) 3 , by (57.1) now obtaining 4 -1 RKM ~ bfr + RKM• Rkm ~ b~- ykm• (57.)) Hence (57.4) This last equation states that the axis of rotation is a unit vector in the direction of R, i.e., A K = ± RKJR. If we attempt to calculate the angle of rotation from (57.)), we get only 0. Instead, we go back to (54.9) 3 , obtaining where the last step follows by {31.10). By {)7.21) 2 and (36.12) we get {}= ±R: (57.5) (57.6) When the displacement gradients are small, the vector R points along the axis of rotation and is of magnitude equal to the angle of rotation, in radians 5• 1 We do not follow the argument of NovozHILOV [1948, 18, § 14], according to which the two terms with minus signs are dropped. We note also that NovozHrLov's equation (1.105) is false. 2 CAUCHY [1827, 2, Eq. (41)]. 3 Implied by [1841, J, Part II], but not explicitly stated. 4 To obtain (54.10)3, 4 , we held the rotation fixed and Iet the extensions approach zero. The apparently contradictory formula (57.3) results from the Iimit UK;M--*0, implyingthat both the extensions and the rotations approach zero tagether and at certain relative rates implied by their relation to the quantities uK;M· This illustrates the remarks in Sect. 531. 5 E. and F. CossERAT [1896, 1, § 11]. Other proofs are given by BoussrNESQ [1912, 1, §§ 9-10) and ÜDONE [1933, 9]. Sect. 57. Small displacerr:ent gradients. )07 A different interpretation of ii, due to CAUCHY1, is more conveniently presented in Sect. 86. Also by (5 7.1) and (34.8) it follows at once 2 that when the displacement gradients are small, the conditions of compatibility assume ST. VENANT's form: (5 7.7) These equations possess an extensive literature 3 • Since the co-ordinates at X and at x are independently selected, no simple relation between E and e can be expected. From (19.5), however, we conclude that if the uk;m are small, so are the uK;M• and conversely, and moreover (57.8) That is, when the displacement gradients are small, uk; m and uK; M are components of the same tensor at x and at X, respectively, and consequently the same holds ot ii and r, il ande: (5 7.9) By (57.1), analogaus relations hold between the RKM and the Rkm• the EKM and the ekm· As a corollary, it follows that jor small displacement gradients, corresponding components of R + 1, R, and r, and of E, e, E, ande in the common frame are numerically equal: 1+Rxx~1+Rxx~Rxx~rxx, l Rxy ~ R"Y ~ !!xY ~._r"Y, ... , Exx ~ e""~ Exx ~ e"", Exy ~ exy ~ Exy ~ exy> •.•• (57.10) For a curvilinear co-ordinate system, no such result as (57.10) holds without further assumptions. 1 [1841, 1, Ths. S-9]. 2 It follows, that is, if we add to our definition of "small" in Sect. 53 the convention that the derivative of a small quantity is small. Such a convention may be achieved by restricting all comparisons to specially selected sequences of functions, e.g. by a perturbation series such as that of Sect. 59. While we follow the custom of writers on elasticity in presenting the conditions of compatibility in this way, we feel compelled to point out that we do not share the confidence many authors put in these formal procedures. The mathematical questions at issue are these. Given a family of displacement vectors Bu depending upon a parameter Bin such a way that BuK;M-+0 as B-+0, isittrue that R}fkPQ~ö~:Äf(j~~ ERT;SU as B-+ 0? Conversely, if E KM satisfies (34.8), Iet u be the corresponding displacement; then does there exist a family Bu which for each B satisfies (34.4) and also satisfies Bu~u as B-+0? CASTOLDI [ 1954, 2] derives forms of the conditions of compatibility for the case of small extensions and for the case of small extensions with small gradients of the extensions. For the latter case he obtains precisely (57-7); hence there is a vector v suchthat EKM= V(K;M), and he finds that this vector v is either the displacement vectoru or eise v K = uK + t (uM uM); K. 3 They were given by KIRcHHOFF [1859, 2, §§ 1-2], but without a clear statement of their meaning, which was first explained by ST. VENANT [1864, 3, § 32]. The classical arguments do not deal with the question raised in the preceding footnote but in fact concern the necessity and sufficiency of (34.8) as conditions on the e!ongation tensor E; cf. BoussrNESQ [1871, 2, §I], KIRCHHOFF [1876, 2, Vor!. 27, § 4], BELTRAMI [1886, 1, Note at end] [1889, 2], FADOVA [1889, 7], E. and F. CossERAT [1896, 1, § 13], CESARO [1906, 2], VaLTERRA [1907, 7, Chap. I]. Explicit forms of (57-7) in curvilinear Co-ordinates are given by OngvrsT [1937, 8] and VLASOV [1944, 12]. 20* 308 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 58. The foregoing theorem is a special case of a general principle for replacement of the derivatives ojoXK by the derivatives ojoxk. By (19.4) we have in general (57.11) Hence if the displacement gradients are small, ofjoXK and ofjoxk are components of the same quantity at X and at x: (57.12) The most important property of the case of small displacement gradients is the superposability of a succession of strains and rotations. Beginning with strain, we consider first the more general case analysed in Sect. 39, in which an arbitrary deformation from X to X is followed by a second from X to x, yielding the exact formula (39.6). When the second displacement has small gradients, it is immediate from (39.6) 3 and (57.1) thatl (57.13) If the gradients of the first displacement are small, independently of those of the second, by (39.7) follows (57.14) while if the gradients of both displacements, taken together, are small, we get (57.15) Analogaus formulae for the mean rotation R follow easily, and we may assert Cauchy's superposition theorem2 : In the common frame, the extensions, shears, and mean rotations corresponding to the succession of two displacements whose gradients are small may be obtained by adding tagether the extensions, shears, and mean rotations corresponding to each individual displacement. 58. Small displacement. If we refer the co-ordinates at x and X to the same system (e.g., the common frame), for continuous f we have (58.1) for small displacements, the dimensionless error being of the order uk (log f) k. That is, for small displacement, any continuous function of x may be taken as the same function of X, and conversely, provided the co-ordinate systems at x and X be the same 3. For small displacement we have (58.2) again in a single co-ordinate system. Hence when both the displacement and the displacement gradients are small, (57.11) yields ofjoX1 ~ ofjo x1, and (57.9) shows that the tensors measuring strain, rotation, and elongation at X are numerically equal, component by component, to their Counterparts at aJ. 1 TRUESDELL [1952, 21, § 19]. An apparently more complicated result is obtained by CI SOTT! [ 1944, 4] for the case when the first deformation is rigid. 2 While not stated by CAUCHY, this theorem is obvious from his work [1841, 1, Part II]. 3 Otherwise, weshall not have x1 -+X1 as U-+0. Note also that the above statement does not apply to the displacement vector when considered as a function of both ;r and X, but does apply to it when considered as a function of ;r only or of X only. Sects. 59, 60. Physical motivation for the theory of oriented bodies. 309 59. Perturbation series. A concept of "small" more restricted than that used in the previous sections was formalized by E. and F. CossERAT1 and has been used by many authors subsequently. We consider a family of displacement vectors Bu given by the formal power series (59.1) where the functions u .. are given and kept fixed and B is a parameter to which no meaning need be attached. Various powers of B occur in the various measures of deformation and rotation calculated from (59.1). In each of these, the terms multiplied by B" are said to be "of order n ". In such a scheme it is impossible, for example, to have mean rotation ii and elongation E which are not of the same order. Moreover, the first, second, third, ... , derivatives of the n-th order approximation to Bu cannot decrease in order of smallness. Such a scheme, therefore, cannot include a displacementsmall in the sense used in earlier sections yet yielding strains which are not small, though it is easy to construct examples of families of displacements depending on a parameter in such a way that as the displacement vanishes the accompanying strain does not. The use of perturbation series restricts the dass of functions which are considered in the limit process which always stands behind the concept of "small ". By so doing, it facilitates formal calculation and perhaps also the proving of approximation theorems, but at the same time it limits the breadth and usefulness of results dependent upon it. f) Oriented bodies. 60. Physical motivation for the theory of oriented bodies. All the foregoing analysis in this subchapter has concerned the changes of length and direction undergone by an arbitrary differential element dX in a deformation from X to a;. In some physical problems more than this much information is needed. The additional data most commonly relevant are the fates of certain preferred directions. First, in some physical materials as they come to band physical anisotropy is observable. For example, the stress required to effect a given extension in an elastic crystal differs with the direction in which the extension is to occur. For the description of the deformation of crystals, then, a mathematical model should carry not only the particle X to the place a; but also directions at X into directions at a;. In fact, all relevant analysis for an arbitrary element at X has already been given. For the intended interpretation, however, the directions selected at X will not be arbitrary, and a dual formalism is not appropriate. All that is needed is a notation suited for distinguishing directions given at X by some rule known a priori. It is a different matter, however, with thin bodies. Thin bodies are mathematical models for physical objects having one or two dimensions much smaller than the third. For definiteness, consider a circular cylinder. The object of a theory of thin cylinders is to produce directly the same predictions as would result from a corresponding three-dimensional theory in the limit as the diameter of the cylinder approaches zero while the resultant loads and the resistance of the material to deformation approach finite and non-vanishing limits. For results of the kind wished, obviously it is not enough to represent the rod only 1 [1896, 1, § 8]. Equivalently, SrGNORINI [1943, 6, § 18] systematically calculated the n-ih derivatives at B = 0 of quantities associated with a family of displacements Bu. 310 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 60. as a straight line which may be lengthened and bent, since such a theory does not describe the twisting of the cylinder about its axis. A proper one-dimensional theory, then, cannot be restricted to point deformations, but must also include directions which can suffer rotations independent of the deformations of the points with which they are associated1• An adequate theory ofthin bodies cannot be merely the intrinsic deformation theory for a Riemannian space of one or two dimensions, but rather must consider such spaces imbedded in a Euclidean space of three dimensions. In the eighteenth century special theories of rods and shells were proposed with little or no reference to phenomena in three dimensions. At the beginning of the nineteenth century, after general three-dimensional theories were accepted in certain domains of mechanics, CAUCHY and PmssoN sought to obtain theories forthin bodies by averaging over a cross-section the results from a three-dimensional theory and then letting the cross-sectional area approach zero. From this point of view, no special geometrical apparatus is necessary for the description ofthin bodies. The twisting of a circular cylinder, for example, may be measured in a three-dimensional description by the parameter K introduced in Sect. 48; if we Iet the diameter of the cylinder approach zero, the value of K remains unchanged and emerges as the measure of twist in the limiting formulae appropriate to the cylinder represented as but a line. It has turned out, however, that in the case of most if not all of the specific theories of materials in common use, the behavior of solutions for a body of thickness a in the neighborhood of a = 0 + is highly singular 2• While the three-dimensional viewpoint is certainly the more fundamental, the mathematical problems to which it has given rise are among the most difficult in analysis. The majority of modern studies of thin bodies pretending to adopt the three-dimensional approach in fact fall to establish any rigorous limit formulae but instead rest on concealed assumptions equivalent to those for an independent theory. In any case, independent theories of thin bodies are of interest and currency sufficient to justify discussion of their foundations. Returning once more to three-dimensional bodies, even for them we may discern possible interpretations for an arbitrarily oriented medium. Suppose, for example, that the dielectric polarization vector is to be taken into account in the deformation of a crystal. This vector may be thought of as attached to a particle and suffering deformation, a deformation, however, which is not in general the same as that of a material element such as an axis of aeolotropy. In a similar way, a continuum model in which directions attached to a particle suffer deformation differing from that of material elements may be equivalent, as far as gross behavior is concerned, to a molecular model whose molecules have internal structure. That physical bodies should be presented as assernblies not only of points but also of directions associated with the points, in brief, as oriented bodies, was suggested by DUHEM 3• Theories based on this idea were constructed by E. and F. CossERAT4, but in the half century since their profound work was published, 1 This was first remarked explicitly, though not very clearly, by ST. VENANT [1843, 3, § 2]. 2 Cf. E. and F. CossERAT [1907, 1] [1908, 2] [1909, 5, § 4]. The typical phenomenon is evanescence or confluence of boundary conditions in the limit. Of course the remark and those immediately following do not apply to exact averagings such as that presented in Sect. 213. 3 [1893, 1, Chap. Il]. 4 [1907, 1] [1909, 5]. Sect. 61. Theoriented bodies of E. and F. CossERAT. 311 scant attention has been given to itl. Sects. 61 to64 present the theory of oriented bodies in the generaland invariant form given to it by ERICKSEN and TRUESDELL2• 61. The oriented bodies of E. and F. CossERAT 3• To the point X assign a set oft> vectors D0 (X), a = 1, 2, ... , tJ, the directors of the body at X. According as X ranges over a region of positive volume, a surface, or a curve, we may speak of a solid, a shell, or a rod. By a deformation of the body we shall mean a transformation carrying X into x and the directors Da at X into directors da at x. In equations, X= x(X), (61.1) in geometrical terms, a deformation consists in a displacement of the points and independent rotations and stretches of the directors. In the special case when d~ = x7KD{;, the directors arematerial elements, and their presence adds nothing to the description already given in this chapter. In the special case when d~ = g'kD{;, the directors are invariable elements and add nothing to what has been explained above 4• In general, the directors are neither material nor invariable; i.e., d~ =f= x7K D{; and d~ =f= g'kD{;. In an oriented body, strain and rotation are defined from (61.1h as for the ordinary bodies considered earlier in this subchapter. What is new is the relation between the directors. For the full possibility of interpretation mentioned at the end of Sect. 60, it should be possible to use an arbitrarily large number of directors. However, an appropriate description of strain has never been constructed in this degree of generality. Therefore we add the assumption that there are but three directors Da, and that these are linearly independent. This number suffices for theories of rods, shells, and anisotropic · solids. Let the reciprocal directors Da be defined as the triad reciprocal to Da, so that where Note that (61.2) (61.3) (61.4) where {)ab is the angle between D 0 and Db. When the directors form an orthogonal unit triad, we thus obtain det Gab= 1. The director triads may be used to define anholonomic components. For example, if we set X~ = D'k X{)., then X~ =Dif X~, and from (26.1) 2 follows C~m =Gab X~ X~, Now set 5 etc. (61.5) (61.6) (61.7) 1 There is an exposition by SuDRIA [1935, 8]. In his § 9, SuDRIA notes an error in the CosSERATS argument and gives a different proof of invariance. We do not follow either the CossERATS or SuDRIA in detail, and we do not present the material on time rates given in SuDRIA's Chap. II. 2 [1958, 1, Part II]. 3 E. and F. CossERAT [1909, 5, §§ 48- SO], ERICKSEN and TRUESDELL [1958, 1, §§ S-6]. Cf. also the discussion of infinitesimal strain by GüNTHER [1958, 4, § 2]. 4 Selecting three orthogonal and invariable directors, we may use them as a fixed an-· holonomic frame and so obtain invariant forms for the results given in many of the older treatments of finite strain, where all quantities are referred to the common frame. 5 A definition of this kind is the starting point of the theory of non-integrable strain constructed by KRÖNER [1960. 3, § 4, Eq. (13)]. 312 C. TRUESDELL and R. TouPIN: The Classical Field Theories. where we use the definition (App. 20.2). Hence D~M = WKPM n:, From (61.7) and (61.2) we find that 2U{KM)P = Gab;PD'J.:D"l.t. Therefore Gab= const is equivalent to ~MP=- WMKP· Sect. 61. (61.8) (61.9) (61.10) In this case, also, if we transform the Da at all points by the same orthogonal transformation, the components WKMP are invariant. From these results it follows that if the lengths of the directors and the angles between them are fixed, as is the case, for example, if the directors are chosen as an orthogonal unit triad, and if the point Xis made to traverse the XP co-ordinate curve at unit speed, the quantities WKMP are the components of angular velocity of the director frame Da carried by X. We do not need to use (61.10), and therefore we do not impose the restriction Ga 0 = const, but this special case serves to motivate our calling WKMP the wryness of the director frame in the undeformed material. Dual results hold for the deformed material, but the dual wryness tensor, since it refers only to the relative configurations of the director frames at different points in the deformed material, does not afford a comparison between the deformed and undeformed conditions. What we wish, in the kinematic terms used above, are generalizations of the angular velocities of the director frame at x relative to those of the director frame at X when x traverses the curve into which the path of Xis deformed. To this end, introduce the relative wryness at x: (61.11) Here, however, we encounter the quantities d~;K, the director gradients, which appear in the theory of deformation of oriented bodies along with the deformation gradients x~K as primary local variables. In general there are 27 director gradients; when the directors form an orthogonal unit triad, only 9 director gradients are independent. Set then d~=AkKD{f, D{f=aKkd~, a=A-1 . From (61.11) follows (61.12) (61.13) (61.14) A deformation of an oriented body is rigid if not only C = 1 but also the directors at x may be obtained from those at X by a uniform orthogonal transformation. In this case the tensor A defined by (61.12) is a covariantly constant orthogonal tensor, the first term on the right-hand side of (61.14) vanishes, and we see that F is orthogonally equivalent to W. To make use of these results, we consider the anholonomic components of W with respect to the directors at X, the anholonomic components of F with respect to the directors at x: WaoP== WKMP D{f Df! = D{f DbK;P' I'aoP== F.mPd~d'b = d~dbk;P' so that by (61.8) and (61.11) 3 we have (61.15) (61.16) Sect. 61. Theoriented bodies of E. and F. CosSERAT. 313 lf we put g00 == g,md~ dg', then (61.14) assumes the form1 FaoP = AkK;PDf! dak + gac cecwebP· (61.17) Also 2F(ab)P = gab;P· Now Ais an orthogonal tensor if and only if gab= Gab· When Ais a uniform orthogonal tensor, (61.17) reduces to FabP = WabP· (61.18) Conversely, suppose (61.18) holds. From (61.15) follows gkmd~db;P = gKMD: D~p; Since this holds for all choices of a and &, we deduce gab;P = Gab;P· Hence (61.19) (61.20) (61.21) where the Kab are constants of integration representing constant differences of length and angle. Thus (61.18) asserts that the lengths and angles of the two sets of directors differ by constants, so that the tensor A is orthogonal everywhere if it is orthogonal at one point. From the foregoing analysis we conclude that necessary and sufficient conditions for rigid deformation of an oriented body are 1. CKM = gKM• l 2. FabK = WabK• f 3. At some one point, Ais orthogonal. (61.22) The tensor W thus appears as analogous to the metric tensor g, while F is analogous to GREEN's deformation tensor, C. However, we must bear in mind that while (61.22) 1 is a general tensorial condition, (61.22) 2 is not, since the anholonomic components FabM and WabM are calculated with respect to different frames. The relative wryness F is thus a measure of strain of orientation, as contrasted with the strain of position analysed earlier in this chapter. J ust as the CKM are certain quadratic combinations of the deformation gradients x~K, the FabM are certain linear combinations of the director gradients d~;K· In general, the numbers of independent quantities xkK, C KM, .. d~. K, and FabM are, respectively, 9, 6, 27, 27. , - , It would be possible, in analogy to Sect. 32, to define and characterize a general measure of strain of orientation. One such measure is given by (61.23) 1 Since the anholonomic components of W and F are defined with respect to different anholonomic frames, the usual rules for manipulating anholonomic components do not always apply. For our purposes the particular choices (61.15) for the anholonomic components are essential. For example, if instead we set W\p"" WKMPD'k Df;l, F0 bp =cFkMpd~ dg', we obtain from (61.14) F\p = AkK;P d~ D{f + W0 bP· Therefore a necessary and sufficient condition for A to be covariantly constant is F0 bp= W0 bP• but this is not at all the same as condition (61.18). 314 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 62. corresponding to GREEN's tensor E (Sect. 31). Necessary and sufficient for a rigid deformation are the conditions E=O, J=O, (61.24) along with the orthogonality of A at one point. We do not take up the question of characterizing all possible measures of strain of orientation. A principle of duality analogaus to that of Sect. 14 may be formulated. The reader will easily construct analogues of c, e, etc. Not pausing to explore the structure whose traits have just been presented, we note only a formula for the gradient of relative wryness. By differentiating (61.11) we get (61.25) Hence F;,bM;K = F"mM;K d~db + F"mM (d~;K dg' + d~ db;K), } = dbk;MKd~ + gre (FaeMF'rbK + FebMF'raK - FacM-FebK) · (61.26) There is no doubt that with the apparatus constructed here much of the work on special theories of oriented media could be unified and correlated. In the following sections we present only the siruplest and most immediate applications. In view of what we have explained already, the theory of deformation of an oriented body can be organized as follows: I. Strain of position A. Intrinsic theory B. Imbedding theory II. Strain of orientation. The intrinsic theory, based on relations between the fundamental tensors g and C, for three-dimensional boJies has been studied in detail earlier in the chapter. For bodies of one and two dimensions it is somewhat simpler but not essentially different. While curved rods and shells are not Euclidean spaces, the analysis based on the mapping (16.2) and the formulae (26.1) is valid in a Riemannian space of any nurober of dimensions 1. However, the conditions of compatibility given in Sect. 34 are no Ionger appropriate, since the curvature of a rod or shell may be changed arbitrarily by deformation. The theory of finite rotation given in Part c continues to be meaningful, but only when the rod or shell is regarded from the Euclidean three-dimensional space in which it is imbedded. In treating the imbedding theory, however, it is fitter to use the concepts of differential geometry: The curvature and torsion of a curve, the second differential form of a surface. In the above program, then, Part I A has been given in principle already, while Part I B can be gotten with little labor from geometry, so in what follows the major attention will be given to Part II. 62. Anisotropie solids. To represent anisotropic solids, the directors are chosen as material elements 2 and hence are related by (20.3): d~=x\D{!, x~K=d~D=O, and thus, provided rp=f=O, from (63.18) we obtain (63.19) That is, the twist is the excess of WMK NK ßM over the torsion. The scalar WMK NK ßM is itself an anholonomic component of W with respect to the principal frame of '??. If we let D* be a unit vector suchthat D*, D, and T form a right-handed unit 318 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 63. triad, sinceD*K=-NKsin q;+BKcos q;, then WKMBKNM=D*K~MDM, so that by (63.1üh and (63.19) follows (63 .20) This formula is the basis of the classical attempts to construct a theory of strain for bent and twisted rods (for references, see the appendix at the end of this section). Since the rotation of D may be prescribed in any smooth way as we traverse~' there can be no general connection between the twists of two different directors. In the applications of the theory to elasticity, it is customary to think of a rod as a line furnished with normal cross-sections; these are represented, for the purposes of the theory, only by their principal axes of geometrical inertia and perhaps one or two constants such as their geometrical moments of inertia about these axes. The twist of the unstrained rod is then defined as the twist of either of these axes relative to ~. Since these axes are normal to one another, a unique twist is obtained. Now consider the deformation of ~ into c, defined not only by (63.3) but also by a relation setting directors da along c into correspondence with the directors Da along ~: (63.21) Pursuing the interpretation mentioned above, we might choose for one director d a unit vector along the direction into which one of the directors D, considered as material, is deformed in the three-dimensional deformation experienced by the finite rod whose strain our aue-dimensional theory is intended to represent. Such a director, d 0 , would not in general be orthogonal to c. In practice, d is selected as the normal to the unit tangent t which lies in the plane of d 0 and t. The triple orthogonal directions so determined are called the principal torsionflexure axes in the deformed rod. The twist in the deformed rod is then defined as the twist of d relative to c. Obviously this classical procedure is motivated only by formal simplicity and furnishes an inadequate description of the strain of a rod. In the first place, twist is defined unsymmetrically with respect to the strained and unstrained rods. While the twist of the unstrained rod is uniquely determined, two different twists can be obtained for the strained rod, depending on which of the principal axes of inertia of the cross-section is selected for D in the unstrained rod, and by the above remarks, these two twists are in general entirely unrelated to one another. Moreover, the insistence that d be normal to c is merely artificial and does not represent any geometrical requirement. For a more precise description, we need only adapt to c what has already been done for ~. Since the operation "~" defined by (63.6) is a derivative with respect to S, not s, the analysis, while strictly parallel, is not merely dual to that given above. W e set F a _ dad~k b = k b· (63.22) The Cab are the components of deformation; pab is the relative wryness of c. By analogy with what was said earlier in connection with ~ it follows that if t0 , Cab' and pab are given functions of 5 subject to the conditions that Cab be symmetric and positive definite and that (63.23) Sect. 63A Appendix. History of the description of strain in a rod. 319 the rod <-" is determined to within a rigid motion combined with a reflection, its arc length s being obtained by integrating the equation (63.24) A. being the stretch. What has been shown is summarized in the following fundamental theorem on the strain of a rod: Given a rod~ with arc length S and directors Da, prescription of the eighteen scalars ta, Cab, andFab as functions of S subfect to the aforementioned conditions determines a second rod c with arc length 6 and directors da (s), uniquely to within a rigid motion combined with a reflection. In other words, the quantities ta, Cab, and pab furnish a complete differential description of the strain of a rod. It suffices to specify only the twelve scalars ta, Cab, and Ca[bF0 , 1 since one can then calculate pab using (63.23). The apparatus constructed is very general, enough so to include what would be regarded physically as an anisotropic rod. To represent a physically isotropic rod, let D1 be the unit tangent to ~ and d 1 the unit tangent to c. In this case, T1 = t1 = 1, T2 = T3 = t2 = t3 = 0. The remaining two directors, both along ~ and along c, may be assigned arbitrarily. Again we consider a general director frame. To determine the finite rotation of a rod, it is necessary to regard the rod from the Euclidean space in which it is imbedded. So as to motivate the definition shortly to be given, we first consider the case when ~ is a material curve in a three-dimensional body and the directors Da are material elements, this being the case that one-dimensional theories of rods are intended to idealize. Then, as remarked in Sect. 62, in (61.13) we have AkK=x~K· aKk=X~. By the theory given in Sect. 37, we can decompose xk;K uniquely into a stretch and a rotation, and the rotation so determined is the rotation of the principal axes of strain at X into the principal axes of strain at x. In a purely one-dimensional theory, we do not have available the x7K• but from the two sets of directors we can define AkK by (61.12), and, since the linear independence of the directors assures us that AkK is non-singular, from the polar decomposition theorem (Sect. App. 43) we can write (63.25) where 0 is an orthogonal double tensor and P and p are positive definite symmetric tensors. The tensor 0 then represents the local rotation of the rod. What we have proved regarding it is summarized as follows: Given two rods with directors Da and da, a unique local rotation is defined by (63.25); the rotation is independent of the choice of the directors to this extent, that if we imagine and then leave fixed a deformation of a three-dimensional body in which ~ is a materialline and the given directors Da are carried materially into the given directors da, then we may choose as directors any other sets of linearly independent material vectors and obtain the same rotation. When the directors are chosen as above, (63.22) 2 becomes (63.26) the quantities Cab are anholonomic components of the three-dimensional deformation tensor C with respect to the directors of ~. 63A. Appendix. History of the description of strain in a rod. The concept of twist may be traced back to ST. VENANT [1843, 3, ~ 2], who called it "gauchissement". He obtained (63.20) [1845, 3], having given an approximation to it earlier [1843, 3, ~ 15]. BrNET [1844, 1] claimed that he had introduced the concept of twist shortly aftcr 1815. The introduction of the principal torsion-flexure axes is due to ST. VENANT [1843, 3, ~ 2]. Other attempts to formulate a general theory of strain in a rod were made by KIRCHHOFF [1859. 2, § 2] 320 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 64. [1876, 2, Vor!. 28, § 2] and CLEBSCH [1862, 2, §§ 48-49, 55]. All this early work employs more or less hidden approximations and is difficult to follow with confidence. The first Straightforward analysis is that of LovE [1893, 5, § 233]. Our Sect. 63 follows ERICKSEN and TRUESDELL [1958, 1, §§ 7-14]; their treatment is freely adapted from that of E. and F. CoSSERAT [1909, 5, § 21] (cf. also [1908, 2]), whose eonsiderations, in our opinion rather fragmentary, are restricted to the common frame. The only previous analysis approaching the generality of Sect. 63 is that of HA Y [ 1942, 8, §§ 2-3]. In effect, he takes the Da as any unit triad suchthat D 1 is the unit tangent to '6', and he chooses co-ordinate systems such that not only D{f = l5{f but also d~ = (j~. Hence (61.12) yields AkK = (5~, aKk = t'lf, but it must be remernbered that the co-ordinates at x are not generally orthogonal. In these Co-ordinates we have from (63.22) 2 (63A.1) thus the components of the metric tensor at x are numerically equal to the components of stretch, this being a generalization of HAY's Eq. (3.9). From (63.22) 3 we get in these Co-ordinates (63A.2) where the rod c is taken as the x1-curve with x1= s. The skew-symmetric part of this equation is essentially HAY's Eq. (3.5); hence our relative wryness F includes and generalizes HAY's rotation vector w. 64. Shells 1• Let the undeforrned surface Y be given by XK = XK (VI, V2) = XK (VE)' the deformed surface ~ by (64.1) (64.2) where, as in the rest of this section, Latin indices run from 1 to 3, Greek indices from 1 to 2. The deformation is specified by the functional forms of the righthand sides of (64.1) and (64.2), augmented by V= V (V). (64.}) For full generality, it is sufficient but not necessary to take ve=J~ VE. The surface metric tensors are given by dS2=At1EdVL1dVE=A edv d~e,} ds2 = a6 e dv6 dve = a.-1 EdV.-1 dV"', where (64.4) (64.5) (64.6) and where ";" denotes a partial derivative. If A (V) and a (v) are known, and if the parameter transformation (64-3) is given, we may calculate for any material element the changes of length and angle occasioned by the deformation. Also, we may calculate the total curvatures R1212 of Y and ~- Corresponding to any assigned real symmetric tensor a(v), with A(V) and the relation (64.3) regarded 1 The general theory of strain of position for a surface was discussed by UsAI [1909, 9]. Strain of orientation was considered but scarcely analysed by E. and F. CossERAT [1908, 1] [1909, 5, §§ 30-32]. Ünr treatment fol!ows ERICKSEN and TRUESDELL [1958, 1, §§ 15-20]. Sect. 64. Shells. 321 as given, there exists a surface d into which Y is deformed, and any two such surfaces d1 and d2 are applicable. In principle, this completes the intrinsic theory of strain of position of a shell. In terms of the three-dimensional equations (64.1) and (64.2}, by (App. 19.7) and (App. 21.5) we may calculate the fundamental forms A, B, and a, b: ALI =gKMXfLIX~, a~.;=gkmx7.,x1. (64.7) BLI =NKX~ , b6 .;=nkx7"<• where N and n are the unit normals to Y and d, and where the total covariant derivatives are defined as in Sect. App. 20. We have BLI 8 dVLldV8 =B".;dv"dv<, b";dv6 dv<=bLI 8 dV"'dV8 , (64.8} where (64.9} The tensors a and b are related by (App. 21.8}, the Gauss and Mainardi-Codazzi equations; A and B, by the duals of those equations. (In applying here the results of Sect. App. 21, we are to replace Greek majuscules by Greek minuscules throughout.) When Y and a parameter mapping (64-3} are given, assignment of arbitrary symmetric tensors a and b satisfying (App. 21.8) determines a surface d into which Y is deformed, uniquely to within a rigid motion1• The surface d is obtained by solving (App. 21.7} with xk and nk assigned at one point, and Yis obtained by solving the dual equations. In principle, this completes the imbedding theory of strain of position. In approximate theories of elastic shells it is customary to take the changes of principal curvature as measures of strain. To do this in the exact theory would introduce complications analogous to those resulting from using the displacement vector and the strain tensors in finite strain of three-dimensional bodies. The theory of strain of orientation for a shell may be constructed in analogy to what was done for three-dimensional bodies in Sect. 61. At the points X of the undeformed shell .9, assign a set of three directors Da, and put so that X~= n: X~, D:;LI = D~Wb01 • Then by (64.7h follows ALls= Gab X~ X~. while by (64.7} 3 follows BL!s= NKD:(x~;E+ wabEX~). From the relation (64.11}, XfsLI = Db' (X~;LI + WbaLI X~), D~ELI = Df (W\LI WbaE + W'aE;LI} · We also have the integrabi:ity conditions 1 E.g. EISENHART [1940, 9, § 39]. Handbuch der Physik, Bd. III/t. 21 (64.10) (64.11} (64.12) (64.13) (64.14) (64.15) (64.16} 322 C. TRUESDELL and R. TüUPIN: The Classica] Field Theories. Sect. 64. whence follows axf.d Wb X" - ·av.E"l + a[.E" .d]- 0, (64.17) wc ".Wb 'LI- wcb.d Wb ". + 2 ~wcc:[.d = o. b- a a- av.:oJ (64.18) From (64.10h and (64.'11) 2 (64.19) Eqs. (64.11) may be regarded as a differential system for the equation X(V) of the shell and for the assignment of directors upon it. When X~, Gab= Gb a, and Wba.d are prescribed as functions of VL1 subject to the conditions (64.17) to (64.19), the system (64.11) is completely integrable. Locally there will then exist a unique solution X (V) and Da (V) taking on prescribed values at one value V0 of V. Provided that not all the quantities X?.d X~1 vanish, that Gab be positive definite and that the prescription of Da (V0) be consistent with (64.10) 2 , the solution represents an oriented shell and the relation (64.10) 1 holds. Using the uniqueness of solutions, it isasimple matter to show that X~, Gab• and Wba.d determine Y and its direcfors to within a rigid displacement combined with a reflection. At the points JJ of the deformed shell 6, assign the directors da, and, as suggested by (63.22), put (64.20) so that (64.21) The Cab are the components of deformation; F\Ll is the relative wryness of J; the x~ are certain tangent vectors. Then a6;= C ab XLI a X,s- b VLl V"" } ;6 ;~· b61; = nk d~ [F\ s V:f x~ V:1 + x~;.E" V:1 V:1 + x~ V:1öJ. (64.22) Thus we see that knowledge of v, x~. Gab• and Fab.d and d~ as functions of V suffices to determine the first and second forms of the deformed shell 6. By analogy with (64.17) to (64.19), we have (64.23) (64.24) and (64.25) i Further, when v, P\s, x~. and Cab are given, subject to the requirements (64.23) to (64.25), lv";.s-J =f= 0, and the conditions that Ca b be symmetric and positive definite and that not all the quantities x[.d x~1 vanish, the differential system (64.21) determines an oriented shell J to within, a rigid motion combined with a reflection, and (64.20) is satisfied. What has been shown, then, is summarized in the following junda·mental theorem on the strain of a shell: Given a shell Y with fundamentalforms A (V) and B (V) and with directors Da, prescription of the 32 quantities vö, F\ s, x~, and Cob Sect. 64. Shells. 323 as functions of V subfect to the aforementioned conditions determines a second shell ~ with equation x =X (v) and directors da (v), uniquely to within a rigid displacement combined with a reflection. In other words, the quantities v~. F\z, x~, and Cab furnish a complete description of the strain of a shell. A theory of rotation is easily constructed by analogy to what was done for rods in Sect. 63. We now consider resolution of these results in terms of normalandtangential components. Since x:;_, X~, and NK are three linearly independent vectors, we may write where D~==AJED{fXK;E, Da=D{fNK. (64.26) (64.27) Then by (App. 21.6) follows whencc D~.1 = (D~;.1- DaB~) X~+ (D~ Bs.1 + Da;.1) NK, NKD~-1 = D~ Bs-1 + Da;-1, XK;ED~.1 = DaE;.1- Da BE.1· (64.28) (64.29) (64-30) From the fact that B.1E and A.1E determine X(V) to within a rigid motion and that D~ (V) is uniquely determined by D~ and Da when X (V) is known, it follows that B.1 s, A.1 3 , D~, and Da determine Y' and its directors to within a rigid motion. Other than the duals of (App. 2!.8), there are no compatibility conditions to be satisfied by these quantities. It involves no restriction to require that ~ and one director not tangent to ~. say d 1 , be material with respect to an unspecified three-dimensional deformation. That is, if x(v), d1 (v), v(V), X(V), and D 1 (V) are given subject to the condition that d1 (D1) be not tangent to ~(Y'), there will exist infinitely many mappings x (X) such that x(v) =x(X(V(v))), X(V) =X(x(v(V))) (64-31) and d1 = x~K Df, (64.32) the quantities x~K being uniquely determined as functions of V by (64.32) and x~6 = x7KX~ V;1. (64.33) Since the remaining directors can be assigned arbitrarily, they will not necessarily be material with respect to this deformation. The formalism just given is easily specialized so as to give results depending on particular choices of Co-ordinates for the strain of position. We consider here only the most general form of the usual approach, that given by SYNGE and CHIEN1. The intended in:terpretation is that ~ and d 1 are material, so that (64.31) to (64.33) apply, with D 1 varying with the deformation x(X) in such a way that d 1 =n. Choose co-ordinates and _parameters such that x(X)=X, X(V)=(Vl,V2,0), x(v)=(vl,t 2,0), gk 3 =bka· (64.34) Then and N = b (Gaa)- i !ff 'M3 ' (64.3 5) (64.36) 1 [1941, 9]. Our formula (64.38) is essentially their formula (69). Cf. also CHIEN [1944. 2, Eq. (6.13)]. 21* 324 C. TRUESDELL and R. TouPrN: The Classical Field Theories. From (64.7)s, (64.34) 2 , (64.35)s, and (64.36)1 we have 1 2BsA = 2NK(a~:~~A +{:Ax~x~- {;Ll}x~). = 2 ( caa)-1 {/K} ~f ~~, =(GSS)~(oGaJ_~J+ oGKa_~K- oGJK ~J~K)+ avA z avz A axa z A + caa)-~GaK ~A (ßAzA + oAAA _ oAzA) K avA avz avA ' which we may write in the form oGJ~ M~K = oGJa ~.{,. + ~~!!__ 1 + (Gaa)-1 caK ~A x ) ()X3 .::. A ()VA - oVZ Lf K ( oA z A + oA A A_ _ oA ~-) _ 2 (Ga a) _ * B _ X oVA oVZ oVA .::.A· From (64.27) and (64.35), Sect. 64. (64.37) (64-38) DJA=GKa~~. Dl=(G33)-~. (64-39) Now, using (64.34)1 , (64.35)1 , and (64.39), we obtain X DK - G Xl ( anlf { K l XM np) 2 K;Z J;A-2 JK ;Z ()VA+ MPf ;A 1 • = 2G K~f{:A~~ ~:, From (64.30), (64.39), and (64.40), we have 2XK;(zDfA)= ~:-~f~~ =DJZ;A + DJA;z- 2DIBZA• oD1z oD1A { A} =avF+ avs--2DIA ELl -2DIBZA• = oGJa !5-l.+ oGJa !51- G ~JA AI (oAz_~ + oAAI- oAzA_) avA "" avs "' Ja A ovA oV8 avi -2 (G33)-l BzA' (64.40) (64.41) which is similar to, but not identical with (64.38). The apparent discrepancy is resolved by noting that the equations AEA AA A = 15~, 0 = b~ = GSK GKJ ~~ = 33 J!5~ + GSK !5-k AsA (64.42) imply that (G33)-1G3K ~'k = _ 31 ~~AAs. (64.43) For ~. we have by analogy with (64.38) tagether with (64.34)1•4 (64.44) SYNGE and CHIEN find it sufficient to introduce nine measures of strain. The set a a if Y >O, expressing the fact that each particle moves aside sufficiently to let the cylinder pass by. Equivalent to (71.12) is the system • a2 . y =V 2 sm20. r (71.15) From (71.10} 2 and (71.14h follows Ö= 5_ (1 + ~) = __!"__ VY2 + 4a2 sin2 0 y2 y2 y2 ' (71.16) and hence from (71.15) dx a2 cos 20 dy a2 sin 20 -dO- t:Y2+4ä2sin20 ' dii VY2+4a2sin20 ' (71.17} where Y is kept constant. Ast varies from - oo to + oo, the parameter 0 varies from 0 to :n;. In the notation of Jacobian elliptic functions, put k __ 2a ll cos u = - sn v,. VY2+4a2 ' (71.18) so that the range of the parameter v is -K(k) to +K(k). By integrating (71.17) we get the equations of the paths 1 in terms of the parameter v: y(v) = Y + -~- (dnv- k'), l x(v) =X+-~ [(1- ~ k2) (v + K)- E (am v)- Ej. (71.19) To get the corresponding times, we observe directly from (71.15) and (71.16) that d • l -di ( x - y cot 0) = - i sec 2 0 + y cs::2 0 0, 2 (71.20) = - V-a + V (1 + a2_)· = V. y2 y2 ' hence V t = x - y cot 0, (71.21) where t = 0 corresponds to x = 0, 0 = t n, y = Ym. Thus the cylinder drives the particles ahead and, except for those situate upon its course, aside. In so doing it causes each particle not directly before or behind it to traverse a loop which is symmetrical about a line normal to the path of the cylinder. The maximumnormal displacement dm, which occurs when the cylinder is abreast of the particle, follows from (71.14) 2 : (71.22) The greatest such displacement, dm =a, results from the limit Y --*0, showing that the particles nearest the cylinder's path are those most pushed aside, yet in amount barely sufficient to let the cylinder by, while if Y = 0 the particle is carried straight along ahead or behind the cylinder. Each path is a loop. For very distant particles, it is approximately a small circle of radins a2/2 Y. When the cylinder is gone past, each particle has suffered a permanent forward displacemen t dt, the amoun t of w hich ma y be read off from ( 71.19) 2 : dt=x(K)-X= 2 ka [(1--}k2)K -E], (71.23) 1 As was shown by RANKINE [1864, 2, § 18], the curvature of the path line is 4 (y- t Y)fa2• A very simple expression for the path in terms of arc length was obtained by HAVELOCK [1913, 3]. Cf. also MILNE-THoMSON [1938, 9, .§ 9.211. 336 Fig. 13 a. C. TRUESDELL and R. TouPIN: The Classical Field Theories. ",.,.,.- -........ / --~-- ....... a / / / / ---- / / / Yz I I I Fig.13 b. Sect. i1. ----- Fig. 13 a. MAXWELL'S sketch of the path.; of the particles which at t = - oo were situatfd upon a line perpendicular to the direction of motion. Fig. 13b. DARWtN's accurate drawing of particle paths. The broken lines at left and right are the loci at t = 'f oo of the particles which are abreast of the cylinder as its center crosses the centralline of the drawing. The numbers indicate the times of passage to the points marked, in units in which the cylinder moves a distance a. . I I ' I I . I I I . . I I I I I . . . I I I I I I n I I I I I I I I I . . : I I I I . I I ~ \ t I . +-Ii" I l J I~ . I I 1--- i \ \\\\~'""' \\ \~ ...-: ~~ \ """'"'"" ~ 7 ". ~~:~, ~ \ \ \ ---------~~ -- ~1 I - -- - -- -------- ------;-;--/7 -~~ ~Vr- ~- ~- -~-- ------- 1117~ I I I 1 111!/. f--t-./7 I -r-, r--~ I 1 I .I I I -"!-I t+-: I I lf-+t-+ I I I I I I I I I I I . I I I I I I I I . . I . I I I I I I I I I I I I I ' I ' I I I l I I ' I . I I I I I I I ~ I ! ~ I I I Fig. 13 c. MAXWELL's sketch of the potential flow past a cylinder. One set of heavy lines are the stream lines of Fig. 11. The other set are the present loci of the particles which at timet= - oo were situated upon perpendiculars. and for particles near the path of the cylinder this is approximately a [log (8a/Y) -2]. MAXWELL's and DARWIN's drawings of this motion are reproduced as Fig. 13. Sect. 72. Material derivative. 337 b) Material systems. 72. Material derivative. In the common frame, Jet A::: be a tensor expressed as a function of Z and t only. Then the partial derivative oA:Jot is the rate of change of A::: apparent to an observer situate upon the moving particle Z. For general co-ordinates X, we write DA::: _ aA::: (X. t) I ~ = ~-ot-- x~const' (72.1) but this formula docs not in general define a tensor. Now consider the double tensor A =A(x, X, t), x and X being general co-ordinates. Then the tensor dA:Jdt, or A:::, defined by dA"'···flk ... m • oA··· [ oA··· { k} y ... 6n ... p = A"' = _ ... + _ .. ._ + Aa; ... ßr ... m+ ••. _ \ dt ... ot oxq qr y ... bn ... p -- {;JA~:J~:::P'- ... J_iq, = aA::: + A" .. iq (it ... 'q ' (72.2) is the material derivative 1 of A:::. In (72.2), A":: is a function of x, X, and t, and ofot is executed with both x and X held constant, while ofoxq is executed with t, all xk except xq, and X all held constant. The field x is regarded as given. If we suppose x and X related through a motion x =X (X, t), X =X(x, t), with x obtained from (67.1), then (72.2) is an invariant time derivative whose value is independent of whether x be replaced by x (X, t) in some or all of its occurrences in A:::. There are two major special cases. First, when the material description is employed, we have A =A (X, t), and (72.2) becomes dA~·:J~·::.p = A_rz ... ßk ... m = __DA~:J!~_::_t + [{ k} Aa; ... ßr ... m + _l dt y ... bn ... p Dt qr y ... bn ... p '· • (72.3) _ { r }Acx ... ßk ... m_,,.Jxq· nq y ... 6r ... p ' in this case, Ais the intrinsic derivative of A with respect to the field x. Second, when the spatial description is employed, we have A =A (x, t), and (72.2) becomes dA::: = A.· .. = oA::~ + A" .. iq or A = o~At + x · gradA, (72.4) at ... ot ... ,q • u where iJA::: __ aA:::(;r, t) I 8t - == __ O_t __ z~const ' (72.5) The essential term A:::, q xq in (72.4) is called the convection of A. Its non-linearity gives rise to many of the celebrated difficulties of hydrodynamics. We have introduced two notations, dfdt and a superposed dot, to stand for the material derivative. Except for rather complicated expressions, we shall 1 The concept of material derivative and the formula (72.2)a in the common frame were used implicit!y by EULER [1770, 1, § 6] and LAGRANGE [1783, 1, §§ 10-11] [1788, 1, Part Il, § 11, ~~ 11-12] and were formalized by CousiN [1786, 1, § 1], PIOLA [1836, 1, §VII, ~ 77] [1848, 2, ~ 14], and STOKES [1845, 4, § 5] [1851, 2, § 49], with the notations ojot, ', and D/Dt, respectively. In the classicalliterature general results are usually derived in the common frame, where there is no distinction between the Operators denoted in the present work by D/Dt and d/dt, but in the cz.ses concerning rates of change along a curved path the classical notation D/Dt often really stands for what we here call dfdt. Handbuch der Physik, Bel. Ill/1. 22 338 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 72. prefer to use the dot, whose extent of application will be indicated by an overbar rather than parentheses, thus: A.IY. d (AIY.) ,k = dt ,k' Ä ~k - ( d:tfY. L . j A; bk - d (AIY. bk) t k cm = dt k cm, e c. (72.6) Since ~ A".. _ ( aA:::) at ... ,k- ot ,k' (72.7) from (72.2) we get the commutation rule d A... (dA:::) _ A... ·m dt ... ,k- -----;{/ ,k-- ... ,mx,k. (72.8) This refers to the partial covariant derivative with respect to the spatial Coordinate xk. No simple rule holds, in general, for the partial covariant derivative with respect to the material co-ordinate XIY.. Recalling that in (72.2) ofot is executed with both x and X held constant, we have for the total covariant derivative (App. 20.2) the following commutation rule: (72.9) This holds for double fields A(x, X, t), but in carrying out the operation on the left we must regard A as a function of X and t only, and in carrying out the operation on the right we must regard x as a function of X and t only. To prove this rule, it suffices to establish it in the common frame. Verification in general co-ordinates is possible but more elaborate, requiring use of a commutation rule for the derivatives of the Christofiel symbols which follows from the vanishing of the Riemann tensor. Also, from (App. 20.3), (App. 20.4), and the fact that the co-ordinate system is stationary, we get (72.10) In problems of motion, it is unnecessary and often inconvenient to consider double fields A::: (x, X, t) as such. Henceforth we adopt the convention that either the material or the spatial description will be used, but not a mixture of both. That is, the tensor A::: will always be assumed a function of X and t, or a function of x and t, but not a function of x, X, and t. This occasions no loss in generality and results in economy of description. E.g. when we write A:,ßk we think of A;:' as a function of X and t to get A:,ß, then of A:,ß as a function of x and t to get (A:,ß),k· The two different partial time derivatives are then distinguished by the notations (72.1) and (72.5). Alsoweshall employ only the partial covariant derivatives ", k" and ",IX" rather than the total covariant derivative, and for uniformity of notation we set oxk Xk -xk _ ,cx= ;cx= oxa.' where t is held constant. X IY. -XIY.- oXIY. ' ' k= ·k=~k' uX (72.11) Under the foregoing conventions, the commutation rules (72.7) and (72.8) hold a fortiori, but (72.9) does not generallyhold if the total covariant derivative is replaced by the ordinary covariant derivative 1. However, for a purely material field viewed in the material description we have ÄIY. ... tJ -AIY. ... ß y ... d,e- y ... lJ,e ' (72.12) 1 TRUESDELL [1954, 24, § 18]. Sects. 73. 74. Material surfaces. 339 as follows from (72.3), whence it is equally plain th~t this rule cannot be extended to non-trivially double fields: A_ =X·"' (76.2) _ [ oi':_ + { k }xm] oxP _ _ik xm - oxP pm oX"- ,m •"' which follow from (72.3), (72.1), (App. 20.2), our convention in Sect. 72, and (App. 22.4). By differentiating the identity x~" X~m = 15~ and using (76.2) we get also X" X" Xß 'm X" ·m k = - m kX·ß = - mX k • , ' , ' ' J From (20.3) and (76.1} we have1 From (20.6} 4 follows similarly equivalently 2, We set (76.3) (76.4) (76.5) (76.6) (76.7) (the notation Id will be motivated in Sect. 83). The rule for differentiating a determinant gives 1 ;.viXI = x~~x~k I ;.vJXI = i~m x:n .. x~k I ;.vJXI, l (76.8) = Id I ;.vjXI, where we have used (76.1). Equivalent formulae are j = Id], dv = Iddv =i\dv; (76.9) the former follows from (16.5) and (72.10h, the latter, from (20.9). The quantity Id is called the expansion and is of fundamental importance. Each of the results (76.8) and (76.9) as well as the equivalent forms . . Id =i~k = divp =log J = logdv (76.10} is called Euler's expansion formula 3. An equivalent form involving an arbitrary function B such that B ·= 0 is the spatial equation of continuity: log.Bj + Id = 0, or o(:/) + div (Bjp) = 0 .. (76.11) Any quantity of the form B j, where B = 0, is called a density for the motion. The properties of such functions are developed in Subchapter III. 1 EDLER [1761, 2, §§ 13, 55] [1757, 2, § 11]. 2 LAMB [1877, 4]. 3 [1757. 2, §§ 10-15] [1770, 1, § 14] [1862, 4, § 156]. Sect. 77. Isochoric motion; steady motion with steady density. Since div(pp)=P+Pifl, by (App.26.1) 2 we have1 Jpdv = ~da·pp- fplttdv. V 4 V 343 (76.12) This identity, an alternative form of (76.10}, expresses the total velocity in a region by means of the normal velocity Pn on the boundary and the expansion ltt at interior points. The formulae of this section make it plain that from the velocity gradient tensor .i~m the material rates of change of all elements of arc, surface, and volume are determined. Pursuit of this idea fumishes the subject of Part c of this subchapter. 77. Isochoric motion; steady motion with steady density. A motionsuchthat the volume occupied by any material region is unaltered, however that region may change its shape in the course of time, is isochoric (cf. Sect. 40). By (76.9) follows as a local and instantaneous condition for isochoric motion Euler's criterion 2 : (77.1) That is, a motion is isochoric if and only if its velocity field is solenoidal. From the theorem of HELMHOL TZ in Sect. App. 31 we conclude that a motion is isochoric if and only if, at any given instant, the strength of each stream tube is the same at all of its cross sections. This assertion was taken, in one form or another, as the defining expression of the principle of continuity in hydrodynamic researches before 1752 3 and is still in frequent use in engineering treatments. Of course, it is but a special case of the defining property of a solenoidal field, which in the present connection assumes the following form: A motion is isochoric if and only if the flux of velocity out of every reducible closed surface is zero: ~da·p=O. (77.2) ' Since steady motion is defined by (67.5) alone, in a steady motion the quantity f need not be steady. However, by (76.10) the quantity logf is steady. Moreover, if B is any steady quantity which is constant on each stream line, we have B = 0, and also div (i B p) = i jJ · grad B + B div (i p) = 0. (77.3) From this formula and (76.11) we see that in a steady motion, by assigning the quantity B a constant value for each stream line we obtain a steady density i B. Since such a choice of B is not necessary, we distinguish the case when it is made by speaking of a steady motion with steady density. In two important cases, then, the velocity is proportional to a solenoidal field: 1. Isochoric motion, in which p is solenoidal, 2. Steady motion with steady density, in which Bfp is solenoidal. For these two classes of motion, we may read off from the results in Sects. App. 31 and App. 32 a sequence of special properties. These properties are extremely important, and it is only for economy that we refrain from repeating them here. 1 MuNK [1936, 7]; more obscurely in [1922, 6]. 2 [1761, 2, § 36]. Special cases are due to D'ALEMBERT [1752, 1, §§ 45, 73]. a E.g. NEWTON [1687, 1, Lib. II, Prop. XXXVI], D. BERNOULLI [1738. 1, Chap. 111, § 2]. Perhaps the earliest recorded statementisthat of LEONARDO DA VINCI [1923, 5, Book VIII, §§ 37-43]. 344 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 78. In particular, we emphasize (App.31.12), which gives the totalsquared speed m the isochoric case, the total of j 2 B2p2 in the steady case. In an isochoric motion with steady stream lines, we may take the divergence of (75.7) 3 , by (77.1) thus obtaining ( 77.4) This is a statement that C = const on each stream line at each fixed time. But if we write i k = i: dk, where d is the field of unit tangent to the stream lines, we get ai~ = ai dk at at · since d is steady. By (75.7) 2 , therefore, whence __(l_l_og i = C. at (77.5) (77.6) (77.7) By (77.4) follows a theorem of CAUCHY 1 : In an isochoric motion with steady stream lines, 8log ifot is constant on each stream line at each instant. The converse is not generally true. In fact, it is easy to show that in a motion with steady stream lines a necessary and sufficient condition that C be constant along each stream line at each fixed time is that Id/i: be steady. 78. Invariant directions. If the element dx is to be instantaneously constant in direction, we must have dxm = kdxm, (78.1) where k is a real factor of proportionality, which may be zero. By (76.4) 2 follows (im - k om) dxP = 0 ,p p ' and hence det (k b;' - .i:p) = 0. (78.2) (78. 3) The analogy to (38.2) is immediate, and the analysis proceeds in the same way as in Sect. 38. First, since (78. 3) is a real cubic, we have at once the theorem of Bertrand 2 : In any motion, at each point there is at least one real direction suffering no instantaneous rotation. Second, let D be the discriminant of (78.)). Recalling from Sect. App. 37 that a complex proper nurober k cannot yield a real solution dx of (78.2), and that to distinct real proper numbers there correspond linearly independent real solutions dx, from the theory of cubic equations we obtain Cases 1 and 2 of the following: 1. If D>O, there are three and only three invariant directions. 2. I f D < 0, there is one and only one invariant direction. 3. If D =0, there may be one, two, three, or an infinite number of invariant directions. The additional possibilities in Case 3 are most easily seen by example. The condition D = 0 is requisite and sufficient for multiple proper numbers. In the rectilinear shearing motion .i = 0, y = x, i = 0, there is but the single proper nurober k =0, thrice repeated, and any material straight line initially parallel to the y-z plane is carried ever parallel to itself. In the motion .i = 0, y = x, i = z 1 [1823, 2]. 2 [1867, 1]. Sect. 79. Kinematics of Jine integrals. 345 the proper numbers are k = 0, 1, the former being double, and the only lines not being rotated are those parallel to the y axis or to the z axis. For D we have the following expression 1 : D =- 18Idl0 B- 4nB +I~ l0 2 - 4l03 - 27B2 , (78.4) where (78.5) While interpretation of the sign of D is difficult, ERICKSEN 2 has observed that the problern is simplified by introducing the deviator, vkm=xk m-tidö~. From (78.2) it is plain that dx is invariant with respect to ik "' if aild only if it is invariant with respect to vk m, the proper numbers for the latter being k- ~I d, where k is a root of (78.3). If we let D0 be the discriminant for vkm, then by (78.4) follows (78.6) where l00 and B0 are formed from vkm as are lO and B from i~m· Hence there are three distinct proper numbers, at least one multiple proper number, or one and only one real proper number according as (78.7) In particular, l00 > 0 is sufficient that D0 < 0 and hence that there be one and only one invariant direction, while l00<0 is necessary in orderthat D0 > 0 and hence that there be three distinct invariant directions. A kinematical interpretation for the sign of lO will be given in Sect. 91. Since ik m is generally not symmetric, the algebraic theorem of KELVIN and TAIT (Sect. 'App. 37) implies that in the case when three distinct invariant directions exist, they are not usually orthogonal. A condition that an instantaneously invariant direction shall remain invariant has been found by ZaRAWSKia. In ~rder that there exist an element of area which is instantaneously constant 4, we must have dak= o; by (76.6) follows (78.8) a statement that ld is a proper number of i"),. From the characteristic equation of i :'). follows the necessary and sufficient condition ' (78.9) 79. Kinematics of line integrals. Definitions and conventions regarding line, surface, and volume integrals are given in Sects. App. 23 to App. 25 and Sect. 73. Let C(f be a materialline. Then for the rate of change of the flow of 'V along C(f we have 1 TRUESDELL [1954, 24, § 22]. 2 [1955. 5]. 3 [1900, 12]. { 1 - J 'V dxk= J 'V dxk, (79.1) 'W 'W 4 BELTRAMI [1871, 1, § 9] found also conditions for (1) Iineal elements suffering pure rotation, (2) plane elements all directions in which are experiencing rates of change normal to one direction, (3) plane elements suffering no instantaueaus rotation. These last form a tetrahedron whose edges, possibly imaginary, are the directions determined by (78.2). 346 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 80. since an equation for<6" is X=X(L) and thus the integral has fixed limits in the material description. Hence by (76.4) follows d~ J IV dxk= J ( ivdxk+ IV d~k), <'(/ <'(/ = j(\vdxk+~Vi~mdxm), (79.2) <'(/ = J ( \vdxk+ IV dxk), <'(/ results implicit in the analysis of KELVIN1 . On the other hand, for a spatialline c we have Tt o f d k_ r oW d k IV X-. 8tx. e e (79.3) Choosing the spatial line c which is the configuration at time t of the material line <6", from (79.2), (79.3), and (72.4) we obtain (79.4) 80. Kinematics of surface integrals. For a spatial surface ~ we have ~j~Vd km= r~d km ot a . ct a ' (80.1) but for a material surface Y' a more elaborate formula is necessary. By (76.5) we get (80.2) or, equivalently, -:ft I IV dak =I [W dak +IV(- x~ dam +i; dak)J. (80.3) g' g' When IV is a vector field c, this becomes a formula 2 for the rate of change of the flux of c through Y': :t {da·c= j'da·(c-c·gradp+cdivp). (80.4) g, g, A generalization is given in Sect. 277. From (80.4) follows Zorawski's criterion: In orderthat the flux of c across every material surface remain constant in time, it is necessary and sufficient that (80. 5) Equivalent forms are A],ck- A]ik cm = 0 ,m ' oc l ( . ) . d' ß( + cur c X p + p tv c = 0; (80.6) 1 [1869, 7, §59(a)]. The general ;ormula is given by JAUMANN [1905, 2, §383] and SPIELREIN [1916, 5, § 29). 2 Cf. ZoRAWSKI [1900, 11], JAUMANN [1905, 2, § 383], ABRAHAM [1909, 1, § 2], SPIELREIN [1916, 5, § 29). Sects. 81, 82. The Stretching tensor of EuLER. 347 in the former, which is a consequence of (76.9h, A is any substantially constant quantity. By comparing (80.5) with (75.4), we conclude that in order for the flux of c across every material surface to be constant in time, it is necessary but not sufficient that the vector lines of c be material. When (80.5) holds, the vector tubes of c arematerial tubes whose strength (Sect. App. 29) at any given cross-section remains constant as the motion proceeds. 81. Kinematics of volume integrals. The transport theorem. Fora stationary volume v we have :t J IV dv = J~~ dv. (81.1} Fora material volume "/"", by (76.%, (72.4}, and (App. 26.1} 2 we getl :t f \V d V = f ( W + \V X~k) d V, -r -r = J[ ~~ + (\Vik),k] dv, -r = J ~~ dv+~Wikdak. -r 9' (81.2) Choosing the spatial volume v which at timet is the configuration of the material volume "Y, we derive the transport theorem2: ~- J \V dv = ~t J \V dv + ~Wikdak = J ~';-dv + ~ \VXnda. (81. 3) -r " " Thus the rate of change of the total \V over a material volume "Y equals the rate of change of the total \V over the fixed volume v which is the instantaneous configuration of "Y, plus the flux of ~\V out of the bounding surface. The transport theorem is really an alternative statement of EuLER's expansion formula (76.10). The formula (81.3) expresses, for a given \V, a time derivative of an integral taken over a volume whose bounding surface is in motion at an arbitrary velocity ;i. To consider a volume v(t) bounded by a surface 6(t) moving at a different velocity u, we need only imagine fictitious particles whose velocity is u. The result, then, is ~~ f \V dv = r~a; dv +~\V uk dak, (81.4) v(t) v where dufdt indicates that the volume of integration is material with respect to the velocity u. By taking U=O or U=~. we recover (81.1) or (81.3). c) Stretching and spin. 82. The stretching tensor of EuLER. Let d~1 and d~2 be material elements. Then by (72.10) and (76.4) we have gkmd~1dxB'=gkm(di~dxB'+dx1diB'), ) = ( •k d Pd m+ "md kd P) gkm X,p X1 Xg X,p x1 Xg , = 2dkmd~dxB', • (82.1) 1 REYNOLDS [1903, 15, § 14), JAUMANN [1905, 2, § 383), SPIELREIN [1916, 5, § 29). 2 Asserted by REYNOLDS [1903, 15, § 14], proved by SPIELREIN [1916, 5, § 29], who gave numerous alternative forms and corollaries. 348 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 82. where d is the symmetric stretching tensor or rate of deformation tensor of EULER: (82.2) By (82.1) it is plain that the tensor d serves to measure the instantaneous rat es of change of length and angle of material elements in the moving material. First, if d;x1 =d;x2 , (82.1) reduces to (82.3) or (82.4) If we write d(n) - lim J = lim log l l->0 l l-->0 (82.5) for the stretching [cf. Sect. 25, especially (25.5)] of a material line of length l emanating from ;x with unit tangent n, then we have from (82.4) d(n) = dkmnknm. In particular, if n points along the x1 co-ordinate curve, we have d dll (n) = -gll Second, if ~ is the angle between d;x 1 and d;x2 , from (82.1) we get - sin ~ ~ = 2dkm n1 n;'- cos ~ (d(n,) + d(n,)). (82.6) (82.7) (82.8) For ~ =0 this reduces to (82.4). If ~=j=O, (82.8) gives us the shearing, - ~, of the directions of d;x 1 and d;x 2 • For the case of orthogonal directions we get 1 p d k m -21012 = kmnlnz. (82.9) We have shown, then, that in an orthogonal co-ordinate system the physical components of d equal the stretchings and the halves of the shearings in the co-ordinate directions: d(l) - t~12 d(2) - t~/3 -t~23 d(3) The physical dimension of d is [T-1]; thus d is a pure rate. (82.10) The linearity of d in x is important: The stretchings and orthogonal shearings corresponding to the vector sum of two velocity fields are the sums of the stretchings and shearings of the two constituent fields. The results just given, as weil as those to follow in this part of the subchapter, could be derived by reinterpretation of the formulae concerning small deformation given in Part e of Subchapter I. Because of the definition of "small" there employed, there would be no loss of rigor in such a treatment. It seems to us clearer, however, to make the theory of rates of change entirely independent of the theory of finite strain. In so doing, in this section we have given a compact, modern version of the original argument of EuLER. The rather elaborate exact connection between stretching and rate of finite strain will be presented in Sect. 95. Sects. 82A, 83. The invariants of stretching. 349 The rate of change of a material element of area is given by (76.6) and hence is not determined by the stretching alone. For the rate of change of the magnitude da, however, we readily calculate1 from (76.6), (82.2), and (82.6) (82.11) 82A. Appendix. History of the theory of stretching and shearing. All the analysis just given, except for the last equation, is due to EuLER [1770, 1, §§ 9-12]. His form of the argument, based on consideration of the infinitesimal displacement :i: dt, is often given in engineering texts today. In reference to infinitesimal strain (cf. Sect. 33 A and Sect. 57), the material was re-created by CAUCHY [1823, 1] [1827, 2] [1828, 3], who added the contents of Sect. 83. Cf. also P!OLA [1836, 1, §VII,~~ 70-71]. All this was discussed afresh in terms of rates by STOKES [1845, 4, § 2], whose kinematical description is more immediate. The formula (82.4) in a less symmetrical form was derived by P!OLA [1848, 2, ~ 15] in connection with rods. From this, as weil as some later work, we must conclude that the equivalence of the theory of small deformation to that of rates of change was not as plain as now it seems; the difficulty, perhaps, lay in a certain vagueness in the concept of "small" (cf. Sect. 53). The explicit equation (82.3), in terms of rates, is due to BELTRAMI [1871, 1, § 4] and DURRANDE [1871, 4]. Further properties of d were derived by KLEITZ [1872, 2, § 8] [1873, 4, §§ 30-32, 38-40]. He analysed the ellipsoid of d-2 and the properties of the vector dkmnm, where n is a unit normal. His results have been presented in moregeneralform in Sect. App. 46. He constructed also a geometrical interpretation for the shear components of d in terms of certain curves associated with the stream lines. A formula for d according to the material description, which fol!ows at once by substituting Xk,m=xk,CJ.X~m and (17.3) into (82.2), was written out by DuHEM [1903, 3]. 83. The principal stretchings, the invariants of stretching, and the quadric of stretching. Since dkm is a real symmetric tensor, it has many properties which may be read off from results in Sects. App. 3 7 to App. 39, Sect. App. 46 and Sect. 21. I ts real and orthogonal principal axes are the axes of stretching; its real proper numbers da are the principal stretchings, which may be distinct or multiple, positive, negative, or zero. Therefore the quadric of d, which is called the quadric of stretching 2 , may be an ellipsoid or a hyperboloid or any of their degenerate cases. If it is a hyperboloid, elements along its asymptotic cone are suffering no extension, and the cone divides elements whiCh are being shortened from elements which are being lengthened. If da >O for a = 1, 2, 3, the quadric is an ellipsoid and all elements are waxing; if da< 0, the quadric is again an ellipsoid, but all elements are shrinking. The stretchings along the principal axes are the extremal stretchings. For two orthogonal directions suffering extremal stretching, the shearing is zero. The orthogonal directions suffering extremal shearings are those bisecting the angles between two of the axes of stretching and normal to a third, while the amounts of these extremal shearings are ±(da- db)· The principal invariants of d are called the invariants of stretching and are written as Id, IId, IIId. Since Id=O in an isochoric motion, from (App.39.7) we obtain Ild ~ o. In any motion, the invariants Vü~ and vu--;; are measures of the intensity of stretching and of the intensity of shearing, respectively. 0d, the deviator of das defined by (App. 38.12) with d =a, is often used in works on plasticity. In the sense of the maximal decomposition (App. 43.1), 0d may be regarded as a measure of instantaneous change of shape without change of volume. By (82.9), a necessary condition for every orthogonal shearing to be zero is that dkm n/ntr = 0 whenever gkm n~ nz = o. Hen.ce d =IX 1, where IX is 1 This result is due in principle to PIOLA [ 1825, 2, § 236] [ 1848, 2, ~ 15]. 2 A very detailed study of the quadric of stretching was made by ZHUKOVSKI [1876, 5, Gl. I]. 350 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 84. a scalar; hence d = ! Ia 1; equivalently, 0d = 0. If 0d = 0, it follows by (82.8) that ~ =0 if ~ =f:O; that is, every shearing is zero. We have shown, then, that a necessary and sufficient condition for alt shearings to vanish is that 0d = 0. Accordingly 0d is often called the distortion tensor. The principal axes and the principal shear components of 0d coincide with their Counterparts for d; however, in general 0d is not the stretching tensor of any motion 1. Since the proper numbers of a deviator cannot be of one sign, the quadric of 0d is always a hyperboloid or one of its degenerate cases 2• Further properties of 0d may be read off from results in Sect. App. 38. When d is spherical, or, equivalently, when the three principal stretchings are equal, the motion is a dilatation. Equivalent invariant conditions are (-iJa) 3 = (!IIa)l =lila and 0d=O. (83 .1) The most generaldilatationwill be determined in Sect. 142. PAILLoux3 has derived the condition IIIa= o ( 83.2) as necessary and sufficient that there exist a material surface whose spatial configuration is instantaneously invariant. The direction in which the stretching is zero is the normal to the surface. 84. Rigid motion. A motion is rigid if all materiallengths remain unchanged by it. From (82.3) follows as a local and instantaneous necessary and sufficient condition for rigid motion Euler's criterion 4 : dkm = 0. If (84.1) holds throughout a region, its general solution is 5 i4 = wkmpm + ck, (84.1) (84.2) where c and w depend only upon t, where w is skew symmetric, and where P~m = b!,. Therefore p is the position vector of ilJ with respect to a certain origin and w is the angular velocity tensor of a certain frame. Since:i- c is perpendicular both to wand top, c is the linear velocity of the origin from which p is directed, and with respect to the frame whose angular velocity is w the velocity of the material is zero. The origin is arbitrary, but w is uniquely determined by (84.2). That is, all the possible frames determined by (84.2) have the same angular velocity. We may think of these frames as rigidly attached to the material, and we call w the angular velocity of the motion. (Cf. the general treatment in Sect. 143.). If w =0, the rigid motion is a translation. A rigid motion is isochoric, since i\ = o. In a rigid motion, the invariants IO and B defined by (78.5) have the values IO = , B = 0, w being the angular speed, and hence by the theorem of Sect. 78 it follows that in a rigid motion which is not a translation there is one and only one invariant direction. The foregoing statements are geometrically trivial and are made here only so as to illustrate the general criteria established earlier. Forageneral motion, six functions dkm assigned arbitrarily cannot in general serve as the components of stretching. Rather, in order that there exist a field 1 In order that 0d be the stretching tensor of a motion, by (84.3) it is necessary and sufficient that Ia gk m be the stretching tensor of a motion. At the end of Sect. 142 it is shown that only special forms for Ia are possible. 2 BELTRAMI [1871, 1, § 3]. 3 [1938, 10]. 4 [1761, 2, §§ 75-77] [1770, 1, § 13]. A generalization of this result in differential geometry was obtained by KrLLING [1892, 6, p. 167]. 5 EULER [1770, J, § 13]. Sect. 84. Rigid motion. 351 x such that (82.2) holds, we have the condition of integrability\ (84.3) If (84.3) is satisfied, let two velocity fields corresponding to d be x1 and x2 ; for their vector difference i*-i 1 - x2 , we calculate the stretching tensor d* and by the linearity of (84.2) conclude that d* = 0; therefore, the vector difference of two velocities having the same stretching tensor is a rigid motion. In view of (84.2), this furnishes proof for the assertion (34.11). The problern of determining corresponding conditions in more general spaces of n dimensions is a difficult one, not yet fully solved. Given a tensor dkm• we are to find necessary and sufficient conditions that there exist a vector ck such that dkm = C(k,m) ( 84.4) where, as usual, - ock p ck,m= oxm- TkmcP. (84. 5) Fora given field ck, let dkm be defined by (84.4), and put wk».""" C[k,ml· Fora symmetric connection r we have the identities Wkm,p+ Wmp,k+ Wpk,m= 0, l ck,mp- ck,Pm = CqRqkmP• hkm,ps- hkm,sp = hkqRqmps + hqmRqkps· (84.6) Since ck,mp=dkm,p+wkm,p• by forming ck,mp-ck,Pm and using (84.6lt, 2 we find that Wmp,k = dkm,p- dkp,m- Rqkmp Cq· (84.7) Now forming Wmp,ks-Wmp,sk and using (84.6)3 , we obtain the identity 2 dkm,ps- dkp,ms- dsm,pk + dsp,mk + l + (Rqsmp,k- Rqkmp,s) Cq + Rqsmpdqk- Rqkmpdqs- - Rqkmp Wqs + Rqsmp Wqk + Rqpks Wqm- Rqmks Wqp = 0. (84.8) For a flat space, (84.8) reduces to (84.3). In more general spaces, we are to derive from (84.8) and from the geometry of the space further identities, by means of which cq and wkm are to be eliminated. PALATINr3 has indicated how the calculation can be initiated in the case of a metric space. First, we replace sums of the type Rq . .. aq by corresponding sums Rq · · · aq, then use the relation Rkmps=Rpskm and the Bianchi identities (34.5) to obtain (Rqsmp,k-Rqkmp,s)cq = -Rmpsk,q cq. Thus (84.8) becomes (jkt rl!,:'s drt,qu- Rmps k,q Cq- Rmpsq d'J. + Rmpkq d~ + } + RmpkqWqs- RmpsqWqk- RkspqWqm+ RksmqWqp = 0. (84.9) Raising the index k and then contracting upon k and m yields 4 I,ps + dp/,k- 2drp,s)q- 2d(pRs)q- 2wrPRslq- Rps,q cq= o, (84.10) 1 ZoRAWSKI [1911, 14, § 4]. Cf. (34.8). A geometrical interpretation for these conditions in terms of the shearing required to render compatible a given tensor d has been constructed by L. FINZI [1956, 7, § 7]. 2 Rrccr [1888, 7, Eq. (20)]. a [ 1943, 3]. The method was sufficiently indicated by his earlier analysis of a space of constant curvature [1916, 4]. Cf. also ANDRUETTO [1932, 1 and 2], AGOSTINELLI [1933, 1], GRAIFF [1958, 3, §§ 6-9]. 4 This identity was given in a special co-ordinate system by PALA TINI [ 1934, 3]; an equivalent generalform is due to GRAIFF [1958, 3, Eq. (22)]. 352 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 84. where (84.6)3 has been used, where Rpq == Rhpqh• and where we have written I for ld. Raising the index p in (84.10) and the contracting upon p and s yields 1 where R == R~. I·~p- dkP,kp- Rpq dpq = R,q cq, (84.11) Eqs. (84.10) aretobe regarded as a systemoft n (n + 1) non-homogeneaus linear equations for the t n (n + 1) unknowns cq and wk5 in terms of the tensord and its second derivatives. Substituting the result into (84.9) yields, in principle, a set of differential equations for d alone2. The geometrical meaning of the condition of solubility expressed in terms of a determinant Formel from the components Rpq and Rpq,s is not known; of course, it is not satisfied in a flat space. More generally, directly from (84.4) we see that the nature of the solution depends very strongly upon the geometry of the space considered. In spaces admitting a group of "motions ", i.e., instantaneously rigid velocity fields, the Solution of (84.4), if compatible, is determinate only to within such a motion. In spaces where instantaneously rigid motions are not possible, the system (84.4) will determine c uniquely from a given d, if compatible. GRAIFF3 has initiated a discussion of the problern in terms of the principal invariants of the tensor Rpq• according as these are linearly independent and non-constant or not. The problern must still be regarded as open. In a space of constant curvature, we have n (n- 1) Rqk m p = R (gkm (jb- gk pi5'!,.), where nRkm = Rgkm• R = const = R~. The terms involving c and w in (84.9) vanish. The resulting conditions, easily seen to be sufficient as weil as necessary, are 4 (84.12) When n = 2, there is but one linearly independent non-vanishing component of (84.12), which may be written in the equivalent forms erxßeY~da.y,ßd + KI = 0, } I.rx,rx.- drxß,rx.ß + K I= 0, (84.13) where K is the total curvature, K = - t R. More generally, when n = 2 and K is arbitrary, we have Ra.ßy~ = K e,.p eyd and Ra.p = - K arx.ß• where a is the metric tensor. Again the terms containing w in (84.9) are annulled, and we have (84.14) Further differentiations are needed to eliminate c. B. FrNzr5 has shown that for a surface applicable upon a surface of revolution the final condition assumes the form ( 84.15) For a general surface, he obtained a system of three conditions of compatibility, each being a differential equation of fourth order. TRUESDELL6 showed indirectly that they must be equivalent to a single condition of fifth order. By a method of infinitesimal variation applied to a complete set of differential invariants of the surface in question, GRAIFF 7 has givena definitive treatment of the problem. She has found two identities relating the quantities aceurring in B. FrNzr's conditions, but she has not effected the elimination explicitly. She has characterized the case when the least possible order of the single scalar condition is 4; in this case, as for a surface of revolution, the lines K = const are geodesie parallels, but K,rx rx. is not a function of K only. ' GRAIFF 8 has discussed also the various possibilities when n = 3 and n = 4. 1 GRAIFF [1958, 3, Eq. (23)]. 2 This seems tobe a correct substitute for the method of PALATIN! [1934, 3], who notes that it is possible to choose co-ordinates so that at a fixed point (84.10) with p = s becomes a system of n linear equations for the n components cq but fails to note that the solution of this system is not sufficient, in general, to obtain the derivatives of cq and so to determine wkm. 3 [1958, 3, § 10]. 4 PALATIN! [1916, 4] [1934, 3], ANDRUETTO [1932, 2], FINZI [1934, 2] including a simp!ification when n = 3. 5 [1930, 1]. 6 [1957, 17]; the reasoning is presented in Sect. 229. [1957, 5]. [1958, 3, §§ 13-14]. Sects. 85, 86. The spin tensor of CAUCHY. 353 Enlightenment is cast upon these results by the analysis in Sect. 229. For another approach to the conditions of compatibility, see Sect. 234. 85. Relative spin. Let dx be a material element, and Iet n be a unit vector fixed in space. Then the angle (/J(di~!,n) between these two vectors is. given to within a multiple of 2 n by and the right-hand rule. By (72.10) and (76.4) follows - sm . cp cp • = -v·=::-··_· gkmdxknm - cos cp d(di~!) • I g,.pdxndxP • kNm d =Xk,mn - cos cp (N)• (85 .1) (85.2) where N is the unit vector dxfdx. The angular rate cp is the spin of dx relative to the fixed direction n. In particular, for the spin relative to an orthogonal direction we ha ve .Tz • • kNm -cp(N,n) =Xk,mn • (85.3) The elegant formula (85.3) enables us to give a ;p12=-i J,z geometric interpretation for the shear components xm, k, k =t= m, of the velocity gradient in an orthogonal co-ordinate system. For Iet N and n point along the m and k co-ordinate curves respectively, and for the corresponding cp write tPkm; then • Xk m -cpk = . . ' ····-· m Vgmm Vgk;. (85 .4) tP~J=-iz,J Fig. 14. Relative spin. for orthogonal Co-ordinates, xl,2/Vk11 yg22 is the 1'ate at which a material element instantaneously pointing along the x2 axis is turning toward the xl axis1• From this result and (82.9) we se that - 12 =- cp12 - cp21 , as is obvious: the shearing of two orthogonal directions equals the negative sum of their spins relative to fixed orthogonal directions (Fig. 14). In interpreting this result, due attention must be paid to the convention of sign. In a right-handed orthogonal system, then,-~ 12 is the excess of the rate of right-handed rotation of an element in~tantaneously pointing along the x2 axis over that of one pointing along the xi aXlS. If we put cp =0 in (85.2), we recover {82.6). 86. The spin tensor of CAUCHY. Set . -. - <>' /"' m] Wkm = X[k,m]- uX[k uX . (86.1) With the interpretation of xk.m as proportional to the relative spin, constructed in the previous section, we an! able to identify the components wkm in the orthogonal case as the halves of the differences of the relative spins of the elements in the co-ordinate directions. Here again due caution regarding the convention of sign is necessary. For a right-handed system, the quantity - w12/fgn Vg22 is one half the sum of the rates of right-handed rotation of elements in the x1 direction and the x2 direction. w, like d, is a pure rate, having the physical dimension [T-1]. 1 This result is implied though not actually stated by CAUCHY [1841, 1, Th. IX]. Handbuch der Physik, Bd. III/1. 23 354 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 86. The axial vector (86.2) is the vorticity vector; its direction is the axis of spin, and its magnitude w is the vorticity magnitude. Wehave (86.3) From the vectorial character of w follows, as another wording for the proposition demonstrated above, the first theorem of Cauchy 1 : The length of the projection of w upon any given direction is the sum of the rates of right-handed rotation about that direction of elements in any two directions perpendicular to it and to each other. The theorem just proved contains a statement of invariance: The component of vorticity is the sum of rates associated with any two perpendicular .directions in a plane normal to the component. This invariance may be investigated by means of the skew projections introduced in Sect. App. 4. Let Greek majuscule indices run from 1 tö 2, and write aWLIS for the skew projection of Xk,m onto the x3 direction. Then (App. 4.2) yields llaw~ll=' 1 ~~~1,2 -~1,111· ~detg!I>'P x 2, 2 -x2, 1 (86.4} while from (App. 4.3) follows LI 2Wu - rl(ktgkm ~LI= V ~. Vdetg!I>'P - Vdetg!I>'P (86.5) The result is of interest mainly for the case when the x3 direction is normal to the other two co-ordinate directions. Then (86.6) That is, the physical component of the vorticity vector in a given d~rection is the trace of the skew projection of the velocity gradient onto the plane normal to that direction. Moreover, since the skew projection w~ is a plane tensor, its symmetric part may be represented by a quadric, as usual, and this quadric is a conic section. By (86.4) and (85.4), the normal components of w~ are pro~ portianal to the relative spins of elements normal to the x3 direction. Hence we obtain the second theorem of Cauchy 2 : The rates of right-handed rotation of elements normal to the x3 direction are inversely proportional to the square roots of the lengths of the corresponding radius vectors to the conic section w~ nLI ns = const. (86.7) Let the constant be given any nop.-zero value which renders the locus real. There are then three possibilities. ( 1}; the conic is an ellipse; the rates of rotation of the elements in the directions of its axes areminimaland maximal rates, and all elementsnormal to the x3 direction are rotating in the same sense. (2), the conic is a pair of parallel straight lines; again all elements are rotating in the same -sense, except that the elements in the direction of the lines are not rotating at all. (3), the conic isapair of conjugate hyperbolae; its asymptotes divide the plane normal to the x8 direction into two portions, elements in one of which are rotating in one sense, while elements in the other are rotating in the opposite sense; 1 [1841, 1, Th: IX]. 2 [1841, J, Th. VII]. Cf. also BERTRAND [1867, 1]. Sect. 86. The spin tensor of CAUCHY. 355 elements in the directions of the asymptotes are suffering no rotation whatever, while the rates of rotation of elements parallel to the axes are maximal and minimal rates (in absolute value, both are maximal). Reinterpretation of (86.6) now yields the third theorem of Cauchy 1 : The length of the projection of the vorticity vector upon a given direction equals the sum of the greatest and least rates of right-handed rotation of normal elements. Returning to (86.4), we use reetangular Cartesian co-ordinates and observe that by the tensor law of transformation we have .it, 1 = i 2,1 cos2 tp - i 1,1 cos tp sin tp + i 2, 2 sin tp cos tp - i 1, 2 sin2 tp, (86.8) where tp is the angle between the x1 and x1 * axes. This formula expresses the relative orthogonal spin of an element subtending an angle tp with the x1 axis. The mean of all these spins is thus 2n 1 f .• d - 1 ( . . ) - - 1 - x. 1 '" - - x2 1- xl 2 - - wl2 - - Wa. 2:n; • ~. T 2 ' ' 2 (86.9) 0 This is the fourth theorem of Cauchy 2 : The length of the projection of the vorticity vector upon a direction is twice the mean of the rates of right-handed rotation of all elements perpendicular to that direction. Cf. Sect. 36 and Sect. 56. From (84.2) it is immediate that the spin tensor of a rigid motion is its angular velocity. In making this Observation, STOKES 3 suggested that in general the spin tensor of a deforming medium could be regarded, at each place and time, as a local angular velocity. STOKEs's interpretation was extended by BELTRAMI 4• Imagine a vanishingly small massy element whose center of mass is at ~ and whose principal axes of inertia are the principal axes of stretching at ~ at time t; let this element suddenly be solidified into a rigid body, and at the same time all the surrounding material shorn away; then this rigid body will continue to rotate indefinitely with angular velocity w. An elegant reformulation of STOKEs's result, free of dynamical concepts, is due to GosiEWSKI 5• Since material elements along the principal axes of extension are instantaneously suffering no rotation with respect to one another, their motion as lines, no account being taken of the motion of particles along them, is instantaneously rigid; hence the spin is the angular velocity of the principal axes of extension, in the ordinary sense of rigid motions. A formal analysis equivalent to the above argument was also presented by GosrEWSKI, who noted that by (76.4) the rate of change of a unit vector n parallel to d~ is given by6 (86.10) 1 [1841, 1, Th. IX]. 2 [1841, 1, Ths. V, VI, VII]. 3 [1845, 4, § 2]. A similar idea had been put forward by CoRIOLIS [1835, 1, pp. 100-101] in connection with systems of mass-points. 4 [1871, 1, § 10]. Cf. HADAMARD [1903, 11, '\[ 63]. STOKES had considered only a spherical element. 5 [1890, 4, §§ 2-3]. APPELL [1921, 1, § 706] attributed the result to BoussiNESQ. It was asserted also by LEVY [1890, 7, § S]. A somewhat obscure treatment based on the equations of relative motion (Sect. 143) was given by INGRAM [1924, 5]. We do not follow the analysis of VALCOVICI [ 1946, 9], who, on the ground that the material lines which at a given instant coincide with the principal axes of extension no Ionger do so after an infinitesimal time, objects to w as a measure of rotation and proposes a substitute. 6 GosiEWSKI [1890, 4, § 8], ZoRAWSKI [1901, 17, § 2]. 23* 356 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 86A,87 If n isaproper vector of d, then (d:,- d(nlb:,) nm =0, and hence (86.10) reduces to (86.11) which is an analytical statement of the result to be proved. To get a formula for w in the material description, we begin with the evident (86.12) From (17.3) it then follows thatl (86.13) The vorticity magnitude is related to other scalar invariants of the velocity gradient by identities which follow at once from (App. 38.1 5) : (86.14) where 10 is defined by (78.5h. Also 4w,.m WmP dkp = - Id w2 + dkm w" wm. (86.15) 86A. Appendix. History of the theory of spin. The components w,.m occur in the earliest analyses of the velocity field, having been introduced in special cases by D' ALEMBERT ( 1749) [1752, 1, §§ 45-49] and generally by EuLER (1752-1755) [1761, 2, §§ 46-47], whose theorems concerning spinwill be presented below. LAGRANGE (1760) [1762, 3, Chap. XLII] and CAUCHY (1827, 5, 2nd Part, § 1, Subsect. 4] were the first to introduce single letters to stand for the components. All this early work is purely formal and somewhat mystifying. While in 1770 EuLER presented all the apparatus necessary to obtain what we have called CAUCHv's first theorem in Sect. 86, he did not take the final step, and the components Wkm• which appear fluently in eighteenth-century works on hydrodynamics, remained uninterpreted symbols. The beginning of the theory is the proof by MAcCuLLAGH in 1839 that the components of the curl satisfy the vectoriallaw of transformation. Before this work was published [1848, 1, § li, Lemma 2], however, CAUCHY, more than a decade after he had completed in all detail his theory of strain, constructed the theory of rotation. MAcCuLLAGH's result is included in CAUCHY's great paper [1841, 1, § II, Eq. 12] from which we have reproduced the major theorems in the foregoing section. The innovations of STOKES, HELMHOL TZ, and KELVIN will appear in the following sections. The terminology of the subject, always a stumbling block, gave rise to a controversy between BERTRAND and HELMHOLTZ [1867, 1] [1868, 2-4, 8-10]. The term spin is due to CLIFFORD [1878, 2, Bk. II, Chap. II, pp. 122-123, Bk. III, Chap. II, pp. 193-194]; vorticity, to LAMB [1916, 3, Preface and § 30]. 87. Circulation. KELVIN's transformation (App.28.1) when applied to the velocity field reads ~i,.dx" = Jw,.da", or ~da:·p=fda·w: (87.1) e e The mean value of the tangential component of velocity around a closed circuit, multiplied by the length of the circuit, equals the flux of vorticity through any surface entirely bounded by the circuit. It was in this connection that KELVIN's transformation was discovered (Sect. App. 28). The line integral~ d~ · p is the circulation of c. Let Wn be the length of the projection of w onto the normal n to a surface d at the point a:. Letting 9 be the area of d, we have by (87.1) ~ x,.dxk = 9 Wn + 0 (s) (87.2) e 1 BELTRAMI [1871, 1, § 6]. BoNVICINI [1932, 3] resolved w with respect to the principal axes of finite strain. Sect. 88. Irrotational motions. as c is shrunk down to the point x. Hence ~;kdxk Wn = lim _e _____ . S--->0 S 357 (87.3) That is, i/ 5 is the area inclosed by a circuit c upon a surface <1, then the ultimate ratio of the circulation of c to the area 5 as c i$ shrunk down to a point P equals the component of vorticity in the direction normal to <1 at P. Hence the direction of the vorticity vector at P is perpendicular to the plane in which the circulation per unit area is greatest, and the vorticity magnitude is the circulation per unit area in that plane 1. The striking interpretation just presented for the vorticity vector is of a quality essentially different from those of the previous section, which involved the rotations of materiallines and hence the fact that x is the velocity of a motion. Interpretations in terms of the circulation, on the other hand, are but especially vivid statements of the properties of the curl of any vector field ( cf. footnote 1, p. 277). 88. lrrotational motions. A motion for which the spin vanishes is irrotational 2 : Wkm=O, or W=O. (88.1) Motions of this kind form the subject of most of classical hydrodynamics. Motions not satisfying (88.1) are called rotational. In the terminology of Sect. App. 33, a motion is irrotational if and only if its velocity field is lamellar, and (88.1) is necessary and sufficient that xk dxk be the exact differential of a velocity potential V: xk =- V:k or p =- grad V. (88.2) The velocitypotential is a discovery of EuLER (1752)3. In simply connected regions of irrotational motion the velocity potential is single-valued; its application in multiply connected regions, where it is generally a cyclic function, was initiated by HELMHOLTZ and KELVIN 4• In the former case, we have ~2 ~? - f xk dxk = f dV = V(x 2) - V(x 1). (88.3) WJ ii!J If we call J xkdxk the flow along. c, then (88.3) asserts that in a simply connected e region of irrotational motion, the value of the velocity potential V at a point P is the negative of the flow along any curve connecting P with some arbitrary fixed Point where V is assigned the value zero. Taking x 1 and x 2 as the same point and the closed circuit of integration as lying within a simply connected region, we find the right-hand side of (88.3) to be zero and hence derive Kelvin's kinematical 1 TRI CO MI [ 1934, 9] has constructed an interpretation of wn in terms of the angular velocity of a ring in the plane normal to n. 2 The term is due to KELVIN [1869, 7, §§ 59-60]; earlier authors, from 1757 onward, had spoken of these motions as "those in which i dx + ydy + z dz is an exact differential". Nowadays irrotational motions are often called potential flows. 3 [1761, 2, §§46-48, SO, 55-56] [1757, 2, §§26-27]. While D'ALEMBERT [1752, 1, §§59, 86] based his theory of plane and rotationally symmetric fluid motions on the Contention that i k dxk is exact, he did not make direct use of the velocity potential, whose existence follows from this assumption. The term Geschwindigkeitspotential is due to HELMHOLTZ [1858, J, lntrod.j. 4 [1858, J, §§ S-6]; [1869, 7, §§ 60(s)-64]. An elaborate analytical treatment is given by LICHTENSTEIN [1929, 4, Chap. 3, § 4]. 358 C. TRUESDELL and R. TouPIN: The classical Field Theories. Sect. 88. theorem1 : A motion is irrotational if and only if the circulation about every reducible circuit is zero. In case the region in question is multiply connected, let it be rendered simply connected by the imposition of suitable barriers. Then from two reconcileable but not necessarily reducible circuits may be formed a single reducible circuit (Fig. 15) by adding one connecting path, to be traversed in opposite senses, upon each barrier crossed by the circuits. By applying KELVIN's kinematical theorem to the resulting single reducible circuit and noting that the net flow along the twice traversed paths on the barriers is zero, we conclude that a motion is irrotational if and only if the circulations about any two reconcileable circuits are equal. More generally, if we let c,, f = 1, 2, .... , t:J, be the circulations about any lJ circuits such that none is reconcileable with any circuit formed by the connection of any nurober of the others by barriers, then (88.3) must be replaced by zz P - f xkdxk= V(x2)- V(a:I) + Ln,c,, (88.4) ~ ~1 where the n1 are integers. The Cr are a set of cyclic constants of the motion. The cyclic constants are not uniquely defined, but any member of one possible Fig. t s. Circulation in a multiply connected region. ity potential at a point dilfer by a linear constants of the motion. set may be expressed as a linear combination, with integral coefficients, of the members of any other possible set. The coefficients n1 in the sum on the right-hand side of (88.4} depend on the path of integration in the integral on the left-hand side. By choosing :~: 1 =a:2 , we conclude that any two determinations of the Veloeintegral combination of the cyclic Since ik m is symmetric if and only if wkm =0, by applying the algebraic theorems ol CAUCHY and KELVIN and TAIT in Sect. App. 37 we conclude that a necessary and sulficient condition for a motion to be instantaneously irrotational at a point is that at that Point there exist three mutually orthogonal directions suitering no instantaneous rotation (cf. Sect. 78). This result follows also from CAUCHv's conics of rotations (Sect. 86) : A necessary and sulficient condition that a motion in. which ik m =f= 0 be instantaneously ir~·otational at a point is that the conics of rotations in 'three ditferent planes through the point be equilateral hyperbolae. The proof lies in the fact that the asymptotes of a hyperbolic conic of rotations are suffering no rotation, so that if these are orthogonal, the sum of the rates of rotation of two orthogonal directions in the plane of the conic is zero, and hence by the first th~orem of CAUCHY (Sect. 86) the component of vorticity normal to that plane is zero. That in an irrotational motion there are at least three mutually orthogonal invariant directions was established above, but a more spedfic statement holds. Since ik m is symmetric, its proper vectors are mutually orthogonal if its proper numbers are distinct. By the analysis in Sect. 78 it follows that in an irrotational motion there are at each point either three or an infinite number of instantaneously invariant directions, according as the three principal stretchings are distinct or not. 1 [1869, 7, §§ 59(e), 59(f)]. The result follows immediately from analysis given by HANKEL [1861, 1, § 8] but was not stated by him. Sect. 88. Irrotational motions. 359 Comparison of the above theorem with the fact that in a rigid motion which is not a translation there is but one invariant direction (Sect. 84) yields the following characterization: A motion is locally a translation if and only tf it is irrotational and rigid1• By (88.2) we have for the derivative of V in the direction of the velocity vector (88. 5) Bence the velocity potential can never increase in the direction of motion along a stream line, and in steady irrotational motion a particle always moves toward a region of lower velocity potential. A stagnation point is a stationary point for the velocity potential along the stream line on which it occurs. Thus a closed stream line lying wholly in a simply connected region of irrotational motion can never exist. In these circumstances it follows also from the continuity of the velocity field that the stream lines in a simply connected region cannot return arbitrarily near to themselves, as in general they may in a rotational motion 2• By (88.5) and (72.4) we have ~-v·- "2> öt . -X = 0. (88.6) Hence the squared speed is the excess of the local over the material time derivative of the velocity potential, and in particular the rate of increase of the velocity potential as apparent to an observer moving with a particle can never exceed the rate apparent to a fixed observer. In an irrotational motion, from (88.2) and (82.2) we have 3 Hence in particular (88.7) (88.8) By (77.1) we derive Euler's {heorem on irrotational motions 4 : An irrotational motion is isochoric if and only if 172 V = 0. (88.9) Thus all properties of isochoric irrotational motions follow from potential theory, and, conversely, every result of potential theory has a kinematical interpretation in terms of isochoric irrotational motions. From (88.8) we see more generally that V is subharmonic, harmonic, or superharmonic according as Id~ 0, Id = 0, or Id ~ 0. Therefore in the interior of a region where ma~erial volumes are not decreasing, the velocity potential cannot suffer a minimum; not increasing, a maximum. GaslEWSKI 5 has noted a connection between irrotational motions and a special property of spin. If a principal axis of d is invariant, then a unit vector n along it must satisfy both Hence 1 EULER [1761, 2, § 77]. 2 Cf. HADAMARD [1903, 11, ~ 67]. (88.10) (88.11) 3 Further properties of the stretching tensor of an irrotational motion have been discussed by REIFF [1887, 4]. 4 [1761, 2, § 67] [1770, 1, § 93]. The Eq. (88.9) is often erroneously associated with the name of LAPLACE, who was three years old when EuLER read the paper deriving it in this connection and obtaining all its polynomial solutions. f> [1890, 4, §§ 4- 5]. 360 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 89. since wkm= -w".k implies wk".nkn"'=O, it follows from (88.11) that C-d(n)=O, and putting this result back into (88.11) yields (88.12) From this formula we may read off the theo'J'em of Gosiewski: lf a principal axis of stretching is invariant, eitheY it must coincide with the direction of the vorticity vector or the motion is irrotational; if two principal axes of stretching are invariant, the motion is irYotational; in an irrotational motion, aU three axes of stretching are invariant. The last part of the theorem is obvious from the last interpretation of the spin in Sect. 86. 89. Two centrat examples: rectilinear shearing and rectilinear vortex. The generalnature of stretching, shearing, and spin is greatly illuminated by two special classes of motion. A rectilinear shearing1 is a steady motion in which the paths of the particles are parallel straight lines and the speed of each particle is constant in time. Thus in a suitable reetangular Cartesian system we have Y i = f(y, z), y = 0, z = 0. (89.1) L------------------------------X Fig. t6. Simple shearing, a rotational motion with straight stream lines. In fact, by (68.2) and (77.1) we see that a pseudo-lineal motion of the first kind is isochoric if and only if it is a rectilinear shearing. The stretching and spin are given by d= 0 0 ' 0 !!., -i '·· I W= 0 o !!,, i t .• 0 0 ·I (89.2) Elements parallel or normal to the direction of motion are not being stretched. Elements parallel to the direction of motion remain so and thus never change in length. Elements normal to the direction of motion, however, in general are being rotated 2, and thus at a later instant are no Ionger normal; hence their exemption from stretching is only temporary. Entirely typical is the plane homogeneaus case, called simple shearing (Fig. 16): i = K y, y = 0, z = 0. (89. 3) Forthis special case· the quadric of stretching is the hyperbolic cylinder K xy =const, (89.4) the principal axes of stretching are the z axis and the bisectors of the x and y axes, the principal stretchings are ± i K, the maximum orthogonal shears are experienced by elements in the x and y co-ordinate directions, the axis of spin is the z axis, and both the vorticity and the maximum shearing have the value K. This example is instructive 3 in that although every particle travels in a straight line at uniform speed, in general the spin is not zero. Rotational motion 1 While this class of motions was introduced and analysed by EuLER [1757, 2, §§ 48- 49], it is usually named after CouETTE or PorsEUILLE. 2 ST. VENANT [1869, 6, § 6, footnote]. 3 It was used by BERTRAND [1867, 1] as a basis for objection to HELMHOLTz's theory of vorticity. Cf. [1868, 1]. Sect. 89. Two central examples: rectilinear shearing and rectilinear vortex. 361 of a continuum is thus different in nature from rotation of a rigid body. The particles need not circle about in orbits like planets. Rather, if we allow a small object marked with a cross to be carried along in a motion of a continuum, and if we suppose the object partakes of the tangential motion, so that its rate of turning is a measure of the circulation\ then in the rotational case the arms of the cross will turn, while in the irrotational case they will not. Spin is a local property of a motion. Shearing, like shear (Sect. 45) has been defined by a particular set of geometrical objects. To replace it by an invariant concept, we observe that at any given point in order for there to exist a co-ordinate system in which d assumes the form 0 tK 0 d= 0 0 ' (89.5) 0 it is necessary and sufficient that d1=-d3 , d2 =0. Butthis is not enough, since in (89.3) the spin is parallel to the second principal axis of stretching, and its magnitude satisfies w =2d1 . Therefore, in order that a motion may be regarded locally as a simple shearing, it is necessary and sufficient that: 1. Id = IIId = 0; ) 2. w is parallel to the principal axis of stretch along which the stretching is zero; 3. w= V-4II~ = the amount of shearing. (89.6) For some purposes it is preferable to have a definition independent of the spin. We shall say that a motion is a shearing if (89.6) 1 holds. An equivalent condition is (89.7) From (89.2) it is plain that the rectilinear shearings (89.1) are included as a special case. Also, any shearing may be regarded locally as a simple shearing superposed upon a rigid rotation. A rectilinear vortex 2 is a motion in which the paths of the particles are circles, whose planes are parallel and whose centers lie upon a straight line, while the speeds of the particles upon a given circle are the same at any one time. Thus in a suitable cylindrical polar system we have r=O, Ö=f(r,z,t), z=O. The circulation of the circle r = const, z = const is given by 2" • • f r () · r d () = 2 n r 2 () = 2 n r 2 w, 0 the physical components of the vorticity vector by w =-r ae;az, w = 0, w = r-1 o(r2 0)jor. (89.8) (89.9) (89.10) 1 The arms of the cross must not be confused with material elements, since even in an irrotational motion material elements are generally rotating (cf. Sects. 78, 86, 88). 2 The idea of a vortex goes back to DESCARTES or earlier. The mathematical theory was initiated, rather incompletely if not inaccurately, by NEWTON [1687, 1, Bk. II, §IX], D. BERNOULLI [1738, 1, § 11] and D'ALEMBERT [1744, 1, Bk. III, Chap. IV]. The theory as given here is due to EuLER [1757. 2, §§ 30-33] [1757, 3, §§ 57-61] [1770, 1, §§ 75-87]. See also Sect. 98. C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 99· If Ö =0 at r=O, (89.9) and {89.10) are related through (87.1). A rectilinear vortex is a pseudo-plane motion of the first kind; typical is the plane special case when Ö = Ö (r, t). Whatever be the function Ö, so long as it is not zero when r > 0, the particles circle about in orbits. Nevertheless, the motion need not be rotational. As EuLER observed, the spin vanishes if and only if (89.11) In this case, the circulations (89.9) all have the same value, 2nf(t), called the strength of the irrotational vortex. Whether or not the motion be irrotational, or indeed the value of the vorticity altogether, in this example has nothing to do with the paths followed by the particles. Again, it is a local matter. Fig. 17 represents the irrotational and rotational cases schematically by means of an object marked with a cross 1• Fig. 17. Two motions with the same stream lines. At the left, an irrotational vortex; at the right, a rotational one. To describe these two special motions, we have used the spatial description. Corresponding material descriptions are, respectively, x =X +F(Y,Z)t, y = Y, z=Z, r=R, 6=8+JF(R,Z,t)dt, z=Z. 90. The fundamental decomposition of CAUCHY and STOKES. Since (89.12) (89.13) (90.1) and since ik ". =0 defines a motion of uniform translation, from the results of EuLER and 'CAUCHY presented in Sect. 82 and Sect. 86 we may read off the following fundamental theorem, first explicitly stated by STOKES 2 : An arbitrary instantaneous state of motion may be resolved at each point into a uniform translation, three in generat unequal stretchings along mutually perpendicular axes sutfering no shearing, and a rigid rotation of these axes. The most important implication of this theorem is the simplest: At any point, a given motion may be decomposed uniquely into a translation, an irrotational motion, and a rigid rotation. Other decompositions will be mentioned in Sect. 92 and Sect. 142. As noted by ZoRAWSKI 3, the several identities connecting the gradients of d and w follow from the fact that xk,mp=xk,pn:· The simplest of these is {90.2) Various other relations follow by simple combinations of this identity and the symmetry conditions dkm=dmk• Wkm=-w".k; in particular, these in turn imply Xk,pm=Xk,mp· 1 Recall the caution expressed in footnote 1, p. 361. 2 [1845, 4, § 2]. The result is equivalent to the theorem of CAUCHY presented in Sect. 57. APPELL [1903, 2, §§ 1- 5] studied further properlies of ;km· On the basis of arguments we do not follow, HENCKY [1949, 13] replaced ik ". by ik,;. + ckxm, where c is an arbitrary vector, and discussed a resolution analogaus to (90.1). ' 3 [1911, 14, § 4]. Cf. the more general result( 84.7) of RICCI. Sect. 91. The kinematical vorticity number. 91. Tbe kinematical vorticity number. Since there is a unique local decomposition of any motion into stretching and spin, it is natural to seek a measure of the rotational quality or rotationality of a given motion. This question, the answer to which has important applications in hydrodynamics, was first attacked by LEVI-CIVITAl, but the measure he suggested, Jw dt, is obviously unsatisfactory. TRUESDELL2 has introduced the kinematical vorticity number: (91.1) the ratio of the magnitude of w to the intensity of d. :WK is dimensionless; when :WK is large, vorticity predominates over deformation, and when :WK is small, the motion is nearly irrotational; :WK =0 is a necessary and sufficient condition that a motion which is not a translation be irrotational, while :WK = oo is a necessary and sufficient condition that a motion which is not a translation be rigid. Thus all motions which are not translations are assigned a numerical degree of rotation between 0 and oo, with the rigid rotations appearing as the most rotational of all. All that we have just said would apply equally well had we taken k WK, where k is any dimensionless constant, as a measure. To normalize such a measure, we might select a particular rotational non-rigid motion and assign it a measure 1. We prefer, however, to approach the normalization geometrically. A motion in which all the components dk". reduce to zero except for one shearing component, say dzv• may be described entirely in terms of two angular rates. One of these, as usua.J., is the local angular velocity, w = l w; for the other, X· select the rate at which elements in the x and y co-ordinate directions are tuming away from each other, i.e., X= dxy. Then 2w w k}»K = k l!A:2 = k - 1 1 -. v4x2 x (91.2) Thus if we take k = 1, the measure k WK becomes, for this dass of motions, just the ratio of the two characterizing angular rates. Some arbitrariness remains in the choice of this particular pair of angular rates; perhaps the final justification for taking k = 1, as we shall do, is the symmetry and simplicity of the resulting formulae. It is easy to verify that (91.3) where (86.14) has been used. Thus we get the following alternative necessary and sufficient conditions that WK = 1: x"'x" ,n ,m =o ' 2l0 = I~. (91.4) It is obvious that the second of these is satisfied by (89.1); hence in a rectilinear shearing, WK = 1. 1 [1940, 13]. I [1953,_31]. C. TRUESDELL and R. TOUPIN: The Classical Field Theories. More generally, from (91.3) we may assert that WK:; 1 according as x:':. x~m ~ o; equivalently, I~~ 210. Sect. 92. (91.5) These are conditions on the proper r..umbers of the velocity gradient. In fact, we may express (91.5h as follows: lf the proper numbers of x:':. are real, then WK < 1; if the proper numbers are a, b ± ic, then WK:; 1 according as c2 :; a2+ b2• This result is not very illuminating, since the proper numbers of x:':. do not have an immediate kinematical significance. Further properties of WK, including other forms of the condition for the sign of WK- 1, will be obtained in Sect. 102. Now consider the deviator vkm of xk m• already mentioned in Sect. 78, and let a subscript 0 distinguish quantities 'calculated from it. Since w 0 =W, by (App. 39.6) and (91.1 )a follows 1 (91.6) where eguality holds if and only if the motion is isochoric or irrotational or both. We can now put some of the conditions in Sect. 78 regarding invariant directions into a clearer form. Wehave IOo == - ~ V km vmk = ~ (wkm wkm- dokm d~m)' l = : w2 ( 1 - _W1k;) . (91. 7) Therefore 100< 0 if and only if WKo< 1; also, by (91.6), WK> 1 is sufficient that 100 >0. From the remarks following (78.7) we derive a result of ERICKSEN 2 : I f WK > 1, or if WKo = 1 and S0 =f= 0, there is one and only one invariant direction; for there to be three invariant directions, it is necessary that W K 0 < 1. Rigid motions are included in the first alternative, irrotational motions in the second. For a simple shearing (Sect. 89) we have WKo = WK = 1, S0 =0, and as noted in Sect. 78 there are infinitely many invariant directions. 92. Sources and vortices. A decomposition which differs from that of Sect. 90 in that it is cumulative rather than local is expressed by the existence of STOKEs's potentials (App.36.1) for the velocity: X.~=- S,k + ekmqVq,m, or p =- grad 5 + curl V. (92.1) The theory of this representation, like that presented in Sect. 87, is purely geometrical. We haveJ;725 = -Id, V2 v = -w, and (App.36.2) asserts that in afinite region, a velocity field is uniquely determined by its values on the boundary and on any surfaces of discontinuity, by its cyclic constants, and by the values of its expansion and vorticity at interior points. In particular, with the special choice of potentials (App. 36.2), the normal velocity Pn on 9"contributes to the scalar potential S, while the tangential velocity Pt contributes only to the vector potential v. It is not possible to assign Id, w, Pn, and Pt arbitrarily, however. For example, GREEN's transformation (App. 26.1) 2, 3 yields J Id d v = ~ da · p, Jwdv =~daxp, (92.2) " d " d as necessary conditions to be satisfied by the four quantities occurring m (App. 36.2). In Sect. 116 weshall see that (92.2) is not always sufficient. 1 ERICKSEN [1955, 5]. 2 [1955. 5]. Sect. 93. Total squared speed. I. Estimates. The general theory of sources and vortices is given in works on potential theory. We may interpret (92.1) and (App. 36.2) with c = p as asserting that any motion may be expressed as the sum of an isochoric irrotational motion and a motion induced by continuously distributed sources and vortices. 93. Total squared speed. I. Estimates. The results of the foregoing section imply that any quantity associated with a motion in a region may be calculated from the values of the vorticity and expansion in the interior and of p upon. the boundary. As a specimen, we now reckon upper bounds for the total squared speed. First, a f<;>rm~la generalizing (App.31.12) is ~ == f jhiv·~ ~da· (P2P- 2p p ·p) + 2fp · (wxp +fJid) dv, (93.1) .,. . .,. as follows by setting b = c = p and K = 1 in (App. :26.2), then contracting the resulting identity. Since the integrand of the volume integral contains p as well as w and Id, we do not yet have a result of the type desired. However, from (93.1) we can derive an inequality free of interior values of p, as follows. First, I~ da· (P2P- 2p P · p) I~ Klp~ + 2K2Pm Pnm (93.2) • where We note that Pm == Max p on o, =~1da·pl, • Pnm == Max Pn on ==~dap. (93-3) • (93.4) where 5 is the area of o and where to derive the last inequality we have chosen the origin of p at the center of the smallest sphere whose interior contains "· D being the diameterofthat sphere. Also IP·(wxp)+pldi~PP(w+lldl); (93-5) therefore, the Cauchy-Schwarz inequality implies thatl \ J p · (w X p + p Id) dv \ 2 ~ ~ J P2 (w + I Id 1) 2 dv, ) ~(iD)2~f(w + l'Id1)2dv. (93.6) .,. From (93.1), (93.2), (93.4), and (93.6) we get ~~iD5(p~.+2PmPnm)+DV~ l/f(w+lldl) 2 dv. (93-7) .,. Set A == i D 5 (p;,. + 2Pm Pnm), B == D V J{w+I ldl) 2 dv. (93.8) .,. Then (93.7) yields ~ = f p2 d V ~ t ( B2 + 2 A + B V B2 + 4 A ) . .,. Thus the total squared speed in a finite region may be estimated from 1. The diameter of the region and the area of its boundary; 2. The maximum speed on the boundary; 3· The total (w +I Idl) 2 • (93 .9) 1 Inequalities of this type were first obtained by KAMPE DE FERIET [ 1946, 6]; we fo!low and generalize the work of BERKER [1949. 2, § 3]. 366 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 93. Several special cases are interesting. First, if w = Irt = 0, we have an isochoric irrotational motion, and (93-9) reduces to f p2dv-;;;,A, a result which is also an " immediate consequence of the fact that the maximum speed and the maximum of each velocity component of such a motion always occur on the boundary (cf. Sect. 121). Second, if 11 is a stationary boundary of the motion, A reduces to the value t D s p~. Third, if 11 is a stationary boundary to which the material adheres, (93.9) reduces to J p2dv-;;;, B2, or " f p2dv-;;;;;, D2 f (w + 1 Ir:~J)2dv. (93.10) " " If w = Irt = 0, we conclude that p = 0: In a bounded stationary domain to which the material adheres, there exists no isochoric irrotational motion other than a state of rest. Cf. the generalizations and alternative developments of this result in Sects. 111 and 113. We now deduce some stronger results for the case of a simply connected domain, where the explicit formulae (App. 36.2) are available. First we note the identity p2 = p . [- grad S + curl v], } = - div (p S) + S lrt - div (p X v) + v · w, (93.11) where the second step is possible because S is single-valued. By (App. 36.2) follows p2 = div (v x p- p S) + :: [J __I_rt/v - ~ da/} 1 + 1 " d + :: . r Jw :!- _ ~ da: p] . .. d (93.12) We now integrate this identity over the volume v. If we use primes to distinguish quantities evaluated at a second running point while unprimed quantities are understood to be evaluated at the running point occurring in the integrations already indicated in (93 .12), we obtain .. p2 f dv=4n .. " d. dvdv'+rda·(vxp-pS)- · 1 JJ lrt Id + w · w' J. . . I 1 jJ.da·pld+(daxp)·w' , (93.13) - 4n 'j' d dv . " d Under various circumstances the second and third terms on the right-hand side are zero. Such is the case, for example, if 11 is a fixed boundary to which the material adheres, or if v is an infinite region and p = o (r-2) at oo. We then obtain the elegant formula of HELMHOLTZ and A. FöPPL1 : f p2dv =-1-JJ lrtld+w·w' dvdv'. 4n d · 1 [1858, 1, § 4] (in the case lrt=O); [1897. 4, §§ 32-33]. (93.14) Sect. 93. Totalsquared speed. I. Estimates. 367 We now determine abound for the right-hand side of (93.14) in the case of a finite region 1• From the Cauchy-Schwarz inequality we get (J J 1ddrd dvdvJ ~(4nC) 2 J J (Idi;,) 2dvdv' = (4nC){fradvr. (93.15) V V V V V where (4nC)2= JJ dvd~v'. (93.16) "" Since (w. w') ~w w'2, we similarly obtain (! ~w~w' dvdvJ ~ (4nC)2(f w2dvr " " " (93.17) Thus for a finite simply connected region to whose boundary the material adheres we get the elegant bound f p2 d V ~ C f (I~ + w2) d V, (93 .18) " " tobe compared with (93.10). The constant C depends only upon the domain v. We now obtain an estimate 2 for C in terms of the diameter D that was used above. Clearly ff dvdv' J (4n C) 2 ~ ----r1,2 = G dv, (93.19) where v 0 is the smallest sphere that contains v, and G-f~~· (93.20) "• Let R1 be the distance from the argument point of G to the center of the sphere v 0 , and introduce polar co-ordinates with axis along the line connecting these two points. Then c-J r 2 - r2+ sin0d(Jdrpdr R~- 2rR1 cos (J' l "• R2-m R+RI = n ---log~ - + 2n R, R1 R-R1 (93.21) where R=!D. A further quadrature yields JGdv=i n2D4, and hence 3 "• (93.22) Comparison with (93.10) shows that the second estimate is much sharper than the first. However, the first estimate holds without restriction on the connectivity of the domain: 1 BERKER [1949, 2, §§ 4- 5]. 2 BERKER [1949, 2, §§4-5]. 3 For the plane case, KAMPE DE FERIET [1946, 6] obtained a plane analogue of (93.18) with {2nC) 2 = J J(rag·~Y da da'. • • 368 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 94. 94. Total squared speed. II. Extremal theorems. A simple and profound theorem of KELVIN1 characterizes irrotational motions as minimizing the total squared speed. Indeed, let us define the kinetic energy 2 Sl' of a motion as (94.1) V where A is any substantially constant function not identically zero. I.e., Aj is a density for the motion (Sect. 76). We now compare the kinetic energy Sl'' of the motion .V' with the kinetic energy Sl' of an irrotational motion, ik =- V k. If V is single-valued, we have ' A j i' 2 = A j [ i2 - 2 (.i' k- ik) ~ k + (.V' -.V) 2] , l =Aji2- 2[(i'k-ik)AjVJ k + + 2 V[Aj(i'k_ ik)] k + Aj(.V' -i-)2. (94.2) By (76.11), the third term vanishes in two cases: when both .V and .V' are isochoric, or when both motions are steady and have the same steady density Aj. In these cases, integration over a volume t> and use of (App. 26.1) 2 yields Sl'' = sr +! J Aj(.V'-.V)2dv- pda. (.V'-x) AjV. (94.3) V d The surface integral vanishes if both motions are assumed to have the same normal flow Aj(i'-i)n through the bounding surface il. If we assume A>O, the second term in (94.3) is positive unless .V' =.V. Rewording this result yields Kelvin's theorem oj minimum energy: Let the normalflow at each point of the bounding surface of a simply connected region be assigned; then amon~ all isochoric motions, or among all steady motions with an assigned positive steady density, the least possible kinetic energy is attained if and only if the motion is irrotational. For multiply connected regions the theorem still holds, provided fixed values be assigned to the fluxes through a set of barriers rendering the region simply connected a. A variational theorem which is a partial converse of KELVIN's theorem has been obtained by PRATELLI 4 • Consider the velocity fieldas given in terms of its Stokespotentials (92.1). We vary the potentials Sand v, obtaining {jp = - grad /j S + curl IJv. We assume the region simply connected, so that S is single-valued, and we set I""' f [ip2 - SB] dv (94.4) V and restriet the variations to those leaving the quantity B unaltered. Then by (App. 26.1) 2 and easy vectorial identities follows /JI = f[p · IJp - B /j S] dv, V = .f[p · (- grad /j S + curl /jv) - B /j S] dv, V (94.5) = f[- div(p {jS + p x IJv) + (Id- B) IJS + w · /Jv] dv, V =-~da· (piJS + p x IJv) + f[(Id- B) IJS + w · /Jv] dv. d V 1 [1849, 3]. A variational formulation is given by PRATELLI [1953. 24, §§ 1-4], who notes also that of all isochoric motions in a tube with assigned discharge, the irrotational motion has the least total squared speed. 2 The general theory of the kinetic energy is given in Sects. 166ff. 3 LAMB [1895. 2, §55], but the result is really immediate from KELVIN's theory of cyclic potentials. A rigorous proof, based on topological theorems, of a result slightly stronger in that the motions considered for comparison may have any density not greater than that of the irrotational motion, is given by HA YES [ 1960, 2]. '[1953. 24, §§6-7]. Sect. 95. Rate of strain from an initial configuration. If we restriet the variations so that t5 S = 0 and t5v = 0 on J, we get as equivalent to t5l = 0 the condi tions w = 0, I,t= B. (94.6) That is, in order to render l an extreme when the scalar and vector potentials of the velocity are subjected to variations vanishing an the boundary but otherwise arbitrary, it is necessary and sufficient that the motion be irrotational and that its expansion be the quantity B. In the case when only S is varied, from (94.5) follows only the latter, not the former of (94.6). That is, of alt molians having the same vorticity, those such that Id = B give an extreme value to l when the scalar potential is subjected to variations which vanish an the boundary but are otherwise arbitrary. These results may be regarded as variational theorems for the equation of continuity (76.11). In the isochoric case, we may set B=O and reduce 2l to the totalsquared speed; the condition (94.6) 2 then becomes Id= 0. As has been remarked by SERRIN1, DIRICHLET's principle may also be regarded as a converse of KELVIN's theorem, as follows. Let h be given; corresponding to any single-valued function V', set ~·~- t J (grad V') 2 dv + ~ V'h da. ( 94.7) V 6 Let ~ be the value of ~· when V'= V, the potential of the unique isochoric irrotational motion in v such that Pn=- o Vfon = h on J. Since 172 V= 0, from (94.7) it follows by use of GREEN's transformation that ~= ~'+ t Jfgrad(V- V')] 2 dv. (94.7) V Hence ~ > ~· unless V= V'+ const. That is, the potential of the isochoric irrotational motion in v such that Pn = h on J renders ~ a maximum. For an isochoric motion, we have i = 1, and we may take A = 1. In the notation used from Sect. 155 onward, this is equivalent to taking e =Ai= const., so that comparison of (169.7) with (94.7) yields e ~ =- ~ + 2 ~ = ~; (94.8) i.e., the maximum value of e ~ equals the minimum value of ~. namely, the kinetic energy of the isochoric irrotational motion which is characterized as the unique solution of each variational problem. 95. Rate of strain from an initial configuration. Our purpose now is to connect the concept of finite stretch, formulated in Sect. 25, with the stretching introduced in Sect. 82. In terms of motion, if the material element given by dX at time t =0 is carried into d;r at time t, then its stretch at time t is the ratio of its present length to its original length: dx ;._(ix· (95.1) As repeatedly emphasized in Subchapter I, stretch is a function of two different configurations of the material. Not so with stretching, which refers only to the present configuration: d ~~logdxi . dt d:r~dX (95 .2) . But, since for a material element we have dX = 0, from (95 .1) follows • dx A. = = A. d or d = log A.. dX ' (95.3) Thus, in general, the stretching of a material element is its rate of stretch per unit stretch. If we put t = 0 in (95. 3), we get A. = 1, and hence (95.4) 1 Sect. 24 of Mathematical Principles of Fluid Mechanics, This Encyclopedia, Vol. VIII/1. Handbuch der Physik, Bd. 111/1. 24 370 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 95. Tlte stretching of an element is its rate of stretch with respect to its present configuration. Despite their truth, the foregoing simple remarks are of little use 1• The stretch of an element is calculated from its components by means of the components of the tensors C and c, and the manner in which these components change with time is not obvious . . For the tensor C, the calculation is easy, since by (26.2h and the fact that dX«=o we get From (82.3} and (20.3), on the other hand, we get Since dX is arbitrary, comparison of (95.5) and (95.6) yields 2 • k • crxß = 2dkmX,a;X:ß = 2 Ea;ß• Similarly, differentiating (26.1 h yields 0 = c d xk d xm + c (xk d xn d xm + d xk _xm d xn) km km ,n ,n , where we have used (76.4} 2 • Since dxk is arbitrary, we get or, equivalently 3, From (95.10} and (95.7) 2 we get (95.5) (95 .6) (95.7) (95 .8) (95 .9) (95.10) (95.11) as expected from (95.4). However, the rates e and Ein a strained configuration are quite different from d and from one another. This is most easily seen by setting d =0 in (95.10) and (95.7), yielding (95.12) Thus in an instantaneously rigid motion the components of E remain constant, but those of e generally change. This difference reflects an inconvenience of the 1 Further results of this kind are noted by REINER [1948, 23, § 7] and TRUESDELL [1952. 21, § 22]. 2 E. and F. CossERAT [1896. 1, § 15, Eq. (2)]. Cf. also DuHEM [1904, 1, Part I, Chap. I, § 1, Eq. (30)]. An equivalent but more complicated variational for'?ula had been given by PIOLA [1833: 3, § 6] [1848, 2, ~ 36]; in effect, he also calculated c-1• BoNVICINI [1932, 3] calculated Cf. A particularly interesting equivalent form was obtained by DEUKER [1941, 1, §VI] in terms of the velocity ){ rx of space relative to the material (cf. Sect. 65): Erxß = -J{(,.,ß)· Comparison with (82.3) shows that the rate of strain tensor E is the negative of the stretching tensor for the motion of space relative to the body. This is an example of the principle of duality mentioned in Sect. 65. As a corollary, 4=-~-- - IP·gradPI · (100.3) It measures the relative importance of the convective and local accelerations. In a steady accelerationless motion, it is indeterminate. We have i9 = oo if and only if the motion is unsteady and the convective acceleration is zero; i9 = 0 if and only if the motion is steady and the convective acceleration not zero. For an unsteady accelerationless motion, i9 = 1. For small values of iS>, we have (100.4) with an error of magnitude iSII p · grad p I, while for large values of i9 we have .. ap or p R; aT, (100.5) with an error of magnitude I opjotlfiSI. Motions in which (100.5) is regarded as correct are often said to be slow. We note that in an unsteady slow motion we have WnR;O; in an isochoric slow motion, WKR;1 (cf. Sect. 91). Further properties of i9 ha ve been developed by SzEBEHEL Y. To illustrate the use of the measures WK.Wn. and i9, consider the following special case of the rectilinear vortex (89.8): ; = 0, (} • = 2nr2- K ( 1 - e -~) 4 vt ' z = 0, (100.6) where v is a positive constant. The interest in this particular case lies in its being a dynamically possible motion for a viscous incompressible fluid of kinematic viscosity v. As t-+o+, the velocity field (100.6) approaches that of an irrotational vortex of strength K, so that (100.6) represents the decay of such a vortex due to the action of viscosity. Put Ä ==r2/(4vt). 1 TRUESDE"-L [1953, 31, § 13]. 2 SzEBEHELY [1952, 19] [1953. 30] mentions that (100.3) was suggested by analogy to (100.1), at that time unpublished. 380 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Then from {91.1), (100.1), and {100.3) we calculate 1 'WK=----- 1 T (e'-- 1)- 1 [ 16:n:2v2 . ).2e2.! _ ( _ el _ 1 )2]-~ Wn= K2 (e"- 1)2 + 1 2). ' 8:n:v ).2 e-.! ~ = ---y (1 - e .!)2 Sect. 101. (100.7) (100.8) (100.9) To interpret these results we Iet ). decrease from oo to 0, representing the effects either of the increase of time at a fix;ed place or the decrease of rat a fixed time. As would be expected, at ). = oo all three measures vanish. As ). decreases, both 'WK and ~ increase steadily, until at ). = 0 we have 'WK= oo and ~ = 8:n:vfK. At a fixed place, then, the motion becomes more and more rotational and more and more unsteady as time goes an. The measure Wn vanishes ( 16:n:2 v2 9 )-~ at ). = oo; at ). = 0 it approaches the value ~ + 4 . 101. Convection and diffusion of stretching and spin. We show now that the material rates of change of stretching and spin are expressible in terms of the gradient of the acceleration. While it is easy to derive the formulae by calculating the material derivatives of d and w, we prefer to follow CARSTOIU1 in beginning with expressions for finite increments along the path of a particle. Since (101.1) we have (101.2) Integrating along the path of a particle, we obtain the basic identity: (101.3) where ia.,ß stands for the initial value of ik,m in accord with the convention of Sect. 66. The first term on the right, ia.,ß X~k X~ m• expresses the mere transport of the initial velocity gradient to the present location of the particle X, while the remainder calculates the cumulative contributions made by the values of the acceleration, its gradients, the gradients of the speed, and the deformation gradients at each point on the path of the particle X. We say that the mechanism of change associated with the former is convection; with the latter, ditfusion 2. The equation that follows by taking the material derivative of (101.3) is ( 101.4) It is easier, perhaps, to establish this result directly by differentiating (98.1) 2 • In the terminology of the preceding paragraph, the first and second terms on the right of (101.4) are the diffusive and convective rates of change, respectively. Thus the gradient of the acceleration field is the diffusive rate of change of the velocity gradient. By a somewhat elaborate calculation it is possible also to derive (101.3) as the integral of (101.4) along the path of a particle. 1 [1954. 1, § 1]. 2 The terms were defined in this manner by TRUESDELL [1948, 36], who derived them from~ J AFFE [1921, 3]. Sect. 101. Convection and diffusion of stretching and spin. 381 As suggested by (90.1), we now split xk,m into its symmetric and skew symmetric parts : (101.5) The skew symmetric part of (101.4) is BELTRAMI's equation for the rate of change of spin 1 : (101.6) or, equivalently, wk = wU + wm i~m- wk Id, } JwkiJ = wU + wmi~m· wdv = w*dv + wdv · gradp. (101.7) Hence the rate of change of vorticity magnitude is given by 2 if2w2jj2=wtwk+dkmwkwm, or ) i w2 (dV)2 = (w* · w + w. d. w) (dv)2. (101.8) The general solution 3 of BELTRAMI's equation (101.6) is the skew symmetric part of (101.3): (101.9) equivalently, t Qrx.{J = f~{J + f w!px~IXX~{Jdt, (101.10) 0 where Q is defined by Eq. (95.16), and where we write ~ß rather than wrx.ß for the initial value of wkm [cf. the convention (66.8)]; another equivalent form is 4 wk = [w" + jw;dt] x~,J]. 0 where W"==:e1Xf1Y(x xk) =]XIX ekmPx =]XIX w*k. * k ,y ,{J ,k p,m ,k The vectors w* and W* are called the spatial diffusion vector and the diffusion vector, respectively. Turning now to the symmetric part of ( 101.3), we obtain 5 (101.11) (101.12) material (101.13) Comparison of (101.13) with (101.9) shows that the diffusion of stretching is a more complicated process than the diffusion of spin. The diffusion of stretching results not only from gradients of acceleration but also from the existing spin and stretching at each point on the path of the particle. 1 [1871, 1, § 6]. For earlier special cases, see Sect. 130. 2 APPELL [1903, 1]. Special cases were obtained by WARREN [1870, 8] and v. KARMAN [1937. 3]. 3 TRUESDELL [1948, 36]. Special cases are due to APPELL [1917, 3, § 1] [1921, 1, § 814], VILLAT [1930, 8, pp. 10-12], BoGGIO [1935, 1], CARSTOIU [1946, 2], and TRUESDELL [1948, 37]. For further discussion, cf. LICHTENSTEIN [1925, 10, Chap. I, § 5] [1927, 5, Chap. 1, § 2] [1929, 4, Chap. 10, §§ 1-2]. For earlier work along this line, see Sect. 134. A derivation in material co-ordinates is given by TRUESDELL [1954, 24, §§ 84-85]. 4 TRUESDELL [1954, 24, § 84]. 6 CARSTOIU [1954, 1, § 6]; the main step was given in [1953. 3]. 382 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 102. The differential equation whose general solution is (101.13) was given by APPELL!: d• d* .p • km= km-Z,(kZm),P• ( 101.14) as follows at once by taking the symmetric part of (101.4). Various other properlies of the acceleration gradient and its invariants have been worked out by APPELL and CARSTOIU 2 • The divergence of the acceleration, x\, offers particular interest and will be studied in the next section. ' 102. The divergence of the acceleration3. From (101.14), (App.38.15), and (86.14) 1 follows 4 IVp=x,k= k= .d+x,mx,k= d+ d-~w' (102. 1) d. ·· .. k d*k I. ·k ·.m I. II 1 z } = Id + I~ - 2 Ild - -! w2 = Id + I~ - 2 K. Since IId ~ 0, as corollaries follow x~k 1. In particular, !A~J = d~J +im,pi~, ) D (2) -1.A(2)- d dm- d(2) + • 'm- d dm pq- 2 pq pm q - pq xm,pX,q pm q Hj;~ = W~ ~- dpm wmq- Wpm d~ + Wpm Wmq· (104.16) The tensors introduced here are only a few of the many in terms of which a correct and complete description of the higher rates of change of length and angle may be expressed. The possible variety corresponds to that for strain mentioned m Sect. 32. Further properties of the tensors A (N) are developed in Sects. 144 and 15 0. Since the displacement vector u"' is given by the formal power series ( 104.17) one may calculate a series for (~~.;"; substituting this series into ( 104.6) and then putting the result into the formal series equivalent to (104.9) yields a series 2 for Ccxß in terms of the derivatives of the u(N). e) Special developments concerning vorticity. e I) The vorticity field. 105. Vorticity and circulation. Throughout this part of the subchapter, in the main, we employ direct vectorial notations. Our subject is the vorticity vector w, defined by (86.2)3 • 1 Had we started from (37-9) instead of (37.8), we should have obtained a similar result except that the order of the factors wa:.v. v =a:.v. ß; (1o8.2) '(j' '(j' alternatively, we may interpret the second theorem of Sect. 107 in the light of KELVIN's transformation: The D'Alembert-Euler condition is necessary and sufficient that the circulation of every material circuit remain constant in time 5• Classical hydrodynamics is characterized by this one basic statement, and all the main theorems of that subject are consequences of it alone 6• Motions in which the circulation of material circuits does not change in time we shall call circulation preserving motions. They will be frequent subject of remark and illustration in the rest of this part of the subchapter. By (101.12), equivalent to (108.1) is the Hankel-Appell condition7 w. = 0. (108.3) 1 PorNCARE [1893. 6, §§5-6, 150-151], ZoRAWSKI [1900, 11], JAUMANN [1905, 2, § 386], VESSIOT [1911, 12, § 4 ]. Earlier but partially incorrect statements were given by MüLLER [1878, 7] and LEVY [1890, 7, § 10]. 2 [1752, 1, § 86] [1761, 1, §§ 10, 15]. 3 [1761, 2, §58] [1757. 2, § 35]. EuLER noted the connection of this condition with BERNOULLI's equation (see Sect. 120). 4 It was remarked by BRANDES [1806, 1, § 150, footnote] that in a theory where friction is taken into account it will generally follow that w* =!= O; this is borneout by the various theories of fluid friction proposed later. 6 The sufficiency was proved by HANKEL [1861, 1, § 8] and KELVIN [1869, 7, § 59(e)]. 6 LEVY [1890, 7, § 7]. . 7 In connection with perfect fluids, the condition W.=o was derived by HANKEL [1861, 1, § 6]; the condition (108.3), by APPELL [1897. 1, ~ 2]. An equivalent but more complicated formula in material co-ordinates had been obtained by LAGRANGE [1762, 2, §XLIV]. Sect. 109. The acceleration potential. 389 This form of the condition for circulation preserving motion is appropriate to the material description. We may now reformulate the results of Sect. 107 in a fashion so closely connected with the hydrodynamical theorems of HELMHOLTZ (1858)1 that, although they are purely kinematical, we shall call them the second and third vorticity theorems oj Helmholtz: (2) In a circulation preserving motion the vortex lines are material lines, and (3) in a motion such that the vortex lines are material, in order that the strengths of all vortex tubes remain constant in time it is necessary and sufficient that the motion be circulation preserving. The Helmholtz theorems furnish a vivid and full picture of the general character of circulation preserving motions, whose theory may thus be said to be, in a certain sense, closed. Most motions that take place in mechanical theories fail to be circulation preserving, however, and a major objective is to clarify the manner in which a general motion departs from circulation preserving character, or equivalently, to describe the mechanism by which the vortex tubes turn away from coincident material tubes and change their strength-that is to say, to generalize the Helmholtz theorems. This research is presented in Parte IV. The stream-line patterns possible in circulation preserving motions are of a restricted type; a geometrical characterization of those possible in the steady case was given by ZHUKOVSKI 2 • 109. The acceleration potential. A function V* such that xk = - V:t, or p =- grad V* (109.1) is called an acceleration potential 3• The function V* may be single-valued or manyvalued. By a theorem in Sect. App. 33 we may reformulate the D' Alembert-Euler condition (108.1) in the following way: A motion is circulation preserving if and only if it Possesses an acceleration potential. An evident example of a circulation preserving motion is an irrotational motion. By putting (88.2) into (99.14) we obtain p .. d[av 1 ( dV)2l = - gra 81- - 2- gra j ; (109.2) comparison with ( 109.1) shows that if a meaningless function of time only is absorbed into V, weshall have 4 V*=~};--+ (grad V) 2 • (109.3) Another example is furnished by the rectilinear vortices (89.8), which are circulation preserving if and only if they are steady and strictly plane; i.e., fi=Ö(r). In this case 5 V*= JrÖ2 dr. (109.4) 1 [1858, 1, § 2]. The validity of the Helmholtz theorems was extended by KELVIN [1869, 7, §§ 59(d), 60(f)-60(i)]. Cf. NANSON [1874, 5]. That these theorems are essentially kinematical was remarked by ScHÜTZ [1895. 5]. According to LAMB [1895. 2, § 143], LARMOR observed that HELMHOLTz's own proofs of the second and third theorems are open to the same objection asthat raised by SToKEs against certain faulty proofs of the velocitypotential theorem; for discussion of the question at issue, see TRUESDELL [1954, 24, §§ 104-107]. 2 [1876, 7, §§ 30-32]. Cf. the unsuccessful attempts of ToucHE [1895, 6] [1897. 8]. For still more special cases in which formal solutions are available, see Sects. 110, 111, 161. Proof of local existence and uniqueness of steady isochoric circulation preserving motion is approached through consideration of Eq. (108.1) by GoDAL [1958, 2, § 2]. 3 The acceleration potential is a discovery of EuLER [1757. 2, § 35] [1770, 1, § 42]. Cf. also ProLA [1836, 1, Eq. (219)]. Its importance was stressed by LEVY [1890, 7, § 7]. Cf. POINCARE [1893, 6, § 6], APPELL [1921, 1, §§ 729, 753, et passim]. 4 EULER [1761, 2, §§ 79-80]. Cf. VESSIOT [1911, 12, § 4], CARSTOIU [1946, 3, § 3]. 6 EuLER [1757. 2, §§ 30-33] [1757. 3, §§ 57-61]. 390 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 110. 110. Complex-lamellar motions. A motion is said to be complex-lamellar 1 if the velocity field p is complex-lamellar (Sect. App. 33). In this case (99.19) reduces to p=FgradG; equivalently W·p=O. (110.1) (110.2) Hence a rotational motion is complex-lamellar i/ and only if its vortex lines are orthogonal to its stream lines. Equivalently, a motion is complex-lamellar if and only if its stream lines possess normal surfaces. Both plane motions and rotationally symmetric motions (Sect. 68) are complex-lamellar. Complex-lamellar motions possess some of the distinguishing properties of irrotational motions 2• The function - G is somewhat analogous to the potential function V: The formulae -F aG = -i0, the theorem again follows, Q.E.D. We note the following corollaries. First, for a motion in the interior of a finite region the condition at oo becomes superfluous, and the theorem holds unconditionally: There is no isochoric or steady irrotational motion, other than a state of rest, within a finite simply connected region with stationary boundary. Second, if the region of motion is suchthat p-+0 uniformly at oo, then for the isochoric case a theorem on isolated singularities of harmonic functions implies that the three components ·fik in the common frame are single-valued harmonic functions analytic at oo. This fact together with the condition that p da · p = 0 for a • sufficiently large sphere yields ·fik=O(p-2), whence follows V=O(p-1). Thus V ~: = 0 (p-a) = o (p-2), so at oo the order condition is satisfied. That is: In an isochoric irrotational motion in an infinite region such that all bounding surfaces are stationary, if the velocity field vanishes uniformly at infinity and if no material is supplied at infinity, then the motion is a state of rest. I TRUESDELL [1951, 29]. 2 [1889. 6]; [1954. 24, §52]. 3 [1849. 3]; [1858, 1, § 1]. 394 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 113. That the foregoing result cannot be extended to multiply connected regions may be seen from the counter example of the irrotational vortex (89.11), which is irrotational in the doubly connected domain O by j = ~- [ 1 + q sin k t sin r] , . 0 < ~ < 1 , ) . r = ]l'f.~cosktcosr, ()=O, =O, r (113.4) is a motion which is continuous and irrotational within the spherical shell ln~r~in, to whose stationary boundaries the material adheres. However, for the isochoric case there are theorems asserting the incompatibility of irrotational motion with adherence to any finite surface, however small. We begin with the theore1n of Kirchhoff2: The only isochoric irrotational motion in which the material adheres to a finite stationary surface, however small, is a state of rest. Proof: By (69.4), the velocitypotential Visa harmonic function all of whose first derivatives with respect to the space variables vanish upon the finite surface. Upon that finite surface, consequently, the function itself is constant and its normal derivative vanishes. The only such harmonic function is a constant. Q.E.D. As an immediate corollary it follows that the only isochoric irrotational motion in which the material adheres to a finite surface in rigid translation is itself a rigid translation. The results just stated and proved refer to the state of motion at any one instant. Following the analysis of SuPIN0 3, let us now consider the case when the material adheres to surfaces suffering rigid rotation at angular velocity w(t). If we refer an irrotational motion whose velocity potential is V to a co-ordinate frame which is at rest with respect to these surfaces, and whose origin coincides with the origin of a system with respect to which the motion is irrotational, for the velocity with respect to the moving frame we obtain p' =-wxp- grad V, (113.5) 1 [ 1901, 7, 5' partie, Chap. II, § 1]. 2 While the proof of KIReRHOFF [1876, 2, Vor!. 16, § 6] is not rigorous, the result is true, being in fact one way of phrasing a now weil known theorem of potential theory. 3 [ 1949. 29]. Sect. 113. Circumstances when an irrotational motion is impossible. or equivalently x' = 2w.y- P,%, Y' = 2wxz- P,y. z' = 2wyx- P, •• where P, given by p = V + X Wy z + y w. X + z Q)% y, 395 (113.6) (113.7) is also harmonic. At any given instant we may orient the axes in such a way that Wz=wy=O, w.=w, so that (113.6) reduces to x' = 2w y- P,", :V= - P,y, z'=- P, •. (113.8) Since the adherence condition (69.4) now assumes the form p' = 0, the harmonic function P must satisfy the boundary conditions P,z= 2w y, P,y= 0, P,,= 0. (113.9) We suppose w =f= 0 so as to exclude the case treated in the previous paragraph. Consider first a motion within finite rigid boundary surfaces, on each point of which (113.9) applies. Then j/ and .i:' are harmonic functions vanishing upon a closed bounding surface, and hence vanishing identically. Thus P,y=O, P,,= 0; since J72P=O it follows that P=ax+b. Comparing this result with (113.9)1 yields 2w y = a, but since w =f= 0 this condition can be satisfied only upon a single plane y = const, a type of boundary not included in the hypothesis. Hence no such motion exists. Second, consider a motion exterior to finite rigid surfaces, on each point of which (113.9) applies. Let us call regular a velocity field which, relative to the surfaces in question, approaches a definite Iimit at oo and possesses a gradient which is 0 (p-2). Then by the uniqueness of solution to th,e exterior Dirichlet problern the functions Y' and z' must vanish identically, and again there is no such regular motion. These results constitute the following first theorem of Supino: No continuous isochoric irrotational motion of a material adhering to boundary surfaces and completely filling a finite domain whose boundaries form a rigid system exists. There exists no such motion with a regular velocity field filling an infinite domain exterior to a rigid system of surfaces to which the material adheres. In the statement of the above theorem the term "rigid system" denotes a set of rigid surfaces rigidly attached to one another, so that all are endowed with the same angular velocity. Various irrotational motions of materials adhering to rigid bounding surfaces can indeed exist if these are in relative motion, or if the motion relative to them is not regular at oo. A simple example is the plane irrotational vortex already cited, for the stream lines are the rigid concentric circles r = r 0 , z = z0 , rota ting a t angular speeds w = K r-2• Another exam ple is furnished by the rectilinear motion x' = 2w y, y' = 0, z' = 0, or, equivalently, .X =wy, y =wx, where the material adheres to the plane y =0, which is rotating at angular speed w about the z-axis. In general, however, as SUPINO points out, the conditions (113.9) 2 and (113.9) 3 are themselves sufficient to determine P, and hence are not likely tobe compatible with (113.9)1 except in degenerate cases. Another way of putting this same thing is to say that if there exists a harmonic function P satisfying (113.9) on any finite surface, however small, then by the theorem of KIRCHHOFF mentioned above it is unique in all space to within an additive constant. Expressed in kinematical terms, this result becomes the following second theorem of Supino: Suppose there exist an isochoric irrotational motion of a material adhering to a certain rigid boundary surface rotating at a certain angular velocity Then there is no other such motion adhering to any finite portion of that surface, however 396 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 114, 115. small, rotating at the same angular velocity. Applying this theorem to the second example noted in the preceding paragraph shows that the only such motion of a material adhering to a finite plane area rotating about an axis in its plane is the motion induced by the rigid rotation of the entire plane. Similarly, from the first example we condude that the only such motion of a material adhering to a finite portion of a circular cylinder rotating about its axis is the irrotational vortex motion induced by the rotation of the entire cylinder1 . 114. Superposahle motions 2• Two velocity fields p1 and p2 belonging to a certain dass are said to be superposable if their vector sum, p 1 + p2 , belongs to the same dass. If 2p belongs to the samedass as p, then p is said to be selfsuperposable. Consider first the dass of circulation preserving motions. Since from (107.2) we ha ve for all motion 3 curl (p1 + p 2) = curlp1 + curlp~ + curl (w1 Xp2 + w 2 Xih) by ( 108.1) follows the condition for superposability: curl(w1 XP2 +w2 XPJ) =0. (114.1) (114.2) It is trivial to remark that any two irrotational motions or any two screw motions having the same abnormality are superposable, and hence that any irrotational or steady screw motion is self-superposable. In the irrotational case, from (88.2) it is immediate that the velocity potential for the combined motion is the sum of the velocity Potentials of the two original motions 4• Also from {111.10) it follows that any circulation preserving motion with steady vorticity is self-superposable. It is not true in general however that an unsteady circulation preserving motion is self-superposable, that any two screw motions are superposable, or that an irrotational motion is superposable upon a circulation preserving motion. Second, consider the dass of motions whose acceleration is complex-lamellar. From (114.1) and (App. 33-5)1 follows the necessary and sufficient condition P1 · curl Pz + Pz · curlp1 + (PJ + Pz) · curl(w1 Xpz + WzXPJ) = 0. (114.3) For the self-superposable case this becomes p·curl(wxp) =0, since p · curl p = 0; equivalently, .. ow P·-ae=o. ( 114.4) (114.5) Thus a motion with complex-lamellar acceleration is self-superposable if and only of the local rate of change of vorticity is normal to the acceleration. e II) Vorticity averages. 115. lntensity balance. The decomposition theorem of CAUCHY and STOKES (Sect. 90) resolves the local and instantaneous motion into stretching and spin. We now consider spatial averages of scalar measures of these portions. With 1 SuPINO shows further that these two motions are the only possible plane isochoric irrotational motions in which the material adheres to any finite rigid surface. 2 The following analysis, suggested by earlier work of BALLABH [ 1940, 1 and 2], STRANG [1948, 28], and ERGUN [1949, 6], was given by TRUESDELL [1954, 24, § 95]. 3 The dot over the bar indicates, of course, the material derivative based upon the total velocity, P 1 + P2 • 4 STOKES [1844, 4, § 5]. Sect. 116. Linear balance. IO defined by (78. 5) 1 , it is easy to show that div(p · gradjJ- Idp) =- 2IO =- 2II.,- tw2, where the second step follows by (86.14) 1 • Hence J IO d v = - } ~ da · ( p · grad p - Id p) , l V ~ 1 ,+.. d (·· 8p I . ) z'j" a·p-81- dP· • 397 (115.1) (115.2) Hence follows the theorem of intensity balance1: I f all finite boundaries are stationary, and if upon them the material adheres without slipping, while in any portion of the material extending to oo the condition (115.3) is satisfied, then the average value of IO over the entire motion is zero; equivalently, f (4IIa + w2) dv = 0. (115.4) V W e note several corollaries. ( 1) F or a motion of the type described in the theorem to be rotational it is necessary that there exist within it a region where Ha< 0. (2) In an irrotational motion satisfying the hypotheses of the theorem, the average value of IIa is zero. (3) In an isochoric motion of the type described in the theorem the average value of the squared vorticity must equal twice the average value of the squared intensity of deformation: fw2 dv=2fiiadv. (115.5) V V From the third corollary, which is a consequence of (86.14) 2 and (77.1), follows another proof of the theorem on the impossibility of irrotational adhering motions in a finite domain (Sect. 113). 116. Linear balance. The previous section demonstrated that in a broad dass of motions an average balance between the second deformation invariant and the squared magnitude of the vorticity is maintained. Weshall now establish two simpler but kinematically less informative relations of balance connecting the vorticity vector w and the expansion Ia. Let h be any single-valued harmonic gradient: h = grad Q, 172 Q = 0. In (App. 26.2), put K = 0, b = p, c = h. Then there results p[da · ('p h + hp) -da p · h] = f[h Ia- hx·w] dv. (116.1) 6 V By formulating conditions sufficient for the vanishing of the surface integral we obtain the first linear balance theorem 2 : Let h be a single-valued harmonic gradient, and let Ia be the expansion and w the vorticity of a motion in a region v such that 1. Each finite boundary is stationary, and upon it the material adheres without slipping; 1 TRUESDELL [ 1950, 34]. A geometrical construction for the space average of IO had previously been invented by BrLIMOVITCH [1948, 2] and has been discussed further [1950, 34] [ 1953, 1]. From these results follows a characterization of those motions in which IO = 0; this class had been studied previously by HAMEL [ 1936, 4, § § 2-3]. 2 TRUESDELL [1951, 30]. The special case I d = 0 forafinite domain was given by BERKER [1949. 1, Th. IV]. 398 C. TRUESDELL and R. TouPrN: The Classical Field Theories. 2. In any Portion of v which extends to oo, (p h + h P)n = 0 (p-2), p · h = 0 (p-2); then j[ h ld - h X W] d V = 0. " A simple condition sufficient for (116.2) is hp=o(p-2). Sect. 116. (116.2) (116.3) Let 1 be a field whose curl is a harmonic gradient: curll = gradF, V2 F = 0, the harmonic function F being single-valued. Then we have div(Fp) =Fld + p · curll, } =F Id + div(lxp)+l·w. (116.4) Hence by GREEN's transformation follows pda · [Fp + pxj] = f[Fld +I· w]dv. (116.5) 6 " By formulating conditions sufficient for the vanishing of the surface integral we obtain the second linear balance theorem 1 : Let F be a single-valued harmonic gradient, and let curll = gradF; let ld be the expansion and w the vorticity of a motion in a region v such that 1. Each finite boundary is stationary, and upon it the material adheres without slipping; 2. In any portion of v which extends to oo then (Fp + pxl)n = o(p-2); f[F Id +I . w J d V = 0. V (116.6) (116.7) A simple condition sufficient for (116.6) is Fpn=o(r- 2), (PXI)n=o(r 2). An immediate corollary of (116.7), following from the choice 1=0, F=1, is the vanishing of the total expansion : flddv = 0. " Putting h =const in (116.3) and employing (116.8) yields hxfwdv =0, " whence, since h is arbitrary, follows Jwdv = 0: " (116.8) (116.9) (116.10) In a motionsuchthat both the linear balance theorems hold, the total vorticity vanishes. We shall discover a broad generalization of this result in Sect. 118. For the case of a motion in a finite simply connected domain, it is possible to show2 that either (116.7) by itself or (116.3) combined with (105.2) is sufficient that assigned functions Id and w be the expansion and the vorticity of the motion of a material adhering to the boundary of v. That (116.3) is not by itself sufficient is plain from the example Id =0, w =gradp-1 when the bounding surface <1 1 TRUESDELL [ 19 51, 31]. The special case I d = 0 for a finite domain is given by BERKER [1949. 1, Th. I]. 2 TRUESDELL [1951, 30 and 31]. The results were asserted also by VAN DEN DUNGEN [1951, 37] and SYNGE [1951, 25], but their proofs are incomplete. Sect. 117. The theorems of LAMB, PorNCARE, J. J. THOMSON, and BJ0RGUM. 399 is a pair of concentric spheres, since then we have J hxwdv = f hxgradp-1dv = Jcurl(Qgradp-1)dv,) P V V = ~dax Q gradp-1 = o . • (116.11) Forthisspecial case, then, the condition ( 116-3) is satisfied, but upon the spherical op-1 boundaries we have Wn = ---ap = -p-2 =f=O, whence by (105.2) it follows that the material cannot adhere to the boundary. When applied to the special case of a plane motion, both (116.3) and {116.7) yield1 j(GI vanish. The case K =0 has already been derived as (116.10). By formulating more general conditions sufficient for the vanishing of the surface integral on the righthand side of ( 118.1), we obtain the following vorticity moment theorem: Subfeet to the boundary conditions Wn = 0 Or Wn = o(p-K-3), the first K + 1 moments of vorticity vanish1• (118.2) For the special case of a motion enclosed by finite stationary boundaries to which the material adheres, our several investigations in Sects. 112 to 113 and Sects. 115 to 118 have revealed a high degree of regularity: The motion is almost certainly rotational (if isochoric, certainly rotational), the intensity balance theorem holds, the linear balance theorems hold, and all the moments tm ---- iw2 = Wk · More generally, either from {121.4) or from {102.7) we have -J72 5*=id+IId(1-Wi), ( 121.6) {121.7) inspection of which yields the following results: In a non-rigid motion, sufficient conditions for S* to be superharmonic, harmonic, or subharmonic, respectively, are id ~ 0 and WK ~ 1; id= 0 and WK= 1; id ~ 0 and WK ~ 1. In the special case when id=O the value of WK becomes the sole criterion of the character of 5*. Hence follow conclusions regarding various types of motion: first, the ma;rimum theorem for motions in which WK ~ 1 : Given a motion in which the expansion experienced by a particle does not increase (id ~ 0), the greatest value of the scalar potential S* in a region where WK ~ 1 cannot be attained in the interior but must be attained on the boundary; and, second, the minimum theorem for motions in 1 TRUESDELL[1954, 24, § 70]. 2 Special cases were given by RowLAND [1880, 9, p. 267], GosiEWSKI [1890, 3, §§ 3-4], LICHTENSTEIN [1929, 4, Chap. 10, § 6], and LAGALLY [1937, 4]. 3 [1936, 4, § 1]. As had been observed by GosiEWSKI [1890, 3, § 8], from (121.4) it follows that the condition - 17 2 5* = lld - i w2 is necessary and sufficient that a material volume once in isochoric motion remain ever in isochoric motion. 26* 404 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 122. which WK ~ 1 : Given a motion in which the expansion experienced by a particle does not decrease (id;;;; 0), the least value of S* in a region where WK ~ 1 cannot be attained in the interior, but must be attained on the boundary1• The former theorem includes the case of rigid motion (WK = oo, Id = 0); the latter, the case of irrotational motion (WK = 0); both include the case Id = 0, WK = 1, which was mentioned from another point of view in Sect. 91. For irrotational or screw motions, an analogaus result can be obtained for the speedp, generalizing KELVIN's theorem in Sect.102. By (121.7) and (121.1) when w X p = 0 we get 17 2 2 1 P • 2 = ~d • - Tt old + - IId ( 1 - "Wk) , l = p . grad Id + IId ( 1 - "Wk) . (121.8) By formulating conditions sufficient that the right-hand side have a given sign, we derive the theorem of m~imum speed for irrotational or screw motions 2 : In a region of irrotational or screw motion suchthat WK ~ 1 (WK = 1) (WK;;;; 1), while the expansion Id does not decrease (change) (increase) in the direction of motion along a stream line, then at an interior point the speed cannot experience a maximum (maximum or minimum) (minimum). In the irrotational case, of course, WK = 0, so the results concerning minimum speed do not apply. A corollary is that in a region of isochoric screw motion where l»K;;;; 1 there can be no stagnation point. By applying PmssoN's integral from the theory of the potential, it is possible from the Poisson equations given above to write down various expressions for S* or for S* + J:p2 as a volume integrai3. 122. Lamb planes and Lamb surfaces. In a motion which is neither an irrotational nor a screw motion the vorticity w and velocity p differ in direction, except possibly at certain singular points, lines, or surfaces, and hence at each regular point determine the Lamb plane, whose normal is parallel to the Lamb vector w x p. A necessary and sufficient condition for the existence of Lamb surfaces 4 , which are simultaneously vortex surfaces and stream surfaces, is that the Lamb vector wxp be complex-lamellar and non-vanishing. By the EulerKelvin criterion (App. 33-5) it is then necessary arid sufficient that wxp-curl(wxp)=O, wxp=j=O, (122.1) a condition which may be put into the form wxp· (p·gradw-w-gradp)=O. (122.2) By eliminating wxp between (122.1) and LAGRANGE's acceleration formula (99.14), we may obtain a form of the condition for the existence for Lamb surfaces which though lacking the symmetry of (122.2) is nevertheless easier to apply·, viz. w x p · ( w*- 8 87) = o. (122.3) 1 These broad generalizations of results of BauLIGAND [1927, 1 and 2] and of HAMEI. [1936, 4, § 1] were given by TRUESDELL [1953, 31, § 10]. 2 TRUESDELL [1953, 34]. 3 BaBYLEW [1873, 1, § 4], FoRSYTH [1879, 1, p. 139], CRAIG [1880, 5, pp. 223-225] [1880, 7, p. 276], TRUESDELL [1954, 24, § 71]. ' These surfaces were introduced by LAMB [1878, 5] [1879, 2, § 145]. Cf. POINCARE [1893, 6, §§ 22-24], APPELL [1921, 1, § 762]. The name "Bernoulli surfaces" was proposed by CALDONAZZO [1924, 2] [1925, 3, § 2] in a somewhat different sense. Sect. 123. The line integral Bernoulli theorem. 405 By the d'Alembert-Euler condition (108.1) it follows then that Lamb surfaces exist in any circulation preserving motion with steady vorticity. Equivalently, for the existence of Lamb surfaces it is necessary and sufficient that there exist a non-constant scalar B and a non-vanishing scalar C such that w X p = C grad B. (122.4) The surfaces B =const are the Lamb surfaces, and B must satisfy the differential system (w xp) x grad B = o. (122.5) For a very simple example of Lamb surfaces in motions which need not be circulation preserving, consider the case when the convective acceleration vanishes: .. öp p= fit' (122.6) Then by (99-14) follows (122.7) Thus the surfaces of constant speed are Lamb surfaces. Moreover, by (99-7) it follows that if the stream lines are steady, they are straight, and the speed is constant along them at each instant 1. Thus the Lamb surfaces are ruled by the stream lines. Since both stream and vortex lines lie upon the Lamb surfaces (if these exist), these surfaces tagether with the stream surfaces and vortex surfaces normal to them form three one-parameter families of surfaces, and hence serve to define a natural curvilinear co-ordinate system 2 . The following statements are but immediate applications of a classical theorem of DARBOUX 3. (a) In a complex-lamellar motion with Lamb surfaces, the vorticity is complex-lamellar if and only if the vortex lines are lines of curvature both an the Lamb surfaces and an the surfaces normal to the velocity. (b) In a motion in which both the velocity and the vorticity are complex-lamellar, Lamb surfaces exist if and only if the stream lines are lines of curvature an the surfaces normal to the vorticity, while the vortex lines are lines of curvature an the surfaces normal to the velocity. (c) In a motion with Lamb surfaces and with complex-lamellar vorticity, the motion itself is complex-lamellar if and only if the stream lines are lines of curvature both an the Lamb surfaces and an the surfaces normal to the vorticity. In a steady motion, the Lamb surfaces, since they are stream surfaces, are material surfaces (Sect. 74), and since they are also vortex surfaces, it follows that in any steady motion such that curl p is zero or normal to the Lamb vector there exist stationary surfaces which are both stream surfaces and vortex surfaces. 123. The line integral Bernoulli theorem. If Pt uenotes the component of acceleration along any direction t which lies in the Lamb plane, and dfd s1 derrotes the directional derivative in that direction, then from ( 111.9), which is valid only in motions with steady vorticity, it follows that 4 Pt= d~~ ( {- p2 + u). (123.1) A curve which is everywhere tangent to the Lamb planes is a Lamb curve cL. Both stream lines and vortex lines are Lamb curves, and in a motion where Lamb surfaces exist any curve lying wholly upon some one of them is a Lamb curve. If we integrate ( 123.1) along a Lamb curve we obtain the line integral 1 CASTOLDI [1953, 5]. 2 CRAIG [1881, 2, pp. 5-6] used as co-ordinate surfaces thc Lamb surfaces (which he incorrectly assumed always to exist), any independent family of stream surfaces, and any independent family of vortex surfaces. 3 [ 1866, 1, ~ 15]. 4 FABRI [1894, 3]. 406 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 124 Bemoulli theorem: (j P2 + U}ieL = J d:x! · p: (123.2) eL In a motion with steady vorticity, the flow of acceleration along a Lamb curve at any instant equals the diflerence of the values of !P2+ U at the two ends of the curve. In particular, in a motion with steady vorticity the circulation of the acceleration around a closed Lamb curve is zero. In a steady motion, since U = const the line integral Bernoulli theorem gives a direct connection between speed and acceleration. For any particular dynamical model the acceleration is expressed in terms of other quantities (cf. Chap. G), and (123.2) then shows directly the effect of these quantities upon the speed of flow1 . 124. The curvilinear Bernoulli theorem. Now in general it is possible to express any vector, and in particular the acceleration p, as the sum of a gradient plus a second field in an infinite nurober of ways: p =- grad Q + p*. (124.1) The function Q may be the Stokes scalar potential 5*, the negative of the Monge potential H*, or some other function. We shall assume that Q=!=const, so that (124.1) really expresses a decomposition of the acceleration field, and we shall assume furtner that at least one possible choice of Q be such that 2 p*=!=wxp. Wehave w*=curlp=curlp*. (124.2) As suggested by the results in Sect.101, weshall call p* the diffusive acceleration, taking care to recall ever that this field is determined only to within an arbitrary gradient. The numerous theorems to follow in whose statements reference to the diffusive acceleration occurs may be divided into two classes. Those which essentially employ only curl p* are single statements, but those which employ p* itself are really an infinity of statements, one for each admissible choice of p*. In a motion with steady vorticity, comparison of (124.1} with (111.9} yields grad(Q + U +! p2) = pxw + p* =I= 0. (124.3) Suppose p x w =!= 0, and at each point let t be a vector determined by the intersection of the Lamb plane with the plane normal to p*. Then by taking the dot product of (124.3) with t we obtain t ·grad(Q + U + tP2) = 0. (124.4} Now the field t is a tangent field for a certain congruence of Lamb curves, determined by the condition that they be normal to the field p*. From (124.4} follows then the curvilinear Bemoulli theorem3 : In a motion with steady vorticity, let the curves cL be the Lamb curves normal to the diffusive acceleration field. Then (124.5} that is, along any one of these curves at any one instant the expression on the left has a constant value. It is possible that some admissible diffusive acceleration 1 An example was given by CARSTOIU [1947. 3, Chap. VI, § 3]. 2 From a kinematical point of view the foregoing statements are trivial. Dynamically, however, a medium is defined by specifying the acceleration, and thus some one decomposition may have particular physical significance. 3 TRUESDELL [1954. 24, § 74]. Special cases involving the determination of particular Lamb curves in the motion of viscous fluids were given earlier by SBRANA [1931, 9], CASTOLDI [1948, 6], and TRUESDELL [1950, 33]. Sect. 125. The superficial Bernoulli theorem. 407 field p* be parallel to the Lamb vector w X p but unequal to it. In this case the result (124.5) holds for any Lamb curve. In the special case of steady motion, the curvilinear Bernoulli theorem (124.5) assumes a simpler form: Q+iP2 =f(cL). (124.6) Thus upon each of the curves cL there is a finite least.upper bound Q ( cL) for Q, attained ( if at all) at and only at a stagnation point: so that (124.6) becomes (124.7) (124.8) If further there is a finite greatest lower bound Q( cJ for Q on some particular Lamb curve er_, then on that same curve there ;"ust be a finite upper bound p for the speed: iP2=Q -Q. (124.9) An equivalent form for ( 124.6) then is t .(p2- />2) = Q - Q. (124.10) From these last results follow the principal applications of BERNOULLI's theorem in hydrodynamics. One of these, for example, consists in the observation that if Q = const upon one of the curves, then the speed also must be constant upon that curve. 125. The superficial Bemoulli theorem. We consider now a motion in which Lamb surfaces exist andin which also the vorticity is steady. By inserting (122.4) and (124.1) into (111.9) we then obtain If further then grad(Q + U + tp2) = p*- C grad B. (125.1) p* =EgradB, grad ( Q + U + t f• 2) = (E - C) grad B, (125.2) (125.3) whence it follows that Q + U + t p2 is constant upon each of the Lamb surfaces B =const. We may state this result as the superficial Bernoulli theorem1 : In a motion where Lamb surfaces • defined by (118.1)t. Fora material volume "Y we have from (80.3) $(K) =~(da· [W {p(Klp} + Wp(K+t)+ Wp(K-llldiv p]} 9' - [gradp·da]. wp). (131.4) Upon substituting for w from BELTRAMI's formula (101.7) 3, we find that by a happy circumstance two of the terms so introduced cancel the last two terms in (131.4), which becomes simply tm(KJ =~da· 'W {p(Klp} +~da· W* p(K+t). (131.5) 9' 9' By (App. 26.1h and the fact that divw* =0 follows tm(K) = ~da · 'W {p(K) P} + J {p(K) W*} dv. 9' "f'" Now by the transport theorem (81.3) we have m(K)= ~tl_ +~da· p {p(Klw}. 9' From (App. 26.1)1 we get the identities 0 ~ {;:.::[(~;~~~;)~'· l = ~ ~ {p (daxp)} +~da· w* p. 9' 9' 1 TRUESDELL [1951, 34]. 2 The special case "V = p was observed by TRUESDELL [ 1948, 34]. (131.6) (131.7) ( 131.8) 416 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 132. By combining this result and (131.5) with (131.7) we finally obtain1 ~~!K_l =~da. [w {pp}- p{pw}] + ~{p(daxp)}. (131.9) ~ ~ The vorticity moment theorem of Sect. 118 states conditions sufficient that tle• tm(I)• ... , 'lB shall vanish. From (131.9) we may infer weaker conditions sufficient that these moments remain constant in time: Subiect to tke boundary condition Wn=O, Pt=O wpn = o(p-K-2), or } Pt =o(p-x-a), (131.10) tke first K + 1 moments of vorticity are constant in time. In the case when all finite boundaries are stationary and the material adheres without slipping, we have Pt =0 and w" =0, so that~the only conditions of the theorem which remain to be considered are the order conditions at infinity. In the conditions as stated, it is clear from (131.6) and (131.8) that the true acceleration p may be replaced by the diffusive acceleration p*, in accord with (124.2). 132. The generalized convection vector. Complex-screw motions. The properties of motions in which vorticity is transported by convection only. are relatively simple and easy to picture. Kinematical analysis of more general motions usually seeks conditions such that some of the properties of circulation preserving motions can be carried over or adjusted to more complicated circumstances. The theorems of mean value in the previous section are examples. Another example is the following theorem of APPELL2, which we state without proof: Given any family oflines, furnisked witk continuously turning tangents, wkick in a given motion p are materiallines, tkere exists a continuously dilferentiable field v wkose circulation about any material circuit is constant and wkose vortex lines are tke given materiallines. We now construct apparatus for a more fruitful generalization. The underlying idea 3 consists in introducing a dass of vector fields proportional to the velocity: vc=-P, Vo where v0 is any non-vanishing substantially constant scalar: V0 = 0, V0 =l= 0. (132.1) (132.2) Any such field ''c weshall call a generaUzed convection vector, and v0 weshall call the defining parameter. We introduce also the curl of the convection vector: Wc= cnrlvc. (132-3) Then we have identically w = curl v0 Vc = v0 wc+ grad v0 X Vc, (132.4) 1 The special case K = 0 was derived by TRUESDELL [ 1948, 32], generalizing an earlier analysis of ]AFFE [1921, 3]. The general formula was given by TRUESDELL [1951, 28, § 12]. Cf. also HowARD [1957. 8, §V]. 2 The analysis of APPELL is in the material description [1899, 1, §§ 1-10]; a shorter spatial proof was given by TRUESDELL [1954, 24, § 89]. Another proof is given by DROBOT and RYBARSKI [1959, 4, § III.4]. 3 Due to HICKS, GUENTHER and WASSERMAN [1947, 6, lntrod.] and extended by TRUESDELL [1951, 32] [1952, 23,' § 8]. A different class of convection vectors was considered by HICKS [1949, 14]. The definition (132,1) is motivated by work of CRocco [1936, 3], although he considered only the trivial case when v0=const; also the introduction of m in (133.1) is suggested by certain special results of CRocco. Sect. 132. The generalized convection vector. Complex-screw motions. 417 and hence quite independently of the condition (132.2) we obtain 2 • v0 Vc · Wc = p · w. (132.5) By (App. 33-5) it follows that the generalized convection vector is complex-lamellar if and only if the motion is complex-lamellar. The researches of NEMENYI and PRIM1 have drawn attention to motions in which VcXWc = 0, (132.6) the possibility Wc = 0 not being excluded. The special case v0 = 1 is an irrotational or screw motion, and the class of motions satisfying the gen~ralization (132.6) will be called complex-screw motions. The remainder of this section presents results equivalent to those of NEMENYI and PRIM concerning this interesting type of motion. We first establish the connection between complex-screw motions and irrotational or ordinary screw motions. By (132.4) and (132.2)1 we obtain • 2 · . o log v0 pxw = v0VcXWc + p2 gradlogv0 + p-8 - 1 -. (132.7) From this identity it is obvious that a complex-screw motion in which the defining parameter is either uniform or steady is an irrotational or screw motion if and only if the defining parameter is both uniform and steady. This result implies broadly that the class of complex-screw motions is more extensive than that of irrotational and screw motions. We may calculate the angle 1p between the vortex line and the stream line in the following way 2• If Wc X Vc = 0, the two summands on the right-hand side of (132.4) are perpendicular, so that =v~w~+(gradlogv xp) , l = v~ w~ + (grad log v0) 2 :p- (p · gradlog v0) 2 , =V~ W~ + (grad log v0)2 p2- ( o!~~Vo r. Simultaneously (132.7) becomes . . 2 d 1 . o log Vo p X w = p gra og v0 + p - 0 - 1 -; in view of (132.2)1 , the square of this equation is (p X w) 2 = p2 [ p2 (grad log v0) 2 ~ ( 0 ~~~Von. The angle 1p is then obtained by combining (132.8) and (132.10): CSC'IjJ = \/XWW\ = (1 + • v5w~ ( o!o V r) p2(grad log v0)2 - f, whence follows ( 132.8) (132.9) (132.10) (132.11) (132.12) By comparing (132.9) with (122.4) we conclude that in a complex-screw motion whose defining Parameter is steady but not uniform, the surfaces upon which that parameter is constant are Lamb surfaces. 1 [1948, 17, § 5] [1949. 21 and 25] [1952, 16, Chap. V, Sect. C]. 2 We follow and correct the analysis of TRUESDELL [1954, 24, § 90]. Handbuch der Physik, Bd. 111/1. 27 418 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 133- By putting (132-9) into LAGRANGE's acceleration formula (99.14) we obtain •• op • d . ologvo d 1 • ) p = af - p2 gra log Vo - p -o-t - + gra 2p2' ovc 2 d 1 2 =v0Tt + v0 gra 2 vc. (132.13) Hence in a complex-screw motion whose convection vector is steady the acceleration is complex-lamellar, its normal surfaces being the surfaces of constant magnitude of the convection vector. Taking the curl of (132.13) yields 1 .. d ovc owc + 1 d 2 d 2 cur p = gra v0 X Te + v0 ----at 2 gra v0 X gra vc . (132.14) On the assumption that the local time derivatives are zero, we now consider in turn the situations that annul the remaining term on the right. First, if v0 is uniform, then from Wc X Vc = 0 it follows that w X p = 0. Second, we may have v~ = const. Third, if the surfaces v0 = const coincide with the surfaces vc = const, these in turn are surfaces of constant speed. In summary of these results we state that a complex-screw motion whose convection vector is steady is a circulation preserving motion if and only if a) it is an irrotational or screw motion, or b) its convection vector is of uniform magnitude, or c) at each fixed time, the defining Parameter is a function of the speed alone. When the motion itself is steady, we may apply the theorems stated just before and just after (132.13) to replace c) by c') the surfaces of constant speed are Lamb surfaces, and the acceleration is normal to them. Although, as indicated by these results, complex-screw motions usually fail to be circulation preserving, yet in some types of such motions the mechanism of diffusion operates in a fashion closely analogous to convection, as will be shown now. 133. Generalized convection theorems. Consider first a steady complexscrew motion, and let m be any solution of div (mvc) =0. (133-1) Noting that vector sheets of Vc are stream surfaces, we apply the theorem of GROMEKA and BELTRAMI derived in Sect. App. 34 and obtain the convection theof'em jOf' steady comple:x:-scf'ew motions: The surfaces wc = const (133.2) mvc are stream surfaces; in particular, (133.2) holds on each stream line. The analogy to convection is immediate, and the result generalizes the second GromekaBeltrami theorem (112.4). The corresponding generalized convection theorem for complex-lamellar motions lies deeper1• For any function F we readily obtain the identity curl [F (~ Vo ovc Ot + Wc X vc)] = grad _!_ Vo X -~~c ot - wc Vo 8F Ot + ~ Vo oF ot wc + l +vc· gradFwc -Fwc · gradvc+Fwcdivvc- vcdiv(Fwc). (133-3) 1 TRUESDELL [1951, 32]. Sect. 133. Generalized convection theorems. 419 Now let F be chosen as a solution of grad !'___ X ovc = "Wc ~!"- v0 ot v0 ot ' (133.4) and let Mlfv. be any permissible density for Vc, i.e., let M be any solution of or equivalently 1-'- . -logM =- dlVVc, Vo : 0 ~ + div (M Vc) = 0. Our identity (133.3) then reduces to ~ curl [F ( :0 o:ec + WcXvc)] I _ 1 d (Fwc) Fwc d vc d' (F ) - v;;- {[[ ---xr- - ---xr- · gra Vc - M 1v Wc , (133-5) (133.6) ( 13 3 .7) a generalization of BELTRAMI's diffusion equation (101.7}, to which it reduces when v0 =1, F=1, M =J-1. We now form the scalar product of (133-7) with FwcfM, thus obtaining a generalization of (101.8}: _1_ 2v0 !___ dt (Fwc)2 M M = Fwc. grad Vc. Fwc M + _!__Vc. M2 Wc Wc. gradF + I Ftv [ ( 1 ov )] (133.8) + M 2c · curl F v;;- 8tc;_ + WcXVc . By (132.5), the assumption that the motion is complex-lamellar, now employed for the first time, annuls the second term on the right in (133.8). Let us assume further that the third term is zero: Wc · curl [F ( :0 ~c + WcXvc)] = 0. (133.9) We then obtain the equation _1_ !___ ( F wc )2 = F wc . grad Vc . F wc . 2v0 dt M M M (133.10) We now impose the further requirement that the vortex lines be steady. Then it is possible to choose stationary co-ordinates at a single pointl in such a way that the x1 co-ordinate curve is tangent to the vortex line at the point in question: ds2 = h2 (dx1)2 + g22 (dx2)2 + gaa (dx3)2, ) oh (w)2 = w1 w1 , .i1 = 0, Tt = 0. From (133.10) we have then 1 d I F Wc - 1 - ovl:_ { 1 } k ·;;;;- dt og ---xr- - Vc, 1 - o xt + 1 k Vc' (133.11) (133.12) 1 The neighboring xl co-ordinate curves need not be vortex lines. Thus (133.11} does not impose any restriction on the class of motions considered. 27* 420 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 133. whence follows, since v~= 0, ~ lo F wc = { 1 } i2 + { 1 } .xa dt g M 1 2 1 3 ' - ologk "2 + ologk "3 - ~x ----axsx, = 0 log k i1 + 0 log k x2 + 0 log k _i3 + 0 log k ox1 ox2 ox3 ot , (133-13) Hence d (Fwc) dt kM = O. (133-14) The equation just derived expresses a simple conservation law. Going back and collecting the assumptions we have made to derive it, we obtain the generalized vorticity convection theorem for comple;r-lamellar motions: Given a complex-lamellar motion with steady vortex lines, let the x1 Co-ordinate curve at the point in question be tangent to the vortex line and let dx1jd s = h-1 ; let v0 be any substantially constant function, let Vc and Wc be defined by let F be any solution of Vc = k, Wc- curl vc; Vo F ovc wc oF grad-X--=--· Vo Ot v0 ot ' and let M be any solution of __!___ ~ + div (M vc) = o; v0 ut (133-1), (133-3) (133.4) (133.6) if further, it is possible to find among the functions v0 and F satisfying these conditions a pair such that also (133-9) then Fwc kM = const (133-15) for each particle. Of the conditions of this theorem only the requirement of complex-lamellar motion with steady vortex lines and the one Eq. (133-9) are truly restrictive, the others being rather in the nature of definitions of the dass of admissible functions v0 , F, and M. For the simplest special case, put v0 = 1, F = 1. Then Vc = p, and we may satisfy the condition (133.6) by the choice M = J-1• By (107.2) the condition (133-9) reduces to W ·W* =0. (133-16) Hence it follows from the theorem above that in a complex-lamellar motion with steady vortex lines, in order that ]kw = const (133-17) for each particle it is necessary and sufficient that the motion be circulation preserving or that the diffusion Vßcfßl': be normal to the vorticity. In particular the theorem applies both to plane motions and to· rotationally symmetric motions. For the Sect. 134. Characterizations of convection, I. CAUCHY's formula. 421 former, we take h = 1 and obtain D'Alembert's vorticity theorem 1 : In order that a plane motion be circulation preserving it is necessary and sutficient that ]w = const ( 13 3.18) for each f!a1'ticle. For rotationally symmetric motion, we have h =r, where r is the distance from the axis of symmetry. Hence follows Svanberg's vorticity theorem 2 : In order that a rotationally symmetric motion be circulation preserving, it is necessary and sufficient that _lrJJ_ = const r (133.19) for each particle. It was these theorems, stating that J w or J wjr is carried by the motion as if it were some native property of the particles, that originally motivated the name "convection" for the process of transfer of vorticity in a circulation preserving motion. Special cases of ( 133.15) appropriate to certain gas motions which are not circulation preserving have been deve!Dped by TRUESDELL 3. 134. Characterizations of convection, I. CAUcHY's formula. The most useful characterization of convection, the celebrated Cauchy vorticity jormula4, follows at once from (101.9) or (101.11): (134.1) (134.1) 1 is the tensor law of transformation for the covariant components of spin, when the motion is regarded as change of co-ordinates. That is, in order for the motion to be circulation preserving it is necessary and sufficient that vorticity be convected. The main consequences of (134.1) have already been given in generalized form in Sect. 129. Here we add only the corollary that w =0 if and only if W = 0. This is the famous velocity potential theorem of LAGRANGE and CAUCHY 5 : In a circulation preserving motion, a particle once in irrotational motion is always in irrotational motion. Thus a portion of material for which a velocity potential exists moves about and carries this property with it, but the part of space which it originally occupied may in the course of time come to be occupied by material which did not originally possess the property, and which therefore cannot have acquired it. This theorem may be proved in several other ways. We leave it to the reader to see how proofs may be constructed from the results given in Sect. 108, Sect. 130, and the two sections to follow. 1 D'ALEMBERT [1761, 1, §XIII] treated only steady isochoric motion. Cf. also STOKES [1842, 4, Eq. (10)], HELMHOLTZ [1858, 1, § 5], LAMB [1878, 5]. Sufficiency is not proved by these authors. 2 [1841, 5, § 4]. Cf. STOKES [1842, 4, Eq. (21)], HELMHOLTZ [1858, 1, § 6], LAMB [ 1878, 5]. Sufficiency is not proved by these authors. 3 [1952, 23, §§ 9-10 [1954, 24, § 92]. This work was motivated by earlier results of CROCCO [1936, 3] and PRIM [1948, 21] [1952, 16, Eqs. (65), (194)]. 4 [1827, 5, 1e partie, Sect. 1, ~ 4]. CAUCHY's analysis is elaborate and does not make evident that (134.1) 2 is sufficient as well as necessary. Cf. CRUDELI [1918, 1, §I]. That the form ( 134.1)1 is a necessary and sufficient condition for circulation preserving motion in ndimensional spaces was observed by DE DoNDER [1912, 2, § 5]. 5 The statement and proof of LAGRANGE [1783, 1, §§ 17 -19] [1788, 1, Part II, § 11, ~ 16 - 17] are faulty. They were corrected by CAUCHY, loc. cit. An extensive Iiterature presents the proofs, misunderstandings, and controversies that have arisen in this connection: PoissoN [1831, 2, ~ 73] [1833, 4, § 654], PowER [1842, 3], STOKES [1845, 4, §§ 10-13] [1846, 1, §I] [1848, 3], ST. VENANT [1869, 6], BRESSE [1880, 3 and 4], BouSSINESQ [1880, 1 and 2], PolNeARE [1893. 6, § 152], HADAMARD [1901, 8, ~ 4, footnote]. Critical reviews are given by DuHEM [1901, 7, Part 5] and TRUESDELL [1954, 24, §§ 104-107]. 422 C. TRUESDELL and R. TOUPIN: The C!assica! Field Theories. Sect. 135, 135. Characterizatioils of convection, II. The transformation of WEBER. CAUCHY's formula (134.1) is a first integral of the kinematical equations. We now seek a corresponding statement in terms of the velocity field itself. For any motion we have (135.1) From (109.1) follows (135.2) By combining (135.1) and (135.2) we get xk i = (1. i 2 - V*) . ,et k 2 ,Ct (135.3) Iotetration along the path of a particle then yields t I (1 • 2 V*) d k • • 2x - ·"' t=x,C2 = t (gradH) 2 + F grad G. grad H + iF2 (grad G)2. (136.9) Putting these two results into (136.7) yields 3 aH 1 1 V*+ Tt + 2 (grad H) 2 - 2 P (grad G)2 = 0. (136.10) A part of the apparent simplicity of CLEBSCH's transformation is illusory, since in steady motion the functions H and G generally cannot be steady. In fact, comparison of (136.6) with LAGRANGE's acceleration formula (99.14) in the case of steady motion yields wxp . [ oH oG l = grad 8t +F Tt . (136.11) 1 [1859. 1, § 3]. Other proofs were given by HANKEL [1861, 1, §VI], HILL [1881, 3, § III and pp. 168-210], BASSET [1888, 1, §§ 33-35]. LAMB [1906, 4, § 166], APPELL [1921, 1, § 799], and TRUESDELL [1954. 24, § 101]. The last of these is the one reproduced here. 2 CLEBSCH [1857. 1, § 5] derived an equivalent formula from the equations for perfect fluids. See also BELTRAMI [1871, 1, § 13], HILL [1881, 3, § IIIJ. 3 This formula, suggested by similar but generally false equations of CHALLIS [1842, 1] and CRAIG [1881, 2, p. 12), was derived by TRUESDELL [1954, 24, § 101]. 426 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 136. Hence H and G can be steady if and only if the motion is an irrotational or screw motion, in which case CLEBSCH's transformation (136.7) reduces to the same form1 as does the spatial Bernoulli theorem (120.1). Finally we consider the more general case in which the functions F and G arenot subjected to (136.1). By taking the curl of DuHEM's acceleration formula (99.24) and comparing the result with the d'Alembert-Euler condition (108.1), we then obtain grad F X grad G = grad G X grad F (136.12) as a necessary and sufficient condition to be satisfied by the Monge potentials of the velocity in order that the motion be circulation preserving. That is, the vectors gradG, gradF, gradG, and gradF must alllie in the plane perpendicular to the vorticity w, and the area inclosed by the parallelogram whose edges are gradG and gradF must equal the area enclosed by that whose edges are gradF and gradG. Without direct use of the foregoing, we introduce the abbreviation W=V*+__!_p"2 + 8H +F~ (136.13) 2 at ot ' form grad W, and simplify the result with the aid of (99.24) and (109.1) so as to obtain2 Hence grad W = G gradF- F grad G. a~rF,~) =grad w. gradFxgradG =O. x,y,z (136.14) (136.15) Since F and Gare necessarily independent, from this result follows the equation of DuHEM3 : W = W(F, G, t). (136.16) Hence aw aw grad W = oF gradF+aGgradG. (136.17) Comparison with (136.14) now yields the Hamiltonian equations of STUART and LAMB4 : aw · ifF = G, aw · ac;=-F. (136.18) Given the acceleration potential V* of a circulation preserving motion, that F, G, W, H satisfy (136.12), (136.13), (136.16), and (136.18) is plainly not sufficient for F, G, H to be Monge potentials of any given velocity field, since His indeterminate up to an arbitrary steady function. The contribution of any Hin (99.19), however, is only an irrotational motion. Thus a solution F, G, W, H always yields a circulation preserving motion. CLEBSCH's transformation (136.7) results from use of the particular solution W = 0. 1 This same result follows also from the fact that the surfaces G = const are material vortex surfaces, and hence in a steady motion in order to be themselves steady they must be Lamb surfaces. 2 DUHEM [1901, 4, § 2]. 3 [1901, 4, § 3]. ' The result was first published by LAMB [ 1906, 4, § 166]; later [ 1924, 9, § 167] he attributeditto STUART. DuHEM [1901, 4, §§ 4-7] discussed the general theory of the integration of the system (136.1), (136.7) by analogy to JACOBI's method in analytical dynamics. MASUDA [1953. 17] uses the Hamilton-Jacobi theory to obtain some of the theorems given above. Suchgeneral methods, failing to take account of the firstintegral (134.1) or its various equivalents, lead to an elaborate and awkward treatment. Sects. 136A, 137. The Euler-Clebsch reduction of steady circulation preserving motion. 427 The reader may verify that the special potentials (1)5.15) and (1)5.17) satisfy (1)6.12) and (1)6.18). Generalizations of the transformations of CLEBSCH and WEBER to arbitrary motions areeasy to work out but not illuminating1 . 136A. Appendix. Variational form of CLEBSCH'S transformation. CLEBSCH2 observed that for the isochoric case, use of the apparatus of Sect. 136leads to a simple variational principle. The three functions H, F, and G are to be varied so as to yield, through (99.19). a velocity field :i; which is isochoric and circulation preserving. The formula (136.7) is taken as a definition of a function V*, not yet known to be an acceleration potential, in terms of F, G,H. Then -!5V*=i"!5" + 8!5H +F 8!5G + oG !5F . 8!5H 8!5G oG x,. ae at at · l = [!5H,k+ G,k !5F+F!5G,k]xk+ ---a-t+F---a-t+ Tt !5F, (1J6A.1) a ,. · · =[(!5H+F!5G)ik] k+ßt (!5H+F!5G)- (t5H+Ft5G) i ,.-Ft5G+Gt5F. Integrating over a fixed volume • yields -t5fdtfV*dv=f lg lg [ ~(t5H+Ft5G)ikdak ] dt+J(!5H+Ft5G) ,~. dv+ ) e, ,. e, ' ,. e, (136A.2) lz [ • • ] + f f{Gt5F-Ft5G- (t5H +Ft5G) i~,.}dv dt. e, ,. We now impose the condition that t5G and !5H vanish throughout • at t=t1 and at t=t, as well as vanishing on 4 at all times. Then from ( 136A.2) it follows that lg t5 J dt J V* dv = o (1J6A.J) li " is equivalent to the conditions i\=0, F=O, G=O in •· The first, by (77.1), is the condition of isochoric motion; the second and third show that (136.12) is satisfied and hence that the motion is circulation preserving. By CLEBSCH's transformation (136.7), the function V* satisfying (136.7) is in fact an acceleration potential. 137. The Euler-Clebsch reduction of steady circulation preserving motion. After establishing a representation of the form (App. )2.5) for a solenoidal field, EULER 3 applied it to the velocity field of a steady motion. Writing p = J C (A, B) grad A x grad B, (1)7.1) we present EuLER's determination of conditions on A, B, C in order that the motion be circulation preserving. Firstwehave the identity wxp. da:= w · (] C gradA xgrad B) xda:, ) =] Cw · [(gradA ·da:) grad B- (grad B ·da:) gradAJ, (1)7.2) =]Cw· [dAgradB-dBgradA). Using (125.5) as the condition for circulation preserving motion, in the steady case we write it as - d(V* + jp2) =WXp ·da:. (1)7.3) 1 APPELL (1917, 3, § 2] [1921, 1, § 815], CARSTOIU (1947, 2, § 2], TRUESDELL (1954, 24, § 103]. 2 [1859. 1, § 3]. Cf. also BASSET [1888, 1, § 34]. 3 [1757. 3, §§54-56]. His analysis, which was rediscovered by BASSET [1888, 1, §§ 39 to 40] and v. MISES [1909, 8, § 4.3], was put into modern terms by TRUESDELL [1954, 24, § 98]. 428 C. TRUESDELL and R. TouPJN: The Classical Field Theories. Sect. 137. Comparison of (137.2) and (137.3) yields EuLER's formulae o(V* + !.p2) dB } oA 2 = - ] C w · grad B = -] C w (iiJ) , o(V*+tfi2) oB =JC 'W . gra dA =]C w ~ dw ' (137.4) whence follows the condition of integrability o o M (] Cw · gradA) + 8if (] Cw · grad B) = o, (137.5) where w is to be thought of as expressed by w = curl (] C gradA x grad B). We may use this apparatus to determine conditions that the validity of (130.5) for two families of surfaces, 'V = A = const and B = const, be sufficient to ensure that a given velocity field p be circulation preserving. We assume C determined by (137.1), and we assume that A, B, and C satisfy (137.5). Then there exists a function F(A, B) suchthat (137.4) is satisfied with F replacing V*+ tp2, and hence by the identity (137.3) follows - gradF=wxp. Hence by (99.14) (137.6) (137.7) Therefore the motion is circulation preserving with F-tp2 as an acceleration potential. In summary of these results, we have EuLER's characterization of steady convection: In a steady motion, let it be possible to find two independent families of stream sheets A = const, B=const, suchthat (130.5) hold for each; with C determined by (137.1), i/ (137.5) holds also then it follows that the motion is circulation preserving. Conversely, in a steady circulation preserving motion (130.5) and (137.5) hold for any two substantially constant functions A, B. Taking C = 1, as is always possible, CLEBSCH found an alternative form for ( 13 7. 5) in the case when the co-ordinates are reetangular Cartesian. Since in any steady motion i:J1 is a function of the six components A,k and B,k• we have so that [ o~,k ( ~ i 2f]2)J.k = Bkmn B,n im,k = - w · grad B,l [ o!,k ( ~ i 2 JP) ],k = w. gradA. Hence (137.5) may be written ~[1-0 (_!_i 2/P)] =~[1-0 (_!_z 2/P)] . oB oA,k 2 ,k oA oB,k 2 ,k (137.8) (137-9) (137.10) In the isochoric case, the problern of determining all steady circulation preserving mQtions is thus reduced in principle to the solution of a single differential equation for the speed. These results may be used to derive a variational principle of CLEBSCHl for the isochoric case. Again taking C = 1, we vary A and B. Since, as established above, V*+ -!z2 is a function of A and B only, we have ~ J(v*+ ~ )dv=J[o~ (V*+~ )~A+ o~ (V*+~ z2)~B]dv; (137.11) V V 1 [1857. 1, §§ 3-4]. Our derivation follows BASSET [1888, 1, § 41]. Sect. 138. APPELL's theorem on the stretching of vortex lines. 429 since z is a function of A ,k and E,k only, we have (137.12) ~ ( " 1 "2 " 1 "2 ) = ~(JA +~(JE d - oA k aB k ak d ' ' - J [(ati2) M + (atz2) (JE] dv. aA,k ,k aB,k ,k " We restriet the variations so that M =0 and (JE =0 on d. Adding (137.11) to (137.12) then yields (J j (V*+ z2) dv = j {[ ~ (v* + ~ z2)- (~~.z::tJ (JA+ l + [a~ (V*+~ z2)- (~~~)JbE}dv. (137.13) By the identity (137.9), it follows that if there exists a function V* such that (J J (V*+ z2) dv = 0 (137-14) " when A and E are varied as stated, z being given by ( 13 7.1) with J C = 1, then the differential system (137.4) is satisfied. Hence (137.14) is a necessary and sufficient condition that a given solenoidal field be the velocity field of a steady isochoric circulation preserving motion. 138. APPELL's theorem on the Stretching of vortex lines. In any rotational motion, for any material element of arc d:n we have !-__ dt (!__!___) jw - - _1_ 2 [~jwdx ~ d~2 - ___!__!___ (]w)3 (]~)2] ' l =-1-[dkmdxkdxm _ dx(Wkwk+dkmwkwml jw dx w2 ' (138.1) where we have used (82.3) and (101.8). In any motion such that the vortex lines are material, we may choose d:n as tangent to the vorticity, so that dxwk = dxkw. (138.2) In this case, (138.1) becomes :t (5:) = -f;k dx. (138-3) The vortex lines have been assumed material; by (107.3) it follows that wt wk =0 if and only if w* =0. This yields Appell's theorem1 : In a motionsuchthat the vortex lines are material, in order that dx Jw = const (138.4) 1 Following indications by APPELL [1921, 1, § 760], this theoremwas stated and proved by TRUESDELL [1954, 24, § 97], who showed also [ibid., § 96] that it is equivalent to a theorem of LAMB [1885, 5]. 430 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 139· for each material element dx in the direction of the vorticity, it is necessary and sufficient that the motion be circulation preserving. If <'C is a finite portion of a vortex line, then (138.4) is equivalent to f dx ]w = const. (138.5) 'C Like the circulation itself, the integral (138.5) is an integral invariant. While circulation is calculated for any closed material circuit, the invariant (138.5) is calculated for any part of a vortex line. The theorem implies that if ] w increases during the course of the motion, the vortex lines are stretched. As a corollary, in order that J w = const for each particle in a circulation preserving motion, it is necessary and sufficient that the length of each vortex line remain constant1• This result is an interesting supplement to D'ALEMBERT's vorticity theorem (133.18). Let n be a unit vector. Then dx dx dxn = dx · n =] w · n 1 w = ] w .. 1 w , (138.6) where again dx is a material element parallel to the vorticity. Then (138.4) is equivalent to the statement that for a given particle dxn cx. ] w... In particular, we may choose n as the unit normal to a material surface and obtain v. Mises' theorem2 : In a circulation preserving motion, let a particle X be infinitely near to a material surface !/; then the distance of X from Y is proportional to the component of ] w normal to !/. Since, as will be shown in Sect. 184, a bounding surface is always material, v. MISES was able to use this theorem to discuss the behavior of the vorticity near a wall. f) Furtherspecial motions. 139. Summary of special motions previously defined. In the foregoing sections of this subchapter numerous special types of motion have been defined. These will be listed, along with their formally simplest defining equations and references to our earlier discussion of them. First, there are classes defined by a geometrical condition: 1. Lineal motions (Sect. 68): x=x(x,t), y=O, z=O, f=f(x,t). 2. Pseudo-lineal motions of the first kind (Sect. 68): x=x(x,y,z,t), y=O, z=O, i=f(x,y,z,t). 3. Pseudo-lineal motions of the second kind (Sect. 68) : x = x (x, t), y = y (x, t), z = z (x, t), j = j (x, t). 4. Plane motions (Sect. 68): x=x(x,y,t), y=y(x,y,t), z=O, i=f(x,y,t). 5. Pseudo-plane motions of the first kind (Sect. 68): x =x(x, y, z, t), y =y(x, y, z, t), z = o, =f(x, y, z, t). 1 Cf. the more special result of CARSTOIU [1946, 4]. 2 [1909, 8, § 2.5]. (139.1) (139.2) (139-3) (139.4) (139-5) Sect. 139. Summary of special motions previously defined. 431 6. Pseudo-plane motions of the second kind (Sect. 68) : x =x(x, y, t), y =y(x, y, t), z =z(x, y, t). f =f(x, y, t). (139.6) 7. Rotationally symmetric motions (Sect. 68): r = r (r, z, t)' z = z (r, z, t)' (j = 0' f = f (r, z, t) . (139.7) Second, there kinematical classes. First among these are those defined by a condition of steadiness: 8. Steady motion (Sect. 67): ~ =:i(x). (139.8) 9. Steady motion with steady density (Sect. 77): :i=~(x), Bf=f(x). B=O. 10. Motion with steady stream lines (Sect. 75): . op PXae=O. Then there are motions defined by conditions on the stretching: 11. Rigid motion (Sect. 84) : d=O. 12. Isochoric motion (Sect. 77): Id=O. 13. Dilatation (Sect. 83): d1 = d2 = d3 = d. 14. Shearing (Sect. 89): d1 =-d3 , d2 =0. Finally, there are motions defined by conditions on the spin: 15. I rrotational motion (Sect. 88) : W=O. 16. Complex-lamellar motion (Sect. 110): p·W=O. 17. Screw motion (Sect. 112): wxp=O, w=f=O. 18. Complex-screw motion (Sect. 132): Vc=P/V0 , V0 =0, } Wc = curl Vc, WcXVc = 0. 19. Motion with steady vorticity (Sect. 111) : ow ae=o. (139.9) (139.10) (139.11) (139.12) (139.13) (139.14) (139-15) (139-16) (139-17) (139.18) (139.19) 432 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 140, 141. 20. Circulation preserving motion (Sect. 108): w*=O. (139.20) Also, Sect. 89 concerns two very particular motions: rectilinear shearing, which includes simple shearing as a special case and itself a special pseudo-lineal motion of the first kind, and the rectilinear vortex, which includes the irrotational vortex as a special case. The next three sections describe additional special classes of motions. 140. D' Alembert motions. Motions such that p (~. t) = T(t) v (~) (140.1) were introduced by D'ALEMBERT1• These motions have steady stream lines, they are isochoric if and only if v is solenoidal, and their kinematical vorticity numbers "WK are steady. Since p = T'v + T2v · gradv, (140.2) we have w* = curl p = T' curl v + T2 curl (curl v xv). (140.3) If T'JT2=f=const, then (139.20) cannot be satisfied unless w =0. Thus follows a theorem of D'ALEMBERT and LAGRANGE 2 : Apart from the case when A, B = const, AB=f=O, (140.4) a D' Alembert motion is circulation preserving if and only if it is irrotational. In the case when (140.4) holds, the condition for circulation preserving motion is that A curlv=curl(curlvxv); if A =0, this is merely the condition that v itself be the velocity field of a circulation preserving motion. Again, if v is the velocity field of a rotational circulation preserving motion, by {140.3) the D'Alembert motion (140.1) is circulation preserving if and only if T=const. From (140.1) and {100.3), SzEBEHELY's measure of unsteadiness assumes the form ~= IT'i. V T2 lv·gradvl (140.5) If v=f=O and lv·gradvl=f=O, the condition (140.4) emerges as necessary and sufficient that ~ be steady. The theorem of D'ALEMBERT and LAGRANGE may thus be expressed in terms of the steadiness of ~- 141. Accelerationless motions. If i =0, (141.1) then every particle X travels in a straight line at a uniform velocity iJ (X). Despite the apparent simplicity of this fact, accelerationless motions have never been characterized in a form really manageable in the spatial description. lndeed, a steady accelerationless motion, typified by the rectilinear shearing motions (Sect. 89), is rather trivial, but when the stream lines are not steady, they are not straight, and the straight path lines may cross each other in a bewildering variety of ways. A relatively simple example was given in Sect. 71. 1 [1752, 1, § 148]. 2 The faulty statement and proof by D'ALEMBERT [1761, 1, § 10] were corrected by LAGRANGE [1762, 3, §52]. Sect. 141. Accelerationless motions. 433 Following CALDONAzzol, we derive a spatial functional equation for these motions. In the common frame the material description of an accelerationless motion is z =z(Z) t +Z. Since z =f(Z), we may put (141.2) into the form z =f(z- zt) 0 Conversely, if (141.3) holds, put y=z-zt; then (141.2) (141.3) zk = l~mYm = l~m(zm- im--zmt), (141.4) or (o~+tf~m) zm=O. Since o~+tf~m is Singular for at most three values oft, it follows that z = 0. Therefore, the functional equation (141.3) is necessary and sufficient for accelerationless motion. Returning to the material description (141.2), we now calculate the ratio of the elements of volume at time t and 0, by (20.9) obtaining 2 dvfdV =] = det zk" = det (bkiX + t zk,IX), (141.5) =1 +IAt+IIAt2 +IIIAt3 , where we have used theexpansion of the seculardeterminant, and whereA -llzk "II· If We ChOOSe the present instant aS the initial instant, Zk IX beCOmeS the Spahal velocity gradient matrix, and for the invariants defined'by (78.5) we get the in terpreta tions lO = _1_ • _1_ d2 (dv) I B = _1_ • _1_ da (dv) I 2 dV dt2 1=0' 6 dV dta I=O' ( 141.6) For accelerationless motions, these results extend (76.9) 2 • Given an accelerationless motion with velocity field z (Z), in order that it be isochoric we derive from (141.5) 3 the necessary and sufficient conditions IA = 0, IIA = 0, lilA= 0, (141.7) which were obtained by MERLIN 3, who noted that (141.7) is an assertion that the hodograph is degenerate. If the hodograph is a single point, the motion is a uniform translation. MERLIN has characterized the curves and surfaces which may serve as hodographs and in terms of them has shown how to construct all isochoric accelerationless motions. Other properties of accelerationless motions have been noted in Sect. 100 and Sect. 102. l [1947, 1, § 3]. 2 The argument is due in principle to EuLER [1761, 2, §§ 27-35]; cf. LIPSCHITZ [1875, 3], MERLIN [ 1938, 8, § 2). Generalizing this idea, KosTIUK [ 1936, 6] has considered the analytic motion oo Ck) (Z t) z = Z + .L:-3--'- yk+t k=O(k+1)! and has calculated explicitly the coefficients A k in the series 00 jz/ZI = 1;Ak Tk, k=O the first ones being A0 = 1, A1 = ld, and has shown by calculation at length that (k + 1) A.<+ 1 = Äk + A1 Ak. 3 [1938, 8, § 2]. Cf. also [1937, 6]. Handbuch der Physik, Bd. III/t. 28 434 Co TRUESDELL and Ro TouPIN: The Classical Field Theorieso Secto 142o 142. Homogeneous motions. A motion is homogeneaus if in the common frame it assumes the form z = a (t) 0 z + b (t) 0 (142o1) The analogy to homogeneaus strain (Secto 42) is immediate, but most of the results which can be read off from this analogy are of little interesto Much of the work on geometrical kinematics listed in Special Bibliography K concerns the homogeneaus case; cfo also the papers cited in footnote 1, po 331. Since any continuous motion may be approximated in the neighborhood of any one point by an appropriate homogeneaus motion, this special dass is of central importanceo However, we have preferred to develop general properties of general motions, and we leave to the reader their applications to this simplest of caseso In a homogeneaus motion, from (14201) we get (142o2) for the tensors of stretching and spin. It is only an application of the fundamental theorem of Secto 90 to assert that any homogeneaus motion may be regarded, at any given instant, as a rigid rotation superposed upon an irrotational homogeneous motion of stretching along three mutually perpendicular axeso In a suitable co-ordinate system, the motion of stretching assumes the form dl 0 0 llill= . d2 0 ·llzll, . d3 1 0 0 = ~ (dl + d2 + d3) 1 0 ollzll+ . 1 1 0 0 (142o3) +t (dl +d2- 2d3) 0 0 ·llzll+ 0 -1 1 0 0 +t (dl+d3- 2d2) -1 0 ollzll· 0 Now simple shearing is a special case of homogeneaus motiono From (8906) we easily conclude that a plane homogeneaus motion is isochoric if and only if it is a simple shearing superposed upon a rigid rotation about an axis normal to the plane of shearingo Therefore, apart from a rigid rotation, the motions represented by the second and third matrices in (14203) are simple shearings in the planes normal to the second and third principal axes of stretching, respectively. The first matrix represents a dilatation (139.13). One of several ways of expressing the foregoing decomposition, due to STOKES1, is: Any homogeneaus motion may be regarded as composed of a rigid rotation, a uniform dilatation, and simple shearings in two of the planes normal to the principal axes of stretching. Cf. also Sect. 46, No. 2. Accelerationless homogeneaus motions have been characterized by TRuEsDELL2. By using (98.1h to calculate z from (142.1), we get the conditions 1 [1845, 4, § 2]o 2 [1955, 27, § 2]. (142.4) Sect. 142. Homogeneaus motions. 435 In the steady case these conditions reduce to a 2 =0, a·b = 0. (142.5) In the unsteady case, the general solutions a and b may be obtained from the algebraic system (1+At)·a=A, (1+At)·b=B, (142.6) where A and B are the initial values of a and b. If A has positive (negative) proper numbers, write the largest (smallest) as -1/L ( -1/t+); otherwise, write L=-oo (t+=+oo). Then a unique solution of (142.6) exists and is a differentiable function of t in the interval t_ < t < t+. Moreover, if m is the largest among the absolute values of all the proper numbers of A, then a and b are analytic functions oft when I tj < 1/m. Thus in general these motions do not remain continuous indefinitely but develop singularities after a finite time, determined by the initial velocities and velocity gradients. A simple example is given by -(J 0 0 a = k -a 0 , b = 0, (142.7) -1 where in order to satisfy (142.6) we must have k (t) ko (t) - __!_±_kL_ = 1 + k0 t ' a -Go 1- k0a0 t · (142.8) In this motion, a reetangular block with edges parallel to the co-ordinate planes is extended along the z-direction at the rate k (t), contracted transversely in the ratio a (t). The stream lines in the y = 0 plane are the curves z x1fa = const. (142.9) The ratio of the volume v at time t to the initial volume V is (142.10) whence appears that in the cases when the motion develops a singularity, the volume is reduced to zero. This motion is illustrated in Fig. 19. For an accelerationless homogeneous motion both (141.2) and (142.1) hold. Hence z=(a·z+b)t+Z. (142.11) We multiply each side by 1+At and use (142.6) to simplify the result, so obtaining (1 +At) ·Z=A·zt+Bt+ (1 +At) ·Z. Hence Z=(At+1)·Z+Bt. (142.12) (142.13) This gives z=A · Z + B, so that z is linear in Z as weil as in z. There is no essential loss in generality if we set B= 0, since its presence indicates only a superposed uniform translation. If B= 0, the point Z = 0 remains fixed, and if we consider the parallelepiped whose vertices are the points 0, Z1 , Z2 , Z3 at time t = 0, the material volume so defined is the parallelepiped whose vertices at timet are 0, z 1 , z 2 , z 3 • We may now interpret (141.5) as giving the ratio of the volumes of these finite parallelepipeds 1• 1 TRUESDELL [1953, 33]. 28* a) b) c) d) 436 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 142. In a homogeneaus motion, the principal axes of stretching and the principal stretchings themselves areuniform in space at each time. lf we relinquish the latter of these requirements, we seek motions such that in a certain reetangular Cartesian frame we have (142.14) These three conditions are necessary and sufficient for the existence of functions ~ (x, y, z, t), F2 (x, y, z, t), F".! (x, y, z, t) such that r r o2 X=i)X~2 y=ey• z=Bz;2 (142.15) . ~ . ~ . ~ ) oxoz (~+F".!)=O, oyox (.Fz+~)=O, ozoy (Fa+Fz)=O. --------------, I ---------------, I I I I z z z Fig. 19 a-d .. An accelerationless deformation of a reetangular block. a) The undeformed block. The arrows are the velocity vectors of the particles; the light lines are their paths; the dashed straight lines are loci of particles whose velocities are parallel; the dashed ellipses are loci of particles having the same speed. b) Stream lines when k,t = 0. c) Stream lines when k0 t =I. d) Stream lines when k0 t = 2. The general solution of the latter three equations when put into the former three yields • 0 x =fiX [G3 (x, y)- G2 (x,z)], y = :y [G1 (y, z)- G3 (x, y)]. (142.16) . 0 z =Tz [G2 (x,z)- G1 (y,z)], where, as henceforth, the functions occurring may depend also on t. The principal stretchings are (142.17) Sect. 143. Motion relative to a rigid rotating frame. 437 For the isochoric subclass, the additional condition imposed is ()2~ + ()2~ + ()2~ - -- -- ·---- 0 ox2 8y2 8z2 • (142.18) Consider first the possibility Illd=Ü, equivalent to, say, 82F3f8z 2=0. From (142.15) and (142.18} it follows quickly that Hence (142.15} 1, 2 ,3 must reduce to ()3~ --··-··- = o. 8x 8z2 i=f(x+y)+g(x-y)+Cxz, ) y = - f(x + y) + g(x- y)- C yz, z =- tc(x2- y2), (142.19) ( 142.20) where the functions f and g and the constant C are arbitrary. This result was obtained by PRAGERl, who determined also the more restricted motions in which llld =!= 0. While a dilatation may be characterized by specializing the preceding results, it is easier 2 to start by substituting (139.13) into (84.3). whence it is immediate that d is a linear function, and therefore the velocity is given to within a rigid motion by z=(a·z+k)z-iz2 a, (142.21) where the vector a and the scalar k are functions of time only. g) Relative motion. 143. Motion relative to a rigid rotating frame. Let i, j, k and i', j', k' be unit vectors along the co-ordinate directions of two reetangular Cartesian frames, and let b be the position vector of the origin of the primed frame relative to the unprimed frame. Then the position vectors p and p' of a given point relative to the two origins are related by p = b + p' (143 _1) (Fig. 20). If the primed frame is in motion relative to the unprimed, the vectors i', j', lf-' are assigned functions oft, and di'fdt, dj'fdt, d k' fd t serve to specify the rate of rotation. The axial vector w defined by - ., dj' k' + ., dk' ., + k' di' ., (143 2) W=' ae· J dl·'l ae·J . Fig. 20. Position vectors p and p' of the same point with respect to two different frames. is the angular velocity 3 of the primed frame with respect to the unprimed. If p and p' are the velocities of a given particle relative to these two frames, it is easy to calculate directly that p=b+wxp'+p'. (143.3) 1 [1953. 23]. 2 CAUCHY [1829, 3]. 3 Recognition of the vectorial character of the angular velocity is due to EuLER [1765, 3, § 5]. It is sometimes more convenient to replace i', f, k' by any rigid triad of independent vectors a, b, c. Then W= 438 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. In fact, if b =0 we have the operator identity In particular p .. = Tt dp = w xp . + ----rit, d'p l =wx [wxp'+p'] + ~: [wxp'+.P'], =wx(wxp') +2wx.P'+ wxp'+p'. For unrestricted b, this is tobe replaced by1 p = b + ro Xp' + w X (w X p') + 2 W X p' + p', Sect. 143. (143.4} (143.5) (143.6) relating the accelerations p and p'. For the higher accelerations, similarly2 (N) (N) ( d' )N p = b + wx +dt p. (143.7) The formula (143.6) separates into easily identifiable parts the apparent acceleration in the unprimed frame arising from the rotation of the primed frame. The terms w X p' and 2 w x p' are traditionally named after EuLER and CoRIOLIS, respectively, while w X (w xp') is the centripetal acceleration. This last is always a lamellar field, since w X (w xp) = grad [i (w · p) 2 - i ru2P2]. (143.8) To see if the Coriolis acceleration may be lamellar, we observe that curl'(2wx p') =- 2w · gradp'+2w divp'. (143.9) For the right-hand side to vanish it is sufficient, though not necessary, that the velocity p' be that of a plane isochoric motion in a plane normal to w. In this case we have 3 2wxp'=-grad'2ruQ', (143.10) where Q' is a stream function satisfying an equation of the type (161.2}. From (130.1}3 we see that a more general sufficient condition is that the vorticity of the pdmed motion be parallel to w and be substantially constant. Since curl'(roxp') =2w, the Euler acceleration is lamellar if and only if it vanishes. We now give a formal derivation of (143.3}, (143.6}, and certain further results, using reetangular Cartesian co-ordinates and a formalism valid in a Euclidean space of any dimension4• The equations corresponding to (143.1} are (143.11) 1 An attempt of CLAIRAUT [1745, 1, Art. I, §IV] to calculate relative acceleration in a special case is faulty from lack of the term corresponding to 2m xjl. Correct results involving special cases in which 2m X p' * 0, employing special units and angular variables appropriate to hydraulic machines, were first obtained by EuLER [1752, 3, Probl. 1] [1753, 1, §§ 9-10] [1756, 1, § 35], but when he came to use reetangular Cartesian co-ordinates, he obtained a result [1757. 3, § 37] which is erroneous in having mxp' rather than 2mXp'. The complete and correct result (143.6) was obtained, if rather obscurely, by CoRIOLIS [1835. 2]. 2 There is a Iiterature on this subject: LEVY [1878, 6], GILBERT [1878, 3] [1884, 1] [1888, 5] (1889, 3], LAISANT [1878, 4], RIVLIN and ERICKSEN [1955, 21, § 3]. Some of these authors obtain resolutions of <;) along the principal directions of p or in orthogonal curvilinear Coordinates. 3 TAYLOR [1916, 6, PP· 100-102]. ' ZORAWSKI [1911, 13, § 1] [1911, 14, § 1]. Sect. 143. Motion relative to a rigid rotating frame. 439 where we employ the summation convention for Cartesian tensors, where A = A (t), b =b (t), and (143.12) Diff(:)rentiating (143.12) yields Akk,.Ä_km' + Äkk'Akm' = Ü, Akk,.Ämk' + .Ä_kk,Amk' = 0. (143.13) The definition corresponding to (143.2) is wkm = Akk'Amk'• wk'm' = Akk'Akm'• (143.14) whence by (143.12) and (143.13) follows In dyadic form, the third of these relations reads w' =-w: The angular velocity of the unprimed frame relative to the primed is equal in magnitude but opposite in sign to that of the primed frame relative to the unprimed. We notealso that from (143.14) (143.16) In fact 2wkm (N) = 2 Akk,Amk' • (N) = Akk'Amk'-Akk,Amk', • (N) • l _ ~ (N) [(N+l-P) (P) (P) (N+l-P)l - LJ Akk' Amk' - Akk' Amk' , P=O p (143.17) where each summand in brackets is skew symmetric in k and m. Similarly, (N +1) (f"J • (N) Akk' = Akk' = Akm'Amk'Amm', (N) (143.18) (N) • (N +1) (N) (N) Thus w is determined by A, A, ... , A, while A is determined by A, w, w, ... , w. Indeed, from (143.18) it follows that (N+l) (N) Akk' = "Pkm Amk' • where "Pkr:J satisfies the recurrence relation Thus N (N+l) _ (N+l) + "\' (N + 1) (P) (N-p) "Pkm - Wkm LJ p Wkq VJqm • P=O (P+l) (q+l) ) ( ) Akk' Amk' = VJ~, VJJ." (143.19) (143.20) etc. l (143.21) (143.22) 440 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 144. generalizing (143.16)r. Putting this result into (143.17) yields an explicit formula for tf:] in terms of the tfJ's: N 2~) = "(N) [•n(N-Pl 1n(P-1) _ 1n_·-..:'1_ = wk ... m_wk wr ... m_ ... -wm \Vk ... r -w r \Vk ... m_ ... -w r liJk ... m dt p ... q r p ... q r p ... q p r ... q q '~'p ... r · (148.7) For the metric tensor gkm• by (72.10) this gives drgk'!'_ _ _ q _ q _ dt - Wk gqm Wm gkq- 0 · {148.8) Hence raising and lowering of indices commutes with drfdt. Since x' = 0 and w' = 0 at the origin of the co-rotational system, it lS an immediate consequence of the definition of drfdt that dN ·k r X dtN = 0, {148.9) This does not follow, however, from (148.7). In fact (148.7) was derived on the assumption that \V itself is a tensor under 49, while neither x nor w enjoys this property, being subject rather to the transformation laws (143.24) and (144.2) 2 • However, as we have said, the process of extension may be applied to quantities 446 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 149. which transform as tensors only under the subgroup , and this is the case for~ and w. If we carry out the process of extension for :e, we find that we are led through just the same steps as were used to derive (143.24), except that b =0, z' =0, and z' =0, and this yields (148.9h for the case N = 1. Similarly, it is possible to derive (148.9) 2 by differentiating (144.2) 2 and evaluating the result at the origin of a co-rotational system. From (148.9) 2 we get1 dr · dr (d ) drdkm } ([ixk,m= ~t km+w,.m =~, = d,.m + w,.qdqm + WmqdqiH (148.10) where, since (144.2h asserts that d is a tensor under ~. the last step follows from (148.7). This formula illustrates the fact that for tensors under 0 the operation drfdt need notcommutewith thecovariantderivative. By(148.9h, (drx,.fdt),m=O. Further_properties of the extensions of a tensor, defined as those tensors under ~ which reduce to &N w foz/.· az;,., ... ot ... ot at the origin of a co-rotational system, have been noticed by THOMAS2. 149. Irrotational frames. We shall call a frame irrotational of order N1 + N2 if at two times ta, a = 1, 2, its orientations with respect to a common frame are assigned and also for a particle at the origin we have w<•l' = 0, r = 1, 2, ... , Na, (149.1) where w<•l' is defined by (103.4) 2 • For the case when N2 =0, these frames were introduced by RIVLIN and ERICKSEN 3, who selected the orientation at time t1 as that of the fixed spatial frame, the orientation at time t2 as that of the spatial axes of finite strain for the deformation from x(X, t1) to x(X, t2). Using (145.8), we may prove by induction that to satisfy the requirement where ,gc•>==g;(t1); moreover, the solution is to satisfy (143.12). degree N satisfying (149.5), namely • ~ r>Cr> (t- t1)' Akk'Amk'= L.."•4 km--,-. r. r=O (149.5) There is a polynomial of (149.6) Since !J~Jn= -!J~k• any solution A of (149.6) satisfies Au'Amk'= o; hence a solution satisfying (143.12) at one time satisfies it at all times. For such a solution, (149.6) is equivalent to . LN (t-t,)' Akk'= !JkmAmk'---. r! r=O (149.7) This isalinear differential system for A; it has polynomial coefficients and therefore a unique solution A*(t) suchthat A*(t1) =A1 • The matrix A(t) given by {( t-t )N+l } A(t)=A*·exp t,-~ log[(A•)-LA1) (149.8) then satisfies Condition 1 as weil as Condition 3 with .N,= 0. An invariant time flux based upon time differentiation in an irrotational frame has been constructed by CoTTER and RIVLIN1• As might be expected from the difficulties mentioned above, they do not attempt the extension process directly. Rather, they appear to lay down the following requirements for the M-th time flux dt' \V fdtM of an absolute tensor \V : 1. dfl \V fd tM shall be a tensor under ~. 2. In any irrotational frame with origin at the point in question and with N~ M, dt' \V fdtM shall reduce to a linear combination of the material derivatives (p) \V with p =1, 2, ... , M. 1 [1955. 4]. 448 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 150. These requirements do not determine dfl W fdtM uniquely. The particular choice made by CoTTER and RIVLIN, in the special case they consider, coincides with the result to be obtaine~ by a different method in the section following. 150. Convected time flux. If we select three independent families of material surfaces, whose equations may be taken as X"'= const, their configurations at time t define a convected Co-ordinate system for the given motion1. Setting xk= b! X"', we thus describe the motion in terms of a specially selected moving Coordinate system. lt is usual to define body tensor fields as tensors under transformations of the X"' and spatial tensor fields as tensors under transformations of the xk. lt is possible, in any given case, to associate a pair of such fields in such a way that their components are respectively equal. In such a case, the motion of the co-ordinate system xk will generally prevent the time derivatives of the two fields from being equal. In the convected co-ordinate system, lengths are determined from a varying squared element of arc: (150.1) G is related to the tensors denoted by g and C in Sects. 14 and 26: At time t0 the components of G coincide with the components of g, while at time t the components of G coincide with the components of C. These remarks suggest how the formalism of convected co-ordinates may be used to treat the subjects developed in the foregoing sections by other and in our opinion preferable methods. Our purpose in introducing convected co-ordinates here is for the following special application. Let W be a tensor field of weight Wunder the group <1P of all co-ordinate transformations, including transformations to moving co-ordinates. Then the material derivative, ..V, is not generally a tensor under :::;' 8,v"- · · · - \11~:::~ 8,v"' + W\11~:::;' 8,v•. (150.6) Thus (150.5) may be written in the form dc'*" = ~ + f: \II (150.7} dt ot :;, • showing that the Lie derivative is the special case of the convected time flux that results when \II is steady. In Sect. 153 we shall see that a four-dimensional 1 SLEBODZINSKI [1931, 10], VAN DANTZIG [1932, 13, § 3], SCHOUTEN and VAN KAMPEN [1934, 7, § 1], ScHaUTEN and STRUIK [1935, 4, § 12]. A very general exposition of the nature and properties of this derivative is given by ScHaUTEN [1954, 21, Chap. III, § 10]. The German verb associated with a convected co-ordinate system is "mitschleppen"; a virtual transliteration occurs in some works ostensibly written in English. WuNDHEILER [1932, 14, § 2] introduced a differential of the type "v"= dv 11 + r" v"'dxi>+A" v"'dt u mp m • which transforms as a contravariant vector under time-dependent transformations of coordinates. Handbuch der Physik, Bd. III/1. 29 450 C. TRuESDELL and R. TouPIN: The Classical Field Theories. Sect. 150. treatment enabling identification of convected time flux with a more general Lie derivative is possible also for unsteady motions. In works on geometry the invariant character of :E W is established in con- v siderable generality. Our treatment above shows that the right-hand side of (150.5) is a tensor under transformations from a convected to a fixed co-ordinate system, but it is not obviously a tensor under transformations of fixed systems. However, the fixed system has not been restricted in any way, and we may choose it as reetangular Cartesian. In this case (150.5) may be written _c_P.._.'l_ 0 'l'k ... m = Wk ... m+Wk ... m_xs + ... +Wk ... m_xs _ ws. .. m_xk _ ... l ot p ... q s ... q ,p p ... s ,q p ... q ,s - wk ... s _im+ wwk ... m .xs p ... q ,s p ... q ,s. ( 150.8) The right-hand side of ( 150.8) is plainly a tensor under all co-ordinate transformations. Being established for a single choice of a fixed co-ordinate system, ( 150.8) is thus valid in all fixed systems. Alternatively, it is possible to transform (150.8) into (150.5) by writing out the explicit forms of the covariant derivatives. For actual calculation, (150.5) is preferable. If we apply (150.8) to the metric tensor gkm itself, we getl (150.9) The convected time flux of the metric tensor is twice the stretching tensor, and only in a rigid motion do we have dcgfdt =0. This is really obvious from the definition of the convected time flux. However, it has the important consequence that at a point where W does not vanish, dcfdt commutes with raising and lowering of indices of W if and only if the motion is rigid. In fact, since ak=gkmam, we have identically dc(ak- gkmam)fdt =0, and hence (150.10) specimens of the rules which hold for general W. In the derivation of (150.5) and (150.8) we have used the tensor law of transformation ( 150.2) from a convected to a fixed co-ordinate system. This might seem to imply that our results are applicable only to quantities W which transform as tensors under the motion itself. This is not so. Wehave employed (150.2) only at a single instant, and the formulae used in passing from (150.3) to (150.5) need hold only at that instant. At this one instant, we may choose any convected and fixed co-ordinates we please; for example, we may choose the two sets of co-ordinate surfaces as instantaneously identical, as is done by many authors. We thus obtain (150.5) without the necessity of assuming that W enjoys any particular transformation property under the motion itself. It is interesting, however, to investigate properties under the entire motion. If dcakfdt=O and dcbkfdt=O over an interval of time, we have (150.11) 1 HENCKY [1925, 7]. ZELMANOV [1948, 39, Eq. (10)] has ca]culated the flux of the Riemann tensor; the terms involving the rotation tensor in his result seem to us to cancel one another. Sect. 151. Cantrast of the various time fluxes. 451 The general solutions of these differential equations are (150.12) where A and B are arbitrary. Thus d0 akjdt = 0 is equivalent to the assertion that ak is carried by the motion as a contravariant vector, or materialline segment; d0 bkjdt =0, that bk is carried as a covariant vector, or plane area. Since, as just established, we have d0 ( d xk)jdt = 0, it follows that .t~J. aN (as2) aN g d s2 = ~-- = _c __ k_'!!__ d xk d xm atN dtN ' (150.13) where the first step is a consequence of the fact that d0 jdt reduces to the material derivative when applied to a scalar. Comparison with (104.2) yields1 A (N) - d~ gPJ_ pq - dtN • (150.14) Generalizing ( 150.9), this formula offers a new interpretation for the RivlinEricksen tensors A (NJ. 151. Contrast of the various time fluxes. The three time fluxes, drfdt, dJdt, and djdt, are in general different from one another. Since an explicit form for dJdt has not been calculated, we omit it from the discussion. Comparison of (148.7) and (150.8) for an absolute tensor W::: yields ('!.c - ~r) wk ... m_ ds wk ... m+ ... + as wk ... m_ dk ws ... m- ... - dm wk ... s (151.1) dt dt p ... q- P s ... q q p ... s 5 p ... q s p ... q· Consequently, the co-rotational and convected time fluxes equal one another if: 1. W is a scalar; or 2. 'V = 0 at the place and time in question; or 3· The motion is locally and instantaneously rigid. In cases 1 and 2 we have (151.2) Returning to Sect. 147, we see that a flux satisfying Requirement 2 is indeterminate to within an arbitrary tensor which vanishes when d = 0. The problern of constructing the mostgeneralinvariant time flux is then equivalent to finding the most general tensor under 49 which vanishes in a rigid motion. We do not know of any solution to this problem. Invariant time fluxes are required for the formation of correct constitutive equations for materials (Chap. G). Special circumstances may make one or another flux appear preferable. From a purely kinematical standpoint, however, the most that can be done is to make the meaning of each flux clear. d0Wjdt is a measure of the instantaneous departure of the rate of change of 'V from the value it would assume if W were carried as a tensor under the motion itself. drWfdt and diWjdt are, in different senses, the rates of change of 'V as apparent to an observer whose frame of reference is carried by the medium and turns with it. It is clear that any fluxes defined in terms of the motion of the material will depend upon w when referred to a common frame. The co-rotational flux drWfdt is thus the siruplest possible, in that it depends on w only, not on d or on higher 1 ÜLDROYD [1950, 22, § 3a]. 29* 452 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 152. derivatives of ~. The most striking advantage of d.Wjdt is expressed by (148.8) and the statement following it. Since in particular problems in Euclidean space there is usually no circumstance indicating contravariant or covariant character, rather than merely tensorial character, for the quantities occurring, a flux operator that commutes with the raising and lowering of indices seems most natural. Finally, the simple way in which extension of non-tensorial quantities from their values in a co-rotational frame may be achieved is a further recommendation for drWfdt. 152. World invariant kinematics. In the preceding sections of this subchapter we have studied various properties of a motion within the mathematical framework of 3-dimensional tensor analysis. We now present essentially the same considerations within a unified formalism of 4-dimensional tensors1 • As preparations we introduce the following notions from geometry. rx) Same geometrical preliminaries. An n-dimensional space S is a set of points p such that to every point there corresponds a subset 'YJ (p) containing p which can be placed in one-to-one correspondence with the ordered sets of n real numbers xr = ( xl, x2, ... , xr.) lying in some interval x{;- hr < xF < x{; + hr, hr> 0 and such that p corresponds to x{;. A Co-ordinate transformation xr· = xr· (a:), xr = xF (a:*) is regarded as replacing the one-to-one correspondence p-(xr) by another one, p-(xr•). The geometry of the space S may sometimes be determined by a group ~ of allowable co-ordinate transformations. Elements of the group ~ relate the preferred co-ordinate systems of the space 5. A field (figure) in the space S is represented by an ordered set of functions 1/.J = (1 invariant, then the geometry of a figure l/>2 with respect to ®2 is identical with the geometry of the set of figures ( lP 1 , l/>2) with respect to ®1 . Let us illustrate the application of KLEIN's principle we intend to make by the following familiar example. Suppose we have given a tensor field /!.::: in ordinary 3-dimensional Euclidean metric space where ®2 is the orthogonal group, i.e., /':,.::: is a Cartesian tensor. Let ® 1 be the group of general analytic co-ordinate transformations in 3 dimensions. ®2 is a subgroup of ®1 • Let gkm(;x) be an absolute symmetric positive definite tensor fieldunder ®1 suchthat its Riemann curvature tensor vanishes. Then in the space with group ®1 we know that there exist preferred co-ordinate systems zk such that gkm (z) = 15km· Furthermore, any such pair of preferred co-ordinate systems are related to each other by an orthogonal transformation. Thus the group @2 can be defined as the subgroup of @1 which leaves the canonical form 15;; of the Euclidean metric field g;1 invariant. Let e':n-::. (;x) be any field in the space with group ®1 having any law of transformation under ®1 such that e':n-::. (z) = t':n·::. (z) ( 152.6) in every preferred Co-ordinate system in which g;;= 15;1• According to KLEIN's principle, the theory of the invariants of the field f':n·::. under the group @2 is identical with the theory of the joint invariants of the fields (e':n-:: .• gpq) under the group @1 of general analytic transformations or under any group containing @2 as a subgroup. With these ideas in mind, consider the group @~ of general analytic transformations of the four Co-ordinates xr of events. Our objective is to define two sets of fields {lf>}G: and {lf>}ili having assigned transformation laws under @~ such that (1) there exists a subdass of preferred co-ordinates zr in which the fields { lf>}o; and { lf>}o; assume certain canonical forms, and (2) the subgroups @G: and ®m consist in all the transformations of @~ which leave invariant the canonical forms of the sets of fields {lf>}lt and {lf>}}~) and (ZK(xr), {lf>}0, (152.8) for all Vr$ 0 and not parallel to t11 • The condition (152.7) is necessary and sufficient for the existence of an absolute scalar field t (;x) such that (152.9) The field t (;x) is uniquely determined to within an additive constant. Let us introduce t(;x) as a co-ordinate surface by setting z4=t and impose on the non1 ScHaUTEN [1954, 21, p. 65]. Sect. 152. World invariant kinematics. 455 vanishing components gkm (xP1 z4) in such a system of co-ordinates the conditions (152.10) These conditions are necessary and sufficient that we be able to make a further transformation zk=zk(xP1 z4) of the first three co-ordinates such that in the co-ordinate system zr the fields gr-1 and tr have the canonical form = [c5rs 0] g 0 0 I t = (01 01 01 1). (152.11) Applying the assumed tensor law of transformation to these canonical formsl we then see that the Euclidean group of transformations (152.1) is the subgroup of ~~ which leaves these canonical forms invariant. Thereforel Euclidean kinematics is the theory of the joint invariants under the group ~~ of the set of fields ZK(;x) 1 grL1(:x) 1 tr(:X) 1 (152.12) -1 where the definition (152.3) of Cl invariant under ~-1 =Oe g'~>Ll +F:fA gALl + FrfAg<~>A = 0 1 ) 17,1 t'l> = 8,1 t'l> - r~ te = 0' r (152.15) jointly with the connection r<~>~. That isl the covariant derivatives of g<~>-1 and tr based on the connection rctt vanish identically. From (152.14) follows the existence of preferred co-ordinate systems in which all of the components of the connection vanish1 . Any two such systems are related by a linear transformation. It follows from (152.15) thatl in any of the co-ordinate systems in which the connection vanishes 1 the components of gn and tn are constants. Set z4 = t(:x). This will be a linear transformation leaving the connection zerol and t will assume its canonical form t = (01 01 01 1). (152.16) From (152.8) it follows that gr-1 is reduced to the form = [gkm 0] g 0 0 I (152.17) 1 EISENHART [1927, 4, § 29). 456 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 153. where gkm is a constant symmetric positive definite matrix. Thus by a further linear transformation of the first three co-ordinates not involving z4, preserving the condition ( 15 2.16) and the vanishing of the connection, we can reduce g to its canonical form (152.11h. It is then an easy matter to verify that the Galilean group ( 152.2) is the subgroup of ~lt which leaves invariant the canonical forms (152.11) and rtJ~ =0. Therefore, Galilean kinematics is the theory of the joint invariants under ~~ of the set of fields (152.18) where the zx again satisfy the invariant condition (152.15). The preferred Coordinate systems in Galilean space-time in which we have (152.11) and F~=O will be called Galilean frames, and r;tJ will be called the Galilean connection. Galilean frames may be identified with the inertial frames of classical mechanics (cf. Sect. 196). Thus we have succeeded in formulating Euclidean and Galilean kinematics as theories of the joint invariants of a motion and a suitable set of fields under the group ~~~ of unrestricted general transformations of the co-ordinates in spacetime. Quantities transforming as a tensor under ~~will be called world tensors. An affine connection under ~~ will be called a world alfine connection. The Galilean connection is a world affine connection, grtJ and tr are world tensors, and the zx (~) are world scalars. Weshall also consider the theory of the joint invariants of an arbitrary world tensor field 'P.f::: and a motion as part of the subject matter of kinematics. In regard to this aspect of the theory, we shall be especially interested in quantities like the stress tensor tkm of mechanics (cf. Sect. 203) which are required to transform as the components of a 3-dimensional Cartesian tensor under the restricted group ~a:· We shall represent quantities of this type in the 4-dimensional formalism by contravariant world tensors tprtJ. · · which satisfy the invariant conditions (152.19) Contravariant world tensors satisfying (152.19) will be called space tensors. We easily verify that, in every Euclidean or Galilean frame, each component of a space tensor with any index equal to 4 vanishes, and the non-vanishing components have the transformation law ljlk'l' = Ak'kAI'z ... tpkl, (152.20) under the restricted groups ~lt and ~(fj. Thus, according to KLEIN's principle, the theory of the joint invariants of tpii and a motion under the restricted groups ~(f and ~()) is equivalent to the theory of the joint invariants of tprtJ and the lists of fields (152.14) and (152.18), respectively. 153. World invariant Euclidean kinematics. In this section we consider certain of the invariants of a motion in Euclidean space-time. As previously mentioned, since ~Gi is a subgroup of ~a:, any invariant of a motion in Euclidean space-time is also an invariant of a motion in Galilean space-time, so that all of the results and considerations of this section apply equally well to motions in Galilean space-time. IT.) W orld velocity field of a motion. Consider the world scalar of weight 1 defined by (153.1) Sect. 153. World invariant Euclidean kinematics. and the world vector of weight 1 defined by br = _!___ eAA.z:r 8 ZK 8 ZL 8 ZM e - 3! LI A l: KLM• (153.2) In a Euclidean frame, ~ = det ok ZK = ± Vde;6 =F 0. Since the law of transformation for ~ is ~*=Jx*fxJ- ~ and Jx*fxl is never zero, ~=f=O in any coordinate system. Thus we can define the absolute world contravariant vector field (153.3) called the world velocity vector of the motion. The form which any world tensor or other type of world invariant takes in every Euclidean or Galilean frame will be called its canonical form. The canonical form of the world velocity vector vr is (153.4) Since ~ is never zero, we can always solve for any system of general co-ordinates xr in terms of the material co-ordinates ZK and the time T=t(x). Thus we always have relations of the form xr=xr(zK,T). (153.5) In terms of these relations, we have The result (153.5) serves to promote the geometric interpretation of a motion as a congruence of lines in space-time which are nowhere tangent to the surfaces t (x) = T = const. Such a surface is called an instantaneous space. The material co-ordinates ZK serve as names for the lines of the congruence, and T is an admissible parameter whose value is never stationary as one moves along a line of the congruence. It is clear that the first three components of the world velocity in a Euclidean frame coincide with the usual definition (67.1) of the velocity vector of a continuous medium. ß) Invariant definition of a rigid motion. A motion in Euclidean or Galilean space-time is called rigid if and only if (153.7) To see that this definition corresponds to the customary one, note that from (153.1) and the fact that ~ =f= 0, we can always introduce the ZK and T= t(x) as a system of co-ordinates in space-time. The components of the space metric g in such a system of convected co-ordinates will have the values gK4=g44=0 and gKL where -1 gKL = CKL (convected co-ordinates). (153.8) -1 The result ( 15 3 .8) follows immediately from the definition of the CKL in a general system of co-ordinates. Let gKL (ZK, T) denote the inverse of gKL. The distance between two neighboring material points ZK and ZK + dZK at time T is given by d52 = gKLdZK dZL= CKLdZK dZL. (153.9) Thus a motion is rigid if and only if dS = 0 for every pair of neighboring material points. 458 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 153. y) W orld tensor of stretching. Consider the contravariant absolute world tensor 11 defined by1 V (153.10) .1rx = _ -i 1: grx, } =-t (vAoAgr.E- gAL'(JA vr- gAT(JA vX), where i: denotes the Lie derivative (150.6) with respect to the world velocity V vector vr. One sees by inspection that the Lie derivative of any space tensor is a space tensor. The canonical form of the space tensor 11 is = [dkm O] 11 0 0. (153.11) The quantities dkm are the familiar Cartesian components of the stretching tensor (82.2). We call 11 the world tensor of stretching. The field g-1 defined by (153.12) is a world scalar of weight - 2 having the constant value g-1 = 1 in every Euclidean frame. The familiar absolute scalar invariants of the stretching tensor dkm are given in world invariant form by the formulae I _ g 'PD AII ATL' S ~ d-28T'l'AB8L'!JII~g g LJ. V V 'J II - g 'PD L1AII L1TL' vE ~ d-2f8T'l'AB8L'DII~g V, IIId = L, Br'l'AB BxDII~ L1'l'D L1AII L1rx v3 v~. 3· (153.13) The canonical form of these scalars (cf. Sect. 83) is Comparing the definitions of the Lie derivative with respect to vr of a world space tensor and the convected time flux ( 150.5) of a Cartesian tensor under 49~t, we see that .. d ~i; ... f: '*' •1· •• == _.::_C -;-:-- V dt 1 (153.15) where, as henceforth, the symbol"-'-" indicates that equality holds in a Euclidean frame and, in general, only in such a frame. From this result and from (150.6) we seethat in thespatialdescription, where the velocity field is basic and is regarded as given, the convected time flux ~fdt defines an infinitesimal transformation of ~- As was remarked by ZoRAWSKI 2, the central problern of the kinematics of the spatial description is to characterize all invariants of this infinitesimal transformation. ~) W orld alfine connections defined by a motion. The structure of Euclidean space-time implied by the underlying group 49~t of Euclidean transformations does not of itself lead to an affinely connected space in the tour-dimensional sense as in the case of the linear Galilean group. However, as was shown by DEFRISE 3, given the basic tensors gsx and ts of Euclidean space-time and a 1 WUNDHEILER [1932, 15, § 8]. 1 The analysis of ZoRAWSKI [1900, 11 and 12] [1911, 13 and 14] [1912, 8] refers only to steady motion described in the common frame. 3 [1953. 8]. Sect. 153. World invariant Euclidean kinematics. 459 world velocity vector v5 , we can construct from these quantities and their partial derivatives a set of quantities .Q~!P having the law of transformation of a world affine connection under ~~. The method of constructing the components of .Qf:!P from the g5 I, t8 , and v5 is analogous to the method of constructing the Christoffel symbols based on a non-singular symmetric absolute tensor of rank two, familiar in Riemannian geometry. Let 17 !P denote covariant differentiation D based on the world affine connection .Q~I· The equations used by DEFRISE to determine the components .Q~ x are equivalent to the following: (153.16) These equations have a unique solution for all64 components of the world connection .Q~I· The canonical form of these quantities is . av• . . .Q'« = - iiZ' + v' 8; v•. (153.17) As a variant of DEFRISE's procedure for determining a world affine connection in terms of a motion in Euclidean space-time, we can solve the equations 17 !P gBI = 0, vii7x v5 = 0, 17x ts = 0, 17 !P v!B gil!P = 0, (153.18) A A A A for the components A~x of a different symmetric world affine connection. The canonical forms of the components of the connection A~x are Afk=O, Ab=O, Ats=(wPs-dPs)vs-~~· (153.19) where wP s == t ( 8 s vP - 8 P vs) are the components of vorticity. It is an immediate consequence of the law of transformation for affine connections that the difference between two affine Connections is an absolute tensor. In the case of the two world affine connections .Q~x and A~x. this difference turns out to be Uffx== A~x- D~x = Ps!PiJ!PIItx + Px!PiJ!PII ts, where Ps• is the absolute tensor which has the canonical form p = [ ~k;m vP::] · (153.20) (153.21) e) Invariant time fluxes in the 4-dimensional formalism. We now note an interesting relation between the covariant derivatives of world space tensors based on the two connections .Q~x and A~x and the two invariant time fluxes dJdt and drfdt defined in Sects. 150 and 148. Let psx ... be a space tensor, and consider the space tensors given by v!P17!PlJISI ... and V!P17!PlJISI.... (153.22) n A As one easily verifies from the canonical forms ( 15 3 .17) and ( 15 3 .19), in a Euclidean frame we have I Tl':' • dc ITJ:" -" lTF' • dr lTJ:" v!P !{ !P r•t ... = dt r•l .. ·, V"' f!P r•l· .. = dt r•t .... (153.23) In an arbitrary system of space-time co-ordinates we have v!P 17.lJIBI ... - v!P 17(P lJISI... = v!P U/rrlJIIII ... + v!P U/rrlJISII ... + ... (153.24) A n 460 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 153 which generalizes the relation (151.1} contrasting· theinvariant time fluxes dcfdt and drfdt. If one adds any symmetric absolute tensor S!/x=S~s to the world affine connection determined by either of the sets of Eqs. (15).16) or (153.18}, one obtains yet another symmetric world affine connection. Many tensors S!/x can be defined in terms of the basic set gsx, t8 , and v5 • Thus, many world affine connections can be constructed from these quantities and their partial derivatives, and the absolute derivatives of world tensors can be defined in terms of them. The question as to which of these methods of absolute differentiation in Euclidean space-time defined in terms of a motion in that space is the most natural or useful is analogous to that raised in Sect. 151 concerning invariant time fluxes. CJ Convected co-ordinates. In a convected system of co-ordinates x3= (ZK, T), the world velocity vector and the covariant space normal have the convected form V= (0, 0, 0,1}, t = (0, 0, 0,1), and, as noted previously, the space metric has the form _ [CKL ol Y- 0 0. (15).25) (153.26) Thus it is a simple matter to determine the convected form of the various world tensors L1 5 x, v w'x ... = v(E r:h vll' (154.6) since by (152.15h we have 2g.z> vE gi:tJ> = V(f>vS gi_ Vnvsv vll gZ } r r r r (154.20) =V<~> a5 g.E<~>_ Vnv V<~> vli g.r<~>. r r r Let D5 .E and Q5 .E denote the symmetric and antisymmetric parts of the acceleration gradients 17 a5 g.E<~>. Taking the symmetric and antisymmetric parts r of ( 154.20} then yields Li s .E = w <1> 17 (f> vll. l r r iJs .z: = ßrr.uJ = QS .z:- 17 n vrs g.El <~> 17 <~> vli. r r (154.21) The classical problern of finding the equation for the rate of change of vorticity in a rotating, deforming co-ordinate system is thus solved by introducing the expressions (154.9) and (154.10) into (154.21} 2 2• 1 E.g., that of McVITTIE [1949. 18]. 2 This equation generalizes one derived by McVITTIE [1949, 18, Eq. (6.8)] by different means. There is an extensive earlier literature devoted to special cases, mostly in meteorological contexts. 464 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 155. 111. Mass. a) Definition of mass. 155. The meaning of mass. In classical mechanics, each body is assigned a mass, a positive real nurober expressing the quantity of matter in the body, according to the requirements: 1. The mass of a whole body is the sum of the masses of its parts; 2. The mass of a body never changes, no matter how that body is moved, accelerated, or deformed; ). The mass of a body is not in general determined by its size. These requirements are translated into mathematical form as follows: 1. M ass is a measure. 2. M ass is invariant under motion. ). Mass bears a physical dimension [MJ, independent of [L] and [T]. W e discuss these in reverse order. No. 3. The dimension of mass. This is the only property of mass that distinguishes it from other measures, such as the probability in phase of statistical mechanics or any other kind of probability or measure. While the dimensional independence of mass has definite and essential consequences in certain parts of mechanics, notably in the theory of modelling, for the developments presented in this treatise no use is made of it. No. 2. The conservation of mass. In classical mechanics, the quantity of matter in a body generally does not change, while in more recent physical theories there are laws governing the change of mass. These physical principles must not close our eyes to the simple fact that it is always possible to define an invariant measure. Indeed, let the motion in the interval of time from t0 to t be regarded as a transformation T/ which carries the particle X to the place ~ = Tl X, and the set of particles 'g> into the set of places d = 7;! .9'. From the axioU: of continuity of motion, if .9' is measurable then d is measurable. Assign any measure m ( .9') to the measurable sets of particles; then the definition !.TR(Il, t) = !m(7;!.9', t) = !.JR(.9') (155.1) induces a measure of sets of places, and this measure is invariant in time for a given set of particles. That is, the measure of the set of places occupied by the set of particles .9' is the same at each time. This trivial construction is valid in the greatest generality. For any sort of motion, whether or not a possible one for a body obeying the laws of classical mechanics, we may have conservation of mass by definition. When we assert that in classical mechanics mass is conserved but in relativistic mechanics it is not conserved, we mean that the foregoing construction is useful in classical mechanics, not useful or at least not appropriate in relativistic mechanics. The distinction is a physical one. In relativistic mechanics, it is equally possible to define an invariant mass, but such a mass does not enter simply into the dynamical equations of the theory and does not correspond to the physical idea of quantity of matter; it is not fit to be compared with the result of an experiment designed to measure quantity of matter. Even in classical mechanics there are cases when conservation of mass is not appropriate. The constituents of a mixture undergoing chemical reactions may lose or gain mass, though the mass of the mixture is constant. A formalism for such exchanges of mass will be presented in Sect. 158. Another example is furnished by the moticn of a burning body in a theory which does not take account of the motion of the combustion products. That in both these cases the apparent changes of mass result from confining attention to only a part of all the matter "really" present does not reduce their cogency as examples: Mechanics must be general enough to admit models for limited situations (cf. Sect. 6}. Sect. 155. The meaning of mass. 465 An invariant measure is at our disposal: Conservation of mass results not from an axiom but from a definition, a definition we may wish to use, or may not. Kinematics is neither more nor less general after the introduction of mass. This subchapter presents topics in kinematics whose usefulness is connected with mass. It would be possible, though less interesting and less fruitful, to develop all this material without mentioning mass. No.l. Massas a measure. A positive, finite mass IDl(X) may be assigned to the particle X. In this case, the mathematical statement of Property 1 becomes 1disc· The measure IDl (X) is discrete, and the particle X is called a mass-pointl. The mass of any finite number of mass points is finite, but the mass of an infinite nurober of Iike mass-points is always infinite. Therefore, if we are to represent finite physical bodies by a model consisting in an aggregate of mass-points having only finitely many different masses, only a finite number of mass-points may be used, and thus necessarily almost all of space is void of matter. In the model of matter as continuous, to which this treatise is devoted, finite mass is assigned not to individual particles but only to sets of particles having positive volume. Moreover, if we have an infinite decreasing sequence of bodies whose common part, if any, has no volume, the masses of this sequence of bodies must approach zero. In mathematical terms, 1cont· The measure Wl(9") is an absolutely continuous function of volume. This requirement is more stringent than the physical motivation No. 1 suggests. The difficulty is typical of infinite models (cf. Sect. 4); it is just the same as a classical difficulty in the experiential foundation of probability theory; thus we pass over it here. The requirement of absolute continuity modifies No. 3 to this extent, that a body of zero volume has zero mass. Requirements 1disc and 1cont may be combined so as to permit mass-points and continuous masses simultaneously: 1mixed. Let IDl ( 9") be the sum of an absolutely continuous measure and a discrete measure defined over a finite number of points. For application to particular cases, 1mixed may be the most convenient. More general definitions of mass are suggested by measure theory, but these do not seem useful in mechanics. According to the field viewpoint, indeed, 1mixed is unnecessarily general, since, as we shall see in Sect. 167, the concept of masspoint emerges derivatively in the theory of continuous bodies. Integration over mass. Since mass is a measure, a Lebesgue-Stieltjes integration may be defined over it. By the fundamental theorem on absolutely continuous measures, from 1mixed follows n JfdiDl=JefdV+L,IDlafa, (155.2) !/' -r a=l where la=f(Xa), where IDla is the mass assigned to Xa, and where (!, the mass density, satisfies dim e = [M L-a], (155-3) 1 EuLER [1736, 1, §§ 98, 117, 134]. The concept of "body" used by NEWTON and other earlier authors was vague. Handbuch der Physik, Bd. III/1. 30 466 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 156. 9.R (9') being the mass of the set 9' whose volume is ~ (9'). One way of asserting (155.2) is to write d9.R = e dV, (155.4) except at the points bearing discrete masses. By definition, (! is an absolute scalar under co-ordinate transformations, since dV, as defined in the line following (20.9), is the absolute scalar element of volume rather than the scalar capacity dX1 dX2 dX3• While the theory of measure is needed to justify rigorous deductions from (155.2), an equivalent idea was used regularly by EULER, LAGRANGE, and other savants of their time. In results based on (155.2), theorems for systems of mass points emerge by taking (! =0; for continuous media, by taking 9.Ra =0. We prefer to follow the latter course, adopting 1cont rather than 1mixed, and confining our attention to continuous bodies alone. Especially in view of the results to be presented in Sect. 167, this seems conceptually the cleanest course. However, the reader who desires to employ the mixed model may do so by trivial modification of the analysis in the following sections. The concept of body. Thus far we have used the term "material" to refer to any set of particles. We now introduce the term body to describe a set 9' of particles such that: 1. 9' has positive mass. 2. No subset of 9' which has positive volume has zero mass. This makes body synonymaus with portion of matter occupying a finite non-zero volume and suggests a model of the universe consisting in certain bodies moving through a massless aether. A finite portion of a body is itself a body, if its volume is positive. The requirement of absolute continuity asserts that a set of volume zero cannot be a body, but leaves it possible for an infinite volume to have finite mass, or for a finite volume to have infinite mass. In the former case, it is necessary that (! -+ 0 at oo except possibly on a set of points of volume 0; in the latter, that e-+ oo at one or more points. It is also possible that (! = 0 over a set of volume zero within a body. Points where (!-+ oo usually require special attention for other reasons, so we classify them as singularities and strengthen the axiom of continuity (Sect. 65) so as to exclude them: o~e 1; if sr = 1' they will be called simple. For heterogeneaus media, German capital subscripts are used to distinguish quantities associated with the individual motions. The individual velocity x~1 is 1 FERRARI [1913, 1, § 2]. 2 For the case W = f%2, WIEN [1892, 14] discussed the form taken by certain choices of i in certain particular field theories. Cf. also MATTIOLI [1914, 7]. 3 The term "conservation law" is used in a different sense by ÜSBORN [1954, 17], who " restricts it to a formula of the type~ oakJoxk=o; he attacks the problern of determining k=l all such laws which follow from a given set of partial differential equations. 4 [1855. 1]; [1871, 6]. Cf. HILBERT [1907, 3, PP· 43-47] [1907, 4, PP· 42-45]. REYNOLDS [1903, 15, § 35] employed a similar mathematical resolution of turbulent flow. 470 defined by C. TRUESDELL and R. TouPrN: The Classical Field Theories. 'k oxk I x~c=-- - ct X~=const • Sect. 158. (158.2) The spatial description of the motion consists in the S'r functional relations :V2l = :V~(;,:, t). Each constitutent has its individual density e~, the total density e being given by .R e = l: !?2l· (158.3) ~=1 The absolute concentration1 c~1 of the constituent ~ is defined by c~== e~fe, (158.4) so that (158.3) is equivalent to .R LCU1=1. (158.5) ~=1 The mean velocity ;E of the mixture is defined by the requirement that the total mass flow be the sum of the individual mass flows: .R .R e;E- L !?Ut:l:~r. or x = L c'1l:i:~ • (158.6) \ß, we may replace (159A.1) by ~ = 1, 2, ...• ~- The indestructibility of the atomic substances is expressed by the postulate .!\ (159A.2) Ln~18 c~/M~ = o. (159A.3) ~=1 c~ being defined by (159.1). If we multiply this relation by M 18 and sum on 18, by (159A.2) we derive (159.4). which we have already seentobe equivalent to the conservation of total mass. The postulate (159A.3) implies a pregnant identity connecting an arbitrary function g~ to g~. the difference between g'lt and its equivalent in atomic substances: .!\ • '\' M!B g~== g~- LJ n~!B --g!B• lB=1 M~ (159A.4) implying that g~r= 0 for ~::;;\ß. Then from (159A.3) follows .!\ .!\ Lt~g~ = L c~g~. m=1 m=\1!+1 (159A.5) This identity asserts that the total rate of production of the quantities gm through creation of mass equals the total rate of production of the quantities g~1 for the compound substances only. The case g~= 1 again yields (159.4). In the study of chemical reactions, it is customary to divide the mass supply cm of the constituent ~ into portions contributed by the f reactions taking place 2 : r ecm= l.: N~ala• (159A.6) a=1 where fa is the reaction rate of the a-th reaction, and where the pure numbers N~a may be called stoechiometric coefticients. The reaction rates are assumed to be variable independ1 Not attempting to trace the origin of the ideas used herein the early chemicalliterature, we follow EcKART [1940, 8]. In a celebrated work which has given rise to the enormous Iiterature on the phenomenological theory of molecular relaxation, EINSTEIN [1920, 1] applied this structure to study reaction rates in a chemically pure gas, some of whose molecules are dissociated. Unfortunately the loose language in many of the more recent studies of relaxation phenomena might mislead the unwary reader into regarding the analysis as belanging to the kinetic theory, while in fact almost all of it employs field concepts, even if not very accurately. 2 DE DoNDER [1927, 3, pp. 10-11] [1938. 3]. The reaction rates are often written in the form ]a = d~afdt, where ~a is called "the degree of advancement" of the a-th reaction. RAw and YOURGRAU [1956, 18] propose to call dfafdt the "acceleration" of the a-th reaction. 474 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 160,161. ently. Hence the condition (159.4) for conservation of total mass in the mixture becomes .ll L N'llo. = o, a = 1, 2, ... , f. (159A.7) 'll=l If we replace (159.4) by the stronger condition (159A.3), then (159A.7) in turn is tobe replaced by .ll L n'll!BN'llo.fMo.= 0, 'll=l n = 1, 2, ... , L b) Solution of the equation of continuity. (159A.8) 160. The need for a solution of the equation of continuity. The spatial equation (156.5)1 is in effect the result of differentiating the material equation (156.1) with respect to time1 . Thus (156.1) is the general solution of the spatial equation of continuity 2• However, (156.1) is expressed within the material description, and in order to render it explicit we must know the motion. A spatial form, for application of which the motion itself need not be given, is preferable 3• 161. Plane motion and related cases. We begin with the special case of an isochoric plane or pseudo-plane motion, defined by (139.5) or (139.6). Then (156.5) 2 becomes (161.1) This equation is a necessary and sufficient condition that there exist a function Q(x, y, z, t) such that . oQ X=--oy, . oQ Y=ax· (161.2) Since dQ =y dx-i dy, the curves Q =const satisfy (70.4) and hence are stream lines. The function Q is D'Alembert's stream function 4• To interpret it, let x vary along a curve c in the x-y plane. The vector d Xn = - i d y + j d x is normal to c and oriented by the right-hand rule. Since dQ =y dx-i dy =X· dxn, it follows that for two points x 1 and x 2 lying in a simply connected region in the sameplane z = const, the difference Q(x1 , t)- Q(x2 , t) is the flow per unit height across any curve c connecting them5• The independence of the flow from the choice of the curve c is a statement that the motion is isochoric. The classical argument just given establishes the single-valuedness of the stream function Q in a simply connected region, but in fact the result holds in much greater generality6• All that is required is to show that the line integral ~2 ~2 J X · dxn, or J Xn d Xn, is independent of path for any pair of points x 1 , x 2 1 A formal proof was given by LAGRANGE [1762, 3, § 51]. 2 A formal proof was given by EuLER [1770, 1, § 112]. 3 Most of the stream functions whose properties are presented in the following sections are included in the listing of KRZYWOBLOCKI [1958, 6]. Interpretations of the equation of continuity or of similar relations satisfied by the acceleration as equalities among certain areas or volumes have been constructed by PoMPEIU [1929, 7], VALCOVICI [1933, 12], J ACOB [1944, 6, § 3], CARSTOIU [1944, 1] [1948, 4] [1954, 1, §§ 9-12], BILIMOVITCH [1948, 2] [1953, 1], and TRUESDELL [1950, 34]. 4 [1761, 1, § 12]. A moregenerat type of stream function for pseudo-plane motions had been introduced by EuLER [17 57, 3, § 62], but he did not note the Sj)ecially simple properties of Q. 5 RANKINE [1864, 2, § 2]. 6 NoLL [1957, 12]. Sect. 161. Plane motion and related cases. 475 interior to the motion. For this, it is sufficient that the boundary consist in a finite number of closed curves moving rigidly, and, in the case of an infinite region, that there be no flux out of the portion of the region of flow that lies within a sutficiently large circle. To prove this, we begin noting that the condition satisfied upon the finite boundaries is (69.1); since Vn is the normal component of the velocity of a rigid motion, we have ~ Vn dxn=O for each closed rigid bounding ( curve l. Let c be any simple circuit, not necessarily reducible, in the interior. Then the regions of flow interior and exterior to c are bounded by c and by a curve ~. finite or infinite, such that ~indXn=O. Since the condition of-isochoric motion is " 0 = ~XndXn- ~XndXn, (161.3) e " it follows that (161.4) for all circuits c, whether reducible or not. Q.E.D. Fig. 21. MAXWELL's construction for stream lines. Any family of plane curves Q(x, y, t) =const may be the stream lines of a plane isochoric motion. Given the stream function Q and Q* of two such motions, Q + Q* is thus also a stream function; if we sketch the stream lines Q = 0, ± 1, ± 2, ... and Q* = 0, ± 1, ± 2, ... , in any system of units, stream lines corresponding to Q + Q* may be sketched approximately by connecting appropriate points of intersection of the two sets Q =const and Q*=const (Fig. 21)1. From (161.2) we have at once 2 (161.5) where V2=o2fox2+o2foy2. Hence in an irrotational motion Qis a plane barmonie function a: (161.6) More generally, D'ALEMBERT's vorticity theorem (Sect. 133) asserts that a necessary and sufficient condition for a plane isochoric motion to be circulation preserving is that w be constant for each particle. From (161.5) we see that an equivalent statement is' V2Q = O; (161.7) 1 According to RANKINE [1864, 2, Appendix], this Observation is due to MAXWELL. 2 This was implied by D'ALEMBERT [1761, 1, § 12]. 3 D' ALEMBERT [ 1768, 1, § II, ~ 7]. ' This result has been misinterpreted by MEISSEL [1855, 3]. 476 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 161. in the steady case, w is constant on each strearn line, and equivalent forrns of (161.7) are1 o(I72Q, Q) - o o(x,y) - · (161.8) N ow we seek a necessary and sufficient condition that a given family of curves F(x, y) = const be possible stream lines of a steady plane isochoric circulation preserving motion. First, this is equivalent to the existence of a function Q that satisfies (161.8) 2 and also has the property Q = Q(F). That is Q"(F) o(F, jgradFj2) + Q'(F) o(F, V2F) = O. (161.9) o(x, y) o(x, y) Now Q"/Q' is a function of F; hence it is constant on each strearn line. For such a Q to exist, then, it is necessary and sufficient that o(F, J72F) ;o(F,jgradFj2) = G(F). o(x, y) o(x, y) (161.10) This criterion is due to STOKES 2• For the irrotational special case, (161.10) is to be replaced by J72Ff!gradFi 2 = G(F). (161.11) The condition (161.8) results from elimination of w from (161.5) by means of the condition of steady plane circulation preserving motion. As was noticed by HILL3 we may instead eliminate Q, even in the unsteady case. The condition Ül= 0 may be written (161.12) If w = w (x, y, t) is known, this isalinear partial differential equation for Q; its characteristics satisfy ~ = ___!_]__ = _!CL . ow ow ow (161.13) 8Y -8-x ot Therefore the characteristic curves are given by w (x, y, t) = const, Jow at Q- ow dx= const, oy (161.14) where in the second equation y is supposed eliminated by aid of the first. The general solution of (161.12) is or Jow at Q = ow dx + G(w, t). oy 1 LAGRANGE [1783, 1, § 21], STOKES [1842, 4]. 2 [1842, 4]. 3 [1885, 2, §I]. (161.15) (161.16) Sect. 162. Stream functions for two-dimensional motions. 477 Substitution in (161.5) yields &•[f i~ 4x +G(w,~]- w, (161.17} a differential functional equation for w alone. Fora steady plane motion with steady density, (156.5}1 becomes aax (ei) + aay (ey) =O. (161.18) Hence there exists a function Q' such thatl • oQ' ex=-a-y· (161.19) The interpretation of Q' as a measure of mass flow is immediate from the interpretation of Q as a measure of flow. Q' is related to the vorticity through the equation (161.20) Neither Q nor Q' is a generalization of the other. In a steady plane homochoric motion, both Q and Q' exist, and Q'=eQ. However, in a steady plane motion in which the density is not uniform, while both Q' and Q exist, they may be related by an arbitrary functional relation F(Q, Q') =0. For example, in a motion such that x= V =const, y=O, e =e(y), we have Q =-Vy, 'Y Q'=- V f e(u) du. 0 For pseudo-lineal motions, defined by (1)9.2) and (1j9.j), the continuity equation (156.5)1 becomes 0(! 0 ( ') ae+a; ex =0. Hence there exists a function Q such that 2 oQ e=--ax· (161.21) (161.22) 162. Stream functions for two-dimensional motions. The equation of continuity (156.5) may be written in the explicit forms (162.1) These suggest various extensions of the results given in the foregoing section. In the isochoric case and in the case of steady motion with steady density, respectively, suppose ~a-(Vgi ) =O, a:a (ygei3) =O. (162.2) 1 CRocco [1936, 3]. We do not follow the analysis of BIRKELAND [1916, 1], who claims to derive a result of the form (161.2) even when the expansion is not zero. 2 W. KIRCHHOFF [1930, 2]. An equivalent result had been given by EuLER [1757, 3, §§ 48-49]. 478 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 162. These conditions are necessary and sufficient for the existence of stream functions Q(xl, xs, xa, t) and Q'(x1, x2, x2, t) such that V- "1 oQ vg- i2 = oQ • lla "1- oQ' lla "2- oQ' (162.)} g X = - ox2 ' oxl ' f6 (!X - - ox2 ' f6 (!X - ();~1 ' respectively1• The factor yg in general depends upon x3 as well well as on x1 and x2• In the special case when Q or Q' does not depend upon x3, knowledge of the motion in any one surface x2 = const suffices for construction of the motion in any other surface x3=const, since the factor Vg is specified by the choice of co-ordinates. In this case, let F =F(x1, x2, t), and we have by (162.)} -il'F =-1 o(F,Q) 1 o(F,Q') ( 6 ) ,k Vi o(x1, x2) or e ]fg o(xl, x2) • 1 2.4 Hence the curves F=const, x3=const are stream lines if and only if F=F(Q) or F =F(Q'), respectively. In particular, the curves x3=const, Q =const or Q'= const are stream lines. The simplest assumption leading to (162.2) is i 3=0. In this case, a particle once upon a given surface x3=const remains ever upon it. Such motions are strictly two-dimensional and may be approached alternatively as motions in a curved space of two dimensions. If the co-ordinates are chosen so that g 3 k = c5 3 k, then, since Vg w3=i2,1 -i1, 2 , we have, respectively, w3 = ,..kmQ 5 ,km (162.5) When the surfaces xll= const are parallel surfaces, the interpretation is particularly simple. For example, if they are concentric spheres of radius r, we may take x1, x2, x3 as the polar co-ordinates (J, cJl, r; then (162.3h and (162.3) 2 become . o. oQ rsm x(O) =---, ocJl • oQ rx(cf>) = ao· (162.6} where x(6) and x(cf>) are the physical components of velocity. This case was considered in detail by ZERMELO. In a rotationally symmetric motion (Sect. 68) the surfaces x3=const are planes, and we may take x1, x2, x3 as the cylindrical Co-ordinates r, z, 0. Then (162.))x and (162.)} 2 become 2 . oQ . rr = ---oz ' • oQ rz=-· 01' ' an explicit form for (162.S)x in the rotationally symmetric case is aaQ 1 oQ aaQ ____ -- -- +- -= r 2 w3 = rw. or2 r or oz2 (162.7) (162.8} By SVANBERG's vorticity theorem (Sect. 133), it follows from (162.8) that a necessary and sufficient condition for a steady isochoric rotationally symmetric motion to be circulation preserving is that Q satisfy 2' [Q] = _.!__ [!!_Q_- _.!._ ~ + 02Q l = /(Q); (162.9) r 2 or2 r or oz2 1 In essence, we follow ZERMELO [1902, 10, Chap. I, §§ 2-4], who used the intrinsic twodimensional approach mentioned below. In this spirit ZERMELO obtained two-dimensional counterparts of some of the classical theorems on potential motion and circulation preserving motion. Cf. SBRANA [1934. 6]. 2 STOKES [1842, 4]. Cf. BASSET [1888, 1, § 306], SAMPSON [1891, 5]. The results that follow by specialization of (162.3)3 and (162.3), in this case were noted by CRocco [1936, 3]. Formulae appropriate to a rotating cylindrical polar system were obtained by v. MISES [1909, 8, § 4.4]. Sect. 163. that is, Stream functions for certain three-dimensional motions. o(.ECQJ. Q) o(r,z) =0. If Q= Q(F), substitution into (162.10) yields Q' o (.E[F], F) + Q" o (J grad FJ2 ,.-2, F) o(r, z) o(r, z) =0. 479 (162.10) (162.11} Hence the curves F = const may be stream Iines of a steady isochoric rotationally symmetric circulation preserving motion if and only if o(F,.E[F]) I o(F,JgradFJ 2 r 2 ) =G(F) o(r, z) 0 (r, z) (162.12) along theml. 163. Stream functions for certain three-dimensional motions. If the motion is isochoric or steady, the field p or the field e'Pis solenoidal; hence by (App. 32.7) and (App.32.9) we have in the former case p = C(A, B) gradA Xgrad B = curl v; in the latter e p = C' (A', B') gradA'xgrad B' = curl v'. (163 .1) (163.2) The reader may verify that all the foregoing results in this part of the subchapter except (161.22} are included as special cases of these four formulae 2• In EULER's representation (163.1}1 the potentials A and B may be selected as any independent functions such that the surfaces A = const and B = const are stream surfaces 3 ; a similar statement may be made about ( 163 .2)1 • By suitable parametrization of these surfaces, we may take C = 1. A kinematical interpretation analogous to that in Sect. 161 is then easily obtained4, for if ~ =~(v) is a parametric representation of a closed surface d, for the mass flux out of d we find from (163.2h ~ e p · rl a = ~ (grad A' x grad B') . ( ;~ x ;~) rl v1 rl v2 , ' f = J.. o(A', B') rl 1 rl 2 'f o(vl,v2) V V' (163.3) = I dA' rlB'' f' where it is assumed that it is possible to select the area d' in a plane with coordinatesA', B' suchthat d'is mapped onto dbythetransformation A' =A'(~(v)), B' = B'(~(v)). Under these assumptions, then, we may decussate space into tubes bounded by the surfaces A' = 0, ± 1, ± 2, ... , B' = 0, ± 1, ± 2, ... , and the number of these unit tubes crossing d equals the mass flux out of d. The representations (163.1h and (163.2h are awkward to use because they are non-linear. While STOKEs's representations (163.1} 2 and (163.2) 2 are linear, the components of v have no simple kinematical interpretation. Thus each of the alternatives in (163.1) and (163.2) possesses one, but not both, of the properties which make the stream function of a plane motion convenient. 1 STOKES [1842, 4]. Cf. also CLEBSCH [1857, 1, § 7). HILL [1885, 2, § II] gave analysis parallel to (161.12} to (161.17) for the rotationally symmetric case. 2 Cf. ABRAHAM [1901, 1, § 7]. 3 A kinematical interpretation of A and Bis given by PRATELLI [1953, 24, § 5]. EuLER's use of (163.2h has been discussed in Sect. 137. Cf. also the sources cited in footnote 2, p. 823. ' ERTEL and KöHLER [1949. 9], GmsE [1951, 10]. 480 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 164. In the representations (163.1) 8 and (163.2) 2 , one of the three components of the vector potentials v and v' may be eliminated. For example, if v is any vector potential of p, so that yg _xr. = ekmf> vp,m = ekmf> (ovpfoX"'), set L ==- v2 + J ::: dxa + F(xt, x2) ') M = v1 - J ::~ dx3 + G (x1, x2). (163.4) Then by differentiation follows1 oL 1/a "1 oM lr "2 oL oM 1/a ·a 8x8 = fgx' or = vgx' (Jxl + (Jx2 =- vgx' (163.5) provided oFfox1 +oGfox2 =0. Thus the velocity :i: may always be expressed in terms of two potentials. 164. The equation of continuity in space-time; its generat solution. Let. v be the world velocity vector (153-3) of a motion, and consider the contravariant world vector density e0 defined by (164.1) where e is the mass density and g is the world scalar density defined by (153.12). The equation (164.2) is invariant under general transformations of the four space-time coordinates. In every Euclidean frame (cf. Sect. 152), we have Q = (e z", e), (164.3) so that (164.2) reduces to 0(! 8 ( "k)- Tl + ()zk (! z - O' (164.4) which is the Cartesian co-ordinate form of (156.5)1• Thus (164.2) is a world tensor form of the continuity equation. It can also be written in the alternative world tensor form (164.5) where the absolute world scalar invariant Id is given by (153.13h. and i! = v0 x~ in accord with (154.18). Equations of the type (164.2) were solved by EuLER2 for spaces of any number of dimensions. For the four-dimensional case, his solution is (164.6) since, trivially, F=F,rvr=o, the function F(a!, t) is substantially constant, as are G and H. Conversely, (164.6) yields a solution of (164.2) when F, G, and H 1 That these formulae furnish a solutionwas noted by LAGRANGE [1783, 1, § 13] for the case of reetangular Cartesian Co-ordinates; our proof of sufficiency is adapted from one given by GEIS [1956, 8, § 3] under an unnecessary restriction. Note that (163.5) yields the contravariant components, not the physical components of velocity, in arbitrary curvilinear co-ordinates. 1 [1770, 1, §§ 44-49]. Cf. the treatment of this and related problems by FINZI and PASTORI [1949, 11, Chap. IV, § 7]; cf. also our Subchapter D IV. Sects. 165, 166. Momentum, moment of momentum, and kinetic energy. 481 are any substantially constant functions. This result is fairly obvious, since it amounts to a statement that the most general velocity field may be obtained by mapping vectorially onto x-t space a field of vectors in X space tangent to the curves of intersection of three appropriate families in surfaces which are stationary and hence consist always in the same particles. The mass in 1"' is • X, y, Z ..". ..". given byl m = J (! dv = r at(. G_._!!)l dxdy dz' l = jdFdGdH (164·7) = (F; - F1) (G2 - G1) (H2 - H1), provided 1"' is small enough that the mapping x, y, z- F, G, H be one-to-one. This generalizes ( 163.3). EULER's solution (164.6) includes as special cases all those given above in Sects. 161 to 163, and the interpretation just obtained likewise includes the foregoing. It is tobe noted that the derivatives occurring in (164.6) are ordinary partial derivatives, even though the result is valid for arbitrary co-ordinates in spacetime. However, it is valid only locally, and its non-linearity makes it difficult to use with profit. c) Momentum. Throughout this part of the subchapter, the Co-ordinate system is assumed to be reetangular Cartesian. Generalization to curvilinear co-ordinates may be achieved by the method of Sect. App. 17. 165. Center of mass. The center of mass of body 1"'is the point whose position vector c is defined by 2 9'Rc=fpd9'R=fePdv. (165.1) ..". ..". In a frame such that the origin of the position vector p' is at the center of mass, we have J p' d9'R = o. (165.2) ..". Hence a body may not lie entirely on one side of any plane through its center of mass 3• The center of mass of a deformable body generally moves about within the body in the course of time. In order for the center of mass to reside in a particle and move with it, it is necessary and sufficient that the velocity of the motion, p, at the center of mass shall equal c as calculated from (165.1). Any homogeneaus motion (142.1) has this property4• 166. Momentum, moment of momentum, and kinetic energy. The momentum ~. sometimes called the linear momentum or impulse or inertia, of the body 1"' is defined by l.ßk = J zk dm, ~ = J p d9'R. (166.1) 'f'" ..". 1 Ym [1957. 19, §VII]. 2 In researches donein 1758-1759, EuLER [1765, 2, §§ 7-10] [1765, 1, §§ 285-287]. after criticizing loose ideas on the "center of gravity" of earlier writers, introduced (165.1) and the name center of mass as well as an alternative, "center of inertia", and established the vectorial invariance of the definition [1765, 2, §§ 11-16]. 3 (165.2) and the above inference from it were given by EuLER [1765, 1, § 286] 1765.2, § 18], the latter somewhat obscurely. 4 LEVI-CIVITA [1935, 3]. We do not follow the analysis of SCHWERDTFEGER [1935, 5]. who claims a broad extension of this result. Handbuch der Physik, Bd. III/1. 31 482 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 166. The moment of momentum ~["Pl, sometimes called the angular momentum, of the body "Y with respect to the point *z is defined by N"!:1- J (z[k- *z[k) (im1- *zmJ) dWL (166.2) '"Y The axial vector ~[•pJ corresponding to the skew-symmetric tensor {)1"!:1 is called by the same names and is given by ~[•pJ == j (p- *p) X (p- *p) diJJl. (166.3) "'' Sometimes, but not always, there is no loss in generality in taking *p = 0. The superscript *p is omitted in cases where no confusion should result. The kinetic energy Sl' of the body "Y is defined by Sl'==tfP2diJJL (166.4) [Cf. (94.1).] 2 Sl', or sometimes also Sl', is called the vis viva or live force ofthe body. The concepts of momentum, moment of momentum, and kinetic energy of a finite body are peculiar to Euclidean space and are the stuff of which classical mechanics is made. They deserve the most minute analysis 1. In non-Euclidean spaces, the Iack of an invariant finite parallel transport makes it impossible todefinesuch volume integrals meaningfully 2 -at least, impossible without introducing new concepts not present in Euclidean mechanics. Since all physical experience necessarily concerns finite bodies, the impossibility of forming geometrically invariant mcasures of the inertia and energy of such bodies reduces mechanics in general spaces to an uneasy formalism. Two approaches are used: (1) Euclidean arguments are applied to infinitely small volumes, leading to formal analogues of all local equations in classical mechanics, or (2) the non-Euclidean space is supposed imbedded in a Euclidean space of higher dimension, where the usual measures and laws of mechanics are valid, thus inducing certain laws in the non-Euclidean subspace. The results of these two approaches are not the same. For the former, see Sect. 238; examples of the latterare given in Sects. 212 to 214. From the standpoirrt of mere postulation, there is no error in either of these approaches. For the conceptual foundations, they are both inadequate in that they use Euclidean concepts as a crutch. (Of course, the objection to the latter method is not relevant to the theories of rods and shells, where the Euclidean imbedding space is the usual three-dimensional one.) In the more general case when discrete as well as distributed masses are present, by (155.2) we get n ~ = .r (p- *p) X (p- *p) edv + 2: (Pa- *p) X (Pa- *p) ID1a, (166.5) '"Y a~l n 2 Sl> =I eP2 av+ 2: mal>~. "'' a~I In particular, for a single mass-point we have q3=ID1p, ~=ID1(p-*p) X (p-*p), 2Sl'=!JJ1p2. (166.6) The quantities !JJ, ~. and Sf are special cases of general moments of the type mk, ... k,.m="'fzk,···zk,.imdm,) Sfk, ... k,.mq ="'; zk,· .. zkn im iq am. (166.7) 1 The history of these quantities has never bcen written; on the basis of the information known to us, we rest content with stating that the concept of linear momentum seems to originate in the middle ages, while kinetic energy and moment of momentum were introduced by LEIBNIZ and EuLER, respectively. It is certain that the generat definitions of all three, and some of their major properties, first appear in the later writings of EuLER. 2 Cf. VAN DANTZIG [1934, 10, Part IV, § 1]. Sect. 167. Theorems on the center of mass. 483 etc., where we have set *z = 0. In this notation ( 166.8) While the introduction of the higher moments is suggested by the occurrence of analogaus quantities in statistical mechanics and in the kinetic theory of gases, their dynamical significance is as yet unknown, and their properties remain to be learned. Dynamical laws concerning them will be proposed in Sect. 205 and Sect. 232. The quantities ip, ~. and ~ are defined in a particular frame. Their invariance under change of frame will be investigated in the two following sections. 167. Theorems on the center of mass. If we take the volume 'f'"as material and differentiate (165.1) materially, from (166.1), (81.2)r, and (76.9), whether or not mass is conserved, we get ip = wi c - f p (log e + Id) d m, l = IDl c +~(c-p) (loge + Id) diDl. ..y (167.1) When mass is conserved, by (156.5) 2 the volume integral vanishes, and we get the fundamental theorem on the center of mass1 : The momentum of any body, no matter what deformation it be undergoing, equals the momentum of a masspoint located at the center of mass and having the same mass as the body. In particular, in a frame such that the origin is constantly at the center of mass, the momentum vanishes. The moment of momentum ( 166.3), being a function only of differences of position, is invariant with respect to change of frame, so long as the primed frame be not rotating: ~ = ~' - j (p' - *p') X } X {:p'- *p') diDl, ( 167.2) Fig. 22. Center of mass. it being understood that *p and *p' are the position vectors of the same point in the two frames. Sometimes, however 2, it is preferable to consider the relation between the moments of momentum with respect to the origins of the two frames: ~coJ_ fpxpdiDl, ..y Putting p = b +p', we obtain ~'[O'J_ f p' xp' diDl. ..y ~[O] = 9Jl [bX b + bxc'- bXc'J + ~'[O'l, (167.3) (167.4) where c' is the position vector of the center of mass in the primed frame (Fig. 22). When there is a mere static shift of origin, b = 0, and we get ~[OJ = b X ip' + ~'[O'J = b X ip + ~[O'J, (167.5) 1 KELVIN and TAIT [1867, 3, § 230]. A closely related, much older statement is given in § 196 below. 2 PAINLEVE [1895, 4, 2e Le~on]. 31* 484 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 167. showing that the difference of the moments of momentum with respect to two different points is simply b X~ (or b X~'), where bis the vector joining the two1 • When the origin of the primed frame is taken as the center of mass, (167.4) reduces to ~[oJ =cxmc + ~'[ 'l: (167.6) The moment of momentum of any body equals the sum of the moment of momentum of a mass-point having the same mass as the body and located at the center of mass, plus the moment of momentum with respect to the center of mass. From (167.4) it is plain that no such simple decomposition is valid if the relative moment of momentum is taken with respect to a point other than the center of mass. In order for (167.6) to hold, it is necessary and sufficient that Similarly, when we have bx c' = bxc'. 2~'= JP' 2 dW1, "Y ~ = t m b2 + m b . c' + ~·. (167.7) (167.8) (167.9) If the origin of the primed system is taken at the center of mass, this becomes (167.10) whence follows ~~ ~', where ~ = S'r' if and only if the center of mass is stationary. This same formula may be used to compare the relative kinetic energies with respect to the center of mass and any other point. Hence follow the theorems of König 2 : If we compare the kinetic energies calculated in a class of frames not in relative rotation, then 1. The relative kinetic energy with respect to the center of mass is the least possible; 2. The kinetic energy of any body is the sum of the kinetic energy of a masspoint having the same mass as the body and located at the center of mass, plus the kinetic energy relative to the center of mass. The foregoing theorems enable us to give a rational position to the classical mechanics of point masses without recourse to the strict concept of mass-point (Sect. 155) or any other mention of the infinitely small: We may consider a body of mass m, whatever its size, to be in motion as a mass-point of mass m, located at its center of mass, if (1) Our knowledge is sufficient to determine the motion of its center of mass, and (2) W e do not require knowledge of its moment of momentum ~' and kinetic energy S'r' relative to the center of mass. According to this view, the mechanics of mass-points appears not as the fundamental descipline of mechanics but rather as a degenerate special case when we have but an incomplete description of the actual motion of an extended body. Since wehavenot said that it is an infinitely small body, or a geometrical 1 In (167.5) the symbol _f)[O'J stands for the moment of momentum taken with respect to the origin of the primed frame but calculated in the unprimed frame. Since the frames arenot in relative rotation, we have p = b +p', and hence _f)[O'] = f (p - b) X (p - b) dlm = _f)'[O'] · "Y 2 [1751, 1, PP· 172-173]. Cf. MASOTTI [1932, 91. Sect. 168. Momentum, moment of momentum, and energy in a rotating frame. 485 point, we suffer no conceptual revulsion in treating the earth as a mass-point on occasion; we are aware of its extension, its spin, its geometrical irregularity, and the motions of its winds and tides, which in some problems may be the focus of our interest, but for its motion with respect to the sun they are of little or no importance. 168. Momentum, moment of momentum, and energy in a rotating frame. We generalize the formulae of the last section by allowing the primed frame to be in rotation with respect to the unprimed frame. By substituting (143.3) into (166.1) we get ~ = m (b + C' + w X c') = m c I in conformity with (167.1). By substituting (143.3) into (166.3) we get where ~ = ~~ + ~~:r~ ~~~~~ = f (p- a) X [w X (p- a)J d9R. ?"" (168.1) ( 168.2) (168.3) The relation ( 168.2) generalizes ( 167.2)1 ; the corresponding generalization of ( 167.4) is elaborate but easy to work out. Since p- a = p'- a', primed quantities may be used to calculate ~~:fl, but the angular velocity w is that of the primed frame with respect to the unprimed. To discuss the form of ~~~(, we introduce EULER's tensor (.i[a] and the tensor of inertia ~[al by the definitions1 (.i[al=J(p-a)(p-a)d9R, l ~[a] = ~ j (p- a)2d9Jl- (i[al; {168.4) 1 The great researches of EuLER on these tensors deserve special notice because they are never cited in reference to the much later work on second-order tensors and matrices they in part anticipated. In 1741 or earlier, EULER introduced the name moments of inertia for the diagonal components of .3; these quantities were already familiar. In [1765, 2, §§ 26-29] he derived the tensor law of transformation for them. In [ibid., § 34] he defined the principal axes by the extremal properties of the moments of inertia, and he gave these axes the name they have retained ever after; in [ibid., § 37] he derived the characteristic equation of .3, which he proved to have three real roots, to which correspond at least three mutually perpendicular principal axes [ibid., §§ 38-42]. For these axes, the tensor .3 assumes diagonal form, and the corresponding components are named principal moments of inertia; any moment of inertia is expressedas a linear combination of these [ibid., §§ 43-45]. If the principal moments are distinct, there are only three principal axes; if two principal moments coincide, any axis normal to the one corresponding to the third moment is a principal axis; if all three principal moments are equal, every axis is a principal axis [ibid., §§ 46-47]. These results, obtained in 1758-1759, are presented even more clearly and fully in [1765, 1, Chap. 5]. Cf. also [1776, 2, §§ 33-34]. The principal axes themselves have a short prior history. In [ 1746, 1, § 77] EuLER bad found that in order for a Iamina to rotate freely about an axis, the two products of inertia with respect to that axis must vanish. In [ 17 52, 2, § 48], EuLER had introduced the full tensor of inertia and bad shown that permanent rotation of a rigid body is possible only about an axis suchthat the appropriate products of inertia vanish [ibid., §§ 14-16, 53]. but he bad not deterl:nined the existence of such axes. SEGNER [1755, 1, pp. 16-30], who acknowledged having seen EuLER's paper [ibid., p. 15], then proved at Jength that at least three such axes exist and are mutually orthogonal, etc.; in particular, he derived the characteristic cubic satisfied by the principal moments of inertia. 486 C. TRUESDELL and R..TouPrN: The Classical Field Theories. Sect. 169. these symmetric tensors are associated with the body "'" and with a particular point a. W e find that ~~:1 = j {(p- a)2w- [w. (p- a)J (p- a)}dim,) =W .~[a] =~[al.w. (168.5) In component form, (168.5) may be written c;[a]_ C'><[a] "e'km- ekmp BqrsWrs "'qp • ( 168.6) Since ~}:l = ~l':J, there exist three real mutually orthogonal principal directions for ~[aJ; these are the principal axes of inertia. From ( 168.4) the normal components of ~[aJ are J [(y- ay)2 + (z- az) 2] dim, etc., and hence the proper numbers of il[aJ -r are positive if the body has positive volume. Thus the quadric of ~[a] is an ellipsoid, the ellipsoid of inertia1• The normal components of il[aJ are called moments of inertia; the shear components, products of inertia. The tensor of inertia, its principal axes, and its proper numbers are all defined in terms of a particular point. The importance of the formula ~~:1 = ~[aJ. w lies in its linearity in w: Once a point a is selected, the tensor ~[aJ can be evaluated from a knowledge only of the shape of the body "'"and of the distribution of density within it; ~[oJ depends on the orientation of the primed frame only through the tensor law, and it is independent of the angular velocity of the primed frame. In the special case when the body is at rest with respect to the primed frame, we have ~' =0, and (168.2) shows that the entire moment of momentum of the body is ~~:rl. In the case of rigid motion, it is possible to choose a primed frame of this kind, and in this frame ~[*pJ is constant in time; w is the angular velocity of the body (Sects. 84, 86, 143), and the geometrical meaning of (168.3) is simple. To compare the moments of momentum ~[OJ and ~'[O'J with respect to the two origins, by substituting (143-3) into (167.3) 1 we easily calculate ~[O] =im [bx b + bx (wxc') + b xc' + c'x b] + ~~?t'J + ~'[O'J. (168.7) If the origin of the primed frame is the center of mass, three of the four punctual terms vanish; also in the case when the origin is stationary there is a corresponding simplification. In a fully general motion, (168.5) remains valid, but since ~~ [O'l generally does not vanish and the components of ~[O'J are continually changing, the result is seldom useful. By substituting (143-3) into (166.4) we get where g:r = im [ t b2 + b · c' + w · c' X b J + W • ~~ [O'J + g:r~~'l + g:r' 2 sr~~'J- w . ~[O'] • w = ~~?t'J . w = w . ~m'l. (168.8) (168.9) 169. Interpretation of certain previous results as theorems on momentum and energy. Various formulae derived earlier may be regarded as expressions for momentum and energy. First, in a homochoric motion we have ~Ne= f p dv, ~rolfe = J pxpdv, 2S'e/e =fp2dv, (169.1) -r -r -r and since p is solenoidal, we may apply the results of Sect. App. 31. By (App. 31.23), (App.31.25), (App. 31.21), and (App. 31.12) we thus get formulae 1 CAUCHY [1827, 3]. Sect. 169. Interpretation of certain previous results as theorems on moment relating ,, ~. and Sl' to certain vorticity averages: 2'/e = fpxwdv +~ (daxp) xp, r • 2~[ lfe =-J p2wdv +~daxp p, r • 3 ~[ lfe = f px (pxw) dv- ~ [(daxp) xp] xp, r • Sl'/e = fp · wxpdv +~da· [iP2P-PP· p]. r • 487 {169.2) In the formulae for ' and ~[OJ, the surface integrals vanish when the velocity is normal to ~. In an isochoric irrotational motion, the volume integrals all vanish, and thus ,, ~[Ol, and Sl' are determinate from the boundary velocities alone. In a steady motion with steady density, e p is solenoidal, and in just the same way we derive 2'=fpxcurl(ep)dv+~[(daxep)xp], ) r • 2~[0J =-f p2 curl(eP) dv +~daxeP p, r • 3 ~[OJ = J p x [px curl (e p)] dv- HC(dax e p) xp] xp}. (169.3) However, if we apply (App.31.12) to e p we getan expression for J e2P2 dv rather than for 2 Sl' itself. The formula r 2 Sl' =~da· p (e p · p)-f p · div (e p p) dv, ( 169.4) r valid for all types of motion, is easy to verify directly. In this formula the surface integral vanishes when ~ is a fixed boundary. It is easy to generalize {169.2) and (169.3) by expressionsvalid for any motion, but these involve various combinations of the derivatives of e which render them difficult to apply. For an exception, consider the simpler formula (76.12), which in a homochoric motion yields {169.5) for a general motion, a corresponding formula is ' = ~ da · e p p - f p div (e p) d v,) • r =~da. e p p + J p ~;- dv. • r (169.6) Hence we see that in a motion with steady density, if all finite boundaries are stationary and if in any infinite regions e p p = o (p-2), then the momentum is zero. In particular, it is sufficient that e p = o (p-3). For a steady irrotational motion, whether or not it is isochoric, in (App. 31.5) we may put c = (!p, '*' = V, where V is the velocity potential (Sect. 88), and obtain1 2Sf= ~ eV-- da dV dn (169.7) 1 KELVIN [1849, 3, § 7]. 488 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 170. for the case when the region bounded by 6 is simply connected. In the multiply connected case, appropriate multiples of the mass fluxes through the barriers sufficient to render the region simply connected must be added to the result of using (169.7) with any particular determination of the cyclic function V. For plane homochoric motion, by applying AMPERE's transformation to the pair of functions Q oQjax, Q oQJoy and then using (161.2) and (161.5), we obtain 2Sf0/e = ~ Q ~~ ds-J Qwda, (169.8) where sro is the kinetic energy per unit height normal to the plane of motion. Various other formulae for sr or Stfe in various circumstances may be derived from the representation given in Sects. 161 to 164. The kinetic energy ~was defined by (94.1), which is formally equivalent to (166.4), and KELVIN's fundamental theorem of minimum energy was given in Sect. 94. In some cases the total squared speed, to which ~/(! reduces for hornoehone motion, has properties more illuminating than those of ~ itself; the general formula (93.1) as weil as estimates in terms of vorticity and expansion were given in Sect. 93· Since the density of ~. per unit mass, is tp2, or tx2, every theorem concerning the speed is an energy theorem. Indeed, we have called tx2 the specific kinetic energy and have obtained many of its properties in Parte III of Subchapter II. Note also that the results cited in footnote 2, p. 822, may be regarded as energy theorems. 170. Rate of change of momentum, moment of momentum, and kinetic energy. By applying (156.8) to the definitions (166.1), (166.}), and (166.4) we obtain1, for any body "/'", p = f pdiDL ~=I (p- *p) X (p- *p)diDL if: = f p. pdiDl. (170.1) ~ ~ ~ For the higher moments (166.7), formulae of such simplicity do not generally hold. For example, with *z =0 we get mkm = f (zk zm + zk zm) d9R, (170.2) ~ whence (170.1) 2 follows by taking the skew-symmetric part. If in the general transport theorem (81.3) we substitute successively \1! =efJ, \1! =pX(!p, \1! =ieP2, we obtain '= ~~ +~da·eftp, d • ro1 ac..coJ J, . . ac..coJ J, .. ~ = -""-8t-+'j'da · ppx e p = -""-0t-+ 'j'px (e p p· da), (170.3) These formulae express the rate of change of ~. ~C l, and ~ for a given body as the sum of a local or apparent rate and an appropriate flux through the bound1 (170.1)3 was noticed by STOKES [1851, 2, § 49]. (170.1)1 and (170.1) 2 , in principle, are still older, but the classical writers, beginning with EuLER ( 1776), were content to regard the right-hand sides a priori as measures of force and moment. As purely kinematical formulae, apparently (170.1Jt, 2 were first derived by v. MrsEs [1909, 8, § 9], who obtained also (170.3Jt. The Straightforwardtreatment given here derives from CrsoTTI [1917, 4, §§ 3-4]. Cf. also SERINI [1941, 6]. Sect. 170. Rate of change of momentum, moment of momentum, and kinetic energy. 489 ing surface1 . From them we conclude that in a steady motion with steady density, if no material enters or leaves v, then the momentum, the moment of momentum, and the kinetic energy of the material occupying v are conserved. The truth of this statement is obvious, but special cases of it are sometimes proved at length. In the formulae for ~ and ~ro1, the flux is proportional to ePP· This important tensor, whose contravariant components are eick Jcm, is called the momentum transfer. It has appeared already in the identities (99.18) and (156.7), which may be used for an alternative derivation of (170.3)1 and (170.3) 2 ; those identities assert that the divergence of the momentum transfer may be regarded as the supply or source strength for creation of momentum per unit mass, to be added to the apparent rate of change of momentum in order to yield the total rate experienced by a moving particle. Cf. also Sect. 207. We record a variational formula which follows at once from (97.4): Jxkrhkd9Jl= :tJxk<5xkd9Jl-<5~. (170.4) "I'" "I'" where the variation of density is so adjusted that <5 diDl = <5 (e dv) = 0. We now calculate the apparent rate of change of momentum2 and energy due to a difference of observers. Differentiating (168.1), (168.2), and (168.8) with respect to time yields ~ = 5m [b + c' +wxc' + 2wxc' +wx (wxc')] = 5mc, (, = ~· + ~~:r1 + w x ~·. ~ = m [b . b + b . e + h . (c' + w x c') + + w · c' X b + w · ( c' + w X c') X b + w · c' X b J + + w · ~'[ '] + w · ((,'[O'l + w X ~'[O'l) + ~~~'J + ~', (170.5) where a dot superposed upon a primed quantity stands for d'jdt. The simplicity of (170.5) 3 is somewhat deceiving, since ~[•pJ stands for d ~[•Pljdt, not d' ~[•plfdt. The relation between (,roJ and ~·[O'l is still more elaborate since in general the two origins are in motion with respect to one another 3• By (170.1) 2 , the two quantities tobe compared are given by ~roJ = f pxpdiDl, ~·ro' 1 = J p' x p' am. (170.6) "I'" "I'" Using (143.6) in (170.6h, we get ~[OJ = f (b +p') X (b + WX p'+ 2wX p'+ w X (wxp') + p') d9Jl. (170.7) "I'" Before evaluating all the terms, we note first that if ~·[O'J is the tensor of inertia with respect to the origin of the primed frame, for its material rate of change as apparent to an observer in the primed frame we have by (168.4) ~·ro'] = 2 t J p'. p' a m - J (p' p' + p' p') a m. (170.8) "I'" "I'" 1 It is easily possible to study the material rate of change of other quantities associated with the mass flow (!P· For example, V. BJERKNES [1898, 1, § 18] calculated the rate of change of the circulation of eP about a material circuit, but the result is not illuminating. 2 v. MrsEs [1909, 8, § 9.2] constructed an interpretation for the relative rate of change of momentum in a stream tube. 3 Cf. PAINLEVE [1895, 4, 2° Le-;:on], HUNTINGTON [1914, 6], KELLOGG [1924, 8]. 490 Hence Also Co TRUESDELL and Ro TOUPIN: The Classical Field Theorieso f p' x (2m xp') d9Jl =2m 0 f [1 (p' 0 p')- p' p'] d9Jl 1 "/" "/" = m 0 f [21 (p' 0 p')- (p' p' + p' p') + "/" + (p' p'- p' p') J d m I = m 0 ~'[0'] + m X ~'[O'J. J p' x [m x (m xp')] d9Jl =-f (p1 "/" "/" • m) (m xp') d9Jll l = - m x f p' p' d m . m I "/" = m X ~'[O'J. m. Using (170.9) and (170.10) in (170.7) yields1 where Secto 170. ( 170.9) (170.10) (170.11) - ~c = b X m [ b + w X c' ~ 2m X c' + m X (m X c') + c'] + c' X m b I l =bx9Jlc+c'x9Jlbl (170.12) - ~r = W. ~'[0'] + m. ~•[O'J + m X ~'[0'] + m X ~•[O'J. m 0 The simple formula (170.5) 2 compares for two different observers the rates of change of moment of momentum calculated with respect to the same point, which may be moving in any way. (170.11) 1 on the other hand, compares the rate of change of moment of momentum when each observer calculates it with respect to his own frame and origin. In (170.12), all dots on primed quantities indicate time rates as apparent in the primed frameo Thus an observer in the primed frame by combining appropriate quantities he hirnself observes may calculate the rate ~[OJ apparent to an observer in the unprimed frame. In rigid motionl it is possible to choose the primed frame rigidly attached to the body1 and then we have both ~·[O'J =0 and ~'[O'J =01 so that ~r assumes the classical simple form used in rigid dynamics (cf. Sect. 294). Further simple cases will be mentioned in Sect. 197. The formula (17001) 2 has been used by NYBORG 2 to obtain an interpretation for the spatial diffusion tensor w* defined by (101.5) 2 0 Writing quantities evaluated at the center of mass with a superscript c, we have by (16501) J (zk- ck) Zm di.JR = J (zk- ck) [z:i, + (zq- Cq) z:;.,q + O(r2)] di.JR, l "/" "/" = a;kq z:i,,q + O(r6 ), (170.13) where r is the diameter of -rand where ~ is the Euler tensor defined by (168.4)1 , with a =Co Hence a; J (zk - ck) Zm di.JR ··c Lo qk Lo "'" ( ) zm,q ,~~ ---ys = ,~~ y6 170.14 lf the ellipsoid of inertia is a sphere, we have a;mk = -l~ <5mk• where ~ is the polar moment of inertia about Co In this case ( 1 70o14) reduces to J (zk- ck) Zm di.JR z:;, k = Lim -'-"~"----.--:~-- ' r-o -l~ (170.15) 1 In principle, this calculation is due to EuLER [1765, 3, §§ 16-24] [17760 3, §§ 31-34]. 2 [1953, 20]. Cf. TRUESDELL [1956, 22]. The analysis presented above is more general than that in the two sources citedo Sects. 171, 172. Galilean invariance. 491 hence, by ( 170.1 )2 , * L" SJkm wkm = 1m---,:---"'-, r---+-0 av (170.16) where we have omitted the superscript c. This result asserts that the curl of the acceleration at ~ is proportional to the rate of change of moment of momentum experienced by a vanishingly small sphere centered1 at ~- 171. Galilean invariance. We now determine the dass of frames in which the rate of change of momentum of every body is the same as its rate of change of momentum in one given frame. Since (170.5) 1 may be written (171.1) in orderthat '= ,, for arbitrary im and arbitrary c', i.e., for all bodies, whatever their masses and locations, we must have b =0 and w =0. This result is sometimes called the Galilean principle of relativity: In orderthat every body shall appear to have the same rate of change of momentum to two observers, it is necessary and sutficient that their frames be in relative translation at constant velocity. That rate of change of momentum fails to beinvariant under rotation or acceleration of the frame of the observer is a very old remark, whose history we do not attempt to relate here. Given any frame, the totality of frames in uniform translation with respect to it constitutes its Galilean class; quantities invariant under change of frame within this subgroup of the group of rigid motions are called Galilean invariants. By (167.2), ~ is a Galilean invariant, and so is ~. Of course ~l l, ~l l, Sf, and ~ are not Galilean invariants, and also steadiness of motion (Sects. 67, 146) fails of Galilean invariance. Cf. the general treatment of the kinematical basis of Galilean invariance in Sect. 154. C. Singular surfaces and Waves. 172. Scope and plan of the chapter. We organize and describe those properties of surfaces of discontinuity, such as vortex sheets, shock waves, and acceleration waves, as are common to all media. First we derive conditions that hold at any given instaut and express the fact that the discontinuity is spread out smoothly over a surface, not isolated at a point or a line. The second subchapter presents a generaldifferential description of moving surfaces. Then we prove kinematical conditions expressing the persistence of a surface of discontinuity. The fourth subchapter classifies the various kinds of discontinuities associated with the motion of a material and proves simple theorems characterizing them. Finally, we obtain the general form taken on by a conservation law when applied at a surface of discontinuity. Most of the major ideas of the subject derive from the work of CHRISTOFFEL (1877) and HuGONIOT (1885), extended in the classical treatise of HADAMARD (1899-1900)2. 1 The variation of density within the sphere need not be considered, since the distance of the center of the sphere from its center of mass is 0 (r2) and hence may be neglected in the interpretation of (170.16). 2 [1903, 11, Chap. II]. There arealso the expositians of ZEMPLEN [1905, 9] and LICHTENSTEIN [1929, 4, Chap. 6]; the latter labors the analytical side but is confined to a narrower range than the older works. Same major additions are contained in a paper by KoTCHINE [1926, 3]. Recent studies of discontinuities have emphasized calculation of solutions in particular theories of materials and have not extended the general theory. There is no modern treatise on the subject. Surfaces at which the derivatives of the velocity are discontinuous first appear in the acoustical researches of EuLER ( 1764-1765); the possibility that the velocity itself may be discontinuous was first remarked by STOKES ( 1848). 492 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sects. 173, 1 74. I. Geometry of singular surfaces. 173. Definition of a singular surface. Intentionally we begin our analysis on a very general plane; the application to the three-dimensional Euclidean space, which furnishes our main interest in this work, will follow shortly. Consider a regular surface " which is the common boundary of two regions [Jl+ and [Jl- in any real space (Fig. 23). Let \II (;.c) be a function which is continuous in the interiors of [Jl+ and [Jl- and which approaches definite limit values \II + and \11- as ;.c approaches a point re0 on " while remaining within fJl-+ and [Jl-, ""..---....... / ' // \ / cll+ ~-__,,., respectively. At re0 , \II need not be defined. The jump of \II across " at re0 is denoted by 1 I I I (173.1) [cf. (App.36.3)]. The sign of the jump isamatter of convention, but, since all our considerations are local, it will occasion no trouble. The quantity [\II] is a function of position upon "· I Fig. 23. Singular surface. When (173.1) is applied to a tensor T, the jump [T] is a tensor defined as a function of position upon "· If [T] is normal to "· the discontinuity of T is said to be longitudinal; if tangent to "· transversal 2• In a metric space, the jump of any tensor may be resolved uniquely into longitudinal and transversal components, such a resolution being given for three-dimensional vectors 3 by an identity such as (App. 30.6). If [\II] ::j=O, the surface " is said to be singular with respect \11. 174. HADAMARD's Iemma. The entire theory of singular surfaces rests upon Hadamard's lemma4 : Let \II be defined and continuously differentiable in the interior of a region [Jl+ with smooth boundary "· and let \II and o" \II approach finite limits \II + and o" \II + as " is approacked upon paths interior to [Jl+. Let re = re (l) be a smooth curve upon "• and assume that \II + is differentiable on this path. Then (174.1) In other words, the theorem of the total differential holds for the limiting values as 11 is approached from one side only. The function \II need not be defined upon 1 The notationwas introduced by CHRISTOFFEL [1877, 2, § 6]. 2 HADAMARD [1901, 8, § 3], [1903, 11, ~ 115]. 3 In some three-dimensional applications it is convenient to use the notations of EMDE (1915, 2., § 1] (cf. SPIELREIN (1916, 5, §§ 10, 26]): Gradu =n[u], Divu = n · [u]. Curlu = nx[u], where n is the unit normal pointing into 01~. Thus Divu and Curlu are the longitudinal and transversal jumps of u, and the resolution of [u] is [u] = nDivu- nxCurl u. While Grad u, Divu, and Curl u are sometimes called the "surface gradient, divergence, and curl" of the discontinuous field u, they must not be confused with the differential invariants of a field defined intrinsically on the surface. EMDE's notations are motivated by (App. 36.2). ' HADAMARD [1903, 11, , 72] gives two proofs, the first of which isthat reproduced above. The Iemma was used tacitly by earlier authors. A more elaborate proof is given by LicHTENSTEIN (1929, 4, Chap. 1, § 9]. Sect. 175. Superficialand geometrical conditions of compatibility. 493 the other side of d. If it is, and if the corresponding limiting values w- and 8"\1!- exist and have the required smoothness, a similar result holds for them, but dw-·jdl is in general unrelated to d\f!+jdl. To prove (174.1), we need only remark that for two sufficiently near points on d, of the polygonal paths employed in the classical proof applicable to interior points at least one is interior to ßi+, so the classical argument may be applied to it. For the case of two dimensions, the two paths are indicated in Fig. 24. The diagram is merely heuristic: HADAMARD's lemma is a proposition in differential calculus, independent of any geometry that may hold in the spaces where we choose to apply it. In particular, in an affine space it may be applied to the individual components T::: of a tensor field. By adding to each side of (174.1) in this case suitable expressions made up from connection symbols and from T:::, we thus obtain f···+==T··; dvr =T''+ dxk ··· ... ,r dt ····" dt (174.2) where f·.::+, the intrinsic derivative of T.::+ along the x Path ~ =~ (v (l)) on d, is defined as the quantity to the Fig. 24. Diagram for proof right of the identity symbol, and where T:::~" stands for of HADAMARD'S lemma. the limiting value of the covariant derivative T:::." as the point in question on d is approached from within ßi+. 175. Superficial and geometrical conditions of compatibility. We now apply HADAMARD's lemma to singular surfaces as defined in Sect. 173. When, hereinafter, we derive a condition conceming the derivatives of a quantity \f!, we presume without further explicit statement that all derivatives of \1! of orders up to and including the one considered exist and are continuous in ßi+ and f,iand approachdefinite limits at 0 along paths lying wholly in ßi+ and wholly in Si-. These limit values are assumed continuously differentiable functions of position on d. Thus HADAMARD's lemma (174.1) may be applied to the limiting values on each side of the singular surface d: d~+ dxk a~- dxk -([l = 0" \1! + dT, -ii1 = 0" \1!- dT. (175.1) Subtracting the second of these equations from the first yields d dxk [ dx"] dY[Wl=[o"wl----;a= a"wdr · (175.2) The entire differential theory of singular surfaces grows from applications of this formula, which asserts that the fump of a tangential derivative is the tangential derivative of the fump. Since the values of \1! in ßi+ and f,i- are in general entirely unrelated to one another, the limiting values of the normal derivatives of \1! on the two sides of the singular surface need be connected in no way: [ ::] is unrestricted. ( 175.3) To express the fact that the discontinuity is spread out smoothly over a surface of exactly n - m dimensions, not isolated upon a surface of lower dimension nor varying abruptly from one part of d to another, we need only apply (175.2) to n - m independent families of curves upon d. If we choose these as co-ordinate curves vr = const, we have (175.4) 494 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 17 5. Equations (175.3) and (175.4) are the superficial conditions of compatibility. Like HADAMARD's lemma, from which they follow at once, they presuppose no geometry in the space. In the important special case when W is continuous, or, more generally, when [W] =const, from (175.4) we obtain [okW] x7r= o. (175.5) Thus [ ok W] lies in the manifold normal to (J. If the dimension of (J is but one less than that of the space, there is but one linearly independent normal vector, which may be selected asthat given by taking n-N =1 in (App.19.4). Therefore (175.6) where B is a factor of proportionality. We have intentionally kept the argument on the highest level of generality. As soon as geometrical structure is introduced, more explicit formulae become possible. In an affine space, for the jump of a tensor W we have the following important identity valid when W is continuous or when W is a scalar and [W] = const: (175.7) since W k- ok W is a continuous quantity. In a metric space, we may introduce the unÜ normal n and evaluate explicitly the factor B in ( 175 .6), so obtaining Maxwell's theorem1 : When [W] = 0, or when W is a scalar and [W] = const, then [W,k] = Bnk, B =[nkokW] =[nkW,k] = [~~]. (175.8) We now consider applications to vector fields in the space of three dimensions. Generalization to tensor fields in n-dimensional spaces is immediate and is left to the reader. If we take W =C, a vector, (175.8) becomes [ck,,J = Bknm, Bk= [nmck,m] = [::J · (175.9) Hence [curlc] = nxB, [div c] = n · B, B = n[div c]- nx[curlc]. (175.10) These formulae embody Weingarten's jirst theorem 2 : The longitudinal and transversal iumps of the gradient of a continuous vector are the iumps of its divergence and curl, respectively. It follows as a corollary that the only possible jump in the gradient of a continuous lamellar field is normal; of a continuous solenoidal field, transversal. Results concerning the jump of the field itself can be inferred by assuming a representation in terms of a continuous potential. First, suppose we have a lamellar field c = - grad P with a potential P [ cf. (App. 3 3 .2) J which is continuous or suffers a constant jump on the singular surface. It is but reinterpretation of MAXWELL's theorem (175.8) to say that only the normal component c" can suffer a jump; the tangential component must be continuous 3• Second, suppose we 1 [1873. 5, § 78] [1881, 4, § 78a]. Cf. also WEINGARTEN [1901, 15]. HADAMARD [1903, 11, ~ 73] called the result "the identical conditions". MAXWELL seems to have been the first to derive compatibility conditions by differentiating a jump relation along a path lying on a surface. 2 [1901, 15]. 3 MAXWELL [1873, 5, § 78] [1881, 4, § 78a], FERRARIS [1897, 2, ~ 42]. This seems tobe the content of the obscure statements and proofs of BROCA [1899. 2], [1900, 2]. Sect. 175. Superficial and geometrical conditions of compatibility. 495 have a solenoidal field c = curl v with a solenoidal vector potential v which is continuous or has a constant jump on the surface. From (175.10) we have at once [c] =[curl v] =nXB, and therefore [cn] =0. That is, only the tangential component nxc can suffer a jump; the normal component must be continuousl. These results are used frequently in electromagnetism. In a metric space, the more general conditions (175.4) and (175.3), which do not presuppose [W] =const, can be included in a single formula, which we now derive for the case of a surface in three-dimensional space. W e simply multiply both sides of ( 175 .4) by au x'('.1, use the identity (App. 21.4h, and obtain [BkW] = [n"' B ... W] nk + gk",aux'('.1 Br[W],} [W ,k] = [n"'W ,".] nk + gkma.1F x~[W];r. (175.11) where for the second form W is assumed tobe a tensor, and the sernicolon denotes the total covariant derivative defined in Sect. App. 20. These geometrical conditions of compatibility 2 are no more than formal alternative expressions of the validity of (175.4) for two independent families of curves upon d. Indeed, taking the scalar product of (175.11h first by nk and then by ~r yields (175-3) and (175.4). An easy calculation using (175.11) and (App. 21.3) 2 yields the following identity for the magnitudes of the jumps: [W ,k] [W·k] =[nk W ,k]2 + [W];r[Wlr, (175.12} which might easily have been predicted from our remarks above concerning the nature of the geometrical conditions. In the case when W is continuous, by comparing (175.12) with (175.8) we obtain [W ,k][W ,k] = B2, or I BI = j[W ,k]j. (175.13) While we have written these last two results in notations appropriate only when W is a scalar, results of the same kind hold when W is a tensor of any order. There is a curious special condition of compatibility which is more easily derived directly than by application of the above results and those in the next section. For any vector field c in !Jl+-, we begin with the identity dck = {c(k,r) - (pq - P(nl (c(k, r),q- c(q,r),k)} d xk + } + d {c[k,q] (pq- P(n)}, (175.14) where p and P(I) are the position vectors of a variable and a fixed point on the curve along which dc is calculated, so that dp = d;E, dp(1)= o. By HADAMARD's Iemma, this identity still holds for a curve on the + side of d, provided all the covariant derivatives be interpreted as limit values from the + side. We now consider the special case when c(k m) and c(~ m) p are continuous across d, while ck and c[k m] may suffer jump discontinuities. 'By writing an identity of the type (175-14) for each side of the surface and subtracting the results, we obtain (175.15) Integrating from some point P(o) to P(I) on d, and then dropping the subscript {1), we obtain [ck] = [ck]o + [c[k,qj]o (pq- P(o> l, } [c] = [c]0 + t[curl c]0 X (p- P(o)l · (175.16) 1 FERRARIS [1897, 2, 'I! 55], with an inadequate proof. A result of this kind had been given by HELMHOLTZ [1858, 1, § 4]. 2 As has been remarked by KoTcHINE [1926, 3, § 1] these conditions are included, if not very obviously, in much more general ones given by CouLON [1902, 3, §§ 3, 8, 46]. They were rediscovered by THOMAS [1957, 1.5, § 3]. The elegant formal proof in the text is due to KANWAL [1958, 5, § 5]. 496 C. TRUESDELL and R. TouPIN: The C!assical Field Theories. Sect. 176. These formulae express Weingarten's second theorem1 : The discontinuity of a vector field c across a surface whe~e C(k, m) and C(k,m),p are continuous is determined, ~s a function of position upon the surface, by ~ts value and the value of [curl c] at any one po~nt. The theorem arose in the context of small displacement and small strain (Sect. 58). So interpreted, it asserts that a displacement corresponding to continuous strain and continuous strain gradient can suffer at most a discontinuity corresponding to a rigid motion of the material on one side of d with respect to that on the other. The more general problern of characterizing those discontinuities of a vector c across which C(k,m) is continuous is easily settled2• By (175.11) 2 we have [c(k,m)] = [nf> n(k Cm),p] + xfLI aLIF gp(k[Cm)];r. By (App. 21.3) 2 it follows that [ nknm C(k, m)] is unrestricted, l 2 [c(k,m)] nk x7A = [nk c",, k] x7A + nk[ck];LI, [c(k,m)] X~Lf x~ = [ck]; (LI x~A) · (175.17) (175.18) These formulae express the resolution of (175.17) into normal and tangential components. Therefore, necessary and sufficient conditions that [c(k,m)] = 0 are [n"n"'c(k,m)] = 0, l [nk Cm,k] x7A + nk [ck];.d = 0, [ck];(LI x7A) = 0. (175.19) These conditions do not suffice for the truth of WEINGARTEN's result ( 1 7 5 .16), which presumes that [ c(k, m), q] = 0 as weil. 176. Iterated geometrical conditions of compatibility. Since the geometrical conditions of compatibility (175.11) are mere identities resolving the jump of an arbitrary derivative into the jump of the normal derivative and the tangential derivatives of the jump of the function, it is clear that iteration will yield an expression for the jump of the second derivatives, [okom W], in terms of the values of [W], a;, [WJ, and of the geometry of the singular surface. An analogous result holds for the derivatives of any order. This was perceived by HADAMARD 3 and applied in special cases; the general reduction for the second derivatives in a Euclidean space of three dimensions was worked out by THOMAS 4, whose analysis we present now. It is possible, but cumbrous, to carry through the work using partial derivatives, but an elegant form results if we suppose W to be a tensor and employ covariant derivatives throughout. Writing A-[W], Ak = [W k], l B=Akn"=[n"W,k]: Bk-[W,km]n"', C Bßnk=[n"n"' W,km], (176.1) 1 [1901, 15]. The proof is a simplified versionofthat given by CESARO [1906, 2]. The original treatment and those in textbooks do not make it sufficiently clear that C(k,m) q is assumed to be continuous. As may be seen from the formulae of the next section, it is only this assumption that makes so definite a conclusion possible. 2 The problern is related to one solved by SoMIGLIANA [1914, 11, § II], who considered the case when c(k m) is the strain tensor of a linearly elastic equilibrated body. The continuity of the stress vector, as required by (205.5), implies that (175.19h 2 are satisfied, so that only ( 17 5.19)3 remains, and this is SoMIGLIANA's result. ' 3 [1903, 11, '1['1[119-120]. 4 [1957, 15, § 5]. Much more general results are given in schematic form by CouLON [1902, 3, §§ 4, 46]. Sect. 176. Iterated geometrical conditions of compatibility. 497 from (175.11) 2 we have ['V,k] =Bnk+gkma"'rx~A;r=Ak,} (176.2) ['V,kml = B",nk + gkpa"'r xfAAm;r· Since the left-hand side of (176.2) 2 is symmetric in the indices k and m, it follows that B", nk + gkp a"'r xfAAm;r= Bk n", +gmp a"'r xfAAk;r. Taking the scalar product of this equation by n yields B",= Cn", +g",pa"'rxf"' nkAk;r, where we have used (App.21.3) 2 and (176.1) 6 • From (176.1) 2 and (176.2)1 we have at once (176.3) (176.4) Ak;r = (Bnk+gkma"'Ax?'AA;A);r= (Bnk);r+gk",aAA (x?'AA;A);r. (176.5) By use of (App. 21.6)t, 2 follows Ak;r = nk B;r- gkp bj. xf"' B + gkm aLIA x~ A; Ar+ nk bj.A ;A. (176.6) Hence by use of (App.21.3) 2 we have nk Ak;r = B;r+ bj.A;A· (176.7) Substituting from (176.7) into (176.4) and then putting the result and (176.6) into (176.2)z, we obtain THOMAs's iterated geometrical condition of compatibility: ['V ,km]= Cnk n", + 2n(kgm)p a"'r xf"' (B;r+ br.z: aiA A;A) + ) + gkpgmqxf"' x!ra"'A ari(A;(IA)- bxA B) • (176.8) = Cnknm+ 2 n- brAB). The scalar product of (176.8) 2 by gkm, simplified by use of (App.21.33) 2 , (App.19.7), and (App.21.10) 6, becomes ['V•k,k] = C +A;~r-KB = [~ n~] + ['V];~r- K [ ~:], (176.9) which may be set side by side with (175.12). While the simple condition of compatibility (175.11), or (176.2}1 , expresses the 3 jumps ['V,k] in terms of the 3 quantities Band A;r defined on the surface, the iterated condition (176.8) is in a measure redundant, since it expresses the 6 jumps ['V,k",] in terms of the 9 surface quantities C, B, B;r• A;A• and A;(I)· Since, however, the 3 quantities A;A and Bare determined by A;(I) and B;r• there are but 6 independent jumps, as expected. Conditions of this kind, being pure identities, merely rearrange the variables. Only when they are used in connection with some further hypothesis, such that some particular quantity is continuous, may fruit be gotten from them. In the important special case when both 'V and 'V k are continuous1, so that A = B = 0, ( 176.8) reduces to the form 2 ' ['V,k",] = [oko",'V] = Cnkn",, } (176.10) C = [nPnq 'V,pq] = [nPnq op oq'V] = ['V,j,P]. 1 If we suppose merely that ~ ,k is continuous and [~];r= O, we obtain [~,kml = Cnknm, C = [nknm~,km], but not all the other forms included in ( 1 76.10) remain necessarily valid. 2 HADAMARD (1901, 8, § 1], (1903, 11, ~ 74]. Handbuch der Physik, Bd. III/1. 32 498 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 177. More generally1, if \V and its derivatives of orders 1, 2, ... , p -1 are continuous, we have [w,,.,,. •... ,."J = [8,., 8,., ... 8,." w], = [n"'' n1111 ••• n"'P 81111 8m, ..• 81111> \V] nk, nk, ... nkp> = [n""n"" ... n"'P \V, 11111111 ••• mp] nk, nk, ... nkp, (176.11) { [\V•"'•,..,,'"'',m, .. .'"'~'r',..."r,] n,., n,., ... n,." if p is even = [nq \V•"'• ·"'• •"'IP-•>I•q] n,. n,. ... n,. if p is odd. ,m1 •"'•··· 1 t p That is, the jumps in all the p! derivatives of p-th order are determined uniquely by the jump in the completely normal p-th derivative and by the unit normal n. By using an argumentsuch asthat leading to (175.8), we see that (176.11) holds in a metric space of any dimension; a corresponding generaliiation of (175.6) holds in any kind of space, irre~pective of what geometrical structute it may have. A second iteration would enable us to derive a general resolution of [IV,,...,p] in the Euclidean three-dimensional case, but the formal complexities encountered in deriving ( 176.8) render the details of such an analysis forbidding and the result too complicated to be useful. II. The motion of surfaces. 177. The speed of displacement and the normal velocity of a moving surface. Consider a family of surfaces given by :xJ = :xJ (V, t) , (177.1) where V stands for a pair of surface parameters VLI identifying what we shall call a surface point. V, in general, is not to be confused with a material particle of any motion that may be occurring; indeed, the considerations in this section should be regarded as independent of the motion of substances, although as an aid to visualization it is often convenient to picture the moving surface as consisting of identifiable particles. The representation (177.1) gives the places :xJ occupied by the surface point V as the time t progresses; thus it describes the motion of a surface. The velocity of the surface point V is defined by az I U = Tt V=const' (177.2) If we eliminate the parameters V, we may write (177.1) in the form f(:IJ,t) =0. (177.3) Conversely, however, from a spatial representation (177.3) it is not possible to calculate a unique form {177.1). This is easy to understand: Given a moving surface, there are infinitely many ways of identifying the points on its successive configurations in such a way that all those configurations are swept out smoothly by the surface points constituting any one of them. Supposing, now, that we have any one parametrization (177.1), by differentiating (177.3) with respect tot we get at ae+u·grad/=0. (177.4) 1 CouLON [1902, 3, § 46], HADAMARD [1903, 11, '1174]. Contrary to the implication of THOMAS [1957. 15, § 1], the analysis of HADAMARD is not restricted to any special choice of CO-ordinates. Sect. 177. The speed of displacement and the normal velocity of a moving surface. 499 Writing n for the unit normal to the surface, by (177.4) we have Of grad t ot Un=U·n=U· !gradfl =- V f,k ,,k ' (177-5) since the right member is determined by the spatial equation (177-3) alone, it is independent of our choice of the parametrization (177.1). That is, alt possible velocities u of the moving surface have the same normal component un, which is called the speed of displacement of the surface1. Cf. Sect. 74. For some purposes it is convenient to make the particular choice of surface points implied by requiring u to be normal to the surface; ( 177.6) This velocity will be called the normal velocity of the surface. The identification of surface points may be visualized by erecting normal vectors of magnitude un dt from each: point on the configuration of the surface at some one time t; the termini of these vectors then sweep out the configuration at time t +dt. When un = f(t), the surfaces so generated are parallel surfaces. Suppose now that we have any parametrization X =X (v, t) ( 177.7) which is consistent with (177.3). Then both of these equations may be used simultaneously, so that (177.3) becomes f(x(v, t), t) =0. Differentiation with respect to t yields of oxk . Te+ l,k8t = o. (177.8) From (177.5) it follows that {177.9) Conversely, (177.5) follows from (177.9), so that these two equations furnish equivalent definitions of the speed of displacement according as the representation (177.3) or (177.7) for the surface is preferred. The parameter v in (177.7) may, but need not, be identified with what was called a surface point V above. In any case, given a particular parametrization (177.7), an observer moving with the velocity (177.6), which we have called the normal velocity of the surface, will encounter points on the surface (177.7) having surface co-ordinates v which vary in time. Their rates of change ur, which we shall call the tangential velocity of the parametrization, may be calculated 2• Such a velocity must satisfy oxk r . .k - k 81 + u X";"r- unn . (177.10) Taking the scalar product first by nk and then by gkmx7'LI yields (177.9) and 3 (177.11) l This quantity was introduced by STOKES [1848, 4, p. 353], who called it "the speed of propagation"; cf. also KELVIN [1848, .5]; for general surfaces it first appears, unnamed, in the work of CHRISTOFFEL [1877, 2, § 1]; also HUGONIOT [1885, J, p. 1120] first called it "vitesse de propagation" but immediately thereafter [1885, 4, p. 1231] distinguished "deux vitesses de Propagation", the other being that we consider in Sect. 183. The term "vitesse de deplacement de l'onde" was introduced by HADAMARD [1901, 8, § 1], [1903, 11, ~ 100]. 2 THOMAS [1957, 1.5, Eqs. (52) to (54)]. 3 Cf. ( 177.2). Eq. ( 177 .10) thus furnishes a resolution of the particular choice of surface velocity specified by ( 1 77 .6). 500 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 178. Conversely, if we choose to define un and ur by (177.9) and (177.11), we have by (App. 21.4h oxk r k - ()zk rLl oxP m _ _k ) -----;)~ + u X; r - -----;)~ - a gp m ----a"t X; Ll x;-r (177.12) - oxk oxP ( mk m k) - -----;)~- gpm ----a"t g - n n , whence (177.10) follows. A necessary and sufficient condition that the parameter v correspond to a surface point V for the normal motion (177.6) is Ur=O. 178. Differential description of a moving surface1• Supposing that a, b, un, and ur are given as functions of v and t, we ask if there exists a representation (177.7) suchthat these quantities belong to it. In other words, in order to define a moving surface does it suffice to assign as functions of t and the parameters v its first and second forms, its speed of displacement, and the tangential velocity of the parametrization? The answer, in general, is negative. In addition to the Eqs. (App. 21.8) of MAINARDI-CODAZZI and GAUSS, it is necessary that other conditions of compatibility be satisfied. First we differentiate (177.11), obtaining ( ox~Ll m + oxk m ) Ur;Ll =- gkm Be X;r -----;)~ X;rLJ · (178.1) By (App.19.7) and (App. 21.6h it follows that oarLl b -o-t-+2u(r;Ll>=- 2un rLJ· (178.2) From the definition of the total covariant der~vative it is easy to verify that ( oxk;r) = ox~rLl ~ ~{A} ot ;Ll ot + ;A ot Llr · (178.3) By (App.19.6h we then have (178.4) We now differentiate (177.9) and by use of (178.4), (App.21.6) 2 , (App.21.7h, (App.21.6)1 , (177.11), (177.9), and (App.19.7) obtain oxk OX~Ll) OX~rLl Un;rLJ = nk;I'Ll-----;)1 + 2nk;(r-8-t-+ nk-8-t -, OXk (bA m +bAb m) = -gkm -----;se r;LI X; A r ALl n - A 0 X~ Ll) 0 ( k b ) - 2gkmx~Abcr- ~+nkßt n rLI, (178.5) - bA bA b - bA oaLI)A obrLI -UA r;LI-un r ALl (r-o-t-+-8-t-. From (App. 21.8)1 and (178.2) it follows that obrLI Ab bA bAb - 0-t-+u FLI;A+ (I'ULI);A=Un;rLl-Un r ALl· (178.6) 1 The results ( 178.2) and ( 178.6), though not the proofs given here, were disclosed to us by J.L. ERICKSEN. Sect. 179. The displacement derivative. 501 Equations ( 178.2) and ( 178.6) are conditions of compatibility to be satisfied by a, b, un, and ur. ERICKSEN has shown that conversely, if the quantities a, b, un, and ur satisfy (App. 21.8), {178.2), and (178.6), then they are derivable from a relation of the form {177.7) with an assigned spatial metric g; that is, the conditions of compatibility here derived are also sufficient for the existence of a moving surface. Therefore any other condition satisfied by a, b, un, and ur will be a consequence of the relations already derived. 179. The displacement derivative. Given a function F(v, t) defined upon the moving surface, its rate of change tJFjtJt as apparent to an observer moving with the normal velocity (177.6) of the surface is1 !_!___=~+ ro F IJt at u r ' (179.1) where ur is the tangential velocity of the parametrization, given by (177.11). Suppose that G(x, t) be a function such that on the surface j we have G(x (v, t) t) = F(v, t). Then aF ac axk -81 = 81 +-8TokG, 8rF=x7rokG. {179.2) Hence by (177.10) we have ( 179-3) If Fand G are tensors, tJFjtJt and tJGjbt as defined by (179.1) and (179-3) 3 generally fail to be tensors. For a double tensor WL:~~·.:·.~(x, v, t), we define the displacement derivative (Jd 'l! jbt as that double tensor under the group of transformations x*=x*(x), v*=v*(v, t) which reduces to bWjbt when the spatial co-ordinates arereetangular Cartesian and the tangential velocity ur vanishes. To calculate (Jd lj! JM, we first introduce the "Lie derivative" f: lj! when any spatial indices of 'l! or dependence of 'l! upon x is ignored, viz. " f: lj!k ... mr ... Ll _ u<~> 0 lj!k ... mr ... Ll _ lj!k ... m

+ .. . p ... q

lj!k ... m r ... Ll lj!k ... m

+ .. . p ... q <~> ... r ,A · ( 179.4) Then we have {179.5) where both x and v are held constant when olj!jot is calculated. Forthis formula to be meaningful, it is not necessary to use any equation x = x (v, t) for the surface whose normal and tangential velocities are un n and ur. To verify its correctness, however, let us eliminate x; by {177.10) we have ~ bd '*' - aw ( oxk r k ) - 81 +w,k at+u x;r +;w, = !'*'_ + oxk oklj! + ~_x~ [wq ... { p} + ... ] + t:w +urx~r'l! at at at ... k q .. , , k' {179.6) = -~j + fW + ~ [wq ... { P} + ... ]. ot V=COnst U Ot ... k q . ------ 1 This definition, stated in words by HAYES [1957, 7, p. 595], seems to give the sense intended also by THOMAS [1957, 15, § 4]. 502 C. TRUESDELL and Ro TouPrN: The Classical Field Theorieso Secto 179° where ~ IV is obtained by replacing " ~" by "0 ~" in ( 179.4) 2 o When ur= 0 and .. ' ' the spatial co-ordinates arereetangular Cartesian, by (179.6)3 and (17901) we have d~: = ~~~v=const = ~~ · (179.7) Since the right-hand side of (179o5) is a double tensor under transformations ~· =~* (~). v* =v* (v, t), it fumishes the required expression for the displacementderivative in all co-ordinates systems. Fora spatial tensor IV~:::;'(~. t), (17905) reduces to Öd't' i.l't' " -Ü- = -fJt- + IV,k Un n , (179°8) and it is this form that is most useful. From (179.8) we have at once ddgkm = 00 t5t ' (179°9) hence raising and lowering of spatial indices commutes with tJdjllto Not so, however with surface indices, for the conditions of compatibility (17802) and (17806) assume the forms (179o10) By differentiating the relation ar.:~ a.:!A =ll'J. we see that (179o10h is equivalent to ddar.d - + 2 br.:! dt - Un • (179°11) Also Öda - Te=-2UnaK, (179.12) whence it follows that tJdajtJt =0 is a necessary and sufficient condition that a moving surface be and remain a minimal surface. From (179.11) and (179o10) 2 we have ~ ddb~ --Un ;r ;.1 + Un br.:! b A.d> l Ödbr.d - ;r.d+ 3 v.:~ b.:! - 15-t-- Un Un A• (179°13) From these results and (Appo 21.10) it is easy to show that ~ ödR (K-2 K) + or =Un -2 Un;r' , ~ ddK- KK-+ (K- r.:~ br.:!) -Un a - Un;r.:J, (179.14) ddb - - ~ =-UnaK K + a(K ar.:~_ bF.:!) Un;r.d· ddnk We now calculate 1 ~· From the relations (Appo 19.6)1 and (App.21.3)1 it follows that (179.15) 1 The result is stated by HAYES [1957, 7, Eqo (22)] without proof; a proof different from ours is given by THOMAS [1957, 15, Eqo (60)]o Sect. 180. Kinematical condition of compatibility. 503 By differentiating (177.9) we obtain (179.16) where we have used (177.10), (App.19.6) 1 , and (179.15) 3 . Since (177.9) presumes that X =X (v, t), the time derivative ojot is taken Oll the understanding that X is eliminated. Hence Ar k A q p bE r A k ( kq k q) 0 n,l ) a X;AUn;r=U gqpX;AX;E ra X;A- g -n n 81-, ~ k ( 179.17) - A bA k un -U AX;A-7)t• . where we have used (App.21.6b (App.21.4) 1 , (App.19.7), and (179.15) 1 . By (A pp. 21.6h it follows that ( 179.18) where, as stated above, x has been eliminated before onkjot is calculated. By use of (179.6) 2 we thus obtain the desired formula: (179.19) For an alternative derivation, we may suppose that ur= 0 and the spatial Coordinates are reetangular Cartesian; most of the terms in the above calculations are then absent, and we quickly obtain a formula recognizable as the appropriate special case of ( 179.19), which is a tensorial equation. From (179.19) and (App.19.7) we have ( 179.20) a result that might have been expected directly from the definitions, since it asserts that the length of the projection of the displacement derivative of the normal onto a given direction on the surface is just the negative of the gradient of the speed of displacement in that direction. In particular, a necessary and sufficient condition for parallel propagation is un = f (t), as was already remarked in Sect. 177. III. Kinematics of singular surfaces. 180. Kinematical condition of compatibility. We now consider a moving surface d (t) which divides a varying region Bi+ (t) from another, :?~- (t). The moving surface is assumed to satisfy the conditions stated in Sect. 177 and at each instant t to be a singular surface with respect to a quantity \V, as defined in Sect. 173; the conditions laid down for \V are now supposed to hold for each t. Assuming also that the limiting values \V+ and \V- are continuously differentiable functions oft in Bi+ and &l-, respectively, we derive a condition that the discontinuity in \V persists in time rather than appearing and disappearing at some particular instant. In a general space this temporal persistency is expressed by superficial conditions analogaus to those discussed in Sect. 175 but applied when one more dimension, that of t, is added both to the surface and to the space in which it 504 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 180. lies, i.e., we need only regard f(~, t) =0 as a single surface in the ~-t-space. In this degree of generality, nothing new results. In a metric space, however, the existence of a definite speed of displacement un for the moving surface makes possible results of a more concrete kind. The essential step, again, is fumished by HADAMARD's lemma (174.1), butthistime we apply it to a particular tangential path on the n-dimensional surface f (~, t) = 0 in the n + 1-dimensional ~-t-space, namely, the path tangent to the vector un n, 1. The derivative 1 on the + side of the surface is the quantity öw+Jöt as defined by (179.1). Thus (180.1) where, as in previous formulae, okw+=(okw)+. Writing a similar equation for the other side of the surface and subtracting the result from ( 180.1), we obtain the kinematical condition of compatibility 2 : [~~] = -un[nkokW] + :t [W]. (180.2) The jumps occurring on the right-hand side are those occurring also in the geometrical conditions (175.11). Thus the fumps of the derivatives 8kW and 8Wf8t across a persistent singular surface are determined by the quantities un, [ nk 8k W], and [W]. A condition equivalent to (180.2) but expressed in terms of tensors is easy to obtain by using the displacement derivative (179.5). Considering a spatial tensor field W k.. · m we ha ve p ... q ' [o'ii] _ k t:5d 8t --un[n W,k] +Tt[W]. (180.3) This is so because ( 1) it is a tensorial equation and (2) when the spatial co-ordinates are reetangular Cartesian, it reduces to (180.2). In the important special case when W is continuous, (180.2) reduces to the form ( 180.4) Thus, in particular, across a stationary surface that is singular with respect to 8k W but not with respect to W, the time derivative 8W Jot is · continuous, as is evident also directly from the definitions. We may write (175.8) and (180.4) as the system [W,k] = Bnk, [~~] =- unB. (180.5) 1 There are n independent paths at any one point of d (I). but we use only a particular one. If all n are employed, we may derive the geometrical and kinematical conditions of compatibility simultaneously, as is done in a special case by HADAMARD [1903, 11, "if97] and more generally by CouLON [1902, 3, § 46]. We prefer separate treatment of the two sets of conditions so as to separate the underlying ideas. In many cases useful in continuum mechanics, the geometrical conditions are satisfied when the kinematical is not; e.g., at the instant a portion of material splits in two, or two parts are joined together. 2 The essential content of this condition seems to be contained in the "phoronomic conditions" of CHRISTOFFEL [1877, 2, § 7], but these are not easy to use, and certainly the general concept of compatibility is due to HUGONIOT. First [1885, 3, p. 1119] he used "propagation" to mean compatibility, but soon thereafter [1887, 1, §§ 3. 5] he introduced the terms "compatibilite" and "conditions de compatibilite". HADAMARD [1903, 11, "if97] and LovE [1904, 4, § 7] obtained the system (180.5); HADAMARD's term is "conditions de compatibilite cinematique". The full condition (180.2) is implied by the results of CouLON [1902, 3, § 46]; cf. also KorcHINE [1926, 3, § 1]. Our presentation follows THOMAS [1957. 15, § 4]. Sect. 181. Iterated kinematical conditions of compatibility. 505 Of all the forms of the conditions of compatibility, it is these, which presume W itself to be continuous, that are most often used. If we square (180.5) 2 and use (175.13)1 , we obtain1 ~] 2 _ 2 2 _ 2 ,k [ at - un B - un [W,,.][W ], (180.6) whereby the magnitude of the speed of displacement is shown to be the quotient of the magnitude of the jump in 8Wj8t by the magnitude of the jump in W k· This last result is expressed in terms and notations appropriate to the case wh~n W is a scalar, but generalization is easy. Since HuGONIOT's time 2 it has been stated that a singular surface upon which the kinematical condition of compatibility does not hold will instantly split into two or more singular surfaces or will become singular with respect to a different quantity, such as a derivative of W. To substantiate a statement of this kind, the theory of some particular material is needed. In the generality maintained here, all that can be said is that the singular surface will not persist. 181. Iterated kinematical conditions of compatibility. Amplifying the notations (176.1), set · =[~;], B1=[n,.a;;k]. (181.1) We may then write (180.3) in the form A l =- B 6dA ( ) Un +~· 181.2 By replacing W by 8Wj8t in (176.2h and (181.2) we obtain [ ----a"t aw, k] - _ BI n,. + g,.ma 4 X;4 ;r• r m A 1 l [ß2'l/] 1 ddA (181.3) ---ai2 = - Un B + (ft . By (181.2), the quantity A 1 is already expressed in terms of A and B. We now obtain a like expression for B1 • To this end, we differentiate (176.1) 3 and use ( 176.2)1 , obtaining ddB _ 4 r m A '6dnk + k 6d['ll,k] (181.4) 6t - g,.ma X;4 ;r 6t n 6t • Now by another application of the argument leading to (180.1) we can show that :t (o,.w+)=(:t akwf+unnmomakw+, (181.5) since the various derivatives are assumed continuous in f!Jt+; consistently with our practice, we have put 8m8kW+=(8m8kW)+. Hence 6d - [aw,k] m[ Tt[W,k]- 8t- +unn W,mkl· (181.6) Substitution of this result into (181.4) yields a result which when simplified by use of (176.1) 6, (181.3) 1 and (App.21.3) 2 becomes BI_ C 6dB 6dnk 4r m A ) --Un +~-gkm~a X;4 ;F• =-unC+~~ +un;rA;r• where the latter form follows by use of (179.19). 1 DUHEM [1900, 3]. 2 [1887, 1, § 14]. Cf. HADAMARD [1903, 11, 'if 108]. (181.7) 506 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 182. The identities (181.3), with A' and B' replaced by right-hand sides of (181.2) and (181.7), are THOMAs's iterated kinematical conditions of compatibility 1• In the case when \V is continuous, we have A = 0, and the conditions ( 181.3) reduce to the simpler forms ( 181.8) When not only \V but also o\Vfot and ok\V are still simpler, for (181.8) and (176.10) yield 2 continuous, the results are where [okom\V] =[\V,km] =Cnknm, l [a~kt\V] = [a;;k] = -unCnk, [~~~] = u~C, (181.9) (181.10) More generally3, if \V and all its derivatives of orders 1, 2, ... , p- 1 are continuous, we have [ok, ok, ... ok, otP-s 0P-s \V] = [ otP-s 0P-s \V,k,k, ... k,] l (181.11) = (- un)P-s [nm, nm, ... nmp om, o~, ... omp \V] nk, nk, ... nk,, = (- un)P-s [nm, nm• ... nmp \V ,m,m, ... mp] nk, nk, ... nk,, the result being valid in any metric space. In particular, choosing s =0 we have (181.12) interpretation of which yields the Hugoniot-Duhem theorem4 : The speed of displacement of a singular surface across which \V and its derivatives of orders 1, 2, ... , p -1 are continuous but at least one p-th derivative of \V is discontinuous is determined up to sign by the ratio of the jump of (}P \V/ o tP to that of the fully normal p-th derivative, dPWfdnP. IV. Singular surfaces associated with a motion. 182. Material and spatial representations of a surface. So far in this chapter our considerations have been independent of the motion of any material medium. We now suppose that a medium consisting of particles Xis in motion through the space of places x according to (66.1). For the time being, we shall assume 1 [1957, 15, § 6]. THOMAS obtains also an alternative form for orA'; see his Eq. (51) as corrected. As regards the history of these conditions, remarks similar to those at the beginning of Sect. 176 may be made. 2 HUGONIOT [1885, 4, p. 1231], HADAMARD [1901, 8, § 2]. 3 DuHEM [1901, 6], CouLoN [1902, 3, § 46], HADAMARD [1903, 11, '1!97]. 'Given in the special cases P=1 and P=2 by HUGONIOT [1885, 3, p.1120] [1885, 4, p. 1231], in general by DuHEM [1900, 3] [1901, 7, Part II, Chap. li, § 2], but expressed by means of Laplacians (cf. (176.11) and (181.10)) rather than normal derivatives. Sect. 182. Material and spatial representations of a surface. 507 that the functions occurring in (66.1) are single-valued and continuous; modifications appropriate to motions suffering discontinuities will be given in Sect. 185. We consider a surface d(t), given by a representation of the form (177.3). and we set F(X, t) = f(x(X, t), t), so that f(x, t) = F(X(x, t), t), (182.1) identically in x, X, and t. Alternative representations of the moving surface are thus 1 f(x,t)=O, F(X, t) = 0. (182.2) In the latter representation, which we denote by Y (t), we may conceive the particles as stationary and the surface Y (t) moving amongst them, being occupied by a different set of particles at each timet. The two representatives (182.2) are the duals of one another in the sense of Sect. 14. The analysis given earlier in this chapter did not presuppose any particular choice of Co-ordinates, so long as they be independent of time, and is equally applicable to both representations. It is easier to visualize in terms of the spatial variables x, t, and from now on we agree to regard all the foregoing equations as so expressed; where we wish to employ a material counterpart, we shall invoke the principle of duality. Thus in the special case when ( 182.2)1 reduces to the form f(x) = 0, (182.3) weshall say that the surface d is stationary; when (182.2) 2 reduces to the form F(X) = 0, (182.4) that Y is material 2 [cf. (73.4)]. In the former case, the surface consists always of the same places; in the latter, of the same particles. Although (182.2)1 and (182.2h are but different means of representing the same phenomenon, the two surfaces so defined are, in general, entirely different from one another geometrically. The surface f(x, t) =0 is a surface in the space of places, while the surface F(X, t) = 0 is the locus, in the space of particles, of the initial positions of the particles X that are situate upon the surface f (x, t) = 0 at time t. Such connections as there are must be established by use of the transformations (182.1). For example, from the assumption that f(x, t) =0 has a continuous normal it follows that F(X, t) =0 also has a continuous normaP. We assume, in fact, that (182.2) are sufficiently smooth astopermit any number of differentiations and functional inversions. The theory we construct is local. Some aspects of the foregoing theory, whil,e not losing their validity, lose their intuitive appeal when applied to the material variables. The shape of Y (t), including its first and second differential forms, and its unit normal, have no immediate interpretation, for they do not correspond to any geometrical properties that an observer of a singular surface in space would perceive. The material representation, rather, is of the nature of a diagram for the moving surface. It is only one of many such diagrams, for by choice of the initial instant, or of the co-ordinates or parameters X corresponding to given initial positions, the particular functional form that results from ( 182.1 )1 will differ 4• As an example of the above remarks we consider the unit normals n and N to f = 0 and F = 0, given by 1 HUGONIOT [1885, 4, p. 1231]. 2 French: stationnaire. (182.5) 3 This is proved under weaker assumptions by LICHTENSTEIN [1929, 4, Chap. 6, § 1]. 4 The effect of such changes is discussed somewhat by HADAMARD [1903, 11, ~~ 79 to 84]. 508 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 183. where, as usual, f•m=gmqf,q and p,ß=gßYF,y. The former has an immediate geometric significance as a unit vector normal to the surface d (t) in the space of places x. The latter, dual to the former, applies only to one of many possible diagrams for the moving surface in various possible spaces of particles X and has no immediate interpretation. Since F,a. = f,k x~ rx.• by the dual of (17.3) we have 1 F,rx. p,a. = I xfXI 2 gßY. t ekrp eß6e f,k x~, Xfp. t emsq eycr,l,m xfs X?q· (182.6) a result which in a common frame assumes the form VF.a.F,a. =I zfZI ( 8(/, Y,Z) )2 + (8(X,f,Z))2 + (8(X, Y,"7))2 a (x, y, z) a (x, y, z) a (x, y, z) . For the unit normals themselves we have the relations ( 182.7) F a.X'\ VFpF.ß n = · · = Na.xrx. · • (182.8) k Vt,mf'm ;k Vt,mf'm' where VF.pF,ß and X~k are thought of as expressed in terms of spatial gradients by means of ( 182.6) and ( 17 .3), respectively. It is a natural requirement that the moving surface d (t) shall have a continuous and nonvanishing gradient vector f,k· Such a requirement if put upon the gradient vector F,rx. of .'7' has no immediate appeal. From (182.7), we see that F,rx. if continuous can vanish at a point if and only if either the radical or the Jacobian on the right-hand side vanishes. For the radical to vanish in a neighborhood, it is necessary and sufficient that f be functionally independent of X, Y, Z: that is, by (182.1), that F=F(t), and such an equation cannot represent a surface. Thus F rx. vanishes only with the Jacobian I zfZI. By the Axiom of Continuity in Sect. 65, it follows that for a material diagram F(X, t) = 0 obtained from a surface f(z, t) = 0 in a continuous motion, Fa. does not vanish except possibly at isolated points or lines. ' 183. Speeds of propagation. Waves. The dual of the speed of displacement, defined by (177.5), is the speed of propagation 2 UN: aF at UN== ---==· VF.a.F•" ( 183 .1) This speed is a measure of the rate at which the moving surface 9'(t) traverses the material. In particular, a necessary and sufficient condition that F = 0 be a material surface in an interval of time is that UN = 0 throughout the interval. A surface that is singular with respect to some quantity and that has a nonzero speed of propagation is said to be a propagating singular surface or wave 3 • Now it is evident that the value of UN for a given surface f(x, t) =0 in a given motion, unless UN = 0, depends in general upon the choice of the instant regarded as the time t = 0 for the motion of each particle 4• Thus there are infinitely many different speeds of propagation. Often it is most convenient to take the instant t=O as the present instant. Then Xrx.=6~xk, 8f8Xa.=6~8f8xk, and the functional forms of F and /, at this one instant and qua functions of x or X, are the same, but of course the time rates calculated with x held constant do not generally coincide with those calculated when X is held constant. In particular, with this choice of X the speed of propagation UN wiii be written as U and called the local speed of Propagation 5 of the surface. This speed, which 1 HADAMARD [1903, 11, '\[82, footnote], TRUESDELL [1951, 35]. 2 This quantity, for a general surface, first appears in the work of CHRISTOFFEL [1877, 2, § 1]; it is the second of the "deux vitesses de Propagation" introduced by HuGONIOT [1885, 4, p. 1231]. Cf. also HADAMARD [1901, 8, § 1] [1903, 11, '\[98]. 3 HADAMARD [1901, 8, § 1] [1903, 11, '\[91]. 4 HUGONIOT [1887, 2, Part2, § 7], HADAMARD [1903, 11, '\[99]. 6 HuGoNroT [1885, 3, p. 1120], "vitesse de propagation rapportee au fluide lui-meme". RANKINE [ 1870, 6, § 2] in dealing with a one-dimensional case called U "the linear velocity of advance of the wave". Sect. 184. Boundaries. 509 is the normal speed of the surface with respect to the particles instantaneously situate upon it, is related to the speed of displacement as follows: at oF U 7ft= UnTt• {183.2) More generally, we have from {182.5), {183.1), (177.5). and (74.1) the alternative forms UN V F,a.F;a. = Un Vf,k /•k- ik /,k l = (un- Xn) Vf,k /•k =-I (183-3) (d. (74.6)). If we choose the present configuration as the initial one, {183.3)3 becomes U=---i Vt,k f.k ' while {183.3) 2 reduces to the moreelegant form1 U = Un-Xn, (183.4) {183.5) expressing the evident fact tha:t the normal speed at which the particles now comprising d (t) are leaving it is the excess of their normal speed over the normal speed of the surface. 184. Boundaries. Recalling that a body P4 is a set of particles X having positive mass, we define its boundary 2 at time t as the set of places x whose every neighborhood contains two places distinct from x, one of which is occupied by a particle of P4 and one is not. In kinematical terms, the bounding surface is adjacent to P4 but not crossed by any particle of P-4. In general, it is a moving and deforming surface. While an axiom of continuity was laid down in Sect. 65, we now replace it by the weaker requirement that the motion (66.1) be a topological transformation, i.e., a transformation that puts open sets in a region of X-t-space into one-to-one correspondence with open sets in x-t-space. In particular, for each fixed t the transformation of X into x will then be topological. Hence the boundary of a set in X-space is mapped, at each t, into the boundary of the corresponding set in x-space. Therefore, the boundary surface of every body in a topological motion is a material surface 3• Conversely, any material surface permanently divides the material (if there is any) on one side from that on the other, and thus consists in boundary points of the bodies (if any) upon each side. From these results and LAGRANGE's criterion in Sect. 74, it follows that in a continuous motion, a necessary and sufficient condition that a surface f = 0 be a portion of the boundary of the material ( if any) instantaneously lying upon either side o f it is i=o. ( 184.1) When the motion fails to be topological, these results hold no longer, and boundaries may be instantly created or destroyed. General transformations 1 HADAMARD [1903, 11, ~ 100] but given in effect by HUGONIOT [1885, 3, p. 1120] in a sentence which is confusingly mispunctuated. 2 Same properties of boundaries have been given in Sect. 69. 3 HADAMARD [ 1903, 11, ~ 48] uses the differentiability of the motion to prove this result. The result was asserted by LAGRANGE [1783, 1, §§ 10-11] (cf. Sect. 74), but his discussion of a bounding surface is insufficient. 510 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 184. are not of interest in field theories. There are important cases, however, when motions fail tobe topological at one certain instaut only, or upon certain isolated surfaces, curves or points. An example of the former is fumished by the fracture or welding of a solid (Fig. 25) or by the formation or coalescence of drops in a fluid. In these cases, 0 0 0 0 Fig. 25. Fracture and welding. Fig. 26. Tear. interior particles suddenly find themselves on the boundary of two disjoint motions, or boundary particles suddetlly become interior ones. A still more complicated singularity is a tear such as that shown in Fig. 26, where the singular line .'!/ is propagating into the material, splitting each particle X in its progress Anyconlinuous moltun a c Prisms rolling upon one onoll!er Anyconlinuous motion b d Fig. 27 a-d. Examples of non·topological motions. into two particles x+ and x-, one of which stays upon the boundary fJB+ and the other upon the boundary !!4-. Singularities located upon lines seem not to have been studied from a general viewpoint. The latter type of nontopological motion allows us to represent motions in which particles enter and depart from a boundary surface, possibly quite smoothly along tangential paths1 . Let Fig. 27 (a) represent, say, a rigid cylinder or a fluid vortex spinning just below the plane free surface of a fluid and tangent at the top, and let the region outside the cylinder be endowed with any topological motion such that the cylinder and the plane constitute its boundary 2• The combined motion fails to be topological at the line of tangency, and we may say that particles on the cylindrical stream surface continually rise into the plane boundary and fall away from it again. In this example, however, as in all others formed by piecing together topological motions upon portions of their boundaries, the 1 PorssoN [1831, 2, § 12] [1833, 4, § 652]. 2 This example and that in Fig. 27 (d) were given by KELVIN [1848, 5]. Sect. 184. Boundaries. 511 complete boundary, consisting of the union of the constituent boundaries, is again a material surfacel, though a material surface on which the motion fails to be topological. Other examples are shown in Fig. 27. In the case of motions that fail tobe continuous in the sense defined in Sect. 65, the condition ( 184.1) is neither necessary nor sutficient that I= 0 be a material surface or consist in boundary points of the material, if any, that it instantaneously separates. From (183.3) it is plain that UN =0, for all choices of the initial configuration, is equivalent to f=o provided F "F·"=I=O. Now F "F·" is not determined by the instantaneous shape of the mo~ing surface alone, but is influenced also by the motion. The two effects are in some measure separated in the identity {182.7). The quantity under the root sign on the right assumes the values 0 or oo if and only if the equation F(X, t) = 0 reduces to the form F(t) = 0, which does not represent a surface. Thus F,"F·", for a surface F =0, is singular only with Jz/ZJ; if the Axiom of Continuity in Sect. 65 holds, from {183.3) we thus read off a formal proof of LAG RANGE' criterion. More generally, by ( 156.2) we have VF.(XF·" oc ~, and hence UN eo oc (! f. (184.2) We consider particles suchthat eo=I=O, oo. Then from (184.2) 2 we conclude thatl: 1. II (! = oo upon the surlace I =0 and UN=I= oo, then (184.1) is satisfiea, but the surface may or may not be material. 2. II e =0 upon the surface I =0 over an interval of time, then the surface is material, whether or not (184.1) is satislied. The condition {184.1) remains sullicient, but not necessary, that I= 0 be a material surlace. 3. Bothin Case 1 andin Case 2, the condition (184.1) is necessary, but not sullicient, that the surlace I =0 be an admissible boundary. The case when e = oo upon f = 0 is illustrated by the following example 1 : x=X-ct, y=(Y}-kt)3 , z=Z. (184.3} Since ( 184.4) the velocity field is plane, single-valued, steady, and irrotational 2, and the stream lines are the similar cubical parabolas y = [: (x- X)+ y~r z = z. {184.s) which cross the y = 0 plane tangentially (Fig. 28). The density is given by _I!Q_ = o(x, y, z) = (1-)i e o(X, Y,Z) Y ' so that I!= oo upon the plane y = 0. For this plane we have the equations I = y = F = ( y!.- k t) 3 = o, and hence (184.6} (184.7} {184.8} Although (184.1) is satisfied, the plane y=O is neither a material surface nor a boundary, since the particles are continuously crossing it. Neither is y = 0, while indeed a persistent surface of discontinuity, a singular surface in the sense defined in Sect. 180, since no jump discontinuity occurs across it. 1 TRUESDELL [1951, 35]. 2 Hence this motion, despite its artificial appearance, is dynamically possible (in theory) for an ideal gas subject to no extrinsic force; the motion is tobe conceived as occurring in a channel bounded by cylinders erected upon two of the curves shown in Fig. 28. 512 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 185. The speed of displacement, Un, is zero, since the plane y = 0 is stationary. The speed of propagation UNis given by (184.9) Thus the surface y = 0 is suffering propagation with respect to all the particles except those instantaneously situate upon it. The local speed of propagation is zero, U = 0, but for no other choice of the initial instant or of material !I variables can UN vanish. These observations are X Fig. 28. The plane y = 0 is not a material surface even though y = 0 upon it. also evident from Fig. 28. The nature of the discontinuity of the motion is made clearer if we calculate the distance d at time t between the two particles X, Y, Z and X, (Y!+n!) 3, Z, whereD>O. From (184.3) we have d = D + 3(YL kt) n! + 3 (Yl- kt) 2 n!. (184.10) Thus D is the distance between the two particles at the instant they cross the plane y = o. From (184.10) it follows that no matter how small is D, there exists a time t0 such that both after t = t0 and before t = - t0 the distance d is arbitrarily !arge. Thus no material volume remains bounded. 185. Slip surfaces, dislocations, vortex sheets, shock waves. The first kind of singular surface tobe used in continuum mechanics was the slip surface of HELMHOLTZ1. On such a surface, the inverse motion X =X(~. t) is discontinuous. Each place ~ is simultaneously occupied by two particles, X+ and x-. Two different masses thus slip past one another without penetration. The surface f (;r, t) = 0 is a material boundary of each motion. Fig. 29. Vortex sheet of order o. Fig. 30. Dislocation. A surface across which the velocity suffers a transversal discontinuity, [ :i:] =f= o, [in] = o , (185.1) is called a vortex sheet. Slip surfaces are vortex sheets; sometimes they are called vortex sheets of order 0. Such vortex sheets are easily constructed by placing adjacent to one another two motions having a common boundary. Unless it happens that [x] =0, that boundary will be a vortex sheet of the composite motion. Thus, alternatively, a slip surface may be regarded as a material surface on which the functions occurring in (66.1)1 are double-valued (Fig. 29). The dislocations of VaLTERRA 2 are singular surfaces intended to represent the deformation corresponding to removal or insertion of one mass within another, or the welding of boundaries (Fig. 30). There results a surface upon which X(~. t) is double-valued. Such surfaces need not be material; they may propagate, and they may bear any kind of discontinuity in the velocity. 1 [1858, 1, § 4], "diskontinuierliche Flüssigkeitsbewegung". 2 [1905, 6], "distorsioni". Cf. the example given in Sect. 49. Sect. 185. Slip surfaces, dislocations, vortex sheets, shock waves. 513 A shock surface1 is one across which the normal velocity is discontinuous: (185.2) Dislocations may be shock surfaces. Little of a generalnature may be said regarding such discontinuities, especially since they may represent removal or insertion of material. The considerations of Sect. 182 regarding the material diagram must now be modified, since to a single spatial surface f (:.c, t) = 0 there correspond two distinct diagrams F+=o, F-=o, viz., 0 =f(:.c,t) =F+(X+(:.c,t),t) =F-(X-(:.c,t),t), (185.3) the functions X+(:.c, t) and x-(:.c, t) being the two inverse functions to the singlevalued equation :.c =:.c(X, t) defining the motion of the medium. Thus the principle of duality fails to hold for these singularities. Moreover, we cannot apply the conditions of compatibility given in Sects. 175 and 180, since the function :.c (X, t) is not defined, in general, upon both sides of either one of the diagrams F+=o, F-=o in the space of particles. HADAMARD's lemmastill holds, however, and may be used to derive some meager information 2• For example, we can calculate the normal n in terms of the normals N+ and N-, using (17.3) and (182.8) with appropriate limiting values from one side only. Thus we may put the condition (185.1) 2 for a vortex sheet into the following form 3 : [xkl e ea.fly F± xm± xP± = 0. J kmp ,a. ,{J ,Y ( 185 .4) Each of the diagrams F+=o and F-=o has its own speed of propagation, u,: and U;J, given by (183.1) applied to F+ and to F-. Each of these speeds of propagation has the indeterrninacy described in Sect. 183. By using (185.3) we may repeat the analysis leading to (183-3) and so obtain ti+ J<+.a: · V"F-::-p-. a: • u,:Vfk ,k+x;=un=U;JVt~f.k +x~. (185.5) For this identity to hold, the derivatives X~a. and xk need not exist upon the singular surface; by HADAMARD's lemma, limit values from the + side of F+=o and the - side of F-=o, respectively, are employed throughout, it being assumed, as usual, that these limit values are continuously differentiable functions of position upon the surfaces. This relation connects the two different speeds of propagation, whatever be the choices of the initial configurations corresponding to the two diagrams F+=o and F-=u. We are at liberty to choose the present configuration as the initial one, both for the particles on the + side of F+ = 0 and for those on the - side of F-=o. By (185.5), the corresponding local speeds of propagation, U+ and u-. satisfy the relations 4 (185.6) 1 RIEMANN [1860, 4, §§4-5], "Verdichtungsstoß"; cf. STOKES [1848, 4], "a surface of discontinuity ". 2 E.g., the Counterpart of (175-5) is (xk a. Xa.LI)+ = (xka. Xa.LI)-, , J J J where the curves vr = const on F+ = 0 and F- = 0 are selected by identification of the points X+ and x- that occupy the same place x. lt does not seem possible to make any use of this relation. 3 BJERKNES et al. [1933, 3, § 18]. . 4 CHRISTOFFEL [1877, 2, § 1], HADAMARD [1903, 11, '1\102]. Handbuch der Physik, Bd. lll/1. 33 514 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 185. The jump in the speed of propagation of a singular surface is the negative of the jump in the normal velocity of the material, and a necessary and sufficient condition for the speed of propagation to be continuous is that the normal speed of the material be continuous, i.e., that the singular surface not be a shock surface. From (185.6) we read off the following theorems concerning shock surfaces and vortex sheets, subject to the assumption that x (X, t) be single-valued and continuous, though X(x, t), in general, is double-valued: 1. The local speed of Propagation cannot be continuous across a shock surface. 2. There are no material shock surfaces, i.e., shock surfaces are always waves 1 • 3· The local speed of Propagation of a vortex sheet is always continuous. In particular, a vortex sheet which is material with respect to the particles on one side is also material with respect to those on the other. In the case of singular surfaces where the motion itself, or the velocity, is singular, the dual of the kinematical condition of compatibility (180.3) must be modified. Retracing the steps leading to that condition, we first write the dual of (180.1) for paths on each of the two material diagrams that may represent the singular surface f (x, t) = 0, thus obtaining ~+w+ ='f++U.+N""8 w+ !5_w- =w-+U.-Nß8 wM N "" ' !5t N ß ' (185.7) where (J+j(Jt and (J_.((Jt as defined with respect to the normal velocities of the two diagrams in the space of particles X. Even when the two diagrams coincide, the speeds of propagation appropriate to the two sides are in general different, as already noted. We now choose the present configuration as the initial configuration, both for the + and for the- sides, and subtract the second of (185.7) from the first. Thus follows 2 ['f] = [~~]- [Unk8kW], = [~~]- u+[nkokw] -[U]nkokw-, = [~~]- U+[nk 8kW] + [.iJ nk okw-, ( 185 .8) where we have used (185.6). In these formulae it must be remernbered that on the two sides of the surface 'f is calculated on the basis of ~+ and ~-. while the normal velocities used in calculating (Jj(Jt are U+ n and u- n, respectively. 1 A shock surface may be material with respect to the particles on one side but not with respect to those on both sides. 2 A result of this kind is asserted by KoTCHINE [1926, 3, § 1], but his statement has !5[W]f!5t rather than [!5W/!5t] and thus is incorrect unless the operator !5/!51, referred to the material diagram, is continuous, i.e., unless the singular surface is a vortex sheet. Fora check on (185.8), put W =ik. The dual of (179.3) yields ~-X !5+xk - 'k+ + u+ '"" k+ N1V X,rx • When the present configuration on the + side is taken as the initial state, this formula reduces to o xk _+_=ik++ M u+nk· , hence [ ~ !5xk] = [.~k] + [U]nk. Therefore (185.8) is satisfied. Sect. 186. Material vortex sheets. 515 Whenx is continuous, (185.8) reduces to the dual of (180.2). Another special case will be discussed in the next section. 186. Material vortex sheets. In the case of a material singularity, irrespective of whether or not m(X, t) is continuous, we have U+= U-=o, andin virtue of the duals of {179.1) we may reduce (185.8) to the form (186.1) where d+fdt and d_jdt are the material derivatives calculated with x+ and xheld constant, respectively. Hence1 ['i!l = d+J:l +[x"]\V~", = d_[w]_ +[xk]\V+ dt ,k, = __!_ (~+_ + d-_) [\V]+~ [x"J (\V\+ \V-"). 2 'dt dt 2 • • (186.2) It was suggested by HELMHOLTZ2 that the velocity ol the vortex sheet be defined as the mean of the velocities on each side: (186.3) Only when the sheet is steady is this velocity tangent to it. Writing dfdt for the material derivative following this velocity u, we have (186.4) and (186.2) becomes (186.5) where we write grad \V == i (grad \V)++ i {grad \V)-, ( 186.6) etc. Eqs. {186.1) and {186.5) arealternative forms of the kinematical condition of compatibility lor material singular surlaces. In particular, if \V and grad \V are continuous, from (186.5) we have [ 'i!] = [x] . grad \V. (186.7) Since the singular surface has been assumed material, we have d+lfdt =d-lfdt =0, and hence [j] =0. Putting \V =I in (186.7) yields [x] · grad I= o. (186.8) Hence it follows that il the velocity is not continuous across a singular surface that is persistently material with respect to the motion on each side, that surface is necessarily a vortex sheet 3• Thus we have another proof that shock surfaces are always waves. 1 The first of these forms is given by KOTCHINE [1926, 3, § 4]. 2 [1858, 1, § 4]. 3 HADAMARD [1903,11, ~94]. 33* 516 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 186. For any material derivative, we have identically T;. = i k-x"' k Im. d+lfdt = 0, this equation yields d+f.ddt = -x"'+,k l,m, ~nd hence ' · Since d-f,k _ 1 ( 'm+ 'm- ) I ~dt- - -2 x ,k +x ,k ·"'' (186.9) or dgrad/ -~. ~(jj- = - grad;v · grad I. (186.10) Therefore, dn -d. . -d. ) -d1-=-gra X·n+n(n·gra ;v.n. (186.11) This result shows that the manner in which the unit normal to the material singular surface changes as viewed by an observer moving with the mean velocity u is definitely determined. The same is true of the jump in the acceleration, for if we put \!! =;r, in (186.5), we obtain (186.12) a result which bears a formal similarity to (98.1) 2 for the acceleration of a continuous motion. ( 186.12) is an iterated kinematical condition of compatibility for the special case of a material singular surface. It is a simple corollary that [or] =0 implies [.Z.] =0; that is, il the velocity is continuous across a material singular surlace, so is the acceleration1• We now interpret (186.12) more closely by resolving it in directions normal and tangential to the surface. From (186.12) 1 and (186.9) we have I ,k [xk] =I ,k ä[ dt xk] + _1_ 2 [x"'] (xk+ ,m + xk-) ,m I ,k' l - k - (186.13) =I d[i 1 _ [ 'k] df,k • k dt X dt ' By (186.8), we may write this result in the alternative forms 2 I ,k [xk] = -2 [ik] ät.k dt = + 21 . k aukl dt ' l [x] = 2n. ä[x]_ " dt ' (186.14) whence it follows that the normal acceleration is continuous if and only if the time rate of change of the velocity as apparent to an observer moving with the mean velocity is tangent to the surface. The more difficult reduction of the tangential component of ( 186.12) has been achieved by MüREAU 3• By (186.12) 2 and (186.11} we have 1 HADAMARD [1903, 11, '\[ 9+] has noted an interesting variant: By differentiating twice the equation i = 0, we conclude that [x nl = 0 if [x] = 0. That is, if at some particular instant the velocity is continuous across a material singular surface, then at that instant the acceleration may suffer a transversal jump but not a longitudinal one. The italicized result in the text above (due also to HADAMARD) refers to persistent continuity of ;i;. 2 KorcHINE [ 1926, 3, § 4]. 3 [1949, 19] [1952, 13, §Sc]. Sect. 187. General classification of singular surfaces. -fe (n X (;r]) = (x] X (grad X· n) - n X ([x] · grad x) + + (nx[x]) (n·gradi:·n) +nx[x], = (nx[x]). gradx + [x] (n. w)- - (n x [i:]) (div x - n · grad ;i: · n) + n x [x], 517 (186.15) where we have used vector identities and ( 186.8), and where w == curl x = ! (w+ + w-), the mean of the vorticities on the two sides. Now by (App. 21.4)1 we have ( 186.16) Thus the quantity on the left-hand side is the divergence, calculated intrinsically upon the surface, of the projection of the mean velocity onto the surface. If, therefore, we imagine on the surface itself a fictitious motion with the velocity field n x u, an element of area da that is carried by this motion will change according to the formula 1ft dda (-d. . -d. ) d = 1v x - n · gra x · n a (186.17) [ cf. (76.6), which holds in any metric space]. Putting ( 186.17) into ( 186.15) yields ~ (nx[i:] da)= nx[x] da+ (nx[x] da). gradx+[x] (n. w) da } [ =n '']d [ '] -. ( [. ) (186.18) x a+nx x da·gradx-(n·w) nx nx x] da. This is MoREAu's result. Its significance is easier to assess when we use EMDE's notation (footnote 3, on p. 492): W= Curl x, W* = Curlx, for then we have d(':tdt1_ = W* da+ W da. grad i: - (n · w) n X W da. (186.19) Except for the last term, this equation has the same form as BELTRAMr's vorticity Eq. {101.7) 3 , establishing an analogy with between the convection and diffusion of the vorticity w dv of a material element of volume in a continuous motion and the transport of surface vorticity W da in an element of area which is material with respect to the mean motion on the vortex sheet. In many cases of vortex sheets in fluid mechanics, the spatial vorticity w vanishes or is tangent to the sheet on each side; in such cases, n · w = 0, and the analogy becomes precise. The analogue of the circulation preserving case is W* = 0. For this case, as MOREAU remarks, superficial analogues of all the classical theorems such as the Helmholtz theorems on spatially circulation preserving motions exist. 187. General classification of singular surfaces. In Sect. 173 a singular surface was defined with respect to an arbitrary quantity \II. DuHEM1 proposed to regard all quantities associated with a motion as functions \II (X, t) of the material variables X, t and to define the order of a singular surface with respect to \II as the order of the derivative ( ~J aa, aa, ... aCI.p \j! of lowest order p + q suffering a non-zero jump upon the surface. Here, as in all that follows, we assume that in regions fll+ and f]l- on each side of the singular surface !/'(t) in 1 [1900, 3]. 518 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 187 the space of material variables X, the function \1! (X, t) and all its derivatives up to the highest order considered exist and are continuously differentia ble functions of X and t, while on f/(t) they approach definite limits which are continuously differentiable functions of position. Thus we may apply the geometrical and kinematical conditions of compatibility ( 175.11) and ( 180.2) and use differential manipulations freely. There is no compelling reason to allow only discontinuities of this special type. J ump discontinuities upon surfaces are not the only ones that occur in physical problems; e.g. in Sect. 184 we have examined a simple and otherwise smooth motion in which the Jacobian I :x:fXI increases steadily to oo as a certain surface is crossed and decreases steadily thereafter. Boundaries, studied in Sect. 184, and slip surfaces, dislocations, and tears, studied or mentioned in Sects. 185 to 186, are excluded as not being defined by sufficiently smooth jump discontinuities in functions of the material variables. Singularities at isolated lines or points are common; some of these are described in works on potential theory. In the case of jump discontinuities on surfaces, there is no a priori ground to expect that the limit values on each side of the surface be continuously differentiable on the surface, as we have assumed. The reasons for considering here only singularities of this kind are, first, that for more general singularities other than those analyzed above, scarcely any definite results are known except in very particular cases, and, second, that singular surfaces of the above types are frequently found useful in special theories of materials. Clearly the definition of the order of a singular surface may be expressed 1 . 1 . f h . d . . (q) N d'f' . a ternahve y rn terms o t e covanant envahves W;a.,a., ... a.p· o mo 11cahon in the results of Sects. 175 to 176 and 180 to 181 is needed to allow us to substitute double tensors of the type TL:~ ~:J in the various jump conditions. Many of the singularities of greatest interest are included in the case when \1! = :x: (X, t), (187.1) i.e., are surfaces across which the motion itself, or one of its derivatives, is discontinuous. By the order 1 of a singular surface henceforth we shall mean, unless some other quantity is mentioned explicitly, that we are taking \1! =:1:. Thus surfaces across which at least one of the functional relations (66.1) defining the motion itself is discontinuous are singularities of zero order; those across which some of the derivatives ik and x~a. are discontinuous are of first order, etc. In the classification of LICHTENSTEIN 2, the definition of the order is based upon the derivatives of the velocity field, ..C. Since i~m=X~a.X~m· a singular surface of order p in LICHTENSTEIN's scheme is also one of order p in that of DuHEM and HADAMARD, but the converse does not hold, for it is possible that a gradient such as x~a. may be discontinuous without there being any discontinuity in ik, etc. Thus the DUHEM-HADAMARD scheme includes a greater variety of singularities. Most researches on singular surfaces in fluids follow LICHTENSTEIN's classification, since in hydrodynamics it is possible largely to avoid consideration of the material variables. In retaining the DuHEM-HADAMARD classification we recognize its more fundamental scope 3 and its necessity in contexts such as the theory of 1 HADAMARD [1901, 8, § 1] [1903, 11, ~ 75]. 2 LICHTENSTEIN [ 1929, 4, Chap. 6, ~ 2]. 3 E.g., LICHTENSTEIN [1929, 4, Chap. 6, ~ 2] concludes that there are no material singularities of first or second order. While this is true according to his definitions, it is true only because those definitions offer no possibility of considering discontinuities in xka. and x\.ß, unaccompanied by discontinuities in time derivatives of x. This example illust~ates th~ insufficiency of LrcHTENSTEIN's scheme. Sects. 188, 189. Singular surfaces of order 1: Shock waves and propagating vortex sheets. 519 elasticity, but we take care to derive from it, among other consequences, the spatial formulae that have found use in hydrodynamics. In the following sections we find the kinematical properties of singular surfaces of finite order1. At a singular surface of order 0, the motion x = x (X, t) suffers a jump discontinuity. This must be interpreted as stating that the particles X upon the singular surface at time t are simultaneously occupying two places x+ and xor jump instantaneously from z- to x+. Such discontinuities have not been found useful in field theories up to the present time. Therefore, in what follows, we study singular surfaces of orders 1 and greater. 188. Material singular surfaces. Material vortex sheets have been studied in Sect. 186. The results derived there remain valid for material singular surfaces of all orders. Fora singularity of order 1 or greater, X (z, t) is continuous, d+fdt = d_fdt, and (186.1) reduces to [Ii!] =['V]. (188.1) This is the generat kinematical canditian af campatibility far material singularities af arder greater than 0. Its major use is to show that ['V] = 0 implies [ W] = 0: Continuity of 'V implies continuity of '!'. In other words, the derivative af lawest arder that is discantinuaus acrass a material singularity is always a purely spatial derivative, never a time derivative 2• This is the dual of the theorem stated just after (180.4). In particular, across a material singularity of first order, since x (X, t) is continuous, so is ~. That is, not only shock surfaces but also vartex sheets af first arder are waves, while for the material vortex sheets described in Sect. 186 it is impossible that the motion itself be continuous across them. Vortex sheets are thus divided into two distinct categories: those of order 0, which are material, and those of order 1, which propagate. Across a material singularity of first order, ~ is continuous, but at least one of the deformation gradients x~" suffers a discontinuity. 189. Singular surfaces of order 1: Shock waves and propagating vortex sheets. Forasingular surface of order 1, we put 'V = x!' in the duals of (180.5) and obtain 3 [~ .. ] = sk N .. , sk =[Nß ~p]. [X"]=- UN sk. (189.1) 1 A singular surface of infinite order is defined by HADAMARD [1903, 11, ~ 76] as one such that on each side, the function occurring in (66.1) aredifferent analytic functions, yet all their derivatives are continuous across the surface. Such singularities seem not to have been studied. They offer interesting possibilities. For example, a one-dimensional motion starting very smoothly from rest at t = 0 is furnished by x =X for t ~ 0, x =X+ f (X) e-c/1' for t > 0, where c > o. In works on physics we often encounter discontinuous solutions regarded as limits of continuousones; cf., e.g., the definition of adiscontinuity given by MAXWELL [1873. 5, §§ 7-8], [1881, 4, § 8]. Perhaps seeking to justify such a treatment, LicHTENSTEIN [1929, 4, Chap. 6, §§ 6-7] proves that any motion with a singular surface of order 1 or 2 may be obtained as a limit of analytic motions. Such theorems, however, do not reflect the physical situation. In electromagnetic theory, for example, physicists are wont to regard the solution of a problern for a material whose dielectric constant changes abruptly from K 1 to K 1 upon a certain surface as the limit of the solutions of "the same" problern for a dielectric suchthat K varies smoothly from K 1 to K1 in a thin layer containing the surface. In gas dynamics, a flow of an ideal gas with a shock wave is regarded as the limit of solutions of a corresponding boundary value problern for viscous, thermally conducting fluids as the viscosity and thermal conductivity tend to zero. Problemsofthis kind presuppose some definite theory of materials and have no meaning in the generality of the present treatise. Even in relatively simple definite cases, they offer the highest mathematical difficulty. Cf. Sect. 5. 2 HADAMARD [1903, 11, ~~93, 104). 3 HADAMARD [1903, 11, ~ 101]. 520 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 189. The vector s is the singularity vector; while (189.1) 2 shows it to be parallel to the jump of velocity, its magnitude varies with the choice of the initial state and thus does not furnish a measure of the strength of the singularity. Rather, guided by the result given in Sect. 188, it is convenient to divide singular surfaces of order 1 into two classes: 1. Material singularities, which affect only the deformation gradients ~"'. 2. Waves, including both shock waves and propagating vortex sheets. For the former, the choice of the initial state is of prime importance. For the latter, it is not, and the nature of the waves is best specified in terms of the jump ofvelocity itself, [:V], which may be arbitrary bothin direction andin magnitude. Indeed, if we adopt a strictly spatial standpoint, we may say the only geometrical and kinematical requirement is that discontinuities in velocity be propagated, both the amount of the discontinuity and the speed of propagation being arbitrary. Even here the adherents of a strictly spatial standpoint are closing their eyes to one of the phenomena occurring, since from (189.1) it follows that a fump ,·n velocity is impossible unless it is accompanied by fumps in the deformation gradients ~«· The results derived in Sect. 185, since they presume less regularity than is here assumed, remain valid for singularities of order 1. In particular, the speed of propagation satisfies (185.6), and the first and second theorems derived from it remain relevant. The third theorem is irrelevant, since, as just shown, vortex sheets of first order are always waves. Since the functions occurring in (66.1) are continuous and single-valued by hypothesis, there is only one material diagram corresponding to a given initial state, and the principle of duality now holds. However, it is necessary to proceed with caution, since the fact that xk« experiences a jump on " implies that if we regard a succession of initial states X corresponding to different initial times t0 , these states jump discontinuously at t0 = t ± 0. In particular, choosing the states on the + and - sides of the singular surfaces leads to different local speeds of propagation, u+ and u-, satisfying (185.6), as already stated. Writing the corresponding vectors s as s+ and s-, we have hence By (185.6) follows [Us]=O. un[s] =[ins]. In the case of a vortex sheet, this relation reduces to in the case of a stationary shock wave, to [ins] = 0. (189.2) (189.3) (189.4) (189.5) (189.6) The nature of first order shock waves is illustrated by the very simple example1 furnished by the one-dimensional motion defined by the equations y = Y, z = Z, and X= l X+ 2vt when X ~0, l X + v t when X :::;; 0, X + v t when X ~ 0, 2X + 2vt when X ~0, 1 LICHTENSTEIN [1929, 4, Chap. 6, § 8]. - oo 0, remain a distance D apart until the latter encounters the shock. After the shock has passed, their distance apart is t D. The shock thus effects not only a sudden drop in velocity but also a sudden condensation. We now prove that this is representative of the general case. To determine the jump experienced by a volume element through which a wave of first order passes, we calculate the jump in the Jacobian V?J/VG = xjX. By {189.1h we have By the dual of (17.3) follows _E_ -1 + PN XY- 1- - s y ,p· (189.11) Now we may apply (182.8) and (189.1) 3 , writing Uii to recall that the derivatives X~"k are used in the calculation2 ; using also (185.5) 2 we thus obtain (189.12) or, by (185.6), Xri-un u+ Xn -Un = U~' {189.13) Thus the passage of a shock wave of first order 3 causes an abrupt change of volume, the ratio being that of the local speeds of Propagation. 1 We are not obliged to use initial positions or reetangular co-ordinates as material co-ordinates. 2 The calculation may be shortened by taking the state on the - side of the surface as the initial state from the beginning. 3 The qualification, "of first order", refers to the possibility of shocks in which :r(X, t) is discontinuous. About such shocks, mentioned in Sect. 185, little definite can be said. 522 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 189. This result is usually presented in terms of the density e; from ( 156.2) then follows the Stokes-Christoffel condition: (189.14) These forms are of frequent use1 . From (189.14) we see that shock waves offirst order areimpossible in an isochoric motion, and that the passage of a vortex sheet offirst order leaves the volume unchanged. By combining (185.6)a and (189.14)1 we obtain the identity e±U±[xn]=-[eU2]=(U-) 2 ~= [e]. (189.15) Since, by hypothesis, J > 0, from ( 189.13) it follows that in- un is of the same sign on each side of the singularity. That is, if to an observer situate upon the surface the material seems to approach on one side, it departs upon the opposite side. The side on which the material approaches is called the front side, and the sign- will be assigned to it; the other side is the rear. The strength ö of a shock wave of first order is defined as .ll = _hl = _[:_- 1 = [in] u- (F (F u+ . (189.16) If ö > 0, the shock effects condensation; in the contrary case, rarefaction 2 • By putting \V= xk in the duals of (176.8) and (181.8) we may calculate the forms of the jumps of x:rxß• i:rx• and xk across a shock wave. The results are complicated. Weshall give them only for the quantity of greatest interest, the acceleration fi. Putting (189.17) from (181.8) and (189.1) 2•3 we obtain 2 t5ct sk t5ct UN ) [xk] = UN ck - 2 UN --- sk --- M M ' _ U? k <5d[ik] _ [ 'k] <5ctlog UN - N c + 2 t5t X 0t ' (189.18) where the displacementderivative t5ct/M is defined in terms of the motion of the samematerial diagram F(X, t) = 0 as is used to calculate the speed of propagation, UN· Since the quantity ck, the jump of the fully normal second spatial derivatives, is essentially arbitrary, there is no immediate interpretation for the result ( 189.18). Indeed, since the jump in velocity is arbitrary, so are the jumps in its derivatives, and there is no restriction in general upon [ik,m] or [xk]. It may be convenient, nevertheless, to resolve these jumps by means of the identities (175.11) and (180.3). The results are 3 [ik,m] = [nPik,p] nm+ gmpa.dr xf.d [ik];r,l [ 8 i k] - [ p • ] + t5ct [ i k] 8t -- Un n xk,P _t5_t_' (189.19) 1 The direct proof given in the text simplifies and generalizes that of KoTCHINE [1926, 3, § 3. ~ 1]; the customary proof, resting upon an integral form of the principle of conservation of mass, will be given in Sect. 193. For one-dimensional motion, the result is due to STOKES [1848, 4, Eq. (2)], RIEMANN [1860, 4, § 5], and RANKINE [1870, 6, §§ 2-4], the general case, to CHRISTOFFEL [1877, 2, § 1] and JouGUET [1901, 9]. The classical treatment isthat of HADAMARD [1903, Jl, ~~ 109-110). 2 There is no kinematical or dynamical reason why rarefaction shocks cannot occur, although special thermodynamic conditions, including those usually assumed in gas dynamics, may forbid them. 3 The somewhat different results obtained by HADAMARD [1903, 11, ~~112, 113 bis, 119-120] are equally inconclusive. Sect. 190. Singular surfaces of order 2: Aceeieration waves. 523 where the displacement derivative 15d/15t now refers to the spatial equation of the singularity, f(:JJ, t) = o. 190. Singular surfaces of order 2: Aceeieration waves. For singular surfaces of order 2, the analysis is simpler, since the speed of propagation UNis continuous. Substituting 1,1! = xk into the dual of ( 176.1 0) yields [~cxlll = [8px~"'] = [8"'x~11] = sk N"'N11 , } sk =[NY N~ o~x~y] =[NY N~ x~y.,]. (19°·1) From the dual of (181.9) we have similarly1 (190.2) These formulae show that a singular surface of order 2 is completely determined by a vector s and the speed of propagation, UN. In particular, material discontinuities of second order affect only the derivatives ~"' 11 , while discontinuities in the acceleration and in the velocity gradient are necessarily propagated, and conversely, every wave of second order carries jumps in the velocity gradient and the acceleration. Waves of second order are therefore called acceleration waves. Since X~m is continuous, by (190.2h and (182.8) we have [i~m] =- UNskNcxX"',m• l =- UN sk VVt.t>_t__t:,_ nm. F,pF.II (190.3) The left-hand side is independent of the choice of the initial state; therefore so also is the coefficient of n upon the right-hand side. Thus if we put Us0k= UNsk Vt,pt:P , (190.4) VF.pF.II the vector Us0 is independent of the choice of the initial state. We may choose to regard U as the local speed of propagation and s 0 as the value of s corresponding to a choice of the material co-ordinates as being equal to the spatial Coordinates at the instant the wave passes. In this notation, (190-3) and (190.2) 3 become 2 [~m] =- U s~ nm, [xk] = s~. From (190.5) we have a number of corollaries. First, - U [nm i~m] = [xk], [()ik]- ["k]- [ ·~ ] •m ot - X X ,m X ' = s~ + U S~Xn, = Uunst (190.5) (190.6) where we have used (183.5). Thus the local acceleration is continuous across an acceleration wave if and only if the wave is stationary. Second, [Ic~J = [divi:] =- Us0 • n, } (190.7) [w] = [curli:] =- Us0 X n; l Eqs. (190.1) to (190.2) are due essentially to HUGONIOT [1885. 4, p. 1231] [1887, 2, Part II, § 9)]. Cf. HADAMARD [1901, 8, § 2] [1'903, 11, ~ 102]. 2 These results and (190.6) were given by HUGONIOT [1885, 3, p.1120] [1887, 2, Part I, § 9] in forms with U s~ eliminated. 524 C. TRUESDELL and R. TOUPIN: The Classica] Field Theories. Sect. 191. interpretation of these identities yields Hadamard's theorem1 : A longitudinal acceleration wave carries a fump in the expansion but leaves the vorticity unchanged, while a transverse acceleration wave carries a fump zn the vorticity but does not affect the expansion. By (156.5) 2 , we may put (190.7) 2 into the alternative form [lo~ e] = U s 0 · n = -& [xn]. (190.8) Since log e is continuous across an acceleration wave, by putting lj! =log e in the dual of (180.5h we have [lo~el =- u [a 1 ;!e]. (190.9) By ( 190.8) follows the important relation [Xn] =- U2 [d~!e]. (190.10) 191. Singular surfaces of higher order 2• For a singular surface of order p, the results are easy generalizations of those for the case when p = 2. Putting lj! = xk into the dual of ( 176.11) yields [xk•"''"'•···"'P] = [8"', 8"', ... 811Pxk] = sk Na, Na, ... N"'P' } sk = [Nß, Nß• ... Nßp 8p, 8p, ... 8ppxk], (191.1) while the dual of (181.11) yields [xk ] = - U. sk N N N ,cx1a:2···!Xp-1 N a1 a2 • •• cxp-I' ( 191.2) [(P_;Ilk] = (- U. )P-I sk N ,o:l N CXt' [~lk] = (- UN)P sk = (- U)P s~. Thus a singularity of order p carries a jump in the p-th acceleration if and only if it is a wave. Now by (156.5) we have -·- 'k "' 'k - log !? =X , a. X, k =X, k, (191.3) so that _(h_)_ (h) -(log e),a.,a., ... a.p-1-" = x k,a.a., a., ... a.p-1-" x~k+ ... , (191.4) where the dots stand for a polynomial in derivatives of orders less than p. By ( 191.2) we thus obtain _l!L_ -[(loge),a.,a, ... ap- _,.]=~- UN)hNa.,Na.,···Na.p-1_,.Na.xa..•sk,} ( 191.5) h - 0, 1' ... 'p - 1 . Choosing the initial state as that on the singular surface yields _l!L_ -[(log e),k,k, ... kp-I-A] = (- U)hson nk, nk, ... nkp-1-h' ( 191.6) ------ 1 HADAMARD [1903, 11, ~~111-115]; announced in part in [1901, 8, § 3]. In part, these results are equivalent to WEINGARTEN's formulae (175.10) and are foreshadowed by a theorem of HuGONIOT [1887, 2, Part I, § 12]. 2 HADAMARD [1903, 11, ~~88, 103,111-111 bis]. Sect. 192. The transport theorem for a region containing a singular surface. 525 In particular, from this result and (191.2) 4 we have __lE::!L 1 ( p) [(logg)] = u [xn]. (191.7) This relation enables us to attach a meaning to the sign of a discontinuity. [Cf. (189.16) and (190.8).] If the propagating singular surface of p-th order carries (p-1) with it an increase in log g, it may be said tobe a compressive wave; in the contrary case expansive. From (191.7) we see that a wave is compressive or expansive d. . b . . d . h 1 (p) accor mg as It nngs an mcrease or ecrease m t e norma component Xn of the p-th acceleration 1• In particular, in an isochoric motion, acceleration waves of all orders are necessarily transversal, and, conversely, material singularities and transverse 'li/aves of all orders leave the density and all its derivatives continuous. In the case of a surface which is singular with respect to 'iJ and also a singular surface of order 2 or greater with respect to the motion itself, the principle of duality when applied to ( 180.3) yields ['f] =- UN[N" 'il,)d•+~pxei>i>·da- J(l+pxf)d!JJl. I -r .9' -r Thus iJ.9' and ~.9'. the force and torque exerted by the material outside !/ upon that inside, may be calculated from the fields 0 ~(! p) ' ~1!_ X e p, I and l within r and the field e p p on !/. öt t First, in steady motion within a stationary outlet vessel of any form, by (69.1) wegetfrom (202.1) iJ.9'=- fldiDl, ~.9'= _!"J(l +pxl)diDl. (202.2) -r Thus the steady motion of a materialfilling a inlet closed stationary vessel has no reaction upon the Fig. 33. Material in steady llow through a pipe. vessel 2• Fora less trivial example 3, consider a material which is being forced in steady flow through a stationary pipe (Fig. 33). Suppose that l =0 and that I is the field of uniform gravity. If we take the normal to the inlet cross section .9{ inward, retaining that on the outlet cross section ~ as outward, then (202.1) reduces to il.9' = f e PP· da- f e PP· da-~. ) ~ .Y'i ~.9' = f p x e p p · da - f p x e p p . da - m c x ~, 9;, .Yj (202.3) where $ is the weight of the material in the pipe. Thus, no matter what the shape of the pipe or the nature of the material in it, from a knowledge of Im and of the density and velocity of the material at the inlet and outlet only, we can determine the total reaction on the material instantaneously occupying the pipe. Writing g;, for the surface constituting the pipe itself, by (202.3) and (202.1) 1 In principle, these formulae are old. Apparently the first general discussion is that of V. MISES [1909, 8, § 9). A clear treatment is given by CISOTTI [1917, 4, §§ 1-4). Cf. LELLI [1925, 8 and 9], MüLLER [1933, 8]. 2 CrsoTTI [1917, 4, § 7]. 3 CISOTTI [1917, 4, § 8]. 540 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 202. we obtain also Jt(n)da = J (t(n)da- e p p. da)- J(t(n)da- e PP· da)-~. 9p .9'i .9"0 f (m(n) + pXt(n))da = f [m(n)da + p X (t(n)da- (! p p ·da) - .9"p .9"1 (202.4) - f[m(n)da + px (t(n)da- e p p. da)- 9Jl cx ~ . .9"0 Thus a knowledge of ~ and (!, p, t(n), and m(nl at the inlet and the outlet alone suffices to determine the reaction the pipe exerts upon the material. Consider next a single finite closed rigid boundary submerged and fixed in a continuous medium, and let .5I;; be an imagined control surface sufficiently large as to include all of [/ (Fig. 34). For simplicity, take f = 0, l = 0. So as to calculate the reaction exerted by the rigid obstacle on the material, we apply /-----......... / ' / ' (D y \ I '- I ' / ............... ____ ,.../ Fig. 34. Reaction on a submerged rigid object. (202.1) to the material between [/ and 9';;, thus obtaining for the force and torque exerted by the obstacle the expressions1 iJ = :t J pd!Dl +~e p p ·da- ~t(n) da, " .9'C .9'C f= Jpxpd9Jl+~PXepp·da- " .9"c -~pxt(n)da. .9'C (202.5) The first terms represent the force and torque arising from local changes in velocity. In steady motion, they vanish, and (202.5) shows that then the reaction of a submerged body may be determined from the values of the stress vector, the velocity, and the density on a control surface far from the body. To calculate this reaction in the steady case, let V be any constant vector; then by mere algebraic identity we have ~ e p p · da = ~ e (P - V) (P- V) · da + l ~ ~ + ~e(P- V) da· V+ V~eP·da. ~ ~ (202.6) Since e p is solenoidal in a steady motion, and since p is normal to the fixed boundary !/, application of GREEN's transformation to the integral of div(e p) over the region between [/ and .5I;; shows that the last summand vanishes. Thus (202.5h may be written iJ = ~ e(P- V)(p- V)· da+~ e(P- V) da· V-~ (t(nl + Pn)da, (202.7) ~ ~ .9'j; where P is an arbitrary constant. This formula may be applied to the case when at great distances from the obstacle the material is moving at a uniform velocity V and the stress is hydrostatic. Indeed, from (202.7) we read off the 1 For the classical special case of isochoric irrotational motion subject to hydrostatic pressure, see e.g. MILNE-THOMSON [ 1938, 9, § § 17.10- 1 7. 51] and the more general result of RASKIND [1956, 12]. Sect. 202. The reaction on bounding surfaces. 541 generalform of the Euler-D'Alembert parado~ : Let a stationary rigid body be immersed in an infinite material in steady motion past it; if there exist constants V and P such that e (p - V) (p - V) = o (p-2), e (p - V) = o (p-2), t + Pn = o (p-2). (202.9) The result is extremely general: lf the velocity approaches its constant limit at oo sufficiently quickly, and if the stress vector approaches a uniform pressure at oo sufficiently quickly, the submerged body exerts no force. However, the "paradox" has limited application, as the order conditions (202.8), which are the very essence 2 of the assumptions underlying it, are rarely fulfilled in particular theories of materials. The classical example is homochoric irrotational flow of a perfect fluid, where (202.9) follows easily from theorems of potential theory. It is often asserted that in fact it is the adherence of materials to solid boundaries that accounts for the resistance to steady motion in real fluids. This may be so, but it does not enter the present argument directly. Indeed, if the stronger condition (69.4) were to replace (69.2) here, the most we could expect would be the annulling of further surface integrals, not the addition of extra terms. Rather, so far as is now known, taking account of adherence in conjunction with any dissipative mechanism has the effect of transmitting stronger disturbances to oo, sufficient to violate the order condition (202.8). There are various generalizations. If the material is confined by a stationary canal, from (202.6) we see that instead of integrating over all of ~, for the first two integrals in (202.7) we need consider only the cross sections of the canal at great distances, and again (202.8k 2 are sufficient to make these integrals vanish in the limit. To make the third integral vanish also, in addition to (202.8) 3 we supply an appropriate assumption regarding the value of the stress vector on the canal walls 3• lf there are surfaces on which p is discontinuous, provided they do not extend to infinity, the result still holds 4 ; if, however, there are infinite shocks or slip surfaces, the resultant force in general is not zero 5• To calculate the torque, we note first that da · p (p Xe p) =da · (p- V)[p X(! (p- V)] + } ) . • (202.10) +(da· V [pxe(P- V)]+ da· e ppx V. Now by GREEN's transformation and (69.2) and (156.6) we have f da· ePP = fdiv(epp)dv = Jpd!JR = 14§ (202.11) .9"c -r -I'" 1 The result was asserted and proved correctly by EuLER [1745. 2, Satz I, Anm. 3] for steady plane flow of a perfect fluid, on the assumption that the stream lines straighten out at oo. D'XLEMBERT [1752, I, §§ 66-69] rediscovered or appropriated it; later [1768, 2, §I] he reasserted it in sensational terms and proved it by an argument assuming the body to consist in eight congruent parts. It has given rise to misunderstandings lasting over centuries. For the hydrodynamical case, a correct proof based on integral transformation was given by CtSOTTI [1906, 3]; we present, essentially, the reformulation by BOGGIO [1910, 2]. A variant argument based partly on the energy balance was suggested by DUHEM [1914, 2] and worked out by B. FtNZI [1926, 2]. 2 In the older treatments this is glossed over, but it is made clear by CtSOTTI [1917, 4, § 9]. 3 CISOTTI [1909, 4]. & DUHEM [1914, 2 and 3], PICARD [1914, 9], MANARINI [1948, 14] [1949, 17]. 5 Examples were given by HELMHOLTZ and others; a general discussion is presented by JouGUET and Rov [1924, 7]. 542 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 203. where !f3 is the momentum of the material between !I' and ~. Substituting (202.10) and (202.11) into (202.5) 2 yields )! = p p Xe ('p- V) (p- V) ·da +V· f da p Xe (p- V) + l 9: 9: c c (202 12) + !f3x V- pdapx (t(n; +Pn). · 9'c Therefore if we strengthen (202.9) to read p-V=o(p-3), e=0(1), t+Pn=a(p- 3), we conclude not only the Euler-D'Alembert paradox but also1 (202.13) )! = !f300 X V, (202.14) where !f3oo is the relative tnomentum 2 of the material exterior to the obstacle. Under the severe conditions (202.13), then, a stationary rigid body will generally exert a torque perpendicular to the direction of motion. In certain cases of symmetry we shall have lf300 = 0; the body then exerts no torque. CISOTTI 3 extended these results to the case of an obstacle spinning at angular velocity w, provided the motionrelative to the obstacle be steady. He found that then (202.15) the expression for )! is more complicated. All our results are expressed in terms of the reaction of the obstacle on the material. That this is equal in magnitude and opposite in direction to the reaction of the material on the obstacle is to be expected and follows from CAUCHY's lemma, to be proved in the next section. 203. The stress tensor 4• The field of stress vectors t(n) is not an ordinary vector field. Rather, since the stress vectors across two different surfaces through Fig. 35. CAUCHY's construction to prove bis fundamental theorem. the same point are generally different, at any given time t(n) is a function both of the position vector p and of the direction n. The first problern is to delimit the dass of t(n) forfixed p as n varies. We apply (200.1) to a tetrahedron (Fig. 3 5), three .Xz sides of which are mutually orthogonal, the fourth having outward unit normal n. Let the altitude of the tetrahedron be h; the area of the inclined face, s; the projections of n onto the orthogonal faces, n1, n2, and n3, so that the areas of the inclined faces are snl, sn2, and sn3. Assurne that the fields eP and e/ are 1 CISOTTI [1910, 4], BoGGIO [1910, 2]. 2 I.e., $ 00 =o f(p- V) d'.m, where by (202.13)1 the integral, taken over the infinite space exterior to the obstacle, is convergent. 3 [1910, 4]; a vectorial derivationwas given by BoGGIO [1910, 2]. 4 The idea and the results here are due to CAUCHY [1823, 1] [1827, 1]. In a sense, the fundamental theorem (203.4) is contained in a memoir written by FRESNEL in 1822 but not published until much later [1868, 7, §§ 3-4]; however, FRESNEL did not disentangle stress in general from purely elastic stress. Cf. also HoPKINS [1847, 1, § 2]. Sect. 203. The stress tensor. 543 bounded, that t(n) is a continuous function both of p and of n. We may then estimate the volume integrals in {200.1) and apply the theorem of mean value to the surface integral: {203 .1) where K is a bound and where t(n) and t~ are the stress vectors at certain points upon the outsides of the respective faces. We cancel 5 and let h tend to zero, so obtaining (203.2) where all stress vectors are evaluated at the vertex of the tetrahedron. From their definitions, the quantities t 1 , t 2 , t 3 do not depend upon n. In the case when n,.=1, n 2 =n3 =0, we have t 1 =t(-n)• since the outward normal to the face 1 is -n. Although the construction of the tetrahedron fails for this case, we may nevertheless suppose nc~1, ~0, ~0, and by the assumed continuity of t(n) as a function of n infer from {203.2) that {203.3) This is Cauchy's lemma: The stress vectors acting upon opposite sides of the same surface at a given point are equal in magnitude and opposite in direction. Now select a reetangular Cartesian co-ordinate system whose planes are the three orthogonal faces. With the convention that tkm is the k-th component of the stress vector acting upon the positive side of the plane zm = const, by {203. 3) we infer that -t1 x=l:.x, -t1 y=tyx• -t1 z=tzx, etc., and hence (203.2) becomes t(n)k =tkmnm, where the quantities tkm are independent of n. By the quotient law of tensorsl, the quantities tk m form a Cartesian tensor of second order. The tensor equation tk - tkmn (n)- m> {203.4) having been established in a special co-ordinate system, is valid in all co-ordinate systems. This proves Cauchy's fundamental theorem: From the stress vectors acting across three mutually perpendicular planes at a point, all stress vectors at that point are determinate; they are given by {203.4) as linear functions of the stress tensor tkm. Application of a parallel argument to (200.3) yields m ) ,h J (205.18) = 'j'tk1 ... knqda + fk 1 ... /nqdiJJl. f/' j/' The corresponding boundary conditions and differential equations are, respectively, tk .... knq= tk .... knqPnP, } (205.19) (! Zk 1 ... kn Zq = tk1 ... kn q p, p +e /k" ... kn q · They may be recast by writing tk .... k.q as the sum of the n-th moments of all stresses of lower order plus a tensor of excess; e.g., in the notation introduced above, (205.20) The assigned moments fk .... knq are subject to a law of transformation such as to render {205.20) invariant with respect to change of the origin of Co-ordinates. 205A. Appendix. Notations for stress. The following table, shortened from that of PEARSON [1886, 4, § 610], presents the notations for stress found in the classical works and in some cases still in use today. The blocks stand for the matrix of tkm in reetangular Cartesian co-ordinates, or possibly for physical components in curvilinear Co-ordinates. CAUCHY's earlier work A F E F B D E D c F. NEUMANN, KIRCHHOFF, RIEMANN Xx Xy xz Yx Yy Yz Zx Zy Zz PoisSON Pa Qa Ra ~ Q2 R2 ~ Ql R1 KELVIN p V T V Q s T s R CoRIOLIS, CAUCHY's later work, ST.-VENANT, MAXWELL Pxx Pxy Pxz Pyx Pyy Pyz P.x Pzy Pzz CLEBSCH tll 112 113 121 122 123 131 Ia 2 l33 ~ Ta I; Ta ~ :z;_ I; :z;_ Na PEARSON XX .iY fi jiX yy yz Zi iY Zi German engineering works usually employ ~. N,;, or r1x for lxx and :fxy, :Z:,, l'xy• or Tz for txy• etc. French authors usually follow LAME, though some adopt the more luminous notation of CoRIOLIS. The definitions sometimes differ in sign. The notation lkm may be remernbered by the word tension; Pkm• defined by (204.5), by the word pressure. In this work we always use PEARSON's notation k;n, already introduced in Sect. 204, when employing physical components in orthogonal curvilinear co-ordinates. By (App. 14.6} and (App. 14.7), CAUCHY's first law (205.2) then assumes the explicit form ex = ef + ± {-v1 a!m (liiJ;;n). + m~l g V gmm +-·-------·----- ~ 1 ayg,;;. mm 1 ayg;;.~} Vfkl. yg:~ oxm Vgmm yg,.-,; oxk I {205A.1) Sects. 206,207. The equivalence of stress, extrinsic Ioads, and transfer of momentum. 549 (LAME [1841, 4, §§VII-VIII] [1859. 3, § 14, Eqs. (14) (15)]; cf. MoRERA [1885, 6], ANDRUETTO [1931, 1]; for generat curvilinear Co-ordinates, cf. Rrccr and LEvr-CrvrTA [1901, 14, Chap. 6, § 3]. ToNOLO [1930, 7]). 206. Stress impulse. For the laws of impulse in continuum mechanics, we apply to ( 198.1) the general results given in Sect. 194. The influx of im pulse is the negative of the stress-impulse tensor i; the supply of im pulse is a vector s; and as the counterpart of CAUCHY's first law (205.2) we get from (194.11) (206.1) while the balance of moment of momentum assumes exactly the form (205.10) with i replacing t and with appropriate new symbols replacing m and l. In the special case when i is spherical and s = 0, the covariant form of {206.1) is (206.2) Hence follows trivially a celebrated theorem of LAGRANGE and CAUCHY1 : The mass flow produced by impulsive hydrostatic pressure is lamellar; in particular, a homochoric motion initiated by impulsive hydrostatic pressure is irrotational in the first instant. More generally, by comparing (205.2) and (206.1) we derive the following analogy: The correspondences (206.3) at a given instant enable us to construct a dynamically possible impulse, Stressimpulse, and supply of impulse from a dynamically possible acceleration, stress, and force, and conversely. 207. The equivalence of stress, extrinsic loa:!s, and transfer of momentum. As follows from the remarks in Sects. 15 7 and 199, in any given problern the stress may be replaced by equipollent extrinsic forces, or the extrinsic and mutual forces by a stress. Indeed, CAUCHY's first law (205 .2) asserts that the resultant force exerted by the stress is tkm m per unit volume, while (205.10) asserts that the stress and the couple-stress exe~t a resultant couple of magnitude mkPq,q_tlkPl per unit volume. Conversely, to replace mutual and extrinsic loads by stresses we must integrate tkm,m=efk with jk given. Any solution of this equation may be added to the stresst in (205.2), resulting in a combined stress under which the material moves as if subject to no extrinsic or mutual force. In general, to calculate an equivalent stress is elaborate, but it turns out to be simple in the special case when ef is lamellar: For then (207.1) (207.2) so that when the assigned force per unit volume is lamellar, it is equipollent to an appropriate hydrostatic pressure 2• This equivalence of various loads is generally artificial and useless, however. The course of the discipline of mechanics is to prescribe functional dependences for fand t (cf. Sect. 7). With a single prescription fort, a host of different motions 1 The statement and proof of LAGRANGE [1783, 1, § 20], motivated perhaps by earlier remarks of D'ALEMBERT [1752, 1, §§SO- 51], were corrected by CAUCHY [1827, 5, Part 2, Sect. 1, §§ 4- 5]. 2 In effect, this observation is due to EULER [1757. 1, §§ 25-31]. 550 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 207. of the body, corresponding to different initial and boundary conditions, are possible; for each of these different motions, a different equivalent f will result. There is a simple and natural equivalence of stress to transfer of momentum, obtained1 by substituting (156.7) into CAUCHY's first law (205.2): :t (e .i") = (t""'- e .i" x"') ,". + e I". (207.3) This equation asserts that if the tensor - e .i" .im is added to t""', the true rate of change of momentum may be replaced by the local apparent rate 8(e~)f8t; therefore - e~ ~ is called the tensor of apparent stress due to trans/er of momentum 2• I t is the negative of the momentum transfer, which we have discussed in Sect. 170. Very little is known regarding the transformation of systems of stresses appropriate to one problern into those appropriate to another. An elegant result of this kind is due to ÜLDROYD and THOMAsa and NoLL 4 • We present NoLL's form of the argument. For an incompressible medium, we consider two velocity fields d:1 and re, which differ only by a rigid rotation at constant angular velocity w (cf. Sect. 143). We assume further that, apart from arbitrary hydrostatic pressures p1 1 and p2 1, the stress tensors in the two motions are the same functions of place and time 5 : p11 + tl = p,1 + t,. (207.4) Let the frame in which the velocity is d:1 be inertial, so that CAUCHY's first law (205.2) is satisfied. By the results in Sect. 197. the second motion, if it is tobe dynamically possible, must then satisfy an equation of the same form but including the apparent forces arising from the rotation with respect to an inertial frame. Thus we have (! fi1 = div t1 + (! f, } (207.5) (! fi1 = div t2 + (! (f + g), where - g is the sum of the CoRIOLIS and centrifugal accelerations. Subtracting (207.5) 1 from (207.5) 2 and using (207.4), we see that (! g = grad (P2 - P1 ). (207.6) In order that the motion d:2 be dynamically possible, it is thus necessary and sufficient that g possess a single-valued potential. From the results in Sect. 143, a sufficient condition for g to be lamellar is that the motion be plane and that w be normal to the plane of motion 6 • Conditions sufficient to ensure single-valuedness of the function Q' in (143.10) have been stated in Sect. 161. Directly from (207.6) we see that the reaction per unit height exerted upon any cylinder perpendicular to the plane of motion and of cross-sectional area ~ differs, in the two cases, only by the reaction of a hydrostatic pressure equalling the potential of the centrifugal and CoRIOLIS accelerations. The moment is thus zero, and the resultant force is easily shown to be that requisite to impel a mass-point of mass (! ~ located at the centroid to move with the velocity of the centroid. Alternatively, the difference of the forces is that which would be exerted upon the region occupied by the cylinder if it were filled with the surrounding material. Hence if a rigid homogeneaus cylinder immersed in an incompressible substance of the same density as the cylinder is made to undertake any motion in a plane normal to its generators, then a uniform rotation of the whole system about any axis parallel to the generators will not alter the motion of the cylinder relative to the surrounding substance. This interesting discovery of G. I. TAYLOR, in the special case of a fluid, has been verified by experiments. 1 Given for the hydrostatic case by GREENHILL [1875. 2, § 22], for the general case by MATTIOLI [1914, 7, § 2]. 2 While equations of the form (207.3) in the case when t is hydrostatic were given by earlier writers, the tensor (!d: re is often named after REYNOLDS; it was discussed by LORENTZ [1907, 5, § 11]. 3 [1956. 16]. 4 [1957. 12]. 5 There is reason to believe that all material constitutive equations for incompressible media must satisfy this requirement; cf. NoLL [1955, 18, §§ 4, 10]. 8 This result, for a perfect fluid, and also the result below for the force exerted upon an immersed cylinder, are due to TAYLOR [1916, 6, pp. 100-104]. Casesofintermediate generality are treated by DEAN [1954, 4] and }AIN [1955, 15]. Sect. 208. Principal stresses and stress invariants. 551 208. Principal stresses and stress invariants. In the non-polar case, CAUCHY's secondlaw (205.11} asserts thattissymmetric. Bythe firsttheorem in Sect. App. 37, it has real proper numbers, called the principal stresses 1, and real orthogonal principal directions, which define the principal axes of stress. Elements normal to the principal axes of stress are free of shearing stress, being subject to normal tension or pressure according to the sign of the corresponding principal stress; these are extremal values of the normal stresses. The scalar invariants of stress, It, Ilt, Illt, are symmetric functions of the principal stresses. When t2=t3=0, t 1=f=O, the state of stress is called simple tension, and the principal direction corresponding to t1 is called the axis of the tension. In the engineering literature, when one and only one principal stress vanishes, the state of stress is often called bi-axial; similarly, if no principal stress vanishes, the state of stress is tri-axial. A state of non-vanishing plane symmetric pure shearing stress (Sect. 200) is called simple shearing stress; in a suitable reetangular Cartesian system 0 txy 0 lltkmJI= 0 0 , txy=f=O; (208.1) 0 an invariant condition is (208.2) so that txy = t 1 = - t2 =V-Ilt, and the principal axes of stress lying in the plane of stress bisect the angle between the elements suffering pure shearing stress. In the non-polar case, the condition (204.4) that all stress vectors have equal magnitude reduces to (t1 ) 2= (t2) 2= (t3) 2, as is obvious. An exhaustive study of the general state of stress at a point was made by KLEITZ2• Some of his results, as weil as the fundamental theorem of CouLOMB and HoPKINS on the maximum shearing stress, have been given in more general form in Sect. App. 46. In particular, (App. 46.14} shows that the magnitude of the maximum shearing stress for a pair of directions, one of which is kept fixed as the other swings perpendicularly about it, is a function only of the differences of the principal stresses, and hence is independent of the mean pressure. For the overall maximum and minimum shears, this independence holds a fortiori. From results of BoussrNESQ3 given in more general form in Sect. 23 it follows that among the planes through the principal axis corresponding to the principal stress t2 when t1 ;;; t2 ;;; t3 occur 1. The planes across which the magnitude of the stress vector is greatest and least; 2. The planes across which the magnitude of the normal stress is greatest and least; 3. The planes on which the magnitude of the shearing stress is greatest; 4. The planes across which the angle subtended by the stress vector is greatest and least. Since theorems of this kind are no more than verbal adjustments of theorems on an arbitrary symmetric matrix, for economy we refer the reader to Sects. App. 3 7 -App. 50. Perhaps the most important function of the present section is to give warning that these results follow only from the assumption that there are no extrinsic or mutual couples or couple stresses. 1 The theory is due to FRESNEL (1821-1822) [1868, 5, § 28] [1868, 6, §§ 1, 8-9] [1868, 7, § 1]. 2 [1872, 2, § 6] [1873, 4, Chap. II]. 3 [1877. 1. § 2]. 552 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 209. 209. Geometrical representations of stress. Geometrical representations for the stress tensor and stress vector may be read off from the results of Sects. 21 to 24. For the non-polar case, the following quadrics were introduced by CAUCHY and by LAME and CLAPEYRON 1 : 1. The quadric oft, called Cauchy's stress quadric. The normal stress across any plane through its center is inversely proportional to the squared length of that radius vector of the quadric which is normal to the plane. 2. The quadric of t-1, called Lame's stress director quadric. The radius vector from the center to any point of the surface is in the direction of the stress vector across a plane parallel to the tangent plane at the point. J. The quadric of t 2, called Cauchy's stress ellipsoid. The central radius vector in any direction is inversely proportional to the magnitude of the stress vector across the plane at right angles to that direction. 4. The quadric of t- 2, called Lame's stress ellipsoid. The magnitude of the stress vector across any plane is proportional to the central perpendicular on the parallel tangent plane of the ellipsoid. The asymptotic cone of LAM:E's stress director quadric, called Lame's cone of shearing stress, is the locus of elements subject to pure shearing stress. When it is real, it separates the planes across which the normal traction is a tension from those across which it is a pressure. When the cone is imaginary, the normal traction across allplanes is a tension or a pressure according as lt>O or lt= ef + ---=c- -k + -----. vgkk ox r!kq (!km (209.1) where the three indices k, m, q are unequal, and where the (!km are the principal radii of curvature of the surface xk = const. While these equations have limited correctness in three dimensions, their specialization to plane stress is always valid and often useful. In view of the results in Sect. 208, in the case when x = 0 and f = 0 these formulae express the rate of variation of a given principal stress along its trajectory in terms of two principal shearing stresses. Further results concerning the lines and sheets associated with the stress field may be read off from the analysis in Sects. App. 47 to App. SO. 1 See Sect. 21 for references. In 1821-1822 FRESNEL [1868, 5, § 28] [1868, 6, §§ 1-7] [1868, 7, §§ 1-4] constructed but did not publish a theory of elasticity based on postulating the ellipsoidal distribution of stress and the coincidence of the principal axes of stress and of infinitesimal strain. For plane stress, a method of representing both the magnitude and the direction of the principal stress fields on a single diagram was constructed by TESAR [1933, 11]. 2 [1948, 19]. 3 While the existence of these trajectories is weil known, we have been unable to trace their history or to find Iiterature concerning them except in very special cases. 4 LAME [1841, 4, §XI] [1859. 3, § 149]. Cf. also WARREN [1864, 5], MoRERA [1885, 6, § 3]. Sect. 210. The equations of motion expressed in terms of a reference state. 553 Other trajectories can be associated with a state of stress. Rather arbitrarily, VaLTERRA 1 chose to consider the trajectories of the field of stress vectors across the elements normal to x. These trajectories, which he called lines of tension, are indeterminate for static problems. 210. The equations of motion expressed in terms of a reference state. We now give forms of CAUCHY's laws in which the independent positional variable is not the place ;I! where the stress is experienced but rather is a reference point X functionally related to ;I! through (66.1) [or (16.2)]. The apparatus of Subchapter BI is at our disposal. Thus far we have considered tkm as a function of ;I! and t only; for such a tensor field, the definition (App. 20.2) yields tkm,m=tkm;m· In this section we prefer to consider t as a double tensor field. CAUCHY's laws (205.2) and (205.11) then assume the forms (210.1) Since tkm;m=tkm;MX~m• by substituting (17.8) into (210.1h we obtain 2 (!xk=tkm;Mofloxm;M+efk, (210.2) where e == (! ]. When the differentiations are carried out, the first term Oll the right becomes a sum of three J acobians 3 : TkK = ]tkmXfm, From (203.4) 2 and (20.8) we have tfnl da= TkK dAK. (210.3) (210.4) (210.5) Hence II TkK II gives the stress at ;I! measured per unit area at X; the quantity TkK is the component along the XK co-ordinate of the component of the stress vector along the xk Co-ordinate, multiplied by the ratio of the area at ;I! to the area at X. Thus the quantities TkK, sometimes called pseudo-stresses, are awkward to interpret. Moreover, in terms of TkK CAUCHY's second law (210.1) 2 assumes the elaborate form 5 (210.6) By (210.4) 1 and (18.2) we have TkK;K = (] Xfm);K tkm + J X~mtkm;K' } =Jtkm;m• (210.7) 1 [1899, 4]. VaLTERRA calculated the flux of mechanical energy across vector tubes of this field. 2 BouSSINESQ [1872, 1, §I, Eq. (3)]. The result may be read off from a transformation given by CLEBSCH [1857. 1, § 2]. Cf. E. and F. CossERAT [1896, 1, § 17]. 3 Due essentially to E. and F. CossERAT [1896, 1, § 43, Eq. (117)]; cf. TRUESDELL [1952, 21, § 26]. The result in the hydrostatic case was given by EuLER [1770, 1, § 119]. 4 [1833. 3, Eq. (42)] [1836, 1, Eq. (73)] [1848, 2, ~m 34-38]. Cf. also KIRCHHOFF [1852, 1, pp.763-764], C.NEUMANN [1860, 3, §§2, 4-5]. E. and F.CossERAT [1896, 1, §15, Eq. (33)]. KIRCHHOFF remarked that the matrix IITkKII is not generally symmetric, as indeed is manifest from the present notation. Although PoiNCARE [1892, 11, § 40] gave a clear explanation of this fact, based upon (210.5) and the observation that three orthogonal elements daa at :r do not in generat arise from three orthogonal elements dAa at X by (20.8), nevertheless his elaborate, confusing, and unnecessarily restricted manner of stating [ 1892, 10, § 35] that for infinitesimal displacement gradients the matrix II TkKII is approximately symmetric gave rise to an unnecessary discussion of this "paradox" and an incorrect notion that it is connected with the presence of initial stress (E. and F. CossERAT [1896, 1, § 26], COLLINET [1924, 3], }OUGUET [1924, 6]). • The corresponding form of the more general law (205.10) is obtained by E. and F. CossERAT [1909. 5, §53]. 554 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 211. Substituting into (210.1h yields an expression1 for CAUCHY's first law in terms of T"K: !!x" = PK;K +et". PIOLA 2 introduced also the tensor TK M defined by TKM == xK;k T"M = J X5,Xf'!,tkm, In terms of it, CAUCHY's laws (210.1h 2 assume the forms 3 -""-( k TKM) +-!" TKM=TMK. (!X - X;K ;M (! ' (210.8) (210.9) (210.10) While the expression of the first law is simple in terms of T"K, the second law is complicated; for TK M, the reverse holds. There is no simple and exact form 4 with X as independent variable. 211. CAUCHY's laws in space-time; the stress-momentum tensor. The results of Sect. 205 express the balance of momentum and moment of momentum in an inertial frame. Apparent forces and torques on a body were calculated in Sect.197, but for the local equations in a frame which is not inertial it is easier to use the transformations (143.3) and (143.6). Since neither :il nor i: occurs in (205.10) or (205.14), we see that the equations for local balance of moment of momentum are valid for all observers. This follows only because, as emphasized in Sect. 205, linear momentum is assumed already in balance and the quantities m and l represent couples only, not torques combined with forces. In the condition (205.3) we find not:il buti:; however, from (143.3) follows [i:] =[i:'], and hence the condition for conservation of linear momentum at a surface of discontinuity, when written in terms of the velocity of propagation of the surface relative to the material, is valid for all observers. Thus among all the dynamicallaws, only the differential equation (205.2) for balance of linear momentum is affected by the apparent forces. To obtain a form of CAUCHY's first law valid in a rotating frame, we have only to replace :il by the expression on the right-hand side of (143.6). The result shows that in order to calculate the balance of linear momentum in an arbitrary frame, in addition to the metric tensor we must know the linear acceleration b and the angular velocity w of that frame with respect to an inertial frame. From the discussion of Sect. 197 it is clear that such dependence on the observer's motion relative to an inertial frame is unavoidable. 1 PIOLA [1833, 3, Eqs. (22), (29)] [1836, 1, Eq. (56)] [1848, 2, ~ 36, Eq. (26)]. 2 [1833, 3, Eq. (45)] [1836, 1, Eq. (132)]. Cf. also KIRCHHOFF [1852, 1, p. 767], E. and F. CossERAT [1896, 1, § 15, Eq. (31)]. L. BRILLOUIN [1928, 2, § 11] [1925, 2, § 7] [1938, 2, Chap. X, § X] and RIVAUD [1944, 10] have remarked that (210.9) is a statement that the quantities TK M and tk m are the components at X and at a: of a tensor density under the deformation (16.2). Since (210.9) is a mere definition, motivated only by the relative simplicity of some of the resulting forms of CAUCHv's laws, this observation does not seem to have mechanical significance. 3 PIOLA [1833, 3, Eqs. (33), (35)]. Cf. also SrGNORINI [1930, 5, § 4] [1943, 6, Chap. Il, § 4], TOLOTTI [1943, 7]. 4 Other forms are given by SIGNORINI [1930, 5, § 5] [1930, 6, § 2], ZELMANOV [1948, 39, Eq. (4)], and CASTOLDI (1948, 5, § 7]. The form given by DEUKER (1941, 1, Eq. (8.7)] was shown tobe false by TRUESDELL (1952, 21, § 2612]. The forms given by PLATRIER (1948, 20, Eq. (13)] and GLEYZAL [1949, 12, Eq. (5)] also seem dubious. Various approximate forms of (210.10) or (210.8) occur in the literature; e.g., NovozHILov [1948, 18, §§ 21-22]. Explicit forms for (210.8) in orthogonal curvilinear co-ordinates are worked out by YosHIMURA [1953, 37]. Sect. 211. CAUCHY's laws in space-time; the stress-momentum tensor. 555 However, it is possible to obtain a more elegant formalism1 for the balance of momentum by using the ideas and methods of Sect. 152. First we recall that the stress tensor t was introduced through use of the balance of linear momentum in an inertial frame; the contravariant components t"m occurring in (203.4) therefore are defined in all inertial co-ordinate systems. We now introduce the agreement that what we shall mean by the stress components t"m in any frame obtained from an inertial frame by a time-preserving transformation (154.8) is "'k' <> m' t"' m' = _u_x __ u_x- tP q - oxP oxq • (211.1) That is, in the terms used in Sect. 152, the components t"m may be regarded as the non-vanishing components of a space tensor tD.d having the canonical form (211.2) in any frame related to an inertial frame by a time-preserving transformation. The agreement (211.1) is consistent with the intuitive notion that assignable forces (including the stress vector 2) are invariant under the Euclidean group of transformations (152.1). The action of one part of the material upon another is thus assumed to be the same to all observers. From (211.2) it follows that the quantities T 04 defined by (211.3) where e is the mass density, taken as a world scalar, and where v is the world velocity (153.6), also constitute an absolute contravariant world tensor, which we shall call the stress-momentum tensor. The name is suggested by the result (207) 3, since in a Euclidean frame T"m = t"m _ e i" zm, ) P" = T"4 = - e i". T44=- f!· (211.4) Similarly, we define a space vector F suchthat in every Euclidean frame F n = (jk, 0). N ow consider the world tensor equation (211.5) where V(/) denotes the covariant derivative based on the Galilean connection r. r (Cf. Sect. 152.) In every Galilean (inertial) frame, (211.5) with .Q =k is equivalent to CAUCHY's first law; the equation that results from putting .Q =4 is the equation of continuity (156.5h- Thus Eq. (211.5) expresses the balance of mass and of linear momentum in world-invariant form. From the world-invariant form (164.2) of the continuity equation and from the definition (154.1) of the world acceleration vector, it follows that an alternative form of (211.5) is (211.6) 1 Special cases have often been noted; e.g. LEVI-CIVITA [1928, 5, pp. 67-81], FINZI [1934, 2], KILCHEVSKI [1936, 5, §§ 1-2] [1938, 5, Part!,§ 10], PAILLOUX [1947, 11], MANARINI [1948, 13], ARZHANIKH [1952, 1]. Our approach differs basically from that of CARTAN [1923, 1, §§ 15-17], who makes the connection F!p depend upon the dynamicallaw. 2 This concept is developed and emphasized as a postulate by NoLL [1957. 11, § 9]. 556 C. TRUESDELL and R. TouPIN: The classical Field Theories. Sect. 212. By substituting the components of the Galilean connection ( 154.1 0) into (211.6), we easily obtain an explicit form for CAUCHY's first law in an arbitrary rotating and deforming co-ordinate system 1 : (211.7) where uk - 0 xk ~~m, t) , ~ = ~ (z, t) being the transformation giving the general co-ordinates xk in terms of the co-ordinates zk in a Galilean reference frame (cf. Sect. 154), where xk is the acceleration as apparent to an observer in the ~-system, and where the comma denotes covariant differentiation based upon the time-dependent metric g (~. t) in the rotating, deforming ~-system. 212. Stress and couple resultants for shells. I. Direct theory. From the dynamical standpoint2, a shell may be regarded in one of two ways: as a surface il, or as a region between two surfaces il1 and il2 • In both cases, the shell is subject to normal forces as well as to tangential forces, and therefore its theory is that of a surface or body in three-dimensional space, not a merely intrinsic theory. To follow consistently the second view, in which the shell is regarded as a region in space, we must derive from the general theory of three-dimensional stress the nature of the forces and couples acting upon the shell. To follow consistently the first view, we cannot use the momentum principle and the stress principle as stated in Sect. 200, since the forms given there are appropriate only to bodies of non-zero volume. Rather, for the first approach we must Postulate new forms of the stress principle and the momentum principle. In the older work 3 on shells as models for solid bodies the two approaches are often confused, while theories of shells as two-dimensional models for soap films, water bells, etc., are so specialized as hardly to be typical. We present the two alternatives independently, beginning with the first. Consider a portion of a surface il bounded by a circuit c, and let this surface be in equilibrium subject to forces F and couples L per unit area. Fand L are three-dimensional vectors which may point in any direction in space. We Postulate 4 a stress principle for shells : The action of the part of the shell outside c 1 Inability to read Ukrainian prevents us from following in detail the related work of KILCHEVSKI [1936. 5, §§ 1-4] [1938, 5, Part I, § 10]; like the corresponding result of McVITTIE [1949, 18, § 4], it seems to rest upon the unnecessary andin general unjustified assumption that there is a four-dimensional metric. The meteorologicalliterature abounds in special cases obtained by laborious transformation. 2 Just as the theory of stress in three-dimensional bodies is independent of kinematics, so also for the dynamics of shells we do not need to mention the theory of deformation given in Sect. 64. 3 While the dynamical equations are implicit in the pioneer work of LovE [1888, 6, '\[ 8], he did not disengage them from special elastic hypotheses and approximations, and they were first given in the forms (212.13), (212.14), (212.20), in special co-ordinates, by LAMB [1890, 6, § 4]. Later, LovE [1893. 5, § 339] remarked that LAMB's dynamical equations are valid for finite deformations if referred to the strained shell. Among the numerous repetitions of essentially the same argument as LAMB's we cite only the efficient vectorial derivation of E. REISSNER [1941, 5, §§ 3-4]. Fundamentally sharper reasoning has been applied in the two special cases when 1. there are no cross forces or moments, the shell being then called a membrane, or 2. the surface is plane, the shell being then called a plate. For the former, a geometrically intrinsic theory is easy; for the latter, the trivial geometry makes a rigorous treatment easy. The history of membranes and plates, which goes back to the eighteenth century, we make no attempt to trace; cf. TRUESDELL [1960, 4, §§ 48-49]. 4 Our treatment follows ERICKSEN and TRUESDELL [1958. 1, §§ 24-26], who shortened the argument of SYNGE and CHIEN [1941, 9, pp. 104-111]. SYNGE and CHIEN were the first to obtain the equations of equilibrium in the fully generalform (212.6). Sect. 212. Stress and couple resultants for shells. I. Direct theory. 557 on the part inside is equipollent to a field of stress resultant vectors S(n) and couple resultant tensors M(n) located on c. The subscript n refers to the unit outward normal to c; of course, n is a vector defined intrinsically in ~. but S(n) and M(n) are three-dimensional fields. In analogy to (200.1) and (200.3), a mathematical expression of this postulate is ~S(n)·ds+ JFda=O,) c d ~(M(nJ+P X S(nJ)ds+ f(L+p X F)da=O, c d (212.1) where d s is arc length along c and p is the three-dimensional position veetor. According to the convention of Sect. App. 13 these vector integrals are understood to be written in reetangular Cartesian co-ordinates. The argument in Seet. 203 can be adapted to two dimensions if we replace the tetrahedron by a curvilinear triangle on ~. The results, analogous to (203.3), (203.4}, and (203.5) are S(nJ =-S(-nJ• Sk _ Sk6n (n)- 6• (212.2) (212.3) In (212.3) the quantities n6 are covariant components of the unit normal to c in any curvilinear co-ordinate system vl, v2 on ~- Greek minuscule indices run from 1 to 2 in this section and the next one. By hypothesis, S(nJ is an absolute vector and M(n) is an axial vector, for given n; while to derive (212.3) we employed reetangular Cartesian co-ordinates, the results are tensorial equations within the scheme of double tensors of Seet. App.15, and hence arevalid in all co-ordinate systems. The double tensors Sk 6 and Mk~ are the fields of stress resultants and stress couples. Wehave dim F = [M L -l r-2], dim L = [M 7-2], } dimS = [MT-2], dimM = [ML r-a]. (212.4) There are six components Sk~ and six components MH; the latter we may sometimes wish to replace by the components of the equivalent skew-symmetric tensor MkP 6• In the classical treatrnents of the theory of shells the vectors L and M(n) are assumed tangent to ~; this assumption, which is analogous tothat defining the non-polar case in three dimensions, reduces the number of nonvanishing components Mk 6 from six to four. Again we suppose the space co-ordinates reetangular Cartesian, we consider S and M as funetions of v only, and we substitute (212.3) into (212.1). Byreasoning strictly parallel to that in Sect. 205 we obtain as analogues of (205.2) and (205.10) the differential equations SH, 6 +Fk=O,} MkP6 + z[k SPJ6 + LkP = 0 (212.5) ,d ,d ' where Mkpd and LkP are absolute alternating tensors equivalent to the axial veetors Mk 6 and Lk. In these equations the subscript comma indicates covariant differentiation with respeet to the surface metric a, except that z~ ozkjov6, where z = z(v} is a reetangular Cartesian equation of the surface ~. Now consider the equations Sk~ +Fk=O,} MkPIJ;d + x[k; 6 SPl d + LkP = 0 , (212.6) 558 Co TRUESDELL and Ro TouPIN: The Classical Field Theorieso Secto 2120 where the space co-ordinates and the surface Co-ordinates are arbitrary independently selected generat curvilinear systems, where x =x(v) is an equation of ~ referred to these two systems, where x':6 = oxkjov6, and where other occurrences of the semi-colon indicate the total covariant derivative (Appo 2002) 0 These equations are in double tensor form; when the space co-ordinates arereetangular Cartesian, they reduce to (21205), which have been proved valid for such co-ordinate systems; therefore (21206) are the generat differential equations of equilibrium for shellso We may continue to regard Sand M as functions of v only, or we may consider them to be functions of x also, as we pleaseo The elegant simplicity of this derivation should not conceal the complexity of the resulto When (21206h is written out, it assumes the form asH + { k } xf 5m 6 + { ; } 5k6 + p = o ovd mp ·6 t5; ' (21207) where {n:p} and {la} are Christofiel symbols based upon the space metric g and the surface metric a, respectively, the two metrics being related as usual: a6 E = gkm x':6 x~ o When {21206) 2 is written explicitly in terms of the axial vector Mk 6, it assumes the form (21208) These formulae involve doubly contravariant tensors considered as functions of v onlyo In terms of physical components, or of components allowed to depend on x as weil, they would take on still more complicated formso To assume L and JU tangent to ~ is equivalent to assuming the existence of surface fields U and Me~ such that {21209) To obtain equations of motion instead of equations of equilibrium, we may assign momentum to d and express its rate of change in terms of a surface density Ak, then replace pk by pk- Ak in (212o6h 0 However, the vector A does not bear any simple relation to the accelerations of points on d, and we prefer to postpone determining the effect of inertial force until the general discussion of relations between three-dimensional and surface variables in Secto 2130 We now resolve all quantities into components normal and tangent to 11: Fk=F6 x~ +FNk, Lk=L6 +LNk, } 5k6 = 5Yd ~7" +56 N"' Mk6 = MYd.x7" +Md Nk' (212o10) where N is the unit normal to l1o The following table connects the components occurring in (212010) with the terms usually employed inshell theory: F, L = normal components of specific applied force and coupleo F6, L6 = specific applied force and couple tangent to the shello 56 = cross force resultanto 5Yd = membrane stress resultanto M6 = cross moment resultanto M"d = couple resultanto According to the usual assumptions (21209), L =0 and Md =Oo The normal and shear components of 5" 6 in an orthogonal co-ordinate system are called normal and shear membrane stress resultants; the normal and shear components of MY 6 in such a system are called bending and twisting couple resultants, respectivelyo Sect. 212. Stress and couple resultants for shells. 559 To express the equations of equilibrium in terms of tangential and normal componentsl, we use the identities (App. 21.6), (App. 21.2), and (App. 21.4) 2 to obtain from (212.1 0) 3 the following resolution: 5k6;6 = (5Y~;ß- aar ba6 56) x7r + (5f6 + br6 5Y 6) Nk. (212.11) Substituting this result and (212.10) 1 into (212.6)1 yields (5Y6;6- aarbad 5 6 + F6) X~y + (56;6 + by6 5Y~ + F) Nk = 0. (212.12) Taking the scalar product of this equation by N yields as the condition for equilibrium of normal forces 5°;o+byo5r~+F=O; (212.13) taking the vector product by N, the condition for equilibrium of tangential forces: 5Y0;0 -araba656+FY=O. (212.14) Similar resolution of (212.6) 2 yields as the condition for equilibrium of bending moments 2 (212.15) for twisting moments, MYO;o- ara ba6 Mo+ aro e~a 5a + LY = 0. (212.16) In these formulae, it is legitimate and natural to regard all fields as functions of v only and to interpret "; r5" as covariant differentiation based on a. Under the usual assumption L = 0, M 0 = 0, the total number of independent components of S and M is reduced from 12 to 10, the second term in (212.16) vanishes, while (212. t 5) reduces to an algebraic equation expressing the difference of shear resultants 5[121 as a linear combination of the four couple resultants Mr~. In works on shell theory it is customary to use in place of M the dual tensor B, defined as follows: (212.17) so that the physical components of these tensors in an orthogonal co-ordinate system satisfy the relations M=-B<21), M<12>=-B(22), M(21)=ß, M(22)=ß(12). (212.18) Eqs. (212.15) and (212.16) representing the balance of moments may be expressed in terms of B: M;oo_ erob~ Boa+ ero5ro + L = 0, } Bvo;o- ava.ea.r b~Mo- 5" + C = 'o, (212.19) where C=ava.ea.rLY; when the classical assumption (212.9) is adopted, these equations reduce to (212.20) the former of which is especially interesting because it shows the tensors S and B to be non-symmetric except in special circumstances. 1 This resolutionwas effected by SYNGE and CHIEN [1941, 9, p. 109] by use of the special co-ordinate system we explain in Sect. 213. 2 The fully general equations (212.15) and (212.16) were given by E. and F. CosSERAT [1909, 5, §§ 35-37]. who derived also forms in material co-ordinates. Cf. also HEUN [1913, 4, § 20]. 560 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 213. The dynamics of shells is not a mere two-dimensional analogue of the dynamics of threedimensional bodies (cf. Sect. 238). As is evident from the ideas used to formulate it and from the occurrence of the second fundamental form b in every one of its equations, shell theory concerns properties of two-dimensional idealizations of three-dimensional bodies. For an intrinsic analogue to the three-dimensional case, we should have to have a tensor p/; 6 satisfying (212.21) where G is assigned. From (212.14), we see that an equation of this form emerges from shell theory if the cross force SY is known. In particular, when SY = 0 we obtain from (212.14) and (212.13) the equations St; 6,6+Ft;=O, b~; t;+F=O. (212.22) When, in addition, the couple resultants and assigned couples are zero, (212.15) reduces to (212.23) Eqs. (212.22) and (212.23) are said to describe a state of membrane stress. The four membrane stress resultants satisfy a system of two linear partial differential equations and two linear algebraic equations with coefficients which are determined by the surface 6. There exists an extensive theory of integration of this determined system. See also Sect. 229. 213. Stress and couple resultants for shells. II. Derivation from three-dimensional theory. If we choose to regard a shell as a portion of material between .... .... 4 ........ .... ' Fig. 37. Shell regarded as a three-dimensional body. two surfaces !l 1 and !l2 , the theory of equilibrium and motion of a shell is derivable as a consequence of the three-dimensional theory. According to this view, the stress and couple resultants, instead of being introduced through a postulated twodimensional stress principle as in Sect. 212, should be defined in terms of the three-dimensional stress tensm: t. Moreover, a shell need not now be a body by itself: All results we are now going to derive hold equally for a shell which is but a part of a three-dimensional body, though this interpretation is unlikely to be useful. · The resultants Sk 6 and MM are defined by the condition that their action upon a curve c lying on a reference surface !l shall be equipollent to the action of the three-dimensional stress tensor tkm upon a finite surface h ( c) intersecting !l along c. E.g., (213.1) where n* is the unit outward normal to h ( c), and where our usual convention regarding integrals of vectors in curvilinear co-ordinates is understood (Sect. App. 17). For each curve c, the surface h ( c) is to be fixed once and for all, subject to the understanding that to a curve which is a part of c there corresponds a surface which is apart of h. The requirement (213.1) then defines SH uniquely. In the practice of shell theory, h is always taken as a surface swept out by the normals to !l along c, and it is always assumed that the surfaces !l1 and !lz are given by equations x0 = h1 (v) and xD = h2 (v), where x0 is the normal distance from !land where v1, '1!8 are curvilinear co-ordinates upon !l (Fig. 37). The quantity 1 h2 - h1 1 is then the thickness of the shell at the point v on !l. In many applications the two surfaces are supposed given by the equations xD = ± h (v); in this case !l is called the middle surface of the shell. For the general theory, however, no such restriction is necessary, and the reference surface !l need not even lie within the shell. Sect. 213. Stress and couple resultants for shells. 11. Derived theory. 561 It is natural to use a co-ordinate system1 in which one family of co-ordinate surfaces consists in the surfaces x" = const, which are parallel to .1. If a (v) and b (v) are the fundamental tensors of .1, then the spatial metric g (v, x0) assumes the form goo =t' 0 = 1, goa: =t'"' = 0, } ga:ß = aa:ß + 2x0 ba:ß + (x0) 2 ba:r b;; (213.2) i.e., the superficial components of g are of the form g = a · (1 + x0 b) 2• In most of the older researches, the Iines of curvature on .1 are chosen as co-ordinate curves, so that gn=an1+x ( OK )2 1 - ( 0 K )2- 1 ( ) 1 =11· g12=0, g22=a221+x 2 -22, 213.3 g g where K 1 and K 2 are the principal curvatures of .1. From (213.2) we have where 2 and also {oko} = {k0o} = 0 • {cx0 ß} = [O, since the parameters v, x 0 are not admissible as co-ordinates if (1 + x° K1) ~ 0 or (1 + x°K )~o. Handbuch der Physik, Bd. 111/1. 36 562 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 213. tensors are the quantities usually employed in shell theory; e.g., if we make the special choice of co-ordinates leading to (213.3), we have 1 hz 5(11) = «n 5 11 = «n J t11 (1 + x° KJ) 2 (1 + x° K2) dxO' hJ ht~ = J 11 (1 + x° K2 ) dx0 , hJ h2--.. 5(12) = J 21 (1 + x° K1) dx0 , hl hz~ 5(21)= J 12(1 +x°K2)dx0 , hl hz~ 5(1) =- f01(1 +x°K2)dx0 , h, hz ~ ß(ll)=- J x0 11(1 +x°K2 )dx0 , hJ hz ~ ß(I2)= - J x0 21 (1 + x° K1) dx0 , kJ etc., where, as usual, k~ is the matrix of physical components oft. (213 .8) We now find the reflection of CAUCHY's second law, in its narrower form (205 .11), upon the stress resultants and couples. Integrating the algebraic identity (213.9) across the shell and expressing the result in terms of the definitions (213.7), we obtain a formula identical to (212.20) 1 . This should not be surprising, since CAUCHY's second law is itself a condition for the balance of moments. The interesting thing about this result is its indication that the classical assumptions (212.9) of shell theory are consistent with the three-dimensional non-polar case, for it is the absence of three-dimensional applied couples and couple-stress that CAUCHY's second law in the narrower form (205.11) asserts. We now derive Eqs. (212.13) and (212.14) for a shell by integrating the corresponding components of CAUCHY's first law (205.2), thus showing that the balance of momentum for a shell as a whole is a consequence of the balance of momentum of the three-dimensional body with which we identify it, as indeed is physically plain 2• Since there are some formal difficulties in using general co-ordinates on the surface ~' weshall use the special metric components (213.3), one advantage of which is the form assumed by the Mainardi-Codazzi identities, VIZ., 8Vgu - 0 [11-( OK )] - ( OK) o}'all ----a;2- - [}V2 yall 1 + X 1 - 1 + X 2 ov2 ' (213.10) and a similar formula obtained by interchanging 1 and 2, 1 and 2. 1 The curvature factors were mentioned by LAMB [1890, 6, § 2] and were used in special cases by BASSET [1890, 1, §§ 5, 18]; the full set (213.8) was given by LovE [1893, 5, § 399], and the general equations (213.7) were formulated by GREEN and ZERNA [1950, 9, § 3]. For discussion of the mechanical significance of the resultants, cf. ZERNA [1949, 39, §§ 3-4]. Definitions not obviously equivalent to these were given by KILCHEVSKI [1938, 5, Part II, § 3]. 2 Üur derivation follows that of NoVOZHILOV [1943, 4] and of NOVOZHILOV and FINKELSTEIN [ 1943, 6, § § 1, 4 ], which is more general than that given independently by TRUESDELL [1945, 6, § 8]. Derivations in general co-ordinates have been sketched by CHIEN [1948, 7] and GREEN and ZERNA [1950, 9, § 3]. Secto 2130 Stress and couple resultants for shellso Ho Derived theoryo 563 As the normal component of CAUCHY's first law (205 o2), from (205 Ao1) we have ,, \1 { <>81 cva22 (1 + X° K2) 01] + -:.8 fall yaz2 vV uV -2 [yall (1 + X° KJ) ozJ} -l - K 1 (1 + x° K 2) f.i- K2 (1 + x0 K 1) 22 + (213°11) + a~o (K* 00) + (! K* (f - x) = 0 0 Integrating this equation from xO =h1 to x0 =h2 and taking account of (21308) yields lau , __ 1l'c.={ a 22 uv <>81 (ya;; 5<1>) + uv <>82 (yall 5<2>)} + l + K1 5<11> + K2 5(22) +F = 0, providing we put h2 ~ h F = - f (! K* (f - x) oxO - K* 00 I 2 0 ~ ~ (213012) (213°13) The result (213012) is identical in form with (212013), in the co-ordinates employed andin terms of physical componentso As the tangential component of CAUCHY's first law (20502) corresponding to vl, from (205Ao1) we have , \,- {-;~dVa22 (1 + xO K2) 11] + -<>8 2 [Va11 (1 + x° K1) 12J} + tau ya22 vV uV + 1 8 logV"ll (1 + xO K) 21- ~ 8 logj'a;; (1 + xO K) 22 + (213°14) Va22 8v2 2 Vau 8vl I + Kl (1 + x° K:~) 01 + 8~0 (K*01) + (! K* (/(1)- x(l)) = 0, where (213o10) has been usedo lntegrating this equation from xO =h1 to xO =h2 and taking account of (21308) yields (213°15) provided we put (213°16) The result (213015) is identical in form with (212014), in the co-ordinates employed and in terms of physical components, when y = 1o To complete the identification of the results obtained by integration of the three-dimensional equations with those of the direct shell theory of Sect. 212, we need to replace (213 013) and (213 016) by invariant formulae valid in general co-ordinates on the surface ~, as follows: 36* 564 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 214. Thus far, except in deriving (213.9) and its direct consequence (212.20)1, we have not assumed the three-dimensional stress tensor to be symmetric. To obtain the equation of moments, however, we rest content with the non-polar case and simply multiply the three-dimensional equation (213.14) by x0, then integrate across the thickness of the shell1• Thus follows l' a11 1v~{ a22 uv "o I(ya;; B) + uv "o2 (~B)} + l + 1 o log Vau B< 1 o log ya;; B< -v- n>- -~- "vl 22)- a22 ov2 lall u - S<1> + C<1> = o, (213.18) where, in general co-ordinates upon 6, we have put C"' =-f2 e K* xo (c5~ + xO b~) (f -xY) dx0 - K* xO (c5~ + xO b~) t0Y1::. hl (213.19) The result (213.18) is identical in form with (212.20)2 , in the co-ordinates employed and in terms of physical components, when ct = 1. In comparing the results of this section with those in the preceding one, we must understand that it is impossible to prove that the quantities entering the two systems are identical, since the difference of basic assumptions and definitions in the two cases makes a statement of isomorphism the best that can be hoped for. What we have shown isthat no error can result if we choose to regard the surface fields S"'fl, SY, and B6 •, defined in terms of the three-dimensional stress tensor t by (213.7), as equivalent to the fields denoted by the same symbols in Sect. 212, where they were defined in terms of the double tensors Sand M, introduced a priori. If we accept this identification, then the results of this section show how the equilibrium theory of the previous section can be generalized to the case of motion. In (213.17) and (213.19) appears not only the assigned forces f but also the acceleration ~. From (213.17) we see that the effective force of inertia, per unit area and surface mass on 6, is not necessarily the acceleration of any particle on 6; rather, at a given point P on 6 it is a certain weighted mean of the accelerations at all points on the normal to 6 through P. Moreover, inertial forces occur also in (213.19) and hence affect the balance of moment of momentum-an unusual phenomenon in mechanics. Finally, even in the static case the effective surface Ioads F and C which enter the equations of equilibrium for shells are not merely the vector differences of the Ioads in the interior, but rather areweighted averages and differences, influenced by the thickness and the curvature of the shell aswell as by the forces and couples applied. 214. Stress and couple resultants for rods 2• If we consider a rod simply as a curve c which may be the seat of dynamical actions, by considerations anal1 In the general case, the definitions (213.7)3 are no Ionger adequate, since additional resultant couples are produced by the couple stress m. Also, instead of simply multiplying the equations of linear momentum by 0 and then integrating, we should integrate (205.10). 2 For the plane case, the stress principle for rods and the appropriate special cases of (214.1), (214.2) and (214.7). independent of any hypothesis regarding the constitution of the material, were first given by EuLER [1771, 2, §§ 1-11, 35-40] [1776, 4, § 17]. For the history of the earlier special theories of rods and flexible lines by PARDIES, ]AMES BERNOULLI, and others cf. TRUESDELL [1959, 8, §§ 2-3, 7-14, 20-21, 25). ST. VENANT [1843, 3,, 3] was the first toremarkthat six equations are needed to express the equilibrium of rods which are twisted as weil as bent, but he did not succeed in obtaining them without special simplifying hypotheses. The general equations were given in principle, but very obscurely, by KIRCHHOFF [1859, 2, § 3), explicitly by CLEBSCH [1862, 2, §50). These Sect. 214. Stress and couple resultants for rods. 565 ogous to those at the beginning of Sect. 212 we are led to postulate a stress principle for rods: At each point on a rod, the action of the material to one side upon the material to the other is equipollent to that of a stress resultant vector Sand a couple resultant M. These quantities have the physical dimensions [M L T-2] and [M L 2 T- 2], respectively. Properly, we should define them as acting on the opposite sides, + and -, of a cut through the rod; then analogously to (212.2) and (212.3), (203.3) we have (214.1) as the first consequences of the principle of equilibrium. Dropping the subscripts + and - but adopting an appropriate convention of sign, as the definitive condition of equilibrium we obtain (214.2) where "~" is the dual of the intrinsic derivative defined by (63.6), where p is the position vector with respect to a fixed origin, and where Fand L are the applied force and couple, per unit length. By substituting (214.2)1 into (214.2) 2 we obtain M+txS+L=O, (214.3) where t is the unit tangent to the rod c. A rod such that M = 0 if L = 0 is said to be perfectly flexible; such rods are often called strings. By (214.3), a necessary and sufficient condition for perfect flexibility is that the stress resultant S always be tangent to the rod. Since the two Eqs. (214.2)1 and (214.3) are in vectorial form, they are valid in an arbitrary curvilinear Co-ordinate system. It is customary, however, to refer them to a particular frame defined with respect to the rod c. Retaining full generality at the start, in the scheme of Sect. 61 let us assign any three linearly independent directors and reciprocal directors da and da to c. With Sk and Fk as the contravariant components of S and F, ~ and Lk as the covariant components of M and L, in general curvilinear Co-ordinates, we define corresponding anholonomic components: sa = d~Sk, P=d~Fk, Ma = d~Mk,} La- d~Lk. By (214.2)1 and the result dual 1 to the reciprocal of (63.10) 3 , we have !d~a_ = sa = d~ Sk + J~ Sk' l =- d~Fk- d~wmk Sk, (214.4) (214.5) and other early treatments are difficult to follow, sometimes imparting the impression that some approximation is made. E.g. LovE [1906, 5, § 254] says "the extension of the central line may be disregarded". In fact, as was noted by BASSET [1895, 1, § 2] (cf. also [1892, 1, § 4]), Eqs. (214.7), analogaus to CAUCHY's laws, are exact when referred to the actnal position of the rod; just as in the three-dimensional theory, no question of approximation appears unless we attempt to refer the equations to a configuration assumed by the rod prior to its being loaded by the forces under which it is in equilibrium. The derivation given in the text is that of ERICKSEN and TRUESDELL [1958, 1, §§ 21-23], patterned on earlier work of E. and F. CossERAT [1909, 5, § 10] and HEUN [1913, 4, § 19]. 1 Note that w is not the F of Sect. 63 but rather the dual of W; neither is it tobe confused with the vorticity vector, which is denoted by the same kerne! index. 566 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 214. where w is the wryness of the directors along c. Similarly dMa - M~ - - dk ( tP Sq + L ) + dm k M. -dS- a- a ekpq k a W m k' (214.6) Hence the statical equations in anholonomic components are 1 dsa + wa Sb + pa = 0 ) ds b · ' dMa b b c _ ds - W a Mb + e0 b c t S + La - 0. (214.7) Thus far the director frame has been arbitrary. We now require it to be a unit orthogonal triad suchthat d1 =t, the unit tangent. By the dual of (63.9) 3 , the wryness w then satisfies Wab =-wba and may be interpreted as an angular velocity 2• The component 5 1, which is the projection of S onto the tangent to c, is called the specific tension in the rod; the components 5 2 and 5 3, the specific shearing forces; M 1, the specific twisting couple; M 2 and M 3, the specific bending couples. This special choice of directors, while not simplifying (214.7) 1 , implies that t1 = 1, 2 =t3 =0 and hence reduces the three components of (214.7) 2 to the following explicit forms: (214.8) In analogy to the reasoning in Sect. 213, it should be possible to derive the equations for rods by integrating the three-dimensional equations or the equations for shells; using power series expansions, GREEN 3 has given a derivation of the former type. If F = 0, from (214.2h it follows that S = const; (214.3) then becomes a statement that M is a prescribed function of s. Since M is an axial vector, such a statement is a formal analogue of (196.3), the general condition for balance of moment of momentum, provided s and I are made to correspond. Such a correspondence may be carried further by taking the director frame of the rod as an orthogonal unit triad, so that the wryness w becomes the analogue of the angular velocity m. This observation forms the basis of KIRCHHOFF's 1 Since by definition we have eabc = + Vi cabc det d~. Now hence g(det d~)2 = det gk m d~ df' = det ge( eabc = ± Ydetgef 10abc• where the sign is to be selected so as to agree with that of det d~. The quantity detgef is evaluated by the dual of (61.4). In particular, for a right-handed unit orthogonal triad we have eabc = 10abc· 2 The classical notation for the component w2 a is T or - r; the other two independent components are written as ±" and ± "'. lt is important to remernher that in the exact theory all these quantities refer to the loaded rod. Cf. the footnotes on p. 565. 3 [1959. 7]. Various earlier authors, e.g. KIRCHHOFF [1876, 2, Vorl. 28, § 5] and LovE [ 1906, 5, § 254 ], had given definitions of the stress resultants and couple resultants in terms of three-dimensional stresses, but their subsequent arguments rest on unnecessary and unrigorous limit processes rather than exact integration such asthat given for shells in Sect. 213. The method of power series expansionwas initiated by HAY [1942, 8, § 6]. Sect. 215. Partial stresses in a heterogeneaus medium. 567 celebrated analogy between the motion of a rigid body and the deflection of an elastic rod 1• The result as usually presented takes on an appearance of greater complexity because (214.8) rather than (214.3) is used as the starting point. 215. Partial stresses in a heterogeneaus medium 2• To discuss the transfer of momentum in a mixture, we employ the formalism of Sects. 158 to 159. Each constituent ~ is regarded .as being subject to partial stress t is normal, the third condition for both theorems is satisfied in virtue of (69.2). Also, it is easy to formulate assumptions under which the surface integral in (218.4) or (218.9) may be expressed as the time derivative of another surface integral. For example, if we apply the stress boundary condition (203.6), assume that the surface Ioad s 1 Suggested by the somewhat vague analysis of GREEN [1839, 1, pp. 248-250] and KELVIN [1855. 4, § 187], given in essentially the above form by LovE [1906, 5, § 125]. 2 HADAMARD [1903, 11, ~ 265]. 3 Due in principle to HELMHOLTZ [1858, 1, § 4], though it may be traced back to D'ALEMBERT in restricted cases. Sect. 219. The virial theorem. 573 satisfies sk=- Bk where B = B(x); then when .9 is stationary we have p t(n)kxkda = p skxkda = -lh, where 5B = p B da. {218.15) f/' f/' f/' In this case, from (218.4) follows ~ + Ü + ~ =-J P dv. (218.16) r We leave it to the reader to formulate conservation theorems appropriate to this case. While the requirements 1 and 1~ may always be satisfied trivially, and while there are many cases where requirement 3 is relevant, the other requirements are satisfied only in restricted elastic and hydrodynamic situations. Our purpose in giving these theorems here is to make it clear that such a conservation law as (218.10) is not to be expected in any typical situation in continuum mechanics, where dissipation of energy is the rule, not the exception. 219. The virial theorem1 . Put IDmk==fzmikdm, 2S'rmk=fzmzkdm, (219.1) r r the quantity- 2S'rmk being the total apparent stress due to transfer of momentum (Sect. 207). Then (216.4) may be written {219.2) The skew-symmetric part of this equation was considered in Sect. 216. To interpret the symmetric part, notice that W(mk) = t ~mk• where ij is EULER's tensor (168.4) 1 with a =0. From (219.2) follows t (i:mk = 2S'rmk + P Z(m tk)q daq + J (e Z(m fk)- t(km)) dv, f/' r the trace of this equation being 2 (219.3) (219.4) (219.5) where we write G: for G:k k, the polar moment of inertia of the body about the origin, where S'r is the kinetic energy and where p is the mean pressure {204.7). The typical application of these results is to obtain time means by integration with respect to t, often under the added assumption that certain terms are periodic. 1 The quantity 1: Fa· Pa was introduced into statics by MöBius [1837, 3, § 123] and a studied by ScHWEINS [1849, 2] [1854, 2], who called it the "Fliehmoment" of the forces. Its introduction in the dynamics of mass-points is due to }AcOBI [1837, 2, § 6] [1866, 2, Vierte Vorl.]. Cf. also LIPSCHITZ [1866, 3] [1872, 3], CLAUSIUS [1870, 1], VILLARCEAU [1872, 4]. Herewe follow a generalization due to FINGER [1897, 3, §I] and elaborated by PARKER [1954, 18, § § 1, 3]. Cf. also the hint of MAXWELL [1874, 2, p. 410]. 2 CISOTTI [1923, 2, § 3] [1940, 6, § 6] [1942, 3]. 574 C. TRUESDELL and R. TOUPIN: The Classica! Field Theories. Sect. 220. 220. SIGNORINI's theory of stress means. A more useful application of FINGER's virial formula (216.4) has been found by SIGNORINI 1• Set 'oakm-PZmtkqdaq- fzm("ik-fk)dim, (220.1) d V where 'o is the volume of v. If we use a superposed bar to denote a mean value over the body, (216.4) is equivalent to {220.2) In the static case, the quantities ak m may be calculated from the applied loads only, so that (220.2) furnishes the mean stresses directly in terms of known quantities. Simple as is the reasoning used to derive (220.2), the result is important: While CAUCHY's first law (205.2) in itself constitutes an underdetermined system and is thus insufficient to yield unique values for the stresses, its corollary (220.2) - T - T Fig. 38. Body subject to intemal and extemal normal pressure. Fig. 39. Body subject to tensile Ioad. enables us to calculate the mean stress uniquely and independently of the physical constitution of the material. We now give SIGNORINI's examples, all for the static case. 1. Torque. If m=O on ~ and l=O in v, a[kml is the torque acting on v. If a[kmJ=O, (220.2) yields tkm=tmk· This is consistent with CAUCHY's second law (205.11), which holds under the stronger assumption that m =0 throughout v. If akm = 0, the load on v is said to be astatic. From (220.2), a necessary and sufficient condition for astatic load on v is (220.3) 2. Hydrostatic pressure. Suppose v is the region between a surface ~o subject to hydrostatic pressure Po and a surface ~; subject to hydrostatic pressure P; (Fig. 38), and suppose f = 0. Then if we write c for the volume of the cavity, from (220.2) follows (220.4) This shows that hydrostatic loading always gives rise to a stress system which is hydrostatic in mean 2• Moreover, if Po~P; and Po>O, the mean normal stress is a pressure. 3· Tensile loading. Let a body be subject to two equilibrated concentrated forces T and -T, acting at points a distance L apart (Fig. 39). Choosing the 1 [1932, 13, §§ 1-2]. In [1939, 11], SIGNORINI applied these results to the motion of rigid bodies. 2 NARDINI [1952, 14] has shown that when a body is subject to equalloads applied at the vertices of a regular polyhedron and directed toward its center, the mean stress is hydrostatic. He has obtained a similar result for Ioads applied at the vertices of a regular polygon. Sect. 220. SIGNORINI's theory of stress means. 575 axis of z1 parallel to T, from (220.2) we get - LT tn = -b-, (220.5) while all other tk m vanish. Thus tensile loading gives rise to simple tension in mean. 4. Body rotating steadily about a principal axis of inertia. Consider a body in steady rotation at angular speed w about the z1-axis, so that z1 =0, z2= -w2 z2 , z3 = -w2z3 • In order that the force and torque acting on the body be zero, we must have [zkdm =O l and J zl zk diDl = 0, (220.6) k =2,3; that is, the axis must pass through the center of mass and coincide with a principal axis of inertia. From (220.2) we get t _ ~kk ) kk- b ' (220.7) k = 2, 3 (unsummed), where Cfkm is EULER's tensor (168.4h with a=O, and all Fig.40. Heavysolidonanaxle. other tk m vanish. 5. Heavy solid on an axle 1• Consider a heavy solid on an axle which makes an angle 'P with the horizontal, being supported by a hinge at one end, a bearing at the other (Fig. 40). Choose the zcaxis along the axle, the z2-axis normal to it, the origin at the hinge, and the plane of zcz2 vertical. The loads are equilibrated; the reaction S of the hinge, being located at the origin, makes no contribution to (220.1); the reaction B of the bearing, being directed along z2 at a point where z2=z3=0, contributes only to a21 , but need not be mentioned in the non-polar case, since then a21 = a12 , and ll:t 2 can be calculated without knowledge of B. By (165.1), from (220.2) we calculate tJ t11 =- jffi c1 sin 1p, tJ t12 =- jffi c2 sin 'P = tJ t21 , tJ t22 = jffi c2 cos 'P, (220.8) where c is the vector from the hinge to the center of mass and jffi is the weight of the body and axle. All other mean stresses vanish. For the mean value of the mean pressure p, we have (220.9) where h is the height of c above the hinge. Thus p is a pressure or a tension according as c is above or below the hinge. When the axle is horizontal and c2 =f= 0, t2 2 is the only stress which does not vanish in mean; when the axle is vertical and the center of mass lies upon it, we have the case of a body balanced upon a single point, and the resulting expression for t11 , the only non-vanishing mean, agrees with (220.5). 1 This example and the next are due to TEDONE [1942, 13, §§ 2-3]. 576 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 221. 6. Pendulous body. Consider a body swinging rigidly about a fixed point (Fig. 41) located at distance L from its center of mass. So that the motion be possible, we assume the support pin is parallel to a principal axis of inertia, while the motion takes place in a plane containing the other two; we take these axes as co-ordinate axes. Since the co-ordinates of the center of mass are c1 =L, c2=0, the only contribution made by the weight Q9 to the _,...".Zz - bak". is made to ba11 and is of amount ~L cos cp, where cp is the angle of inclination to the vertical. From (143.6), for the components of acceleration in an inertial frame instantaneou~lycoincidingwith the co-ordinate frame we get- ~ z2, -~ +~z , 0. From (220.2) then follows bt11 _ ~L coscp + 11 • •• bt22 22 ,} (220_10) bt12 - cp 22 • bt21 -- cp (~11 • t3k- 0. Fig. 41. Pendulous body. That t12 =I= t21 except when 11 = 2 2 = 0 or when the angular acceleration vanishes is because a pendulous body not devoid of rotary inertia must be provided with a restraining torque from the support if it is to execute an accelerated rotation. 221. Moments of stress. Setting bbpqr= ~zpzqt,".da".- J zpzq(z,-f,) diDl = ~bqpr• f " in (216.2) we put \II =ZpZq and obtain1 Therefore t,p Zq + t,q Zp = bpqr• ------1 t(pq) z, + t[pr] Zq + t[qr] Zp = 2 (bqrp + bprq- hpqr} = Cpqr• (221.1) (221.2) (221.3) Henceforth we consider only the non-polar case. From (221.3) follows then (221.4) so that the second moments of the load determine the mean values of all the first moments of the stresses. Since bpqr=c,pq+c,qp in the non-polar case, the first moments of the stress determine the second moments of the Ioad. It is impossible that this one-to-one correspondence between moments of the Ioad and moments of the stress can continue indefinitely, for if it did, the stress would be determinate from the Ioad, while in fact CAUCHY's laws form an underdetermined system. lndeed, we have tpq=apq from CAUCHY's first law alone; to obtain (221.4), we use the second law as weil, reducing the number of independent stress moments tpq z, from 27 to 18, the number of independent Ioad moments. For higher moments, the number of independent Ioad moments of a given order is much less than the number of independent stress moments of the next order, and thus the full set of Ioad moments is insufficient to determine all the stress moments. To calculate the higher moments 2, set tJ b!:'bc == ~~ z~ 4 t,".da".- f ~ z~ 4 (z, -/,) diDL (221.5) f " 1 SIGNORlNI [1933, 10, § 2]. A special study of the bpq is made in [1932, 12]. 2 GRIOLI [1953, 12, § 1] [1952, 8, § 1]. Sect. 222. Estimates for the maximum stress. 577 The bn, for which a + li + c = n are the load moments of ordern, being-! (n + 1) x (n+2) in number; for n=1 they reduce to the a,p; for n=2, to the bpqr· By putting W =z~z~z~ in (216.2), we obtain an equation satisfied by the )n(n+1) stress moments of order n: at, z~ zgz~+lit, z~z~ z~+ct, z~zgz~ =li~~'' (221.6) on the understanding that any term in which the exponent " - 1 " appears is to be annulled. From the choice a = n, li = c = 0, we get t,sz~- 1 = ~ Pi'~ (s unsummed, n ~ 1), (221.7) where p'f~ denotes b~~c with exponent n for zs and with the other two exponents taken as 0. From the choice a = n-1, li = 1, c =0, we get (n- 1) t,k z~ 2 zs + t,sz~- 1 = qJ:Jn (k unsummed), (221.8) where qJ:J n denotes b~~, with n -1 for the exponent of zk, 1 for the exponent of zs, and 0 for the third exponent. When k =r, (221.8) and (221.7} yield in the nonpolar case t zn 2 z = - 1 - [q''l - ..!_ prsl] (r unsummed) . " , s n- 1 rsn n rn (221.9) From (221.7) and (221.9) we see that when n ~ 3, the values of the load moments of order n determine unique values for at least 15 of the stress moments of order n in the non-polar case. 222. Estimates for the maximum stress. SrGNORINI was the first to observe that a lower bound for the maximum stress is determined by the loading. Here we present generalizations and extensions of his work by GRIOLI 1• Considering only the non-polar case, write ti-tn, t2 = t22• t3 = taa. } t4=t2a=ta2• t,=ta1=t13• t6=t12=t21; (222.1) let Q'l1, W. = 0, 1, ... , m be a set of functions orthogonal over v, with mean norms W1~ gi ven by b W1~ = f Qfu d v ; let kb, , li, c = 1, 2, ... , 6 be the symmetric coeffi- V cients of any constant positive semi-definite form; and let Cb 21 , li=1, 2, ... , 6, W. = 0, 1, ... , m, be any constants. Then 0~~ J L kb,(tb-Cb 21 Q~ )(t,-C,!BQ!B)dv,) V b, m:,~ = I. kb, tbt, + I. kb, cb'll (W?fu c,'ll- 2 Q'l1 t,) . b, c b, c, ~~ (222.2) Choosing the constants C, 21 so that c,'ll = Q2>"\' ~(lb lbmax=lb=L_, 2 ~( rnl'll (222.15) Other estimates for the maximum stress 1 follow immediately from (221.7) and (221.9): lBIP1')1 l 1 rslmax ~ nJiz.r s~ dv' I m I q\!A- PrMn I (222"16)· ltrrlmax ~ ( n-1 )JI 2 1 · z~ z5 dv V From the identity L: km ~~ rs max = -=L;=----Jcc-l-k~~Qi8l dv • !llv (222.18} GRIOLI 2 has shown that in each case there exists a particular choice of the constants k~( rendering the bound (222.18) sharper than (222.15). In fact, with r and s held fixed, put k the means and first moments of the stress, and vice-versa. By (223.2), the conditions (223.1} become (223 .4) The former of these is equivalent to (223.5) Conversely, if (223.5) and (223.4) 2 are satisfied, the linear stress defined by (223.3) and (223.2) will be a null stress such that the stress vector on 6 iss. Thus the conditions (223.4} 2 and (223.5), along with the condition e(z -I) =const, constitute necessary and sufficient conditions to be satisfied by the loads in order that a linear solution to Cauchy's laws, with a prescribed load on the boundary, may exist. In Sect. 208 we have derived a condition that all stress vectors at a point have the same magnitude. The hydrostatic special case, t = - p 1 and x = 0, is statically determinate (cf. Sect. 208). The corresponding problern for plane stress yields t3= 0 and (t1) 2= (12) 2 and is statically determinate even in the non-hydrostatic alternative 1, since t1=-t2 implies tl = - t~ in all co-ordinate systems in the plane of stress, so that the equations of equilibrium become tl,l + ti,2= et1. ~~.1- tl,2= f.!/2. with (1-gl2g12) t~= (gllg21-gl2gn) tl+gllg22ti. (223.6) Statically determinate problems also follow from assumptions regarding the stress trajectories. For example, from (209.1) we see that in plane stress with f- x = 0, if one family of stress trajectories consists in straight lines, the other principal stress does not vary along its trajectories 2 . In this same case, the angle in (App. 6.3) is of course constant along the straight trajectories. TRUESDELL 3 has found the most general stresses compatible with the contravariant Velocity components given in cylindrical CO-ordinates r, (), Z by r=rR(t), 0 = z A (t), .i = z Z (t); (223.7) this motion is a simple type of torsion, expansion, and extension of a circular cylinder. For the physical components of d we have R 0 0 d = R trA z By solving (156.5} 2 we obtain e = eo S(t), S = exp [- J (2R + Z) dt]. The contravariant components of acceleration are We restriet attention to stresses such that 1 THEODORESCO [1937, 9]. 2 HEYMANS [1924, 4]. 3 [1955, 28, § 11] okm 871 =0, rz =0, (223.8) (223.9) (223.11) 582 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 224. and we put /==0; the dynamical equations (205A.1), in cylindrical co-ordinates, become a or rr + rr - r ifo = <::O n s r [R + R2 - z2 A 2] ' l a;{) + aifz + 2 ;o = Sr z[i + A(2R +Z)] or oz r (!o ' I an · a-;- = (!o S z [ Z + Z2J . (223.12) The general solution is fTz = - r~ :r (r2 J r7J d z) + ~ (!o r z2 S [ Ä + A (2 R + Z) J + Q, - 0 • () () = 7iY (r ,-,) + (!o r 2 S [ z2 A 2 - R - R2J , (223.13) 1 • zz = 2 (!o z2 S [ Z + Z2] + P, where P and Q are arbitrary functions of r, t. The terms proportional to (!o express the effect of the inertia of the material. The most interesting statically determinate problems arise in the theory of shells, where a state of membrane stress Ieads to the determined system (212.22), (212.23). The general solution of these equations will be developed in Sect. 229. IV. General solutions of the equations of motion. 224. Steady plane problems. So as to make clear the approach to stress functions in general, we begin with the siruplest case, when the motion is steady and plane, the stress is plane 1, and the assigned forces satisfy (207.1). Then in reetangular Cartesian co-ordinates CAUCHY's first law (205.2) assumes the form (txx- ex2 - V), ... + (t..-y- ex:V).r = o, } (224.1) (tyx- e:Vx),x + (tyy- (!Y2- V),y = 0. Since each of these formulae is a condition of integrability for a differential form, the pair is equivalent to the existence of functions F and G such that txx-ex2-V=F.r· t..-y-ex:V=-F,,..} (224.2) lyx- (!YX = G,)'' tyy- (!Y2- V=- G,x• When the stress is not symmetric, no simplification is possible, but in the nonpolar case, to which this subchapter is restricted, the left-hand sides of (224.2) 2 and (224.2) 3 are equal, so that F,,. + G,y = o. (224.3) This, in turn, is a condition of integrability for the existence of a function A such that F=-A,y, G=A,r (224.4) Substitution in (224.2) yields txx- ex2 - V=- A.rr• trr- e:Y2 - V=- A,xx• t,.y- exy = A,xr. (224.5) When t-(!XX- V 1 is twice continuously differentiable, the theorem on exact differentials implies that the existence of a function A satisfying (224.5) is 1 These results hold also when 1zzoF 0 but lzz z=O, as for example when the stress is a hydrostatic pressure independent of z. ' Sect. 224. Steady plane problems. 583 necessary and sufficient that (224.1) hold. Accordingly, (224.5) gives the general solution of the equations of motion for the case considered. The function A is the celebrated stress function of AIRY1. Our argument, since it rests upon the theorem of the exact differential, implies that A is single-valued in simply connected regions, generally multivalued in multiply connected regions 2• Since a unit normal to the element of arc i dx + j dy is - i dyfds + j dxfds, from (203.4) and (224.5) we obtain the components of the stress vector across the arc in the forms t (ra)x - (! i. Pn + V !!_t ds -A - ,yy !:J'_ ds + A ,xy !!.!_- ds - dA,y ds ' l (224.6) t _ · · _ V dx __ A !_}'-_ _ A dx __ dA,x (raJy eYPn ds - ,yx dx ,u ds - ds · Therefore the normal stress and shear stress on the element of arc are given by t - p'2- V=- dA,y !.!.._- dA,% ~ n (! n ds ds ds ds ' __ !____ (A dx + A !_}'-_) + A !____ (dx) + A _!_ (dy) - ds ,x ds ,y ds ,x ds ds ,y ds ds ' - - d2A + "(- A !!t + A !!.!_) - ds2 ,x ds ,y ds ' (224.7) d2A dA = - ds2 + "!in' • • d2A dA tt- (! Pn Pt = ds dn + "ds' where Pn and Pt are the normal and tangential components of velocity. These results are due to MICHELL 3, for the case of equilibrium. If the element is a stationary boundary, the momentum transfer vanishes upon it, and (224.7) yields the stress vector directly. To find the dynamical significance of the intermediate function F, we integrate along a curve c from ;x:1 to ;x:2 , obtaining Fl:~: Fi:r:1 = f dF = f[(- txy + eiy) dx +(tu- ei2 - V) dy ], l c c (224.8) = f[i (tu- ei2 - V) + j (txy- ei:Y)] [i dy-j dx]. "' If we include the apparent stress due to assigned force and to transfer of momentum, by (203.4) the integrand is the x-component of the stress vector actinc upon a cylinder of unit height based upon the curve c. Thus the difference of 1 AIRY [1863, 1] considered only the case when e:i::i:- V 1 = 0, and he did not prove necessity; the generality of the solutionwas asserted by MAXWELL [1870, 4, pp. 192-193]. The first fully satisfactory treatment for the case of equilibrium was given by MICHELL [1900, 6]; among other things, he included V [ibid., p. 100] and gave an explicit form of (224.5) in orthogonal curvilinear co-ordinates [ibid., p. 111]. E.R. NEUMANN [1907, 6, § 1] obtained (224.5) for motion in which t is hydrostatic and V=O. Cf. also BRAHTZ [1934, 1], BATEMAN [1936, 1, § 2], CRocco [1950, 5]. VorGT [1882, 4, pp. 297-298] remarked that (224.1) continues to hold when all components of stress are constant in the z-direction; in this case, which is often appropriate to torsion of a cylinder, the z-component of CAucHv's first law yields a function W such that ty:- e yz = l~x· t,..- exz =- ~Y· when (V+ ez2),z =0. 2 After making this observation, MrcHELL [1900, 6, p. 103ff.] determined the nature of this multi-valuedness for the case of linear elasticity theory. a [1900, 6, p. 110]. An incorrect formula of this kindwas given by NEUMANN [1907, 6, § 3]. 584 C. TRUESDELL and R. TouPIN: The C!assical Field Theories. Sect. 225. the values F at ;r1 and x 2 equals the x-component of the force 1, including inertial force, acting upon c. When c is a stationary boundary, the momentum transfer terms contribute nothing to the integral, which then yields the force alone. A similar interpretation holds for G. To interpret2 the function A, we see that for a curve c connecting x 2 to x 1 we have Alor2 - Alor1 = f dA= f (A,xdx + A," dy) = f (Gdx -F dy), " " " = (x2- xl) Gior2 - (Y2- YI} Fizz+ + J {(x- x 1} [(t"y- (}'y2 - V) dx- (tx"- eiy) dy J + (224.9) " + (y- y1) [(tu- ei2 - V) dy- (tx"- eiy) dx ]}, = (xz- xJ} Gior2 - (Y2- YJ} Flor2 + 2 where 2 is the torque about x 1 exerted by the stress, including the apparent stress due to rate of change of momentum and to applied force, acting upon the right-hand side of c, thought of as directed from x 1 toward x 2 • While we have used reetangular Cartesian co-ordinates, the extension of (224.5) to arbitrary curvilinear Co-ordinates is immediate 3 : tkm- eikim- V gkm =- ekP emq A,pq• (224.10} where gkm is the contravariant metric tensor in the plane, ekP is the absolute alternating tensor, and "," denotes covariant differentiation based on g. Since (224.10} is a tensor equation, and since it reduces to (224.5) when the co-ordinates arereetangular Cartesian, it is valid in all co-ordinate systems. 225. Generalized Iineal motion. Consider a motion in which t, iJ, (!, and V all depend only upon x and t. Using (161.22) and the x-component of (205.2), we readily infer the existence of a function U such that 4 o(t;,, L;xl U i= _ Z:.:x 1 t =V+ o(x,t)- e = ,XX' L;xx ' XX L;xx • (225.1) The second of these equations may be written in the form i =- (t;~l,t . (225.2) (L; rl,, ' thus the velocity equals the slope of the curve U.x = const in the t-x plane. The remaining components of (205.2) yield the existence of functions V and W such that Z = W xfUxx, In the lineal case, V= W = 0. (225.3) 1 This result is due to MAXWELL [1870, 4, pp. 192-193], who considered only the static case; he called the vector F, G "the diagram of stress ". His derivation was criticized and corrected by MrcHELL [1900, 6, p. 107]. In rediscovering this result for the case of hydrostatic stress, BATEMAN [1938, I, § 1] called Fand G the "drag and Iift functions". 2 PHILLIPS [1934, 4] based his proof of the existence of AIRY's function on this interpretation. Cf. also SoBRERO [1935, 7]. An energetic interpretation for A and for stress functions of all types considered below has been given by L. FINZI [1956, 7, § 6]. 3 B. FINZI [1934, 2, § 1]. 4 The result was worked out by W. KrRCHHOFF [1930, 2, § 1], following a suggestion of E.R. NEUMANN [1907, 6, § 6]. It is rediscovered by McVITTIE [1953, 18, § 3]. Sects. 226,227. General solution for a flat space. 585 226. Conventions for the remaining general solutions. All general solutions are obtained by essentially the same reasoning as in Sect. 224, although the details may be more elaborate. At bottom, we are to integrate1 skm =0 ,m ' (226.1) that is, to find the most general symmetric null stress. It is the additional condition of symmetry that makes the problern interesting and different from that solved in Sects. 161 to 164. The variations in the answers from case to case arise because of different numbers of dimensions and different metrics on which the covariant differentiation is based. For the Euclidean case, whatever the number of dimensions, the entire analysis consists in repeated use of the fact that a continuously differentiable tensor whose divergence vanishes may be expressedas the curl of another tensor. The tensor potentials so determined exist subject to various conditions that may be found in the literature on the theory of the potential. We do not remind the reader again that when all co-ordinates are space Coordinates we may take skm as tkm- exkxm with an additional term -V gkm when f satisfies (207.1). For general f, the linearity of CAUCHY's laws enables the general solution to be gotten by adding to any particular solution the general null stress. The problern of finding a particular integral is relatively trivial in a flat space. In a convex region, quadratures suffice. More generally, for any Riemannian space, Iet u be any solution of the linear differential equation (226.2) where R::O is the Ricci tensor. Then it is easy to verify that a particular solution pskm of the system skm,m+efk=o, skm=smk is given by2 (226.3) We now turn our attention to determining the most generalnull stress 3. 227. Generalsolution for a flat space. We suppose s tobe twice continuously differentiable. In order that skm m = 0 in a flat space of any number of dimensions, it is necessary and sufficient tliat there exist a tensor b such that 4 (227.1) The condition skm =smk may now be written in the form (bkmP _ bmkP),p = 0. (227.2) 1 Cf. the treatment of this class of problems by FINZI and PASTORI [1949, 11, Chap. IV, § 7]. 2 This result is suggested by the special case for flat spaces given by ScHAEFER [1953, 28, § 5]. 3 Most of the material in the rest of this subchapter is taken from a work of TRUESDELL [1959, 11]. ' As observed by GWYTHER [1913, 2], for a stress tensor which is not symmetric the analysis breaks off at this point. In the more general viewpoint allowing couple stress as weil as ordinary stress, all differential conditions of equilibrium are prescriptions of divergences, as shown at the end of Sect. 205. Stress functions for the system (205.2}, {205.10) are given by GüNTHER [1958, 4, § 3]. 586 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 227. This condition, in turn, is equivalent to the existence of a tensor c such that where ckmpq = _ ckmq p = _ cmkPq 0 Therefore 2bkmP = (cPkmq + cmPqk + ckmPq),q, so that (227o1) becomes where skm = kPkmq ,pq• kPkmq = i (cPkmq + cmqPk), } kPkmq = _ hkPmq = _ kPkqm = hmqpk 0 (22703) (227.4) (227o5) (227.6) (227o7) The foregoing derivation, given by DORN and SCHILD1, shows that (227o6) furnishes the gener•al solution of Cauchy's laws in a flat space of any dimension. In the three-dimensional case, set a - 1 e e hPkmq rs = 4 rpk sqm • so that hPkmq = ekrp emsq a, 50 a,. = a.,. Then (227.6) becomes the Gwyther-Finzi general solution2 : skm = ekrp emsq a,s,pq. (227o8) (227.9) (227o10) B. FINZI noticed that the tensor a is indeterminate 3 to within an arbitrary symmetric tensor· a0 satisfying ekrp emsq a~s,pq =0. As we have mentioned in 1 [1956, 4]. The basic idea was suggested by BELTRAMI [1892, 2]. Cf. also MORERA [1892, 10]. A variant ofthis derivation had been given earlier by GüNTHER [1954. 8, § 1]. He begins by introducing the skew-symmetric dual tensor of fourth order, 1/.mpq = Bkmn Bpqs T"s' (*) which he interprets as a "transversal stress tensor". An explicit solution for 1/.mpq in terms of stress functions is simply obtained. In DoRN and ScHILD's proof, the duals appear in (227.8)0 While GüNTHER's solution for the dual tensor is indeed valid, as he says, in a flat space of any dimension, to derive from it the ordinary stress tensor we must use the inverse of (*) and hence presume the nurober of dimensions to be three. Of course GüNTHER's proof can ·be adjusted to the n-dimensional case also, but DoRN and ScHILD's proof is equally simple and natural in all cases. 2 While the Cartesian tensor form of (227.10) is almost obvious from an equation of KLEIN and WIEGHARDT [1905, 3, Eq. (33)], they did not infer it. GWYTHER [1912, 3] obtained (227.10) in orthogonal curvilinear co-ordinates, writing out the special cases appropriate to reetangular Cartesian, cylindrical polar, and spherical polar co-ordinates (cf. also [ 1911, 5]). His steps are such as to imply the necessity of the result; the sufficiency is immediate. B. FINZI [1934, 2, § 3] observed that (227.10) yields a symmetric tensor satisfying skm ". = 0; for a proof of completeness he was content to refer to the previously known fact that the special cases (227.12) and (227.13) are complete. While KRÖNER [1954, 11, § 1] does not prove completeness, he begins from an invariant decomposition of symmetric tensors that might be made the basis of a rigorous proof. 3 This fact is used by PERETTI [1949. 23] to show that it is possible to choose a in such a way that from its components may be obtained simple expressions for the resultant force and moment of the stresses on a surface. Cf. also BLOKH [1950, 2]. GüNTHER [1954. 8, § 3] gives simple expressions for the resultant force and torque on a body in terms of integrals of stress functions around particular curves. ScHAEFER [1955. 22] discusses the nature of null stress on this basis. In a later work, SCHAEFER [1959. 10] interprets the components of a as dynamical actions upon the bounding surface, conceived as being that of a plate and of a slab simultaneously. Sect. 228. Two applications: rotationally symmetric case, and plane unsteady motion. 587 Sects. 34 and 84, such a tensor is of the form a~, = b(m,r)• where bis an arbitrary veetor. For a given a, in reetangular Cartesian co-ordinates we may choose b so as to satisfy one or the other of the conditions b(m,r) = - amr> bm,m =- amm r=f=m } (unsummed). (227.11) These two choices of b show that for reetangular Cartesian co-ordinates there is no loss in generality in assuming in the first place that a is diagonal, or that the diagonal components of a are zero. The former alternative yields etc. (227.12) with a1 a,."'"' a3 = aYY, a3- az z; the latter alternative yields (227.13) with a4 a23 , a5- a31 , a6- a12 • These two forms of the general solution were obtained by MAXWELL and MoRERA1, respectively. As follows from the special choices of a made to derive them, these special forms are not invariant under transformations even of reetangular Cartesian co-ordinates. The explicit form for (227.10) in reetangular Cartesian co-ordinates, with no restrietions on the six potentials, may be obtained by adding together the right-hand sides of (227.12) and (227.13). Other special choices of the potentials are possible 2, but it by no means follows that a solution obtained by imposing three arbitrary conditions on the six potentials akm remains complete 3 • An attempt to adjust the stress funetions so as to satisfy the stress boundary condition (203.6) 1 on a reetangular parallelepiped has been m.ade by FILONBNKOBoRODICH4. 228. Two applications: the rotationally symmetric case, and plane unsteady motion. If we write (227.1 0) explicitly in cylindrical polar co-ordinates, at the same time supposing that all derivatives with respect to the azimuthangle are zero, we 1 [1868, 12], [1870, 4]; [1892, 9]. Using results given by BELTRAMI [1892, 2], MaRERA [ 1892, 10] modified his derivation so as to yield (227 .12) alternatively to (227 .13). Cf. also GwYTHER [1913, 2]. A Iiterature devoted mainly to rediscovery of known results regarding this subject has arisen recently; cf. KuzMIN [ 194 5, 3], WEBER [ 1948, 38], MaRINAGA and NöNa [1950, 19], ScHARFER [1953, 28], LANGHAAR and STIPPES [1954, 12], ÜRNSTEIN [1954, 16]. 2 Cf. MaRINAGA and NöNa [1950, 19, § 3]. BLOKH [1950, 2] lists the essentially different forms which result from such special choices: 5 in reetangular Cartesian Co-ordinates, 20 in general co-ordinates, 18 in cylindrical co-ordinates, 10 in cylindrical Co-ordinates for rotationally symmetric problems, 19 in spherical Co-ordinates. 3 What reductions are possible is not obvious. In writings on stress functions there is a deplorable custom of inferring completeness by merely counting the number of arbitrary functions. Apart from the logical gap in such inference, its danger is illustrated by the solution written down without proof of completeness by PRATELLI [1953, 25, § 1]: (A) While indeed a solution for any choice of the three arbitrary potentials F, H, K, it is not general. In fact, by using the expression for the product skpq smrs as a determinant of t5~'s, we may put (A) into the form 5km= p.q,qgkm_ p,km, (B) where P ~F- H + K, and it is easy to exhibit solutions of skm m = 0 which cannot be expressed in the form (B). ' 4 [ 19 51, 7 and 8] [ 19 57, 4]. The work rests on trigonometric series or special functional forms. 588 obtain1 C. TRUESDELL and R. TouPIN: The Classical Field Theories. --8+13 25 rr-a,. -a,--a,, r 1' ' r , - 8 2 8 1 1 zz=a"+-a,--a" ' r ' r , rz=- a -j--a --a - [ ll 1 8 1 1] •' r r ,z' 0-- 4 1 4 1 4 6 + 2 z--a"--a,+- 6 2 a +a., -a., • r ' r ' r • Sect. 228. (228.1) where commas denote partial derivatives, and where we have set a1 = a,,, a8=a66fr 2, a3=a .. , a4=a61fr, a5=a," a6=a,6fr. The potentials a1, a8, a3, and a5 occur only in the first four members of Eq. (228.1); the potentials a4 and a6 , only in the last two. Rotationally symmetric stress distributions in which rO = 0, Oz = 0 are often called torsionless; the most general stress of this kind is obtained by setting a4 = 0, a6 = 0 in (228.1). In any case, the six potentials may be reduced to three in a variety of ways 2• For example, if we set L = a8 + __1_ a8 - __1_ a1 ) ,r •' r r ' M =a,.+-a,--a,- - 8 1 .~ 2 5 L .,, , r ' r ' , .. (228.2) then the first four members of (228.1} become 3 ;"Y =L, .. +M, Oo = (rM),,+L,..,) - 1 - zz=L"+-L,, rz=-L", , " . ' (228.)) furnishing the generat solution for torsionless rotationally symmetric stress. Similarly, if we put (228.4) the last two members of (228.1} become 4 - 1 rO=----.-W,, r• • - 1 0z=-2 W,, r ' (228.5) fumishing the generat solution for purely torsional stress. A more interesting application, resting upon a device to be used more strikingly in Sect. 229, begins with the observation that in the case of plane motion, 1 BRDil:KA [1957. 2, § 6]. A more symmetrical expression is given by MARGUERRE [1955. 16], but his potentials are connected by a condition of compatibility, and there is no proof of completeness. 2 Cf. the work of BLOKH [1950, 2]. 3 This solution, whose completeness is easy to prove also directly from the equations of equilibrium, was obtained by LovE [1906, 5, § 188]. Variants are given by BRDil:KA [1957. 2, § 4]. ' This isanother form of the solution of VoiGT mentioned in footnote 1, p. 583. Cf. also MICHELL (1900, 7, p. 133], PHILLIPS [1934, 4]. Sect. 229. Stress functions for membranes. 589 Eqs. (211.5} with F=O are of the form skm,m=O in a flat space of three dimensions with reetangular Cartesian co-ordinates x, y, t. Restriding attention to these special Co-ordinates, we apply the solution (227.10). After time differentiations and time components are written explicitly, the result turns out to be in tensorial form under transformation of general Co-ordinates in the plane: ) (228.6) where a prime denotes 8j8t and where the range of indices is 1, 2. In the steady case, this reduces to Amv's solution (224.10} with A =-a33 . The six potentials may be reduced to three in various ways. For example, if we choose a to be diagonal in a particular reetangular Cartesian co-ordinate system, we obtain 1 - (! = a~yy + a~:w (!X= a~xt• (!Y= a~yt• ) ·2- A 2 ·2 _ A 1 txx=(!~.--=_- ,yy+a,tt• tyy-(!Y -- ,xx+a,tt> (,~, (!XY-A,xy• (228.7} generalizing Amv's solution to the case of arbitrary plane motion. 229. Stress functions for membranes. From the result at the end of Sect. 212, the problern of equilibrium of a membrane leads to a system of the form (226.1); explicitly, we are to find the most general symmetric tensor s~ ~ satisfying os~~ { ~ } M { .; } 6A _ Tue-+ ;..; s + ;..; s -o, (229.1) where the {;.~.;} are Christofiel symbols based on the positive definite surface metric tensor a6 ;, and the range of indices is 1, 2. There have been several attacks upon this intrinsic problem, which turns out to be more difficult than those considered in the preceding sections because forcurvedspaces an analogue of STOKES's representation (App. 32.9) of a solenoidal field, which yields (227.1) in a Euclidean space of arbitrary dimension, is not known. A different approach is needed. STORCH! 2 has obtained a solution in geodesie co-ordinates by a direct and laborious reduction. At the present writing, a general invariant solution is not yet known, but we present what results are available. The similarity in form between the conditions of compatibility (84.3} and the general solution (227.10} has been remarked for half a century. This simi1 A similar but not obviously identical solution is obtained by KILCHEVSKI [1953, 14]. 2 [1950, 28]. STORCH! [1950, 27] observed that if for an arbitrary surface we choose co-ordinates x, y so that ds2 =). (dx2 + dy2), then a solution is furnished by the formulae ).2sxx = o2Afcy2, ).2sxY = ).2sY"' = - o2Afoxoy, ).2sYY = ()2Afox2, provided 82 Afox2 + 82 Afoy2 = o. For such solutions the mean pressure vanishes: Ä(s"'"'+sYY)=s~=O, but it is not shown that all solutions such that s~ = 0 are included. STORCH! treated the case of a surface of revolution in [1949, 28]; in [1952, 18] [1953, 29], the case when a general solution involving derivatives of orders no higher than the fourth is possible. Cf. the counterpart expressed in terms of the conditions of compatibility which we have mentioned in Sect. 84. A solution for minimal surfaces is initiated by CoLONNETTI [1956, 3]. We do not discuss the older solutions for special cases defined by conditions of in extensi bility. For Riemannian spaces of 2, 3, and 4 dimensions, a special solution containing an arbitrary function of the total curvature is obtained by STORCH! [1957, 13]. 590 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 229. larity may be used to yield a direct solution1 of (226.1) based upon the classical principle of virtual work. While that principle will be developed in Sect. 232, here we need only a special case which may be verified at once, namely, that s satisfies (226.1) if and only if f skmd dv-O km - (229.2) " for all fields d suchthat dkm= c(k,m) for some vector c vanishing upon the boundary. Here, as henceforth in this section, we systematically neglect surface integrals as having no effect on the differential equations we seek; then the condition (229.2) follows at once from GREEN's transformation and hence is valid in any space where covariant differentiation is defined. Now suppose the conditions of compatibility (Sect. 84) for the system dkm=c(k,m) have the explicit form 0 _ kmd + ßkmrd + kmrsd + S:kmrstd + -(X km km,r Y km,rs U km,rst · · ·' (229-3) where the tensors IX, ß, ... , are symmetric in the first two indices. Introducing a multiplier A, we replace (229.2) by J [skmdkm- A (1Xkmdkm + ßkmr dkm,r + .. ·)] dv = 0' " where the variation of d is now unrestricted. Equivalently, J[skm_IXkmA +(ßkmrA),,- ···]dkmdv =0. Hence for any s satisfying (226.1) there exists a function A such that skm = IXkm A - (ßkmr A),, + (ykmrs A),,.- .... (229.4) (229.5) (229.6) In this general solution, the stress function A appears as a multiplier for the constant expressing the compatibility of the field d with virtual displacements in the space considered. To apply to the two-dimensional case the local result obtained, consider first a surface of constant Gaussian curvature K. By comparing (229. 3) with {84.13) we have (229.7) Substitution into (229.6) yields the generat solution for surfaces of constant curvature 2 : s60 = e6a eO'~' A,aq; + KA a60 . (229.8) Second, consider a surface applicable upon a surface of revolution. By comparing (229.3) with (84.15) we have IX6E = K,AK,Aa6E + K·6 K·"' ß6Ea=KK•aa6<, } b6EarpVJ =- ea (6 e<>'P K·"'. (229.9) 1 Given by TRUESDELL [1957, 17], revising work of L. FrNzr tobe described in Sect.224. Earlier ScHARFER [1953, 28, § 4] had introduced multip!iers in just the same way, concluding that "Jeder Verträglichkeitsbedingung ist eine Spannungsfunktion zugeordnet", but his presentation employs results of a kind valid only in flat spaces, and he did not mention any further possibilities. GüNTHER [1954. 8, § 2] had in effect noted the method and had remarked that the tensor of stress functions in a three-dimensional flat space may be interpreted as Lagrangean multipliers expressing the reactions agairrst the geometrical constraints but had concluded that "no new point of view results". 2 B. FrNzr [1934. 2, § 2] verified that (229.8) satisfies (226.1) when K is constant but did not prove the completeness of this solution; his result is rediscovered by LANGHAAR [1953. 15]. B. FINZI [1934, 2, §§ 4, 7] conjectured also corresponding general solutions for spaces of three and four dimensions with constant curvature; TRUESDELL [1959, 11, § 12] establishes their completeness by this method. Sect. 229. Stress functions for membranes. 591 Substitution into (229.6) yields the generat solution for a surface applicable ttpon a surface of revolution of non-constant curvature 1 : s ,A ,Aa

K·"' JA + (229.10) ·"'" ,q>lp ,;. + [2e"'(6 e eA S'F!J Ax." 'PD, A s' where (227.7) imply the following conditions of symmetry for A: Al:." 'PD=- A."l:'F!J =- Ax."D'F• A Idl'FD = A 'FDI!f>. (230.4) Inspection of (230. 3) shows that only the second of these sets of symmetry conditions is essential; the first set may be abandoned without impairing the solution. The tensor Ais indeterminate to within a tensor A 0 suchthat erA Elf> eA S'FD A'l-."'FD, A x = 0. The mostgeneralsuch tensorisalinear combination of tensors of the type B Idl 'I', D; to satisfy the essential symmetry condition (230.4)8 , we may choose Bx."'F D + B'FDI,."; if, finally, we wish to satisfy also the condition (230.4)1, 2 , we have A~'dl'F!J: 4B[Idl]['l',!J] + 4B['FS}][I,dl]• l - B l:dl'F, !J- B Edl!J, 'I'- B."l:'F,!J + B."l:!J, 'I'+ + B'l'Dl.',."- B'F!Jdl,E- Bu'Fx,<~> + BD'Fdl,x· (230.5) Since classical space-time need not be regarded as a flat tour-dimensional space, these formulae do not necessarily have invariant significance for it (but cf. Sect. 153). However, in reetangular Cartesian co-ordinates in an inertial frame (211.5) with F=O do reduce to the form yrA,A = 0, yrA = pr, (230.6) For these special co-ordinates, then, the solution (230.3) is general. In this solution, we may write time differentiations and time components explicitly. The resulting formulae, derived in reetangular co-ordinates, turn out to be of tensorial form under transformations of the space co-ordinates alone. These formulae, valid for all curvilinear co-ordinate systemsinan inertial frame, are 2 : - (! = s'' = e•mnePqr Amnqr,sp• ) _ nxk = 5k4 = _ 6ksmePq•(A' + 2A ) o: smqr,p 4mqr,sp (230.7) tkm _ exk ,im= 5km = 4ekPq emsn A,p,s, qn + +2(ekPqem•"+eksnemPq)A' + ekPqem•nA" 4npq,s pqsn• 1 B. FINZI [1934, 2, § 5] wrote down (230.3) and symmetry conditions consisting in (230.4) and a further requirement which we do not verify; he was content to infer completeness by counting the number of assignable arbitrary functions. A somewhat involved proof was given by MaRINAGA and NöNo [1950, 19, § 4]; we do not follow the argument whereby they claim [ibid., § 5] to establish the alternative form sl'Lf = el'Adl'FeASI'l'Adls,A:I:; they give corresponding results in n dimensions. 2 B. FINZI [ 1934, 2, § 6]. In reetangular Cartesian co-ordinates, this result is rediscovered by ARZHANIKH [1952, 1] (while he uses 21 potentials, obviously one may be eliminated). KILCHEVSKI [1953,' 14] observes that if RrA is the contracted Riemann tensor based on the Riemannian metric tensor G l'A, then in any Riemannian 4-space the quantities TrA == RrA - tGrA R"'." satisfy TrA A = o. Putting GrA = drA + eHrA, he calculates TrA = eQrA+O(e2); hence follows öQFAfo~A = o, so that Ql'Lf, in reetangular Cartesian co-ordinates, gives a solution of the type presented in the text above. A similar approach, involving a detour through relativity theory, is presented by McVITTIE [1953, 18, § 2]; that his solution is not general is remarked by WHITHAM [1954, 26], who obtains what appears tobe a special case of FINZI's solution in a special co-ordinate system. Rediscoveries of Gther special -cases are made by MILNE-THOMSON [1957, 9] and by BLANKFIELD and McVITTIE [1959, 1 and 2). Handbuch der Physik, Bd. III/1. 3 8 594 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 231. where primes denote ofot, and where the symmetries of the tensors Apqmn• A 411 Pq• and A4 p4 q may be read off from (230.4). This is Finzi's general solution of the equations of balance of mass and momentum in an inertial frame. In the case of equilibrium, this solution reduces to (227.10) with 4A4 p4m=apm· In a particular co-ordinate system, by means of (230.5) we may impose 14 conditions upon the 20 independent potentials occurring in the solution (230.7). For example, we are tempted to take A,p"'' as diagonal, A 4mqr as zero, and Amnqr as zero when m =!= q andn =!= r. We have been unable to prove that it is possible to choose Hin (230.5) in such a way as to justifythis special choice of A; if it is legitimate, then, writing A 1 = 4A4141 , ••. , A 4 = 2A 2323 , ... , in this case we reduce (230.3) to the form - e = A~:u + A~yy + A:.,, ei = A~xt .... , } Iu- ei 2= A~u+ A:yy+ A~tt • ... , lxy- exy=- A~xy • ... , (230.8) extending MAXWELL's formulae (227.12) so as to yield a simple yet general solution in terms of six potentials. When the field theories were discovered in the eighteenth century, solutions in arbitrary functions such as those presented in this subchapter were sought eamestly, but, for the most part, sought in vain. In the nineteenth century, researches on partial differential equations tumed away from such general solutions so as to concentrate upon boundary-value problems. When, in the twentieth century, the general solutions were at last obtained, scarce attention was paid to them, and to this day they remain virtually unknown. Though so far they have been used but rarely, they might turn out to be illuminating in .studies of underdetermined systems, where the conventional viewpoint of partial differential equations has gained little. V. Variational principles. 231. On the qualities of variational principles. In favor of variational principles as expressions of physicallaws it is commonly alleged1 : 1. They are statements about a system as a whole, rather than the parts that it comprises. 2. Since they refer to the extremum of a scalar, they are invariant, and may be used to derive the special forms appropriate to any particular description. 3. They imply boundary conditions and jump conditions as well as differential equations. 4. They automatically include the effects of constraints, without requiring that the corresponding reactions be known. 5. They have heuristic value for suggesting generalizations 2• No. 2 is outmoded, now that the principles of tensor analysis offer usasimpler and more direct method, used throughout this treatise, for obtaining invariant statements. No. 3 is shared by the direct statement of physical laws in integral form as equations of balance 3, as shown by the development of continuum mechanics given earlier in this chapter (cf. especially Sects. 203 and 205). Moreover, the boundary conditions ernerging from a variational principle depend upon what boundary integrals, if any, are included in the statement of the principle, and the selection of these boundary integrals is not always dictated by the physical idea which the variational principle is assumed to express. No. 4 is a somewhat dubious blessing, since only a special kind of constraints is included 1 Cf. HELLINGER [1914, 4, § 1], who speaks of "the pregnant brevity". 1 A sixth, the use of direct variational methods to calculate or prove existence of solutions, may.be added in cases where a determinate system is considered but in the present treatise, devoted to underdetermined systems, is not relevant. a Cf. footnote 4, p. 232. Sect. 232. Virtual work and the Lagrange-D' Alembert principle. 595 in each case, namely, those constraints having no effect on the quantity being varied. In mechanics, these are typically constraints which do no work. Not all constraints are of this kind, and for those which are not, the variational approach requires as direct a statement as does any other. No. 5, while having a basis in the physics of this century, is largely an expression of taste. There remains only No. 1, along with the elegance that variational principles sometimes exhibit. Both these are reduced if not annulled when the variational principle itself is awkward of unnatural. This is usually the case in continuum mechanics. The lines of thought which have led to beautiful variational Statements for systems of mass-points have been applied in continuum mechanics also, but only rarely are the results beautiful or useful. For completeness, we now present variational principles and related topics, but we regard them as derivative and subservient to the principles of mechanics already developed. In particular, no variational principle has ever been shown to yield Cauchy's fundamental theorem (203.4) in its basic sense as asserting that existence of the stress vector implies the existence of the stress tensor1 . For the statements henceforth we take the stress tensor rather than the stress vector as given. Our purpose, in each case, is to learn the role of the effective force of the stress, tkm,m, in modifying the classical theorems concerning mass-points 2 • Our presentation concerns solely the formal problern 3 of setting up expressions such that the vanishing of their first variation is equivalent to CAUCHY's laws. Analytical questions and, except in Sect. 23 5, the existence of minima are not discussed. 232. Virtual work and the Lagrange-D' Alembert principle. The principle of virtual work, which when the reaction of inertia is included may be called the Lagrange-D'Alembert prin,ciple, is the oldest general variational form of the equations of mechanics. We begirr by following but extending the traditional development 4 of the principle. Some variants are presented afterward. Corresponding to a sequence of independent variations ~xk, ~xkm• ~xkmp• ... , that is, a set of arbitrary covariant fields, we define the virtual work done on a 1 The derivation given by HELLINGER [1914, 4, § 3a] fails through petitio principi, since the stress components appear in the original variational principle. We do not understand the remark attributed to CARATHEODORY by MüLLER and TrMPE [1906, 6, footnote 32]. Existence of the stress tensor can be proved from variational principles which assume the existence of an internal energy having a special functional form. Such results are presented in Sects. 232 A, 236, and 262. 2 It is possible simply to transcribe the theorems for systems of mass-points, written as Stieltjes integrals over material volumes, and then add tkm, m to the contravariant force vector. Cf. EICHENWALD [1939, 7]. a We do not attempt to discuss variational principles from the axiomatic or conceptual standpoint. For the difficult question of the contrast between the momentum principle and the Lagrange-D'Alembert principle, cf. HAMEL [1908, 4, Kap. 2, §§ 1, 3]. 4 HAUGHTON [1855, 2, pp. 99-100], KIRCHHOFF [1876, 2, Eilfte Vorlesung, § 5], BELTRAM! [1881, J, pp. 385-388], HELLINGER [1914, 4, §§ 3a-b]. A variant, bringing in explicitly the dependence of general curvilinear Co-ordinates on reetangular Cartesian co-ordinates, was given by MoRERA [1885, 6, § 1]. We do not Iist the many sources that formulate a principle of virtual work for special systems such as rods or membranes, nor do we trace the origin of the principle for continuous media in general through the studies of LAGRANGE on perfect fluids, of GREEN and KELVIN on elastic bodies. Thc firsttreatmentvalid for general continua isthat of ProLA (1833), tobe given below. The "general formula" of LAGRANGE [1788, J, Seconde Partie, Seconde Sect., ~ 7] is valid only for systems of mass-points and certain other special systems. For D' ALEMBERT's principle, see above, p. 532, footnote 1. All the foregoing references concern only the classical case, where the terms in Oxkm• oxkmp, ... , are absent from (232.1). The general theory given here is suggested by an intermediate case due to HELLINGER [1914, 4, § 4b]. 38* 596 C. TRUESDELL and R. TOUPIN: The Classica] Field Theories. body "f/ as the linear form 1 ~ = j [sk~xk + skm~xkm + skmP ~xkmp + .. ·]da+ I + J {e [jk ~xk + fk"' ~X km+ fkmP ~Xkmp + · · ·] - r - [tkm ~xk,m + tkmp ~Xkm,p + .. ·]} dv. The quantities sk, skm, ... ,er. efkm, ... , tkm, tkmP, ... are defined simply coefficients of the form. By GREEN's transformation follows ~ = p [(sk- tkmn") ~xk + (skm_ tkmPnp) ~Xkm +···]da+ ) + j [(e fk + tk"',m) ~Xk + ((> fk"' + tkmP,p) ~X km+···] dv · The principle of virtual work is the assertion f (> [xk t5xk + ixmt5xkm +pkpmf:P Oxkmp + · · ·J dv = ~. r Sect. 232 (232.1) as the (232.2) (232-3) p being, as usual, the position vector from an arbitrary origin. Eq. (232.3) is to hold for all variations consistent with the constraints. For the time being, we leave aside the effect of constraints and assume that the virtual fields may be varied arbitrarily, Then, by (232.2), the principle of virtual work is equivalent2 to the system (205 .19), with boundary conditions sk = tkm nm, etc. The classical special case of (232.3) is Jexkoxk=~ pskoxkda + f[efkoxk-tkm()xk,m]dv. (232.4) r f/' r This special case, for unconstrained variations, is equivalent to CAUCHY's first law (205.2). We now consider some alternative variational formulations for one or both of CAUCHY'S laws. If we restriet the variations considered, we may avoid using the stress components directly in the principle of virtual work. Set )ffi - p sk oxk da +Je jk oxk dv. (232.5) f/' r The theorem oj Piola 3 asserts that the condition Je xk ~xk dv = )ffi (232.6) r 1 Here and the sequel we set o:rk m == (o:rk) .,., etc. 2 For the non-polar case, PIOLA [1848, 2, '1[48] and HELLINGER [1914, 4, §3d] provcd the equivalence by adjusting the variations so that (232.2) reduces to (200.1). We prefcr the simpler argument (232.3) ~ (205.19) ~ (205.20). The result we have established showsalso that the common claim that symmetry of the stress tensor does not follow from the principle of virtual work is misleading. lndeed, it does not follow naturally. The principle must be adjusted so as to imply that the virtual work of the torques is exactly the virtual work of the moments of corresponding forces. As follows from (205.21), this may be done by adding a priori the assumption that the coefficients in (232.1) satisfy t[km]p =P[k tm]P• i[kp] =P[k /p], and only skew-symmetric o:rkm need be considered. 3 The pioneer work of PIDLA [1833, 3] [1848, 2, '\['\[ 34-38, 46- 50] is somewhat involved. -1 First, PIOLA used the material variables, and his condition ofrigidityis oC KM =0 or oCKM = 0, so that the outcome is (210.8) or (210.10) rather than (205.2); (205.11) and (205.2) are then proved by transformation. Second, he seemed Ioth to confess that his principle employed rigid virtual displacements; instead, he claimed to establish it first for rigid bodies only. In the former work, he promised to remove the restriction in a later memoir; in the latter, he claimed to do so by use of an intermediate reference state. He was also the first to derive the stress boundary conditions from a variational principle [1848, 2, '\[52], and he formulated an analogaus variational principle for onc-dimensional and two-dimensional systems [1848, 2, Chap. VII]. Sect. 232. Virtual work and the Lagrange-D'Alembert principle. 597 for virtual translations is equivalent to Cauchy's first law (205.2); for rigid virtual displacements, to Cauchy's second law (205.11) as well. In these statements, a virtual translation is a field (Jx suchthat bxk.m =0, while a rigid virtual displacement is a field (Jx suchthat (Jx(k,m)=O. To prove ProLA's theorem we first set up the nine side conditions (Jxk m=O, and we write -tkm for the corresponding multipliers1• Then (232.6) is 'equivalent to f (! 3(k (Jxk d V = ~ sk (Jxk da+ .r ((! jk (j Xk- tkm (Jxk, m) dv (232.7) j' Y' j' for arbitrary variations bx. By applying GREEN's transformation, we derive both (203.6) and (205.2); conversely, these latter imply (232.6). To derive the second statement in PIOLA's theorem, we set up the six side conditions (j xq,q) ~x[k,Pl dv + +f (tx satisfying m f o ck ... m " pn ... q __(l_n . ._._q_ (l>x') dv = 0 L.J a k ... m oxr ,s , a~l "f" ,s (233·3) Consider only those (233 .4) where aPL:~. is a tensor of multipliers. The same procedure leads to boundary conditions and equations of motion in which tkm is replaced by m 0 cu ... s tk m + gk r " pn .. . q __(1____1<_..:: • L.. a u ... s oi' a~l ,m (233.5) The fact that in general there exist no motions satisfying the fully general constraints (233.3) does not invalidate the procedure. Sometimes there are constraints depending on the deformation gradients x7K from a reference state 1 . For simplicity, consider a single equation of the form c ( x\ K, t) = 0 . (233 .6) The variations are now subject to the constraint oc (~ k) - -k- uX ;K-0. OX;K (233.7) To apply this side condition, it is convenient to transform (232.2) by introducing the variables appropriate to the description in terms of a reference state. By (210.4), (210.6), and (20.9), in the classical non-polar case we get 12! = p (skda- PK dAK) l>xk + J (PK;K +?! jk) l>xk dV. (233.8) g' "f" Now introducing a multiplier p corresponding to the constraint (233.7) and proceeding as before, we obtain the boundary condition corresponding to (210.5) and the equations of motion in ·ProLA's form (210.8), except that in both the double vector PK is replaced by TkK -pgkm~. (233.9) OXm;K Equivalently, the stress tensor t is replaced by (233-10) From the results (233.5) and (233.10) it is evident that general constraints such as (233.3) or (233.6) yield a non-symmetric contribution to the stress 2• As is shown by the example of isochoric motion, certain special constraints may be maintained by symmetric stresses. 1 PoiNCARE [1889, 8, § 152] [1892, 11, § 33]. Cf. HELLINGER [1914, 4, § 4c], who considers constraints involving higher derivatives. ERICKSEN and RIVLIN [1954, 6, § 4] work out the explicit form of the result which is implied because c is a scalar. 2 This was remarked by ERICKSEN and RIVLIN [1954, 6, § 3]. 602 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 234. All the constraints considered in this section are of the type called holonomic1• Non-holonomic constraints will be mentioned in Sect. 237. 234. Converse of the principle of virtual work. DoRN and ScHILD2 have formulated an elegant converse to the principle of virtual work in the static and non-polar case: Suppose that, corresponding to any vector u given on Y, there exists a symmetric tensor c such that JOtkmckmdv = pOtkmukdam (234.1) -r [/' for all null stresses 0t; then the vector field u can be extended throughout f in such a way that (234.2) In other words, if u =X it follows that in fact c = d. The statement of the theorem is limited to null stresses only for simplicity, the extension to general equilibrated stresses being easy, but for motion rather than equilibrium the formulation is awkward. The proof of the theorem is divided into two stages: (a) we show that c satisfies the conditions of compatibility for d, and thereafter (b) we prove (234.2). Proof of (a). By (227.10), we may write the hypothesis (234.1) in the form J ekrs e'lmn a,m,sn Ckq dv = p ekrs eqmn a,m,sn UR daq, (234.3) -r [/' where a is an arbitrary symmetric tensor. Two applications of GREEN's transformation put (234.3) into the form -r (234.4) J ekrs eqmnckq,snarmdv ) = j ekrs eqmn {ckq,n a,m da5 - Ckq a,m,s da"+ a,m,sn Uk daq} · We choose a as zero outside a small region surrounding a given point, thus annulling the surface integral on the right; since a may be arbitrary, to within requirements of smoothness, we conclude that ekrseqmnckq,sn• since it is symmetric, must vanish. By the remarks following (84.3), or by those at the end of Sect. 34, there exists a vector v such that (234.5) Proof of (b). From (234.5), GREEN's transformation, and the fact that 0t is a null stress, we have Comparison with (234.1) yields p 0tkm (uk- vk) dam = 0 [/' (234.6) (234.7) for arbitrary null stresses 0t. This condition asserts that the virtual work of an arbitrary equilibrated stress in the virtual displacement uk- vk is zero. Hence the motion u- v is rigid. That is, there exist constants wkm and bk such that 1 HELLINGER [1914, 4, § 4c] discusses also holonornic constraints applied only on surfaces or lines, as weil as constraints which are inequalities. 2 [1956, 4]. The staternent (a) and its proof are due to LocATELLI [1940, 15 and 16], who considered also the case when the assigned force does not vanish. Sects. 235,236. Lagrangian and Haroiltonian principles. 603 (234.8) on f/'. Now v is defined through "f/". Therefore we may take (234.8) as extending the definition of u throughout "f/". Since u(k,m) =v(k,m)• (234.2) follows from (234.5). Q.E.D. L. FINZI 1 has remarked that the process leading from (234.1) to the conditions of compatibility for (234.2) is entirely general and may be applied in any space where covariant differentiation is defined. For example, let the general null stress have the form Otkm = akm A + bkmr A,, + ckmrs A,,. + .... (234.9} Substitution in (234-3) and proceeding as in the lines following yields 0 = akm ckm- (bkmr ck".},, + (ckmrs ck ... ),,.- . . . (234.10} as the conditions of compatibility. The reader may use this result to derive (84.5) from (229.8), and to derive (84.4) from (229.10). The process is easily modified so as to apply to the case when the general null stress is given in terms of a tensor of stress functions Akm or Ak,.Pq• etc. 235. Principle of minimum stress intensity. Given a symmetric tensor field skm in "f/", we shall say it is given a potential increment when it is replaced by skm + bck,m)• where bis a vector field which vanishes upon the boundary f/'of "f/". If sk"',".=O, we have [skm + b(k,m)] [skm + b(k,m)] = Skm skm + (2bk sk"'),m + b(k,m) b(k,m). (235.1) By GREEN'S transformation follows I [skm + b(k,m)J [skm + b(k,m>] dv =I [skm skm + b(k,m) bCk,ml] dv, ) ..".. ..".. (23 5 .2) ~ Isk".skmdv, ..".. where equality holds if and only if b(k,m) = 0. The analysis is valid in a Riemannian space of any number of dimensions, so long as the metric be positive definite. We apply the foregoing result to a body subject to no assigned force, both in the three-dimensional and the four-dimensional cases, so obtaining PRATELLI's theorems of minimum stress intensity 2 : 1. In steady motion, the total intensity of a stress tensor t- (!~~ satisfying Cauchy's laws is a minimum with respect to potential increments. 2. In a reetangular Cartesian co-ordinate system in an inertial frame, the total intensity of a world stress-momentum tensor (211.2) satisfying the equations of continuity and momentum balance is a minimum with respect to potential increments. From the analysis, it is evident that no result of this kind can be expected to hold for general variations of stress. 236. Lagrangian and Hamiltonian principles. In this section and the next we derive principles related to (232.4). Thus we are restricting attention to principles equivalent to CAUCHY's first law. 1 [1956, 7, § 10]. Indeed, the stateroent above roay easily be inferred froro the work of LOCATELLI [1940, 15 and 16]. 2 [1953, 25]. PRATELLI, who uses variational calculus, does not obtain an absolute roinirouro in space-tiroe because he uses the energy-rooroenturo tensor of special relativity rather than the classical tensor (211.2). 604 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 236A. By (170.4), we may write (232.4) in the form :t J ex,.~x"dv = ~~ + ~. (236.1) "'" where ~ is the kinetic energy and where the variation of density satisfies ~ (ed v) = 0. Eq. (236.1), which is merely a rewriting of the principle of virtual work, is called the Lagrangian central equation1• Integrating (236.1) from t=t1 to t=t2 , we impose the condition that ~:JJ=O at t=t1 and at t=t2 so as to obtain Hamilton's principle 2 : tz f (~~ + ~) dt = 0' (236.2) tl und conversely, by the identity (236.1), if (236.2) holds for every pair of times t1 and t2 , (232.4) for all admissible ~:JJ follows. By further restriction of the variations, it is easy to derive the principle of least action from (236.2). The elegant and useful forms that these principles assume for systems of masspoints do not carry over to general continuum mechanics. 236A. Appendix. HAMILTON'S principle in the case when there is a strain energy. In the special case when the formula (232A.S) holds, HAMILTON's principle (236.2) may be written in the more familiar form tg t5 J 53dt = 0, tl (236A.1) where 53= f J.dV+ ~s,.ukda, ;. =e0 [!x2 - •]. "'" 9' (236A.2) Here the tildes have been dropped, for the reference configuration must be a fixed one. While the equivalence of this variational principle to CAUCHY's first law, the associated stress boundary condition, and the stress-strain relations (232A.4) follows from results already given in Sect. 232A, (236A.t) is our only example of a variational principle of the conventional simple type to which the Euler formulae may be applied, so we give an independent verification 3 • Recalling that in the material description_ik = 8 xkf8t, we think of the time integral of the volume integral in (236A.t) as an integral over a four-dimensional manifold, and to this integral we apply the usual rules of the calculus of variations. Hence, when there are no constraints, the spatial differential equations following from (236A.t) are (236A.3) equivalent to CAucHv's first law when there is a strain energy and a potential energy. 1 For continuum mechanics, this equation was introduced in a somewhat more general form by,HEUN [1913, 4, § 21] and HELLINGER [1914, 4, §Sb]. These authors consider also an alternative form in which the time as weil as the path is varied. 2 According to HELLINGER [1914, 4, §Sb], the extension of HAMILTON's principle to continuous mediawas first obtained byWALTER [1868,15]; it was given also by KIRCHHOFF [1876, 2, Vorl. 11, § S]. 3 KIRCHHOFF [1859. 2, § 1]. While in [1852, 1] it was assumed that the strain energy is a quadratic function, as far as concerns the variational formulation this restriction was not used. The second variational principle of CLEBSCH, which we have presented in a purely kinematical form in Sect. 137, may be interpreted as a special case of HAMILTON's principlc. ZEMPLEN [190S, 7] [1905, 9, § 3] used (236.1) to obtain (236A.3), (210.5). and (20S.3). The general formula has been rediscovered by DE DONDER and VAN DEN DUNGEN [1949, 5] and by E.HöLDER [19SO, 11, §§ 1-3], who discuss alternative forms. Sect. 237. The principle of extreme compulsion. 605 In the above formulation of HAMILTON's principle, l?o is taken as an assigned function of X, and any additional parameters upon which the function T may depend are kept fixed in the variation. We may set eo= ef by ( 156.2), and then, as has been remarked by HERIVEL1, we may vary I? and the additional parameters, at the same time setting up as side conditions the continuity equation and the constancy of the additional parameters for each particle. The result is unchanged. It is also possible, though more elaborate, to formulate HAMILTON's principle in terms of the spatial variables and to vary the velocity field rather than the displacement2. 237. The principle of extreme compulsion 3• In the principle of virtual work (232.4), the variation IJ;r may be selected as any field satisfying the constraints, presumed holonomic. Precisely, as we have seen in Sect. 233, this means that if there are constraints 4 aC(x,X,x~K,x7KM• ... ,t) =0, (237.1) then (j;r ranges over the dass of vectors v satisfying 8aC vk+~~vkK + _Jla~vkKM + ... = 0. 8xk 8x7K ' 8x7KM ' (237.2) For the actual motion, differentiating (237.1) materially yields 8aC xk+ 8aC X~ +_Jlo~_xk + ... + 8aC =0 8xk 8 k ,K 8 k ,KM 8t ' x;K x;KM (237-3) 8aC "k+ 8aC "k + 8aC "k + +F _ --X ··~·-X.K ---X·KM ··· -0 8xk 8x7K ' 8x7K M ' ' (237.4) where Fis a function of JJ, X, x7K' x7K M' ... '.Xk, x7K' .... This identity suggests a means of constructing variational fields which conform to the constraints. Consider the dass of variationssuch that, at each fixed instant, the varied motion has the same displacement and the same velocity as the original motion, but not necesarily the same acceleration: (j;r = 0, I'Jx = o. (237.5) For such variations, by (237.1) we have IJ(oaCfoxk)=O, ... , and IJF=O. From (237.4) follows --uX 8aC ~"k+ --uX.K 8aC ~"k + ... - . _0 8xk ox7K ' (23 7.6) This is an equation of the form (237.2). What we have proved is that if IJx is any variation satisfying (237.5), then IJx is an admissible virtual displacement field. We may therefore replace IJ;r by I'Jx in (232.4), obtaining f exk IJxkdv = 9i sk IJxkda + J [elk IJxk- tkm I'Jxk,m] dv. (237.7) r Y' r Hence, supposing IJ(e dv) =0, we get 1Jfiex2 dv= pskiJxkda+ f[efkiJxk-tkmiJxk,mJdv. (237.8) r Y' r 1 [1955. 13, § 1]. 2 Unsatisfactory special cases are given by EcKART [1938, 4, § 3] and HERIVEL [1955. 13, § 2]; their result is corrected by LrN in a work not yet published. We make no attempt to cite the !arge hydrodynamical Iiterature on special variational principles, some of which is discussed by SERRIN, Sect. 15 of The Mathematical Principles of Fluid Mechanics, this Encyclopedia, Vol. VIII/1. 3 For systems of mass-points, this principle is associated with the names of GAuss, LrPSCHITZ, GrBBS, and APPELL. For continuum mechanics, we follow the development of BRILL [1909, 2, § 18] and HELLINGER [1914, 4, §Sc]. 4 The tensorial character of the equations of constraint is irrelevant here. 606 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 238. If we add the convention that at points and times in the varied motions the same force f is assigned as at the corresponding points in the actual motion, so that ()j=O, we may replace (237.8) by (j f l e(~- /) 2 dv = ~ s" ()x"da- ftkm(jxk,mdv. ~ ~ ~ (237-9) The integral whose variation stands on the left is the totalsquared effective force, or the compulsion, of the motion, and the variational equation (237-9) is the principle of extreme compulsion1• The principle may be extended so as to hold for non-kolonomic constraints of the more general form C ( X k .. k "k "k ) - a :JJ, ,X;K•X";.KM•···•X,X;K•···•t -0. (237.10) For the actual motionundersuch a constraint, we have oaC "k + oaC ··k + + oaC "k + + oaC _ --X --X.K · · · --X · · · ---0 oik oi~K ' oxk ot . (237.11) In order that varied motions satisfying (237.5) be compatible with (237.11), it is necessary and sufficient that (237.12) Thus for constraints of the type (23 7.1 0) we may still infer the principle of extreme compulsion, but the variations besides satisfying (237-5) are subjected to (237.12) as an additional condition. 238. Remarks on mechanics in generalized spaces. There have been many discussions of mechanical principles appropriate to non-Euclidean spaces 2• Except for relativistic mechanics, which is outside the scope of this treatise, these developments seem to consist mainly in observations that certain parts of the theory do not require Euclidean three-dimensional space, but may be carried over bodily to more general ambients. For example 3, while the momentum principle in the form {196.3), since it requires that the integrals of vector fields over bodies enjoy invariance, is restricted to Euclidean spaces, or at least to spaces with distant parallelism, CAUCHY's laws (205 .2) and (205 .11) are meaningful in any space where covariant differentiation may be defined 4• Sometimes they are derived for Riemannian spaces by assuming that the momentum principle applies to infinitely small volumes, but in essence such a derivation is no more than a direct postulation of the desired result. There are infinitely many possible "laws" of mechanics in generalized spaces if we demand no more than that they be invariant and intrinsic equations which in the Euclidean case reduce to CAUCHY's laws. For example, if R",.. is the contracted curvature tensor of an affine space, we may replace x" by x" +KR" ,..x"' in (205.2), and for all values of K the resulting equation reduces in Euclidean spaces to the usual form of CAUCHY's first law. Another formal generalization, the analogue of common practice in masspoint mechanics, replaces sr in (236.1) or in (236.2) by (238.1) 1 As if English vocabulary were insufficient to supply two different words to translate "Nebenbedingung" and "Zwang", (237-9) is often called "the principle of least constraint". 2 See Special Bibliography M at the end of this treatise. 3 VAN DANTZIG [1934, 10, Part IV, § 1]. ' At bottom, thisis the content of BELTRAMI's observation [1881, 1, p. 389] that Eq. (232.4) is meaningful and may be applied in curved Riemannian spaces. Sect. 239. Scope and plan of the chapter an energy and entropy. 607 where a""' di" dx"' is an arbitrary quadratic form, not necessarily reducible to a sum of squares, and not necessarily the metric tensor of space. A third generalization1, more interesting from the mechanical point of view, replaces the principle of virtual work (232.4), after converting it as in {232A.2) to an expression in terms of a material reference state xcx, by a tour-dimensional equation /dt { fsk t5x" da+ f [eo {f" t5x" + bk t5~")- Tkcx t5x7cx] dv} = 0, I I f/' 'lVg (238.2) putting the impulsive coefficients b" on a par with force and stress as coefficients in a linear variational form. As in HAMILTON's principle, it is supposed that t5~=0 at t=t1 and at t=t2 • Equivalent to {238.2), in the case when there are no constraints, are the equations skda=T ... cxdAcx on Y,} eobk=eoi ... +T,.cx;cx in -r. {238.3) The case when b = ;i gives the classical equations in the form (210.5), {210.8); the case when bk =akmx"' gives the generalization described in the paragraph preceding. It is easy to extend (238.2) to a mechanics of moments which generalizes that following from {232.3). E. Energy and entropy. 239. Scope and plan of the chapter. We attempt to collect here everything of a general nature concerning the balance of energy in continuous media and the mathematical properties of entropy. There can be no doubt of the relevance of the first subchapter, which defines the specific internal energy and derives differential equations and jump conditions expressing the balance of total energy in such a way as to reflect the interconvertibility of heat and work. The second subchapter concerns the more special and more dubious subject of thermodynamics. The specific entropy is regarded as a static defining parameter entering a caloric equation of state which is supposed to regulate the specific internal energy, regardless of deformation and motion. The formal content of this subchapter coincides with that customarily said to describe "reversible" processes in a substance obeying an equation of state depending upon any finite number of parameters, except that our considerations are phrased in terms of a particle in a general continuum and that we do not take up the special thermodynamic properties distinguishing fluids from solids. The subchapter ends by deriving a differential equation for the production of entropy and by discussing relations between the rate of change of total entropy and inequalities governing local changes of entropy. The last subchapter considers several definitions of equilibrium and mentions connections between stability of equilibrium and inequalities restricting thermodynamic equations and quantities. In historical origin the balance of energy and the theory of the equation of state, logically independent and in fact relevant to different levels of physical 1 Due to HELLINGER [1914, 4, § 5d], generalizing a form proposed by E. and F. CossERAT [1909, 5, §§ 61-67, 76-80]. 608 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 240. generality, are intertwined not only with each other but also with a third concept, that heat and temperature are mean manifestations of molecular motion. To CARNOT (1824-1832) is due not only a general understanding and statement of the equivalence of heat and work, formulated independently by JOULE ( 1843) and WATERSTON (1843), but also the concept of entropy. While owing much to numerous earlier researches, the first thermodynamic writing to achieve a modicum of clarity is that of GIBBS (1873-1875), upon which our treatment is based, though we do not enter many of the details and applications he developed, nor do we fail to elaborate the mathematical structure rendered possible and natural by his approach. There exists no other comparably general and complete exposition of the material given in this chapter1• I. The balance of energy. 240. The law of energy balance. In Sect. 217 we have seen that kinetic energy st' generally fails to be conserved. Using the apparatus of Sect. 157, we could set up an influx and a supply of kinetic energy and so obtain a general equation of balance for it 2• Such a balance, while not incorrect, does not lead to a fruitful theory, because it fails to reflect the physical principle of interconvertibility of heat and mechanical work. This principle, which is too broad and too vague for us to attempt a general formulation, as well as too old for us to include its history3 in this treatise, suggests that any equation of energy balance should contain terms which can be identified with non-mechanical transfer of energy. These terms may, but need not be dependent upon changes of temperature. To keep full generality, we refrain from defining temperature specifically until the next subchapter, but its effects remain in our minds to motivate the introduction of the non-mechanical power, 0. At the same time, the special theorems on conservation of energy in Sect. 218, in the circumstances when they hold, should not remain outside the general for1 Wehave been unable to derive much help from any of the numerous treatises except that of PARTINGTON [1949, 22, Sect. II], distinguished for its concise and clear statements of practice and for its critical references. 2 Such is the approach of MATTIOLI [1914, 7]. 3 That heat is a mode of motion was widely believed in the eighteenth century, and both EuLER ( 1 729, 1782) and DANIEL BERNOULLI ( 1738) constructed kinetic molecular models in which temperature may be identified with the kinetic energy of the molecules. The generat and phenomenologi<;al principle, independent of a molecular interpretation, is more recent. That it was known to CARNOT by 1832 is proved by his notes [1878, I], which calculate the mechanical equivalent of heat and project the porous plug experiment. The contention of CLAUSIUS, still reproduced in textbooks, that CARNOT's celebrated treatise [1824, I] obtained correct results from an incorrect axiom, is shown tobe false, as PARTINGTON [ 1949. 22, Part II, § 33. last footnote] has remarked, by the presentation of LIPPMAN [1889, 5, pp. 76-78]; to render CARNOT's work in accord with later views, translate "calorique" as "entropy" (cf. CALLENDER [1910, 3, §§ 16, 20, 22], LAMER [1949, I6]), but is unlikely that anyone would grasp the principle of conservation of energy from reading CARNOT's treatise. In our opinion, the first clear statements to be published are those of JouLE [1843, 2] [1845, I and 2] [1847, 2] and WATERSTON [1843, 5], the latter's being more restricted in scope because derived exclusively from a kinetic molecular model. Forerunners are MoHR [1837. 4], SEGUIN [1839. 2, Chap. VII, § 1], and J.R. MAYER [1842, 2]. The history of the "first law of thermodynamics" is discussed by JouLE [1864, I], TAIT [1868, 14, Introd. and Chap. I] [1876, 6, §§ Il, III, and Introd. to 2nd ed.], CHERBULIEZ [1871, 3], HELMHOLTZ [1882, I], MACH [1896. 3, PP· 238-268], SARTON [1929. 8], EPSTEIN [1937,I. § 11], BOYER [1943, I]; the most nearly complete account is given by PARTINGTON [1949. 22, Part Il, §§ 10-12]. Sect. 241. The equation of energy balance for continuous media. 609 malism. The existence of a strain energy in some cases suggests that in the most general motions we introduce an internal energy G:, an additive set function such that the total energy Sl' + @ is balanced. The fundamental energy balance is then 1 (240.1) where ID3 is the mechanical power or total rate of working of the mechanical actions upon the body. This basic law, sometimes called the "first law of thermodynamics"2, is tobe set alongside the laws of momentum (196-3). 241. The equation of energy balance for continuous media. For a continuum, the mechanical power ID3 is the rate of working of the stress vector and the couple stress on the boundary, plus the rate of working of the assigned forces and couples in the interior (cf. Sects. 217, 232); the non-mechanical power 0, as tobe expected from Sect.157, is expressed in terms of an efflux of energy 3 h and a supply of energy q. Thus (240.1) becomes 4 ~ + ~ = p (tP' ip- mPqrwpq) da,+ J (fP.iq-lPq wpq) dl)R + ) [/' -r + p hP d ap + J q d>JR. [/' -r (241.1) The internal energy @, being an additive set function 5, may be expressed in terms of a specific internal energy e: (241.2) Thus (241.1) is an equation of balance (157.1). In regions where t.~. m, w, and h are continuously differentiable, while ä-!, i,f, l, and q are continuous, we may apply (157.6) and obtain exPip+ ei =tP',,xp + tq dpq + k~p + eq, (241 A.1) as follows at once from (218. 7). In particular, ifthe right-hand side is zero, we get e = 11 + f (X), showing that when there is no dissipative stress, no flux of energy, and no supply of energy, the internal energy and the strain energy differ by a constant for each particle. Further formal simplification results when we assume 3 an 0 tPq = -- odpq • (241A.2) side of the shock ensures conservation of energy also at the shock. This is not so. That a distinct additional condition is needed was first seen by RANKINE [1870, 6, §§ 7-9], who obtained a spl(cial case of (241.9) for lineal motion, as did HuGONIOT [1887, 1, § 149]. According to a note added in the 1883 reprint of [1848, 4], KELVIN and RAYLEIGH were aware of the matter. Cf. the·discussion by HADAMARD [1903, 11, 'i[ 209]. A special case for general motionwas derived by JouGUET [1901, 9] and discussed by DuHEM [1901, 7, Part 2, Chap. I, §§ 7-8] and ZEMPLEN [1905, 9, § 8]. A fairly general case was obtained by CouRANT and FRIEDRICHS [1948, 8, §§54, 118]. We are unable to follow the physical arguments sometimes adduced to infer (241.9) directly from remarks concerning the enthalpy. Indeed, if p = n, the thermodynamic pressure (Sect. 247) in a fluid obeying a caloric equation of state of the forme= e(1j, v), then (241.9) assumes the form [X+ l U2] = 0, but in more general circumstances (241. 7) bears no apparent relation to the enthalpy defined by (251.1)3 . Indeed, to derive (241.7) no thermodynamic formalism is used, and in the degree of generality maintained here, the thermodynamic tensions need not be defined, and hence an enthalpy need not exist. 1 Note that the condition [J:1] = 0, which follows from (205.6) 2 in the present case, has been used here; cf. the remark after {241.7). 2 Contrary to the implication of McVITTIE [1949, 18, § 7]. solution of this problern requires no commitment as to the form of the energy equation in relativistic theories. McVITTIE obtains a special case of (241.11). in which e has a special functional form. 3 RAYLEIGH [1873, 6, § Il]. 39* 612 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 242,243. where Dis a homogeneaus function of degree r in the components dpq• since (241A.1) then becomes e(e-ä)=htp+eq+rD. (241A.3) This result is an equation of balance ( Sect. 1 57) : If we set ). == e- a, then h is the efflux of ). and e q + r D is the supply of ).. The function D is called a dissipation function. This same result, with D == o, holds a fortiori when the dissipative stresses vanish. Thus it is clear that energy equations of the type encountered in classical elasticity, hydrodynamics, and thermal conduction, while consistent with the general scheme of energy balance, result only from the special assumptions peculiar to those disciplines and would be false guides in any general approach to energetic theory. 242. Energy impulse. Equations for energy impulse may be gained by considerations parallel to those of Sect. 198 and Sect. 206, using the results of Sect.194. Apparently, however, this has never been done, and when we set about it, we find an unexpected complication. Suppose a stress impulse given by the tensor iP' on a boundary .9 gives rise to a velocity im pulse -{ .ip}· Then there certainly results an energy impulse on Y, but it is not~ iP' {.ip} da,. Indeed, since iP' da, [/' is a diflerence of momenta, while -{ .ip} is a difference of velocities, the difference of energies resulting will not in general bear any simple relation either to i or to {x}. Thus the causes of energy impulse in general we cannot easily separate into a purely energetic portion plus a portion arising from the impulses of velocity, momentum, and stress. In full generality we have (242.1) where k and n are the total influx and the total supply of energy impulse, but for the reasons given above we cannot generally write k and n partially in terms of i, s, and {x}, and thus analysis parallel to that leading from (241.3) to (241.4) is not possible. An exception is the case when the motion is generated from rest, for then we have -{ x }- = i:, {e ;i:} = e i:, and in the non-polar case kP = f iqP .iq + OkP, n = f sq .iq + 0n, (242.2) where 0k and 0n are the non-mechanical influx and supply of energy impulse. In this case (242.1) may be reduced by use of (206.1), so that (242.3) 243. Balance of energy in a heterogeneaus medium 1• To discuss the transfer of energy in a mixture, we employ the formalism of Sect. 158, and we proceed 1 Equations of the type (243.1). (243.2), and (243.3), with special forms for eme and m arise in MAXWELL's kinetic theory of mixtures of monatomic gases [1867, 2, pp. 47-49]. There, however, it is customary to define all quantities in terms of molecular motion relative to :i:, not to i-m; thus the partial heat flux vector qm of HIRSCHFELDER, CuRnss and BIRD [1954, 9, Eq. (7.2-25)] is not to be identified with our -hm but rather, subject to the special assumptions of the kinetic theory, with the negative of the whole expression in brackets on the right-hand side of (243.2). Our ef'm includes as a special case what CHAPMAN and CowLING [1939, 6, § 8.1] denote by nm L1 Em- Cm · nm L1 mm Cm; our ee, as defined by (243.1), - ~ - includes their nE, our Eq. (243.6) corresponds to the sum of their equations L nmLl mm Cm = 0 ~ _ m=I and L n'MLl Em = o, etc. m=l The continuum theory, while more general and simpler in concept, developed only later, imperfectly, and apparently in oblivion of what had been done long before in the kinetic theory. Differential equations for the internal energy of a mixture of continua have been given Sect. 243. Balance of energy in a heterogeneaus medium. 613 as in Sects. 159 and 215. Each constituent m has its own partial internal energy e'll and is subject to flux of energy h 111 and supply of energy q111 • The total internal energy e is the sum of the partial internal energies plus the kinetic energies of diffusion: R e= L c'1!(e'1!+-!u~). 111=1 (243-1) The total flux of energy h arises from three sources: The constituent non-mechanical fluxes of energy h'Jl, the rates of working of the partial stresses against diffusion, and the fluxes of total constituent energies by diffusion: (243.2) The total energy of the constituent m need not be balanced in itself, as energy may be transferred from one constituent to another. Thus, restricting attention to the non-polar case, so that ~"' = t;tk, we define the supply of energy e~1 by (243-3) the condition e111 = 0 is then necessary and sufficient that the energy of the constituent m be in balance by itself. for various special cases and under various special hypotheses by REYNOLDS [1903, 15, § 39]. }AUMANN [1911, 7, §IV], HEUN [1913, 4, § 24c], LOHR [1917, 5, Eqs. (108), (109)] [1924, 10], VAN MIEGHEM [1935, 9, § 2], MEISSNER [1938, 7, § 3], EcKART [1940, 8, p. 272], MEIXNER [1941, 2, Eq. (12)] [1943, 2, Eq. (2,8)], VERSCHAFFELT [1942, 14, §§ 15-16] [1942, 15, §§ 7-8], PRIGOGINE [1947, 12, Chap. VIII, § 3], KIRKWOOD and CRAWFORD [1952, 12, Eqs. (12) and (18)], and later writers. These authors write down their differential equations essentially by inspection, without derh.•ing them from the equations governing the constituents, and without unequivocal specification of the total internal energy in terms of constituent energies. ECKART [1940, 8, p. 271] hinted at the definition (243.1) but in the end neglected the kinetic energies of diffusion; they are included by KIRKWOOD and CRAWFORD [1952, 12]. An equation of the form (243.3) with EIJl = 0 was proposed by LEAF [1946, 7, Eq. (14)]; from the kinetic theory it is clear that such an assumption is usually false. Thermodynamic writers often pass over mechanical aspects rather cavalierly; typically (e.g. DE GROOT [1952, 3, § 44]) they replace or supplement the mechanical power term t~ "'dk ". by expressions containing some or all of the thermodynamic power terms in (255.15). Thus it is not surprizing that the results do not always agree with one another and do not contain all the terms in (243.9) or any Counterpart of (243.6). A discussion similar to ours is given by PRIGOGINE and MAZUR [1951, 21, § 3] [1951, 17, § 2], but their definitions differfrom (243.1) and (243.2) in including terms depending on potential energy and in employing the velocities ~'l! rather than the diffusion velocities Ul]l; thus their definitions of e and h do not reduce to e1 and h1 when S't =I. Also, they do not derive any counterpart of the condition (243.6), but it may be implied in the equation they write down for the balance of potential energy. Cf. also the special case considered by NACHBAR, WILLIAMS and PENNER [1957, 10, §V]. Our treatment follows TRUESDELL [1957. 16, § 7]. In view of the divergences among thermodynamic writers, it appears necessary to emphasize that all results in the text stand in detailed consistency with their counterparts in the kinetic theory. In particular, that the correct energy equation for the mixture should be of just the sameform asthat forasimple medium, viz., (243-7), has long been known; cf., e.g., the careful derivation of HIRSCHFELDER, CURTISS and BIRD [1954, 9, Eqs. (7.2-49) and (7.6---7)]. The difference in results obtained by thermodynamic writers arises only partly from their apparent use of e1 rather than e (cf. our analysis in Sect. 259) but seems tobe inherently unresolvable because of their failure to define t and h in terms of quantities associated with the constituents. 614 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Summing (243.3) on ~. we obtain where .R .R ' '\' A ~{ (' 12) "k (! .wE~= 2..,. (!'l( Eil(+ 2U'll - (!'llU~kX~I- (!'l(q'l(- 'll~1 ~~1 - (t~m- (!'llU~u~i)xk,m- - [ht + t~m u'llm- (!'ll (EIJl + i- Ufu) utJ.k + + t~':'ku'llm- [eiJl(EIJl + i- Ufu) u~J,k}, = e i-tkmxk,m- hk,k- .R - 2::{e c~ (E~+i-ut) + u~k(e'l(x~- ttml + e~q~}, ~~1 = (!E- tkmdkm- hk,k_ (!q- .R - e ~ [c'll(E~ + i- ut) + Nru'llk], 'll~1 .R q ==:= L Cll( (q'll + f;t UIJlk) · &~1 Sect. 244. (243.4) (243.5) To derive (243.4), at the first equality we have used only algebraic rearrangement, (158.7), and (158.11); at the second equality we have used (243.1), the fundamental identity (159.5), and (215.1) and (243.2); the last equality follows by (215.2). From (243.4) we see that a necessary and sufficient condition for the non-polar case of (241.4) to hold for the mixture is .R '\' [A A k A ( 1 2 )] .w E~+P'llu'llk+cm E~+2u~ =0. (243.6) &~1 This result, analogous to (159.4) and (215.5), asserts that the energy supplied by an excess internal energy rate, plus the energy supplied by the work of the excess inertial forces against diffusion, plus the energy supplied by the creation of mass, must add up to zero for the mixture. Wehave shown that for the non-polar case the condition (243.6), along with the definitions (243.1), (243.2) and (243.5), Ieads to an energy equation of the form (243.7) for the mixture. For later use we require this same equation put in terms of the inner parts Er, hr, and qr of E, h, and q: .R er= LC~E~, 'll~1 .R qr == L c~ q~. ~1~1 (243.8) Either by transforming (243.7) or by summing (243.3) on ~ and then using (159.5), (215.7), and (243.6), we obtain (!Er=t1m dkm + .R 1 t~m U~:,m + (h1- 'll~ .R 1 (!& E~ u~) ,k + I + e qr - e L [Pt u2tk + c~. t Ufu]' 'll~1 (243.9) where tr is defined by (215 .6). 244. Remarks on the generat energy balance. The interconvertibility of heat and ~ mechanical work is expressed only indirectly in (241.4) and (241.6), or, for Sect. 245. Thermostatics and thermodynamics. 615 that matter, in the basic equation (240.1). In the interpretation, the variables :0, h, q, and s are to be associated at least in part with thermal phenomena, yet they are measured in mechanical units. In fact, dim :0 = dim ~ = [M L 2 T-3], l dim s = dimi2 = [L 2 T-2], dim h = [M T-3], dim q = [L q-3]. (244.1) In order that h may be connected with a temperature gradient, for example, the mechanical equivalent of heat must be used. Only in its tacit assumption, necessary for its intended applications, that all thermo-energetic phenomena may be measured in mechanical units, does the energy balancebring in any new physical idea beyond those used in pure mechanics. Indeed, with an equation such as (241.4) we appear to be further from solving any problems concerning energy than we were in Sect. 217. To secure balance of energy, we have introduced three new quantities s, h, and q, and only one condition connecting them. In any given motion, (241.4) may be used for each particle X as a definition of s to within an additive constant, as a definition of h to within an arbitrary solenoidal field, or as a definition of q, when two of these quantities are assigned arbitrarily. But this is not the way in which (241.4) is used in practice. Rather, its new variables correspond to physical ideas, and the simple structure set up in this subchapter serves as a framework (cf. Sect. 6) within which more special considerations concerning changes of energy may conveniently be expressed. Some of these special assumptions will be discussed in the remainder of the chapter. II. Entropy. a) The caloric equation of state. 245. Thermostatics and thermodynamics. The reader who has no preconception of thermodynamics may pass over this section, entering at once into the theory in Sect. 246. 1. The classical difficulties. Thermostatics, which even now is usually called thermodynamics, has an unfortunate history and an unfortunate tradition. As compared with the older science of mechanics and the younger science of electromagnetism, its mathematical structure is meager. Though claims for its breadth of application are often extravagant, the examples from which its principles usually are inferred are most special, and extensive mathematical developments based on fundamental equations, such as typify mechanics and electromagnetism, are wanting. The logical standards acceptable in thermostatics fail to meet the criteria of most other branches of physics; books and papers concerning it contain a high proportion of descriptive matter to equations and results. The obscurity of its concepts is witnessed by the many attempts, made alike by engineers, physicists, and mathematicians and continuing today in greater number, to reformulate them and to set the house of thermostatics in order. The difficulty of the subject lies partly in its task of comparing different equilibria without describing the intermediate states whereby bodies may reach equilibrium. At the outset, the reader is told to imagine a system changing so slowly as to be in equilibrium at all times, for such paradoxical "quasistatic processes" are to furnish the main subject of the theory. The critical student must long have realized that some kind of linearization is involved; in the shadows behind classical thermostatics must stand a better theory including motion and 616 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 245. change of motion, and from this theory the classical quasistatic process should result when kinetic energy, diffusion, and inhomogeneity are neglected. In addition to quasistatic or "reversible" processes, the classical theory attempts to deal with certain "irreversible" processes, but rather than describing their course in its formal structure, it merely lays down prohibitions regarding their outcomes1• Here too the traditional obscurities arise from an incomplete description 2 of the phenomena the theory is envisioned as representing. 2. "Irreversible" thermodynamics. Both these difficulties are cleared by a simple expedient: The basic equations of classical thermostatics are applied to elements of volume in a moving material, or in a mixture of materials. This device of a local thermostatic state 3, when employed in conjunction with the general principle of energy given in Subchapter I, leads to a theory in which equilibrium is but a special case of motion and change, and through which many processes considered "irreversible" in thermostatics are described easily and naturally, at least in principle. This theory, usually called "irreversible thermodynamics ", we prefer to call simply thermodynamics 4• That so great a difference in scope can result from transferring long familiar ideas to a local scale should not be surprizing. It is typical of the success of the jield viewpoint (cf. Sect. 2). Similarly, problems of fluid motion which could be treated only grossly by the mechanics of "bodies" (i.e., mass-points) were solved successfully 200 years ago by hydrodynamics, which applied the principles of mechanics to the continuous field. What is surprizing is that, granted occasional exceptions, the dynamical theory of heat should have remained for nearly a century in a stage of dElvelopment analogaus to the theory of masspoints in mechanics. Even though both theories employ parallel concepts on different scales, beyond the first few steps we cannot expect simply to read off results for the thermodynamic field from counterparts in thermostatics, any more than hydrodynamics can be read off from mass-point mechanics. ). The concept of entropy. Much of the wordiness of the traditional presentation grows from its insistence on justifying the basic assumptions by experience, and in particular on developing the concept of entropy in terms of heat and 1 Cf. the remark of PARTINGTON [1949, 22, Sect. li, §51]: "The thermodynamics of irreversible processes is entirely qualitative and of little interest in physical chemistry." This remark applies to the traditional view, to which the text alludes, but not to the more recent studies mentioned in footnote 1, p. 618, andin Sect. 306. 2 Cf. the remarks of DuHEM [ 1904, 2]: "La Thermodynamique ne possede pas de moyens qui suffisent a mettre completement en equations Je mouvement des systemes qu'elle etudie ... ", etc. Hence arise the peculiar difficulties of the theory of thermodynamic stability (Sects. 264 and 265). 3 The earliest examples are cited in Sect. 248 in connection with the thermal equation of state for gas flows. The first systematic treatments of the energy and entropy fields in a deformable medium were given by J AUMANN [1911, 7] [1918, 3] and LoHR [1917, 5] [1924, 10]; their work is difficult to study because its main object, the explanation of a set of linear constitutive equations intended to describe all physical phenomena known to the authors, has lost what interest it may have had (cf. Sect. 6), and from the maze of calculation, which is highly condensed despite its length, the reader can scarcely disengage the physical principles. It is clear, however, that J AUMANN and LOHR deserve great credit for realizing the nature and importance of the production of entropy and for being the first to derive differential equations for it. Cf. also the early exposition of DE DONDER [1931, 4, § 6]. 4 A description of "irreversible thermodynamics" in classical terms would be: Valurne elements are assumed to suffer only reversible changes, possibly resulting in irreversible changes for the body as a whole. Such terms can be rather confusing, as when PRIGOGINE [1947, 12, Chap. IX, § 1] interprets (255.1) as an assertion that the mean motion of a mixture does not produce entropy and is thus a "reversible phenomenon", even if accompanied by viscosity, diffusion, and the conduction of heat. We prefer to cleave to equations and eschew verbalisms. Sect. 245. Thermostatics and thermodynamics. 617 temperature 1• In the more highly developed parts of theoretical physics, such discussions do not ordinarily form a part of a treatise on the theory itself, but belong rather to works on the physical foundations and on the connection between theory and experiment. While it is true that the physics laboratory does not contain an entropy meter, the concept of entropy is not more difficult than some others, such as electric displacement 2 ; even temperature and mass prove elusive to critical inquiry. A glance at the equations of the theory, once the preliminary words are past, shows that thermodynamics is the science of entropy. This is true even more of recent works on irreversible processes than of the classics on thermostatics. 4. The nature and scope of our presentation. An axiomatic development, deriving entropy from heat and temperature, would be desirable, but in our opinion there exists no acceptable treatment of this kind 3. As in the other domains presented in this treatise, we are content to explain the formal structure of the theory as it is practised (cf. Sects. 3, 196). Thus we take entropy as the primitive concept in terms of which thermodynamics is constructed 4• Surely, any future axiomatic treatment if successful will lead to the same equations as those from which our presentation begins. For readers who prefer arguments concerning steam engines, we cite the original memoirs on thermostatics 5• Neither do we 1 Hence results an apologetic tonein many recent works on "irreversible thermodynamics ", which often find it necessary to discuss whether or not it is "meaningful" to speak of temperature and entropy for systems not in equilibrium. Often included are arguments from statistical mechanics. While a rigorous development of equations governing entropy and temperature from general statistical mechanics would be most illuminating (cf. Sect. 1), all attempts thus far rest on formal approximation procedures in the kinetic theory of monatomic gases or on the theory of small perturbations from statistical equilibrium, so that their validity is confined a fortiori to physical situations far less general than those which the results they claim to derive are intended to represent. In any case, it is not right to single out thermodynamics as the only branch of physics where such arguments are in order. Rather, the development of field theories from statistical theories constitutes a general program of inquiry (cf. Sect. S). Such a program is outside the scope of the present treatise, the purpose of which is to explain the field theories as such. 2 The parallel is good. Entropy cannot be measured except in terms of other quantities, such as energy and temperature; the same is true of dielectric displacement, but the constitutive assumption ~ = e E (cf. Sect. 308) is so common that we are often led to regard ~ as closer to experience than in reality it is. 3 Special mention must be made of the celebrated work of CARATHEODORY [1909, 3], [1925, 4]; cf. BORN [1921, 2], EHRENFEST-AFANASSJEWA [1925, .5], LANDE [1926, 4], MrMURA [1931, 7 and 8], IWATSUKI and MIMURA [1932, 7], }ARDETSKY [1939, 9], WHAPLES [1952, 24], FENYES [1952, 6]. CARATHEODORY succeeded in deriving the concepts of absolute temperature and of entropy from a suitable formalization of the idea of equilibrium and the assumption that for any state, there is an arbitrarily near state that the system cannot reach without work's being expended. Despite the mathematical elegance and success of this approach, we cannot regard it as fundamental for a theory intended to describe arbitrary changes of energy, where thermal equilibrium is as little to be expected as is mechanical equilibrium in dynamics. For thermodynamics, it is not equilibrium that is basic, but entropy production. Cf. also the criticisms expressed by LEAF [ 1944, 8, p. 94]. 4 This method is due to GrBBS [1873. 2, p. 2, footnote] [1873, 3, p. 31] [1875. I, pp. 56, 63] who did not attempt to justify it; it was recognized at once by MAXWELL [1875. 4, p. 195] and was adopted by HrLBERT [1907, 4, pp. 435-438]. Among modern authors who follow it, we cite EcKART [1940, 8] and MEIXNER [1941, 2, § 3] [1943, 2, § 2). 6 The traditional theory was developed by CLAPEYRON [1834, I], KELVIN [1849, 4] [1853, 3), CLAUSIUS [1850, I] [1854, I) [1862, J) [1865, J), REECH [1853, 2], RANKINE [1853. J], and F. NEUMANN [1950, 2I] (deriving from 1854 to 1855 or earlier). A particularly careful discussion of the dozens of assumptions, mostly tacit, on which the traditional development restswas given by DuHEM [1893, 3]. A variant system, Jargely unpublished until 1928, was devised by WATERSTON from 1843 onward; it is recommended and developed by HALDANE [1928, 3, Chap. II]. Other variants or extensions are proposed by BRONSTED [1940, 4 and .5] and LEAF [1944, 7]. Brief histories of thermostatics are given by MAcH 618 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 246. consider it the province of any treatise on mathematical theory to explain how measurements testing the theory are to be made. Finally, and also in defiance of the tradition of the subject, we do not consider it any more appropriate here than in the theory of stress to work out elementary examples based on special assumptions. This treatise presents the generat theory of the entropy field 1 . The theory of entropy, like any other theory, has limitations. That there are many physical occurrences to which it does not apply may be taken for granted. To justify its inclusion in this treatise, it is enough that there are many physical situations to which it does apply, and even in the rather limited special cases now being explored its relevance seems to be waxing. 246. Entropy 2 • At a given place and time in a body, let there be given f parameters Va which are regarded as influencing the internal energy e. The assignment of these parameters 3 is made a priori; their totality is the thermodynamic substate. The physical dimensions of the Va are made up of mechanical and electromagnetic units but are otherwise arbitrary. The most familiar case is when the thermodynamic substate consists in a single scalar parameter, the specific volume v. In another common example, the Va are the nine deformation gradients x7a from a material reference state. In still another, they are the densities or the concentrations of the constituents of a mixture. For all general [1896, 3, pp. 269-301], DUHEM [1903, 10], CALLENDER [1910, 3, §§ 20-23], and PARTINGTON [1949, 25, Part Il, §§ 23-37] [1952, 15]. In the system of DuHEM [1893,1, Eqs. (43bis) and (56)], [1911, 4, Chap. XIV, §§ 1-2], the caloric equation of state (246.1) is regarded as only approximate; DuHEM adds a mutual internal energy resulting from the interaction of all pairs of elements of mass. The system of FINK will be described in Sect. 253. Herewe mention the system of REIK [1953. 26] [1954. 19], which replaces the traditional "second law" by an axiom governing the time change of entropy. This axiom seems to us tobe a constitutive relation, generalizing the conventionallinear ones (cf. the next footnote), and thus we do not attempt to present REIK's theory here. We recognize the difficulty of drawing a firm distinction: Eq. (246.1), on which the rest of the chapter is founded, is itself, in a strict view, a constitutive equation, and the chapter should stop after Sect. 244. 1 The majority of recent studies on "irreversible thermodynamics" rest on the special constitutive assumption (cf. Sect. 7) that the affinities arelinear functions of the fluxes. In the present treatise, these special developments would be as inappropriate as classical linear elasticity. The Iiterature is too extensive to cite, but we mention the expositians of PRIGOGINE [1947, 12], HAASE [1951, 11], DE GROOT [1952, 3] and MEIXNER and REIK, this Encyclopedia, Vol. III/2. Although special cases of such linear relations are old, apparently the first proposal of a general theory was made by DE DoNDER [ 1938, 3]. While great emphasis is currently laid upon the so-called "Onsager relations ", we are unable to see in them, at least for the present, anything more than an indication of a special choice of variables; cf. CoLEMAN and TRUESDELL [ 1960, 1 A]. Future analysis may show that they result by linearization from some as yet undiscovered general principle of invariance. For the now generally accepted approach to the theory of the entropy field, see the SUmmary by DE GROOT [1953, 9]. 2 Since the developments of Sects. 246-249 are parallel to those of classical thermostatics, we do not cite referehces beyond those for Sect. 245. except toremarkthat the early papers concern only the case f = 1, corresponding developments for equations of state with arbitrarily many variables having been given by GIBBS [187 5. 1], SCHILLER [1879. 4] [1894, 8], HELMHOLTZ [1882, 2, § 1], DuHEM [1886, 2, Part II, Chap. II] [1894. 2] [1891, 2], and OuMOFF [1895. 3]. Some of the results of GIBBS are special in that they rest upon a condition of homogeneity (Sect. 260). The general view of the subject is due to HELMHOLTZ: "Der Zustand des Systems sei durch () und eine Anzahl von passend gewählten Parametern Pa vollständig bestimmt." Some of the identities are derived anew and interpreted in linear thermo-elastic contexts by TING and Lr [1957. 14]. That X may enter the equations of state, so that the thermodynamic behavior of one particle may differ from that of another, was noted by HuGONIOT [1885, 4] and emphasized by DUHEM [1901, 7, Part II, Chap. IV, § 1]. 3 In classical thermostatics they are often divided into two classes, called "extensive" and "intensive", but this distinction is not necessary here. See, however, Sect. 260. Sect. 246. Entropy. 619 developments, the parameters Va are left unspecified. They are tensor fields of arbitrary order, functions of place ~ and timet, or, if we prefer, functions of time for each particle X. In following the more recent custom of the subject, where, except in the case of fluids, the variables Va are arbitrary, we feel compelled to caution the reader that the physical meaning of the results is likewise left uncertain. Results depending only on the possibility of differentiation are in the main rather insensitive to the choice of variables, but results following from inversion of functional relations are applicable, in most cases, only to particular choices of the variables va in any given physical system. Our aim here is to present certain mathematical features common to all the simpler thermodynamic theories. Compared to the contents of the other chapters of this treatise, the matter here is not concrete; in a satisfactory treatment, the variables va should be identified with definite physical quantities, as they always were in the sturlies of GIBBS. We have said that the thermodynamic substate is regarded as influencing e. The basic assumption of thermodynamics is: The substate plus a single further dimensionally independent scalar parameter sulfices to determine e, independently of time, place, motion, and stress. That is, we assume that it is possible to assign a priori a function f such that e = f('fJ, v1 , v2 , •• • , vr. X) = e('fJ, v, X). (246.1) The parameter 'fJ is called the specific entropy 1• Its physical dimension, postulated to be independent of [MJ, [L ], [T] and the electromagnetic units, is traditionally left unnamed, but the dimension given by dim e [L 2 r-2] [G] = dim 7J = -dim 7J (246.2) is called the dimension of temperature. In any given motion, of course we have e =g(X, t). In a different motion, a like functional relation of different form, e =h(X, t), will hold. The first implication of the postulate (246.1) is that we can determine e without knowing the particular motion occurring, and without regard to the time. In other words, the value of the internal energy can be ascertained from information which is static and universal. This information consists in 1. The value of the substate, v. 2. The value of the entropy, 'f/· 3· The functionalform of the relation (246.1). Thus the role of entropy is that of a specifying parameter. The mechanical and electromagnetic information expressed by the substate is in itself not enough, but, for any given substance, the value of the one additional quantity 'fJ suffices to yield the internal energy of each particle, whatever the motion it is undergoing or has undergone 2• Adjoining the entropy 'fJ to the substate v, we obtain the 1 Entropy is to be identified with the "calorique" of CARNOT (1824). While used by others, notably by RANKINE (1854), its distinction from the caloric of CLAPEYRON and earlier writers was emphasized by CLAusrus, who invented the name [1865, 1, § 14]. That thermodynamics is, at bottom, the science of entropy was first made clear by the researches of GIBBS (1875). 2 This striking property of entropy results only from the field viewpoint. Typically of field theories, the greater generality obtained by introducing a field which may vary from point to point is gained at the cost of closeness to experiment. HADAMARD [1903, 11, ~ 107] remarked that experimental verification of equations of state for !arge masses in equilibrium gives no indication whatever that local equations of state hold in deforming media. As in any other field theory, the experimental justification must result indirectly by comparing the solution of specific problems with measurements. 620 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 246. thermodynamic state: 1], v. The relation (246.1) is the caloric equation of state. Choice of the form of f in (246.1) defines different thermodynamic substances. When X does not appear in (246.1), the substance is thermodynamically homogeneous. In the present treatise, the form of the caloric equation of state is left arbitrary, except for some inequalities to be laid down in Sects. 263 and 265. The meaning of (246.1) is most easily seen from Gibbs' diagram1, where e is represented by a height over the space of thermodynamic states. For the case when f=1, such a diagram is shown in Fig. 43. The relation {246.1) is then represented by an energy surface, whose form is fixed once and for all for each V I v*, TJ* Fig. 43. GIBas' diagram. substance considered. Suppose we have a motion in which a particle X is carried from a state v, 17 at timet to a second state v*, TJ* at timet*. During the motion, e, v, and 'fJ will vary: e =e(t), v =v(t), 'fJ =TJ(t). By (246.1), the motion in space is mapped onto a curve on the energy surface, and the changes in energy and thermodynamic state during the motion must be consistent with this fact. Conversely, however, the energy surface does not determine the motion, and many different motions may be mapped onto the same curve on the energy surface. Moreover, if two motions carry the particle from the state v, 'fJ to the state v*, TJ* by different paths on the energy surface and over different intervals of time, the energy increments are the same, being determined by the form of the energy surface. The dimensional independence of 'fJ fumishes the avenue by which thermal phenomena are to be connected with mechanical and electromagnetic actions. Were it not for this dimensional independence, 'fJ would be but another parameter in the set of Va. An essential part of the physical content of the caloric equation of state is that thermal information must be added to mechanical and electromagnetic information if we are to determine the internal energy without knowing the motion. Since e and 'fJ are dimensionally independent, any relation between them must involve constants e0 and 'fJo having the same dimensions as e and 'fJ, respectively. Thus the caloric equation of state {246.1) must really be of the form .!____ = t (l, v, x). ~>o fJo (246.3) Dimensional invariance imposes some restriction upon the manner in which v enters (246.3) also, but until the physical dimensions of the Va have been rendered explicit, nothing definite can be concluded. Even after the introduction of reduced variables, it is not clear what geometric structure, if any, is to be adduced for the thermodynamic space of points B-TJ-V. In particular, there is no reason to think of it as metric. But until the space itself is fully defined, we are at a loss to know what properties of the energy surface can. have physical significance for a given substance, apart from the accidents of its representation 2• Invariance requirements for thermodynamics have never been established. 1 The method of GIBBS [1873, 3, pp. 33-34] was adopted and made known by MAxWELL [1875. 4, PP· 195-208]. 2 Cf. the remarks of GIBBS [1873. 3, pp. 34-35] and L. BRILLOUIN [1938, 2, Chap. I, §§VIII-IX]. Sect. 247. Temperature and thermodynamic tensions. 621 247. Temperature and thermodynamic tensions. The temperature () and the thermodynamic tensions Ta are defined from (246.1) by (247.1) Hence for any change whatever in the thermodynamic state of a given particle X we have1 ! ds = Odrj + LTadva. a=l (247.2) The temperature and the tensions are the slopes of the curves of intersection of the energy surface with planes parallel to the co-ordinate planes in the diagram of Sect. 246. The temperature measures the sensitivity of energy to changes in entropy; the tensions, to changes in the corresponding parameters. When v1 is the specific volume, - T 1 is called the thermodynamic pressure, :n:. In the case of a homogeneaus mixture, when the substate includes both the total volume and the masses of the constituents, and when the caloric equation of state is taken as referring to the whole mass, then the tension Ta corresponding to the mass of the constituent 58 is called its chemical potential 2, ,u~ (cf. Sect. 260). When Va is the deformation gradient x~,.. we may set ,Tt"- Ta/eo and obtain a double vector of elastic stress analogous to (218.6). See Sects. 25 5 to 256A for development of the properties of these coefficients. In all cases, from (247.1) and (246.2) follow dim () = [8], . [L2 r-2J d1m Ta = -d-. --. Imva (247.3) Thus temperature is dimensionally independent from the thermodynamic tensions. Also, from (247.1) and (246.1) it is immediate that () = () (rJ, v), (247.4) The temperature and the tensions are functions of the thermodynamic state 3• Since (247.2), being the result of differentiating a scalar relation with fixed X, is valid for all paths on the energy surface for a given particle, as a special case it is valid along the curve onto which is mapped the actual motion of the particle X. Hence ! e = ()i; + L TaVa. (247.5) a=l 1 A special equation of this kindwas derived by GIBBS in bis first work [1873. 2, Eq. (4)] but is taken as the starting point of bis later work [1873. 3, Eq. (1)] [1875. 1, Eq. (12)] and hence is often called "the GIBBS equation ". For references to related earlier work, see Sect. 248. 2 GIBBS [1875. 1, p. 63]. 3 The temperature is easier to interpret physically than is the entropy, and for this reason most treatments of thermodynamics prefer to take the temperature as a primitive concept and then introduce the entropy as a defined concept. We have remarked upon this in Sect. 245, No. 4, and further remarks are given in Sect. 250. Our reasons for preferring the present course are two: ( 1) it is formally much simpler, and (2) in our opinion no existing treatment along classical lines is logically clear. A possible formal alternative would be to take free energy 1p rather than internal energy e as primitive, define TJ through (251.4)3 , then define e through (251.1) 2 , but to us this seems indirect and more difficult to motivate. This same criticism applies to the work of DuHEM [1893, 1] [1911, 4] who always started from the free enthalpy {;", taken as a function of 8 and the substate. 622 C. TRUESDELL and R. TouPIN: The Classical Field Theories. For arbitrary changes, however, we have f Tt 06 = () Tt OTJ + a-s '\' ia Tt OVa + axa. oe Tt' axa. l f e,k = Oru + L ia Va,k + a~a. x~k· a=l Sect. 247. (247.6) In a homogeneaus material, the last terms in these expressions vanish. We lay down an assumption of regularity: All thermodynamic functional relations are differentiable as many times as needed and are invertible to yield any one variable as a function of the others. The caution stated when the substate v was introduced in Sect. 246 must be borne constantly in mind: a strong restriction not only on the functional forms admissible for the caloric equation of state (246.1) but also on the choice of the variables Va is implied; in particular, it is thus assumed that various partial derivatives occurring are of one sign. While some related inequalities will be derived in Sect. 265, an adequate analysis of the nature of equations of state and of the singularities they may possess is not available. We lay down also a notation for partial derivatives: A subscript denotes the variables held constant, and (247.7) Moreover, we agree to hold X constant in all differentiations unless the contrary is noted explicitly. In this notation, (247.1) reads (247.8) In all such expressions it is understood that the quantity being differentiated is regarded as a function only of X and of the variables actually written in the denominator and the subscripts. Rates of change subject to the condition 'YJ = const are called isentropic. Thus the thermodynamic tensions are the rates of change of the internal energy in isentropic changes of the substate. Other isentropic rates are defined by 1 ( 07:a) ab- OVb q,ub' (247.9) As a consequence of the assumption of smoothness, we may invert (246.1) and obtain 'YJ = 'YJ (s, v). Hence f d'YJ = (:~lds+ t;):~J. dva, l (247.10) = (:~t[()d'YJ+ t/adva] + at):~J,uadva. Equating coefficients of differentials yields (247.11) 1 When v1 =V, we have v2 cjl11 = (onfoe) 11 = yU2 , where V0 is the Laplacean speed of sound. Cf. (297-13). Sect. 248. Thermal equations of state. 623 248. Thermal equations of state. From the assumption of smoothness it follows also that we may invert (247.1) 1 and obtain 1J = 'YJ (0, v). (248.i) Substituting this into (246.1) yields a functional relation of the form e = e(O,v). (248.2) But also we may substitute (248.1) into (247.4) 2 and obtain Ta= Ta (0, v). (248.3) Similarly, Va = Va (0, T) · (248.4) The relations (248.)) and (248.4) are the thermal equations of state 1• They are conveniently represented by surfaces over the subspace 0-v; a set of such surfaces may be called Euler's diagrams for the particle or the subspace 2• The following coefficients may be calculated from the thermal equations of state: ~ - ( ~~a )T • ßa = ( ~~a )., • ~ab = ( ::: \,.,b V ab = ( ::; )o,Tb • (248.5) so that l f dva=LVabdTb+ocadO, dTa=L~abdvb+ßadO. (248.6) b~l b~l Since the coefficients (248.5) are rates of change of measurable quantities, they are useful for inferring the forms of the thermal equations of state by experiment. When v1 is the specific volume, oc1fv 1 is called the coeflicient of thermal expansion or isobaric compressibility, -v11fv1 is called the isothermal compressibility, and -ß1 is called the Pressure coeflicient. Th(coefficient((248.5) are related by the identities ßa + L r ~abOCb = 0, OCa + L r Vabßb = 0, l b~l b~l (248.7) LVac~cb=t5ab• L~acVcb=t5ab• c~l c~l As compared with the caloric equation of state, the thermal equations of state offer the advantage of connecting easily measurable quantities, the disadvantage of being insufficient, despite their number, to determine a11 the thermodynamic properties of the material. The latter statement will be proved in Sect. 251. In a theory in which entropy is not regarded as a primitive, the thermal equations of state are often taken as the first postulate, and from them an argument leading to the caloric equation of state is constructed. The heat increment ( Q), the excess of the increment of internal energy over thermostatic work is defined by3 l (Q)==de-~Tadva. (248.8) a~l 1 For a perfect gas in equilibrium, the thermal equation of state :n;v = A (0- 00) was established by combining the experimental results of BoYLE (1662) and of AMONTONS (1699), rediscovered many times subsequently. This equation, along with the more general equation n=/(0, v) and other of its special cases, was used regularly by EULER [1745, 2, Chap. I, Laws 3. 4, 5] [1757, 1, §§ 17-18] [1757, 2, § 21] [1757. 3, §§ 29-31] [1764, 1] [1769, 1, §§ 24-30, 90-108] [1771, 1, Chap. V], but did not reappear in field mechanics until the work of KIRCHHOFF [1868, 11], after which it quickly became universal for studies of gas flows. 2 EULER [1769, 1, § 28]. Cf. also GIBBS [1873, 2]. 3 Special equations ofthisform appear as derived results in the work of CLAUSIUS [1850, 1, Eq. Ila] [1854, 1, Eq. {2)]. 624 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 249. ( Q) is a differential form in the variables v, 6, since both Ii and the Ta, according to this approach, are assumed to be known functions of these variables. In general, this differential form is not integrable. Special arguments and added assumptions are required in order to conclude that ( Q) is integrable and that an integrating factor is 1/6. Under these assumptions, entropy is defined by integrating d7J = (~) = ~ [dli- ±Ta dva], (248.9} a=l identical with (247.2), and the consequent theory is the same as that we have discussed in Sects. 247 to 248 and will develop further in the rest of this chapter. It is also possible to consider a theory in which the thermal equations of state hold but there is no entropy or at least no caloric equation of state. 249. Thermodynamic paths. Specific heats, I. Treatmentbasedon entropy1• For a given particle X a tkermodynamic patk is a path 'YJ = 'YJ (Ä), v = v (Ä} in the space of thermodynamic states 'YJ, v. On any path P, at a given point, any differentiable function g of the thermodynamic state has a differential, which we may write as dgp. Isentropic paths are defined by 'YJ=const (Sect. 247}; isothermal paths, by () = const. The specific keat "P on the path P is defined by d7Jp "P = () (i"O . p (249.1) For an assigned path, the ratios of differentials dvapfd'YJp are assigned, and from (249:1), by use of (247.4}1 , results adefinite value of the specific heat. Wehave dim "P =dim 'YJ = (L2 T-2 9-1]. By (247.2) we may write ! dep-1:TadVaP a=l "p= d6p (249.2} From the assumption of regularity in Sect. 247, we may regard e as a function of () and u, and thus (249.2) becomes (~t d6p+ ~[(~)B,u« dvap- TadVaP] l "P = d6p ' ~ (!;t + tJ( :a .. -··I·:::· I (249.3) In particular, if the path is one on which u = const, "P is called the specific heat at constant substate and is written "u· Forthis quantity, (249.3) yields 2 _ ( a11) _ ( ae) "u= () 7fii u- 7fii u' (249.4} Hence (249.3) becomes ! "' [( ae ) ] dvaP "P = "u + L..J -a- a- Ta ~6 . a=l Va O,u P (249.5) From the assumption of regularity, we may regard v0 as expressed in the form (248.4). Writing ""' for the specific heat at constant tensions 3, by holding 't 1 REECH [1853, 2, Chap. IV], for the case of a fluid. 2 CLAUSIUS [1854, 1, p. 486) for the case f = 1, HELMHOL TZ [1882, 2, § 1) for the general case. 3 DuHEM [1894. 2, Chap. IV, § 7]. Sect. 250. Specific heats and latent heats, II. M. BRILLOUIN's general theory. 625 constant m (249. 5) we obtain """- "u = ± [(;-) a- Ta] OC:a, a=l Va O,u where OC:a is defined by {248.5) 1 • The ratio of specific heats, {249.6) (249.7) is an important dimensionless scalar. It is connected with the coefficients is the differential form {248.8). Similarly, the latent heats Ara and Ava are defined by f f e- ~ TbVb i.QL e- ~ TbVb (Q) A-ra- b=l Ava= b=l (250.2) ia d-ra ' Va dva In general, such specific heats and latent heats will be functions of time even for a given path. Definitions equivalent to these were used long prior to the concept of entropy and the theory of thermostatics. While they serve to record the results of measurements, they are too generaltobe the basis of a mathematical treatment. M. BRILLOUIN 2 proposed assumptions which, while far more general than those used in the preceding sections, are yet definite enough that the major classical formal properties of specific and latent heats remain valid. In order to represent the possibility of permanent deformation, he suggested discarding even the thermal equations of state, replacing them by the differential forms {248.6), where the coefficients '~'ab•OC:a,~ab•ßa are given functions of V, T, and e. The resulting differential forms in all 2 f + 1 variables v, T, () are not necessarily integrable. The identities (248.7) remain valid, but the interpretations {248.5) are correct only for the integrable case. The idea is easiest to picture in the case of three variables, d v = v ( v, -r, 0) d -r + IX ( v, -r, 0) d 0. Consider a closed loop in the -r-0 plane, described by T=-r(t), 0=0(1). As the system is carried through this closed cycle, the point -r, 0, v traverses a curve on the right cylinder whose base is the loop. This curve is obtained by integrating v = v(v, T(l), O(t)) i(t) + ~X(v, -r(t), 0(1)) Ö(t). (250.3) 1 An attempt to study specific heats in something approaching this degree of generality was made by M. BRILLOUIN [1888, 2], who discussed possible restrictions on the dependence of" on the substate. 2 [1888, 3, §§ 3, 8-9]. Our treatment is somewhat more compact and general than BRILLOUIN'S. BRILLOUIN also studied a generalized entropy 1)(V, T, 0) [1888, 3, §§ 12-25]. The criticism of BRILLOUIN's theory expressed by DuHEM [1896, 2, Introd. to Part I] refers to this generalized entropy, not to the developments presented above. DuHEM [ibid., Chap. I, § 1] proposes a generalized free enthalpy. Handbuch der Physik, Bd. III/1. 40 626 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 250. If the argument v does not actually appear in v and et, as is the case when there is a thermal equation of state, this curve is closed, and v returns to its initial value when the loop is traversed; indeed, the only possible curve is given by the intersection of the cylinder with EuLER's diagram. If the form (248.6h is not integrable, however, v will not generally return to its initial value, and after the system is carried through the closed cycle of values of (} and T, a change in v will result. But all these changes are reversible to the extent that if a path may be traversed in one sense, it may also be traversed in the opposite sense, since the form (248.6h is linear and homogeneous. We now add the assumption that an internal energy exists, but we replace (248.8) by allowing the more general assumed functional relation B = B (t', T, fi). (250.4) Then from (248.6)1 follows (250.5) where the quantities Ba and Ca are arbitrary. With the choices 1 Ba=TJ- 8 8 e , 8e Va Ca=- - 8-, we get Ta f (Q) =a~/TadTa +x"dO, l = L Ava d Va + Xu d e' a~l where the latent heats and specific heats are given by Hence (250.6) (250.7) (250.8) 1 By other choices of the Ba or Ca it is also possible to eliminate d (} and any f- 1 of the dva and dra. but the result is not illuminating except in the special case f = 1, for which it is given at the end of this section. Sect. 251. Thermodynamic potentials and transformations. 627 a result which reduces to (249.6) when (250.5) is replaced by the more special assumption (248.8). From (250.7), (250.8), and (248.7) we obtain r r x .. - "" = L IXa Äva =-L ßo. Ä.To. • (250.9) 0.=1 o.=1 These identities express the difference of specific heats as bilinear forms in the thermal coefficients 1X0 , ßo. and the latent heats Ä.To, Ä.v0 . Their form is precisely the same for Brillouin's theory as for the classical special case when there arethermal equations of state. Thus if one specific heat is known, the other may be calculated directly from the measurable quantities defined as coefficients in the forms (248.6) and (250.6). Moreover, the relation (250.9) does not serve as a test for the existence of a thermal equation of state. When f = 1, we write ß for ß1 , ct for ct1 , etc., and obtain "T- "v = - ß l. = ctÄ", so that tke latent keats are proportional to tke differentes of specific keats, and (250.6) become 1 1 (Q)=- ß '"T- "v) dT + "Td0 = -;x- ("T- "v) dv + "vd0, (250.10) with ß + ct~ = 0, ct + ßv = 0, ~11 = 1. Also we have the alternative form (Q) ~ ~:d:+T :;T :; ) '' + (:: + t :; ) dT, l ct ß (250.11) Therefore (250.12) Thus when f = 1 the ratio of specific heats may be calculated from ct, ß, and the ratio dTjdv in a process where ( Q) = O, or, equivalently, all energy changes are balanced by thermostatic work. Such a process generalizes the notion of "isentropic path" introduced in Sect. 249. 251. Thermodynamic potentials and transformations. We return to the classical theory based upon the caloric equation of state (246.1). The thermodynamic potentials1 are named and defined as follows: internal energy: e , free energy : 1p - e - 'Y} () , enthalpy: r X-e- LTa.Va, 0.=1 r free enthalpy: C - X - 'Y} () = e - 'Y} () - L Ta Va · They are related through the identity e-"P+C-x=o. 0.=1 (251.1) (251.2) Each of the potentials has certain specially simple properties. From (247.2) and (251.1} it follows that f f de = fJ d'Yj + a~f /a dva, d1p _=- 'Y} df) + a~f /a dva, l (251. 3) dx = Od'Yj- L VadTa, dC-- 'Yjdf)- L vadTa· 0.=1 o.=l 1 Proportional functions were introduced by MASSIEU [1869, 4 and 5] [1876, 3, §§I and IV], who was motivated by the desire to express all thermodynamic properties in terms of functions of 0, u and of 0, 't". Cf. Eqs. (251.7). The theory was elaborated by GrBBS [1875. 1, p. 87]. Cf. also HELMHOLTZ [1882, 2, § 1]. Further comments on the interpretations of the potentials are made by NATANSON [1891, 4] and TREVOR [1897. 9]. 40* 628 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 251. Hence o =(~L· 1] =-(::t. 0 = (:~)T, (251.4) 1] =- (!~t. the first two of these having already appeared as {247.1). From {251.1) and (251.4) follow the relations f u a=l Ta 'I,T {251.5) 'P- e = 0 (::) , X-e= L Ta( :x) a,) C - e = 0 (~CO) + ± Ta ( !' ) a • u T a=l uTa 8,T From {251.3) it appears that we may use any one of the alternative sets of independent variables 1], v; 0, v; 1], T; 0, T (251.6) and yet be able to calculate all energy changes. The functions appropriate to the four cases are the four potentials e, 'P· x. C. In particular, (251.4) 3 shows that from the equations'P='P(O, v) andC=C(O, T), theentropymaybe calculated as a change of free energy per unit temperature. The name free energy is appropriate for 'P because, as follows from {251.3)2 , it is the portion of the energy available for doing work at constant temperature. Similarly, by (251.3) 3 it follows that the entkalpy X is the portion of the energy which can be released as heat when the thermodynamic tensions are kept constant. When f =1, the free enthalpy is called the "Gibbs function" or "thermodynamic potential". When f>l, the usage of GIBBS, followed in most texts today, was somewhat different from ours. In the case when v1 is the specific volume v, GrBBS set (251.6A) even wken f > 1; i.e., in the enthalpy he left out of account the energy corresponding to any parameter other than v1 • This results in a different physical interpretation for X and C if f > 1; in particular, GrBBS' function C is useful in situations where the temperature is controlled and a uniform hydrostatic pressure is maintained while other thermostatic parameters vary. In adopting here the generalized enthalpy given by (251.1) 3 , not only do we seek to exploit the neater mathematical development which will be seen below, but also we recognize that in the general case in continuum mechanics the thermostatic pressure does not exist or does not enjoy much importance. An equation giving one thermodynamic variable as a function of one of the four sets {251.6) is said tobe afundamental equation1 iffrom it all thermodynamic variables that are not its arguments may be calculated by partial differentiation, functional inversion, and algebraic operations. In this definition, the set of "thermodynamic variables" is Va, Ta, 1], 0, e. Fundamental equations are e=e(?J,v), 'P='P(O,v), x=x(?J,T), C=C(O,T). (251.7) 1 GIBBS [1875. 1, PP· 85-92]. Sect. 252. Thermodynamic identities. 629 As will be shown in Sect. 253, an example of a thermodynamic relation which is not a fundamental equation is a thermal equation of state such as Ta= Ta (0, v). There is a formal analogy between the equations appropriate to one potential and those appropriate to another. Examples will be given in Sect. 252. For the case when f =1, HAYES1 has constructed a systematic and exhaustive method of permutation to obtain them all. 252. Thermodynamic identities. From the assumption of smoothness in Sect. 247, we may write down the following conditions of integrability2 for the four differential forms (251.3): (252.1) where T" b stands for the set of all the T's except Ta and Tb. These identities are called Ma:xwell's relations or reciprocal relations. The expressions for cxa and ßa are of particular interest in showing that certain rates of change of entropy can be inferred from the thermal equations of state. When f =1, the above identities are easily expressed in terms of Jacobians. For example, (252.1h implies that o(O, TJ) o(T, v) . o(v,fi} o(TJ,Vf • (252.2) hence 3 o(T,v) _ 1 o(TJ, 0) - • (252.3) This may be regarded as a summary of all the Maxwell relations when f = 1, since the same procedure applied to any one of the identities (252.1) yields this same end product, which asserts that a mapping from the plane of 'fJ, 0 to the plane of T, v is area-preserving. Again by the assumption of smoothness, we consider both 'fJ and e as functions of 0 and v, by (247.2) obtaining OdrJ =de- ±Tadv0 , I = (::ta;~ + L [( ::Jo,ua- Ta] dva· a=l (252.4) 1 [1946, 5]. Earlier SHAW [1935. 6] had given tables based on rather Jaborious calculation. It has Jong been noted that the relations (251.1) are contact transformations. CoRBEN [1949. 3] and FENYES [1952, 5] have constructed an analogy to the dynamics of mass-points, whereby each dynamical formula enables us to write down a thermodynamic identity, certain new thermodynamical quantities being thus suggested. 2 The results, when f = 1, are due to MAXWELL [1871, 5, Chap. IX] and, for the general case, to GIBBS [1875, 1, Eq. (272)] and DUHEM [1886, 2, Part II, Chap. II, Eq. (82)]. Cf. also MILLER [1897. 6]. It was NATANSON [1891, 4, §I] who first remarked that they are conditions of integrability. 3 Allowing for the fact that CLAPEYRON did not distinguish properly between entropy and caloric, we may see nevertheless that he gave the essential content of this relation and of (252.1) 8 [1834. 1, §V]. 630 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 252. Hence we again obtain (249.4) 2, but also1 () (~) - (_!_!_) - T ova 6, ,.a- ova 6, ,.a a • (252.5) This last result enables us to express the latent heats in terms of entropy changes. Indeed, specializing the formulae (250.7) 3, 1 to the case when e = e (v, 0), we obtain Ava = 0 ( :7J ) , uVa 6,ua (252.6) By the Maxwell relation (252.1) 4 , these results may be put into the forms 2 f Ava =- 0 ßc., ATa =- 0 L ßb Vba· (252.7) b=l While the last formulae express the latent heats in terms only of the quantities occuring in the differential forms (248.6), the presence of the factor 0 indicates that the differential properties of the entropy have been used, and in fact test of the relations (252.7) serves as an experimental check on the integrability of the form JO. By differentiating (252.7) 4 and using (252.5), (249.4), and (252.1h we obtain the following identity of CLAUSIUs 3 : ( o.ilva) -(o~,.) --ß ----e6,. ova 6, ,.a- a · From (252.7)1 and (250.9) it follows that f ""'- "" = - 0 L Clta ßa • a=l (252.8) (252.9) whereby the difference ""'-"u is given in terms of quantities obtainable from the thermal equations of state. For alternative forms, we use (249.7), (248.7)1 2 , and (252.9) to obtain ' 0 0 f Y- 1 =- L ~abotnotb =- L Vabßaßb· (252.10) ~ .. a, b=l ~ .. a, b=l We now address ourselves to the task of finding an explicit relation between the coef:ficients a b and m,, defined from the caloric equation of state by (247.9), and the coefficients ~ab and ß,, defined from the thermal equations of state by (248.5)s. First we write out the two Hessian matrices whose individual components we seek to relate: ll ~~~~=llab m, llj o(u, 71) m, 0/"u ' o(U,7J) Vab Cltc 118(~11 = II ot, ""joll · (252.11) where we have employed only the definitions (247.9), (248.5)1, 4 , and (249.1). The product of the two matrices on the left-hand side is the unit matrix; there1 CLAUSIUS (1854, 1, p. 486], when f= 1. 2 The first of these relations, for the case when f = 1, is dueto CLAPEYRON [1834, 1, §IV], with the pr-oviso noted in footnote 3. p.629. It was obtained from the present theory by CLAUsrus, first for the case of a perfect gas [1850, 1, Eq. IV] and then more generally [1854, 1, Eq. (13a)]. 3 [1854, 1, Eq. (3)], given earlier for the case of a perfect gas [1850, 1, Eq. II]. Sect. 252. Thermodynamic identities. fore, so also is the product of those on the right-hand side. Thus f L ac '~'cb + W"a IXb = Öab, c~l f L UJ"b 'Vb a + () 1Xa/Xu = 0, b~I f L abiXb + W"ax-./0 = 0, b~I ! L W"a 1Xa + Y = 1 • n~I where for the last identity we have used the definition (249.7). From (252.12) 4 we have f Y - 1 = - L W"a IXa. n~I Multiplying (252.12h by IXn and summing yields f x-.(y-1) =0Lab1Xa1Xb, a,b~I 631 (252.12) (252.13) (252.14) where (252.13) has been used, while a similar process applied to (252.12) 2 yields f Y - 1 = ~u L 'V ab W"a UJ"b · a,b~I Similarly, from (252.12h and (252.12) 4 we have (J f -1 Y- 1 =-L ab W"aW"b. ;e-. a, b~I These formulae for y-1 aretobe set alongside (252.9) and (252.10). (252.15) (252.16) Coming now to (252.12) 1 , we multiply by ~be and sum on 6; simplifying the result by use of (248.7)t, 3 and (252.1) 9 yields ab =~ab+ W"a ßb · From this identity and (252.1) 8, 9 we obtain a reciprocity theorem: W"a ßb = UJ"b ßa • If we write then taking the determinant of (252.12}t yields (252.17) (252.18) (252.19) v = -:- _= det (Ö1 ab- W"a ocb) ·] (252.20) - 1 - L W"a1Xa a~I [cf. the steps used in deriving (189.11)]. By (252.13) it follows thatl =y$. (252.21) 1 For the case when f=I, the identity (252.21) becomes y= (~:)j(~:) . which may be interpreted as a statement that in a perfect fluid, the ratio of the isentropic and isothermal speeds of sound is always y, whatever be the form of the equation of state e = e (TJ, v) (cf. Sect. 297). While this weil known result is often attributed to REECH [1853. 2, p. 414], he did not derive or state it, although he gave related expressions for ;e,. and ;ev. A different generalization is given by DuHEM [1903, 10]. 632 C. TRUESDELL and R. ToUPIN: The Classica! Field Theories. Sect. 252. Looking back at (252.11)1 , we take the determinant of each side and expand the right-hand member according to minors of elements in the bottarn row: 8(-r,O) Ocp f_1 a{u, ") = -;t - L ab IDa lUb, ., " a,b=J Ocp = "[y-(y- 1)]' ... (252.22) = _IJ_t = _~}_! "u = () ~ "·· "..- y V' where we have u:>ed (252.16) and (252.21). The identities just derived enable information about the caloric equation of state to be inferred from properties of the thermal equations of state, which are more easily accessible to measurement. Essential use will be made of them in Sect. 265 in connection with the theory of stability. Returning to the system composed of (252.5) and (249.4) 2 , as its condition of compatibility we derive the following necessary and sufficient condition for the existence of entropy as an integral of the form fO=drJ: ~) = l"a- () (~) =-()2 ( 8(ra/O)) . (252.23) 8va o,ua 80 u 80 u Many treatments of thermostatics contain arguments rendering plausible the validity of (252.23) as a postulate, thus enabling the concept of entropy to be derived from those of temperature and an internal energy assumed given by an equation of the form (248.2), supplemented by thermal equations of state (248.3). While we have developed only the identities following from use of e as a potential, the same processes Iead to corresponding identities if any one of the functions 1p, x. C is used as the starting point. For physical applications, the function C is often the best suited. For a concrete example, consider the van der Waals gas, defined by a caloric equation of state (246.1) given parametrically as follows: e=Jc(u)du-:. 1J=Jc~u)du+Rlog(v- v), v> 0v (252.24) where a, R, and 0v are positive constants, and where c (u) is an arbitrary function. From (247.7) 1 and (249.4) we obtain at once 0 = u, Xv = c (u) = c (0), (252.25) whereby physical interpretation is attached to the parameter u and the function c (u). From (247.7) 2 and (252.25) follows the thermal equation of state: RO a :TC=- T = ---~. (252.26) v- 0v v2 By (248.5). (249.11}, (252.10}, and (252.13) we have ot= R ß=--R-, :rc _ ~ + 2a 0v ' v - 0v v2 1fl a :rc+vz V- 0v 2a v3 ' 1 V=T· y-1 l11=---, ot R (y- 1) c(O) = x"- Xv = ~~~~~~ 2a(v- 0v) 2 1 - RO'Ifl (252.27) Sect. 253. Thermodynamic degeneracy. 633 The conditions of ultrastability, as shown in Sect. 265, are y > 1 and "" > 0, where the former is equivalent, alternatively, to v > 0 and ; > 0. Hence with c (0) assumed positive, a state is ultrastable for this substance if and only if R 0 > 2a (v- 0v) 2 v-3 ; neutral stability occurs when " >" is replaced by " = ". When a = 0 and 0v = 0, the van der Waals gas reduces to the ideal gas, which is a special case of the ideal materials to be defined in the next section. For the ideal gas, if c (0) = const, u is easily eliminated between (252.24h and (252.24) 2 , so that (252.28) where v0 and 1'/o are arbitrary constants connected with the zero of e and the unit of 0. All states are ultrastable for the ideal gas if 0 > 0, provided, of course, that "v > 0. 253. Thermodynamic degeneracy. When f + 1 is the greatest nurober of independent variables occurring in any equation of state, either caloric or thermal, but in one of them less than f + 1 are present, the material is said to be degenerate. The most familiar example of a degenerate substance is one for which the equation of state e = e (0, v) reduces to e = e (0). Such a substance is called ideal. From (252.23) we read off as a necessary and sufficient condition for an ideal material Ta= 0 fa(v) = Oßa =- Äva' (253.1) where the steps follow by (248.5) 2 and (252.7h. This example establishes the contention just following (251.7), since thermal equations of state of the form (253.1) do not yield the functional form of the caloric equation of state (246.1), but only the restriction (253.2) For an ideal material, inversion of (253.1) yields Va=ga(t:/0) and hence by (248.5) f Tb oga - O(l.a =~I T o(Tb/0) = ha (v). (253-3) Substitution of these results into (252.9) yields f f X,.- Xu =-L OCaTa = L fa(v) ha(v) =F(v). (253.4) a~I a~I Thus for an ideal material the difference of specific heats, besides enjoying the special expression (253.4)1 , is a function of the substate only1 . For an ideal material the identity (248.7h becomes a differential equation for determining ßa when all the functions - Orxb or hb (v) are known: f oßa O.l:a-rxb+ßa=O. (253-5) b~I Vb A second familiar degenerate material is one in which the thermodynamic tensions are determinate from the thermodynamic substate alone, and conversely. 1 For an ideal gas, rv = - R 0, so that a. = - R/r and ß = - Rfv, and (253.4) reduces to "T-"v = R [cf. (252.27) 8]. While this celebrated relation is often named after MAYER, the only possibly related specific statement we have been able to find in his paper [1842, 2, p. 240] is the unproved assertion, " ... findet man die Senkung einer ein Gas comprimirenden Quecksilbersäule gleich der durch die Compression entbundenen Wärmemenge ... ". Even the generaus interpretation of MACH [ 1896, 3, pp. 247- 250], who finds MAYER "so unzweifelhaft klar ... , daß ein Mißverständnis nicht möglich ist," does not impute to him the relation in question, which seems tobe due in fact to CLAusrus [1850, 1, Eq. (10a)]. 634 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 254. Such a material, characterized by the relations {253.6) is called piezotropic. From (252.1) 1 follows the necessary and sufficient condition 0=0('YJ)· In other words, the caloric equation of state {246.1) in the piezotropic case reduces to1 e = eo ('YJ) + eu (v), (253-7) so that the internal energy may be split into a thermalportioneo and a substantial portion eu. For the former, by (249.4) we obtain eo=f~ud(), {253.8) where ~u=~u (()). By {248.5) and {247.9) we obtain as alternative necessary and sufficient conditions for a piezotropic material ota=O, ßa=O, Wa=O ~ab=ab· {253.9) From (249.10), or alternatively from (252.10) or any of Eqs. (252.13) to (252.16), follows y =1, {253.10) this condition being necessary but not sufficient unless f = 1. An interesting question regarding apparent degeneracy has been raised by FINK 2 . In all the formal developments of thermodynamics it is assumed that the f parameters va are sufficient to describe the physical body under consideration. If this is not so, the theory may yet be sufficient to describe some, though not all, of the physical behavior of the body. To see the effect of neglecting the additional parameters necessary for a complete description, we consider the theory based on e = f(rJ, V 10 V2 , ••• , Vf, Vf+l• ••• , Vf+m) (25).11) as exact, and we compare the exact results with those which result from treating the material as if it were degenerate: (253.12) where A 1 , ••• , Am are constants. If f did not in fact depend on its last m arguments, the material would be really degenerate and the results obtained from (253.12) exact. Since, by assumption, e does depend on these arguments, quantities such as (J and r 1 , ••. , Tf calculated from (253.12) will depend on the constants A 10 • •• , Am. That is, different behavior as an apparently degenerate material may correspond to different values of the unknown constants A 1 , •.. ,Am. 254. Heterogeneous media I. Compatibility of an equation of state for the mixture with equations of state for the constituents. We now divide the parameters v into two groups: the set of individual densities e21 and the set of additional parameters w 1 , .•• , w1, independent of the densities. We assume that each constituent has its own partial entropy 'YJ~r and its own caloric equation of state ~=1,2, ... ,sr. (254.1) We now define the total specific inner energy er and the total specific entropy 'YJ of the mixture: 5l er L c~re'H, 'H=1 1 COURANT and FRIEDRICHS [1948, 8, § 3]. 5l 'YJ = L C'H'YJ~l· ~1=1 (254.2) 2 [1947, 5] [1948, 11] [1949, 10] [1951, 6]. Most of FINK's work considers the effect of adding one additional parameter in the usual thermostatic theory for fluids so as to describe metastable states, etc. Sect. 254. Heterogeneous media I. 635 By (254.1) and the definition (254.2)1 , which was given earlier as (243.8)1 , E1 is defined as a function of all the 1J'11, all the em, and the set w. It is then a definite mathematical problern to find conditions under which Er depends on the 1J'D. only through the linear combination 1J defined by (254.2) 2• Thus no thermodynamic arguments, but only Straightforward use of the theory of functional dependence, is required to determine whether or not the caloric equations of state (254.1) for the constituents imply a caloric equation for the mixture: (254.3) Indeed, a necessary and sufficient condition for this functional dependence is the vanishing of the following J acobians: o(er·11· 1!1• ... , 1!5\•(1)1• ... ,w,) 0(1j1•1IB• f!l• ... , f!ft• (1)1• ... , Wf) ',',' o(e1•11• (!1, ... , (!ft,Wp ... ,wf) 0(1j1•1/s•1IJ• f!2• "., f!ft• W1, ... , WrJ '' ''' iJ(ei•1/• f!1• .. . , f!ft• W1, .. . , Wr) 0(111•112• 1!1• "., 1!5\•1/a• "., Wr) ' ... , (254.4) Since the variables 171• ... , 1JR• e1, ... , eft, w1, ... , Wr are independent, the only members of the above set of determinants that are not zero by definition are those of the first line, which reduce to 8(e1.1j) F(?h.1js), • .. , (254.5) where e 1 , ••. , eR, w 1 , ••. , w, are held constant. If we set (254.6) for the temperature of the constituent ~. the determinants (254.5) may be expressed in the forms c1 c3 (01 -03), ... ,} c2 c3 (02 - 03 ), ... . (254.7) Therefore we have the following local theorem1 : I f each constituent of a mixture has its own caloric equation of state (254.1), and if the total inner energy E1 and total entropy 1J of the mixture are defined by (254.2), then in order that there be a caloric equation of state (254.3), it is necessary and sutficient that at each place and time the constituents fall into two categories: 1. ~ constituents have the same temperature: (254.8) 1 TRUESDELL [1957. 16, § 10]. The necessity of (254.8) and (254.9) is easily established alternatively as follows: from (254.2) 2 and (254.3) we have ( -- oe1 ) -cm- (OBI) . o11m p,..,- a11 p, ."· but by (254.2h ~) _ cm (oem) 011'11. p.w- 01jiJI p,w· Equating these two results shows that Om = 0 if c'll =I= 0. 636 C. TRUESDELL and R. TouPIN: The Classical Field Theories. 2. The remaining ~- 5} constituents are not present: c~.l!+l = c~.I!+S = · ··c~.l\= 0. Sect. 255. (254.9) The form of the caloric equation (254.3) is thus derived; it depends not only on the form of the caloric equatioris (254.1) for the individual constituents but also on the particular variety defined by the conditions (254.8) and (254.9). 255. Heterogeneaus media II. Explicit form of the Gibbs equation. Although it is only a matter of functional elimination to determine the caloric equation of state according to the results just preceding, we cannot expect to find its explicit form in general. However, in the practice of thermodynamics for continuous media Eq. (254-3) is never used other than through its material derivative following the mean motion: .1\ ~ Er= 0~ -nv + L ,u~c~ + 1: O"aWa· (255.1) ~=1 a=l Of course, this equation, often called the Gibbs equation, is a special case of the differential relation (247.2); the differentials are taken along the mean motion of the mixture, and the parameters Va are rendered partly explicit as v, c1, .. . , c.l\, where the c~ satisfy (158.5). The thermodynamic pressure n, the chemical potentials1 .U'11, and the remaining tensions O"a are defined by n-- (~~)~,c,w' ,U~t=(~;r) ~ +f(rJ,V,c,w), O"a=(:er) a· (255.2) ut 11, v, c , w Wa 11, v, c, w The arbitrary function f in the definition of ,u~ reflects the indeterminacy corresponding to the infinitely many possible functional forms for dependence of 8r upon the variables c~, which are related by the condition (158.5). A possible method of resolving this indeterminacy is to regard the ~-th constituent as the "solvent", so that 8r depends onlyupon c1 ,c2 , ••• , c.I\-J, not uponc51 ; the functionf in (255.2h must then be taken as 0, yieldinguniquevalues for ,u1 ,,u2 , ····1-'!il.-I• but .U!i\ is not defined. Alternatively, (255.1) may be written in the form Er= 0~- (n- L !i\ e'1l,u~) v+ L !i\ v,u~~ + L r O"aWa' ) ~=1 ~=1 a=l !i\ !i\ r = 0~ + ~"'f1 V [,u~ +V (n-!B~l e!B.U!B)] e-n: + a'{;l O"aWa; (255.3) that is, the chemical potentials are related to the tensions corresponding to the individual densities by the following equation: .1\ ( :;~)~,p'l,w =V [,u'H +V (n-!8~/!B.U!B)]. (255.4) The quantity on the left-hand side is a uniquely defined thermodynamic coefficient; the identity is valid for all possible choices of ,U'11 !8 V!ß + ~ U!Bv!B+ ~aa.wa.+ ~1J'; Pr + ( h;) + hPo~.P_ + eoq_, (257.2) ,p by (257.1) and (156.8) 1 we have H-~ hP~ap = f(L1 + endv, (257-3) :7' .y 1 The criticism of TRUESDELL [1952, 21, § 33] is directed toward this assumption, which is usually not stated in discussions of Case 2. In evaluating the theory, it is essential to recall that its object is to relate the total stress t to the thermodynamic tensions Ta. Werewe content, as are most writers on thermodynamics, to accept the Ta as the actual tensions in the material, there would be no problem, since the Ta may always be calculated from (247.8) 2 or from {251.4),. 2 Our presentation in this section and the next follows EcKART [1940, 7]. Cf. also MEIXNER [1941, 2, §§ 2-3] [1943, 3, §§ 3-4]. Earlier authors were inclined to select one or another quantity bearing the dimension of [entropy]/[time] and on the basis of some physical argument call it the "irreversible" production of entropy; this arbitrary procedure has been criticized by DE GROOT [1952, 3, §§ 1, 82]. However, the criticism is not applicable to the work of TaLMAN and FINE [ 1948, 31], who in essence follow EcKART's procedure and on the basis of (257.3) call LI the "irreversible" rate of production of entropy, thereafter deriving other equations in which LI occurs, e.g. e () iJ = hf>. P - kP (log O). P + () LI + e q . Some alternative forms involving the coefficients {248.5) are discussed by HuNT [1955, 14, § 1C, d]. 3 There is always a question as to how the total entropy of a system should be defined The definition {257.1), asserting that what we call the total entropy of a body is the sum of the entropies of the several elements of mass, is unequivocal and isthat customarily introduced in continuum mechanics. We do not attempt to decide whether this definition is consistent with others, such as those used in statistical mechanics. Sect. 258. The entropy inequality. 643 where (25 7.4) Ya being given by (256.10). Eq. (257.3) for production oj total entropy is an equation of balance of the type (157.1). Unlike our earlier examples of equations of balance, (257.3) is not set down by definition 1. Rather, it is derived from the assumptions expressed by previous equations, and the quantities occurring in it have meanings in earlier associations. From (257.3) we see that total entropy may be regarded as flowing into a body through its boundary at the rate -hj(), where - h is the influx of non-mechanical energy. The supply of entropy is ilfe +qjß; in particular, by (257.4), a nonzero flow of non-mechanical energy creates entropy only when it is in the presence of a temperature gradient. Both the supply of non-mechanical energy and the excess of total working over recoverable working contribute to the total entropy in just the same: way as to the specific entropy. In summary, total entropy is changed by the same factors that change:energy, with two modifications: 1. Entropy rate= ( energy rate)/ ( temperature), and 2. In addition to the sources fust stated, there is also an effective supply of energy of amount hP (log ()), P. The supply of entropy iJ is given by the bilinear form (257.4). The terms occurring in this form are the building blocks of recent theories of particular thermodynamic phenomena 2• It has grown customary to name the two terms in each summand a thermodynamic force or affinity and a corresponding thermodynamic flux. For example, we may set up the table: Affinity (1/ß), k Corresponding flux -hk (! Yaf() There seems tobe no compelling reason for assigning any one term to one category or to the other, and usage varies 3• In any case, the terms entering iJ are not uniquely determined, since, as was remarked in Sect. 157, in an equation of balance it is trivially possible to shift any term from the surface integral into the volume integral at will, and conversely, though not always explicitly, to shift any term from volume integral into the surface integral. Thus we are unable to see any physical significance in the interpretation of any one term in (257.3) unless accompanied by interpretation of all the other terms. 258. The entropy inequality4. It is a matter of experience that a substance at uniform temperature and free from sources of heat may consume mechanical work but cannot give it out. That is, whatever work is not recoverable is lost, 1 A formally analogaus treatment can be applied to the calculation of the rate of change of J va di!R but has no interest. Indeed, as is plain from the manner in which entropy was "'I" introduced in Sect. 246, its properties differ from those of the parameters va only in physical dimension. 2 Cf. footnote 1, p. 618. a Cf. the references cited in Sect. 259. 4 The postulate (258.3), which sometimes shares with (247 .2) the name second law of thermodynamics, for the case when h = 0 and q = 0 is due to CLAusrus [1854, 1, p. 152] [1862, 1, § 1] [1865, 1, §§ 1, 14-17]; the surface integral was added by DuHEM [1901, 7, Part. I, Chap. 1, § 6]. 41* 644 C. TRUESDELL and R. TOUPIN: The Classical Fie!d Theories. Sect. 258. not created. Similarly, in a body at rest and subject to no sources of heat, the flow of heat is from the hotter to the colder parts, not vice versa. The two observations, abstracted and generalized, may be put as follows: 1. PE- P1 ~ 0 when q =0 and () =const, } 2. hPO,p ~ 0 when q =0 and PE -P1 = 0. (258.1) In both these cases, by (257.4) we conclude that ()LI~ o. (258.2) When 0>0, by (257.3) we see that (258.2) 1s equivalent to H-f hP~a/>. ~ J e,l dv. (258-3) [/' '"f'" Guided by these special cases, we might set up (258.3), or the equivalent condition (258.2), as a general postulate oj irreversibility1. Unlike the previous assumptions of the field theories, it is an inequality rather than an equation. I t asserts a trend in time for various processes. The most familiar of these is an adiabatic process, defined by the condition that non-mechanical energy flow neither in nor out through the boundary, nor be created or destroyed within the body: h = 0 on !/, q = 0 in "Y. Then (258.3) yields H ~ 0: In an adiabatic process, the total entropy cannot decrease 2• If (258.3) is to hold for all bodies, it is equivalent to the local condition (258.2) restricting the sum of products of affinities by fluxes (cf. Sect. 257). Whether or not this condition may be broken down into a statement that separate parts, such as hPO,p, are to be severally non-negative depends on whether or not the body is susceptible of independent variation of the parameters Va and () or of sets of these parameters, and whether or not the partial sums selected are scalars under appropriate transformations 3• 1 We are aware of the unsatisfactory nature of (258.3) in that h and q are not uniquely defined. However, since q is to include the possibility of arbitrarily assignable sources and sinks of heat, we cannot restriet its sign or its value. 2 Notice that the specific entropy 1] is not necessarily non-decreasing at all places and times in an adiabatic process. Cf. MEISSNER [ 1938, 7, § 11]. 3 Here we touch on a great mystery of the subject. Writers on irreversible thermodynamics appear to select partial sums at will and then demand that each one be non-negative. Certainly, however, an arbitrary choice is not justified: E.g., it is not necessary that h1 0, 1 ~ o, and it would make no sense to require it, since h1 0, 1 is not scalarunder co-ordinate transformations. In dealing with the more complicated situation to be considered in the next section, PRIGOGINE and MAZUR [1951, 21,_ §3d] demand that certain terms in L1 be separately non-negative, "tout coupJage entre quantites de caractere tensoriel different etant interdit ... " but the meaning of this assertion is not clear to us. Cf. also KrRKWOOD and CRAWFORD [1952, 12, p. 1050]: "We must treat scalars, vectors and tensors separately, for entities of different tensorial character cannot interact (CuRrE's theorem)." In the publication of CuRIE [1894, 1] sometimes cited in this connection we are unable to find anything relevant. Interactions between quantities of different tensorial orders are well known in the kinetic theory of gases and are illustrated in Sect. 307. Possibly what the writers on "irreversible thermodynamics" mean to describe is the separation of effects that follows by linearization of isotropic functions (cf. Sects. 2931] and 307). Also we stumble again over the most serious gap in the fundamentals of thermodynamics, that the appropriate group of transformations of the thermodynamic variables and the invariance to be required arenot known. Cf. footnote 2, p. 620. Sect. 259. Production of entropy in a heterogeneaus medium. 645 If we raise (258.3) to the level of a general postulate of mechanics, then it is to be applied also to regions containing surfaces of discontinuity. At such a surface, provided a (erJ)f8t and eqf() be bounded, it is equivalent to 1 (258.4) 259. Production of entropy in a heterogeneaus medium 2• Confining attention to the non-polar case, we eliminate i 1 between (243.9) and (255.15), obtaining ft ft e()'lj=h~,k+ L (e'l!.U'llu;),k- L (!'l!.U'll,ku;+D, (259.1) 'll~l ~[~1 where ft h~ = h1 - L (!'ll e'll ut 'll~l f ft D == t1m dk m + n d~ - (! L O"b Wb + L t;m U'llk, m - (259.2) b~I 'll~l ft - (! L [P~ u'll k + c'l! (,U'l! + t Ufu)] + (! qr · 'll~I Therefore • ( sk) A f! qi erJ- 7r ,k =LJ+e· (259-3) 1 In the case when h = O, this farnaus condition was introduced into gas dynamics by }OUGUET [1901, 9] [1904, 3, § 2] and ZEMPLEN [1905, 8] (cf. also HADAMARD [1905, 1], RAyLEIGH [1910, 8, pp. 590- 591]), where it is used to prove that shocks of rarefaction are impossible, since it may be shown to follow from the conditions of stability (Sect. 265) and from 205.7) that sgn [17] = sgn [e]. For proof, see Sects. 55 to 56 of the article by SERRIN, Mathematical Principles of Classical Fluid Mechanics, this Encyclopedia, Vol. VIII/1. 2 This subject has been discussed by numerous authors, e.g. EcKART [1940, 8], MEIXNER [1943, 2, § 2] [1943, 3, § 4], PRIGOGINE and MAZUR [1951, 21, § 3] [1951, 17, § 2]. Our treatment follows TRUESDELL [1957, 16, § 12]. As mentioned in Sect. 243, detailed comparison of our results with those of thermodynamic writers is not possible, since they employ an equation for balance of energy obtained by intuition rather than derivation. An exception is furnished by the treatment of HIRSCHFELDER, CuRTISS and BIRD [1954, 9, §§ 7.6a, 7.6b, and 11.1]; if we presume that their phenomenological variables are to be understood as including as special cases the corresponding quantities they define explicitly for the kinetic theory of gas mixtures, then our treatment and theirs are in entire agreement up to the point where they introduce a caloric equation of state for the mixture. Instead of our (255.1) they write an equation of similar form containing i rather than i 1 on the left-hand side. It seems to us that such an equation cannot be exact: the total kinetic energy of diffusion cannot be a function of static parameters alone, since any diffusion velocities, at a given place and instant, are compatible with any values of the local thermodynamic state. From our point of view, HrRSCHFELDER, CuRTISS and BIRD neglect the quadratic term in (243.1) when they calculate (255.1), although they do not neglect it when deriving (243.7). This observation accounts entirely for the difference between their equation for production of total entropy and our Eq. (259.4). Perhaps motivated by results in the approximate kinetic theory of diffusion, numerous writers on irreversible thermodynamics (e.g. PRIGOGINE [1949, 24, § 1]) recommend use of a "reduced" heat flux which in our notation is given by From (243.2) and (243.8) 2 we see that this reduced heat flux is precisely our - hr if we assume (as in theories of diffusion) that t21 = - :7l2( 1 and if we neglect the kinetic energy of diffusion. 646 C. TRUESDELL and R. TouPrN: The Classical Field Theories. where 5t sk = h~ + L: e~ ,u~ u~' ~=1 5t = M + L: e~ (.u~r- e~) ~, ~=1 5t (.)Lf == h~ (log ll),k- ll L; e~ (,u~/ll),k ~+D- eqr, ~=1 5t f = ( h~- L; e~e~u~) (logll),k + t1m dkm + nd~- e L; a0 w0 + ~=1 0=1 5t + L: [t~mu~k.m-e~e (,u~;e),k~-eP~ u~k- ec~(.u~ + t ut)J. ~=1 Sect. 259. (259.4) The form of L1 suggests the following possible division of the variables into affinities and fluxes: Affinity Corresponding flux 5t ( 1/ ll) k - [ h1- L: e~ e~ u~] ~=1 dkm ~ [t~m+ngkmJ wo - e aofll U~k,m t~m;e u~ - [e~ (,u~/ll),k + e P~kfliJ ec~ 1 [ 1 2] -0 ,u~+2u~ To the doubts mentioned at the end of Sect. 257 must be added the remark that various alternative forms and regroupings of terms in (259.4) itself are obviously possible and lead to different selection of affinities and fluxes 1• To mention the simplest of examples, if as in Sect. 257 we choose some of the parameters Wa as the x7"', the second and third lines in the above table coalesce into the single line as has already appeared in our treatment of a simple medium. Also, part of the flux on the first line, since it is proportional to u~, could equally well be combined with that on the next-to-last line. For heterogeneaus media the entropy inequality is assumed to hold in the form (258.3), or, equivalently, (258.2). It is customary, though there appears to be no solid reason for it, to infer that various sums occurring in L1 are separately 1 We share the view of EcKART [1940, 8] that the classification is arbitrary. However, most thermodynamic writers disagree with this view and with each other's choices. The point has been discussed from a physical Standpoint by MEIXNER [ 1942, 9, Zusatz bei der Korrektur] and by DE GROOT [1952, 3, §§ 2, 9-13, 18]. MEIXNER [1943. 2, § 3] has also investigated the invariance of the classification under linear transformation, but only in the case when the affinities and fluxes are assumed linearly related. Cf. also PRIGOGINE [1947, 12, Chap. IX, § 2], DE GROOT [1952, 3, §§ 29, 44, 52, 78]. MEIXNER and REIK in § 5 of their article, "Thermodynamik der irreversiblen Prozesse", this Encyclopedia, Vol. 111/2, notice the infinitely many possible ways of rearranging and regrouping terms in (259.3); they suggest that it should be done in such a way as to render the source of entropy non-negative in all circumstances, and they assert that the only possibility of satisfying this requirement is given by the particular form they adopt. Sect. 260. Differential condition for homogeneity. 647 .S\ non-negative. For example, EcKART1 concluded that ~c~,u~~O; by (159A.5), ~=1 we have equivalently .S\ t:... "" c~,u~ ~ * ~ 0. (259.5) ~=\1!+1 In particular, if only one compound ~ is being created, it is forming or dissociating according as its potential difference ,u~ is negative or positive. Thus when but a single compound substance can result, the reaction proceeds so as to reduce its potential difference to zero. EcKART concluded also that from his special cases of (259.3) and (258.3) that c~~,U~l,kut ~ o; (259.6) that is, the diffusion current for the substance ~ carries it toward a region of lower chemical potential. Interesting as are such inequalities, it is not justified at present to regard them as derived from any general principle. 111. Equilibrium. There are several different ideas of the meaning of "equilibrium"; when put into mathematical form, they lead to conditions that are not generally equivalent to one another. These conditions occur frequently in the Iiterature of irreversible thermodynamics, but their interrelations have been studied only subject to the assumption of linear constitutive equations. We confine the following sections to the general definitions. 260. Differential condition for homogeneity. Thus far wehavenot needed to restriet the variables va specifying the sub-state. Now, however, we presume that they are the densities of additive set functions. The set functions themselves are called extensive variables 2• When the thermodynamic state is homogeneous, which here is taken to mean uniform, throughout a body, we have then 'YJ = HJIDl, e = ~JIDl, Va = TaJIDl, where Ta- f VadiDl, ..".. and (246.1) may be written in the equivalent form ~ = m e(~ .~) = ~(H, T, IDl), (26o.1) .S\ 1 [ 1940, 8, esp. p. 924]. The quantity A = (! ~ c~ ,uw was called the "affinity" or "chemical ~=1 affinity" by DE DoNDER and has been studied intensively in connection with chemical reactions; cf. DE DONDER [1927, 3] [1929, 1] [1932, 5], DUPONT [1932, 6]. If we use (159A.6), we may write the chemical affinity A in the form ( .S\ A = ~ JaAa. Aa = ~ N~a.U~· a=1 ~=1 where { is the total number of chemical reactions that may occur. The quantity Aa is the chemical affinity of the reaction a. Further developments follow by assuming that A a is an assignable function of the thermodynamic state, etc. The Iiterature of this subject is obfuscated by the habit of writing all rates as time derivatives of otherwise undefined quantities. 2 The distinction between extensive and intensive variables is due to MAXWELL [ 1876, 4]; the former kind, which he called magnitudes, "represents a physical quantity, the value of which, for a material system, is the sum of its values for the parts of the system ", while the latter "denote the intensity of certain physical properties of the substance". The defining property of intensive variables is expressed by (260.2) and hence is an alternative statement of the homogeneity of (260.1) when the densities of the extensive variables are uniform. In general, then, "intensive variable" is no more than another name for "field ". 648 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 260. where the function on the right-hand side is homogeneous of degree 1 in all its arguments. Since by (247.1) we have 0~ - o(me) - oe - () 0~ oe BH- o(m1J) - BrJ- ' oYa = ova =Ta, (260.2) the temperature and thermodynamic tensions are independent of the amount of material present. Such variables are called intensive. The Euler differential equation expressing the homogeneity of (260.1) is1 G: = 0 H + 1fiaTa Ta +,u Wl, l 8 = O'Yj + L Ta Va + ,u, a=1 where 0~ ,u == om · By (251.1) 4 we may write (260.3) in the form c =,u, (260.3) (260.4) (260. 5) expressing succinctly the relation between the caloric equation of state for the density, 8, and that for the total energy, Gl: [cf. the analogous result (255.14) 3]. While (260. 3) is the basic expression of homogeneity, other forms are more commonly encountered. First, from (260.2) and (260.3) we have l d@ = 0 d H + L Ta d Ta+ ,Ud Wl (260.6) a=l as the counterpart of (247.2). If we subtract this from the differential of (260.3)1 , we obtain the Gibbs-Duhem equation 2 : l 0 = H d() + L TadTa + Wld,u. (260.7) a=1 For an alternative derivation, we need only multiply (251.3) 4 by ill( and then take note of (260.5). In the usual applications the Va are supposed to consist in the sr masses ill(~ of the constituents of a mixture and in a further set S~ of extensive variables Da independent of the partial masses. These quantities, in the case of a homogeneous system, are related to the densities used in Sect. 254 and 25 5 as follows: 9)(2! V I c~ d ill( = c~ ill(' ~=1 .i wc~ = ill( ' I Da= I WadWl = Wa Wl. V (260.8) From (260.3) we have ! ft Gl:r = 0 H + L aaDa + L;,u~Wl~ + ,u Wl, (260.9) a=1 ~=1 where (260.10) 1 GIBBS [1875. 1, Eq. (54)]. 2 GrBBS [1875, 1, Eq. {97)], DUHEM [1886, 2, Part Il, Chap. Il, Eq. (81)]. Sect. 260. Differential condition for homogeneity. 649 and where we have written 1 instead of ~ as areminderthat a mixture is being considered (cf. Sects. 254 and 255). The indeterminacy represented by the occurrence of /(H, rol, .2) is the sameasthat already encountered in (255.2). So as to eliminate this indeterminacy, we may agree to regard 1 as a function of ml, m2, ... , mjl but not also of m; then we may set (260.11) It is these chemical potentials ,u~ which generally are used in works on chemical equilibrium. They have the disadvantage of not being easily interpretable in terms of densities. However, for any f in (260.10), by (260.4), (260.6)a, and (260.8) we obtain ,u~ = ,U'll. + ,u ' while (260.6) and (260.7) become, respectively, d~, ~ 9 d H +,~,"' dQ, +.~f; d!!Jl,,l 0 = H d () + I.J2adaa + L m'l(d,u~. a=1 'li.=I The latter relation may be written in terms of densities: f 5l 0 =1Jd() + LWadaa + L c'll.d,u~, a=1 'll=1 (260.12) (260.13) (260.14) but it is important to note that the differentials of the ,u~, not generally the ,U'll., occur here. However, the custom among thermodynamical writers of expressing all relations in differential form can lead to confusion, since the differentials dc'li. arenot independent. From (158.5) it follows that By (260.12), then, .lt Ldc'll. =0. 'll.=1 5l 5l L f.l'li. d Cl]l = L f.l~ d C'l(. 'l1.=1 '11=1 (260.15) (260.16) Thus the Gibbs equation (260.13) 1 , for homogeneaus conditions, may be written in either of the forms ds1 = () d1J +a~ aa dwa +~r~l~ dem, I f 5l (260.17) = () d1J + L aadwa + Lf.lmdc'l(, a=1 21=1 as well as in other forms, for variations in which the total mass is kept constant. It is customary to take one of the parameters !2a as the volume, 78. The Gibbs-Duhem relation (260.14) then assumes the form 5l f o = 1J d () - v d n + L c~1 d,urr + L wa daa. (260.18) ~1=1 a~1 While writers on "irreversible thermodynamics" sometimes use the relations of this section in problems concerning deformation, we are unable to find any 650 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 261, 262. solid ground for ascribing any relevance to them except in equilibrium. Consequences of (260.13) are ft ( o.u~) 2>~ -0- ~ =O, ~=l C~ O,n,c ,a V~=-- ( o.u~) . o:rr: O,c,a (260.19) Another condition of equilibrium, apparently independent of the foregoing, is the principle of detailed balancing, asserting that in thermostatic equilibrium each chemical reaction is individually in equilibrium. In terms of the reaction rates Ja occurring in (159A.6), this postulate asserts that 1J = const and v = const implies Ja= 0 for all a. 261. Mechanical equilibrium. By definition, a system is in mechanical equilibrium if :V= 0 in an inertial system. For simple media, no special thermodynamic consequences result. For heterogeneaus media, however, by (215.4) and (205.2) we have 5\ tkm,m + L (J~I ~~ == 0. (261.1) ~~1 If now we add assumptions 1 sufficient to validate the Gibbs-Duhem equation (260.18), from its two consequences (260.19) we obtain 5\ ft [ ft o.u*) ~2;1 e~(f~,.- .U~.k) =-tk:m- (J ~2;1 c~ ];:1 ( oc: o,",c~./!!.!,k + + ( :o~)", c,a e,,. + ( ~~ t. o,a n, k+ bt ( ::: t. "· c, ab ab,k]' (261.2) =- (t'l: + n b'/:),m- e .± c~ f( ~) e,k + ± ( :.u~) bab,k] · ~~1 l "· c,a b~1 ab 0, "· c, a Further drastic assumptions are required in order to get a simple result. If () = const and if the parameters w do not appear or if ab = const for all b, and if further t'l: =-nb'/:, then (261.2) yields 5\ L: e~u~!k- .U~,k) = o. '2!~1 In this case, by (158.19), we may apply (158.18) to show that 5\ 5\ L (J'H {f~ k - .U~. k) ut = L !?'H (f'l! k - .U~. k) ut '!1~1 'H~l for all definitions of the diffusion Velocity U'H. (261.3) (261.4) It is also possible to reorganize the terms in (261.2) in a fashion analogaus to that used in Sect. 256. According to EcKART2, it should be possible to arrange the affinities and fluxes (Sect. 259) in such a way that the vanishing of the ones is a definition of equilibrium; the vanishing of the others should then be a provable criterion of equilibrium. This interesting idea does not serve to classify the quantities uniquely. 262. Variational condition of equilibrium. Toward the end of the nineteenth century arose a tendency to regard all gross phenomena as essentially thermodynamical, whence followed attempts to construct a system of energetics including mechanics as a subsidiary part3. The strength and the weakness of this 1 The idea was suggested by PRIGOGINE [1947, 12, Chap. IX, § 3]; we generalize the treatment of DE GROOT [1952, 3, § 47]. 2 [1940, 8, p. 270], "D-factors" and "C-factors". 3 Cf. DuHEM [1911, 4], }AUMANN [1911, 7] [1918, 3], LoHR [1917, 5] [1924, 10] [1940, 17] [1948, 12]. Sect. 262. Variational condition of equilibrium. 651 approach are illustrated in the special case when the substate v in (246.1) consists in the 9 deformation gradients x~x from a reference state X. The objective is to give a purely thermodynamic delinition ol stress, without reference to mechanical considerations, and to derive CAUCHY's laws (Sect. 205) as special cases of a general criterion of equilibrium 1• This condition is bipartite. First, the principle of virtual work in the form (232.6) when äl =0-i.e., ~ =0-along with the postulate ~=t5fediDl, (262.1) -r is assumed. Second, thermal equilibrium is defined by t5 H = o. (262.2) All variations are assumed to respect the conservation of mass. From (262.1), (232.5), (247.2), and (156.1) follows r Je [0 I5'Yj + L Ta t5 Va] dV = ~ s" t5xk da + Je I" t5xk dV, -r a=I fl' -r (262.3) Je I5'Yjdv= o. -r The second condition is equivalent to the requirement that 'YJ = const in the variations; in this case, or in the alternative case that (262.3) is used as a side condition and the variations are executed with (J = const, the former condition becomes r Je L Ta t5va dV = ~ s" t5xk da + Je I" t5xk dV. (262.4) -r a=I fl' -r Further progress cannot be made unless we specify the relation between the t5va and the t5x". If all variations are independent, we get Ta = 0, sk = 0, jk = 0 (262.5) as the conditions of equilibrium for this trivial case. However, if the Va are in fact the ~K· as we agreed to assume, then we write (262.3) in the form f e ( ~;) t5x~x dV = ~ s" t5xk da + f e jk t5xk dV. (262.6) -r ' [/' -r Since t5~x = (t5x");K• by GREEN's transformation we thus obtain 0 = J. (s" da - e ~:- dAx) t5x" + f [e jk + (e ~1----). ] t5x" dV. (262.7) 'f" ox.K OX ·K ,K [/' ' -r ' lf there are no kinematical constraints, it follows that s"da =e~dA" ox;K o=(-~~) +-1" (! oxk ·K (! ;K' on .91 in "f'". (262.8) 1 The analysis we present is due to Grass [1875. I, pp. 184-190]. The theory based on the more special assumptions E= e(E, X) was initiated by GREEN [1839. I, pp. 248_:_255] [1841, 2, pp. 298-300] and developed by KrRCHHOFF [1850, 2, § 1] [1852, I, pp. 770-772] and KELVIN [1855. 4] [1856, 2, Chaps. XIII, XIV] [1863, 2, §§ 61-67] [1867, 3, § 673] and App. C. § § (c)- (d)]. Cf. Sect. 232. Further attempts along this line have never been successful. The usual approach is to assume e depends on various variables at hand. For example, VAN MrEGHEM [1935, 9, § 3] sets up the equation e = e(n, 0, A, ih where A is the affinity and Eis defined by (31.6) 2 , as supposedly appropriate for a viscous fluid. 652 C. TRUESDELL and R. TouPIN: The Classical Field Theories. By (210.5) and (210.8), the quantities defined by F,K--~ k -(!~k uX;K Sect. 263. (262.9) in virtue of (262.1) and (262.2) satisfy the same equations as those imposed on PIOLA's double vector T,.K in virtue of the conditions of equilibrium in continuum mechanics. Thus it is not inconsistent with mechanical principles if we set (262.10) and replace the laws of statics by the conditions (262.1) and (262.2) of thermodynamic equilibrium. At bottom, this analysis rests on the same idea asthat in Sect. 256A. If there are additional parameters Va beyond the x7K, the corresponding Ta must vanish in equilibrium, provided there are no constraints. The details here are akin to those in Sect. 256. Despite the elegance of the foregoing analysis, it cannot be accepted. Indeed, from a special thermodynamic assumption, CAUCHY's first law has been derived. That CAUCHY's second law cannot be derived by such methods is plain from its generalform (205.10) and from footnote 2, p. 596. In fact, in general the stresst derived from (262.1 0) is not symmetric unless some additional assumption, such as that e depends on the x7K only through E, is added. But there is a deeper objection. The equations of mechanics describe a wider range of phenomena than do the principles of thermodynamics. No one will contest the principles of balance of mass, momentum, and energy, but the existence of a caloric equation of state is an assumption of a more special kind1. CAUCHY's laws arevalid for all sorts of continuous media, but in essence the thermodynamic method just explained is restricted to perfectly elastic bodies 2• If there are viscous or plastic stresses, they must be dragged in by extra assumptions having nothing to do with thermodynamic principles 3 • Finally, to obtain equations of motion it is customary to apply the Euler-D'Alembert principle to the equations of equilibrium, and this is at bottom no different from assuming the balance of momentum to start with. 263. Inequalities restricting the equations of state, I. "Absolute" temperature. Thus far in our formal structure the temperature () has been taken as defined by (247.1) 1 with no other restriction than that the equations of state be such as to permit any number of differentiations and functional inversions. For the validity of one or two of the formulae it has been tacitly supposed that () =f= O, and at one point in connection with the entropy inequality (258.3) we have assumed that () >O. This last requirement is customarily imposed throughout the subject. When the caloric equation of state (246.1) is so restricted that ()>O, 1 This is borne out also by general statistical mechanics, whence, as shown by NoLL [1955, 19], the field Eqs. (156.5), (205.2), (205.11), and (241.4) followin the greatest generality, but the existence of thermodynamical equations for cases other than equilibrium remains in doubt. In the kinetic theory of monatomic gases, there is a thermal equation of state in all circumstances, but a caloric equation of state is valid only in conditions sufficiently near to equilibrium. 2 Within this Iimitation, CoLEMAN and NoLL [1959, 3] have replaced the analysis in this section by a rigorous development based upon a genuine minimum principle; they obtain not only the classical stress-strain relations but also full conditions of· stability. 3 Cf. the method of DUHEM [1901, 7, Chap. I, §§ 1- 5] [1903, 4 and 5] [1904, 1. Chap II. §V] [1911, 4, Chap. XIV, § 3]. who was a devotee of the thermodynamic approach. Only the details, not the principles, of more recent allegedly thermodynamic treatments are different. Sect. 264. Stability of equilibrium. 653 for all allowable values of the thermodynamic state 'fJ, t', then () is said to be an absolute temperature. It does not seem to be possible within the concepts of thermodynamics to give any clear idea of what absolute temperature is1. We may, however, rephrase the assumption as follows: the caloric equation of state (246.1) is such that () has a finite lower bound. By (247.1h this is equivalent to 00 = inf ( ;; )., >- oo. (263.2) For a given function e ('YJ, v), put (263-3) Then . f ( oe') m 87].,=0. (263.4) While, as stated in Sect. 245, the requirements of invariance to which the caloric equation of state is subject are not clear, it seems that e' as an internal energy function is physically equivalent to e; by (263.4), if () is defined from e' rather than from e, we obtain (263.1). Thus for an internal energy function satisfying (263.2) it is possible to find a physically equivalent energy function such that the temperature is absolute. The requirement (263.1) has a simple physical interpretation: Addition of heat to a body increases its internal energy. This is so because 'fJ, while itself not a measure of heat, is to be interpreted as a quantity which increases or decreases according as heat is being supplied or drawn off. Pursuing this same interpretation suggests also the additional restriction lim e=oo: (263.5) If, at a fixed substate 'L', heat is added indefinitely, the internal energy also increases indefinitely. 264. Stability of equilibrium. At the conclusion of a paper on various forms of the "second law of thermodynamics ", CLAUSIUS 2 asserted, "Die Energie der Welt ist konstant. Die Entropie der Welt strebt einem Maximum zu." GIBBS 3 replaced this claim of a trend in time by a definition of thermostatic stability for an isolated system: (L1H) 0. (265.3) When f is twice differentiable, (265.3) implies that f"(w) ~ 0. Conversely, the condition f"(x) >O is sufficient, but not necessary, for convexity. Likewise the more general condition (265.1) implies that II ~w11 u :11 u is positive semi-definite, (265.4) where we have put OB Ga- owa · (265.5) This is the result inferred by GIBBS1 • Suppose that for a given mixture there exists a convex set of local states w such that the homogeneous state corresponding to each is a state of stable equilibrium. It follows then from (265 .1) and (265 .2) that for all states ro', w" in the set we ha ve e (w") - e (w') - (rJ"- rj') e'IJ (w') - l .1\-1 - (v" - v') Ev (w') - ~ (cU( - c~) Ec, (w') > 0, ~=1 ~l (265.5 A) 1 [1875. 1, pp. 111-112]. In his earlier study of simple fluids [1873. 2, p. 29] GIBBS had written, "The condition of stability requires that, when the pressure is constant, the temperature shall increase with the heat received,-therefore, with the entropy .... It also requires that, when there is no transmission of heat, the pressure should increase as the volume diminishes ... " l.e., the "condition of stability" is (on) ~ 0 ov 'lj (equivalently, ""~o. O, are consequences of (265.1). DuHEM's methods did not enable him to derive (265.8); neveriheless, he knew that this equation holds in various applications, so he adopted it, calling it "the postulate of HELMHOL TZ". DuHEM criticized certain inequalities of GIBBS similar to (265.7)3 [1875. 1, Ineqs. (167). (168), (169)]: "une des rares inexactitudes"; also [1893, 2, Part I, Chap. V, §V] [1894, 2, Chap. IV, especially §§ 5 and 7] [1898, 2] [1911, §, Chap. XVI, § 9]. Such results are very sensitive to the choice of variables, and GrBBS is obscure in the detail of argument as weil as reluctant to state just what is being varied. CoLEMAN and NoLL have straightened the whole matter out; they find that DuHEM's criticism is weil taken if his interpretation of what GrBBS meant to say is correct, but they find another interpretation in which GIBBS' inequalities are correct. In summary, there is no conflict between the results of GrBBS and those of DuHEM, which complement one another; all their results and many more of like nature are stated unequivocally and derived precisely in the forthcoming work of CoLEMAN and NoLL. Sect. 265. Inequalities restricting the equations of state, II. 657 where the subscripts to c; denote partial derivatives. I t is clear that (265. 5 A), for the special variables considered here, is a mathematical statement of the physical notions behind (264.7). Thus the approach based upon (264.7), when made clear, amounts to an asserting as a postulate the main result GIBBS sought to derive. For the case of a fluid obeying a caloric equation of state of the form (264.11) -the case to which CoLEMAN and NoLL's analysis applies-we have 1 lJ = ((), -n, t-t1 ,f1z, ... ,ft~- ), and the matrix occurring in (265.4) assumes the form ao ao ao ao OrJ ov ocl OCSt-1 on on on on -8r/ ov 0/l) ocl OCft-1 (265.6) OrJ 0/lft-1 OCSt-1 where all quantities are taken as functions of 'f), v, c1 , ... , cSt_ 1 • In order that this matrix be positive semi-definite, it is necessary that each principal minor be non-negative. In particular, ~) ~ 0, OrJ v, c- (an) ~ O, oe ~.c- (265.7) The first of these, by (249.1), may be written in the form 2 . (265 .8) that is, in order to increase the temperature of a body held at constant substate, it is necessary to supply heat. The second of the inequalities (265.7) may be interpreted as a statement that the Laplacian speed of sound is real [ cf. footnote 1, p. 631 and (297.13)]. But also all principal minors of (265.6) must be nonnegative; in particular, o(n, 0) I ~ O. o(rJ, v) c- (265 .9) Up to now we have identified the variables constituting the substate, conformaply to the caution stated in Sect. 246. For more general thermostatic systems, it is clear that (265 .4) cannot generally hold. For if wb is a suitable parameter for describing the state of a system, so also are -w0 and 1/wb; use of - wb, leaving all other Wa unchanged, would change the sign of 8aaf8wb, and use of 1/wb would change the sign of 8abj8wb. However, it seems likely that for some admissible choice of the wb in any system, (265.4) will result, and its simplest consequence, oaa > O owa = ' has often been taken as a condition for stability. (265.10) 1 Note that /l~ = ll'H- llSt, m = 1, 2, ... , S\'- 1, where /l~I is the chemical potential defined by (255.2) 2 . In virtue of (158.5) we have St St-1 ~ ll'H c'H = ~ ll~ c'1f.. '11=1 '11=1 2 GIBBS [1875. 1, Eq. (166)], HELMHOLTZ [1882, 2, § 1]. Handbuch der Physik, Bd. Ill/1. 42 658 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 265. We now translate the condition of stability (265.1) for homogeneous states of isolated systems into forms more a~cessible to experimental test. We have seen that (265 .4) is a necessary but not sufficient condition for stability; if we replace "positive semi-definite" in (265 .4) by "positive definite", we obtain a condition which is sufficient but not necessary. We agree to consider this stronger condition only1• Also we shall rule out the possibility of thermodynamic degeneracy (Sect. 253), and we exclude from consideration states such that IDa = 0 or oca = 0, for any a. Also we adopt (263 .1). A system satisfying these conditions will be called ultrastable. By a weil known theorem on matrices, from (252.11) we see that a necessary and sufficient condition for ultrastability is that II abll be positive definite and o(T, O)fo(v, 'YJ) >O. From the former statement it follows that ct> >O. By (252.22)a we conclude that the latter statement, when the former is satisfied, is equivalent to x .. > 0. Hence we have the First necessary and sufficient condition for ultrastability: llabll is positive definite, x .. > 0. (265.11) Proceeding in just the same way from (252.11) 2 and using (242.22) 5 , yields the S econd necessary and sutficient condition for ultrastability 2 : u;a bll is positive definite, Xu > 0. (265.12) In this condition, ;ab may be replaced by Vab· From (265.11) and (252.14) or (252.16), or, alternatively, from (265.12) and (252.15), it follows that y> 1. Conversely, if y> 1 and llc!>abll is positive definite, from (252.14) or (252.16) it follows that x .. >O, whence by (265.11) the equilibrium is ultrastable. · Thus we have proved the Third necessary and sutficient condition for ultrastability: llc!>abll is positive definite, y>1. In just the same way, from (265.12) and (252.15) we prove the F ourth necessary and sutficient condition for ultrastability: n;a bll is positive definite, y > 1 . (265.13) (265.14) The stability of thermodynamically degenerate substances requires a separate analysis. The particular kind of degeneracy which defines an ideal material (Sect. 253) does not affect the arguments given above, so the conditions of stability (265.11) to (265.14) remain applicable. For piezotropic substances, by (253.9) and (253.10) it is easy to see directly from (252.11) that a necessary and sufficient condition for ultrastability is llvabll is positive definite, xu>O or x .. >O. (265.15) We do not attempt to find conditions appropriate to other degenerate materials or to investigate the conditions of neutral stability. 1 This was done by SAUREL [1904, 6 and 7], who phrased GrBBs' considerations more formally in terms of differentials. Our text, in effect, replaces the Gibbs-Saurel arguments by more efficient and precise mathematical analysis. It appears that the subtle logical difference between stability and ultrastability was recognized by GrBBS, though he did not emphasize it; in the work of DuHEM, it is completely obscured, and a reader with modern standards of rigor will need to apply some Iabor if he is to disentangle DuHEM's arguments. 3 The condition (265.12h, when f = 1, is traditionally used in analysis of the stability of a VAN DER WAALS gas, as was mentioned in Sect. 252. Sect. 265. Inequalities restricting the equations of state, II. 659 An important consequence of (265.12) has been derived by EPSTEIN1• Consider two tensions, Ga andab; then Ga =aa(wa,Wb,wab), ab =ab(wa,wb,wab), where wab stands for all wa's except Wa and wb. The Maxwellrelations (252.1) may all be expressed in the form (265.16) Inverting the relation giving ab yields wb = wb(ab,Wa,wab) so that Ga= aa(wa, wb (ab, Wa, wab), wab). Differentiating this relation yields But ( oaa ) ( oaa ) ( oaa )' ( owb ) owa oo • ..,ab= owa ..,a + OWb ..,b owa oo • ..,ab' 0 = (:::Lab= (:::)ab . ..,ab+ ( :: )..,b ( ::J..,a' l = (:::)ab, ..,ab+ ( ~= )...b ( ::~ )...b' (265.17) (265.18) where the second step follows by (265.16). Substituting this result into (265.17) yields the identity ( oaa ) ( oaa ) [( oaa ) 12 ( OWb) owa ab, ..,ab = owa ..,a - owb ..,b -aab ..,b · (265.19) By (265.12) we see that the term subtracted is non-negative; hence (_!aa ) < (~) . owa D'b, ..,ab= &wa ..,a' (265.20) equivalently, ~) oaa 00, ..,ab >(~) = oll'a ..,a · (265.21) The foregoing analysis is due to EPSTEIN, who calls (265.20) and (265.21) the restricted Le Chatelier-Braun principle. For the case of a simple fluid, the relation just derived is equivalent to y ~ 1 . Whether all inequalities called "conditions of stability" in the Iiterature are consequences of GIBBs' condition {265.4) is not certain. Further inequalities to be satisfied by the equations of state have been discussed2, but not definitively. It is to be noted that equations of state need not satisfy the conditions of stability for all values of the thermostatic state. Rather, the conditions serve to distinguish stable states from unstable ones. In a theory where mechanical phenomena are of primary interest, it may be natural to seek and impose a require1 [1937. 1, § 143]. 2 Cf. WEYL [1949, 38, §3]. who treats only the case when f=l, and who proposes also the postulate ( ()2~) > 0, i.e., ( :4>) < 0, and derives from it some further inequalities. WEYL notes als:~ha1 from (252.3)rit11 follows that ( :; ) 11 and (:~ ). cannot vanish simultaneously; hence (265.19) implies th.at ( :; )v and ( :~ t are of one sign. CouRANT and FRIEDRICHS [ 1948, 8, § 2] propose the stronger inequality (~::)'I;::::; o. For the effect of these inequalities in gas dynamics, see § 37 and § 56 of SERRIN's article, Mathematical Principles of Classical Fluid Mechanics, This Encyclopedia, Vol. VIII/1. 42* 660 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sects. 266,267. ment of universal stability, but in a theory aiming to determine criteria of stability of equilibrium it is more natural to include the theoretical possibility of unstable states1• F. Charge and Magnetic Flux. I. Introduction. 266. The scope of this chapter. Classical mechanics is founded on the principles of conservation or balance of mass, momentum, moment of momentum, and energy. Alongside these mechanical principles we now set up two further principles of conservation as a basis for the theory of electromagnetism. These are, the conservation of charge and the conservation of magnetic flux. Subchapter li formulates these principles and deduces and interpreis some of their consequences. In Subchapter III we introduce an additional postulate, the aether relations, and in Subchapter IV we consider the mechanics of the electromagnetic and charge-current fields. The sequey1ce of hypotheses and the order of logical development that we adopt here depart from the traditional treatments. A principal objective is to isolate those aspects of the theory which are independent of the assigned geometry of space-time.from those whose formulation and interpretation depend on or imply a particular space-time geometry 2• For example, we regard the conservation of charge as a physical or intuitive concept logically independent of the concepts of rigid rods, uniform clocks, and inertial frames, and we have chosen to express this law in a mathematical form likewise independent of the representation of these extraneous entities. The development of electricity and magnetism is too closely connected with special cases and special phenomena for detailed historical references as in the previous chapters tobe practicable here. The central importance of MAXWELL's work is well known; in FARADAY and KELVIN he had major predecessors, and the classical theory is in part the creation of his successors, HERTZ and LoRENTZ. An excellent history has been written by WHITTAKER 3• 267. Antisymmetrie tensors. Since the only quantities occurring in the conservation laws of electromagnetic theory are antisymmetric tensors, we introduce some relevant specific terminology and notation. Let x''=x''(:x:) (267.1) denote an element of the group of analytic transformations of the co-ordinates of an n-dimensional space. Let U' = u-~u (267.2) denote a transformation of the unit 4 U. As U runs over the real numbers, (267.2} generates the group of unit transformations on the unit U. 1 E.g., in gas dynamics the van der Waals equation may be used for the entire density range only so long alj the temperature exceeds the critical temperature, while in the theory of liquefaction it is tiseful to consider it also below this temperature. 2 Our development. and ordering is similar tothat of KoTTLER [1922, 4]. Forahistory of the mathematics and physics leading to Kot"TLER's formulation of the basic equations of electromagnetic theory, cf. WHITTAKER [1953, 35, p. 192]. This earlier work includes that of HARGREAVES [1908, 5], BATEMAN [1910, l], and MURNAGHAN [1921, 4]. Cf. also the remarks ofW:EvL [1921, 6, § 17]. [1950, 35, § 17]. KOTTLER's ideas were taken upbytheDutch school of geometers and mathematical physicists and culminated in the series of papers by VAN DANTZIG [1934, 11, '\[ 12] [1937, 10 and 11]. Cf. also ScHOUTEN and HAANTJES [1934, 8]. Shortcomings of the metric viewpoint even in strictly mechanical situations are emphasized by KRÖNER [1960, 3, § 18]. 3 [1951. 39] [1953. 35]. ' See Sect. App. 9 for a discussion of units and unit transformations. Sect. 267. Antisymmetrie tensors. 661 A tensor field under the group of analytic co-ordinate transformations having absolute dimension [UJ is a set of functions of the co-ordinates :JJ whose law of transformation has the general form ' ' oxm' oxn' ox' t"f"···(:JJ' U')=I(:JJ'/:JJ)I-wsgn(:JJ'/:JJ)PU-----···------, ···fm"···(:JJ U) (267.3) '··· ' oxm ox" ox' • '··· • • where (:JJ'/:JJ) denotes the Jacobian of the Co-ordinate transformation. If p =W =0, I is called an absolute tensor. If p = 0, w =l= 0, I is called a relative tensor of weight w. If p = 1, w =0, I is called an axial tensor, and if p = 1, w=j= 0, I is called an axial relative tensor of weight w. A covariant or contravariant completely antisymmetric tensor of rank k will be called a k-vector. We can restriet the value of k to be less than or equal to n since k-vectors suchthat k >n vanish identically. 0-vectors are called scalars and 1-vectors are called vectors. Weshall be concemed primarily with the following special types of k-vectors: 1. Absolute covariant or contravariant k-vectors. 2. Contravariant relative k-vectors of weight + 1. 3. Covariant relative k-vectors of weight - 1. 4. Covariant and contravariant axial k-vectors. 5. Contravariant axial relative k-vectors of weight + 1. 6. Covariant axial relative k-vectors of weight -1. Tensors of types 2 and 3 will be called k-vector densities. Tensors of types 5 and 6 will be called axial k-vector densities. If the group of co-ordinate transformations consists solely of unimodular transformations, (:ll' /:JJ) = + 1, then the transformation laws of absolute k-vectors, axial k-vectors, k-vector densities and axial k-vector densities coalesce. If the group consists solely of transformations with positive Jacobian, then the trans~ formation laws of absolute k-vectors and axial k-vectors coalesce, as do the transformation laws of k-vector densities and axial k-vector densities. In the following definitions, Fand Y stand for k-vectors; they may be absolute, axial, densities, or axial densities, but their variance is specified by their indices. The dot product of a contravariant k-vector Fand a covariant m-vector Y, k-:?:. m, is defined by (cf. Sect. App. 3) (267.4) (267.5) Let e'•'•···'" and e,,,, ... , .. denote the permutation symbols such that e12···" = e12 ..... = + 1; these symbols define axial n-vector densities. The duals, dual F and dual Y, of covariant and contravariant k-vectors F and Y are defined by 1 (dual F)'•'•···'"-k = - e'•'•···'n-ks, ... sk F. or k! s,s, ... sk, dualF = E·F, (267.6) dual Y=Y·E. (267.7) These definitions imply that for any type of k-vector we have dual dualF = F. (267.8) 662 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 268. The cross product of a covariant (contravariant) k-vector F and a covariant (contravariant) m-vector Y, k +m~ n, is defined by 1 (F X Y)'•'•···'"-.t-.. = --e'•'•···'"-.t-•••···•.t'•···"" F. y: k! m! ••····~ t, ... lm (267.9) (F X Y) r,r, ... rn-.t-m = - k! m! -F••····~Y •···"" Bs, ... s.tl1 ••• 1mr1 r1 ••• rn-.t-M . (267.10) These definitions generalize that for the cross product of a vector and a tensor given in Sect. App. 3. We have the useful identities relating the cross product, dot product, and dual: Fx Y =(dual Y) ·F, (Fand Y covariant), (267.11) Fx Y = Y · dualF, (Fand Y contravariant). (267.12) A constant tensor field whose components have the same values in all Coordinate systems related by a group of transformations ~ is called a constant invariant tensor ol the group ~- The axial n-vector densities e'•'•···'" and e,,,, ... ,,. are constant invariant tensors of the full group of general analytic transformations. The mixed Kronecker deltas .:5~:~::::~: are examples of absolute constant invariant tensors of the group of analytic transformations. The covariant Kronecker delta d,. is a constant invariant tensor of the orthogonal group. Since the fundamental axial n-vector densities are constant invariant tensors of the group of general analytic transformations, the identity (267.8) establishes a one-to-one correspondence between k-vectors and their dual (n- k)-vectors which is invariant under the group of general analytic co-ordinate transformations. This implies that the transformation laws of a k-vector and its (n- k)-vector dual are indistinguishable. For example, suppose that 11 , 12 , ••• , 16 is a set of six functions of four Co-ordinates whose transformation law is suitably defined by setting l1 = ~~· l2 = ata• Ia = al,. 1, = a2s· ls = a2,. ls = aa,. where a is an absolute covariant 2-vector. The same transformation law for the f's is obtained by setting 11 =(dual a) 34 , 12 =(dual a) 42 , Ia =(dual a) 23 , 1, = (duala)14 , 15 = (duala)31 , 16 = (duala)12 • The point we make is analogaus to the better known fact that, under the orthogonal group, the transformation laws of covariant and contravariant tensors are indistinguishable. The similarity of the two situations can be made even more apparent by noting that a contravariant index may be raised or lowered by the Kronecker deltas d,. or d'" and that this process of association is invariant under the orthogonal group because the covariant and contravariant Kroneckerdeltas are constant invariant tensors of that group. 268. Invariant integral and differential equations independent of a metric or connection. Let o, stand for ojox'. If F is an absolute or axial k-vector field, the quantities defined by (rotF),s,s, ... s.t = (k + 1) o[,.F.,s, ... s.t] (268.1) transform as the components of an absolute or axial (k + 1 )-vector field under general transformations of the co-ordinates. Similarly, if Y is a contravariant k-vector density or axial k-vector density, the quantities defined by (div Y)'•'•···'.t-• = a. Y'•'•···'~-·· (268.2) Sect. 268. Invariant integral and differential equations. 663 transform as the components of a (k -1)-vector density or an axial (k -1)- vector density. The (k+1)-vector field defined in (268.1) is called the natural rotation of the field F, and the (k -1)-vector field defined in (268.2) is called the natural divergence of the field Y. Note that the natural rotation is defined only for covariant absolute or axial k-vectors and not for covariant k-vector densities or contravariant k-vectors of any type. Similarly, the natural divergence is defined only for contravariant densities or axial densities. The dual of the natural rotation of a covariant k-vector is a contravariant (n- k -1)-vector called the curl of F. curl F = dual rot F. (268.3) Wehave the following identity relating the divergence, curl, and dual of a k-vector: curl F = div dual F. (268.4) Consider the oriented k-dimensional surfaces in an n-dimensional space admitting a parametric representation (268.5) by piecewise continuously differentiahte functions of the parameters u,., a = 1, 2, ... , k. We denote such a surface (hypersurface) by ~. If the parameters u are transformed by continuously differentiable one-to-one parameter transformations with positive Jacobian u"' = u"' (u), (u'ju) > 0, (268.6) we obtain another admissible parametrization of the surface 9f given by x' = x'(u') = x'(u(u')). (268.7) A k-dimensional circuit is an ~ topologically equivalent to the complete boundary of a (k + 1)-dimensional interval. Let IDl [ ~, U] denote a quantity defined for every ~. We may think of IDl[~. U] as a physical quantity having the dimension [U] suchthat for every k-dimensional set ~ of events or points in space a value is assigned to it in principle. Thus, for example, we may think of IDl as the total charge in a given spatial region or as the total charge which has passed through a 2-dimensional surface in space in a given interval of time. In the general case, we assume that IDl is an additive set function. More specifically, weshall assume that IDl is expressible in the form IDl[~. U] = J m(~(u), :::)du1 du2 ... du", (268.8) .9'k where the integrand is a polynomial in the vectors ~=:. Theorem: I f IDl [ ~, U] has the transformation law IDl' [ ~, U'] = U IDl [ ~, U] (268.9) under independent transformations of the unit U, generat analytic transformations of the co-ordinates ~. and transformations of the Parameters u, the set function IDl [ ~, U] is expressible in the form a-n = __1_ J ...,_ d Y.'•'• .. ·'k = J m · d .9.. .:u~ k! . .. ......... ~ " '" (268.10) 664 Co TRUESDELL and Ro TouPIN: The Classical Field Theorieso Secto 268. wkere m is an absolute covariant k-vector field, of absolute dimension [UJ, independent of tke vectors ::: , and wkere d ~ is tke absolute contravariant k-vector defined by o x[r, o x'• · o x'k] • d oJ,,r,ooo'k = k I----. 0 • --du1 du2 du" Jk - 0 oul ou2 ouk . . . . (268.11) Proof: First consider the invariance of !In under the subgroup of unimodular parameter transformations: (u'fu) = +1. Under parameter transformations with the co-ordinates held fixed, the quantities ox'fou" transform as a set of n absolute covariant vectors in a k-dimensional space. A known theorem of classical invariant theory1 states that every invariant polynomial function of a set of n covariant vectors in k dimensions: (n ~ k) under the group of unimodular transformations is reducible to a polynomial in the (~) determinants of the vectors taken k at a time. In the present application of this theorem we conclude that the integrand of (26808) must reduce to a polyn?mial in the (;) variables (268.12) The coefficients in the polynomial are at most functions of the co-ordinates x'(u). Under parameter transformations with Jacobian not equal to 1, the variables D of (268012) transform as relative scalars of weight 1, while the coefficients of these quantities transform as absolute !'!Calarso The invariance of m under this larger group allows us to conclude . that . the integrand must reduce to a linear homogeneous function of the variables D. Hence we can write !In in the form (268.10). The invariance of !In under general analytic transformations of the co-ordinates and the quotient rule of tensor algebra allow us finally to conclude that the coefficients m,,,,o 0 o't must transform as a covariant tensor. Only the antisymmetric part of the tensor of coefficients contributes to the transvected sum in (268.1 0); hence, there is no loss in generality by assuming that m is a k-vector. On transforming the unit U, we see that the absolute dimension of m must be [U] provided we assume that the absolute dimension of d ~ is [7]. As explained in Sect. App. 9, c;:o-ordinates and parameterswill always be assigned the absolute dimension [7]. The physical dimension of d ~ may differ from [7], and the physical dimension of m may differ from the absolute dimension of !In; but this problern will be treated in some detaillater in Sect. 277. Corollary: I I !In [ 9k, U] is an axial scalar with tke transformation law !In[~, U'J = sgn (~'/~) U!m[~, UJ, then it is expressible in tke form !m[~;U] =Jm-d~, wkere m(~(u)) is an axial k-vector. The set function !In [ ~, UJ can be written in the dual form !In [ ~. U] = (- 1)"(n-k> J (dual m). d,9!, (268.13) (268.14) (268.15} where d§;. =dual d9k. In an odd-dimensional space, the factor ( -1 )"(,.-k) = + 1 for every value of k; however, in even-dimensional spaces, this factor alternates in sign as k runs over the integers. 1 WEYL [1946, 10, p. 45]0 Sect. 269. Conservative k-vector fields. 665 Let ~ be a circuit given by x' = x'(tt1, u2, ••• , uk), and Iet it form the complete b d f ro • b r '( 1 2 k+I) 'fth t r ox' ox' ox' oun ary o Jk+.l• g1ven y x = x v , v , ... , v ; 1 e vec ors w , 7fl a--z · · · Bk o x' o x' 8 x' . . . u . u u and the vectors ovf, ~ • · · ~+! have the same onentanon, where w 1s a vector directed out of ~+I• we have (Sect. App. 21) pm · d~ = Jrotm -d~+I• (268.16) where m is a continuously differentiable absolute or axial covariant k-vector field in ~+I· The dual form of this fundamental integral identity is 1 (-)"+I p (dual m) · d~ = J div dual m · d~+I (268.17) Substituting from the identity (268.4) we can write (268.17) in the alternative form (-)"+Ip (dualm) · d~ = J curlm · d~+I· (268.18) 269. Conservative k-vector fields. In mechanics, if we are given a force field I such that ,~.. f dx' = 0 ':1' ', (269.1) for every circuit, the field I is said to be conservative. Vector fields satisfying (269.1) play an important role in many physical theories. In other contexts, vector fields satisfying (269.1) are called lamellar (see Sect. App. 33). Hereinthis chapter we prefer to use the name "conservative" owing to the connection we shall establish between the conservation laws of charge and magnetic flux and k-vector fields satisfying equations bearing a formal resemblance to (269.1). If a vector field satisfies (269.1), there exists an infinity of scalar fields h such that zs h(~ )- h(~ ) = J I· d~. (269.2) illJ where 1 and 2 are the end points of the curve ~(u). At a point where I is continuous, we have t, = a,h. (269.3) A scalar field h with these properties is called a potential of the conservative fieldf. The potential h is not uniquely determined by the field I and these requirements. If h is any one field satisfying (269.2), then so also is the field h' given by h' = h + const. (269.4) We shall now extend the above definitions to k-vector fields in n dimensions. Let F denote an absolute or axial covariant k-vector field and Y an absolute or axial covariant (k + 1 )-vector field such that ~F · d~ = f Y · d~+l (269.5) for every k-dimensional circuit. We then call Y the source of F. A source-free k-vector field F, i.e., one for which Y =0, is called a conservative k-vector field. The integral theorem (268.16) states that if F is continuously differentiable, its 1 The factor (-)"+1 appears here and not in the corresponding formula of ScHOUTEN [1954, 21, Chap. II, Eq. (8.14)] because our definitions (267.6) (267.7) of the dual tensors differ slightly from his [ibid., Chap. I, Eq. (7.15)]. If we had followed ScHOUTEN's definitions, we should have had to insert a factor (-)10+1 before the right-hand side of the identity (268.4). 666 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 270. source Y is given by Y = rotF. (269.6) Now let F denote a conservative k-vector field so that ~F-dY',.=O. (269.7) If F has continuous derivatives of all orders up top >O, there exists at least one (k -1)-vector field K having continuous derivatives of all orders up to p + 1 such that (269.8) for every Y;., where 9";,_1 is the complete boundary of 9";. .1 A field K satisfying (269.8) will be called a potential of the conservative field F. The field K is not uniquely deterrnined by the field Fand (269.8). Given any potential field K of F, we obtain any other possible potential by adding to K an appropriate conservative (k -1)-vector field. For example, if G is any twice differentiable (k- 2)-vector field, then K' given by K'=K+rotG (269.9) is an alternative potential of the field F. The gmup of transformations relating the potentials of a given conservative field will be called the group of potential transformations. II. The conservation of charge and magnetic flux. 270. The electromagnetic and charge-current fields and the Maxwell-Bateman laws. Space-time is a 4-dimensional space. A point in this space will be called an event, whose four real co-ordinates we denote by x 0 , .Q = 1, 2, 3, 42• Quantities transforrning as a tensor under the group of general analytic transformations of the ~will be called world tensors (cf. Sect. 152). W e assign an additive set function (!; [ 9;;, QJ called the charge to every 9;; in space-time. We assume that the charge is expressible in the form (270.1) where lJ is the charge-current field and where Q is the unit of charge. I t follows from (268.15) that we can also write (!;[~, QJ in the dual form (l;[~,QJ =-J (duallJ) -d~. (270.2) The charge (!;[~, Q] is assumed to transform as an axial scalar having absolute dimension [0]. Therefore, it follows by the corollary (268.14) that dual lJ is an axial 3-vector having absolute dimension [0]. Thus lJ is a vector density having absolute dimension [0).3 Under the group of analytic transformations of the co-ordinates ~ and the group of transformations Q' = o-lQ (270-3) 1 Throughout this chapter we assume that the underlying space is topologically equivalent to the Euclidean space of the same dimension. For such spaces, the theorem embodied in (269.8) follows from WHITNEY's Iemma [1957, 18, § 25]. 2 Henceforth in this chapter Greek indiceswill range over the four values 1, 2, 3, and 4. 3 Our reasons for assuming that the charge transforms as an axial scalar instead of an absolute scalar will be explained in Sect. 283. In Sect. 285 we introduce a more general expression for the charge. Sect. 270. The electromagnetic and charge-current fields. of the unit of charge Q, (J has the transformation law oxO aD' = 01(~'/~)1-1 oxD an. ·667 (270.4) We now postulate the law oj conservation of charge: the world scalar invariant integral (r[ .9;, QJ vanishes for every 3-dimensional circuit, (270.5) In a similar fashion we assign to every 9; in space-time a world invariant additive set function ~ [ 9;, Cl>] called the magnetic flux and assume tha t magnetic flux: is expressible in the form (270.6) where p is the electromagnetic field and where Cl> is the unit of magnetic flux. The magnetic flux and the electromagnetic field have the absolute dimension [ ]. Under the group of analytic coordinate transformations of the ~ and the transformations Cl>'= -1 Cl> (270.7) of the unit of magnetic flux, the electromagnetic field has the transformation law (270.8) We now postulate the law oj conservation of magnetic fluz: the world scalar invariant ~ [ 9;, Cl>] vanishes for every 2-dimensional circuit: ~p·d9;=0. (270.9) Eqs. (270.5) and (270.9) are the Cornerstones of electromagnetic theory. The laws of nature embodied in these postulates are perhaps the most Iasting achievements of the classical theory of electromagnetism. For the moment we regard the charge-current and electromagnetic fields as independent. In Subchapter III we shall see the manner in which they are related to one another by an additional hypothesis. Also, we shall subsequently give a more general mathematical expression for the law of conservation of charge. The intended generalization, however, involves no new physical idea, and we prefer now to consider the simpler Eqs. (270.5) and (270.9), independently of any further physical assumptions or increased mathematical generality. When we are on more familiar ground, these generalizations may be added with less difficulty and abstractness. The world invariant integral equations (270.5) and (270.9) were deduced by BATEMANl, who took as a starting point the differential equations commonly referred to as MAXWELL's equations. Consistently with the program of Sect. 7, we prefer, following KoTTLER2, to announce our basic premises in the stronger form of the integral equations (270.5) and (270.9). The acceptance of these M~well-Bateman laws may be motivated on the grounds of simplicity and the general intuitive notion of conservation. As we shall see, the customary field equations and boundary conditions of electromagnetic theory follow as consequences of these integral equations if we supply appropriate assumptions regarding the continuity of the fields and the nature of the Co-ordinates ~- 1 [1910, 1]. BATEMAN cites the earlier work of HARGREAVES [1908, 5] on invariant integral forms. 2 [1922, 4]. 668 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 271. We trust that our rather abstract postulation of the laws of conservation of charge and magnetic flux will not discourage the reader seeking more concrete and familiar results. Our main reason for this approach is to emphasize the independence of the conservation laws from any geometry of space-time. The ideas of conservation as formulated here have, in a certain sense, topological significance, transcending both the intuition and the mathematics of length, time, and angles. This "metrical independence" of the conservation laws has been noted by VAN DANTZIG 1• As he remarks, the concept of metric and the measurement of lengths, angles, and time intervals is perhaps one of the most sophisticated and complex aspects of any physical-theory. Furthermore, is not the intuitive notion of conservation of charge, for example, quite independent of measurements of length and time? We view the conservation of charge and magnetic flux as independent of ideas like inertial frames, rigid rods, absolute or uniform time, Lorentz transformations, Galilean transformations, etc., and hence as deserving an independent mathematical expression. Since there is a one-to-one correspondence between k-vectors and their duals, the same physical quantity or physical law involving only k-vectors may be expressed in equivalent dual forms. It is a matter of taste and convenience which representation is used. Usually the representation involving the fewer number of indices is to be preferred. Thus the charge-current field is usually represented by a contravariant vector density rather than by the dual covariant axial 3-vector. As far as the electromagnetic field is concerned, both cp and dual cp are of rank 2, and it is a matter of convention that we tend to favor the covariant representation. However, for some purposes, a degree of formal simplification is obtained by using one or the other representation, and in what follows we do not restriet ourselves to any one particular choice. 271. Electromagnetic and charge-current potentials. If the electromagnetic and charge-current fields are continuous, the Maxwell-Bateman laws (270.5) and (270.9) are sufficient conditions for the existence of continuously differentiable fields a; and 11 such that J (J. d,§; =- ~"' 0 d~, fcp. d~= ~a; 0 d!l;_. The integral theorems (268.16) and (268.17) lead to the local relatjons (J = div 11, In terms of components, we have cp =rot a;. (271.1) (271.2) (271.3) (}Q=OA'YJDA, (/JuA=20[.o1XLI]· (271.4) A contravariant 2-vector density 11 satisfying (271.1) will be called a chargecurrent potential, and a covariant absolute vector a; satisfying (271.2) will be called an electromagnetic potential. The existence of charge-current potentials '1/ is a consequence of the law of conservation of charge. In Subchapter 111 we shall subsequently introduce the aether relations fixing a particular charge-current potential in terms of the electromagnetic field. Our procedure here is not unlike a procedure sometimes adopted in mechanics, where the stress t's may be introduced as a solution of the equations fef,dv=~t ,da , J (! Z[r /s] dv = p Z[r tPs] dap. where the force field (! f is regarded as prescribed (cf. Sect. 203). These equations do not uniquely determine the stress field t since addition of a null stress, i.e., any symmetric tensor I [1934, lJj. Sect. 272. Field equations and boundary conditions. 669 satisfying o, t's = 0, to a given Solution of these equations yields another solution (cf. the remarks at the end of Sect. 205). Nevertheless, theories of continuum mechanics generally e_mploy constitutive equations for the stress which arenot invariant under the process of adding a divergence-free symmetric tensor. For the moment we emphasize that the existence of an infinity of distinct charge-current potentials follows from the law of conservation of charge. Similarly, conserva:tion of magnetic flux is a sufficient condition for the existence of aninfinity of distinct electromagnetic potentials provided we assume rather weak continuity properties for the electromagnetic field. The group of transformations of the electromagnetic potentials has been called the group of gauge transformations 1 . It is customary to render the electromagnetic potential unique by imposing upon it additional restrictions in the form of boundary conditions, continuity requirements, and algebraic or differential relations amongst its components. Same remarks on these conditions are given in Sect. 276. 272. Field equations and boundary conditions. If the electromagnetic and charge-current fields are continuously differentiable in a region, by applying the classical argument based on (268.16) to (268.17) (cf. Sect. 157), from the MaxwellBateman laws (270.5) and (270.9) we derive the jield equations div 6 = 0, curl fJ! = 0 =dual rot fJ!. In terms of components, we have Let ou an= 0, c;n.J'l'€1 O,j CfJ'l'B = 0. (272.1) (272.2) (272-3) (272.4) (272.5) be the equation of a 3-dimensional surface in space-time dividing a region f!A into two regions !JI+ and &~-. W e suppose the electromagnetic and chargecurrent fields continuous in the closure of !JI+ and &~- but possibly discontinuous at L'=O. We may think of the surface L'=O as the set of events representing the history of a 2-dimensional surface in space across which the electromagnetic and charge-current fields have finite jumps. Applying the basic integral laws (270. 5) and (270.9) to circuits divided by the surface .E = 0, we conclude that the discontinuities [dual fJ!] and [ 6] are such as to satisfy the restrictions [dual 'f!] · grad .E = 0, } (272.6) [ 6] · grad .E = 0. In terms of components, these equations read (dual 'f!)nc. od .E = 0, } an On .E = 0. (272.7) These are the electromagnetic and charge-current boundary conditions. They imply that the most general discontinuities in the electromagnetic and charge-current fields allowed by the conservation laws are expressible in the form [dual ffJ]= ßxgrad.E, [6] = w X grad .E, or, in terms of components, [(dualffJ)n.1]=en.J'l'eß'P8e.E, } [an] =-lc:D.J'l'ew.J'P8e.E, where ß and w are arbitrary fields defined on .E = 0. 1 BERGMANN [1942, J, p. 115]. (272.8) (272.9) (272.10) 670 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 273. 273. The geometry of space-time, reduction of k-vectors, and the units of length and time. So as to obtain equations and results familiar from conventional treatments of electromagnetic theory, we now introduce a space-time geometry. In Sect. 152 we studied two such geometries based on the idea of the underlying Euclidean and Galilean groups of transformations. We showed that the dass of Eudidean frames could be defined as the dass of Co-ordinate systems in spacetime for which the space metric g (x) and the covariant space normalt (x) assume the canonical forms = [b'• o] g 0 0' t = (0, 0, 0, 1). (273.1) The subdass of these frames for which the Galilean connection r(x) = 0 were identified as the inertial frames of dassical mechanics. The question of inertial frames in electromagnetic theory will be taken up in Subchapter III, where we shall discuss the Lorentz invariance of the aether relations, Lorentz frames, and the relation between these entities and their Galilean counterparts. For the remainder of this Subchapter it suffices that we consider the entire dass of reetangular Cartesian co-ordinate systems in space-time characterized by the canonical forms (273.1). In addition to the co-ordinate transformations considered in Sect. 152, we shall here consider transformations of the units oflength and time L' = L -1 L, T' = r-1 r. (273.2) The space metric g will be assigned the absolute dimension [L -2], and the covariant space normalt will be assigned the absolute dimension [T]. Thus, under transformations of the co-ordinates ;r and the units of length and time, the world tensors g and t have the transformation laws g Q'A'( ;r' ' L') = L-2 ox.Q' OXQ ~xA' oxA g QA( X, L) ' l oxQ (273.3) ttJ' (x', T) = T - 0 xQ' t!J (x, T). Recall that ta=o!Jt, where t(x) is the time. The absolute dimensionoft is [T]. If g and t are to retain their canonical form (273.1) when the units of length and time are changed, the co-ordinates belanging to the two systems of units must be related by a transformation of the form z'' (L') = L[A'',(t) z'(L) + d'' (t)], } z4' (T') = t' = T z4(T) + const = T t + const, (273.4) where Ais an orthogonal matrix. Weshall denote the co-ordinates ;r by z and t when (273.1) holds. That is, z denotes reetangular Cartesian spatial co-ordinates and t = x4 denotes the time. Recall that in Sect. 152 we called a co-ordinate system for which we have the canonical forms (273.1), a Euclidean frame. Let F denote a covariant world k-vector. Let us denote the components of F referred to a Euclidean frame by (273.5) All the components of F are determined from these particular components. With reference to the decomposition (273.5) weshall use the notation F = (a, b). (273.6) Sect. 273. The geometry of space-time, reduction of k-vectors. For a contravariant world k-vector Y we write Y = (c, d), where Note the different position of the index 4 in (273.5) and (273.8). 671 (273.7) (273.8) If Fis an absolute covariant k-vector of absolute dimension [UJ, the transformation laws of its components a and b can be put in the form r;r; ... r_t= r; r;··· r_t s,s, ... sk ( 273 . 9 a U L ) -kAs• As• Ask a l b ,, , -uL-k+lT-lAs~As, Ask-l(b +a ut) '•'•···'k-l- 'i r;. •. 'k-1 s,s, ... sk-1 s, ... sk-lt ' where u1==oijoz4 '. Thus the quantities a transform as a 3-dimensional tensor; but unless a = 0, the quantities b do not. For contravariant absolute k-vectors, the components c and d in (273.8) have the transformation laws c';r; ... rk =ULk A~: A:: ... A~: (cs, ... sk _ kurs,ds,s, ... skl), l d';r; ... r;,_1 = uLk-1 TA'; ... A'; A'.t-1 ds,s, ... sk-1. S1 s1 Sk-1 (273.10) Similar considerations apply in the case of covariant and contravariant axial k-vectors, k-vector densities, and axial k-vector densities. For example, if Y is a contravariant k-vector density we have c';,; ... rk = U u-a r-1 A'; sl A'; Sz . .. A'k sk (cs, ... sk- k urs,d s, ... skl), l d';r; ... •A:-1 = ULk-4Ar; A'; ... A'L1 ds,s, ... sk-1. 51 Sa S.t-1 (273.11) For axial k-vectors and axial k-vector densities a factor, det A =± 1, will occur in the transformation laws of the Euclidean components. The reetangular Cartesian co-ordinates z (L) of a Euclidean frame based on the unit of length L are assigned the physical dimension [L J (cf. Sect. App. 9) and, consistent with (273.4), z4(T) is assigned the physical dimension [T]. The physical dimensions of the components a, b, c, etc., of world tensors referred to a Euclidean frame are determined by the absolute dimension of the corresponding world tensor, its variance, and its weight by writing the transformation laws for its components in the forms (273.9), (273.10), (273.11), etc. Thus if Fis an axial or absolute covariant k-vector of absolute dimension [UJ, then phys. dim. a = [UL-k], } (273.12) phys. dim. b = [U L -k+I PJ. If Y is an absolute or axial contravariant k-vector, then phys. dim. c = [ULk], } phys. dim. d =[ULk-t T]. If Fis a covariant density or axial density, then phys. dim. a = [U L -k+a T], } phys. dim. b = [U L -kHJ. Finally, if Y is a contravariant density or axial density, then phys. dim. c = [U Lk-a r-1], } phys. dim. d = [U U-4] • (273.13) (273.14) (273.15) 672 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. Sect. 274 Consider a world covariant k-vector or axial k-vector F and a world covariant (k -1)-vector or axial (k -1)-vector G such that F =rotG. Let F = (a, b), G = (g,h). We shall then have a =rotg, b =roth+ (-1)k-1 ~~, or, equivalently, duala = curl g, dual b = curlh + (-1)k-1 o(d~~Ig). (273.16) (273.17) (273.18) (273.19) 274. The charge density, current density, electric field, and magnetic flux density. If we assume the geometry of space-time to be Euclidean in the sense of Sect. 152, then we can apply the considerations of Sect. 273 to a reduction of the charge-current and electromagnetic fields into four distinct fields. Refer the electromagnetic field p and the charge-current field to a Euclidean frame, and set p = (dualB,E), lJ = (J, Q), (274.1) where B is called the density of magnetic flux, E the electric field, J the current density, and Q the charge density. From the rules for determining the physical dimensions of these quantities given in Sect. 273 we get immediately, phys. dim. B = [ L-2], l phys. di~. E = [ L -1 PJ, phys. d1m. J = [0 L -2 r-1], phys. dim. Q = [0 L-3]. (274.2) Let vn be the world velocity field of a motion as defined in Sect. 152. In terms of v, the electromagnetic field, and the charge-current field, we can define the space tensors (274-3) Recall that a space tensor was defined in Sect. 152 as a contravariant world tensor for which pnA · · · tn = pAn · · · tn = · · · = 0. A k-vector which is a space tensor has the representation F = (c, 0), (274.4) so that, according to (273.10) and (273.11), c transforms as a 3-dimensional Cartesian tensor. If we refer the components of lf and il to a Euclidean frame, we get lf =~+~X B, 0) _ (lf, 0), } il-(J Q V, 0) - (il, 0), (274.5) where i = E +v X Bis called the electromotive intensity at a point moving with the particles of the motion, and il = J- Qv is called the conduction current relative to the particles of the motion. The electromotive intensity and the conduction current of a motion transform as vectors under time-dependent transformations of the spatial coordinates. However, one should note that the electric field and current density do not transform as vectors under transformations between co-ordinate frames in relative motion. Thus, if the electric field and current density vanish in one Euclidean frame, they need not necessarily vanish Sect. 275. The 3-dimensional integral form of the laws of conservation. 673 in every Euclidean frame. A distribution of charge in one frame will constitute a current density in a frame in relative motion. Similarly, a density of magnetic flux in one frame will be interpreted as an electric field in a frame in relative motion. However, if the density of magnetic flux and the density of charge vanish in one Euclidean frame, they vanish in every Euclidean frame. 275. The 3-dimensional integral form of the laws of conservation of charge and magnetic flux. Consider first the 3-dimensional circuits in spacetime formed by 3-dimensional tubes (Sect. App. 29) of the world velocity field v defined by a given motion closed on either end by surfaces lying in the instantaneous spaces t (x) = t1 and t(x) =t2 • (See Fig. 44.) If we apply the law of conservation of charge (270.5) to such a circuit and refer all quantities to a Euclidean frame, we get t, t, J Qdv] + J dt~!J·da = 0, (275.1) " t, t, !:I' Fig. 44. Tube of integration in Eq. (275.1). where v is a spatial region moving with the particles of the motion and 9" is its complete boundary. Set LI t = t2- t1 and consider the limit (275 .2) If the limits of the two terms exist separately, we get :tJQdv+~!J·da=O, (275.3) which has the traditional form of an equation of balance when sources are excluded (cf. Sect. 157) and puts the law of conservation of charge in terms easy to understand. The conduction current relative to the particles 1 of the motion is the efflux of chargeout of the moving region through its boundary. Consider next the 2-dimensional circuits in space-time formed by segments of 2-dimensional tubes of the world velocity field closed on either end by surfaces lying in the instantaneous spaces t (x) = t1 and t (x) = t2 • If we apply the law of conservation of magnetic flux (270.9) to such a circuit and refer all quantities to a Euclidean frame, we get f 2 ta fB·d(l] + fdt~fi·dX =0, (275.4) !:I' t, t, '{/ where 9" is a 2-dimensional surface in space moving with the particles of the motion and ~ is its complete boundary. Applying a limit argument as in (275.2), we then get (275.5) 1 These particles are defined mathematically by integration of the given velocity field and are a convenient device for visualizing it; they need not be mass-bearing. Handbuch der Physik, Bd. lll/1. 43 674 C. TRUESDELL and R. TOUPIN:The Classical Field Theories. Sect. 276 which is the traditional form of Faraday's law of induction for moving circuits. If we apply (270.9) to a 2-dimensional circuit which lies in the surface t (~) = const (Fig. 45) we get the condition pB-da =0. (275.6) A visualization of FARADAY's law of induction is obtained by introducing the notion of lines of magnetic flux or lines of induction. The number of lines of magnetic flux threading a closed curve in space is measured by the integral J B ·da, where [/ is any surface having the curve for its boundary. Eq. (275.6) states that this measure is independent of the choice of [/ and depends only on the curve. FARADAY'S law of induction states that the time rate of change of the total number of lines of magnetic flux threading a moving circuit is measurde by the negative line integral of the electromotive intensity around the moving circuit. Since we have (275.)), (275.5), and (275.6) for any motion, Fig.45. surtaceofinte· the corresponding laws for circuits which are at rest in some gration in Eq. (275· 6l· Euclidean frame follow as a special case by setting v' = 0. 276. The 3-dimensional integral form of the potential equations. Let the components of the electromagnetic potential and the charge-current potential in a Euclidean frame be denoted by Cl= (A,- V), 1'1 =(dual H, D), (276.1) where A is called the magnetic pote11.tial, V the electric potential, H the current potential, and D the charge potentiall. If we apply the world invariant equations (271.1) and (271.2) to circuits in space-time constructed in a manner similar to those used to obtain (275.)), (275.5), and (275.6), we obtain the equations J~·da=~(H+Dxv)·da- :tJD·da, (276.2) JQdv=pD·da, (276.)} JB·da=pA·da, (276.4) ~ ~ Ji·da=- :tJ A·da+(A·v-V)], (276.5) •• where z1 and z2 denote the end points of the moving curve c. Eq. (276.2) will be called the current equation, and (276.)) will be called the charge equation 2• The potential equations for stationary circuits follow simply by setting v' = 0. For the classical theory, (276.4) and (276.5) are less used than are the current and charge equations, although the introduction of magnetic and electric potentials and their relative importance 1 The axial vector H is generally called the "magnetic field intensity" and the vector density D the "electric displacement". Since the charge-current field and the electromagnetic field have not been related to one another in any way, this customary terminology would be inappropriate for our purposes here. The conventiooal names for the magnetic potential A and the electric potential V of the electromagnetic field are appropriate and suggestive, and we have assigned names to H and D an the basis of their similar relation to the charge-current field. The terms "magnetic field intensity" and "electric displacement" will take an their proper connotation after we introduce the aether relations in Subchapter III. 2 Eq. (276.2) is HERTz's form of the current equation for moving circuits. HERTZ [1900, 4, Chap. XIV], WHITTAKER [1951, 39, pp. 329-331]. Sect. 277. Time derivatives of scalar integrals over moving curves, surfaces, and regions. 675 is a matter of debate. The charge and current potentials Hand D play a more fundamental role in the classical theory than do the magnetic and electric potentials A and V. MIE and DIRAC1 have proposed theories of electromagnetism in which the four potentials H, D, A, and V enter the theoretical structure on a more equal footing. However, in most treatments of the classical theory, the magnetic and electric potentials are introduced as auxiliary fields, determined only to within a gauge transformation. Additional restrictions are often imposed in the form of boundary conditions, continuity requirements or algebraic and differential relations between the four fields A1 , A2 , A3 , and V. For example. many authors introduce the "gauge condition" V= 0, while others impose the "Lorentz gauge condition" div A + c-2 ~ = o. MAXWELL 2 .imposed the condition div A = 0 and called A the electromagnetic ot momentum. Perhaps a suitably "gauged" electromagnetic potential and the ideas of MIE and DIRAC will serve as the basis of a better theory of electromagnetism. We do not take up these questions, resting content with our choice of charge and magnetic flux as the fundamental quantities entering the equations of conservation. 277. Time derivatives of scalar integrals over moving curves, surfaces, and regions. The electromagnetic equations for moving circuits involve the time derivatives of scalar integrals having the form ~ = k\ J F,,,, ... ,kdY{•'•···'k = J F · dY" = J (dualF) ·dfi{, (277.1) where Fis a 3-dimensional k-vector (k =0, 1, 2, or 3) and .9k is a spatial region, surface, or curve moving with the particles of a motion with velocity field v' in a Euclidean frame. Now the motion can be presented in the form z'=z'(ZK, t) where the zK are material Co-ordinates. In general, the k-vector field F depends explicitly on the time t as weil as the spatial co-ordinates z. The surface .9k is given by parametric equations z' = z' (u1, u2, ... , uk, t) = z' (ZK (u), t). (277.2) The limits of integration (277.1) correspond to fixed values of the Z and u independent of the time t. Thus we can write !!Sl_ = ~1 ()FK,K, ... Kk df/'.K,K, ... Kk at k! ot k (277.3) provided that the field F and the motion z (Z, t) are continuously differentiable. In (277.3), the FK,K, ... Kk are defined by oz'• oz'• oz'k P,KK K =----•••---F. ' •··· k- ezK, ezK, ezKk '•'•···'k' (277.4) and d.9J.K,K, ... Kk is the corresponding transform of dSJ.•'•···'k. lf follows from (277.3) and the results in Sect. 150 that d~jdt can be written in the form d;J - ~j~F. df/'.'•'•···'k dt - k! dt '•'•···'k k ' (277.5) where dßfdt denotes the convected time-flux of the k-vector field F. We can also write d~fdt in the dual form d;J }. d ~ dt = d~ (dualF) · dY". (277.6) Now the convected time-flux of an absolute or axial k-vector field has the particularly simple form d 8F 8F -ftF = Tt + v · rotF + rot(v · F) = Tt + curlFxv + rot(v · F). (277.7) 1 [1912, 6]; [1951, 5]. 2 [1881, 4, § 618]. 43* C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 278. Similarly, the convected time-flux of a contravariant k-vector density or axial density has the simple form :; (dual F) =dual (div dual F x v) + curl (dual F X v). (277.8) One can easily verify that dual a~; = :~ dual F. (277.9) Let us set F- dual F. Eqs. (277.7) and (277.8) written in terms of components are The foregoing integral formulae and vector indentities are useful for the study and interpretation of the electromagnetic equations for moving circuits. In what follows we shall also need to consider a moving surface given in the form of an equation E(z,t) =0, (277.12) where z denotes the reetangular Cartesian spatial co-ordinates of a Euclidean frame, and t denotes the time. The vector n defined by 8,E ( ) n, = 277.13 V85E85 E is called the instantaneotts unit normal to the moving surface E = 0, and the quantity s defined by (277.14) is called the speed ( cf. ( 177.5)) of the surface E = 0 relative to the reference frame (z, t). The quantities n and un may be regarded as the Euclidean components of the world vector 8aE va = --::Vrcg=;cLl~e~aLl~Ec===a=ec=:E (277.15) That is, in the notation of Sect. 273 v = (n, -un). (277.16) 278. Field equations and boundary conditions in 3-dimensional form. The 3-dimensional form of the field equations and boundary conditions may be obtained in two ways. One method is to work with the 3-dimensional integral equations (275-3), (275.5), (275.6), (276.2) to (276.5). By assuming the fields in these equations continuously differentiable, one can use the results of Sect. 277 to express the time derivatives of integrals over moving surfaces, curves and regions in terms of the convected time-flux. Then, by appropriate application of the fundamental integral theorem, line integrals can be transformed into surface integrals, surface integrals into volume integrals, etc. Various terms cancel and, by the usual type of limit arguments, certain local conditions in the form of partial Sect. 279. The 3-dimensional form of the aether relations. 677 differential equations follow from the integral equations. Boundary conditions in 3-dimensional form follow by applying these same integral equations to appropriate circuits divided by a surface of possible discontinuity. A simpler and more direct method applies the reduction formulae derived in Sect. 273 to the world tensor field equations and boundary conditions already obtained. By this latter method we obtain as an immediate consequence of the field equations (272.1) and (272.2): d. J oQ lV +Bt =0, aB curlE + aT = o, divB =0. The field equations (271.3} yield the relations Q =divD, an J=curlH-at, B =curlA, aA E = ---grad V. ae (278.1} (278.2} (278.3) (278.4} (278.5) (278.6) (278.7) Fig. 46. Geometry of an electromagnetic discontinuity. Eq. (278.1} is the differential or local form of the law of conservation of charge, and (278.2} is the differential form of FARADAY's law of magnetic induction. The 3-dimensional forms of the boundary conditions follow easily from the world tensor equations (272.6) and the definitions (277.13) and (277.14) of the unit normal n and the speed un of a moving surface of discontinuity1. nx [E]- un[B]= 0, [B]·n =0, [J]·n- un[Q] = 0. (278.8} (278.9} (278.10) Eq. (278.8) implies that the most general discontinuities in the magnetic flux density and the electric field allowed by the law of conservation of magnetic flux are expressible in the form [E] =ln-unk, [B] =kxn, (278.11} (278.12} where the fields I and k are arbitrary fields defined on the surface of discontinuity. We see that conservation of magnetic flux restricts considerably the geometry of any electromagnetic discontinuity. Eq. (278.9) demands that any discontinuity in the magnetic flux density be transversal. For a stationary surface of discontinuity, we have un =0, and from Eq. (278.8} it follows that the discontinuity in the electric field is longitudinal. If the surface of discontinuity is moving, the discontinuity in the magnetic flux density must be normal to the plane determined by the discontinuity in the electric field and the normal n (see Fig. 46). 111. The Maxwell-Lorentz aether relations. 279. The 3-dimensional form of the aether relations. In all that precedes, we have treated the electromagnetic and charge-current fields as independent. We now introduce a fundamental relation between these fields by postulating the Mazwell-Lorentz aether relations. 1 LuNEBERG [1944, 9, p. 21] has obtained these boundary conditions holding at a moving surface of discontinuity in the e!ectromagnetic field by other means. 678 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 279. So as to introduce the aether relations in their simplest and most farniliar form, we shall· begin with a statement of thein in terms of the vector fields E, B, D, and H. First of all, we assume the existence of at least one Euclidean frame in which the Maxwell-Lorentz aether relations (279.1) are valid. The quantities e0 and f1o are fundamental constants depending only on the units of length, time, charge, and magnetic flux. That is, we assume the existence of at least one Eudidean frame in which a charge potential D is proportional to the electric field E and in which a current potential H is proportional to the magnetic flux density B. Now it is easy to show that if the Maxwell-Lorentz aether relations (279.1) hold in one Euclidean frame, they cannot hold generally in every Eudidean frame. To see this, simply consider the transformation laws of the four fields E, B, H, and Dashave been determined in Sect. 273. These can be written in the form D'=D E'=E,+uxB, l B'=B, H'=H-·uXD, (279.2) from which it is immediately plain that the aether relations cannot hold generally and simultaneously in Euclidean frames in relative motion. Thus the M axwell-Lorentz aether relations are not Euclidean nor even Galilean invariants. If we adopt the aether relations we see that the equations of electromagnetic theory will not have the same form in every Euclidean frame, or more importantly, they will not havc the same form in every Galilean frame, as do the equations of dassical mechanics. Thus it appeared, from the dassical point of view, that if the aether relations could be verified in some selected and preferred inertial frame, then by a simultaneaus application of the laws of mechanics and electromagnetic theory one should be able to detect the relative motion of inertial frames. To put it otherwise, the aether relations determine a preferred dass of Euclidean frames all at rest relative to one another. It may also be assumed for our purposes here that this preferred dass of frames are inertial or Galilean. In Galilean frames which are in motion relative to this preferred dass, the aether relations do not generally hold. This gives rise in a natural way to the notion of an" aether". That is, we may think of the preferred inertial frames for which the aether relations are valid as the dass of inertial frames in which the "aether" is at rest. In all other inertial frames, there will be an "aether wind", controverting the validity of the simple relations (279.1). Matter resides in an aether characterized by the relations (279.1). These relations are not to be regarded as constitutive equations for matter. We may think of them as constitutive relations for the aether. Allow us to remind the reader already familiar with the dassical theory of dielectric and magnetic materials that the aether relations apply only to the charge and current potentials of the resultant charge and current distribution of all kinds of charge and current. Charge and current distributions arising from the polarization and magnetization of a material medium are to be induded in the charge-current field 6, and (H, D) is a resultant potential. Sect. 280. The world tensorform of the Maxwell-Lorentz aether relations. 679 We shall enlarge on this point in Sect. 283. Here it suffices that our intuitive motivation for adopting the aether relations {279.1) as valid both inside and outside of "matter" is based on the guiding principle used by LoRENTz [1915, 3]. In LoRENTz's theory, matter is regarded as a collection of charged point particles which exist and move through an aether or vacuum whose properties are unaffected by the presence or motion of the particles. Here we treat matter as a continuous medium but carry over the Lorentz hypothesis that the aether relations are unaffected by the presence of matter. The effects of polarization and magnetization in the Lorentz electron theory were determined by arguments based on a particle model of a dielectric and magnetic material medium. A treatment of these effects regarding matter as a continuous medium is given in Sect. 283. At the same time, we give warning that the aether relations represent an assumption not adopted in every existing theory of electromagnetism1, whereas the conservation laws of charge and magnetic flux are, to our knowledge, common to all. Thus the considerations of this subchapter are of a rather special nature, and we have attempted to present but one point of view. The principal objective of the chapter as a whole is to formulate and develop the conservation laws of electromagnetic theory. These laws hold independently of the aether relations 2, and the contents of this subchapter do not in any way restriet the considerations in Subchapter II. 280. The world tensor form of the Maxwell-Lorentz aether relations. Let the constant c2 be defined by (280.1) The fundamental nature of this constant will become apparent as we proceed; it is called the square of the speed oflight in vacuum. Consider a space-time coordinate system (z, t) for which the aether relations are valid. As we have seen, from the classical view of space-time geometry the frame (z, t) will be one of a restricted subdass of all Galilean and Euclidean space-time frames. Consider a world contravariant, absolute, symmetric tensor of rank two whose components in the frame (z, t) have the particular values (280.2) Now the components of a tensor in a general system of co-ordinates are determined uniquely by its transformation law and its components in any one co-ordinate system, so that (280.2) determines the components of y in every space-time Coordinate system. The determinant of the inverse y of y is given by det y = - c2 (280.3) in the frame (z, t) and transforms as a scalar density of weight 2 under general transformations of the co-ordinates. Now consider the world tensor equation r;!M =V8 o (- dety)fyD'l'yLieiP'l'e· flo (280.4) By the quotient rule of tensor algebra, we easily verify that (280.4) is indeed a tensor equation; i.e., the rank and weight of both sides of the equation agree. 1 ABRAHAM [1909, J] reviews several of the points of view regarding the aether relations in material media. 2 A similar distinction was made in Chap. E, where developments resting on an equation of state were separated from the general theory of energy. Cf. the remarks at the end of footnote 5. pp. 617-618. 680 C. TRUESDELL and R. TouPIN: The Ciassical Field Theories. Sect. 281. Since it is a tensor equation, it will be satisfied in every Co-ordinate system if it is satisfied in one Co-ordinate system. In the frame (z, t) where the r have the values (280.2), we can verify that the tensor equation (280.4) is satisfied if the Maxwell-Lorentz aether relations (279.1) are satisfied. Thus we call (280.4) the world tensorform of the Maxwell-Lorentz aether relations. For some purposes it proves convenient to write (280.4) in a somewhat different form. Consider the world contravariant tensor of weight l defined by -1 - 1DLI @!M =='= y . V-detf (280.5) -1 The determinant of ~ is an absolute world scalar having the value - 1 in every co-ordinate system. We find that the aether relations can be put in the alternative invariant form V --1 -1 'YJ!M == ~ @D'P @LIS !f''PS. flo -1 -1 In the frames where r has the specialform (280.2), ~will have the form ~ = Vc [ d~ • - o ;2]· (280.6) (280.7) 281. Dimensional transformations and the aether relations. The absolute dimension of the world tensor fields a, p, 'rj, and cx are given by abs. dim. a = [0], abs. dim. 'r/ = [0], abs. dim. p = [], } abs. dim. cx =[]. (281.1) From these dimensions and the rules of Sect. 273 follow the physical dimensions of the fields B, E, J, Q, D, H, A, and V. Some of these have been stated elsewhere already, but we shall list all of them now for easy reference: phys. dim. B =[ L -2], phys. dim. J = [OL-2T-1J, phys. dim. D = [OL-2], phys. dim. A =[ L-1], phys.dim.E= [L-1T-1], l phys. dim. Q = [OL -s], phys.dim.H= [OL-IJ-1], phys. dim. V = [ r-1] . (281.2) Thus, in order that the aether relations be invariant under independent transformations of the units of length, time, charge, and magnetic flux, we must have phys. dim. e0 = [ -1 0 L -1 T], phys. dim. #o = [ <1> o-1 L -1 T]. The constant c defined in (280.1) has, therefore, the dimension phys. dim. c = [L r-1]. (281.3) (281.4) The dimension of the constant l [i; occurring in the world invariant forms of v;.; the aether relations (280.4) and (280.6) is given by phys. dim. V eo = [0 -1]. flo (281. 5) The absolute dimension and physical dimension of absolute world scalars and constants are equal. From (281.5) and (280.6) we see that dimensional invariance Sect. 282. The Lorentz invariance of the Maxwell-Faraday aether relations. 681 of (280.6) requires that abs. dim. 6; = 1 , (281.6) which is consistent with the fact that the determinant of this tensor has the value - 1 in all co-ordinate systems and unit systems. Dimensional invariance of the canonical form (280.2) requires that abs. dim. y = [L - 2]. (281.7) Of course, this requires that the absolute dimension of the inverse y be [L 2]. When magnetic flux, charge, length, and time are measured in units of Q = 1 Coulomb, = 1 Weber, L = 1 Meter, T = 1 Second, (281.8) the fundamental constants in the aether relations have the values1 : e = 8_854 X 10_12 Coulomb-Second O Weber-Meter ' = 1.257 X 10_6 Weber-Second flo Coulomb-Meter ' C = 2.998 X 108 _M~ter_ Second ' (281.9) V e0 = 2_654 X 10_3 Cou!omb . P.o Weber 282. The Lorentz invariance of the Maxwell-Faraday aether relations, Lorentz transformations, and Lorentz frames. In Sect. 152 we showed that the Galilean (inertial) frames of classical mechanics could be characterized as the preferred co-ordinates in space-time for which the world tensors g!M, tA, and the Galilean connection r assumed the canonical forms g =[do" oo]' t = (0, 0, 0, 1), (282.1) The Co-ordinates of any two such frames must be related by a Galilean transformation having the form z'' = A'~ z' +ur' z"' + const, } z"'' = z4 + const, (282.2) where A is an orthogonal matrix and the u'' are constants representing the relative velocity of the two Galilean frames. If we also consider transformations of the units of length and time and assign g the absolute dimension [L - 2] and t the absolute dimension [T], the co-ordinates of a Galilean frame based on the units of length L and T are related to the co-ordinates of a Galilean frame based on the units L' and T' by a transformation of the form z''(L') =L(A•;z'(L) +u''z4 (T) +const),} (282.3) z"'' (T') = T (z4' (T) + const). So as to distinguish these more general transformations from those having the special form (282.2), we shall call (282.3) a generalized Galilean transformation. Following the same general procedure outlined above for the case of Galilean space-time, let us consider the co-ordinate transformations and space-time frames 1 These values are quoted from STRATTON [1941, 8, p. 601]. 682 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 282. defined by the canonical form (280.2) of the Lorentz tensor r. A Co-ordinate transformation which leaves the Lorentz tensor invariant must satisfy the equations " .Q' ., <1' -l.Q'<1'(L' T') = L_2 _u_x _ _ v_x_-l.Q<1 (L T) y ' oxD ox<1 y ' ' (282.4) where -1 y (L,T) = [!5" 0 - 0 ~2 l , [ -y, (L', T') = - !5' s 0 __E._ l • 0 c2 L2 (282.5) Since the ·:./ and y' are constants, it follows that the co-ordinate transformation must be linear. A coordinate transformation satisfying the (282.4) will be called a generalizecl Lorentz transformation. One can deduce the general form of these transformations by multiplying the well known special Lorentz transformation 1, Z~= (282.6) by an orthogonal transformation of the spatial Co-ordinates, simple extensions of the co-ordina tes corresponding to transformations of L and T, and by time inversions. The generalized Lorentz transformations have the form z'' (L') = LA'~{Ws + (C- 1) u-2 u' u.] z• (L) -Cu' z4 (T)}, ) z4 '(T') =±TC {z4 (T)- -}u,z'(L)}, ( 282·7) where Ais an orthogonal matrix, C = ( 1 - :: r~. and u2 == u, u' < c2 is the squarecl relative speecl of the two Lorentz frames. Various subgroups of the group of generalized Lorentz transformations have received special attention and special names. If we keep the units of length and time fixed, so that L = T = 1 in (282.7), we get what has been called the extended Lorentz group 2 • If we transform the units of length and time by the same factor so that L = T and c is invariant, we get what BATEMAN3 has called the group of spherical wave transformations. This subgroup is also called the conformal Lorentz group. BATEMAN noted that the aether relations are invariant under the group of spherical wave transformations. If we require that det A = + 1 and disallow time inversions corresponding to the minus sign in (282.7) 2 , we get the proper Lorentz transformations. On writing the aether relations in the form (280.6), we see that they are invariant under the conformal Lorentz group. A Lorentz transformation with ufc <{::: 1 approximates a Galilean transformation. In special relativity theory, the notion of a stationary aether is abandoned. The inertial frames of relativistic mechanics are identified with the Lorentz frames. The equationsand definitions of mechanics are revised so as to be Lorentz invariant rather than Galilean invariant as in classical mechanics. The world invariant form of the aether relations (280.6) provides us with our principal motivation for assuming that the charge ([ [( .9"3 , O)] transforms as an axial scalar rather than as an absolute scalar under general transformations of the space-time coordinates. With magnetic flux an absolute scalar and charge an axial scalar as we have assumed, the -1 transformation law of the tensor density (f) is (282.8) Had we assumed that both charge and magnetic flux were absolute scalars, the charge-current -1 potential 11 would have been an axial density. The transformation law (280.8) for the (f) 1 See, for example, BERGMANN [ 1942, J]. 2 CORSON [1953, 6, Chap. 1]. 3 BATEMAN [1910, J]. Sect. 283. Polarization and magnetization. 683 would then have involved the square root of the Jacobian rather than the square root of -1 the absolute value of the J acobian. Thus, in some coordinate systems, the coefficients 6) would have been imaginary or complex-valued. We have considered this undesirable. Another motivation for assuming charge to be an axial scalar is that the current density J and the charge density Q, being tensor densities under time-independent transformations of the spatial co-ordinates when this assumption is adopted, transform as absolute tensors under the group of time independent orthogonal transformations of the spatial co-ordinates. This is the transformation usually assumed for these quantities in traditional treatments of electromagnetic theory. The distinction between axial tensors and absolute tensors and the distinction between axial densities and densities is made necessary only because we find the consideration of improper co-ordinate transformations to be of some major concern in many physical theories. It is for this reason that we did not restriet ourselves at the outset to the group of proper co-ordinate transformations. 283. Polarization and magnetization. a.) The principle of Ampere and Lorentz. Polarization and magnetization are auxiliary fields introduced into the general theory so as to serve in formulating constitutive relations for special types of materials called dielectrics and magnets or magnetic materials. In the classical theory of dielectrics and magnets, surfaces across which the electromagnetic properties of the medium change abruptly are important for the description of many electromagnetic phenomena. These surfaces are most often associated with surfaces of discontinuity in the polarization and magnetization. Sudaces of discontinuity in the polarization and magnetization are in turn associated with surface distributions of charge and current. The point of view we adopt here is that charge and current are the fundamental entities while polarization and magnetization are simply auxiliary fields introduced as mathematical devices providing a convenient description of special distributions of charge and current in special types of materials. This may be called the principle of Ampere and Lorentz1. Thus, if we anticipate a treatment of surfaces of discontinuity in the polarization and magnetization, we must first introduce surface distributions of charge and current into the conservation law of charge. For mathematical simplicity, at the outset we did not burden the reader with this additional complication. The fundamental physical idea remains the same, conservation of charge, but now we give a slightly more general mathematical expression for the measure of charge. The mathematician will readily infer a general mathematical statement for the physical idea of conservation in terms of additive set functions. ß) Surface distributions of charge and current. Let E (a:) = 0 be the equation of a 3-dimensional surface in space-time representing the history of a 2-dimensional surface in space bearing a surface distribution of charge and current. We now replace (270.1) by the more general assumption that the charge is expressiblein the form a: [ Ya. QJ = J (J. dYa + J w . d~*' (283.1) .9',n 2' where (J is the volume density of charge-current and the world contravariant 2-vector density w is the surface density of charge-current. The field w is defined only at events corresponding to the surface E (~) = 0. In (283 .1), ~ n E denotes the intersection of the arbitrary 3-dimensional surface ~ and the special3-dimensional surface E(~) =0. Here we have assumed that, if the intersection is not empty, Ya n E is a 2-dimensional surface and d~* is its differential element. Generalizing (270.5). we now write the law of conservation of charge in the form ~ (J • d~ + J w . d~* = 0. (283.2) 1 WHITTAKER [1951. 39, p. 88, PP· 393-400] 684 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 283. We know that, for all surfaces .9':; not intersecting E=O, as a consequence of (283 .2) there exist potential fields 117'" such that f6·d.9':i=-~11'f'""d~. (283-3) W e can also construct fields 11.'7' such that ~ 11.'/'" d~ = 0 for every circuit not intersecting E(a:) =0, and suchthat ~ 11.'/'. d~ = - J w . d~* (283.4} (283.5) for every circuit intersecting E = 0. Therefore, we can write the charge-current potential equation in the form J (J. d.9':; + J w. d~* =- ~ 11" d~. (283.6) Y',n.z: where 11 =w-+11.'7'· Eq. (283.6) generalizes (271.1). If the fields are continuously differentiable except perhaps at E (:~:) = 0, at every event not on E (:~:) = 0 we have 6 = div 11· (283.7) Howcver, at E(a:) =0, the potential11 must have a discontinuity consistent with the condition or, in component form [11] ·gradE= -w·gradE, [1JDLI] oLlE=- wDLI oLlE. (283.8) (283.9) The aether relations (280.4) or (280.6) are postulated to hold between the electromagnetic field p and the generalized potential 11 of (283.6). Thus, the electromagnetic field is discontinuous at a surface bearing a distribution of charge and current. The 3-dimensional form of the law of conservation of charge and the charge and current equations taking into account a surface distribution of charge and current can be obtained from the world invariant equations (283.2) and (283.6) by introducing a Euclidean frame. The resulting equations have an interesting but complicated structure. Since we do not make use of these equations here, we leave to the reader the task of deriving them. Having established these preliminary results on surface distributions of charge and current, we are in a position to give a fairly general treatment of polarization and magnetization in moving and deforming material media, allowing that these fields may be discontinuous at special surfaces. y) Polarization charge and current and magnetization current. As an introduction to the more difficult dynamical case, we note first some simpler and better known results of the static theory of dielectrics and magnets. These latter theories are concerned with the determination of the electric and magnetic fields in regions of space containing dielectric and magnetic materials at rest. It is customary in the electrostatic theory of dielectrics to divide the charge distribution into two types: ( 1) the bound charge or polarization charge; (2) the free charge1• The polarization charge contained in a 3-dimensional spatial region v is given by [[v, QJ =-p P · dd (283.10) 1 The polarization charge is also called the "induced" charge, and the free charge is also called the "real" or "true" charge. See, for example, STRATTON [1941, 8, Sec.t. 3.13, p. 183]. Sect. 283. Polarization and magnetization. 685 where 6 is the complete boundary of v and P is the density of polarization. One generally considers problems where P is a continuously differentiable field except at special surfaces where it suffers a jump or discontinuity. Excepting points on these surfaces, we can use the integral theorem to transform the surface integral (283.10) to a volume integral. Thus, if v is a region in which the polarization field is everywhere continuously differentiable, we have C\:[v,QJ =- JdivPdv. (283.11) The scalar field - div P may be called the volume density of polarization charge. If the region v is divided into two regions v+ and v- by a surface 6* across which P suffers a jump [P], and if P is continuously differentiable in the closure of the regions v+ and v-, we can write (283.10) in the form . (290.7) Thus, for example, transformations of the unit of magnetic flux are generally regarded as fixed in terms of the transformations of M, L, T, and Q by the relation (290.8) This is consistent with our having called DxB a momentum density and Ex H a flux of energy without introducing any additional dimensional constants of proportionality. Thus we have phys. dim. D X B = [M L - 2 y-1], phys. dim. EX H = [M T- 3], (290.9) (290.10) in agreement with the customary dimensions of momentum per unit volume, and energy per unit area per unit time. 291. Conclusions. The principal result contained in this chapter is a world tensor invariant formulation of the laws of conservation of charge, magnetic flux, energy, and momentum, independent of special assumptions as to the geometry of space-time. In Subchapter IV, we motivated the law of conservation of energy and momentum by considering the classical energy and momentum equations of a continuous medium interacting with an electromagnetic field. We assumed that the stressenergy-momentum tensor T/} was a 2-event world tensor field. While the chargecurrent potential and the electromagnetic field are connected through the aether relations, no general relation between the stress-energy-momentum tensor and the other fields has been proposed here. The basic content of Subchapter IV is thus independent of the two preceding. Rather, the conservation law of energy and momentum as formulated here is to be regarded as encompassing and generalizing the purely mechanical considerations of Subchapters D II and E I; it also includes and extends the various purely electromagnetic proposals for conservation of energy and momentum. The conservation laws of charge, magnetic flux, momentum, energy and angular momentum constitute an underdetermined system of equations. They must be supplemented by relations between the conservative fields. The aether relations are an example of such relations. The many determinate special theories 700 C. TRUESDELL and R. TouPIN: The C!assical Field Theories. Sects. 292, 293· encompassed by the conservation laws is evidence of the different special ideas of material behavior consistent with them. The conservation laws provide a general framework common to all electrodynamic theories, classical and relativistic, with which we are familiar. Special and general relativity theory entail modification of our concepts of time, mass, energy, momentum, the aether relations, and inertial frames, but have yet to alter in principle· either the formal or the intuitive aspects of the laws of conservation of charge, magnetic flux, energy, and momentum. G. Constitutive Equations. I. Generalities. 292. The nature of constitutive equations. In Sect. 7 we explained that the field equations and jump conditions express the general principles of mechanics, thermodynamics, and electromagnetism, while constitutive equations define ideal materials, which are mathematical models of particular classes of materials encountered in nature. The preceding treatise has developed in detail the properties of motions and of the fundamental physical principles of balance, which imply both field equations and jump conditions. In this final chapter we illuminate the concept of constitutive equation by stating general principles and adducing common examples. 293. Principles to be used in forming constitutive equations. It should be needless to remark that while from the mathematical standpoint a constitutive equation is a postulate or a definition, the first guide is physical experience, perhaps fortified by experimental data. However, it is rarely if ever possible to determine all the basic equations of a theory by physical experience alone. Every theory abstracts and simplifies the natural phenomena it is intended to describe (cf. Sect. 4). Supposing that the theorist has assembled the facts of experience he wishes to use in defining an ideal material, we now list the mathematical principles he may call to his aid when he attempts to formulate definite constitutive equations. We know of no ideal material for which all these principles have been demonstrated to hold, although for the simpler classical theories it is generally believed that they do. rx) Consistency. Any constitutive equation must be consistent with the general principles of balance of mass, momentum, energy, charge, and magnetic flux. This is obvious and easy to say, but to test it in a special case may be difficult. ß) Co-ordinate invariance. Constitutive equations must be stated by a rule which holds equally in all inertial Co-ordinate systems, at any fixed time. Otherwise, a mere change of description would imply a differentresponsein the material. Such a rule may be achieved, usually trivially, by stating the equations either in tensorial form or by the aid of direct notations not employing co-ordinates at all1 . y) I sotropy or aeolotropy. Materials exhibiting no preferred directions of response are said to be isotropic. There are differences of opinion as to how this somewhat vague concept should be rendered mathematically precise. In the common phrase "isotropic material" we are unable to discem any meaning. Only after a kind of material has been defined by a particular constitutive equa1 However, the Iiterature of hydraulics abounds in "power laws" which are not invariant, as was pointed out by KLEITZ [1873, 4, § 22] in a work dating from 1856; the same may be said of rheology. Sect. 293. Principles to be used in forroing constitutive equations. 701 tion does it become possible to state an unequivocal concept of isotropy. Pursuant to this idea, NoLL1 defines the isotropy group of a material as the group of transformations of the material coordinates which leave the constitutive equations invariant. The nature of this group specifies the symmetries of the material. A material is then isotropic if its isotropy group is the full orthogonal group. Various kinds of aeolotropy, which is a symmetry with respect to certain preferred directions, are shown by materials whose isotropy groups are proper subgroups of the orthogonal group, or are other groups. While these definitions reflect broader ideas, up to this time they have been rendered concrete only in pure mechanics, since the general constitutive equations expressing energetic and electromagnetic response are not yet known. In theories not limited to purely mechanical phenomena it is usual to define isotropy in respect to the variables A, B, C, ... entering a constitutive equation by requiring the functional dependence of A, B, C, ... to be isotropic in the mathematical sense. A material which is isotropic in respect to one property, e.g. stress and strain, need not be so in respect to another, e.g., electric displacement and electric field. There is a large current Iiterature 2 concerning means of expressing isotropy and aeolotropy, but there is no adequate survey of the fieldas yet. (J) Just setting. Constitutive equations connecting a given set of variables should be such that, when combined with all the principles of balance affecting these same variables, there should result a unique solution corresponding to appropriate initial and boundary data, and a solution that depends continuously on that data. This principle can rarely be used. With great mathematicallabor, it has been proved to hold only in the simplest classes of boundary-value problems in the simplest classical theories 3• At best, it shifts the problern to another domain, that of finding adequate initial and boundary conditions 4• e) Dimensional invariance. It is essential that included in each constitutive equation should be a full list of all the dimensionally independent moduli or material constants upon which the response of the material may depend. This requirement has often been neglected in recent work, although it is a commonly accepted principle of physics, known at least since GALILEo's day, that any fully stated physical result is dimensionally invariant. (It is tacitly understood that dimensionless moduli need not be listed, since they cannot be specified until a particular functional form is stated.) The use of dimensional invariance to classify constitutive equations and in some cases to exclude inappropriate terms was initiated by TRUESDELL 5• The tool used is the classical n-theorem, a precise statement and simple rigorous proof of which was recently given by BRAND 6• 1 [1948, 8, §§ 19-20]. 2 The basic principles derive froro work of CAUCHY. Nuroerous articles on the subject have appeared in the Journal and Archive for Rational Mechanics and Analysis, 1952 to date. 3 The honored custoro of verifying that the nurober of equations equals the nurober of unknown functions seeros to bring corofort. ' In roany cases the traditional setting is known to fail to lead to a unique solution. E.g., in buckling probleros the specification of the loads on the boundary is not sufficient to insure a unique solution. Such probleros roay nevertheless be regarded as weil set if we aroplify either the boundary conditions or the constitutive relations. E.g., in the case of buckling we roay include as a part of the constitutive equations the requireroent that the elastic energy shall be a roinirouro with respect to coropatible deforroations; a unique solution results. Still better, the solution for static buckling roay be regarded as the liroit of a uniquely soluble dynaroic problero. 5 [1947, 17] [1948, 32 and 33] [1949, 33 and 34] [1950, 32, §§ 7-8] [1951, 27, §§ 22, 25] [1952. 22] [1952, 21, 1 § 47, 62-65. 67-69. 74-75] [1955. 28, § 1]. 6 [1957. 1]. 702 C. TRUESDELL and R. TouPIN : The Classical Field Theories. Sect. 293. C) Material indiflerence. The most important and the most frequently used correct idea for formulating constitutive equations is that the response of a material is independent of the observer. In proposing the first of all constitutive equations for deformable materials, namely, the law of linear elasticity, HoOKE1 gave the first dim hint toward this principle in the suggestion that by carrying a spring scale to the bottom of a deep mine or to the top of a mountain, the change of gravity could be measured. A more striking example is furnished by use of a spring, attached to the center of a horizontal, uniformly rotating table, to measure the centrifugal force acting upon a terminal mass. The assumption implicit in these measurements is that the force exerted by the spring in response to a given elongation is independent of the observer, being the same to an observer moving with the table as to one standing upon the floor. Another example is furnished by FoURIER's law of heat conduction, to be discussed in Sect. 296 below. According to this law, the flow of heat is proportional to the temperature gradient. Since both of these quantities are independent of the motion of the observer, in order that FOURIER's theory satisfy the principle of material indifference the constant of proportionality, or heat conductivity, must also be so invariant and hence is an absolute scalar under general transformations of space-time. This invariance or material indiffere:nce has nothing to do with the coordinate invariance described in Subsection ß, above. Indeed, if we conceive forces and motions directly, without the intermediary of co-ordinates and compon~nts, the requirement of material indifference is not thereby satisfied automatically 2• Neither can it be achieved by a thoughtless tour-dimensional formulation, for the precise meaning of "observer" in classical mechanics, where the time is a preferred co-ordinate, must 'be stated in terms of time-dependent orthogonal transformations, not of more general ones. That this requirement has a fundamental physical meaning is shown by the fact that the laws of motion themselves do not enfoy invariance with respect to the observer. "Apparent" forces and torques, in general, are needed to reconcile the descriptions of mechanical phenomena given by two observers in relative motion (Sect. 197). The principle of material indifference states that these are the only mechanical effects of the motion. For example, the deflection of the spring on the rotating table is supposed to be proportional to the entire force acting parallel to the spring, and thus, since the end is free, to measure precisely the force measurable by an observer at rest upon the table, this force being the "apparent" force that accompanies the rotation of the observer's frame with respect to one in which the spring would suffer no extension 3• All the classical linear theories, and some of the non-linear ones, satisfy the principle of material indifference trivially and automatically. For example, since mutual distances are invariant under change of observer, any theory in which the stress is determined only by mutual distances and differences or derivatives of mutual distances or relative velocities of particles exhibits material indifference. This is the case with most theories of elasticity and fluid motion 4 • 1 [1678, 1]. 2 Cf. THIRRING [1929, 10, § 6]: "Es sei ausdrücklich darauf hingewiesen, daß Unabhängigkeit von der Koordinatenwahl durchaus nicht mit Unabhängigkeit vom Bezugssystem zu verwechseln ist." 3 If the spring is not idealized as massless, a simple correction is made for the centrifugal force acting upon the mass of the spring itself, but otherwise the response of the spring is unchanged. ' It is important not to confuse material indifference with any kind of tensorial invariance. Some constitutive equations relate quantities which transform as tensors under change of Sect. 293. Principles to be used in forming constitutive equations. 703 But as soon as time rates of stressing come into consideration, this invariance holds no longer. The problern was first faced and solved correctly, in a special case, by CAUCHY1 ; the idea was expressed more clearly and given a somewhat different form by ZAREMBA 2• While several recent studies have attacked the problern in one way or another, a general mathematical statement, for purely mechanical theories, was first achieved by NoLL3• Since this principle has not yet been formulated in a scope broad enough to cover all situations envisaged in this treatise, and since in the purely mechanical case it will be presented in the article, "The Non-linearField Theories of Mechanics" (Vol. VIII, Part2), we do not discuss it further here, though we give an example of its use in Sect. 298, below. rJ) Equipresence. In the most general physical situations, mass, motion, energy, and electromagnetism are simultaneously present. In the classical theories, the variables describing these phenomena are divided more or less arbitrarily into classes, the members of each of which are supposed to influence only each other, not the members of the other classes. Stress is coupled with strain or stretching, flux of energy with temperature gradient, electric displacement with electric field strength, magnetic intensity with magnetic induction. Such simple divisions of phenomena, perhaps lingering remnants of old views on "causes" and "effects ", suffice to describe a great range of physical experience, but, as is well known, the members of the different classes often react upon each other, and there are also various special theories of energetico-mechanical, electro-energetic, and electro-mechanical interactions. Here the process of the theorists has been conservative. They have maintained as much of the old separation of variables as possible, relinquishing just enough of it to include the bare existence of a particular phenomenon of interaction. In truth, the separation in itself is unnatural and unjustified by physical principle. Resulting only from the gradual discovery of individual phenomena, it reflects the old opinions that break physics up into compartments. The present treatise is conceived in the belief that the classical field theories, if rightly understood, describe most of the gross physical phenomena of nature. Constitutive equations, then, should not artificially divert these theories into disjoint channels. This same view, for energetico-mechanical effects alone, results also from statistical models of continuous matter. There, stress and flux of energy appear as gross or mean expressions of a purely mechanical process. The separation frame; e.g., the theories of fluids discussed in Sects. 298-299 relate t to d, both of which, by the results in Sects. 144 and 211, are indeed tensors under change of frame. The principle of material indifference requires that any relation between such tensors be a tensor relation under change of frame. But some variables occurring in constitutive relations are not tensors under change of frame, the vorticity w and the deformation tensor C being examples. The classical theory of finite elastic strain (Sect. 302) relates t to C; its constitutive equations satisfy the principle of material indifference, but the elastic coefficients do not generally transform as tensors under change of frame. 1 [1829, 4]. 2 [1903, 20 and 21] [1937, 12, Chap. I, § 2]. 3 [1955. 18, § 4], "isotropy of space"; [1957. 11, § 9] [1959, 9, § 7] [1958, 8, § 11], "principle of objectivity". A noteworthy earlier attempt is given by ÜLDROYD [1950, 22]: "The form of the completely general equations must be restricted by the requirement that the equations describe properties independent of the frame of reference ... Moreover, only those tensor quantities need be considered which have a significance for the material element independent of its motion as a whole in space." ÜLDROYD's process is to write all constitutive equations in convected co-ordinates, extension to spatial systems then being achieved by imposing tensorial invariance. Cf. LonGE [1951, 16]. While this procedure leads to correct results, it is not the only one possible, nor is ÜLDROYD's formulation of the problern unequivocal. TheprinciplesofRIVLINandERICKSEN [1955, 21, § 11 ff], ofTHOMAS [1955, 24and25], and of CoTTER and RrvLIN [1955. 4], which likewise rest upon use of special co-ordinate systems, are special cases of NoLL's. 704 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 294. of stress as arising from deformation and flux of energy from changes of temperature emerges, not without reason, as a first approximation; but on a finer scale, it is illusory. Any attempt to draw precise conclusions from statistical theories show that the classical separation of phenomena is artificial and unphysical1. For energetico-mechanical phenomena, a principle denying any fundamental separation was formulated by TRUESDELL 2 : The stress and the flux of energy depend upon the same variables. Here we propose the following more general principle oj equipresence: A variable present as an independent variable in one constitutive equation should be so present in alt. Let it not be thought that this principle would invalidate the classical separate theories in the cases for which they are intended, or that no separation of effects remains possible. Quite the reverse: The various principles of invariance, stated above, when brought to bear upon a general constitutive equation have the effect of restricting the manner in which a particular variable, such as the spin tensor or the temperature gradient, may occur. The classical separations may always be expected, in one form or another, for small changes-not as assumptions, but as proven consequences of invariance requirements. The principle of equipresence states, in effect, that no restrictions beyond those of invariance are to be imposed in constitutive equations. It may be regarded as a natural extension of ÜCKHAM's razor as restated by NEWTON 3 : "We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances, for nature is simple and affects not the pomp of superfluous causes." This more general approach has the added value of showing in what way the classical Separations fail to hold when interactions actually occur. We illustrate these remarks in Sect. 307, below. II. Examples of kinematical constitutive equations. 294. Rigid bodies. A body is rigid if it is susceptible of rigid motions only: The distance between any two particles of the body is constant in time. The motion of the body is then completely determined when the position of one particle and the orientation of an appropriate rigid frame with origin at that particle are given as functions of time. For any body, rigid or not, the motion of the center of mass c is determined from the total assigned force by integration of (196.6). This fact is of little or no use when deformation occurs, since the center of mass generally moves about and fails to reside in any one particle, but the center of mass of a rigid body is stationary within it; more precisely, in any frame with respect to which the body is at rest, the center of mass c is also at rest. Thus we may say that the motion of the center of mass determines, or, more properly, constitutes the translatory motion of the rigid body. The kinematical statement that there exists a frame with respect to which the body is permanently at rest may be expressed in terms of the tensor of inertia and the moment of momentum as follows: In a suitably selected frame, denoted by a prime, we have il'[O'l = 0, ~'[0'] = 0 (294.1) for all t. These are the constitutive equations of a rigid body. Substituting them into the general equations of balance of moment of momentum (197.3) 4 and 1 M. BRILLOUIN [1900, 1, § 37]. supported by many later results. 2 [1949, 34, § 19] [1951, 27, § 19], "BRILLOUIN's Principle". 3 [1687, 1, Lib. III, Hypoth. I]. Sect. 295. Diffusion of mass in a mixture. 705 (197.4) yields Euler's equations forarigid body1 : w -~'[ '1 +wx~'[ 'l. w = ~[O'J_ c'xWlb. (294.2) Here ~[O'l is the applied torque with respect to 0'. The last term on the righthand side vanishes if 0' is the center of mass: c' = 0; it vanishes also if there is some one particle of the body which moves at uniform velocity in an inertial frame and if 0' is chosen at that particle: b = 0. In these cases and in any others when c' X b is known, the angular velocity w of any frame rigidly attached to the body is determined to within an arbitrary initial value by the applied torque, the mass and the translatory motion of the body, and the tensor of inertia ~'[O'J with respect to the same frame. This conclusion follows from the existence and uniqueness of solution to (294.2), which is a non-linear differential equation of first order for w. Thus the purely kinematical assumption that the motion is persistently rigid renders it simply determinable. This is no wonder, for the infinite nurober of degrees of freedom generally present in a body of finite mass and volume have been reduced to six. While the laws of motion (196-3) are linear in an inertial frame, EULER's equations (294.2) are non-linear, since they state the balance of moment of momentum in a non-inertial frame, the coupling between these frames being that expressed by the principles of apparent torques in Sect. 197. In a rigid body, we are at liberty to imagine internal stress and flux of energy and their respective balances, but since the motion is determinate without these considerations, we invoke ÜCKHAM's razor 2 to excise them. A theorem of balance of mechanical energy follows from (294.2). Choosing the origin of the primed frame at c, by (168.8) and (168.9) we obtain 2~ = Wlc 2 + w. ~[o'J. w. By differentiating this result and using (196.6) and (294.2) we obtain Sr= iJ. c + ~'[O'J. w. (294.3) (294.4) This equation asserts that the entire rate of working of the applied force iJ in producing the translatory motion and of the applied torque ~'[O'J in producing the rotation is converted into kinetic energy. Assumptions analogaus to those discussed in Sect. 218 suffice to derive from (294.4) a law of conservation of total energy. The theory of rigid bodies is presented in detail in the article by SYNGE in this volume. 295. Diffusion of mass in a mixture. A simple theory of diffusion of mass in a non-uniform binary mixture was proposed by FrcK3 : The rate of diffusion of 1 The theory is usually and justly attributed to EULER [ 1765, 1]; it has a lang prior and a short but important subsequent history, which has never been adequately traced. 2 Adopted byNEWTON [1687, 1] as the first ofthe"Hypotheses", calledinlatereditions Rules of Philosophizing", set at the head of Book III. On p. 704, we have quoted in full the statement in the edition of 1687. Here we mention that the usual derivation of EuLER's equations given in textbooks must also be excised by ÜCKHAM's razor. That derivation rests on an argument, summarized in Sect. 196 A, showing that if all the mutual forces between the particles of a body are central and pairwise cancelling, balance of momentum implies balance of moment of momentum. In a rigid body, by definition, the mutual forces never manifest themselves in any way except to maintain rigidity. Rigidity has already been assumed, and it suffices; to hypothecate mutual forces is to multiply causes. In any case, the derivation is absurd from the Standpoint of modern physics, which does not represent the smallest portians of any kind of body as stationary centers of force. 3 FicK [1855, 1, pp. 65-66] gave as his only physical basis an assertion that diffusion of matter in a binary mixture at uniform total density and pressure seems analogaus to flow of heat according to FouRIER's theory (Sect. 296). Handbuch der Physik, Bd. Ill/1. 45 706 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 295. mass of the constituent m: is proportional to the gradient of the density of that constituent and is such as to tend to restore uniformity. In equations, D>O, m:=1,2, (295.1) where the coefficient D, the diffusivity, has the physical dimensions [L 2 r-1]. FICK assumed that f!,k = 0, as indeed is necessary in orderthat (295.1) be compatible with (158.8) 1 • Substituting (295.1) into (159.2) 2 , an expression of the conservation of mass, we have the diffusion equation: (295.2) where (159.4) is assumed to hold. This equation is of the same form as the isotropic case of FouRIER's equation (296.10). FrcK's law was proposed and has been applied under various specializing hypotheses, typically that c~1 =0, that there is no mean motion, and that the pressure is constant. Also, only a binary mixture was envisioned, and while formally (295.1) is meaningful for an arbitrary nurober of constituents, it does not allow for their possible mutual actions. While there have been numerous studies1 1 The main features of a general theory of diffusion might have been found in the kinetic theory of gases; indeed, MAXWELL [1860, 2, Prop. XVIII] [1867, 2, Eq. (76)] emphasized the fact that diffusion arises from equal and opposite forces proportional to the differences of velocities, so that (295-5) is virtually suggested by him. However the later development of the formal theory (e.g. [1939, 6, § 8.4]), emphasizing special features of the binary case, tended to obscure the simple mechanical idea. The effort of HELLUND [1940, 13] to calculate the basic equations for multi-constituent mixtures according to the kinetic theory did not lead to clear results. The general continuum theory based upon (295.5) and (295.8) may fairly be attributed to STEFAN [1871, 6, Eqs. (3)], who gave the special case of (295.6) appropriate to a ternary mixture of perfect fluids, but his work seems to have attracted no attention, and many inferior attempts were published later. DuHEM [1893, 2, Chap. VI, §I] in effect noted that a constitutive equation forafluid mixture should specify the functional form of piJ!, but instead of perceiving a connection with Frcx's law, he proposed to determine the piJ! by assuming the mixture to be such that all constituents may participate in a common infinitely small isentropic motion. We do not understand the meanings of all the terms in the theories of diffusion proposed by }AUMANN [1911, 7, §XXXVI] [1918, 3, §§ 136-141, 150-154] and LoHR [1917, 5, §§ 5, 7, 12, 15]. A thermodynamic theory of diffusion, including thermal diffusion and the diffusionthermal effect, seems first to have been proposed by EcKART [1940, 8, Eq. (49)]. Cf. the earlier and more special theory of ÜNSAGER and Fuoss [1932, 10, § 4.12]. A more fully elaborated theory of the same kind, adopting the "Onsager relations ", was proposed independently by MEIXNER [1941, 2, § 5] [1943, 2, § 2] [1943, 3, § 4]. The later Iiterature in this field does not always take pains to recognize all the conditions laid down by MEIXNER but otherwise seems to diverge from his work only in minor points; cf. LAMM [1944, 7], LEAF [1946, 7, Eq.(48)), PRIGOGINE [1947, 12, Chap. 10, §§ 1 and 5, Chap. 11, §3], DE GROOT [1952, 3, §§ 45-46], KIRKWOOD and CRAWFORD [1952, 12], HIRSCHFELDER, CURTISS and BIRD [1954, 9, § 11.2d]. There is also a more primitive theory of ÜNSAGER [1945, 4, pp. 242-247], resting upon an inverse of (295.4); apparently only special circumstances were envisioned, but it is not clear what they are. In the thermodynamic theories, little if any use is made of mechanical concepts and principles. Meanwhile, clear results from the kinetic theory approximations for a multi-constituent mixture had been obtained byCowLING [1945, 2, Eq.(34)]; since his dl}! is approximately (though not exactly) proportional to our piJ!, his result is equivalent to the inverse of a special caseof (295-5). Hisanalysis was generalized by CuRTISS and HIRSCHFELDER [1949, 4, Eqs. (19), (20), (21)] so as to include thermal diffusion; after allowance is made for the approximations of the kinetic theory, their result is seentobe a special case of (295-5). In the kinetic theory the relation (295-8) is valid in first approximation even if st >2 but has not been shown to hold in a more accurate treatment. The theory of STEFAN was proposed anew by ScHLÜTER [1950, 25, Eq. (3)], [1951, 23, Eqs. (1) to (3)] and by JoHNSON [1951, 14]; the latter gave an argument indicating that F!BIJ!/(!!B(!IJ! is roughly independent of the densities. Sect. 295. Diffusion of mass in a mixture. 707 of more general cases, we prefer to follow an independent argument 1 toward a linear theory of diffusion which reduces to FicK's law under the conditions in which it is usually applied. First, we note that by means of (158.8), we may express (295.1) alternatively in theform 1 em e!B e'2l e!B , , e'll,k= 2n (elßu!Bk-e'Hu'Hk)=-eD(u!Bk-u'Hk) =en(x!Bk- xu), (295-3) still for a binary mixture. This symmetrical expression indicates that it is an excess of the mass flow of one constituent over another's that increases the density gradient of the latter. Thus for a general mixture we might have ~ ~ em,k = L Y!B'H (e'H u!B k - e'll u!Bk) = L .F!B ~~ (x!Bk - Xmk). (295 .4) !B~l !B~l But since the right-hand side of this relation is equal to a linear combination of the excess of the momenta of all constituents above the momentum of the constituent lll:, a still more natural idea of diffusion is expressed by the more general constitutive equation ~ eP'H = L .Flß'll (x!B- x'll), (295.5) !B~l where P'H is the supply of momentum, defined by (215.2) and restricted by (215.5). In this relation the purely kinematical view expressed in FICK's law (295.1) is replaced by a dynamical one: Diffusion gives rise to a force tending to restore uniformity. The dynamical equations corresponding to (295.5) follow at once from (215 .2) : ~ ("k jk) km " ;= ( 'k 'k) e~( x'H - 'll - t'll, m = L.. \ll'll x!B - x'll . (295.6) !B~l Since c'll=O, in order that (295.5) be consistent with the balance of total momentum in the mixture, expressed by (215.5), it is necessary and sufficient that ~ L (.F!B'll- .F'llm) = o. 'H~l When ~ =2, this result is equivalent to 2 .F'llm= .F!B'll· 1 The ideas given here are presented more fully by TRUESDELL [1960, 6]. (295.7) (295 .8) 2 This condition, for the binary case, originated in the work of MAXWELL [ 1860, 2, Prop. XVIII], to whom is due the appeal to "NEWTON's third law" which JoHNSON [1951, 14] phrased more generally: "The force exerted by species ~ on species 18 is equal in magnitude and opposite in direction to that exerted by 18 on ~ .... " The argument in the text above, basically different in that it rests upon the principle of linear momentum, was first suggested by STEFAN [1871, 6, p. 74]: "Da die ... Kräfte aus Wechselwirkungen zwischen den im Element dx dy dz zur selben Zeit befindlichen Teilchen des ersten und zweiten Gases entspringen, so ändern sie die Bewegung des Schwerpunktes dieses Elementes nicht." Indeed, this establishes (295.8) in the binary case, but if .lt >2, only (295-7) follows. For the ternary case, STEFAN assumed (295.8) without analysis or comment. On the other hand, there is twofold insufficiency in the argument of MAXWELL and J OHNSON, for in order to invoke "NEWTON's third law" we have to know that the diffusive term represents the entire force arising from the mutual forces of the two species, which it generally does not, and, secondly, the influence of species GI: upon the mutual action of species ~ and 18 is left out of account. We feel the true content of the Maxwell-Johnson argument does not yield a proof but merely suggests the supplementary assumption (1) in the text above, stating that species GI: has no effect upon the relative diffusion of species ~ and 18. The relations (295.8) for arbitrary .lt may weil be named "STEFAN's relations ". The "Onsager reciprocity relations" for pure diffusion refer to a different set of coefficients. LAMM [1954, llA], who considered STEFAN's relations to be "self-evident for physical reasons ", showed that they are equivalent to ÜNSAGER's relations when .lt = 3; for general .lt, this is established by TRUESDELL [1960, 6] .. 45* 708 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 295· but if ~ >2, such a relation of syrnmetry does not hold without further assumptions. It may be shown tobe sufficient that (1) r~~ shall not depend upon 1?e=O, no wave can travel at the speed given by (297.16) except in the very special circumstances when (J = const. This difference ceases to seem paradoxical when we recall that the conduction of heat is a dissipative mechanism, and that when the second main dissipative mechanism, that of viscosity, is included in the mathematical model, both the kinds of waves considered here become impossible, as is shown in Sect. 298. The Iimit behavior is discussed by DuHEM [1901, 7, Part III, §§ 2- 3] and by SERRIN, § 57 of Mathematical Principles of Classical Fluid Mechanics, this Encyclopedia, Vol. VIII. Part 1. 3 The first analysis of this kindisthat of STOKES [1848, 4, p. 355], faulty from failure to take account of the balance of energy. Fora perfect gas, the full results are due to HuGONIOT [1887, 2, §§ 154, 161-163]; cf. the summary of VIEILLE [1900, 10, p. 185]. The general theory of weak shocks had been given earlier by CHRISTOFFEL [1877. 2, §§ 2-5]. An exposition for general tri-variate fluids is given by SERRIN, Sect. 56 of the article just cited. Sect. 298. Linear!y viscous fluids. 715 a jump of 17 which is not uniform, thus rendering the flow baroclinic and destroying the circulation preserving property, as weil as inducing a jump of vorticity1. Further general theorems of gas dynamics follow from furth~r assumptions; typically, that the flow is steady, that f = 0, etc. 2• 298. Linearly viscous fluids. To consider a substance which in equilibrium has the same behavior as that predicted by EuLER's equation (297.1), viz. grad p = ef, (298.1) yet when in motion can support appropriate shearing stresses, assume that the stress tensor t be a linear function of the velocity x and the velocity gradient. By (90.1) we may write t =g(x,w,d), (298.2) where g is a linear function. The constitutive equations (298.2) define a linearly viscous fluid, it being supposed also that there is no couple stress: m = 0. We now apply the principle of material indifference (Sect. 293 0). The constitutive equation (298.2) is to have the same form for all observers. To an observer in a co-rotational frame, x = 0 and w = 0; for him, (298.2) reduces to a relation giving t as a function of d alone, and therefore, since both t and d transform independently of x and w [cf. (144.3) and (211.1)], it must reduce to such a relation for all observers. l.e. 3, t = f(d). (298.3) N ow consider frames whose axes coincide with the principal directions of d (Sects. 82 and 83), so that (298-3) becomes (298.4) For a definite assignment of the undirected axes different assignments of the positive senses yield four different positively oriented frames, any one of which may be obtained from any other by a rotation through a straight angle about one axis. Under such rotations, it follows from (82.6) that da is invariant. By the principle of material indifference, then, fkm in (298.4) is invariant under these rotations: ftm=fkm• say. But t transforms according to the tensor law; in particular, under rotation through a straight angle about the k-axis we have ttm = - tk m for k =I= m . Comparing these two results yields tk m = 0 if k =I= m. Thus the principal axes of stretching are also principal axes of stress. Since further orthogonal transformations may permute the da in any way, the principal stress tb is a symmetric function of them. Hence the relation (298.3) reduces to one giving 1 Indicated by HADAMARD [1903, 10, Notelll]. A modern exposition is given by SERRIN, § 54 of the article, just cited. Despite the above-stated facts, the speed of sound behind the shock is still given by (297.13). since, as remarked by TRUESDELL [1951, 36], this follows from (297.12) and the more general theorem proved in footnote 4 on p. 712. 2 A development of these principles is given by TRUESDELL [1952, 23]. 3 This equation, in an inertial frame, along with f(O) = - p 1, was taken as the definition of a fluid by STOKES [1845, 4, § 1]. 716 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 298. t as an isotropic function of d. In other words, all fluids included in the definition (298.2) are necessarily isotropic1• The mostgenerallinear isotropic function t of a symmetric second-order tensor d may be written in the form 2 of the Na vier-Poisson law: t=-P1+Ud1+2p,d, } t" m m = - p (Jk + Ä. dq lJk + 2 IL d" q m r m• (298.5) where we have used the requirement that t= -p1 when d =0. From (298.5) it follows that t is symmetric. Therefore CAUCHY's second law in the form (205.11) is satisfied automatically. In other words, it is a consequence of the principle of material indifference that a fluid cannot support extrinsic couples; when there are no extrinsic couples, balance of moment of momentum follows from balance of momentum 3• Substitution of (298.5) into CAUCHY's first law (205.2) yields a system of three differential equations known, at least when subjected to further simplifying assumptions, as "the Navier-Stokes equations ". If the pressure p and the three components i" of the velocity are regarded as unknowns and the coefficients Ä., p, are regarded as given, the system is still underdetermined, but since the means of rendering it determinate are essentially the same as those used in the theory of perfect fluids (Sect. 297), they will not be discussed here. Since the physical dimensions of the coefficients Ä. and p, in (298.5) are [M L -1 r-1], it is plain that our definition (298.2) violates the principle stated in Sect. 293 e. In rectification, we replace (298.2) by the complete definition of a linearly viscous fluid: t = /(~. w, d, p, 0, 00 , fto}, (298.6) where, as before, f isalinear function of ~. w, and d and a continuous function of its remaining arguments, and where 00 and p,0 are material constants such that phys. dim ()0 = [8], phys. dim. fto = [M L -1 y-1]. (298.7) The presence of the first of these constants makes it possible for the fluid's response to deformation to vary with the temperature. Motivation for introducing the secon:i constant may be found in simple experiments on the resistance of fluids in viscometers of various kinds. Since it would be possible, still within the framework of pure mechanics, to lay down a relation like (298.6) but involving four 1 Sometimes encountered are "anisotropic fluids" defined by constitutive relations of the type t~ = C~ d$ + D~, where 0 and D are general tensors of the orders indicated. Such an equation does not generally satisfy the principle of material indifference. Given such a relation in an inertial frame, for example, transformation of t and d by the appropriate laws to a non-inertial frame yields a relation of the same form except that the components of 0 and D in the non-inertial frame depend upon the timet, violating the original postulate (298.2). In order to obtain a properly invariant theory of anisotropic fluids, it is necessary to modify (298.2) by introduction of some vector or vectors specifying preferred directions. Cf. the oriented bodies studied in Sects. 60-64 and the theory recently proposed by ERICKSEN [1960, 1]. 2 The simplest case of this law was proposed by NEWTON [1687, 1, Lib. II, Chap. IX]. For incompressible fluids, equivalent dynamical equations were obtained from a molecular model by NAVIER [1821, 1] [1822, 2] [1825. 1] [1827, 6]. The continuum theory of CAUCHY [1823, 1] [1828, 2, § 111, Eqs. (95). (96)] lacks the term -pt5:_. The general formula was obtained by PorssoN [1831, 2, ~~ 60-63] from a molecular model. The continuum theory, subject to an unjustified but easily removed specialization, is due to ST. VENANT [1843. 4] and STOKES [1845, 4, §§ 3- 5]. 3 The reader is to recall that in Sect. 205 symmetry of t was proved equivalent to balance of moment of momentum only under the assumptions that l = 0, that m= 0, and that linear momentum was balanced. Sect. 298. Linearly viscous fluids. 717 rather than two dimensionally independent material constants, our definition (298.6) implies not only the kinematical restrictions we have already demonstrated but also dimensional ones, which we now proceed to determine. First, the same reasoning as given above suffices to reduce (298.6) to an equation of the form1 t = f(d, p, (), ()o, f.lo) · (298.8) The dimensional matrix of the 11 quantities appearing in any one component of (298.8) is L T M e tkm -1 -2 0 (k and m fixed) p -1 -2 0 d 0 -1 0 0 (6 rows) flo -1 -1 0 () 0 0 0 Oo 0 0 0 The first 2 rows are alike: so are the next 6 and the last 2; the ninth may be obtained by subtracting the third from the first; thus the rank is at most 3; in fact, it is exactly 3· By the n-theorem, (298.8) is equivalent to a relation among 11 - 3 = 8 dimensionless ratios formed from the quantities entering it. A possible set of such ratios is tkm/P, f.lo dfp, fJj()0 • Hence (298.8) is equivalent to a relation of the form 2 tjp = g (f.lo dfp, ()f()o), (298.9) where the function g is a dimensionless function depending linearly upon f.lodfp and continuously upon fJjfJ0 • Thus the requirement of dimensional invariance implies that p may enter the dynamical equations only in a strikingly restricted way. Re-examining the argument used to reduce (298.3), we see that the presence of additional scalar arguments does not affect it. Hence we again obtain (298.5), except that now the coefficients A. and f-l are shown to have restricted functional forms: (298.10) the functions f1 and /2 being dimensionless. While, as shown by (298.9), it has been tacitly assumed that p =f= 0 in the region considered, the assumed continuity of f as a function of p in (298.6) allows this restriction to be removed by inspection of the final result. The material coefficients A. and f.l are the viscosities of the fluid. The dimensional constant flo is a parameter which, in any system of units selected, may be assigned a numerical value representing the amount of shearing stress or other resistance to a given stretching that a particular physical fluid may offer. The dimensional constant ()0 and the dimensionles functions / 1 and /2 are parameters which enable representation of a viscous response varying with temperature. From their definitionvia (298.6), the viscosities are independent 3 of all kinematical 1 In the following argument, we tacitly employ reetangular Cartesian co-ordinates throughout, so that all tensor components bear the physical dimensions shown in the table. The final results are in tensorial form and hence valid in all co-ordinates. 2 TRUESDELL [1949, 33, §§ 3, 6] [1950, 32, §§ 4, 7] [1952, 2], §§ 63-65] [1952, 22, p. 90]. 3 To speak of a "frequency-dependent viscosity" or a "non-linear viscosity", as is not uncommon in the Iiterature of ultrasonics and rheology, is a misleading way of saying that the linear law (298.5) does not hold but some particular (and usually imperfectly specified) non-linear law does. 718 C. TRUESDELL and R. TouPrN: The Classical Field Theories. Sect. 298. variables; according to the results (298.1 0), the viscosities are also independent of the pressurel, being functions of the temperature only. When .?. = fJ, = 0, the dynamical equations for viscous fluids reduce to EuLER's equation (297.1) for perfect fluids. Thus perfect fluids are often called inviscid. That, irrespective of the values of .?. and f-t, the equations of the theory of viscous fluids reduce to the same statical equation (298.1) when d = 0, is a standard example to show that the commonest statement of "D'ALEMBERT's principle" is false 2 ; for this reason, the portion .?. Id 1 + 2ft d of the stress is regarded as arising from internal friction. Since t~=2f-td~ when k=j=m, (298.11) . the quantity fJ, is the ratio of shear stress to the corresponding shearing (Sect. 82) of any two orthogonal elements; hence it is called the shear viscosity. The mean pressure (204.7) is given by . p - p = - (.?. + i fh) d~ = (.?. + i fh) log e . ( 298.12) Thus.?. + if-t, the bulk viscosity, is the ratio of the excess of the mean of the three normal pressures over the static pressure to the rate of condensation. To render this last interpretation definite, it is best to distinguish two cases. First, for an incompressible fluid we have e =0, and from (298.12) follows p = p in all circumstance5. Thus, for an incompressible fluid, the pressure p occurring in the constitutive relation (298.5) is always the mean pressure, and .?. drops out of all equations. Second, for a compressible fluid, in accord with the agreement that (298.1) holds in equilibrium, we take p in (298.5) tobe the same pressure as would hold in equilibrium under the same thermodynamic conditions, i.e., p = n = n (e, 0) as given by the thermal equation of state. This pressure is then determined, independently of the motion, as soon as e and (} are known. (298.12) relates this pressure to the mean of the pressures actually exerted upon three perpendicular planes at the point in question. (A pressure-measuring device, in general, measures some component of the stress tensor; in a viscous fluid, it is not justifiable to identify either p or p with the results of measurement, a theory of the flow near the instrument being required in order to interpret the experimental values in terms of the variables occurring in the theory.) 1 I.e., to obtain viscosities which depend also upon the pressure, it is necessary to start with a definition more generat than (298.6): A constant or scalar bearing the dimension of time or stress must be included, as is done in the following section. Cf. the discussion of this point by TRUESDELL [1950, 30, §§ 4, 7, 11] [1952, 21, §§ 62-63]. 2 I.e., to obtain "the" equations of motion from the statical equations, for a given system, add the "inertia force" -x to the assigned forcelper unit mass. lf (298.1) are the statical equations, then this form of "D' ALEMBERT's principle" is to be supplemented by adding to the "inertial force" any "frictional forces" that may be present. "Frictional forces" are then defined as any forces arising from motion other than inertial force. Putting all these definitionstagether Ieads to the conclusion that D' ALEMBERT's principle asserts that equations of motion follow from statical equations by supplying inertial force plus such other forces as may arise in conjunction with motion. Thus D' ALEMBERT's principle appears to have no content at all. To the reader who finds this confusing we remark that it was not by oversight that from the !ist of guiding principles in Sect. 293 we omitted "D' ALEMBERT's principle ", for we consider it to be either trivial or false in the usual statements in this context. (This does not affect the validity of the different "D'ALEMBERT-LAGRANGE principle" given in Sect. 232.) Acorrect statement, revealing the limited but useful validity ofthisform of the principle, is as follows: Given the statical equations for a material, constitutive equations for a dynamically possible material resuZt if I is replaced by I- x. This special kind of material, being only one of the infinitely many that share the same statical properties, is called "perfect ". This definition applies to several classical theories. Sect. 298. Linearly viscous fluids. The stress power (217.4) assumes the form P =PE= tkmdkm =-Pd~+ J.(d~} 2 + 2fl d!, d'/:, while by the above agreement regarding p we have from (256.4) 1 P1 =- pd~. Hence Eq. (256.6) for production of total entropy becomes e o ~ = cJ> + h~p + e q. where 719 (298.13) (298.14) (298.15) (298.16) [It is plain that t (/> is a dissipation function in the sense of (241 A.2).] If we adopt the corollary (258.1)1 of the entropy inequality, we conclude that (/> as given by (298.16) must be a positive semi-definite quadratic form. An easy analysis of (298.16) shows that this is the case if and only if1 ft-;;;,o, 3J.+2fl-;;;,o. (298.17) These results have immediate mechanical interpretations. By (298.11), (298.17)1 asserts that the shear stress always opposes the shearing. By (298.12), (298.17) 2 asserts that in order to produce condensation ( expansion), a mean Pressure not less (not greater) than that required to maintain equilibrium at the same density and temperature must be applied. These interpretations, showing that the effect of the viscous stresst +P 1 as given by (298.5) is always toresist change of shape or bulk, reinforce the view that the viscous stress is of the nature of frictional resistance. The quantity C/> in (298.15) is the viscous dissipation of energy per unit volume. As (298.15) shows, this energy goes into the increase of entropy, or is carried off by the flux - h, or is drawn away by sinks - q. It is customary torender the theory moredefinite by setting q =0 and adopting FoURIER's law (296.4). The resulting equations furnish an example, somewhat more typical than the perfect non-conducting gas (Sect. 297), of a fully thermomechanical theory, in which the basic principles of mechanics, energetics, and thermodynamics are employed. The presence of viscosity has the effect of rendering propagation of most kinds of waves impossible. We consider here only the simplest case, that of an acceleration wave, across which p, x, and e are continuous. By substituting (298.5) into the dynamical conditions (205.5) we obtain Ank[x;q] +!lnm[im,k] +!lnm[.ik,m] =0, (298.18} where it is assumed that },, fl· and ef are continuous. Writing sk=- U s~ in (190.5)1 and substituting the result into (298.18) yields (J. + fl) nms,. nk + flSk = 0. Taking the scalar product of this equation by n, we have (A + 2fl) nmsm = 0. (298.19) (298.20) It is a consequence of (298.17) that in order for }. + 2fl = 0 to hold, it is necessary and sufficient that 2 = 0 and fl = 0. Hence by (298.20) we conclude that in a viscous fluid nm sm = 0. Putting this result back into (298.19) yields fl sk = 0, and hence sk = 0. What has been proved 2 is that the instantaneous existence of 1 DuHEM [1901, 7, Part I, Chap. 1, § 3], STOKES [Note, pp. 136-137 of the 1901 reprint of [1851, 2]]. 2 KorcHINE [1926, 3, § 3]. but the result is really included in an earlier one of DUHEM [1901, 7, Part II, Chap. III], who uses a different terminology. 720 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 299. a surface upon which x and p are continuous but ik,m suffers a jump discontinuity is incompatible with the law oflinear viscosity (298.5). We recall that kinematical analysis alone (Sect. 190) shows that a discontinuity in ik m in order to persist must be propagated as an acceleration wave. In Sect. 29l we showed that acceleration waves are propagated in a barotropic flow of a perfect fluid at the speed c, determined by the barotropic relation p = f (e). Any thermal conductivity, however small, generally renders the flow baroclinic and results in the Newtonian speed of propagation (297.16) instead of the Laplacian speed (297.13) that holds when there is a non-degenerate equation of state p =P (e, 'YJ) and when there is neither flux nor supply of energy. Now we have seen that in fluid with non-vanishing viscosity, however small, such a surface cannot exist at even at a single instant. This last result is included in a general theorem of DuHEM1 : In a linearly viscous fluid, no waves of order greater than 1 are possible. Since viscous and thermally conducting fluids are regarded as a refined model, superior to the perfect fluid, for the same physical materials, a device must be found whereby wave propagation in some sort may occur. Moreover, perfect fluids are formally a special case of viscous fluids; thus, properties of perfect fluids must be reflected in corresponding properties of viscous fluids. The appropriate device is the quasi-wave of DuHEM, a thin layer in which some of the variables suffer rapid but nevertheless continuous changes. The solutions containing waves that occur in perfect fluids are to be regarded as limit cases of an appropriate solution of "the same" problern for viscous fluids, in the limit as ;., f-l, and " approach 0, and hence the layer becomes arbitrarily thin. The rate at which x-+0, relative to those at which A-+0 and p-+0, influences the results. A full analysis of the plane quasi-wave has been given by GILBARG (1951) 2• A similar difficulty arises in connection with the boundary conditions. For viscous fluids it is customary to impose the condition (69.3) representing adherence of the fluid to the boundary. For perfect fluids, solutions satisfying this condition generally fail to exist, and only the weaker condition (69.1) is employed. In many cases the solutions afforded by the two theories are sensibly the same throughout most of the region occupied by the fluid but differ only in a thin boundary layer near solid objects. The existence ofthissmall region of difference, while perhaps effecting little alteration of the gross appearance of the flow, can yield results which are dynamically of a different kind; e.g., the force exerted by the fluid on an obstacle will generally be very different, as is plain from the results given in Sect. 202. The purpose of the foregoing remarks is to point up a major instance of the ideas sketched in the second paragraph of Sect. 5. 299. Non-linearly viscous fluids. A simple and now familiar theory of nonlinear viscosity is obtained by taking (298.2), but without the restriction that f be linear, as the definition of a fluid. The analysis at the beginning of Sect. 298 made no use of the linearity of f; thus it follows, in full generality, that (298.2) must reduce to a form giving t as an isotropic function of d. By the representation theorem for such functions we thus ha ve (299.1) 1 [1901, 6] [1901, 7, Part II, Chap. 111]. DuHEM asserted also that shock waves arenot possible in a viscous fluid, but this result holds only subject to some qualification. Cf. Sect. 54 of the article by SERRIN, Mathematical Principles of Classical Fluid Mechanics, this Encyclopedia, Vol. VIII, Part 1. 2 For an exposition, see § 57 of SERRIN's article, just cited. Sect. 299. Non-linearly viscous fluids. 721 where N0 (0,0,0) =0. (299.2) This reduction was first given by REINER1. In an incompressible fluid, there is no loss in generality in taking N0 = 0. The scalar coefficient functions generalize the classical viscosities occuring in the linear law (298.5). If we set 2p, = N1 (0, 0, 0) and ). = 8N0j(:Hd when d =0, then (298.5) results from (299.1) by linearization. The coefficient N2 , the cross-viscosity, gives rise to effects of a type not present in the linear theory. For the dimensional analysis, we may proceed just as in Sect. 298, but instead we prefer to replace (298.6) by the more general defining equation t = f(x, w, d, p, fJ, fJ0 , p,0 , to) where t0 , the natural time, is a material constant such that (299.3) phys. dim. t0 = [T] . (299.4) Use of the principle of material indifference and the n:-theorem reduces (299.3) to the form2 (299.5) where, as shown already, the dimensionless function g is an isotropic function of its first argument. Thus in (299.1) the coefficients have the more explicit forms Nr = f!:'!_ • t[- 1 !:lr, (F unsummed) } lo !:lr = fr(to Id, t~ Ild, tg IIId, to Plflo, fJ!fJo), (299.6) where the functions f r are dimensionless functions of their five dimensionless scalar arguments. When the definition of a fluid is narrowed, as it was in Sect. 298, so as to exclude the time constant t0 , the forms (299.6) must be replaced by others, as follows 3 : Nr=P ( p, ; )r-1 !Ir, l ( f.lo ,u~ p,~ ) !Ir= Ir p Id, -:p2 Ild, pa IIId, ()J()o · (299.7) The theory based upon (299.7) is easily seentobe a special case ofthat based upon (299.6). The specialization, however, is one of consequence. The theory devoid of a time constant is nearer to the classicallinear theory in that it possesses no more dimensionally independent material constants. A possible parameter governing dynamical similarity is the truncation number ], given by 4 (299.8) 1 [1945, 5, § 4]; cf. the treatment of the incompressible case by RrVLIN [1947, 13] [1948, 24]. A simple rigorous proof is given in Sect. 59 of the article by SERRIN, just cited. 2 TRUESDELL [1950, 32, § 11] [1952, 21, §§ 68-69] [1952, 22, PP· 89-90]. 3 TRUESDELL [1949, 33, § 4] [1950, 32, § 5], "Stokesian fluid". 4 TRUESDELL [1949, 33, § 7] [1950, 32, § 8]. lt was stated erroneously by TRUESDELL that ] must be controlled independently of the classical scaling parameters. In fact, in two geometrically similar flows having the same Euler and Reynolds numbers the two truncation numbers (299.8) are necessarily also the same. The only further scaling parameters required for Stokesian non-linear viscosity are such dimensionless material constants as are used to specify the functions fr in (299.7). No such remark applies to (299.9). Handbuch der Physik, Bd. III/1. 46 722 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 300. A condition for approximating the constitutive relation (299.1), supposed differentiable with respect to d, by the linear relation (298.5), is 3J< 1; that is, the intensity of stretching (Sect. 83) shall be small in respect to the ratio of pressure to linear viscosity. While the parameter (299.8) may be introduced in the more general theory defined by (299.6), it does not suffice; for dynamical similarity it is necessary to control also a parametersuch as (299-9) The criterion for truncation is now that the product of the intensity of stretching by the natural time of the fluid shall be small. The difference between the two theories is reflected even in the linear case, for in the theory based upon (299.7) the linear viscosities Ä. and ll• as we have seen in Sect. 298, are independent of the pressure, while in the theory based upon (299.6) this need not be so, for they may depend upon to Pl!lo. Further, it can be shown that the stress of the "Stokesian" theory is of a type that occurs also in results following from the kinetic theory of monatomic gases, while the more general theory based upon (299.6) is generally in contradiction with the kinetic theory, in which there is no time constantl. A summary of existing knowledge of the theories of non-linear viscosity is given in "The Non-linear Field Theories of Mechanics ", Vol. VIII, Part 2 of this Encyclopedia. 300. Perfectly plastic bodies. The theory of perfectly plastic bodies is intended to describe an elastic rather than viscous flow in response to stretching and hence, while adopting the constitutive relation (298.3), relinquishes the requirement of consistency with (298.1). Thus we have at once in the linear case 2 t!.=J.d~<5~+2!ld~. (300.1) but Ä. and !l are not material constants, being rather functions of d to be determined by additional conditions. Writing (300.1) in terms of the deviators oft and d (cf. App. 38.12), we have \ -3P=IXIc~. 1X=3.1.+21l·} 0t = 2/.l 0d. (300.2) The additional conditions imposed as an essential part of the theory are (1) the mean pressure is a function of the dilatation only, and (2) some scalar invariant of 0t vanishes. These conditions are said to represent plastic flow or yield. The first amounts to replacing (300.2)1 by a general relation f (p, Ic~) = 0 and is a law of compressibility. Often it is replaced by the condition of incompressibility, Id = 0; in this case p becomes an additional unknown function. The second, characteristic of the theory, may be put in the form y ( II,!_ III,!) = O az ' a3 ' (300.3) where a is an elastic modulus called the yield stress of the material and where the yield function Y is dimensionless. This condition represents a material which responds, or at least responds in the manner indicated by (300.2), only when stresses of a certain kind reach an appropriately !arge value, while the state of plastic flow is assumed such as to maintain this value unaltered. 1 The "relaxation time" in a Maxwellian gas is notamaterial constant such as t0 , being in fact p.fp, a function of temperature and pressure. 2 CAUCHY [1823, 1] [1828, 2, § III, Eqs. (95), (96)] gave these formulae as appropriate for a "soft" body but stated that Ä and p. are constants. Sect. 301. Linearly elastic bodies. 723 A more general theory for incompressible substances1 is obtained by replacing (300.2) by a relation of the form 2u dk = Y~ (300.4) r m ot'k'' where the dimensionless plastic potential P(tfa) is subject to the condition ~k aP _ ) Um ot'k' -0, (300.5 so that (300.4) is consistent with the condition of incompressibility, Id = 0. For isotropic materials, P is taken as a function of II,1/a2 and III,1/a3 only. While (300.4) might seem to determine d uniquely when t is known, this is not so. Indeed, t-tdfa is so determined; conversely, if Pis a sufficiently smooth function, tja is determined as a function of f-t dfa. Substituting this function into the yield condition (300.3), in the isotropic case we obtain a functional relation of the form t(~: nd, ~: IIId) = o. (300.6) Assuming this can be solved for t-tfa, we see that = (J t ( yliid ) . t-t Vnd Vnd (300.7) Thus the factor f-t occuring in (300.4) is determined, not assignable. Comparing a formula of the type (300.4), f-t being eliminated by means of (300.7), with the relation (299.1) for a non-linearly viscous fluid, we see that a still more general theory of perfectly soft bodies, including both viscous fluids and perfectly plastic bodJ.es, results by allowing the coefficients N0 , N1 , and N2 to be discontinuous at d = 0 and by leaving the physical dimensions of the moduli unrestricted. The principal difference between the theories of viscosity and plasticity arises from the different physical dimensions of the material constants. While A. and f-t are viscosities, having the dimension [M L -1 r-1], a is a stress or elasticity, so that phys. dim a = [M L -1 r-2]. The theory of perfectly plastic bodies, as defined by (300.2h and (300.3), is due to ST. VENANT, LEVY, and v. MISES 2• The most commonly employed yield condition is that of MAXWELL and v. MISES 3, viz., II,t =K a2, where K is a constant (cf. the alternative forms of II,t given in Sect. App. 38); this amounts to taking f = const in (300.7). In this theory P = Y. As appears at once by confronting (300.1) with CAUCHY's first law (205.2), the theory of perfectly plastic bodies is a dynamical theory. Nevertheless, almost all of its large Iiterature treats it as if it were statical. Thus, as far as the exact theory is concerned, very little is known regarding it. A survey of the mathematical developments as practised by current specialists in the field is given in the article by FREUDENTHAL and GEIRINGER in Vol. VI of this Encyclopedia. 301. Linearly elastic bodies. The simplest kind of elastic or springy body is one such that the stress arises solely in response to such change of shape as the body has undergone from its "natural" or unstressed state. Considering the strain tobe very small, by the results in Sect. 57 we may take the tensor e as a measure of it, and we assume the stress depends linearly upon it. Moreover, t = o if e = o. 1 GEIRINGER [1953, 11, § 3]. 2 [1870, 7]; [1870, J]; [1913, 6]. 3 [1937. 5, pp. 32-33] (written in 1856), [1913. 6, §§ 2-3]. 46* 724 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 301. There results the constitutive equation t"'k = CkmPq epq• CkmPq = Cmkpq = Ckmqp, (301.1) where for simpler writing we have dropped the tilde from e. A linearly elastic body is isotropic if the relation (301.1) is an isotropic one. For an isotropic body, (301.1) reduces to Cauchy's lawl: (301.2) To the extent that e is an approximate measure of the mutual distances of particles, the principle of material indifference is satisfied by (301.1) and (301.2). If (57.1) is interpreted strictly, however, the principle of material indifference is violated, although (301.1) and (301.2) areinvariant under infinitesimal rigid timeindependent displacements. Linear elasticity theory does not represent exactly the kind of behavior possible in any real material. Rather, it is to be regarded as a mathematical approximation to the properly invariant theories described in the two following sections. The elasticities Ä., f.l, and CkmPq in (301.1) and (301.2) arematerial constants or functions of the temperature or entropy; their physical dimensions are those of stress, [M L -l r-2], and they bear no physical connection with the mathematically analogous viscosities appearing in (298.5). The static theory isalinear one: Uniformly doubled displacements always result from uniformly doubled loads, and, more generally, from displacements u', u" corresponding to stresses t', t", assigned forces j', f", and assigned surface loads t{n)• t(~) we may construct a displacement u=u'-u" answering to the stress t=t'-t", forcef=f'-f", and surface load t(nl =t'. Consider two solutions u' and u" corresponding to the same assigned loads and boundary values. Form a solution u = u'- u" as indicated above; for this solution we have f = 0 in z>, t(n) = 0 on j 1 , and u =0 on j 2 • Substitution of (301.6) 2 into CAUCHY's first law (205.2) yields (;~L=o. (301.8) Hence 0 = J um (8 -:) dv = J [(um~-:) -um k _o-:] dv, öek ,k öek ,k • öek U V (301.9) V =-2JI:dv, V where we have used EuLER's theorem on homogeneaus functions as well as the fact that on <1 either u or t(n) vanishes. Now if J:(e) is of one sign for all values of e, it follows from (301.9) that L'=O in z>. Looking back at (301.4), we see that if 1: is a definite quadratic form ( whether positive or negative), the generat boundaryvalue Problem of static linear hyperelasticity cannot have two distinct solutions. What has been proved is that e = 0; the strains is thus unique, and from the results in Sect. 57 it follows that the displacement u is determined uniquely to within an infinitesimal rigid displacement. This degree of indeterminacy is inherent in the linear theory of elasticity and is to be understood in all statements concerning it, except in cases where this indeterminacy is removed by specification of the displacement on the boundary. There is physical reason to require that }; be a positive definite form, for then in any given small strain from an unstressed state, the stress must do positive work. This idea seems tobe related to, but is not identical with, the requirement (258.1) 1 following from the entropy inequality, which is expressed rigorously in terms of time rates rather than displacements. In the isotropic case, 1: is positive definite if and only if f-l > 0, 3 Ä + 2J.l > 0. (301.10) There is a remarkable principle enabling us, in the case of equilibrium subject to given surface displacements and vanishing assigned force in the interior, to select among all kinematically possible deformations that one which is consistent with the theory of hyperelasticity, a positive definite stored energy function 1 KIRCHHOFF [1859, 2, § 1] [1876, 2, Vor!. 27, § 2]. 726 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 301. being assigned 1 : The displacement that satisfies the equations of equilibrium as well as the conditions at the bounding surface yields a smaller value for the total stored energy than does any other displacement satisfying the same conditions at the bounding surface. To prove this theorem, we writl~ e for the strain leading to a stressthat satisfies the conditions of equilibrium, and e + e' for some other strain, where it is assumed that u' =0 on the boundary 11. Then J .E(e + e') dv = J .E(e) dv + J (e'ZO a~~ej__ + .E(e')) dv, (301.11) since .E(e) isahomogeneaus quadratic form in the components e!,. Now J •m aE(e) d J , tkmd e k aezo V= U(k,m) V, " " (301.12) = ~ u_; tf .. ) da = 0, 4 where we have used (301.6) and (205.2) and the fact that u' =0 on 11. Substitution of (301.12) into (301.11) yields J .E(e + e') dv = J .E(e) dv + J .E(e') dv. (301.13) This identity shows that the energy stored by a deformation corresponding to the vector sum of two displacements, one of which leads to an equilibrated elastic stress and both of which have the same values on the boundary, is the sum of the energies of the constituent displacements. Since the stored energy is assumed to be a positive definite form, the above-stated theorem of minimum energy follows immediately. The two theorems just proved are representative of the many that are known 2 in this classical subject, the theory of which has been brought to a state of analytical completeness second only to that of the theory of the potential 3• Having given some consideration to static theorems, we turn now to the pro· pagation of waves. For a body of continuous constant elasticity C, putting (301.1h into (205.2) yields (301.14) where we have supposed ef tobe continuous. In linear· elasticity theory we have [up,q,J =g~ g~ [xp,apl· By applying the general identities (190.1) and (190.2) for an acceleration wave, when the present configuration is taken as the initia1 one, from (301.14) we thus obtain or (301.15) (301.16) 1 KELVIN [1863, 2, § 62] took the assertion as "the elementary condition of stable equilibrium"; in this sense, that of a postulated variational principle, its history may be traced back to an idea of DANIEL BERNOULLI in respect to elastic bands (1738). As a proved theorem of linear three-dimensional elasticity, it seems first to have been given by LovE [1906, 5, § 119]. 2 A masterly exposition of some of them is given by LovE [1927, 6, Chap. VII]. 3 Despite this fact, there exists no general exposition of the theory from a rigorous mathematical standpoint. Sect. 302. The rotationally elastic aether. 727 Thus in order for an acceleration wave with normal n to exist and propagate, the jump s which it carries must be a proper vector of Ck mpqnqnm corresponding to the proper number e U2• For a body such that the work of the stress in any deformation is positive, as is the case for a hyperelastic body with positive definite stored energy, the tensor C k mpq nq nm is positive definite, its quadric being called Fresnel's ellipsoid for the direction n; therefore allproper numbers e U2 are positive, and therefore all possible speeds U are real. In the general case, then, in any linearly elastic body such that the work of the stress is positive for arbitrary deformations, a wave with given normal n may carry a discontinuity of the acceleration parallel to any one of three uniquely determined, mutually orthogonal directions, and corresponding to each of these directions there is a speed of Propagation determined uniquely by the elasticities of the material and by n. When the proper numbers e U2 are not distinct, the above conclusion must be modified, as is seen most easily by considering the isotropic case, for then (301.14) assumes the more special form [exk] = (J. +,u)[u~pk] +,u[ui.~p]. (301.17) so that for an acceleration wave we have {301.18) specializing (301.15). Taking the scalar and vector products of this equation by n yields {e U2 - (J. + 2,u)} s · n = o, } {e U2 - ,u} s x n = o. (301.19) If s . n =f= 0, the first equation yields e U2 = J. + 2,u, and the second, if we exclude the case when J. + ,u = 0, yields s X n = 0. If s · n = 0 but s X n =f= 0, the second equation yields e U2 =,u. Summarizing these results, we see that in an isotropic linearly elastic body for which A + ,u =f= 0, a necessary and sufficient condition that acceleration waves be propagated at positive speeds is A +2,u >0, ,u >O. This condition is satisfied when the stored energy is positive definite. Two kinds of acceleration waves are possible: longitudinal waves, whose speed of propagation is given by and transverse waves, for which U2 = Jl-.. e (301.20) (301.21) In view of the kinematical interpretation furnished by HADAMARD's theorem in Sect.190, the longitudinal waves are called expansion waves or irrotational waves, while the transverse waves are called equivoluminal waves or shear waves. The foregoing results, which are due to CHRISTOFFEL and HUGONIOT1, illustrate the far-reaching effect of isotropy: instead of three speeds of propagation, for an isotropic body there are only two, but instead of there being only three possible directions for the discontinuity, there are infinitely many, though the possible directions are still far from arbitrary. 302. The rotationally elastic aether. The quest for a mechanical theory of light as a vibration of an elastic medium attracted the attention of many of the 1 CHRISTOFFEL [1877, 3] obtained really all of the above results and more, but he did not present them very clearly, nor did he recognize as such the isotropic case, for which HuGONIOT [1886, 3] gave a very simple treatment. Our proof is essentially that of HADAMARD [ 1903, 11, ~~ 260, 267- 268] and DuHEM [ 1904, 1, Part IV, Chap. I, § V]. 728 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 302. illustrious physicists of the nineteenth century1• The most successful of these mechanical theories as applied to the deduction of the laws of transmission, reflection, and refraction of light in transparent media was devised by MAcCuLLAGH in 1839 2• Some thirty years after the publication of MAcCULLAGH's theory, FITZGERALD 3 applied the new electromagnetic theory of MAXWELL to derive the laws of reflection and refraction of light at the interface between two dissimilar dielectric media. In this memoir, FITZGERALD called attention to the formal analogy between the equations of the new theory and the old equations of MAcCuLLAGH's theory. This seems remarkable, for the physical ideas underlying the two theories are totally different. The constitutive equations of MAcCuLLAGH'S "rotationally elastic aether" do not conform to the principle of material indifference known to be satisfied by ordinary elastic materials. In all other regards, however, the theory is based on the usual equations of mechanics, specialized somewhat by linearization. a.) MacCullagh's equations and boundary conditions. As a starting point we consider the equations of motion (205.2) and boundary conditions (205.5) at a stationary surface of discontinuity which is not a shock: (302.1) Now contrary to the theory of ordinary elastic materials, where the stress arises solely in response to changes in shape, MAcCULLAGH supposed the aether to be a medium in which the stress, while completely insensitive to changes in shape, arises only in response to rotations about its relaxed state. This implies the existence of a preferred dass of reference frames. We may regard these preferred frames as the inertial frames of classical mechanics. The aether in its relaxed state will be at rest or in uniform translatory motion with respect to one of these frames. Let uk(~. t) denote the displacement vector of the medium taken in this sense. Then the quantities w .. = U[, s] = -l (u, s- U5 ,) ' ' ' (302.2) measure an infinitesimal rotation of the medium (cf. Sect. 57). Thus MAcCULLAGH assumed that for small rotations of the aether medium the stress is given by (302.3) Although MAcCULLAGH considered the more difficult case of crystalline media, we shall here treat only the isotropic case, where A••Pq is an isotropic tensor. It follows that, in this case, the relations (302.3) reducc to t, 5 = Kw, 5 • (302.4) The constant K is the gyrostatic rigidity. The aether is assumed to pervade all ordinary material media and to have the same density ein all materials. However, the gyrostatic rigidity of the aether is assumed to have a different value in materials with different indices of refraction. Thus at the interface between two dissimilar isotropic media ofthissimple type we have [e] =0, [K] =1=0. Moreover, at such an interface, the displacement u is assumed continuous, [u] =0. Thus it follows from the results of Sect. 175 that if u is differentiable in each of the adjoining media with differentiable limit values for aujat and grad u on each side of the 1 WHITTAKER [1951, 39] has given us a detailed and fascinating account of the evolution of these theories and their interrelations. 2 [1848, 1]. 3 [1880, 8]. Sect. 302. The rotationally elastic aether. 729 interface, then [ ~7] = 0, [curl u]. n = 0. (302.5) Substituting the constitutive relation (302.4) into (302.1), linearizing the acceleration with respect to u and its derivatives, and collecting our assumptions thus far, we have K curlcurl u + e a;;- = o, I [K curl u] xn = o, (302.6) [e ~] =0, [curlu] ·n =0. These are the basic equations of MAcCULLAGH's theory of light. ß) Fitzgerald's analogy. MAXWELL's constitutive equations for a non-magnetic, linear, rigid, homogeneous, isotropic stationary dielectric were discussed briefly in Sect. 283 and are treated in detail in Sect. 308. For such media we have ~=eE, l "= ~B (302.7) '!I' /1-o ' where the dielectric constant e is a measure of the ease of electric polarization of the medium. If we substitute these relations into the general electromagnetic equations (278.2), (278.3), (278.8), (278.9), and (283.42) to (283.45), we obtain the system aB curlE + -ät = 0, div B = 0, ) ( 302_8) [E]Xn=O, [B] =0, [eE]·n=O, 1 aE - - curl B - e- = 0 div E = 0. /1-o ot ' (302.9) These equations, generalized to the case of crystalline media, formed the basis of FrTZGERALD's derivation of the laws of reflection and refraction of light from the Maxwell theory. We see that, leaving aside the question of physical interpretation of the symbols, if we make the substitutions Kcurlu~oc.E, e ~7 ~oc.B, ) (302.10) K ~ßfe, (! ~ßfto• in (302.6), we obtain (302.8) and (302.9). The quantities cx and ß in (302.10) are arbitrary non-vanishing constants. The last two Maxwell equations (302.9) are a consequence of the identities div curl u = 0, curl ~ - -fft curl u = o. This mathematical equivalence between the electromagnetic equations in ideal media characterized by the constitutive equations (302.7) and MAcCuLLAGH's equations was first perceived by FrTZGERALD1• As remarked by WHITTAKER2, " ••• there can be no doubt that MAcCuLLAGH really solved the problern of devising a medium whose vibrations, calculated 1 [1880, 8]. The analogy between MAcCuLLAGH's equations and MAXWELL's equations based on the replacements (302.10) is discussed briefly by SoMMERFELD in his lectures [1947. 14]. He also considers a second set of replacements which renders the equations equivalent. HEAVISIDE [1893. 4, Chap. 111] gives an elaborate account of electro-mechanical analogies based on the Maxwell equations of dielectrics, conductors, etc., and the mechanical equations of elastic solids, viscous fluids, etc. 2 [1951. 39, p. 144]. 730 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 303. in accordance with the correct laws of dynamics, should have the same properties as the vibrations of light". 303. Finitely elastic bodies. The classical theory of finite elastic strain generalizes the concepts of linear elasticity by considering a body in which the stress arises solely in response to the difference of the present shape from that in an unstressed "natural state". An apparently still more general notion is included in the constitutive equation tkm = fkm(X'L,a.), {303.1) where X'l,a.=oxqjoXa. and where ~=~(X) is the deformation from the natural state to the present configuration. The prominence of the natural state is extreme: neither any intermediate state nor any event in the stress and strain history need be considered in ascertaining the stress. We may say that the body exhibits a perfect memory of its natural state and is entirely oblivious to every other except the present one. The principle of material indifference (Sect. 293 0) makes it possible to reduce (303.1) to the form1 T=f(C), (303.2) where C is the deformation tensor defined by (26.2) and where T is ProLA's stress tensor, connected with t through {210.9). Although both C and T transform as double tensors under time-independent changes of material and spatial co-ordinates, they do not transform as tensors under change of frame. Thus the principle of material indifference does not force the relation {303.2) tobe an isotropic one; indeed, any relation of the form (303.2) satisfies that principle. Herein lies the explanation of why the classical theory of elasticity includes anisotropic as weil as isotropic behavior, in contrast to the theories of fluids described in Sects. 298 and 299. An elastic body is isotropic if (303.2) reduces to an isotropic relation. The principles given in Sect. 32 suffice to show that in this case, alternatively, t may be regarded as an isotropic function of c, allowing a statement of the law of elasticity in terms of spatial tensors only2 : t = N0 1 + N1 c + N2 c2 (303-3) where the coefficients Nr and !:l.r are scalar functions of c, and hence may be taken as functions of Ic, Ilc, IIIc or as functions of Ic, Ic-1 and IIIc or IIIc-1, etc. Thus far we have followed CAUCHY's concept of elasticity. GREEN's concept, which may be stated exactly as in Sect. 301, Ieads to a more restricted theory, as we shall see now. The basic assumption of finite hyperelasticity is that there exists a stored energy E, a scalar function of the material co-ordinates X and of the deformation gradients x". a., such that all the work of the stress is recoverable. We have then, by hypothesis, JLi=t"md !!o km• (303.4) where the inessential factor eleo is added for conformity with standard usage. The principle of material indifference requires that in fact E = E (X, C). A classical argument, which has been given in three different forms in Sects. 218, 232A, 1 NoLL [1955. 18, § 15a]. A weaker theorem had been proved under stronger hypotheses by earlier writers. 2 REINER [1948, 23, §§ 1-2]. Sect. 304. Hypo-elasticity. and 256A, enables us to conclude thatl T,cx- ol: "'- ax"' • ,cx etc. These stress-strain relations furnish a special case of (303.2). 731 (303.5) Within hyperelasticity, a body is isotropic if its stored energy is an isotropic function of C. From the fundamentallemma in Sect. 30, it follows that, alternatively, 1: =l:(c) =l:(Ic-'• Ilc-'• Illc-,)· It is easy to show by using (App. 38.16) that for isotropic bodies any one of Eqs. (303.5) reduces to the form given by FINGER2 : t!, = ~: [ (nc-, a:~_, + Illc-' 81~~-~) <5!, + ) a.1: -1,. al: ,. ] + 8Ic-' Cm- Illc-, olle-' cm • (303.6) Equivalently, the principal axes of stress coincide with the principal axes of strain at z, and the principal stresses ta are related to the principal stretches Äa as follows 3 : a =1,2,3. (303.7) The definition (303.1) is incomplete in that the dimensional moduli are not specified; it should be amplified to read where phys. dim. p. = [M L -1 r-z]. (303.8) (303.9) Thus, as in the theory of perfectly plastic solids, there is but a single independent dimensional material constant, and it has the dimensions of stress. By the n-theorem we see at once that (303.8) must reduce to t"m/p. =h"m(xq,cx), and hence (303.2) may be replaced by T /i =g(C), (303.10) where g is dimensionless. Corresponding modifications are easily made in all equations of the theory. After a quiescence of half a century, the theory of finite elastic strain has undergone remarkable development in the past decade. See "The Non-linear Field Theories of Mechanics" in Vol. VIII, Part 2 of this Encyclopedia. 304. Hypo-elasticity. A different generalization of the classicallinear theory of elasticity is obtained by regarding it as describing approximately a material not necessarily having any natural state but rather experiencing a stress increment arising in response to the rate of strain from the immediately preceding state. Thus no finite memory is ascribed to the material. From the results at the beginning of Sect. 95 it is plain that the appropriate tensor measuring the rate of strain is the stretching, d. From the principle of material indifference in Sect. 293 (} we see that one of the various time fluxes introduced in Sects. 149 to 151 may be used 1 The argument is due in principle to GREEN [1839, 1, p. 249] [1841, 2, pp. 295-296] whose analysiswas corrected by KELVIN [1863, 2, §§ 51- 57]. All essential arguments occur in the treatment of a special case by KIRCHHOFF [1852, 1, pp. 770-772]. The many known equivalent forms are summarized by TRUESDELL [1952, 21, §§ 39-40]. 2 [1894, 4, Eq. (35)]. 3 KöTTER [1910, 6, § 1)], ALMANSI [1911, 1, § 7]. 732 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 304. to measure the rate of stress in a properly invariant way, and that the constitutive equations must be sufficiently general as to render immaterial the choice of a particular flux. We select the co-rotational flux drtfdt as defined by (148.7); allowing the magnitude of the present stress t to moderate the response of the material to stretching, we write the constitutive equation in the form dtt = j(t, d, p,), (304.1) where f-l = const and phys. dim. f-l = [M L -1 7-2]. (304.2) This last requirement asserts that all material moduli shall be elasticities, just as in the theories of linear or finite elasticity or of perfectly plastic materials. Fluid behavior, to the extent that it is accompanied by effects of viscosity or relaxation, is thus explicitly and intentionally excluded. (Effects of temperature differences are easily taken into account if desired but are here omitted for simplicity.) Application of dimensional analysis shows that t may enter (304.1) only in the ratio lff-l and that f, assumed continuous at d =0, must be linear in d: drskm =KkmPqd skm:==tkmj2u, dt Pq• r (304.3) where K is a function of s. Since drsfdt and d are both tensors under change of frame (Sects. 144 and 148), K must alsobesuch a tensor; consequently K is an isotropic tensor function of s. By a representation theorem due to RIVLIN and ERICKSEN1 it follows that the most general constitutive equation satisfying the hypotheses (304.1) and (304.2) is where dt = N0 I.t 1 + N 1 d + N2 I.t s + N3 M 1 + l t + t N4 (d s + s d) + N5 I.t s 2 + N6 M s + + N7N1+!N8 (ds2 +s2 d) +N9 Ms2 + + N10 Ns + N11 Ns 2, M - s~ dk', N s~ srp dt Nr = Nr (18 , II8 , III8 ) F= 1, 2, ... 11. (304.4) (304.5) (304.6) The theory based upon these constitutive equations is called hypo-elasticity 2• In keeping with the expressed aim of elastic rather than viscous response, Eqs. (304.4) areinvariant under a change of the time scale. While the stress at a given time generally depends upon the manner in which the load has been applied during previous instants, it is thus independent of the actual speed of deformation. The exact theory is a fully dynamical one. However, Eqs. (304.4) reduce to those of the linear theory of elasticity (Sect. 301) under the assumptions usually made in formulating that theory. Moreover, every isotropic elastic body is also hypoelastic 3. This result, however, must not be regarded as reducing hypo-elasticity to elasticity, for the converse is generally false, and in particular the simpler special cases of (304.4) are not elastic cases. Finally, it has been shown that most of the common "incremental" theories of plasticity other than that of 1 [1955, 21, §§ 40 and 33]. A result of the sameform had been deduced from a too restrictive definition by TRUESDELL [1951, 27, § 26]. 2 TRUESDELL [1953, 32, §56 (revised)J [1955, 27 and 28]. 3 NoLL [1955, 18, § 15b]. Sect. 305. Visco-elastic and accumulative theories. 73.3 perfectly plastic materials are included as special cases of hypo-elasticity, providing they be first corrected so as to satisfy the principle of material indifference1 . In much of the foregoing discussion, it has been assumed that the body is initially unstressed, but this assumption is not at all necessary in hypo-elasticity. Suppose that in an interval dt a body subject to initial stress s 0 undergoes a displacement having small gradients in the sense defined in Sect. 57. Then we have wk, dt R:j Rk, and dpq dt f'::j epq• where il and e are the tensors of infinitesimal rotation and strain for the displacement considered. Also skmdt R:j skm_ s~m. From (.304._3) and (148.7) we have then (.304.7) The essential content of these equations, which define a theory of small deformation of an initially stressed body, was given by CAUCHY 2• It reflects the fundamentally greater generality of hypo-elasticity in comparison to elasticity. While, as is plain from (303.6), the most general theory of finite elasticity is not capable of representing an elastically isotropic body which in its natural state suffers any but hydrostatic stress, in hypo-elasticity there is, in general, no natural state, and the stress at any given instant may be arbitrary. It was CAUCHY's theory of initially stressed bodies, a special case ofthat defined by (.304.7), that suggested the theory of hypo-elasticity. Hypo-elasticity may be regarded as a theory in which relations of CAUCHY's type are applied in each time interval dt. 305. Visco-elastic and accumulative theories. Studies of elasticity and viscosity, according to the classical theories presented in Sects. 298 and .301, made it natural to attempt to combine the two kinds of phenomena within a single theory. Two ideas immediately present themselves: 1. The total stress is the sum of an elastic stress arising from the strain and a viscous stress arising from the stretching, and 2. The total rate of strain is the sum of an elastic rate arising from the rate of stressing and a viscous rate arising from the stress. Schematically, the two alternatives may be written t=f(e) +g(e) and e =h(t) +k(t). (305.1) Both imply necessarily the existence of material constants having the physical dimensions of elasticity and of viscosity; hence there is a modulus having the physical dimensions of time, and relaxation effects may be expected. The former alternative was proposed by 0.-E. MEYER, VoiGT, and DUHEM 3; the latter, by MAXWELL, NATANSON, and ZAREMBA 4. Theoriesofthis type, including generalizations obtained by allowing time derivatives up to arbitrarily high order to occur on the each side of (.3 0 5 .1 h or (3 0 5 .1) 2 , are called visco-elastic. If e is the infinitesimal strain tensor, the constitutive relations are n (o)k m ( 0 )k ~ Ak 8t t = J:o Bk 8t e, (305.2) 1 GREEN [1956, 9 and 10]. Perfectly plastic·materials are included.formally by a limit process. 2 [1829, 4, Eqs. (36), (37)]. 3 [1874. 3 and 5] [1875, 4]; [1889, 10] [1892, 12 and 13] [1910, 10, §§ 395-396]; [1903, 6-9] [1904, 1, Part I, Chap. II, and Part IV, Chaps. II-III]. 4 [1867, 2, pp. 30-31]; [1901, 11-13] [1902, 7-9] [1903. 12-14]; [1903, 17-20] [1937. 12]. 734 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 306. where the coefficients Ak and Bk are usually assumed constant. These theories possess a large recent literat~re, mostly in one-dimensional andjor linearized contexts. Such linear theories generally violate the principle of material indifference, as was first noticed by ZAREMBA, who was led to his form of the principle in this context. His is the only early attempt in viscoelasticity to yield a theory that is a mechanically possible one for unrestricted motions. A generaland properly invariant theory of the MEYER-VOIGT type, allowing the stress to depend upon the first spatial derivatives of the displacement and of the accelerations of all orders, has been achieved by RrvLIN and ERICKSEN1 . The Maxwell-Zaremba theory is generalized by NoLL's theory of hygrosteric materials 2, which are defined by (304.1) without requiring f to be linear in d, and by a more general theory sketched by CoTTER and RIVLIN 3• BoLTZMANN and VoLTERRA4 proposed a theory in which the stress is determined by the entire sequence of strains undergone by the body in the past. The material is represented as having a weaker memory for older experiences; the stress is obtained by integrating the strains from - oo to t, with a suitable damping or "memory" function. The linear theory, sometimes called "hereditary" but more fitly named accumulative, has a considerable literature. It is still more general to allow the stresstobe determined in any way by the strain history, i.e., 1 t = F [x~cx], (305.3) -oo where F denotes a functional. This extremely general and natural concept of material behavior has been put into properly invariant from by NoLL5 and by GREEN, RrVLIN, and SPENCER6 (1956). The latter authors determined conditions under which the functional in (305.3) may be replaced by a sum of integrals of VüLTERRA's type or by a finite combination of time derivatives as in the theory of RrvLIN and ERICKSEN. NoLL's method, which rests directly upon properties of invariance as the definitions of particular materials, is presented in "The Non-linear Field Theories of Mechanics ", This Encyclopedia, Vol. VIII, Part 2. V. Examples of thermo-mechanical constitutive equations. 306. Irreversible thermodynamics. While even the classical theory of isotropic viscous and thermally conducting fluids furnishes an example where both mechanical and energetic principles must be used in order to solve any definite problem, the constitutive equations themselves, namely, (296.1) and (298.5), embody separate principles, one being purely mechanical and the other, purely energetic. The various theories of "irreversible thermodynamics" attempt to describe the numerous physical phenomena which cannot be separated as belonging to one or the other category to the exclusion of the other. Such phenomena occur especially in heterogeneaus substances. Current practice concentrates upon the production of entropy owing to such interactions and supposes that the "affinities" and "fluxes" which enter into the production of entropy depend functionally, and usually linearly, upon one another. 1 [1955. 21]. 2 [1955. 18, §§ 6-9]. 3 [1955. 4]. 4 [1874, 1]; [1909, 10 and 11] [1930, 9]. 5 [1957. 11] [1958, 8]. 6 [1957. 6] [1959. 6]. Sect. 307. The "Maxwellian" fluid. 735 In Sects. 257 and 259 we have explained the concepts in terms of which such theories are constructed. Since we do not consider that the problern of dynamical and energetic invariance is yet sufficiently understood, we rest content with citing expositians of the subject as it is currently received 1. 307. The "Maxwellian" fluid. As our sole example of the use of the principle of equipresence (Sect. 29317), we mention a fully thermomechanical theory of fluids based on two constitutive assumptions: 1. Both the stress and the flux of energy depend upon the spatial and temporal derivatives of the thermodynamic state and of the velocity, of all orders. 2. The constitutive relations involve material coefficients having the physical dimensions of viscosity, thermal conductivity, and temperature, but no others. These constitutive relations define the M axwellian fluid of TRUESDELL 2• While this theory is known 3 to stand in need of revision so as to be rendered properly invariant, the general forms of the leading terms in the expansions of the stress and flux of energy in powers of the viscosity give an idea of what is tobe expected. From the first of the defining conditions, expressing an application of the principle of equipresence, it might be thought that the expansions for the stress t and flux of energy h would be very similar. However, the different tensorial characters of t and h, combined with requirements of dimensional invariance, force the the counterparts of terms present in the expansion of t to be absent from the expansion of h, and conversely. To avoid long formulae we summarize the forms of the terms of orders 0, 1, 2 in words. The stress t is the sum of TL The terms in the linear law of fluid viscosity (298.5). T2. The quadratic viscous terms according to the theory of RIVLIN and ERICKSEN mentioned in Sect. 305. T}. The mostgenerallinear isotropic function of P,k P,m· T4. The mostgenerallinear isotropic tensor function of P,(k(),ml· T 5. The most generallinear isotropic tensor function of (), k (), m. T6. The mostgenerallinear isotropic tensor function of P,km· T7. The mostgenerallinear isotropic tensor function of O,km· Similarly, the flux of energy h is the sum of H 1. A linear term, where IX is dimensionless, and H 2. The most general linear isotropic vector function of P,k' e,m' and Xp,q. H3. The mostgenerallinear isotropic vector function of ip,q,. (307.1) It is most striking that in these results the separation of effects follows from considerations of invariance alone. For example, despite the much more general definition of a fluid used here, the linear terms T 1 in the stress are exactly the same as occur both in the classical linear theory and in the simpler non-linear 1 PRIGOGINE [1947, 12], DE GROOT [1952, 3], J. MEIXNER and H. REIK in Vol. III, Part 2 of this Encyclopedia. 2 [1949, 34, §§ 2-4) [1951, 27, §§ 19-21). We have altered slightly TRUESDELL'S definition and results. 3 Cf. the remarks of TRUESDELL [1956, 13, § 16). 736 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 308. "Stokesian" theory (Sect. 299). While FouRIER's law (296.4) is slightly generalized by (307.1), the added term is of a thermodynamic type, parallel to the classical thermal one, and in the linear approximation the effects of deformation cannot enter the constitutive equation for h. Similar, though more elaborate, separations of effects are seen in the quadratic terms T2 to T7 and H2 to H3. These quadratic terms allow for interactions of a definite, not arbitrary kind. Restrietions such as these, which follow from principles of invariance alone, are only now coming to be studied. VI. Electromagnetic constitutive equations. 308. The Maxwellian dielectric. The problern of formulating constitutive equations for moving and deforming material media is one of the most difficult and controversial in electromagnetic theory. We shall illustrate some of the relevant ideas by treating a simple example. rx.) The Euclidean invariant constitutive equations of a Maxwellian dielectric. In classical electromagnetic theory as refined by LoRENTZ we have the aether relations (cf. Sect. 279), which can be written in the 4-dimensional tensorform r;nd = 1 ~ ( _ det y)~ yll'Fyde CfJ'I'e V~ or the 3-dimensional vector form H = _!_B. f.lo D =e0 E, (308.1) (308.2) According to the Lorentz point of view, these relations hold in all material media, moving or stationary, but it must be recalled that D and H are the charge and current potentials for the total charge including that due to polarization and magnetization of the material medium. We define the ideal material called a Maxwellian dielectric by the relations P=e0 xE, M=O X= const } (308.3) for the polarization P and the magnetization M, provided the medium is at rest in a Euclidean frame which is simultaneously a Lorentz frame. Recall that such a frame is one for which we have the canonical forms (cf. Sects. 280 and 152) _ 1= [<5rs 0 l = [<5rs 01 1 Y 0 _ ~ , g 0 0 , t = (0, 0, 0, 1), c2 =-. (308.4) ~ ~~ If we introduce the potential T= (~, ~) of free charge and current (cf. Sect. 283) in a polarizable and magnetizable medium, the constitutive relations (308.3) can be put in the 4-dimensional form (308.5) with -)(1 = [ 15,5 0 1' 0 - Bflo e = e0 K, K = 1 +X (308.6) or in the 3-dimensional vector form ~=eE, 1 ~=-B. f.lo (308.7) Sect. 308. The Maxwellian dielectric. 737 To see how these relations are generalized to the case of moving media, it is easiest to employ the world tensor formalism of Sect. 152 as applied to the problern of constructing invariants of fields under the Euclidean group of transformations (rigid motions). It was noted in Sect.152 that if lJI"was a world tensor satisfying the conditions (308.8) then the non-vanishing components prs. · · in a Euclidean frame transform as a 3-dimensional tensor under the group of Euclidean transformations. This special kind of world tensor was called a space tensor. Consider then the electromagnetic field cp and the world velocity vector v of a motion. In terms of these quantities we can define two associated space tensors (i;!.l = gD-1 f/J-1s vS' 58!.1-1 =g!.I'Pg-1Sf/J'l'S· In a Euclidean frame we have (274.1), so that, in such a frame, i = (E + vxB, 0), !8 = (dualB, 0). (308.9} (308.10} (308.11) Consider next the polarization-magnetization world tensor :rt defined in (283.23). In terms of this tensor we defined the space tensors 1.)3!.1 and 9Jl!.l -1 called the world polarization and magnetization densities in (283.24) and (183.25). In a Euclidean frame we have !l3 = (P, 0), IJJl =(dual M, 0). (308.12) Therefore, the world tensor equations 1.)3D = eo Vg X (i;!.l, 9JlDLI = 0, (308.13) reduce to (308.3) when the medium is at rest in a Euclidean frame. Moreover, since these equations involve only space tensors, they are rotationally invariant. Stated more explicitly, the proportionality of polarization and electromotive intensity and the vanishing of the magnetization are invariant conditions under the group of Euclidean transformations relating the dass of reference frames moving rigidly with respect to one another. Now from (283.33), (283.24), (283.25), (279.13), and the aether relations (308.1) we have yn.1 = "1!.1.1 _ ;n;D.1, = 'V*(-dety)![y!.l'~'y-1 q;'l'fJ- ~g-1'Pv!.lvSq;'Ps-J (308.14) - _K_gnSv-1v'P m ] c2 r'PS • or, in a frame for which we have the canonical forms (308.4), ~ = e0 [E +X (E + v X B], l ~ = - 1 [B + ~ v X (E + v X B)]. 1-'o c (308.15) We have arrived at these constitutive equations for the potentials ~ and ~ by demanding that the relations between polarization and magnetization be invariant under the group of rigid (Euclidean) transformations. We are led to this idea of invariance by thinking of polarization and magnetization as quantities characteristic of the moving material medium and carried with it. Handbuch der Physik, Bd. 111/t. 4 7 738 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 308. Thus, on the basis of classical thought, any relation betwcen them and the electromagnetic field should be invariant under the group of rigid motions. This is an application of the principle of material indifference. We wish now to cantrast with the results (308.14) and (308.16) obtained by this classical procedure the corresponding results obtained by MINKOWSKI 1 , whose reasoning was guided by the idea of Lorentz invariance of the constitutive relations in moving media. ß) The Lorentz-invariant constitutive equations for a moving M axwellian dielectric. Suppose a Maxwellian dielectric is at rest in some Lorentz frame which, for our present purpose, we may imagine as coinciding with some Euclidean frame. Recall that the dass of Lorentz frames are those for which the tensor y has the canonical form (308.4)1 . Let us assume that, when the medium is at rest, it is characterized by the relations (308.5). Now suppose that such a medium is set into motion. Any selected particle of the medium undergoing the motion will be instantaneously at rest in some other Lorentz frame. MINKOWSKI reasoned that the relations (308.5) should hold in that Lorentz frame as a consequence of the correct constitutive equations for the moving medium. To put this idea into effect, let LD .1 be the coefficients of a Lorentz transformation ( cf. Sect. 282), so that by definition we have -1 -1 LD.-1 L'~'e yL1e = yD'P. (308.16) Let wD be the relativistic velocity vector of the motion defined by c vD w0 == ~· , Yn.-1 wDw.-1 =- c2 , (308.17) - Y~e v~ve where, as before, v is the classical world velocity vector of thc motion. In the reference framein which the medium is moving we have w-( v 1 ) (308.18) - v1- ~ v1 - v2 • c2 c2 We can choose the coefficients LD.-1 of a Lorentz transformation so that has the form w = (0, 0, 0, 1) (308.19) (308.20) at some selected event. If the medium is in uniform translatory motion, so that v =const, then w will have the form (308.20) throughout a space-time region, but, for general motions, we shall have (308.20) only at a single event. Let if'DL1 and (jjDL1 denote the transformed components of r and Cf!· That is, if'DL1 =LD'PLL1e T'~'e, fPnA = L'Pn Le.-1 CfJ'Pe· (308.21) Thus, in the new coordinate system, MINKOWSKI assumes that -!J.-j_,;~( d t-)1-!JljF-Ltf')- - V f.lo - e """ " CfJ'P&• (308.22) where x has rest values consistent with (308.6). Applying the inverse Lorentz -1 -1 transformation LeA (LD.-1 LL1'P= bfl,) to (308.22) we see that the relation between I [1910, 7). Sect. 308. The Maxwellian dielectric. the unbarred components of r and gJ must be TQLJ = v-~- (- det x)~ XQ 'P xL1 e IP'Pe' flo where, for the moving medium, x!JLl = L~'PLLJ e x_'Pe. 739 (308.23) (308.24) To compute the values of the unbarred components x!JLl it is convenient to decompose x as follows: X,!JLl = yQLl- f-lo (e- eo) w!J wLl' l =yQLl _ Lw!JwLl. c2 It then follows easily from (308.16) and (308.19) that (308.25) (308.26) Substituting this result into (308.23), we get the Minkowski relations in the form T!JLJ =V;:(- dety)~ x I X [JJQ'PyLJe IP'Pe -1dYQ'P wLl we IP'Pe- "J}Ll'P w!2we IP'Pe)] or, in terms of ~ and ~. ~= e0 [E + x 2 (E + vxB)- ~-x- -vv ·E], 1 - !____ c2 (1 - !____) c2 c2 (308.27) (308.28) Eqs. (308.28) are to be compared with the classical expressions (308.15). We see that if one neglects all terms 0 (v2jc2) in the Minkowski constitutive equations, they reduce to the classical ones. MINKOWSKI's method of deriving Lorentzinvariant electromagnetic constitutive relations was extended to the case of moving crystals by EINSTEIN and LAUB1 and by BATEMAN 2• y) M oving surfaces of discontinuity, the velocity of light, and Fresnel' s dragging formula. If we substitute the constitutive relations (308.15) into the electromagnetic boundary conditions (278.8), (278.9) and (283.44), (283.45) at a moving surface of discontinuity, we obtain the following system of equations: nx[EJ-un[B]=O, [B]·n=O, [E+x(E+vxB)]·n=O,l (308.29) nx [B + 4-vx(E+vxB)] + 4-[E + x(E + vxB)]=O. c c The second and third of these conditions are satisfied as a consequence of the first and the fourth provided we assume un =f= 0. Let us suppose that the velocity v and the polarizability X of the dielectric medium are continuous. In this case, the latter two vector equations constitute a system of six homogeneaus linear 1 [1908, 3]. 2 [1922, 1]. 47* 740 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 309. equations in the six quantities E, and B'. Therefore, the system admits nonzero solutions for the jumps [E] and [B] if and only if the determinant of the coefficients vanishes. To simplify matters, let us consider the case where the normal n to the surface of discontinuity is parallel to the velocity of the medium v. In this special case, the vanishing of the determinant requires that the speed of propagation un satisfy the quadratic equation K (u")2- ~ (11..!!.)' + ~~ -1 =0. c c c c2 (308.30) The solution of this equation is x-± V v--V2 K-x-- c c2 -------- K (308.31) N ow when the polarizability of the medium vanishes (x = 0, K = 1) the speed of propagation of the surface of discontinuity is ± c. Thus the fundamental constant c == V1/e~.U~ is the speed of propagation of an electromagnetic discontinuity in vacuum or any medium in which the polarization and magnetization vanish and there is no free charge or current. We see that the motion of such a medium devoid of polarization has no effect on this result. Now set n = VK, where n is the index of refraction. We can then write the solution (308.31) in the form Un = _1_ [± 1 + ~_11_ + 0 (v2jc2)J]' c n n c {308.32) which, when the terms 0 (v2fc2) are neglected, reduces to FRESNEL's dassie formula 1 for the dragging of light by a moving polarizable medium. If one substitutes MINKOWSKI's constitutive relations into the jump conditions, the equation analogaus to (308.30) is (308.33) Again we see that neglecting the terms 0 (v2fc 2) leads to FRESNEL's result. However, the two equations for the determination of the speed of an electromagnetic discontinuity in a moving Maxwellian dielectric based on the classical and Minkowski constitutive equations of the moving medium differ by terms 0 (v2jc2). 309. VoLTERRA's electromagnetic constitutive equations. We next mention VoLTERRA's generalization of MAXWELL's constitutive relations for ~ and ~ in stationary dielectric and magnets 2• In mostreal materials, the magnetization is not a single-valued function of the magnetic flux B, much less a linear function. The magnetization at any instant, however, can be considered as a functional of the history of the field B (cf. Sect. 305). If the "heredity" is linear, VaLTERRA writes 00 B(~.t) =ft~(x,t) + Jif>(-r)~(~.t--r)d-r, (309.1) 0 where if>(-r) is the coefficient of heredity. When constitutive equations of this type are substituted into the differential field equations obtained from the conservation laws, one obtains a system of integro-differential equations which govern the evolution of the physical system rather than a system of partial differential equations as in the simpler theories. I WHITTAKER [1951, 39, p. 403]. 2 [1912, 7], [1930, 9, p. 195]. Sects. 310,311. ÜHM's law for moving conductors. 741 310. MIE's theory. The role of the potential fields fj and a in the classical theory is rather odd and asymmetrical. We see that the electromagnetic potential a is but a subsidiary field which is generally introduced in order to facilitate the solution of problems. The whole system of electromagnetic equations is invariant under potential transformations of the electromagnetic potential (gauge transformations). On the other hand, the charge-current potential fj plays a more fundamental role in the classical theory because of the Maxwell-Lorentz aether relations, which are not invariant under potential transformations of f/· In MIE's theory1, the potentials a and fj are made to enter the theory in a more symmetrical manner. In addition to the conservation laws of charge and magnetic flux, MIE assumes the existence of a "universal" function A such that a oA a = aoc!i. (310.1) The function A is supposed to be a Lorentz-invariant function of the electromagrtetic field (/! and the electromagnetic potential a. MIE's theory is concerned with the fundamental question of the electromagnetic constitution of matter. VAN DANTZIG 2 proposed a theory somewhat similar to MIE's in which the potentials fj and a were assumed tobe linear fundionals of the fields (/! and lJ. We cite these examples of constitutive relations to illustrate the variety of viewpoints which have been expressed as regards the appropriate constitutive relations to accompany the conservation laws of electromagnetic theory. 311. OnM's law for moving conductors. We consider the dass of ideal materials such that, when they are at rest in a Euclidean frame, the current J is a linear isotropic function of the electric field E: J=CE, C = const. (311.1) This relation is called Ohm's law. We generalize ÜHM's law to the case of a moving medium in much the same way as the constitutive equations of a Maxwellian dielectric were generalized in Sect. 308. First we consider the classical or rotationally invariant generalization. In terms of the charge-current vector (J and the velocity vector v of a motion, we define two space tensors :O==anta, } ~n =an- vn aLl tLl. (311.2) In a Euclidean frame we have :0 = Q, ~ = (~, 0)' (311.3) where Q is the charge density and ~ is the conduction current. MAXWELL's generalization 3 of ÜHM's law to the case of a moving medium can then be stated in the form (311.4) Since ~ and ~ are space tensors, the 4-dimensional formalism insures that the constitutive equation (311.4) is rotationally invariant. In a Euclidean frame it assumes the form ~ =C~, } J- Q v = C (E + V X B), (311. 5) 1 [1912, 6]. A summary of Mm's theory is given by WEYL [1921, 6, § 26] [1950, 3.5, § 28]. 2 [1934, 10]. 3 [1873. 5, §609]. 742 C. TRUESDELL and R. TOUPIN: The Classical Field Theories. Sect. 312. where (.! is the electromotive intensity at a point moving with the medium. When the medium is at rest, (311.5) reduces to (311.1). MINKOWSKI's generalization of ÜHM's law is obtained by requiring (311.1) to hold in the Lorentz frame in which the particle of the moving medium is instantaneously at rest. Thus we set (j!i = LQLJ afl' l cpgLJ = L'~' D LeLJ T'Pe• -1 wD = (o, o, o, 1), wD = LD Ll wfl, (311.6) so that the relation (311.1) in the moving Lorentz frame can be put in the form (311.7) Applying the inverse Lorentz transformation, we then get Q + 1 'P e D- C D'P . e a (JY'Pea w w - y T'Pflw . (311.8) The spatial component of this last equation is J + _1_ ('J · v- c2 Q) V = C (E + v X B) c2 1 - v2(c2 1 - v2(c2 . (311.9) Taking the scalar product of (311.9) by v, we obtain an equation which can be solved for J · v: J · v = C V1= v2jc2 v . E + v2 Q. (311.10) We then eliminate J · t' from (311.9) and obtain finally J-Qv=. C (E+vxB-~-VV·E), Vt-v2fc2 c (311.11) which is the relativistic form of ÜHM's law for moving media. Again we see that the relativistic or Lorentz-invariant constitutive equation (311.11) differs from the classical or Euclidean-invariant constitutive equation (311.5) only by terms 0 (v2jc2). VII. Electromechanical constitutive equations. 312. Elastic dielectrics. The phenomena of piezoelectricity, photoelasticity, and electrostriction in elastic solids are closely related. Because of their importance in engineering applications, the classical theories for these effects have become highly specialized disciplines 1. Any such theory must be based on simultaneaus application of the principles of mechanics and of electromagnetism. The laws of conservation of energy and momentum, agumented so as to include the effects of the electromagnetic field, have been formulated in Chap. F. The relevant equations of mechanics, aside from the boundary conditions, are conveniently summarized in the set of Eqs. (288.11) to (288.13) and (286.14). For dielectric media in the absence of free charge and current and of magnetization, the charge and current densities occurring in (288.11) and (288.12) are expressed in terms of the polarization Pas in (283.22). If these expressions for the charge 1 As sources of experimental and theoretical results and references to original and contemporary Iiterature in this field we may cite the following works: VorGT [1910, 10], MASON [1950, 17], CADY [1946, 1], CoKER and FrLON [1931, 3], STRATTON [1941, 8]. Sect. 312. Elastic dielectrics. 743 and currcnt are substituted into (288.11) and (288.12) we obtain the basic field equations of mechanics as applied to the case of dielectric media: (! x' = trs - ~ div P (§;' + -1- (~!'I>_ X B)' ,s yg yg dt ' • • 1 dc P a:. d" h (!e =t'sxs,, + Vfat. ~ + IV , (312.1) 0(! ( "r) - Tt + QX ,r- 0, tfrs]=Q. We have written these equations as they appear in a general curvilinear inertial co-ordinate system. Under general time-independent transformations of the spatial Co-ordinates xk, pr and ßr transform as vector densities of weight 1. The quantities (!, e, t' 5 , h' and E, transform as absolute tensors. A comma denotes, as usual, covariant differentiation based on the ChristoHel symbols of the metric tensor g,,. Eqs. (312.1) are supplemented by the conservation laws of charge and magnetic flux and by the aether relations of electromagnetic theory. The classical theory of piezoelectricity is based on the linearized version of this system of equations corresponding to infinitesimal deformations and weak fields and the linear piezoelectric constitutive equations of VorGT, which can be expressed in the form t's = crsmn e + _1 mn yg yrs m pm ' l 1 -1 E, = yg Xrs ps +rmn,emn• (312.2) where e,s =u(r,s> (u =displacement vector) is the classical measure of infinitesimal strain. Because of the relation D = e0E +P, the Voigt relations (312.2) can be written in various forms corresponding to different choices of independent variables. A thermodynamic treatment of the Voigt relations can be based on the energy equation (312.1) 2 by making the assumptions necessary to yield the equation e ()ij = div h, (312.3) where r; is the entropy density and () is the absolute temperature. Then by assuming that the internal energy s, the stresst, the electric field E, and the temperature () are functions of the infinitesimal strain measure e, the polarization P, and the entropy r;, we obtain VorGT's relations in the linear approximation if the energy equation is assumed to be satisfied identically in the independent variables e, P, and r;. The coefficients in the Voigt relations are then given by mn - lfa oe r s - e v g a;--a:ps . mn -1 oe X - ng--- (312.4) rs- o: oP•(!ps . The classical linear theory of piezoelectricity outlined above has been generalized to the case of finite deformation and large field strengths by ToUPIN1. The non-linear theory stands in the same relation to the Voigt theory as the theory of finite elastic deformations stands in relation to linear elasticity theory. In the non-linear theory, the internal energy e is assumed initially to be a general polynomial function of the displacement gradients xi;.l of a deformation, the 1 [ 19 56, 20]. 744 C. TRUESDELL and R. TouPIN: The Classical Field Theories. Sect. 313. polarization P, and the entropy density 'YJ· One then shows that if the energy is invariant under the group of rigid motions, it must reduce to a function of the variables 1]. (312. 5) Assuming that (312.3) holds for this ideal medium, that the electromotive intensity, the stress, and the absolute temperature are functions of the displacement gradients, polarization, and entropy, and that the energy equation is satisfied identically yields the constitutive relations of the general theory: OE trs = 2 (! --X' X5 = tsr ()CIJV ;IJ ;v ' (!)r- "'-~X"' i}I/IJ ;r• (312.6) 0=~. OTJ Theserelations generalize the Voigt relations (312.2) to the case of finite deformations, large field strengths, and moving media. They reduce to the Voigt relations in the linear approximation and for stationary media. 313. Magnetohydrodynamics. Electrodynamics of continuous media is currently enjoying a new birth of interest. This contemporary work is generally classified under the title, magnetohydrodynamics. The abundance of theoretical and experimental labor now directed toward the novel behavior of conducting fluids moving in a magnetic field should yield progress toward an understanding of the general theory of electrodynamics of continuous media. In the present theories of magnetohydrodynamics, the constitutive relations for the stress generally take the form (313.1) as in the classical theory of viscous fluids. ÜHM's law for moving media (311.5) is generally assumed and a simple type of constitutive relation such as M = kB is introduced to account for the effects of magnetization. The general equations of magnetohydrodynamics are obtained by substituting these rather special constitutive relations into (288.11) and augmenting this set of equations by the equations of conservation of charge and magnetic flux. The linear magnetohydrodynamic equations can then be obtained by casting away the non-linear terms in the general equations. 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EuLER, L.: Theoria Motus Corporum Solidarum seu Rigiderum ex Primis nostrae Cognitionis Principiis Stabilita et ad Omnis Motus, qui in hujusmodi Corpora Cadere Possunt, Accomodata. Rosteck = Opera omnia (2) 3 and 4, 3-293. (165, 168, 196, 294) 746 2. 3. 1766 1. 1767 1. 1768 1. 2. 1769 1. 1770 1. 1771 1. 2. 1776 1. 2. 3. 4. 1783 1. 1786 1. 1788 1. 1806 1. 1821 1. 1822 1. 2. 1823 1. 2. 3. 1824 1. 1825 1. 2. 1827 1. C. TRUESDELL and R. TouPIN: The Classical Field Theories. EuLER, L.: Recherehes sur Ia connaissance mecanique des corps. Mem. Acad. Sei. Berlin [14] (1758), 131-153. (165, 168) EuLER, L.: Du mouvement de rotation des corps solides autour d'un axe variable. Mem. Acad. Sei. Berlin [14] (1758), 154-193. (143, 170) EuLER, L.: Supplement aux recherches sur Ia propagation du son. Mem. Acad. Sei. Berlin [15] (1759), 210-240 = Opera omnia (3) 1, 452-483. (17, 20, 66A, 156) EuLER, L.: Recherehes sur le mouvement des rivieres (1751). Mem. Acad. Sei. 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Petrop. 15 (1770), 381-413 = Opera omnia (2) 11, 37-61. (200, 214) CouLOMB, C. A. : Essai sur une application des regles de maximis et minimis a quelques problemes de statique, relatifs a l'architecture. Mem. divers savants 7 (1773), 343-382. (200, 204) EULER, L.: Formulae generales pro translatione quacunque corporum rigidorum. Novi Comm. Acad. Sei. Petrop. 20 (1775), 189-207. (168, 196) EuLER, L.: Nova methodus motum corporum rigidorum determinandi. Novi Comm. Acad. Sei. Petrop. 20 (1775), 208-238. (170) EuLER, L.: De gemina methodo tarn aequilibrium quam motus corporum flexibilium determinandi et utriusque egregio consensu. Novi Comm. Acad. Sei. Petrop. 20 (1775), 286-303 = Opera omnia (2) 11, 180-193. (214) LAGRANGE, J. L.: Memoire sur Ia theorie du mouvement des fluides. Nouv. mem. Acad. Sei. Berlin (1781), 151-198 =Oeuvres 4, 695-748. (72, 74, 99, 134, 161, 163, 184, 206) CousiN, J. A. J.: Memoire contenarrt quelques remarques sur Ia theorie mathematique du mouvement des fluides. Mem. Acad. Sei. Paris (1783), 665-692. (72) LAGRANGE, J. L.: Mechanique Analitique. Paris. Oeuvres 11, 12, are the 5th ed. Our references are to the first edititon. (6, 15, 20, 72, 74, 99, 134, 196, 232) EULER, L. : Die Gesetze des Gleichgewichts und der Bewegung flüssiger Körper. Leipzig (being a transl., with some changes and additions, by H. W. BRANDES, of [1769, 1], [1770, 1], [1771, 1], and another paper). (108) NA VIER, C.-L.-M.-H.: Sur !es lois des mouvements des fluides, en ayant egard a l'adhesion des moh~cules. Ann. chimie 19, 244-260. (298) FouRIER, J.: Theorie Analytique de Ia Chaleur. Paris = Oeuvres 1. (241, 296) NA VIER, C.-L.-M.-H.: Sur ]es lois du mouvement des fluides, en ayant egard a l'adhesion de leurs molecules. Bull. Soc. Philomath. 7 5-79. (298) CAUCHY, A.-L.: Recherehes sur l'equilibre et le mouvement interieur des corps solides ou fluides, elastiques ou non elastiques. Bull. Soc. Philomath. 9-13 = Oeuvres (2) 2, 300-304. (7, 21, 26, 27, 82A, 200, 203, 298, 300, 301) CAUCHY, A.-L.: Memoire sur une espece particuliere de mouvement des fluides. J. Ecole Polytech. 12, cahier 19, 204-214 = Oeuvres (2) 1, 264-274. (77, 99) NA VIER, C.-L.-M.-H.: Sur !es lois de l'equilibre et du mouvement des corpssolides elastiques (1821). Bull. Soc. Philomath., 177-181. (Abstract of [1827, 7].) (301) CARNOT, S.: Reflexions sur Ia Puissance Motrice du Feu et sur !es Machirres Propres a Developper cette Puissance. Paris= Ann. Ecole Norm. (2) 1, 393-457 (1872). See [1878, 1]. (240) NA VIER, C.-L.-M.-H.: Memoire sur !es lois du mouvement des fluides, en ayant egard a l'adhesion des mo!ecules (1822). Bull. Soc. Philomath. 49-52. (Abstract of [1827, 6].) (298) PIOLA, G.: Sull' applicazione de' principj della meccanica analitica del Lagrange ai principali problemi. Milano: Regia Stamparia, VII + 252 pp. (17, 82, 156) CAUCHY, A.-L.: De Ia pression ou tension dans un corps solide. Ex. de math. 2, 42-56 = Oeuvres (2) 7, 60-78. (21, 200, 203, 204, 205) List of Works Cited. 747 2. CAUCHY, A.-L.: Sur la eondensation et la dilatation des eorps solides. Ex. de math. 2, 60-69 = Oeuvres (2) 7, 82-83. (21, 22, 26, 27, 30, 54, 57, 82A) 3. CAUCHY, A.-L.: Sur les moments d'inertie. Ex. de math. 2, 93-103 = Oeuvres (2) 7, 124-136. (168) 4. CAUCHY, A.-L.: Sur les relations qui existent dans l'etat d'equilibre d'un eorps solide ou fluide, entre les pressions ou tensions et les forees aeeeleratriees. Ex. de math. 2, 108-111 = Oeuvres (2) 7, 141-145. (205) 5. CAUCHY, A.-L.: Theorie de la propagation des ondes a la surfaee d'un fluide pesant d'une profondeur indefinie (1815). Mem. divers savants (2) 1 (1816), 3-312 = Oeuvres (1) 1, 5-318. 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CAUCHY, A.-L.: Sur !es pressions ou tensions supportees en un point donne d'un eorps solide par trois plans perpendieulaires entre eux. Ex. de math. 4, 30-40 = Oeuvres (2) 9, 41- 52. (203) 2. CAUCHY, A.-L.: Sur Ia relation qui existe entre !es pressions ou tensions supportees par deux plans quelconques en un point donne d'un eorps solide. Ex. de math. 4, 41-46 =Oeuvres (2) 9, 53-55. (205) 3. CAUCHY, A.-L.: Sur !es eorps solides ou fluides dans lesquels Ia eondensation ou dilatation lineaire est la meme en tous sens autour de ehaque point. Ex. de math. 4, 214-216 = Oeuvres (2) 9, 254-258. (142) 4. CAUCHY, A.-L.: Sur l'equilibre et le mouvement interieur des eorps eonsideres eomme des masses eontinues. Ex. de math. 4, 293-319 =Oeuvres (2) 9, 243-369. (150, 293, 301, 304) 5. PorssoN, S.-D.: Memoire sur l'equilibre et le mouvement des eorps elastiques (1828). Mem. Aead. Sei. Inst. Franee (2) 8, 357-570. (205, 301) 1830 1. 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(205, 232A) ScHWEINS, F.: Fliehmomente, oder die Summe L (x X+ yY) bei Kräften in der Ebene, und L (x X+ y Y + zZ) bei Kräften im Raume. J. reine angew. Math. 38, 77-88. (219) THOMSON, W. (Lord KELVIN): Notes on hydrodynamics (5), On the vis-viva of a liquid in motion. Cambr. Dubl. Math. J. 4, 90-94 =Papers 1, 107-112. (94, 113, 169) THOMSON, W. (Lord KELVIN): An account of Carnot's theory of the motive power of heat; with numerical results deduced from Regnault's experiments on steam. Trans. Roy. Soc. Edinb. 16, 541-574 = Ann. Chimie 35, 248-255 (1852) =Papers 1, 113-155· (245) 1850 1. CLAUSIUS, R.: Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Ann. Physik (3) 19, 368-398, 500-524 = Abh. 1, 16-78. Trans!., On the moving force of heat, and the laws regarding the nature of heat itself which are dedueible therefrom. Phi!. Mag. (4) 2 (1851). 1-21, 102-119; transl., W. F. 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CLAUSIUS, R.: Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie. Ann. Physik 93, 481-506 = Abh. 1, 126-154. (245, 248, 249, 252, 258) 2. ScHWEINS, F.: Theorie der Dreh- und Fliehmomente der parallelen Seitenkräfte, in welche Kräfte im Raume zerlegt werden können. J. reine angew. Math. 47, 238-245. (219) 1855 1. FICK, A.: Über Diffusion. Ann. der Phys. 94, 59-86. (158, 295) 2. HAUGHTON, S.: On a classification of elastic media, and the laws of plane waves propagated through them ( 1849). Trans. Roy. Irish Acad. Pt. I-Science 22, 97-138. (205, 232) 3. MEISSEL, E.: Über einen speciellen Fall des Ausflusses von Wasser in einer verticalen Ebene. Ann. Physik (4) 5, 276-283. (161) 4. THOMSON, W. (Lord KELVIN): On the thermo-elastic and thermo-magnetic properties of matter. Quart. J. Math. 1 (1855-1857) 55-77 = (with notes and additions) Phi!. Mag. (5) 5 (1878), 4-27 = Pt. VII of On the dynamical theory of heat. Papers 1, 291-316. (218, 262) 1856 1. RANKINE, W. J. M.: On axes of elasticity and crystalline forms. Phi!. Trans. Roy. Soc. Lond. (146) 46, 261-285 = Papers 119-149. (200) 2. THOMSON, W. (Lord KELVIN): Elements of a mathematical theory of elasticity. Phi!. Trans. Roy. Soc. Lond. 146, 481-498. (See [1877, 5].) (262) 1857 1. CLEBSCH, A.: Über eine allgemeine Transformation der hydrodynamischen Gleichungen. J. reine angew. Math. 54, 293-312. (125, 136, 137, 162, 210) 1858 1. HELMHOLTZ, H.: Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J. reine angew. Math. 55, 25-55 = Wiss. Abh. 1, 101-134. Trans!., P. G. TAIT, On integrals of the hydrodynamical equations, which express vortex-motion. Phi!. Mag. (4) 33, (1867) 485-512. (8, 75, 88, 93, 106, 108, 113, 133, 175, 185, 186, 218) 1859 1. CLEBSCH, A.: Über die Integration der hydrodynamischen Gleichungen. J. reine angew. Math. 56, 1-10. (136, 136A) 2. KIRCHHOFF, G.: Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes. J. reine angew. Math. 56,285-313 = Ges. Abh., 285-316. (57, 63A, 214, 236A, 301) 3. LAME, G.: Le~ons sur !es Coordonnees Curvilignes et leurs divers Applications. Paris: Mallet-Bachelier. xxvii + 368 pp. (205 A, 209) 1860 1. DIRICHLET, G. LEJEUNE: Untersuchungen über ein Problem der Hydrodynamik. Gött. Abh., math. Cl. 8 (1858-1859), 3-42 = J. reine angew. Math. 58, 181-216 (1861) = Werke 2, 263-301. (66) 2. MAXWELL, J. C.: Illustrations of the dynamical theory of gases. Phi!. Mag. (4) 19, 19-32 and 20,21-37 =Papers 1, 377-409. (295) 3. 4. 1861 1. 2. 1862 1. 2. 3. 4. 5. 1863 1. 2. 1864 1. 2. 3. 4. 5. 1865 1. 2. 1866 1. 2. 3. 4. 1867 1. 2. 3. 1868 1. 2. 3. 4. List of Works Cited. 751 NEUMANN, C.: Zur Theorie der Elasticität. J. reine angew. Math. 57, 281-318 (210) RIEMANN, B.: Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gött. Abh., math. Cl. 8 (1858-1859), 43-65 = Werke, 145-164. (185, 189, 205) HANKEL, H.: Zur allgemeinen Theorie der Bewegung der Flüssigkeiten. Göttingen. (66A, 88, 108, 128, 136) WARREN, J. W.: On invariant points, lines, and surfaces in space, and their physical significance. Amer. J. Math. 4, 306-310. (38) CLAUSIUS, R.: Ueber die Anwendung des Satzes von der Aequivalenz der Verwandlungen auf die innere Arbeit. Vjschr. nat. Ges. Zürich 7, 48-95 = Ann. Physik 116, 73-112 = Abh. 1, 242-279. (245, 258) CLEBSCH, A.: Theorie der Elasticität fester Körper. Leipzig. (63A, 214) EULER, L.: Fragmentum 97, Opera postuma 1, 494-496 = Opera omnia (1) 29, 437-440. (33A) EULER, L.: Anleitung zur Natur-Lehre, worin die Gründe zu Erklärung aller in der Natur sich ereignenden Begebenheiten und Veränderungen festgesetzt werden (probably written between 1755 and 1759). Opera postuma 2, 449-560 = Opera omnia (3) 1, 16-178. (76) W ARREN, J. W.: On the internal pressure within an elastic solid. Quart. J. Math. 5, 109-117. (21) AIRY, G. B.: On the strains in the interior of beams. Phi!. Trans. Roy. Soc. Lond. 153, 49-80. Abstract in Rep. Brit. Assn. 1862, 82-86 (1863). (224) THOMSON, W. (Lord KELVIN}: Dynamical problems regarding elastic spheroidal shells and spheroids of incompressible liquid. Phi!. Trans. Roy. Soc. Lond. 153, 583-616 =Papers 3, 351-394. (262, 301, 303} JouLE, J. P.: Note on the history of the dynamical theory of heat. Phi!. Mag. (4) 28, 150-151. (240) RANKINE, W. J. M.: On plane water-lines in two dimensions. Phi!. Trans. Roy. Soc. Lond. 154, 369-391 = Papers 495-521. (71, 161) ST. VENANT, A.-J .-C. B. DE: Etablissement elementaire des formules et equations generales de Ia theorie de l'elasticite des corps solides. Appendix in: Resurne des Lec;:ons donnees a l'Ecole des Ponts et Chaussees sur l'Application de Ia Mecanique, premiere partie, premiere section, De Ia Resistance des Corps Solides, par C.-L.- M.-H. NA VIER, 3rd. ed. Paris. (57) ST. VENANT, A.-J .-C. B. DE: Theorie de l'elasticite des solides, ou cinematique de leurs deformations. L'Institut 321, 389-390. (27) WARREN, J. W.: Note on a transformation of the generat equation of wave propagation, due to internal force. Quart. J. Math. 6, 137-139. (209) CLAUSIUS, R.: Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Vjschr. nat. Ges. Zürich 10, 1-59 = Ann. der Phys. 125. 353-400 = Abh. 2, 1-44. (245, 246, 258, 264) THOMSON, W. (Lord KELVIN}: On the elasticity and viscosity of metals. Proc. Roy. Soc. Lond. 14, 289-297. (See [1877, 5].) DARBOUX, G.: Sur les surfaces orthogonales. Ann. Ecole Norm. (1) 3, 97-141. (122) ]ACOBI, C. J. J.: Vorlesungen über Dynamik (1842-1843). Werke, Suppl. (219). LIPSCHITZ, R.: Ueber einen algebraischen Typus der Bedingungen eines bewegten Massensystems. J. reine angew. Math. 66, 363-374. (219) V ERDET, E.: Introduction, Oeuvres de Fresnel 1, IX-XCIX. (301) BERTRAND, J.: Theoreme relatif au mouvement le plus generat d'un fluide. C. R. Acad. Sei., Paris 66, 1127-1230. (78, 86, 86A, 89) MAXWELL, J. C.: On the dynamical theory of gases ( 1866). Phi!. Trans. Roy. Soc. Lond. 157, 49-88 = Phi!. Mag. (4} 35 (1868), 129-145, 185-217 =Papers, 2, 26-78. (215, 243, 295, 305) THOMSON, W. (Lord KELVIN}, and P. G. TAIT: Treatise on Natural Philosophy. Part I. Cambridge. (8, 12, 21, 27, 35, 36, 38, 42, 45, 46, 167, 205, 262) Anonymous: Les Mondes 17, 620-623. (89) BERTRAND, J.: Note relative a Ia theorie des fluides. Reponse a Ia communication de M. Helmholtz. C. R. Acad. Sei., Paris 67, 267-269. (86A) BERTRAND, J.: Observations nouvelles sur un memoire de M. Helmholtz. C. R. Acad. Sei., Paris 67, 469-472. (86A) BERTRAND, J.: Reponse a Ia note de M. Helmholtz. c. R. Acad. Sei., Paris 67, 773-775- (86A) 752 C. TRUESDELL and R. TouPrN: The Classical Field Theories. 5. FRESNEL, A.: Premiere memoire sur Ia double refraction (1821). Oeuvres 2, 261-308. (208, 209) 6. FRESNEL, A.: Supplement au memoire sur la double refraction (1822). Oeuvres 2, 343-367. (21, 208, 209) 7. FRESNEL, A.: Second supplement au memoire sur Ia double refraction (1822). Oeuvres 2, 369-442. (21, 203, 205, 208, 209) 8. HELMHOLTZ, H.: Sur le mouvement le plus general d'un fluide. Reponse a une communication precedente de M. J. Bertrand. C. R. Acad. Sei., Paris 67, 221-225 = Wiss. Abh. 1, 135-139. (86A) 9. HELMHOLTZ, H.: Sur le mouvement des fluides. Deuxieme reponse a M. J. Bertrand. C. R. Acad. Sei., Paris 67, 754-757 = Wiss. Abh. 1, 140-144. (86A) 10. HELMHOL TZ, H.: Reponse a Ia note de M. J. Bertrand, du 19 Octobre. c. R. Acad. Sei., Paris 67, 1034-1035 = Wiss. Abh. 1, 145. (86A) 11. KIRCHHOFF, G.: Ueber den Einfluß der Wärmeleitung in einem Gase auf die Schallbewegung. Ann. Physik 134, 177-193 = Ges. Abh. 1, 540-556. (241, 248) 12. MAXWELL, J. C.: On reciprocal diagrams in space, and their relation to Airy's function of stress. Proc. Lond. Math. Soc. (1) 2 (1865-1969), 58-60 = Papers 2, 102-104. (227) 13. RIEMANN, B.: Über die Hypothesen, welche der Geometrie zu Grunde liegen (1854). Abh. Ges. Wiss. Göttingen 13, (1866-1867) 133-150 =Werke, 2nd. ed. (1892), 272-287. (34) 14. TAIT, P. G.: Sketch of Thermodynamics. London. (240) 15. WALTER, A.: Anwendung der Methode Hamiltons auf die Grundgleichungen der mathematischen Theorie der Elasticität. Diss. Berlin. (We have not been able to see this work.) (236) 16. WARREN, J.: Theorem with regard to the three axes of invariable direction in a strained elastic body. Quart. J. Math. 9, 171-172. (38) 17. WEBER, H.: Über eine Transformation der hydrodynamischen Gleichungen. J. reine angew. Math. 68, 286-292. (135) 1869 1. CHRlSTOFFEL, E. B.: Über die Transformation der homogenen Differentialausdrücke 2ten Grades. J. reine angew. Math. 70, 46-70 = Abh. 1, 352-377 (34) 2. CHRISTOFFEL, E. B.: Über ein die Transformation homogener Differentialausdrücke zweiten Grades betreffendes Theorem. J. reine angew. Math. 70, 241 -24 5 = Abh. 1, 378-382 (34) 3. LIPSCHITZ, R.: Untersuchungen in Betreff der ganzen homogenen Functionen von n Variablen. J. reine angew. Math. 70, 71-102. (34) 4. MASSIEU, F.: Sur les fonctions caracteristiques des divers fluides. C. R. Acad. Sei., Paris 69, 858-862. (251) 5. MAssrEu, F.: Addition au precedent memoire sur les fonctions caracteristiques. C. R. Acad. Sei., Paris 69, 1057-1061. (251) 6. ST. VENANT, A.-J .-C. B. DE: Probleme des mouvements que peuvent prendre les divers points d'une liquide, ou solide ductile, contenue dans un vase a parois verticales, pendant son ecoulement par un orifice horizontal interieur. C. R. Acad. Sei., Paris 68, 221-237. (89, 134) 7. THOMSON, W. (Lord KELVIN): On vortex motion. Trans. Roy. Soc. Edinb. 25, 217-260 = Papers 4, 13-66. (79, 88, 105, 106, 108, 128, 129, 297) 1870 1. CLAUSIUS, R.: Über einen auf die Wärme anwendbaren mechanischen Satz. Ann. Physik u. Chem. (5) 21 = 141, 124-130. (219) 2. LIPSCHITZ, R.: Fortgesetzte Untersuchung in Betreff der ganzen homogenen Funktionen von n Differentialen. J. reine angew. Math. 72, 1-65. (34) 3. L:Evv, M.: Memoire sur les equations generales des mouvements interieurs des corps solides ductiles au dela des limites ou l'elasticite pourrait les ramener a leur premier etat. C. R. Acad. Sei., Paris 70, 1323-1325. (300) 4. MAXWELL, J. C.: On reciprocal figures, frames, and diagrams of forces. Trans. Roy. Soc. Edinb. 26 (1869-1872), 1-40 = Papers 2, 161-207. (216, 224, 227) 5. MAXWELL, J. C.: On the displacement in a case of fluid motion. Proc. Lond. Math. Soc. 3 (1869-1871), 82-87 = Papers 2, 208-214. (71) 6. RANKINE, W. J. M.: On the thermodynamic theory of waves of finite longitudinal disturbance. Phil. Trans. Roy. Soc. Lond. 160, 277-288 = Paptrs, 530-543. (183, 189, 205, 241, 297)) 7. ST. VENANT, A.-J.-C. B. DE: Memoire sur l'etablissement des equations differentielles des mouvements interieurs operes dans les corps solides ductiles au dela des limites ou l'elasticite pourrait les ramener a leur premier etat. C. R. Acad. Sei., Paris 70, 473-480. (300) List of Works Cited. 753 8. WARREN, J.: Note on a fundamental theorem in hydrodynamics. Quart. J. Math. 10, 128-129. (101) 1871 1. BELTRAMI, E.: Sui principi fondamentali della idrodinamica. Mem. Acad. Sei. Bologna (3) 1, 431-476; 2 (1872). 381-437; 3 (1873), 349-407; 5 (1874). 443-484 = Richerche sulla cinematica dei fluidi. Opere 2, 202-379. (78, 82A, 83, 84. 86, 101, 105, 129, 136) 2. BoussiNESQ, J.: Etude nouvelle sur l'equilibre et le mouvement des corps solides elastiques dont certaines dimensions sont tres petites par rapport a d'autres. Premier Memoire. J. Math. pures appl. (2) 16, 125-240. (57) 3. CHERBULIEZ, E.: Geschichtliche Mittheilungen aus dem Gebiete der mechanischen Wärmetheorie. Mittheil. Naturf. Ges. Bern 1870, 291-324. (240) 4. DuRRANDE, H.: Extrait d'une theorie du deplacement d'une figure qui se deforme. C. R. Acad. Sei., Paris 73, 736-738. (82A) 5. MAXWELL, J. C.: Theory of Heat. London. (252) 6. STEFAN, J.: Über das Gleichgewicht und die Bewegung, insbesondere die Diffusion von Gasmengen. Sitzgsber. Akad. Wiss. Wien 632, 63-124. (158, 295) 1872 1. BoussiNESQ, J.: Theorie des ondes liquides periodiques (1869). Mem. divers sav. 20, 509-615. (210) 2. Rapport sur un memoire deM. Kleitz intitule: .,Etudes sur !es forces moleculaires dans !es liquides en mouvement, et application a l'hydrodynamique" (ST. VENANT). C. R. Acad. Sei., Paris 74, 426-438. (82A, 208) 3. LIPSCHITZ, R.: Extrait d'une lettre. Bull. sei. math. astr. (1) 3, 349-352. (219) 4. VILLARCEAU, Y.: Sur un nouveau theoreme de mecanique generale. C. R. Acad. Sei., Paris 75, 232-240. (219) 1873 1. BoBYLEW, D.: Einige Betrachtungen über die Gleichungen der Hydrodynamik. Math. Ann. 6, 72-84. (211) 2. GIBBS, J. W.: Graphical methods in the thermodynamics of fluids. Trans. Connecticut Acad. 2, 309-342 = Works 1, 1-32. (245, 247, 248, 265) 3. GIBBS, J. W.: A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans. Connecticut Acad. 2, 382-404 = Works 1, 33-54. (245, 246, 247) 4. KLEITZ, C.: Etudes sur !es forces moleculaires dans !es liquides en mouvement et application al'hydrodynamique. Paris. (Cf. [1872, 2].) (82A, 110, 204, 208, 293). 5. MAXWELL, J. C.: A Treatise on Electricity and Magnetism. 2 vols., Oxford. Cf. [1881, 4]. (175, 187, 200, 276, 311) 6. STRUTT, J. W. (RAYLEIGH): Some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4 (1871-1873). 357-368 = Papers 1, 170-181. (241A) 1874 1. BoLTZMANN, L.: Zur Theorie der elastischen Nachwirkung. Sitzungsber. Akad. Wiss. Wien 702, 275-306 = Wiss. Abh. 1, 616-639. (305) 2. MAXWELL, J. C.: V an d·~r Waals on the continuity of the gaseous and liquid states. Nature 10, 477-480 = Papers 2, 407-415. (219) 3. MEYER, 0.-E.: Zur Theorie der inneren Reibung. J. reine angew. Math. 78, 130 to 135. (305) 4. MEYER, 0.-E.: Theorie der elastischen Nachwirkung. Ann. Physik (6) 1, 108-119. (305) 5. NANSON, E. J.: Note on hydrodynamics. Mess. math. 3, 120-121. (75, 108) 6. UMov, N.: Ableitung der Bewegungsgleichungen der Energie in continuirlichen Körpern. Z. Math. Phys. 19, 418-431. (217) 187 5 1. GIBBS, J. W.: On the equilibrium of heterogeneaus substances. Trans. Connecticut Acad. 3 (1875-1878), 108-248, 343-524 = Works 1, 55-353. (14, 245, 246, 247, 251, 260, 262, 264, 265) 2. GREENHILL, A. G.: Hydromechanics. Encycl. Britt., 9th ed. (carried through 14th ed., 1926). (99, 207) 3. LIPSCHITZ, R.: Determinazione della pressione nell' intorno d'un fluido incompressibile soggetto ad attrazioni interne ed esterne. Ann. mat. (2) 6 (1873-1875). 226-231. (102, 141) 4. MAXWELL, J. C.: 4th. ed. of [1871, 5]. (245, 246) 5. MEYER, 0.-E.: Zusatz zu der Abhandlung zur Theorie der inneren Reibung. J. reine angew. Math. 80, 315-316. (305) 1876 1. CoTTERILL, J.: On the distribution of energy in a mass of liquid in a state of steady motion. Phil. Mag. 1, 108-111. (125) 2. KIRCHHOFF, G.: Vorlesungen über mathematische Physik: Mechanik. Leipzig. 2nd ed., 1877; 3rd ed., 1883. (57, 63A, 113, 129, 196, 200, 205, 214, 232, 236, 301) Handbuch der Physik, Bd. 111/1. 48 754 C. TRUESDELL and R. ToUPIN: The Classical Field Theories. 3. MASSIEU, F.: Memoire sur les fonctions caracteristiques de divers fluides et sur Ia theorie des vapeurs. Mem. divers savants 22, No. 2, 92 pp. (251) 4. MAXWELL, J. C.: On the equilibrium of heterogeneaus substances. Repts. South Kensington conf. spec. loan coll. sei. app. 144-150 = (with a note added by LARMOR) Phil. Mag. (6) 16, 818-824 (1908). Condensed version, Proc. Cambridge Phil. Soc. 2, 427-430 (1876) =Papers 2, 498-500. (260) 5. 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(33A) KrRKWOOD, J. G., and B. CRAWFORD jr.: The macroscopic equations of transport J. Physic. Chem. 56, 1048-1051. (243, 285, 295) MoREAU, J .-J 0: Bilan dynamique d'un ecoulement rotationnel. J 0 Math. pures appl. (9) 31, 355-375; 32, 1-78 (1953). (117, 130, 186, 297) NARDINI, R.: Sul valore medio dello stress per particolari sollecitazioni. Ann. Univ. Ferrara sez. VII (2) 1, 89-91. (220) PARTINGTON, J. R.: Advances in thermodynamics. Nature, Lond. 170, 730-732. (245) PRIM, R. C.: Steady rotational flow of ideal gases ( 1949). J. Rational Mech. Anal. 1, 425-497. (111, 132, 133) RrcHTER, H.: Zur Elasticitätstheorie endlicher Verformungen. Math. Nachr. 8, 65-73. (37) STORCH!, E.: Le superficie eccezionali nella statica delle membrane. Rev. mat. Univ. Parma 3, 339-360. (229) 19. 20. 21. 22. 23. 24. 1953 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. List of Works Cited. 779 SZEBEHELY, V. G.: Generalization of the dimensionless frequency parameter in unsteady flows. David Taylor Model Basin Rep. 833. (100) ToNoLo, A.: Sopra un problema di Darboux della meccanica dei mezzi continui. Ann. Univ. Ferrara sez. VII (2) 1, 103-109. (38) TRUESDELL, C.: The mechanical foundations of elasticity and fluid dynamics. J. Rational Mech. Anal. 1, 125-300. (12, 14, 17, 29, 33A, 34, 44, 46, 50, 57, 95, 96, 210, 256A, 293, 298, 299, 303) TRUESDELL, C.: A program of physical research in classical mechanics. Z. angew. Math. Phys. 3, 79-95. (2, 293, 298, 299) TRUESDELL, C.: Vorticity and the Thermodynamic State in a Gas Flow. Mem. sei. math. No. 119, Paris. (132, 133, 297) \VHAPLES, G.: Caratheodory's temperature equations. J. Rational Mech. Anal. 1, 301-307. (245) BILIMOVITCH, A.: Sur l'homogenisation des equations de nature velocidique. Acad. Serbe Sei. Pub!. Inst. Math. 5, 29-34. (115, 160) BJ0RGUM, 0., and T. GoDAL: On Beltrami vector vields and flows, Part I I. Univ. Bergen Arbok 1952, Naturv. rekke No. 13. CARSTOIU, I.: Sur la detormation d'une particule dans le mouvement d'un fluide. C. R. Acad. Sei., Paris 236, 2209-2211. (101) CASTOLDI, L.: Teoremi di Bernoulli per fluidi comprimibili viscosi. Atti Accad. Ligure 9 (1952), 215-222. (125) CASTOLDI, L.: Sui moti di fluidi reali per cui si verifica una esatta linearizzazione della equazione dinamica. Atti Accad. Ligure 9 (1952), 222-227. (122, 127) CoRSON, E. M.: Introduction to Tensors, Spinors and Relativistic Wave-Equations (Relation Structure). New York. (282) DARWIN, C.: Note on hydrodynamics. Proc. Cambridge Phi!. Soc. 49, 342-354. (71) DEFRISE, P.: Analyse geometrique de la cinematique des milieux continus. Inst. R. Meteorol. Belg. Pub!. ser. B, No. 16, 63 pp. (12, 152, 153) DE GROOT, S.: Hydrodynamics and thermodynamics. Proc. Symp. appl. Math. 4, 87-99. (245) FoRTAK, H.: Zur Bedeutung der in der Clebsch-Transformation der hydrodynamischen Gleichungen auftretenden Funktionen. Acta Hydrophys. 1, 145-150. (135) GEIRINGER, H.: Some recent results in the theory of an ideal plastic body. Adv. appl. Mech. 3, 197-294. (300) GRIOLI, G.: Proprietä. di media ed equilibrio elastico. Atti 4to Congr. Un. Mat. Ital. (1951) 1, 68-77. (221, 222) HoWARD, L. N.: Constant speed flows. Thesis, MS in Princeton University Library. (111) KILCHEVSKI, N. A.: Stress, velocity, and density functions in static and dynamic problems in the mechanics of continuous media [in Russian]. Dokl. Akad. Nauk SSSR. 92, 895-898. (228, 230) LANG HAAR, H.: An invariant membrane stress function for shells. J. appl. Mech. 20, 178-182. (229) LANG HAAR, H.: The principle of complementary energy in nonlinear elasticity theory. J. Franklin Inst. 256, 255-264. (232A) MASUDA, H.: A new proof of Lagrange's theorem in hydrodynamics. J. Phys. Soc. Japan 8, 390-393- (136) McVrTTIE, G. C.: A method of solution of the equations of classical gas dynamics using Einstein's equations. Quart. appl. Math. 11, 327-336. (225, 230) Mr~rcu, M.: Echilibrul mediilor continue cu deformäri märi. Acad. Repub. Pop. Romane. Stud. Cerc. Mec. Metalurgie 4, 31-53. (12) NYBORG, W. L.: Acoustic streaming equations: laws of rotational motion for fluid elements. J. Acoust. Soc. 25, 938-944. (170) PLATRIER, C.: Conditions d'integrabilite du tenseur de deformation totale dans une transformation finie d'un milieu ä. trois dimensions. Ann. Ponts Chaussees 123, 703-709. (34) PLATRIER, C.: Conditions d'integrabilite du tenseur de detormation totale dans une transformation finie d'un milieu ä. trois dimensions. Bull. Acad. Roy. Belg., CL Sei. (5) 39, 490-494. (34) FRAGER, W.: Three-dimensional plastic flow under uniform stress. Brown univ. tech. rep. No. 95, August = Rev. fac. sei. Univ. Istanbul 19, 23-27 (1954). (142) PRATELLI, A.: Principi variazionali nella meccanica dei fluidi. Rend. Ist. Lombardo (3) 17 (86), 484-500. (94, 163) 780 C. TRUESDELL and R. TouPIN: The Classical Field Theories. 25. PRATELLI, A.: Sulla stazionarieta di significativi integrali nella meccanica dei continui. Rend. Ist. Lombardo (3) 17 (86), 714-724. (227, 235) 26. REIK, H.: Zur Theorie irreversibler Vorgänge. Ann. Physik (6) 11, 270-284, 407-419, 420-428; 13, 73-96. (245) 27. REISSNER, E.: On a variational theorem for finite elastic deformations. J. Math. Phys. 32, 129-135. (232A) 28. ScHAEFER, H.: Die Spannungsfunktionen des dreidimensionalen Kontinuums und des elastischen Körpers. Z. angew. Math. Mech. 33, 356-362. (226, 227, 229) 29. STORCH!, E.: Sulle membrane aventi comportamento meeeanieo eceezionale. Rend. Ist. Lombardo (3) 17 (86), 462-483. (229) 30. SzEBEHELY, V. G.: A measure of unsteadiness of time-dependent flows. Proe. 3rd Midwest Conf. on Fluid Mech., Univ. Minn., 221-231. (100) 31. TRUESDELL, C.: Two measures of vorticity. J. Rational Mech. Anal. 2, 173-217. (Partial abstract in Proc. Int. Congr. theor. appl. Mech. Istanbul 1952.) (91, 100, 121) 32. TRUESDELL, C.: Corrections and additions to "The mechanieal foundations of elasticity and fluid dynamics". J. Rational Mech. Anal. 2, 593-616. (12, 14, 28, 37, 150, 304) 33. TRUESDELL, C.: Generalization of a geometrical theorem of Euler. Comm. mat. Helv. 27, 233-234. (142) 34. TRUESDELL, C.: La velocita massima nel moto di Gromeka-Beltrami. Rend. Lineei (8) 13 (1952), 378-379- (121) 35. WHITTAKER, E. T.: A History of the Theories of Aether and Elastieity, vol. I I. London: Nelson. (266) 36. YosHIMURA, Y.: On the natural shearing strain. Proc. 2nd Japan Mat. Congr. appl. Mech. (1952), 1-4. (33A) 37. YoSHIMURA, Y.: On the definition of stress in the finite deformation theory. J. Phys. Soc. Japan 8, 669-673- (210) 1954 1. CARSTOIU, I.: Vorticity and deformation in fluid mechanics. J. Rational Mech. Anal. 3, 691-712. (101, 160) 2. CASTOLDI, L.: Le .,condizioni di congruenza" per deformazioni infinitesime non lineari. Atti Ist. Veneto, Cl. sei. mat. nat. 112, 41-47. (54, 57) 3. CASTOLDI, L.: Sopra un classificazione dei comportamenti elastici dei mezzi deformabili. Atti Ist. Veneto, Cl. sei. mat. nat. 112, 1 7-30. (55) 4. DEAN, W. R.: Note on the motion of an infinite cylinder in a rotating viseous liquid. Quart. J. Mech. appl. Math. 7, 25 7-262. (207) 5. DuGAS, R.: La Mecanique au XVIIe Siecle. Ed. Griffon, Neuchatel. 620 pp. (3) 6. ERICKSEN, J. L., and R. S. RrvLrN: Large elastic deformations of homogeneaus anisotropic materials. J. Rational Mech. Anal. 3, 281-301. (49, 62, 233) 7. GREEN, A. E., and W. ZERNA: Theoretical Elastieity. Oxford: Clarendon Press. xiii + 442 pp. (12, 66 B) 8. GüNTHER, W.: Spannungsfunktionen und Verträglichkeitsbedingungen der Kontinuumsmechanik Abh. Braunschweig. Wiss. Ges. 6, 207-219. (227, 229) 9. HIRSCHFELDER, J. 0., C. F. CURTISS, and R. B. BIRD: Moleeu!ar Theory of Gases and Liquids. New York: Wiley. xxvi + 1219 pp. (215, 243, 259, 295) 10. KoNDO, K.: On the theory of'the mechanieal behavior of mieroscopieally nonuniform materials. Research notes Res. Assn. appl. Geom. (Tokyo) (2), No. 4, 37 pp. (34) 11. KRÖNER, E.: Die Spannungsfunktion der dreidimensionalen isotropen Elastizitätstheorie. Z. Physik 139, 175-188. Correction, Z. Physik 143, 374 (1955). (227) 11 A. LAMM, 0.: The formal theory of diffusion, and its relation to self-diffusion, Sedimentation equilibrium, and viscosity. Acta Chem. Scand. 8, 1120-1128. (295) 12. LANGHAAR, H., and M. STIPPES: Three-dimensional stress functions. J. Franklin Inst. 258,371-382. (227) 13. LoDE, W.: Tensoren zur Berechnung großer Formänderungen. Kolloid-Z. 138, 28-38. (33A) 14. MANACORDA, T.: Sopra un prineipio variazionale di E. Reissner per Ia statica dei mezzi continui. Bol!. Un. Mat. Ital. (3) 9, 154-159- (232A) 15. MAZZARELLA, F.: Determinazione delle componenti di secondo ordine della deformazione riferite ad un generico sistema di coordinate eurvilinee. Rend. Accad. 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London: xix + 285 pp. (33A) 24. TRUESDELL, C.: The Kinematics of Vorticity. Indiana Univ. sei. ser. No. 19. (12, 66A, 72, 78, 101, 102, 108, 110, 112, 114, 118, 119, 120, 121, 124, 125, 129, 130, 132, 133, 134, 136, 137, 138, 145) 25. TRUESDELL, C.: Rational fluid mechanics 1687-1765. L. Euleri Opera Omnia (2) 12, IX-CXXV. (66A, 120, 130, 200) 26. WHITHAM, G. B.: A note on a paper by G. C. McVittie. Quart. appl. Math. 12, 316-318. (230) 1955 1. ADKINS, J.: A note on the finite plane-strain equations for isotropic incompressible materials. Proc. Cambridge Phil. Soc. 51, 363-367. (47) 2. BILBY, B. A., R. BuLLOUGH and E. SMITH: Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc. Roy. Soc. Lond., Ser. A 231, 263-273. (34) 3. BJ0RGUM, 0.: On the physical boundary conditions in fluid dynamics. Univ. Bergen Arbok, Naturv. rekke Nr. 4, 8 pp. (105) 4. CoTTER, B., and R. S. RrvLIN: Tensors associated with time-dependent stress. 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TRUESDELL, C.: The simplest rate theory of pure elasticity. Comm. pure appl. Math. 8, 123-132. (142, 304) 28. TRUESDELL, C.: Hypo-elasticity. J. Rational Mech. Anal. 4, 83-133, 1019-1020. (223, 293, 304) 29. TRUESDELL, C.: Review of [1954, 23]. Math. Rev. 16, 307. (33A) 1956 1. BERKER, R.: Sur les equations de compatibilite relativesau mouvement d'un gaz. C. R. Acad. Sei., Paris 242, 342-344. (110) 2. BRESSAN, A.: Sulla possibilita di stabilire limitazioni inferiori per le componenti intrinseche del tensore degli sforzi in coordinate generali. Rend. Sem. Mat. Padova 26, 139-147. (222) 3. CoLONNETTI, G.: L'equilibre des voiles minces hyperstatiques (Le cas des voiles de surface minimum). C. R. Acad. Sei., Paris 243, 1087-1089, 1701-1704. (229) 4. DoRN, W. S., and A. ScHILD: A converse to the virtual work theorem for deformable solids. Quart. appl. Math. 14, 209-213. (227, 234) 5. DoYLE, T. C., and J. L. ERICKSEN: Nonlinear elasticity. Adv. appl. Mech. 4, 53-115. (12, 14, 18) 6. 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KoPPE, E.: Methoden der nichtlinearen Elastizitätstheorie mit Anwendung auf die dünne Platte endlicher Durchbiegung. Z. angew. Math. Mech. 36, 455-462. (34) 15. MARGULIES, G.: Remark on kinematically preferred co-ordinate systems: Proc. Nat. Acad. Sei. U.S.A. 42, 152-153. (148) 16. OLDROYD, J. G., and R. H. THOMAS: The motion of a cylinder in a rotating liquid with general elastic and viscous properties. Quart. J.Mech.Appl. Math. 9, 136-139. (207) 17. OswATITSCH, K.: Über eine Verallgemeinerung des Potentials auf Strömungen mit Drehung. Öst. lng.-Arch. 10, 239-241. (125) 18. RAw, C. J. G., and W. YouRGRAU: "Acceleration" of chemical reactions. Nature, Lond. 178, 809. (159A) 19. SPEISER, A.: Einleitung. L. Euleri Opera Omnia (1) 29, VII-XLII. (33A) 20. ToUPIN, R. A.: The elastic dielectric. J. Rational Mech. Anal. 5, 849-915. (14, 19, 37, 312) 21. TRUESDELL, C.: Zur Geschichte des Begriffes "Innerer Druck". Physik. BI. 12, 315- 326. (200) 22. TRUESDELL, C.: Review of [1953, 20]. Math. 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Lombardo 90 (1956), 369-378. (229) 14. TING, T. W., and J. C. M. LI: Thermodynamics for elastic solids. General formulation. Phys. Rev. (2) 106, 1165-1167. (246) 15. THOMAS, T. Y.: Extended compatibility conditions for the study of surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6, 311-322, 907-908. (175, 176, 177, 179, 180, 181) 16. TRUESDELL, C.: Sulle basi della termomeccanica. Rend. Lincei (8) 22, 33-38, 158-166. (158, 159, 215, 243, 254, 255, 259) 17. TRUESDELL, C.: General solution for the stresses in a curved membrane. Proc. Nat. Acad. Sei. U.S.A. 43, 1070-1072. (84, 229) 18. WHITNEY, H.: Geometrie Integration Theory. Princeton Univ. Press. (269) 19. YIH, C.-S.: Stream functions in three-dimensional flows. Houille Blanche, 445-450. (164) 1958 1. ERICKSEN, J. L., and C. TRUESDELL: Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1, 295-323. (60, 61, 63A, 64, 212, 214) 1 A. FERRARESE, G.: Sulla relazione simbolica della meccanica dei sistemi continui vincolati. Rend. Mat. e Appl. (5) 14, 305-312. (233) 2. GoDAL, T.: On Beltrami vector fields and flows, Part III. Same considerations on the general case. Univ. Bergen Arbok 1957, Naturv. r. Nr. 12, 28 pp. (108) 3. GRAIFF, F.: Sulle condizioni di congruenza per deformazioni anche finite. Rend. Lincei (8) 24, 415-422. (84) 4. GüNTHER, W.: Zur Statik und Kinematik des Cosseratschen Kontinuums. Abh. Braunschweig. Wiss. Ges. 10, 195-213. (61, 200, 227) 5. KANWAL, R. P.: Determination of the vorticity and the gradients of flow parameters behind a three-dimensional unsteady curved shock wave. Arch. Rational Mech. Anal. 1, 225-232. (175) 6. KRZYWOBLOCKI, M. Z. v.: On the stream functions in nonsteady three-dimensional flow. J. Aeronaut. Sei. 25, 67. (160) 7. MARSHAK, R. E.: Effect of radiation on shock wave behavior. Phys. of Fluids 1, 24-29. (241) 8. NoLL, W.: A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2 (1958/59), 197-226. (3, 196, 200, 293, 305) 9. TouPIN, R. A.: World invariant kinematics. Arch. Rational Mech. Anal. 1 (1957/58), 181-211. (152) 10. TRUESDELL, C.: Geometrie interpretation for the reciprocal deformation tensors. Quart. appl. Math. 15, 434-435. (29) 11. TRUESDELL, C.: Intrinsic equations of spatial gas flow. Math. Res. Center U.S. Army, Univ. Wisconsin rep. No. 33, July. Cf. [1960, 5]. (99) 12. WASHIZU, K.: A note on the conditions of compatibility. J. Math. Phys. 36, 306-312. (34) 1959 1. BLANKFIELD, J., and G. C. McVITTIE: Einstein's equations and classical hydrodynamics. Arch. Rational Mech. Anal. 2 (1958/59), 337-354. (230) 784 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. C. TRUESDELL and R. TouPIN: The Classical Field Theories. BLANKFIELD, J., and G. C. McVITTIE: A method of solution of the equations of magnetohydrodynamics. Arch. Rational Mech. Anal. 2 (1958/59). 411-422. (230) CoLEMAN, B. D., and W. NoLL: On the thermostatics of continuous media. Arch. Rational Mech. Anal. 4 (1959/60), 97-128. (262) DROBOT, S., and A. RYBARSKI: A variational principle of hydromechanics. Arch. Rational Mech. Anal. 2 (1958/59), 393-410. (132) GRAIFF, F.: Soluzione generate delle equazioni indefinite di equilibrio per una membrana. Rend. Lincei (8) 26, 189-196. (229) GREEN, A. E., R. S. RIVLIN and A. J. M. SPENCER: The mechanics of non-linear materials with memory. Part li. Arch. Rational Mech. Anal. 3, 82-90. (305) GREEN, A. R.: The equilibrium of rods.Arch. Rationa!Mech. Anal. 3, 417-421. (214) KRÖNER, E., u. A. SEEGER: Nicht-lineare Elastizitätstheorie der Versetzungen und Eigenspannungen. Arch. Rational Mech. Anal. 3, 97-119. (20, 34) NoLL, W.: The foundations of classical mechanics in the light of recent advances in continuum mechanics. The Axiomatic Method, with special reference to geometry and physics (1957). Amsterdam: North Holland Co., 266-281. (3, 196, 293) ScHAEFER, H.: Die Spannungsfunktionen des dreidimensionalen Kontinuums; statische Deutung und Rand werte. lng.-Arch. 28, 291-306. (227) TRUESDELL, C. : Invariant and complete stress functions for generat continua. Arch. Rational Mech. Anal. 4 (1959/60). 1-29. (226) ZoLLER, K.: Die Wärmeleitgleichung bei Wärmespannungen. lng.-Arch. 28, 366-372. (256A) 1960 JA. CoLEMAN, B., and C. TRuESDELL: On the reciprocal relations of Onsager. J. Chem. Phys. forthcoming. (245) 1. ERICKSEN, J. L.: Anisotropie fluids. Arch. Rational Mech. Anal. 4 (1959/60), 231-237- (205, 298) 2. HAYES, W. D.: Generalized Kelvin's minimum energy theorem. J. Fluid Mech., forthcoming. (94) 3. KRÖNER, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rational Mech. Anal. 4 (1959/60). 273-334. (20, 61, 201, 266) 4. TRUESDELL, C.: The rational mechanics of flexible or elastic bodies, 1638-1788. L. Euleri Opera Omnia (2) 11, Part 2. (33A, 66A, 200, 212) 5. TRUESDELL, C.: lntrinsic equations of spatial gas flow. Z. angew. Math. Mech. 40, 9-14. (99) 6. TRUESDELL, C.: Mechanical aspects of diffusion, forthcoming. (295) Additional Bibliography K: Kinematics of special motions (geometrical theory). 1872 K 1. DuRRANDE, H.: Proprietes generales du deplacement d'une figure de forme variable. C. R. Acad. Sei., Paris 74, 1243-1247. K 2. DuRRANDE, H.: De l'acceleration dans Je deplacement d'un systeme de points qui reste homographique a lui-meme. C. R. Acad. Sei., Paris 75, 1177-1180. 1873 K 1. DuRRANDE, H.: Essai sur le deplacement d'une figure de forme variable. Ann. Ec. Norm. (2) 2, 81-121. K 2. GROUARD: Sur le mouvement d'une figure, qui se deplace dans l'espace en restaut semblable a elle-meme. Bull. Soc. Philomath. (6) 9, 47-49. Abstract: L'lnstitut (2) 1, 163-164. 1874 K 1. BuRMESTER, L.: Kinematisch-geometrische Untersuchungen der Bewegung ähnlich-veränderlicher ebener Systeme. Z. Math. Phys. 19, 154-169. K 2. Bu:RMESTER, L.: Kinematisch-geometrische Untersuchungen der Bewegung affinveränderlicher und collinear-veränderllcher ebener Systeme. Z. Math. Phys. 19, 465-492. K 3. DuRRANDE, H.: Deplacement d'un systeme de points. Proprietes geometriques dependant des parametres differentiels du second ordre. C. R. Acad. Sei., Paris 78, 1036-1040. K 4. DuRRANDE, H.: Etude de l'acceleration dans le deplacement d'un systeme de forme variable. Ann. Ec. Norm. (2) 3, 151-164. 1875 K 1. BuRMESTER, L.: Kinematisch-geometrische Untersuchungen der Bewegung gesetzmäßig-veränderlicher Systeme. Z. Math. Phys. 20, 381-422. K 2. JORDAN, C.: Sur le mouvement des figures dans le plan et dans l'espace. Bull. Soc. Math. France 1,144-148. Additional Bibliography K. 785 K 3. LrGUINE, V.: Sur le lieu des points d'un systeme invariable mobile d'une maniere generale dans l'espace, dont les accelerations du premier ordre sont constantes. Bull. Soc. Math. France 1, 152-154. 1877 K 1. MüLLER, R.: Über Selbsthüllcurven und Selbsthüllflächen in ähnlich-veränderlichen Systemen. Z. Math. Phys. 22, 369-376. 1878 K 1. BuRMESTER, L.: Kinematisch-geometrische Theorie der Bewegung der affinveränderlichen, ähnlich veränderlichen und starren räumlichen oder ebenen Systeme. Z. Math. Phys. 23, 108-131. K 2. BuRMESTER, L.: Über den Beschleunigungszustand ähnlich-veränderlicher und starrer ebener Systeme. Civilingenieur 24. 1879 K 1. BURMESTER, L.: Über die Festlegung projectiv-veränderlicher ebener Systeme. Math. Ann. 14, 472-497. K 2. FouRET, G.: Sur le mouvement d'un corps qui se deplace et se deforme en restant homothetique a lui-meme. C. R. Acad. Sei., Paris 88, 227-230. K 3. FROMENTI, C.: Movimento delle figure ehe se mantengono simili a se stesse. Giorn. math. 17, 232-243. K 4. GEISENHEIMER, L.: Untersuchung der Bewegung ähnlich-veränderlicher Systeme. Z. Math. Phys. 24, 129-159- K 5. GEISENHEIMER, L.: Die Bildung affiner Figuren durch ähnlich-veränderliche Systeme. Z. Math. Phys. 24, 345-381. 1880 K 1. BuRMESTER, L.: Über das bifocal-veränderliche System. Math. Ann.16, 89-111. 1881 K 1. ScHUMANN, A.: Beiträge zur Kinematik ähnlich-veränderlicher und affin-veränderlicher Gebilde. Z. Math. Phys. 26, 157-179- 1883 K 1. MEHMKE, R.: Über die Geschwindigkeiten beliebiger Ordnung eines in seiner Ebene bewegten ähnlich-veränderlichen Systems. Civilingenieur (2) 29, 487- 508. K 2. MEHMKE, R.: Über den geometrischen Ort der Punkte ohne Normalbeschleunigung in einer Phase eines starren oder affin-veränderlichen Systems. Civilingenieur (2) 29, 581-582. K 3. NrcoLI, F.: Intorno ad un caso di movimento di una figura piana ehe si varia rimanendo simile a se stessa. Mem. Accad. Modena (2) 1, 59-71. K 4. NrcoLI, F.: Intorno ad un caso di movimento di una figura piana ehe si conserva simile a se stessa. Mem. Accad. Modena (2) 1, 171-178. K 5. NrcoLI, F.: Intorno a due casi di movimento di una figura solida ehe rimane simile a se stessa. Mem. Accad. Modena (2) 1, 249-260. 1885 K 1. SoMOFF, P.: Über die Bewegung ähnlich-veränderlicher ebener Systeme. Z. Math. Phys. 30, 193-209. 1888 K 1. BuRMESTER, L.: Lehrbuch der Kinematik. Leipzig: A. Felix. xx + 942 pp. + Atlas. See §§ 329-351. 1889 K 1. SoMov, P.: Some problems on the distribution of velocityinvariable systems [in Russian]. Varshavskia Univ. lzv. 1889, No. 4, 32 pp. 1890 K 1. MoRLEY, F.: On the kinematics of a triangle of constant shape but varying size (with a note). Quart. J. Math. 24, 3 59-369, 386. K 2. SoMow, P.: On the acceleration in collinearly variable systems [in Russian]. Proceedings of the 8th meeting of Russian Natural Scientists and Physicians, St. Petersburg, 1890. Math. and Astron. 41-44. (We have not been able to see this reference.) 1892 K 1. SHEBUEV, G. N.: Application of the theory of quaternions to the mechanics of similar and homogeneously variable systems [in Russian]. lzv. Fiz.-Mat. Obsh. Kazan (2) 3, 111-160. 1894 K 1. MANNHEIM, A.: Principes et developpements de geometrie cinematique. Paris: Gauthier-Villars. ix + 589pp. See pp. 14-53, 457-475. 1897 K 1. DE SAUSSURE, R.: Calcul geometrique regle. Amer. J. Math. 19, 329-370. 1898 K 1. DE SAUSSURE, R.: Cinematique des fluides. Mouvement d'un fluide dans un plan. Arch. Sei. Phys. Nat. Geneve (4) 5, 497-519. 1899 K 1. CAVALLI, E.: Le figure reciproche e la trasformazione quadratica nella cinematica. Atti Ace. Napoli (2) 9, No. 12, 29 pp. 1901 K 1. DE SAussuRE, R.: Sur le mouvement d'une droite qui possede trois degres de liberte. C. R. Acad. Sei., Paris 133, 1283-1285. K 2. SEILIGER, D.: On a fundamental theorem in the statics of a variable system [in Russian]. Papers Univ. Kazan No. 718, 75-82. (We have not been able to see this reference.) 1902 K 1. BuRMESTER, L.: Kinematisch-geometrische Theorie der Bewegung der affin-veränderlichen, ähnlich-veränderlichen und starren räumlichen oder ebenen Systeme, Teil2. Z. Math. Phys. 47, 128-156. Handbuch der Physik, Bd. lll/1. so 786 C. TRUESDELL and R. TouPrN: The Classical Field Theories. K 2. CARDINAAL, J.: Over de beweging van veranderlijke stelsels. Amst. Akad. Vers!. 10, 550-566, 687-691. K 3. CARDINAAL, J.: Over de afbeelding van de beweging van veranderlijke stelsels. Amst. Akad. Vers!. 11, 466-471. K 4. DE SAussuRE, R.: Theorie geometrique du mouvement des corps. Arch. Sei. Phys. Nat. Geneve (4) 13, 425-461; 14, 14-41, 209-231; 18, 25-63 (1904); 21, 36-55, 129-133 (1906). Also issued separately, in parts, Geneva 1902, 1904, 1906. K 5. ScHOENFLIESS, A., u. M. GRüBLER: Kinematik. Enzykl. Math. Wiss. 41, 190-278. K 6. SoMOV, P.: On hinged members with variable elements [in Russian]. Varshavskia Univ. Izv. 1902, Part 8, No. 3, 45 pp. 1903 K 1. SoMOFF, P.: Über einige Gelenksysteme mit ähnlich-veränderlichen oder affinveränderlichen Elementen. Z. Math. Phys. 49, 25-61. 1907 K 1. KoENIGS, G.: Sur !es deformations elastiques qui laissent invariables !es Iangueurs d'une triple infinite de lignes droites. C. R. Acad. Sei., Paris 144, 557-560. 1910 K 1. KRAUSE, M.: Zur Theorie der ebenen ähnlich veränderlichen Systeme. Jber. dtsch. Math.-Ver. 19, 327-339. 1911 K 2. MEHMKE, R.: Analytischer Beweis des Satzes von Herrn Reinhold Müller über K3. K4. K5. K6. K 1. K2. K3. die Erzeugung der Koppelkurve durch ähnlich-veränderliche Systeme. Z. Math. Phys. 58, 257-259. MüLLER, R.: Erzeugung der Koppelkurve durch ähnlich-veränderliche Systeme. Z. Math. Phys. 58, 247-251. MüLLER, R.: Über die Momentanbewegung eines ebenen ähnlich-veränderlichen Systems in seiner Ebene. Jber. dtsch. Math. Ver. 10, 29-89. SKUTSCH, R.: Über die von Herrn Reinhold Müller untersuchte besondere Bewegung eines ähnlich-veränderlichen Systems. Z. Math. Phys. 58, 252-257. STUDY, E.: Die Kinematik der Herren de Saussure und Bricard. Jber. dtsch. Math.-Ver. 19, 255-263. KRAUSE, M.: Zur Theorie der affin veränderlichen ebenen Systeme. Sitzgsber. Akad. Wiss. Leipzig 63, 271-288. KRAUSE, M.: Über räumliche Bewegungen mit ebenen Bahn kurven. Sitzgsber. Akad. Wiss. Leipzig 63, 515-533. MEHMKE, R.: Beiträge zur Kinematik starrer und affin-veränderlicher Systeme, insonderheit über die Windung der Bahnen der Systempunkte. Z. Math. Phys. 59, 90-94, 204-220, 440-442. 1912 K 1. HARTMANN, T.: Zur Theorie der Momentanbewegung eines ebenen ähnlich-veränderlichen Systems. Diss. Rostock, 144 pp. 1913 K 1. DE DoNDER, T.: Sur divers modes de croissance des milieux continus. Bull. Acad. Sei. Belg. 614-621, 642-646. K 2. HERRMANN, E.: Über die einförmige Bewegung des ebenen kreisverwandt-veränderlichen Systems. Diss. Tech. Hoch. Dresden, 93 pp. 1914 K 1. CARL, A.: Zur Theorie der ebenen ähnlich veränderlichen Systeme. Diss. Dresden, 125 pp. K 2. WrNKLER, R.: Über die Bewegung affin-veränderlicher ebener Systeme. Diss. Dresden, 73 pp. 1920 K 1. KRAUSE, M., Assisted by A. CARL: Analysis der ebenen Bewegung. Berlin. 216 pp. 1922 K 1. DELASsus, E.: Stabilite de l'equilibre sur une Iiaison finie unilaterale. Bull. Sei. Math. (2) 46, 283-304. K 2. GAMBIER, B.: Mecanismes transformables ou deformables. Couples de surfaces qui s'en deduisent. J. Math. Pures Appl. (9) 1, 19-76. 1932 K 1. ABRAMEsco, N.: Le mouvement d'une figure plane variable qui reste semblable a elle-meme. Ann. Sei. Norm. Pisa (2) 1, 155-164. K 2. PASCAL, M.: Sul moto di un corpo deformabile ehe si mantiene simile a se stesso. I: Formola fondamentale e proprieta ehe se ne deducono. II: Centro istantaneo di velocita e conseguenze. Rend. Lincei (6) 15, 871-874; 16, 320-324. 1933 K 1. ABGHIRIADI, M.: Sur le mouvement d'une figure plane semblablement variable. Mathesis 47, Suppl., 14 pp. K 2. PASCAL, M.: Sul moto di una figura deformabile piana di area costante e ehe rimane affine a se stessa. Rend. Napoli (4) 3, 71-77. K 3. PASCAL, M.: Sul moto di una figura deformabile piana ehe si conserva affine a se stessa. Rend. Napoli (4) 3, 78-82. K 4. PASCAL, M.: Sul centro istantaneo di velocita nulla nel moto di una figura piana di area costante e a deformate affine. Rend. Napoli (4) 3, 110-113. K 5. PASCAL, M.: Sull'accelerazione nel moto di una figura piana di area costante e a deformate affine. Rend. Napoli (4) 3, 123-126. Additional Bibliography N. 787 K 6. PASCAL, M.: Sulla cinematica affine di una figura piana di area costante. Rend. Napoli (4) 3, 142-144. 1934 K 1. Dr Nor, S.: Considerazioni geometriche sul moto di un corpo deformabile ehe si mantiene simile a se stesso. Rend. Napoli (4) 3, 176-181. K 2. PASCAL, M.: Sul moto di un corpo deformabile di volume costante e ehe rimane affine a se stesso. Atti Soc. ital. Progr. Sc. A 222, 194-195. 1936 K 1. ABRAMESCO, N.: Proprietäti geometrice ale mi~cärii unei figuri plane variabile care rämane asemenea cu ea in sä~i, cand trei drepte ale figurii trei prin trei puncte fixe, sau cand trei puncte descriu trei drepte fixe. Gaz. Mat. Bucarest 41, 409-414. K 2. HARMEGNIES, R.: Sur le mouvement d'une figure plane qui reste homographique a elle-meme. C. R. Acad. Sei., Paris 202, 1323-1324. See also the following items from the "List of works cited": DuRRANDE [1871, 4], and the papers cited in Sects. 140-142. Additional Bibliography N: Non-relativistic kinematics and mechanics in generalized spaces. 1876 N 1. BELTRAMI, E.: Formules fondamentales de cinematique dans !es espaces de courhure constante. Bull. Sei. Math. (1) 11, 233-240. 1878 N 1. LE:vv, M.: Sur la cinematique des figures continues sur les surfaces courbes, et, en general, dans les varietes planes ou courbes. C. R. Acad. Sei., Paris 86, 812-818. N 2. LE:vv, M.: Sur les conditions que doit remplir un espace pour qu'on y puisse deplacer un systeme invariable, a partir de l'une quelconque de ses positions, dans une ou plusieurs directions. C. R. Acad. Si., Paris 86, 875-878. 1881 N 1. BELTRAMI, E.: Sulle equazioni generali dell'elasticitä.. Ann. Mat. (2) 10, 188-211 (1880-1882) = Opere 3, 383-407. 1884 N 1. BELTRAMI, E.: Sull'uso delle coordinate curvilinee neUe teorie del potenziale e dell'elasticita. Mem. Accad. Sei. Bologna (4) 6, 401-488 = Opere 4, 136-179. N 2. HEATH, R. S.: On the dynamics of a rigid body in elliptic space. Phi!. Trans. Roy. Soc. Lond. 175, 281-324. 1885 N 1. KrLLING, W.: Die Mechanik in den nicht-Euklidischen Raumformen. J. reine angew. Math. 89, 1-48. N 2. HrLL, M. J. M.: On some general equations which include the equations of hydrodynamics ( 1883). Trans. Cambridge Phi!. Soc. 14, 1-29. 1888 N 1. CESARO, E.: Sur une recente communication deM. Levy. C. R. Acad. Sei., Paris 107, 520-522. 1889 N 1. PADOVA, E.: La teoria di Maxwell negli spazi curvi. Rend. Lincei (4) 51, 875-880. N 2. SoMIGLIANA, C.: Sopra la dilatazione cubica di un corpo elastico isotropo in uno spazio di curvatura costante. Ann. Mat. (2) 16, 101-115. 1890 N 1. PADOVA, E.: I! potenzialedelleforze elastiche di mezzi isotropi. Atti Ist. Veneto 48 = (7) 1, 445-451. 1894 N 1. CESARO, E.: Sulle equazioni dell'elasticitä. negli iperspazii. Rend. Lincei (5) 32, 290-294. 1900 N 1. DE FRANCESCO, D.: Aleuni problemi di meccanica in uno spazio a tre dimensioni di curvatura costante, I and li. Atti Accad. Napoli (2) 10, Nos. 4 (38 pp.) and 9 (33 pp.). N 2. DE FRANCESCO, D.: Sul moto spontaneo di un corpo rigido in uno spazio di curvatura costante. Atti R. Accad. Sei. Torino 35, 34-38, 231-243. 1901 N 1. BoHLIN, K.: Sur l'extension d'une formule d'Euler et sur le calcul des moments d'inertie principaux d'un systeme de points materiels. C. R. Acad. Sei., Paris 133, 530-532. N 2. DE FRANCESCO, D.: Su aleuni problemi di meccanica, in uno spazio pseudosferico, analiticamente equivalenti a problemi nello spazio ordinario. Rend. Accad. Napoli (3a) 7, 28-38. 1902 N 1. DE F~ANCEsco, D.: Aleuneformole della meccanica dei fluidi in uno spazio a tre dimensioni di curvatura costante, I and li. Atti Accad. Napoli (2) 12, Nos. 9 (18 pp.) and 10 (13 pp.). N 2. STÄCKEL, P.: De ea mecanicae analyticae parte quae ad varietates complurium dimensionum spectat. Libellus Ioannis Bolyai ... ad celebrandam memoriam ... (Claudiopoli), 63-79. 1903 N 1. STÄCKEL, P.: Bericht über die Mechanik mehrfacher Mannigfaltigkeiten. Jber. dtsch. Math.-Ver. 12, 469-481. 50* 788 C. TRUESDELL and R. TouPrN: The Classical Field Theories. 1907 N 1. RrQUIER, CH.: Sur !es systemes d'equations aux derivees partielles auquels conduisent: 1° l'etude des dt\formations finies d'un milieu continu dans l'espace a n dimensions; 20 Ia determination des systemes de coordonnees curvilignes orthogonales a n variables. C. R. Acad. Sei., Paris 145, 113 7- 1139. 1911 N 1. VESSIOT, E.: Sur Ia einematique des milieux continus an dimensions. c. R. Acad. Sei., Paris 152, 1732-1735. 1912 N 1. DE DoNDER, T.: Sur Ia cinematique des milieux continus. Bull. Acad. Roy. Belg. Cl. Sei., 243-251. N 2. ZoRAWSKI, K.: Über gewisse Pfaff'sche Systeme, welche bei Bewegungen kontinuierlicher Medien invariant bleiben. Bull. Int. Acad. Sei. Cracovie A 1912, 436-461. 1913 N 1. DEL RE, A.: Sulle equazioni generali per Ia dinamica negli spazii ad n dimensioni ed a curvatura costante, Ann. Mat. (3) 22, 63-70. 1926 N 1. SYNGE, J. L.: App!ications of the absolute differential calculus to the theory of elastieity (1924). Proc. Lond. Math. Soc. (2) 24, 103-108. 1930 N 1. ToNoLo, A.: Une interpretation physique du tenseur de Riemann et des courbures principales d'une variete V3 • C. R. Acad. Sei., Paris 190, 787-788. N 2. ToNOLO, A.: Equazioni intrinseche di equilibrio dell' elastieita negli spazi a curvatura costante. Rend. Sem. Mat. Fadova 1, 73-84. 1931 N 1. ToNOLO, A.: Sistemi isostatiei dei corpi elastici negli spazi a curvatura costante. Rend. Sem. Mat. Fadova 2, 152-163. 1933 N 1. CARTAN, E.: La einematique newtonienne et Ia theorie des espaces reglees a connexion euclidienne. Ass. Franc. Avancem. Sei. 19-20. N 2. CARTAN, E.: La einematique newtonienne et les espaces a connexion euclidienne. Bull. Math. Soc. Roumaine Sei. 35, 69-73. N 3. FrNZI, B.: Equazioni intrinseche della meccanica dei sistemi continui perfettamente od imperfettamente flessibili. Ann. Mat. (4) 11, 215-245. N 4. TEODORIA, L.: Sur Ia einematique du corps solide dans l'espace euclidien a n dimensions. Bull. Math. Soc. Roumaine Sei. 35, 243-247. 1934 N 1. PASTOR!, M.: Sulle equazioni della meccanica dei mezzi isotropi non euclidei. Rend. Lincei (6) 19, 566-572. N 2. PASTOR!, M.: Sulla dissipazione di energia nei fluidi viscosi. Rend. Ist. Lombarde (2) 67, 823-848. 193 5 N 1. WESTERGAARD, H. M.: General solution of the problern of elastostatics of an n-dimensional homogeneaus isotropic solid in an n-dimensional space. Bull. Amer. Math. Soc. 41, 695-699. 1936 N 1. LaTZE, A.: Die Grundgleichungen der Mechanik im elliptischen Raum. Jber. dtsch. Math.-Ver. 46, 51-70. 1937 N 1. LAMPARIELLO, G.: Varieta sostanziali nel moto di un sistema continuo. Rend. Lincei (6a) 15, 383-387. 1940 N 1. 0RTVAY, R.: The physical implications of some new viewpoints in mathematics [in Hungarian]. Mat. Fiz. Lapok 47, 111-138. N 2. SKOLEM, T.: A little study on transfinite mechanics [in Norwegian]. Norsk Mat. Tidsskr. 22, 5-9. 1942 N 1. BLASCHKE, W.: Nicht-Euklidische Geometrie und Mechanik. Hamburg. Math. Einzelschr. 34. 1944 N 1. BLASCHKE, W.: Nicht-Euklidische Mechanik. Sitzgsber. Akad. Wiss. Heidelberg 1943, No. 2, 10 pp. 1951 N 1. SANTALO, L. A.: On permanent vector-varieties in n dimensions. Portuga!iae Math. 10, 125-127. N 2. SYNGE, J. L.: On permanent vector-lines in n dimensions. Proc. Amer. Math. Soc. 2, 370-372. See also the following items from the "List of Works Cited": CLEBSCH [1857, 1; §§ 1-4], FADOVA [1889, 7], ZORAWSKI [1900, 12], [1901, 17], ZERMELO [1902, 10], ZORAWSKI [1911, 13 and 14], DE DONDER [1912, 2], ZORAWSKI [1912, 8]. Additional Bibliography P: Principles of Mechanics. Any partial bibliography of work on the concepts and axioms of mechanics from the origins through the time of LAGRANGE would be misleading. No adequate critical history has ever been written. The remarks on this subject given in treatises or general histories of physics are often mendacious and usually so incomplete and inaccurate as to be totally misinformative. 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L.: Relativity: The Special Theory. Amsterdam: North Holland Publishing Co.; New York: Interscience Publ. R 5. TAUB, A. H.: Isentropic hydrodynamics in plane symmetric space-times. Phys. Rev. (2) 103, 454-467. R 6. WrNOGRADSKI, J.: Sur Je tenseur impulsion-energie metrique et Je theoreme de Noether. Cahiers de Phys. 67, 1-5. 19 57 R 1. ARNOWITT, R. L.: Phenomenological approach to a unified field theory. Phys. Rev. 105, 735-742. R 2. MARX, G.: Innere Arbeit in der relativistischen Dynamik. Acta phys. Acad. Sei. Hungar. 6, 353-379. R 3. SYNGE, J. L.: The Relativistic Gas. Amsterdam. R 4. TAUB, A. H.: Singular hypersurfaces in general relativity. Illinois J. Math. 1, 370-388. 1959 R 1. SYNGE, J. L.: A theory of elasticity in general relativity. Math. Z. 12, 82-87. R 2. TAUB, A. H.: On circulation in relativistic hydrodynamics. Arch. Rational Mech. Anal. 3, 312-324. Appendix. Tensor Fields. By J. L. ERICKSEN. With 2 Figures. I. Preliminaries. 1. The nature of this appendix. This appendix is a heterogeneaus collection of results on tensor fields. The reader is assumed familiar with the elements of vector and tensor analysis and of matrix algebra. The author is indebted to Drs. W. NoLL, R.A. TouPIN, C. TRUESDELL and M. K. ZoLLER for bringing to his attention many references and helping to correct many errors. Such defects as remain are of course the responsibility of the author. a) Notation. 2. Tensor notation. We employ the methods and notation of the tensor calculus and, to a lesser extent, matrix algebra and GIBBS' vector analysis. Until further notice, write xl, x2, ••• , x" for the co-ordinates of pointsinan n-dimensional space. For abbreviation, we let a: stand forthisset of Co-ordinates. Until further notice, italic lower case letters other than x will be used for the kernel indices and tensor indices of tensors. Also, for a given set of tensor components a"···"'p ... q we write a; for "the components a"···"'p ... q of the tensor a" or "the tensor having the components a"···"'p ... q"• we write "the tensor a"--·"'p ... q"• or, more often, "the tensor a ". W e employ the summation convention for diagonally repeated tensorial indices and indices of Christofiel symbols. The notation a:::;" denotes the covariant derivative, except when otherwise noted. A square bracket enclosing m running indices indicates that the tensor is completely antisymmetrized with respect to these indices. This is done by permuting the indices in all possible ways, attaching a positive (negative) sign to the tensors corresponding to even (odd) permutations, adding these tensors and dividing by m!. Parentheses inclosing m running indices indicate that the tensor is completely symmetrized with respect to the enclosed indices. This is done in the same way except that the positive sign is attached to all permutations. For example W e use the short notation \ IX\ = 8 (xl, x2, ... , x") a:, - o(X1,X2, ••• ,Xn) (2.1) (2.2) when the x" are given in terms of parameters XK by a mapping a:= a: (X), whether this be regarded as a co-ordinate transformation or as a point transformation. Unless greater or lesser generality is asserted explicitly or is evident from the context, the space considered in this treatise is Euclidean 3-space with real Sect. 3. Matrix and vector notations. 795 co-ordinates. The components of associated tensors are regarded as different sets of components of the same tensor; the same kernelleHer is used for all these components, and the context will make clear whether a stands for a particular set of components or is intended as a general symbol for all components. In particular, weshall often write 1 for the metric tensor gkm• its mixed components ö!:,, and its inverse gkm_ Setting g == det gk m and assuming g > 0, we use the notations (2.3) where n is the dimension of the space and the s's are the usual permutation symbols, with sl...n = sl. .. n = 1 in all co-ordinate systems. The e's are examples of axial tensors, which transform as absolute tensors under co-ordinate transformations with positive J acobian, and und er those only. In Euclidean space, a reetangular Cartesian co-ordinate system is available. If we write z or z1, z2, ••• , zn for the co-ordinates of points, this in itself indicates that we are using reetangular Cartesian co-ordinates, and at the sametime we follow the notation and the summation convention of Cartesian tensors, writing all tensor indices as subscripts and summing on repeated indices. Moreover, for the position vector from a fixed point to x, we write pk or pk or p. If the fixed point has reetangular Cartesian co-ordinates 0z, then p is the vector whose covariant and contravariant components reduce to z- 0z in reetangular Cartesian systems. If 0z = 0, then in reetangular Cartesian systems and only in such systems we have x =p =Z. Formulae in which p occur should be invariant when 0z is replaced by a different fixed point 0z*; in most cases, verification of this invariance is left to the reader. In general co-ordinates, p and x are entirely different quantities. For the vector element of arc we write dx in general Co-ordinates, dz in formulae whose validity is limited to reetangular Cartesian co-ordinates. Curvilinear components of p at x generally differ from those at the fixed point. Except where otherwise noted, we use the former. 3. Matrix and vector notations. For second order tensors, we sometimes use matrix or dyadic notation1. For example, we write c =a · b for the equation c!:, = a~ b!;.; a-1 for the tensor, if it exists, which is related to a by a · a-1 =1, the components of a-1 being a;;;lk; while a' denotes the transpose of a, given by a~m =amk· More generally, the single dot operation is defined by the equations a · b == t and b · a = u, where the tensors t and u are given by (3.1) In three-dimensional spaces, also the single cross operation of GIBBS 2 may be defined. If a is any vector, b any tensor, then a X b == c and b X a == d, where the tensors c and d are given by (3.2) The cross or vector tx of a second order tensor t is the axial vector given by tk = ekrs t = ekrs t x rs [rs]· (3 ·3) Thus apart from algebraic sign, the contravariant components of tx are the components of 2 Vg t[rs]• not of Vg t[rs]• andin particular (grad b)x =curl b. Since a 1 Alternatively, our direct notation may be interpreted in terms of linear transformations. Cf. e.g. HALMOS [1942, 1, Chap. II], and, for applications to kinematics and mechanics, NoLL [1955. J]. 2 [1881, J], [1884, J], [1909, 1]. 796 J. L. ERICKSEN: Tensor Fields. Sects. 4, 5. skew-symmetric !ensor of second order in a three-dimensional space has at most three independent components, in any equation where it appears it may be replaced by its cross. When both t and u are skew-symmetric, (3.4) We introduce the symbol '*'· which may be read "trident", to stand for a scalar, vector, or tensor of any order, so long as the formulae in which it appears have a sense. By grad '*'· div W, and curl W we mean the tensors a, b, and c given by a,. ........ = w ........ ,k. b"·""'= '*''""· ... ,,. or c:. ..... =e"" w............ (3.5) where the last applies only in the three-dimensional case. These definitions are so framed that the identities div curl W = 0 curl grad W = 0, familiar when W is a vector, hold for a tensor of any order. For positive integral K, the K-th power lJCK) of a vector bis the tensor b"'• .. . b mk, The definition is extended to all integral K and put in inductive form as follows: lJH)= 0, lJ(O) 1, lJ(K+l) = b(K) b. (J.6) We denote by {bCKl '*'} the expression1 {b(K) W} = b(K) '*' + b(K-l) Wb+ ... + Wb(K). (3.7) 4. Projections. If n is a unit vector, we define the normal projection Wn and the tangential profection W1 of w onto the direction of n by (4.1) the latter formula being restricted to three-dimensional space. When W = c, a vector, we write Cn rather than Cn for the normal projection. Again in 3-space, consider the special Co-ordinate transformations which leave x1 fixed: zU= x1, x•« = x•« (x2, zll), and write 1g= det g .. p. where cx, ß have the range 2, 3. Then, considering only those co-ordinate systems such that 1g > 0, we see that the quantities 1t defined by (4.2) where t is a three-dimensional tensor, transform as components of a tensor 2 an the surface x1 = const. The two-dimensional tensor 1t may be called the skew projection of t onto the x1-direction. Taking the trace of Eq. (4.2) yields a relation3 between the components of tx and the traces of the three skew projections kt: t; = (gfkg)! k t~. (4. 3) b) Use of complex co-ordinates. 5. Use of tensor methods in complex co-ordinates. The formalism of the tensor calculus applies to complex as well as real co-ordinate transformations; for problems in the Euclidean plane, complex Co-ordinates are sometimes convenient. Suppose, for example, that we introduce complex co-ordinates x1 = z1 +iz2 , x2 =XI, where the z,. arereetangular Cartesian and the superposed bar denotes the complex conjugate. The usual formulae of tensor transformation 1 TRUESDELL [1949, 3], [1951, 2], [1951, 3], [1954, 2, Chap. I]. 2 This result and Eq. (4.2) are due in principle to CAUCHY [1841, 1, Th. VIII]. Cf. TRUESDELL [1954, 6, § 2]. 3 When both tx and ,.e are referred to physical components (Sect. 13) in an orthogonal co-ordinate system, the factor (gf1g)! disappears from Eq. (4.3). Sects. 6, 7- Preliminary definitions. 797 yield the covariant components t""' (x) in terms of the Cartesian components t".,.: 4tu (x) = 4t22 (x) =tu-t2 2- i (t12 + t21), } 4t12 (x) =4t21 (x) =tu +t22 +i(t12 -t21). (5.1) For the special case1 when t is the metric tensor, this yields gu(x) =g22 (x) =0, g12 (x) = g21 (x) = ! ; hence g11 (x) = g2 2 (x) = 0, g12 (x) = g21 (x) = 2. All results which do not depend essentially on conditions of reality carry over to complex co-ordinates. As is clear from the above example, the inequalities gu > 0, gug22 -g~ >0 which, in real co-ordinates, express the fact that the g""' are coefficients of a real, positive definite quadratic fom1, do not. Since we occasionally use such conditions, our treatment is restricted to real co-ordinates except where otherwise noted. 6. Complex representation of plane second order Cartesian tensors. Retuming to the example (5.1), we note that a rotation of the co-ordinates z" induces on x" the transformation xu = xu = x1 e•'P, where cp is the angle of rotation, and that the tensor t11 .,. (x) transforms according to the simple laws t12 (x*) = t21 (x*) = t12 (x) = t21 (x), fu (x*) = t22 (x*) = t11 (x) e-2i'P. (6.1) In linear elasticity, t, interpreted as the stress tensor, satifies the equilibrium equations and compatibility conditions for generalized plane stress if and only if 2 112 (:r) = t21 (:r) = tZ = I + J. 111 (:r) = g - x2 /', (6.2) where I and g are arbitrary functions of the complex variable x1• Similar results hold for plane strain. In this context, Iu (:r) is sometimes called the conjugate stress deviator to indicate that it is unaltered if t is replaced by t + K 1, where K is any scalar and 1 is the two-dimensional unit matrix. Comparison of these results with their Cartesian analogues shows the simplicity that can result from using complex co-ordinates. In both applications all three Cartesian components of t may be calculated easily from the two complex components 112 (:r) and Iu (:r), the former of which may be regarded as a scalar. It follows easily from Eq. (5.1) that 112 (:r) = 0 if and only if I is symmetric and traceless, and that in the symmetric case 111 (:r) = 0 if and only if t = K 1, where K is a scalar. When t is symmetric, by Eq. (5.1) 2 we see that t12 (x) is always real. As will be shown in Sect. 48, there exists a real rotation that renders t diagonal. Letting x* =zi +izi, from Eq. (5.1) we see that tu(x*) is real. By Eq. (6.1) we get t11 (x) =fu(x*) i'~', and hence Eq. (5.1) yields 3 2112 tan2cp= ----. 111-122 II. Dimensionsand physical components. a) Dimensionsofatensor and its components. (6.3) 7. Preliminary definitions. Among the transformations to which quantities occurring in physics are subject are those of dimensional units 4. In constructing a system of measurement, in principle one must introduce a unit of measurement for each type of physical quantity to be considered. It is customary to lay down a finite set of units U0 ( a = 1, ... , n) as fundamental units, and to require that 1 For other examples, cf. GREEN and ZERNA [1954, 1]. 2 MusKHELISHVILI [1953, 1, Chap. 5] gives a rigorous derivation of these results, which he attributes to KoLOsov [ 1909, 4]. 3 MAXWELL [1870, 3, PP· 194-195]. 4 Dimensional analysis remains a controversial and somewhat obscure subject. We do not attempt a complete presentation here. 798 J. L. ERICKSEN: Tensor Fields. Sect. 8. every unit be expressible in terms of them in the symbolic form Uf' ... U~n, where the K's arereal exponents. A fundamental unit may be regarded as a measure of a definite type of entity, such as length or time, the "size" of the unit being assigned arbitrarily. To compensate for this arbitrariness, we regard any other system of units, obtainable from the given set by changing the size of the fundamental units independently, as equivalent or indistinguishable. More precisely, we require physical equations to be invariant under the transformations (7.1) hereafter called dimensional transformations, the ua being arbitrarily chosen real numbers. A function Q (U) of the units Ul, ... , Un is called a measure number having the dimension [Uf' ... U~n] provided that it transforms under Eq. (7.1) according to the law Q (U*) = Q (U) Uf' ... U~n. The appropriate units for Q are Uf' ... U~n, in accordance with the usual notion that the product of a measure nurober by its units is unaffected by dimensional transformations. We write [1] for [U~ ... un Fora given group Gof Co-ordinate transformations, a set of functionstZ:::~(;r, U) are said to be the components, relative to the co-ordinates xk and units Ua, of a tensor of weight w having the dimension [Uf' ... U~n] provided that, under Eq. (7.1) and an arbitrary co-ordinate transformation x*k = x*k (x) of G, these functions transform according to the law (7.2) With a fixed choice of co-ordinates, each component of t is clearly a measure nurober having the dimension [Uf' ... U~n]. Statements to the effect that physical equations are "in their simplest form" generally mean only that they reduce to equations relating such tensors. 8. Physical dimensions of tensors. In classical mechanics, it is customary to take G to be the orthogonal group1, to require one of the fundamental units to be a unit L of length, and to assume that the co-ordinate differentials dzk transform as a Cartesian vector having the dimension [LJ, in agreement with the notion that ds2 = dzk dzk is an absolute scalar having the dimension [L2]. Using physicallaws and other devices, one determines the appropriate Cartesian tensor law to represent the entities to be considered. The dimensions so assigned in reetangular Cartesian systems we call physical dimensions to distinguish them from other dimensionstobe introduced below. From this point of view the Cartesian metric tensor bkm can consistently be regarded as a dimensionless tensor, by which is meant a tensor having the physical dimension [1]. In passing to orthogonal curvilinear co-ordinates, one assigns a dimension to each variable introduced as a co-ordinate 2• The common procedure is to use LAM:E's theory of orthogonal curvilinear co-ordinates 3, or some other mathematically equivalent procedure, to obtain curvilinear components of Cartesian tensors. Viewed as scalars, these components have a common dimension, which is the physical dimension of the Cartesian tensor which they represent. Under co-ordinate transformations they do not transform as simply as do tensor com1 For simplicity, we restriet ourselves to time independent co-ordinate transformations. 2 In principle, this dimension may be arbitrary. In practice, each co-ordinate is a homogeneous function of the reetangular Cartesian co-ordinates, so that it is natural to assign it the dimension [L K], where K is the degree of the homogeneaus function. 3 The method is given in [1840, 1], [1859, 1, Les;ons 1-2]; a typical application, in [1841, 2, §§VII-VIII], [1859, 1, §§ 144-147]. Sect. 9. Absolute dimensions of tensors. 799 ponents. Since Rrccr and LEVI-CEVITA1 set down the foundations of tensor calculus, there has been an increasing tendency to use tensor components in place of LAME's, primarily because tensor components transform more simply under co-ordinate transformations. In treating Cartesian tensors having dimensions, one fixes the units and transforms the components according to the appropriate tensor law. Having done so, one can assign a dimension to each component of the tensor consistent with the tensor law, the dimension assigned to the Cartesian tensor, and the dimensions assigned to the co-ordinates. In general, thesedimensionswill be different for different components and from those of the Cartesian tensor. At least for an absolute tensor t, the dimension of the scalar which we denote by the Russian letter ,Il,, n = lit tk ... m ~t- k ... m ' (8.1) will be unaltered by co-ordinate transformations, and it is customary to regard this dimension as the physical dimension of t. Apparently McCoNNELL 2 was the first to give a satisfactory explanation of the relation between the "components" of mathematical physics and tensor components for tensors of arbitrary order. His method might be explained as follows. Given a tensor field referred to an orthogonal curvilinear co-ordinate system, let its physical components 3 at a given point be defined as its corresponding tensor components, at that point, in a reetangular Cartesian system whose axes are parallel to the co-ordinate curves at the point. In mathematical physics, the term "components" usually means these physical components, which henceforth we denote by t =--"-log fg""' uSm and k =m k=j=m, (14.6) k=j=m. To calculate physical components of derivatives, one has only to substitute these formulae into t(k ... m,P>=a:p t(k ... m) +F+ ··· +} (14.7) +F be that portion of the surface of the sphere of radius p with center at P which lies in the interior of t. If \1:' be such that lim I da \I:'P"' = 0, p_.oo,P (24.1) we write 2, respectively, \l:'n=ö(p-m-2), \l:'t=o(p-m-2), \l:'=o(p-m-2). (24.2) For (24.1) to hold it is clearly sufficient that each component of \l:'n, \l:'t, and \1:', respectively, be o (p-"'-8) as P~ oo, which motivates the notation. The symbol o may be read "lower mean order than ". Weshall frequently wish to state boundary conditions of the following type: 1. On each finite boundary of v, \1:' = 0. 2. In any portion of v extending to \1:', \1:' = 0 (p-m-2). As apart of the convention of Sect. 23, we agree that henceforth \1:' = 0 or o(p-"'-8) on ~ (24.3) shall serve as an abbreviation for the statements 1 and 2. Analogons conventions are adopted for statements in terms of \l:'n or '*'t· b) Circulation, flux, total,. and moments. 25. Definitions. Let \1:' be any integrable tensor field. The line integral I dz · \1:' =I dz,. \l:'k...... (25.1) 1: 1: is called the flow of \1:' along the curve c. When c is closed, it is called a circuit; the integral (25.1) is then called the circulation of \1:' around c and is written ~dz· \1:'. {25.2) 1: Two curves are reconcilable 3 in a given point set provided one can be continuously deformed into the other while remaining in the set. A circuit which can be 1 [1929, 1, Chap. IV]. 2 The notations were introduced by TRUESDELL [1954, 2, § 4]. 3 These concepts and names derive from KELVIN [1869, 1, §58 and § 6o(a)]. Topologists use the ward "homotopic" in place of "reconcilable", whereas the latter ward is generally employed in mechanics. KELVIN stated that he applied RIEMANN's [1857, 1] theory of multiple connectivity as presented by HELMHOLTZ [1858, 1], but the concept of reconcilable curves does not appear in these papers. Sect. 26. GREEN's transformation. 815 continuously shrunk to a point while remaining in a given set is said tobe reducible in the set. The surface integral f da· \V (25.3) 4 is called the flux of \V across the surface 6. When 6 is closed and da is directed outward, the integral (25.3) is called1 the flux of \V out of 6 and is written ~da· \V. (25 .4) 4 A closed surface which can be continuously shrunk to a point while remaining in a given region is said to be reducible in it. With the notation (3 .7), the volume integral f {p(K) \V} dv (25.5) " is called the K-th moment of \V with respect to the origin. The zeroth moment, f \Vdv (25.6) " is called 2 the total \V in v-. c) The transformations of GREEN and KELVIN. 26. GREEN' s transformation. We shall employ Green' s transformation s in the forms fgrad\Vdv=~da\V, fdiv\Vdv=~da·\V, fcurl\Vdv=fdax\V, (26.1) " d ". d ". d where it is assumed that \V is single valued and continuous throughout the closure of the finite region v-, bounded by d, that grad \V is continuous throughout a finite number of subregions of v- whose sum is v-, and that the volume integrals are convergent. In the terminology of Sect. 25, Eq. (26.1) 2 states that the total divergence of a quantity in a region is equal to the flux of the quantity out of the boundary of the region. Another form of GREEN's transformation is 4 f [b {cp. The case K=O is due to BuRGATTI [1931, 3]. 816 J. L. ERICKSEN: Tensor Fie!ds. Sects. 27, 28. where b and c are arbitrary continuously differentiable vector fields, p is the position vector, and K is any integer. 27. POINCAR:E's generalization. In n dimensions, the tensor element of area of an oriented m-dimensional surface dm, given parametrically by xk = xk ( u), is d r ... s- I ~[r. 0. ozS] d 1 d m S(m) - m. oul oum U • . . U ' (27.1) which transforms as an absolute contravariant tensor of order m under co-ordinate transformations, as an absolute scalar under parameter transformations with positive Jacobian. Let !lm be bounded by the surface !lm_1 , given parametrically by xk = xk (v). Let v be a vector defined on dm_1 , directed outward relative to !lm. Arrange the parametrizations so that the vectors vk, oxkjovl, ... , axkjvm-1 and 8xkj8u1, 00.' axkjoum, in these orders, have the same orientation relative to (jm. For any continuously differentiable covariant tensor field t of order m, we then have 1 --- t dsru ... s = --t dsru ... s = t dsu ... s f ox' a u ... s (m) Ja ox[r u ... s] (m) ~ u ... s (m-1). (27.2) It then follows from the transformation properties of the integrands that each integral transforms as an absolute scalar. The underlying space need not be Euclidean or have any other geometrical structure. In metric spaces, or affine spaces with a symmetric connection, where covariant differentiation is defined, we have a!ru tr ... s]=trr ... s,u]• which makes the invariance of the integrals more obvious. Various alternative forms of Eq. (27.2) are available in the literature 2 • In three-dimensional Euclidean space, GREEN's transformation may be deduced from Eq. (27.2) with m = 3, KELVIN's transformation (Sect. 28) from Eq. (27.2) with m=2. 28. KELVIN's transformation. KELVIN's transformation, commonly 3 called " STOKEs' theorem ", is J da. curl \j! = cji dz. \j!, (28.1) 6 e subject to the usual right-handed screw convention connecting the sense of dz with that of da. \j! is assumed continuously differentiable in a region whose boundary contains 4 d. It is important to note that the field \j! need not be defined on both sides of 6. In the terminology of Sect. 25, Eq. (28.1) states that the circulation of a quantity araund a closed circuit equals the flux of its curl across any surface bounded by the circuit. It is possible to give forms of Eq. (28.1) which contain dz \j! and dzx \j! on the right-hand side, in the style of Eq. (28.1). Since these are awkward in dyadic notation, we write the former in Cartesian tensor notation as follows; (28.2) Contradingthis equation on k and m yields Eq. (28.1), while replacing km by [km J yields the second alternative mentioned above. 1 This result is due in essence to POINCARE [1887, 1, § 2], [1895. 1, § 7]. 2 See e.g. ScHOUTEN [1951, 1, Chap. IVl. 3 We follow TRUESDELL [1954, 2, § 8] ih departing from usage. See the appendix to this section. 4 See LICHTENSTEIN [1929, 2, Kap. 2, § 3] for references to works proving the result under weaker assumptions. Sect. 29. Definitions. 817 Appendix. History of Kelvin's transformation 1• The plane form of Eq. (28.1) was discovered by AMPERE [1826, E1]. The three-dimensional form was discovered by KELVIN, who communicated it to STOKES in a letter dated July 2, 1850 {LARMOR's annotation to the 1905 reprint of [1854, 1] ). Independent discovery, as weil as priority in publication, is due to HANKEL [1861, E 1, § 7] (cf. also RocH [1863, E 1, § 4]). KELVIN's proofs were published later [1867, 1, § 190(j)] [1869, E1, § 60(q)], andin the second of these publications he claimed priority. The only connection of STOKES with the matter was to set proof of the result as an examination question [1854, T2]. Later KELVIN [1879 ed. of [1867, 1, § 190(j)]J mentioned this fact, which had been noted earlier by MAXWELL [1873, 3, § 24], after whose custom the common name has been adopted. It was KELVIN who first realized the significance of Eq. (28.1). showing how to use it for the proofs of important kinematical theorems. If additional names beyond KELVIN's aretobe attached to Eq. (28.1), they should be those of AMPERE and HANKEL. V. Vector fields. a) Vector lines, sheets, and tubes. 29. Definitions. A vector line of a given vector field c is a curve everywhere tangent to c. Parametrie equations of a vector line are the solutions x" = x" (u) of the differential system or dU d.c XC =0, l !!;~" cml = 0. (29.1) If c is continuous in a closed region, there exists at least one vector line through each interior point of the Fig. 2. Vector tube and cross·sections. region; if c also satisfies a Lipschitz condition 2, there is exactly one through each point where c =f= 0. A surface everywhere tangent to c is a vector sheet of c. Such surfaces f (~) = const are non-constant solutions of c · gradf = 0, or ,. of c oxk =0. (29.2) Equivalently, a parametric representation x =X (u, v) of a vector sheet satisfies o.c o:JJ o.c o.c C·a;-Xav=O, -0u-Xav=f=O. (29.3) A vector line can always be represented locally as the intersection of two vector sheets. Consider a field c possessing a unique vector line through each point of a region r, and let c be a curve which is not a vector line of c; then, locally, the surface swept out by the vector lines intersecting c define a unique vector sheet of c associated with c. When c is a circuit, this vector sheet is called a vector tube of c. Let 6 be a vector tube of c, bounded by two simple circuits c1 and c2 , each embracing the tube once. Let 61 and 62 be two surfaces whose complete boundaries are c1 and c2 respectively (Fig. 2); when the unit normals to 61 and 6 2 are given senses so as to subtend an acute angle with c, d1 and 62 are cross-sections of the tube. Assuming c is integrable on the cross-sections 61 and 6 2 , we call its flux through them the strengths of the vector tubes at these cross-sections. Letting 63 = d + 61 + 62 , we have ~da· c = J da· c + J da· c, (29.4) •• ., •• 1 Cf. TRUESDELL [1954, 2, § 8}. 2 See e.g. KAMKE [1930, 1, §§ 15, 16] for proof of uniqueness under weaker·conditions. Handbuch der Physik, Bd. III/1. 52 818 J. L. ERICKSEN: Tensor Fields. Sect. 30. since da· c =0 on ~. In Eq. (29.4), the normals to 1 and 2 are both taken outward. If c is directed inward relative to d on :1 1 and we take the unit normal on :11 tobe directed inward, we have ~da· c = J da· c- J da· c. (29.5) da d:a d1 Hence the flux of c out of a surface consisting of a vector tube of c and two crosssections equals the ditference between the strengths of the tube at these cross-sections. Let v be the region bounded by :13 • Assuming sufficient smoothness for GREEN' s transformation (26.1) to hold, from (29.5) we have fdivcdv = J da ·C-J da· c. (29.6) V 62 61 H ence the total divergence of the field c in a region bounded by a vector tube of c and two cross-sections equals the ditference between the strengths of the tube at these crosssections. The above definitions may be extended to n-dimensional spaces which need not possess a metric tensor. lf c is any relative or absolute contravariant vector, the vector lines of c are the curves obtained as solutions of Eq. (29.1) 2 • As before, in a closed region where c satisfies a Lipschitz condition there exists a unique vector line through each interior point at which c =1= o. The vector lines intersecting an rn -1-dimensional surface c, when 1 < rn< n, then sweep out an rn-dimensional vector sheet d of c, this being defined as an rn-dimensional surface given parametrically by xk= xk(ul, ... , um), where ();x[k ox' --···--c•l-o 8u1 aum - ' (29.7) or, equivalently, as the set of points satisfying fa (::~:) = const, a = 1, ... , n-rn, where the f's are functionally independent solutions of Eq. (29.2) 2 • lf c be closed, 6 is called a vector tube of c. For an n-dimensional vector tube of a relative vector of weight one, generalizations of Eqs. (29.4) and (29.5) are immediate, and a generalization of Eq. (29.6) is obtained by setting tr ... s=ck Ekr ... s in (27.2). 30. Invariants of vector lines. Consider a region in which the magnitude c of a vector field c is non-zero. Let t = cjc 'denote the unit tangent to the vector lines of c, and put Then Now so that A =t·curlt, D = divt. A = t · curl (cjc) = (tjc) · curl c = c- 2 c · curl c. grad c = grad (c t) = (grad c) t + c grad t, (30.1) (30.2) (30. 3) div c = t · grad c + c div t = ~; + c D, (30.4) curl c = grad c X t + c curl t, (30.5) where djdt denotes differentiation with respect to arc length along the vector line1 • For any unit vector e and any vector v, we have the identity v=ee·v-ex(exv), (30.6) so that curl c = t t · curl c - t X (t X curl c) . (30.7) 1 Here and elsewhere, we denote by dfde the directional derivative in the direction of a unit vector e. Sect. 31. Solenoidal fields I: Integral properties. 819 Using Eqs. (30.5) and (30.6) with e = t, v = grad c, we get txcurlc =tx(gradcxt) +ctxcurlt, } = grad c - t t · grad c + c t X curl t. (30.8) Introducing the unit principal normal n and binarmal b of the vector lines, we have grad = t t · grad + n n · grad + b b · grad, which enables us to rewrite Eq. (30.8) as t X curl c = n n · grad c + b b · grad c + c t X curl t, dc dc = n !in + b Tb + c t x curl t, dc dc = n dt + b Tb - c t · grad t, ( dc ) dc =n dn- cx + bdb, where x is the curvature of the vector line. Hence dc ( dc) -tx(txcurlc)=libn + cx -!in b. (30.9) (30.10) (30.11) From Eqs. (30.2), (30.7), and (30.11) we then obtain an intrinsic relation for 1 curl c: dc ( dc) curl c = c A t +Tb n + c x - dn b. (30.12) The special case c = 1 yields curl t = A t + x b. (30.13) Following LEVI-CIVITA 2, we call theinvariant A the abnormality of the field, this name being suggested by the following geometric interpretation, attributed to BERTRAND by LECORNU 3. Consider any regular surface with unit normal N such that N = t at some interior point P. On this surface, let d be any region containing P and bounded by a circuit c, reducible on the surface, and let s be the area of d. Then by KELVIN's transformation (28.1) s-1 ~da:· t = s-1 f N· curltda. {30.14) e • Reducing c to the point P, so that s --+0, N --+t, by using Eq. (30.2) we get Ajp = lims- ~da:·t S------)>0 c {30.15) lf there exists a congruence of surfaces with t as normal, we may take d to be a region on one of these, obtaining from Eq. (30.15) A = 0. Thus A may be regarded as a measure of the departure of c from the property of having anormal congruence of surfaces. b) Special classes of fields. 31. Solenoidal fields 1: Integral properties. An integrable vector field c is called solenoidal provided its flux out of every reducible closed surface 6 in its 1 MASOTTI [1927, 2]. Cf. BJ0RGUM [1951, 4, § 2.3], CoBURN [1952, 1, § 2]. GILBERT [1890, 2] (cf. also KLEITZ [1873, 3, §§ 2, 8, 40]) had given formulae for the gradient and divergence of a tensor in terms of the radii of curvature of the members of a triply orthogonal system. 2 [1900, 1]. The names "torsion of the curve system" and "torsion of neighboring vector lines" have also been proposed; cf. BJ0RGUM [1951, 4, § 2.4]. 3 [1919, 1]. 52* 820 J. L. ERICKSEN: Tensor Fields. Sect. 31. region of definition is zero. From Eq. (26.1) 2 follows Kelvin's 1 characterization: A continuously differentiable field c is solenoidal if and only if its divergence vanishes divc = o. (31.1) There is also the characterization oj Helmholtz 2 : A field c, continuously differentiable 3 in a closed region, is solenoidal if and only if the strength of every vector tube is the same at all cross sections. The necessity of this condition follows immediately from Eq. (29.6). If the strength of every vector tube is the same at all cross sections, by Eq. (29.5) it follows easily that Eq. (31.1) holds, hence that c is solenoidal. In Sects. 31 and 32 we assume c is a continuously differentiable solenoidal field. Then by Eq. (13.7) we calculate div (cp(K+Il) = c · gradp(K+I) = {p(K) c}; hence by Eq. (26.1) 2 we obtain 4 for the moments (25.5) of c C(KJ= f {p(Klc}dv =~da· cp(K+I) = ~dacnp(K+IJ. .. d d (31.2) (31. 3) Hence the moments of a solenot.dal field are determined by the boundary values of its normal projecrion cn. I f cn = 0 on ~, then all moments of c over the volume u bounded by ~ vant"sh. More generallys, div (c \V)= c · grad \V, (31.4) whence follows J c · grad \V d v = ~da· c \V = ~ da cn \V. (31. 5) .. d d Hence, if c and \V are continuously differentiable and c is solenoidal, then the total c · grad \V in a region u is determined by the boundary values of cn \V. I f cn = 0 on ~. then the total c · grad \V in u is zero 6• 1 [1851, 1, § 74]. 2 [1858, 1, § 2]. From the fact that the vorticity field is solenoidal, HELMHOLTZ [1858, 1, § 2] concluded that "Es folgt hieraus auch, daß ein Wirbelfaden nirgends innerhalb der Flüssigkeit aufhören dürfe, sondern entweder ringförmig innerhalb der Flüssigkeit in sich zurücklaufen, oder bis an die Grenzen der Flüssigkeit reichen müsse. Denn wenn ein Wirbelfaden innerhalb der Flüssigkeit irgendwo endete, würde sich eine geschlossene Fläche construiren lassen, für welche das Integral f q cos {} dw nicht den Werth Kuli hätte". TRUESDELL [1954. 2, § 10] has noted that this statement is misleading and incomplete. It is not trivial to reformulate it as a theorem susceptible of rigorous proof. One point glossed over in HELMHOLTz's statement is the fact that a vector tube of a field continuously differentiable in a compact region need not approach the boundary, be closed, or end in the interior. For example, in the plane, it may be bounded by curves having as a common limit set a limit cycle. One can show that this particular situation cannot occur if the field be solenoidal. Cf. Sect. 32, footnote 1. 3 TRUESDELL [1954, 2, § 10] remarks that it suffices to assume c continuous and piecewise continuously differentiable, but that the result need not hold for piecewise continuous fields. 4 TRUESDELL [1949, 3], [1954, 2, § 10]. The case K=O was discussed by FöPPL [1897. 1, § 4]. 5 TRUESDELL [1951, 2, § 9] has remarked that the more general result is implied by an example due to KELVIN [1849, 1, § 7]. 6 Using KELvrN's transformation (28.1}, one can show that this result is equivalent to the lemma 1 of WEYL [1940, 2]. A special case is given by BERKER [1949, 5, Chap. III]. We follow TRUESDELL [1954, 2, § 10]. Sect. 31. Solenoidal fields I: Integral properties. 821 Setting b =C in Eq. (26.2) and employing Eq. (31.1), we obtain ~ [da · c c p(K) - -i c2 da p(K)] l ~ = J [c {cp(K-l)}-! c2 grad p(K) + curl c X cp(K)] dv. " (31.6) The case when K = 0 is PoiNCARE's 1 identity ~ [da· c c-! c2 da J = J curl c X c dv. (31.7) 6 " Recalling the convention of Sect. 24, we formulate conditions sufficient for the vanishing of the integral over il and conclude that if a field c which is continuously differentiable and solenoidal in a region v satisfies the boundary conditions then cn=O or ccn=o(p- 2) c2 = const or c2 = o (p- 2) J curlcxcdv = 0. " on il, } on il, Sufficient for Eq. (31.8) to hold are the simpler conditions c = 0 or c = o (p-1) on il. For K = 1, Eq. (31.6) becomes ~ [da · c c p - -i c2 da p J = f [ c c - ! c2 1 + curl c x c p J d v. d " (31.8) (31.9) (31.10) (31.11) Contrading Eq. (31.11) yields an identity of LAMB and J. J. THOMSON 2 ! f c2 d v = ~ da · (! c2 p - c c · p) + f p · cur 1 c X c d v . (31. 12) V 6 V from which it follows that i/ a field c which is continuously differentiable and solenoidal in a region v satisfies the following conditions: 1. c=O or c2 =o(p-3), p-ccn=o(p-2) on il, (31.13) 2. Throughout v, curl c x c = 0; then c = 0 throughout v. Sufficient for (31.13) to hold are the simpler conditions c = 0 or c = o (p- ;) on il. The alterna ting part of Eq. (31.11) yields 3 ~ [da-cpxc +ic2 daxp] =fpx (curlcxc)dv, 6 " (31.14) (31.15) (31.16) from which we conclude that if a field c which is continuously differentiable and solenoidal in a region v satisfies the boundary conditions cn = 0 and c = const or p X c cn = ö (p- 2), (31.17) then fpx (curlcxc) dv = 0. Sufficient for Eq. (31.17) to hold are the simpler conditions (31.15). 1 [1893, 1, § 5]. See also TRUESDELL [1954, 2, §§ 10, 64]. (31.18) 2 [1879, 1, § 136], [1932, 6, § 153]; [1883, 1, § 6]. Cf. also TRUESDELL [1954, 2, § 10] for the results given in italics here. 3 PoiNCARE [1893. 1, § 115]. Cf. TRUESDELL [1954, 1, § 10]. 822 J.L. ERICKSEN: Tensor Fields. Sect. 32. In Eq. (26.2), put K =1, b =p; then j[pc+cp-p-c1+(3c+curlcxp)p]dv } =~[da· (pc + cp)p- da (p · c)p] . • (31.19) Contracting this relation yields 2fp·cdv =~da·cp , (31.20) " . while the alternating part is equivalent tol 3 J pxcdv = J px (pxcurlc) dv -~ " . - ~ [(daxc) xp] xp . • (31.21) J.J. THOMSON 9 derived two similar integral formulae which, while not restricted to solenoidal fields, are of interest mainly in the solenoidal case. First, in Eq. (26.2) put K =0, b p, obtaining J(curlcxp+3c+pdivc)dv ) " = T [da· (cp +pc)- da (p. c)], = ~ [(daxc) xp- da· cp] . • Since div (cp) =p div c +c, application of Eq. (26.1h yields 2 J cdv = J pxcurlcdv +~ (daxc) xp. " . . Second, if we integrate the identity curl (p2c) = p2curlc + 2pxc over v and then apply Eq. (26.1)s, we obtain 2fpxcdv =-J p2 curlcdv + ~daxp c. ". ". . (31.22) (31.23) (31.24) (31.25) For the various integral theorems we have given here, the region need not be simply-connected. 32. Solenoidal fields II: Differential properties. lf c is any continuously differentiahte field, there exists an infinite number of scalar functions m such that mc is solenoidal. To see this, one need only note that div (mc) = 0, (32.1) being a first order differential equation for m, with continuous coefficients, admits an infinite nurober of solutions. Since c and mc have the same vector lines, the vector lines of any continuously ditferentiable field c are also the vector lines of an infinite number of solenoidalfields 3• This result may mislead in seeming to imply that vector lines of solenoidal fields have no special properties. Distinctions can arise, for example, because of the fact that solutions m of Eq. (32.1) 1 MUNK [1941, 1, p. 95]. 2 [1883, 1, §§ 4- 5]. The second of these is usually presented in a non-invariant form; cf. e.g. LAMB [1932, 6, § 195], where it is asserted that an error in J. J. THOMSON's formula was corrected by WELSH. 3 APPELL [1897, 2, § 5]. Sect. 32. Solenoidal fields II: Differential properties. 823 can possess singularities at points where c =0, even if c be analytic. For example, in the plane, one can show that an isolated point where c = 0 is never a spiral point for the vector lines of a solenoidal field c, whereas it can be if c is not solenoidal 1. Any continuously differentiable solenoidal vector field c may be represented locally in EULER's form 2 c =gradFxgradG where F and G are scalar functions which satisfy C·gradF=O, C·gradG=O. (32.2) (32.3) The surfaces F = const and G = const are thus vector sheets of c. One of the functions, say F, may be taken as an arbitrary non-constant solution of Eq. (32.3); the other may be taken as the function given by G Jexc·dx =- e·gradF' (32.4) where e is an arb itrary continuous vector field such that e · grad F =!= 0, and where the path of integration lies on one of the surfaces F = const. Combining the results of the two foregoing paragraphs, we see that any continuously differentiable field may be represented locally in the form c =H gradFxgradG, (32.5) the surfaces F = const and G = const being vector sheets of c. In Eq. (32. 5), F and G may be chosen as arbitrary functionally independent solutions of Eq. (32.3), and the function H is then determined uniquely. If c is solenoidal, Eq. (32.5) yields d. o(H,F, G) 0 = lVC = "( 1 2 3); u X, X, X (32.6) therefore H =H(F, G). From Eqs. (32.5) and (32.6) follows Euler's theorem on solenoidal jields 3: I f F and G are any independent functions such that the surfaces F = const and G = const are vector sheets of a continuously ditferentiable solenoidal field c, then c can be represented locally in the form c =H(F, G)gradFxgradG. (32.7) From Eq. (32.2) and the identity curl (F grad G + grad H) = grad F X grad G, (32.8) it follows that any continuously differentiahte solenoidal field c may be represented locally as the curl of a vector v, c = curlv, (32.9) the field v being indeterminate to within an additive gradient. Conversely, if Eq. (32.9) holds, div c =div curl v =04• 1 In connection with HELMHOLTz's characterization (Sect. 31, footnote 2), many expositians assert that the vector lines of a solenoidal field cannot end in the interior. KELLOGG [ 1929, 1, Chap. II, § 6] has remarked that this is false. 2 [1770, 2, §§ 26, 49]. [1806, 1, § 142]. Derivations are given in works on vector analysis, e.g. BRAND [1947, 2, § 104, Chap. 3]. 3 [1757. 1, §§ 47-49]. ' More generally, any analytic field v yields theinfinite sequence of solenoidal fields curlK v (K = 1, 2, .... ). For K~ 3 we have curlK v =- 172 currK-2v, whence follow volume integrals yielding currK-2v in terms of cur!Kv. Thesefacts were noted by RowLAND [1880, 1] and FABRI [1892, 3] who attempted a kinematical interpretation of cur!Kv. For the case K = 2, cf. BOGGIO-LERA [1887, 3]. 824 J.L. ERICKSEN: Tensor Fields. At points where c =!= o, Eq. (30.4) shows that div c = 0 if and only if dlogc = _ D dt ' which may be integrated along the vector lines of c to give c = c0 exp [- J D dt], Sect. 33. {32.10) {32.11) this characterization being due to BJ0RGUM1• It follows immediately that a non-zero solenoidal field is determined by its vector lines and by its magnitude at one point on each. By successive differentiation of (30.4), we conclude that if c and D are analytic functions of arc length on a given vector line, c vanishes at one point if and only if it vanishes all along the vector line through the point. 33. Lamellar and complex-lamellar fields. Following BJ0RGUM2, we call any vector field proportional to a . . . field a complex - ... field, wherein any name may be inserted for . . . From results given in Sect. 32, an arbitrary continuously differentiable field may thus be called a complex-solenoidal field. A field c is lamellar 3 in a region provided ~ d~ · c = ~ dJ!' c" = 0 (33-1) e e for any reducible circuit c in the region 4• It follows that a field c, continuous in a region v, is lamellar if and only if there exists a scalar P, called the potential of c, such that c =- gradP. (33.2) The potential is single-valued if v is simply connected, not so in generat if v is multiply-connected. A lamellar field is thus everywhere normal to the equipotential surfaces P = const. A complex-lamellar field, being by its definition representable in the form c = QgradP, (33-3) where Q is a scalar, also is normal to a family of surfaces. From KELVIN's transformation (28.1) it follows that a continuously ditlerentiable field c is lamellar if and only if curl c = 0, or 8crkfox"'1 = 0. (33.4) Similarly, a continuously ditlerentiable field c is complex-lamellar if and only if c. curl c = 0, or c[k oc,j ox"l = 0. (33-5) These results were established by KELVIN 5, a result equivalent to Eq. (33-5) having been obtained much earlier by EULER6• From (33-5h it follows that c is complex-lamellar if there exists a co-ordinate system in which (33 .6) A field c is plane if Eq. (33.6) holds in some reetangular Cartesian co-ordinate system, rotationally symmetric if Eq. (33.6) holds in cylindrical co-ordinates, x3 l [1951, 4, § 3.1]. 2 [1951, 4]. The reader will not confuse this usage of "complex" with "complexvalued". 3 The names lamellar and complex-lamellar derive from KELVIN [1850, 1], [1851, 1, §§ 68-69, 75]. 4 WEYL [1940, 2] generalizes the definitions of lamellar and solenoidal field given here. 5 [1851, 1, § 75]. VELTMANN also discussed these fields [1870, 1, pp. 453-456]. For a modern proof that Eq. (33-5) is sufficient as well as necessary, see e.g. BRAND [1947, 2, § 105]. Therepresentation c = Q grad P + grad F(Q, P) is discussed by CASTOLDI [1955, 5]. 6 [1770. 1, § 1]. Sect. 33. Lamellar and complex-lamellar fields. 825 being the angle variable. It is essential that the covariant components of c be used in Eqs. {33.4) 2 and (33.5h. For any vector field c, any curve ;x; =J:(u) satisfying the equation (33.7) is orthogonal to the vector lines of c which it intersects. A theorem of CARATHEODORY 1 asserts that in a closed and bounded region where c is non-zero and satisfies a l ipschitz condition, c is complex-lamellar if and only if in every neighborhood of an arbitrary interior point ;x; there exist a point ;x;* such that no curve satisfying Eq. (33.7) foins ;x; to x*. If c is complex-lamellar, it follows from Eqs. (33.3) and (33.7) that ;x; and ;x;* may be joined by a solution of Eq. (33.7) if and only if they lie on the same surface P = const, from which the necessity of this condition follows. We omit the more complicated proof of sufficiency. With the exception of the definitions of plane and rotationally symmetric fields, all these results are easily extended to n-dimensional spaces which need not possess a metric tensor, c being taken to be a covariant field 2• Eqs. (3 3.1) to (33.3), (33.4) 2 , (33.5) 2 and (33.7) require no alteration, while Eq. (33.6) is tobe replaced by cl = f(xl, x2), c2 = g(xl, x2), Ca= ... = c,. = 0. (33.8) It is clear from Eq. (33·3) that the magnitude of a complex-lamellar field is in no way restricted by its vector lines. Indeed, it is immediate that a non-vanishing field is complex-lamellar if and only if its unit tangent is complex-lamellar. However, the condition that a field be lamellar is essentially different in that it connects the magnitude of the field with the geometric properties of its vector lines. In the three-dimensional case, it is easytorender the above remarks explicit. In the first place, either directly from Eq. {30.1h or by substitution in Eq. (33.5), or from Eq. (30.15), we conclude that a non-zero continuously ditferentiable field is complex-lamellar if and only if its abnormality A is zero. For a lamellar field, we have from Eq. (30.12) the much stronger necessary and sufficient conditions of BJ0RGUM 3 : (1) A =0 when c=j=O, (2) c is constant along the vector lines of the binormal field b, and (3) c = c0 exp J xdn, (33.9) the integration being performed on the vector lines of the principal normal n, which lie on the equipotential surfaces, as do those of b. It follows that, in a sufficiently small region, a lamellar field is determined by its vector lines and by its magnitude at a single point on each equipotential surface. When a field is complex-lamellar, its unit tangent t is the unit normal of the surfaces P = const. Hence the mean curvature K of those surfaces is given by 4 K- d' t D d log c 1 d' ( ) =- lV =- =dt--c lVC. 33.10 Therefore, if we are given a set of vector lines having a normal congruence, we may obtain all non-vanishing complex-lamellar fields having these vector 1 [1909, 5]. 2 In situations to which CARATHEODORY's sufficiency condition has been applied, it is artificial and unnecessary to introduce a metric tensor. Cf. CARATHEODORY [1909, 5], ERICKSEN [1956, 1]. 3 [1951. 4, § 3.4]. 4 Cf. e.g. BRAND [1947, 2, § 131]. The result is due to CHALLIS [1842, 1], whose derivation was criticized and corrected by TARDY [1850, 1]. The geometry of the surfaces P= const is studied by CALDONAZZO [1924, 1], [1925, 1] and PASTORI [1927, 2]. 826 J. L. ERICKSEN: Tensor Fields. Sect. 34. lines by prescribing the scalar field div c everywhere and the value of c at one point on each vector line. Also, for a non-vanishing complex-lamellar field, any two of the following three conditions imply the third1 : 1. The normal surfaces are minimal surfaces. 2. The field is solenoidal. 3. The magnitude of the field is constant on each vector line. 34. Screw fields. From Eq. (33-5), a field c is complex-lamellar if and only if it be normal to its curl. The opposite extreme is fumished by a screw field 2 , defined as being parallel to its curl: equivalently, cxcurlc = 0, curlc =j= o; k_o c[k,m] c - , c[k,m] =j= 0. (34.1) (34.2) In either of these equations covariant derivatives may be replaced by partial derivatives. It follows from Eqs. (30.12) and (34.1) 1 that c is a screw field if and only if curlc=Ac=j=O, (34-3) so the abnormality of a screw field c is the factor of proportionality between the curl and the field. Since c and curl c have the same vector lines when Eq. (34-3) holds, and since the abnormality depends only on the vector lines, we may replace c by curl c in Eq. (30.2), obtaining3 A _2 l curl c · curl curl c =c C·cur C= . curl c · curl c If m is any scalar function suchthat mc is solenoidal (cf. Sect. 32), 0 = divmc = div(~ curlc) = grad ~ · curlc =Ac· grad ~, (34.4) (34.5) so the surfaces mjA =const are vector sheets of c and of curl c. Conversely, if, for some scalar function f (x), the surfaces f = const are vector sheets of a screw field c, we may set m=fA, where A is the abnormality; reading Eq. (34.5) backwards, we obtain div mc = 0. These results constitute the following theorem 4 : For a twice continuously differentiahte screw field c, a necessary and sufficient 1 A special case is due to CALDONAZZO [1924, 2, § 6]. We generalize an argument of PRIM [1948, 1, § 3], [1952, 1, Chap. V]. Cf. also CASTOLDI [1947, 5], BYUSHGENIS [1948, 2, § 2.1]. 2 Much of the literature follows CrsOTTI [1923, 3] in calling these fields "Beltrami fields", after the researches of BELTRAMI [1889, 1], but in fact nearly all of BELTRAMI's results were included in the prior and more extensive work of GROMEKA [1881, 2]. BELTRAM! hirnself called them "helicoidal". We revert to the more descriptive term screw field introduced by CRAIG [1880, 2, p. 225]. [1880, 3, p. 276], [1881, 5, pp. 5-6], the first person to remark them. Earlier STOKES [1842, 2, p. 3] had concluded that Eq. (34.1h implies C=O, but later he realized his error (footnote 1, p. 3, 1880 reprint of [1842, 2]). 3 LECORNU [1919, 1]. If we set B ==I curl clfc, then c is a screw field if and only if B =A, the proof being immediate from Eq. (34.3); necessity was proved by APPELL [1921, 1, § 763], sufficiency by CARSTOIU [1946, 4]. A differential system for a screw field is given as Eq. (35.4) below. Another follows by taking the curl of Eq. (34.3) and then eliminating A by Eq. (34.4) (BALLABH [1948, 3, § 4]), but a simpler one may be obtained by putting Eq. (34.4) into Eq. (34.3) directly. Other differential equations are derived by BJ0RGUM [1951, 4, Sect. 5], [1954, 7]. 4 TRUESDELL [1954, 2, § 12]. This generalizes theorems of NEMENYI and PRIM [1949, 6, Th. 1], BELTRAMI [1889, 1] (see also MORERA [1889, 2]), and GROMEKA [1881, 2, GI. 2, § 9], the references being arranged in order of decreasing generality. Sect. 34. Screw fields. 827 condition that the surfaces A (curl eh = _!<:url c) 2 = (curl c)3 = const mc1 mc2 mc3 (34.6) m be vector sheets is that mc be solenoidal. It is true a fortiori that mfA is constant an each vector line of a screw field c if mc is solenoidall. Setting m = 1, we obtain the corollaries: The surfaces of constant abnormality of a screw field c are vector sheets if and only if div c = 0; the abnormality of a solenoidal screw field is constant along each vector line. Thus from the vector lines alone, without knowledge of the magnitude of the field, we can determine whether or not a screw field is solenoidal. More generally, by putting m = 1 in Eq. (34.5) we may derive c-1 div c = - ~o~! AI • (34.7) A relation between a screw field and its vector lines may be read off from Eq. (30.12). This relation results from that given in Sect. 33 for lamellar fields if we replace "A = 0" by "A =!=O" 2• Indeed, from Eqs. (30.4) and (34-3) follows c A = c0 A 0 exp (- J D dt). (34.8) BJ0RGUM 3 has shown that for a given screw field c, it is always possible to choose a co-ordinate systemsuchthat the x3-lines are the vector lines, g13 = 0, g23 = x1 g33 , and cg33 = 1. Conversely, in a Co-ordinate system such that g13 = 0 and g23 = x1 g33 , if a vector field c be tangent to the x3-lines and of magnitude c so adjusted that cg33 = 1, then c is a screw field. From Eqs. (34-3) and (34.4) we conclude that o < curl c · curl c = c · curl curl c, (34.9) so a screw field and the curl of its curl always intersect at an acute angle. If this angle is zero, curl c is also a screw field. I t follows from taking the curl of Eq. (34.3) that the curl of a screw field c is again a screw field if and only if the abnormality of c is uniform. Then c is solenoidal, and successive curls of c are screw fields having the same abnormality 4• In the case when A is uniform, that c is solenoidal follows alternatively as a corollary of the first corollary following Eq. (34.6). Hence taking the curl of Eq. (34.3) yields 172 c +Ac= o. (34.10) This equation was derived by GROMEKA 5, who based upon it a theory of determining a screw field of constant abnormality from appropriate boundary conditions. The same problern has been taken up by BJORGUM and GoDAL 6 ; besides constructing many interesting examples, they have shown that such a field c can be represented in the form c = A gradH x e + eA2 H + e · gradgradH, (34.11) where e is a fixed unit vector and 172 H + A 2 H = o. For the many more known properties of screw fields, the reader is referred to the treatise of BJ0RGuM7, 1 TRUESDELL [1954, 6, § 527] attributes to VAN TuYL the remark that this theorem is an immediate consequence of the fact that any two solenoidal vector fields with common vector lines are proportional along them. 2 BJ0RGUM [1951, 4, § 3-3]. 3 [1957. 4, § 5]. 'NEMENYI and PRIM [1949. 6, Th. 3]. 5 [1881, 2, GI. 2, § 9]. Cf. also STEKLOFF [1908, 2, §§ 39-52], TRKAL [1919, 3], BALLABH [1940, 5, §§5-7]. 6 [1953, 3], [1958, 2]. Cf. also BJ0RGUM [1951, 4, § 6]. 7 [1951, 7]. Cf. also TRUESDELL [1954, 2, §§ 12, 52]. 828 J. L. ERICKSEN: Tensor Fields. Sects. 35. 36. In n-dimensional metric spaces, screw fields may be defined by Eq. (34.2). There is no immediate extension to spaces which are not metric since associated components do not exist. One natural generalization of Eq. (34.2) is obtained by requiring a covariant vector b and a contravariant vector c to satisfy C!b[kfaxmJ c" = o, ob[kfoxmJ =!= o. (34.12) Suchpairs of vectors occur in studies on relativity1 • c) Potentials. 35. MoNGE's potentials. If c be a twice continuously differentiable field, the field curl c, being solenoidal, has a representation of the form (32.2), namely, curlc = gradFxgradG, (3 5.1) from which follows curl(c-FgradG)=O. Thus c-FgradG is lamellar, so there exists a scalar H such that c =gradH +FgradG. (35.2) In general, this representation is valid only locally 2• The three scalarsF, G, and H, called Monge potentials 3 of c, are not uniquely determined, but in most applications there is no need to specify one particular set rather than another. From Eqs. (35.1) and (35.2) follows I o (H, F, G) ( 5 ) c·curc= 8 (xi,xz,x2)' 3-3 whence, by Eq. (33-5), c is complex-lamellar .if and only if its three Monge potentials are functionally dependent. Directly from (35.1) we see that cislamellar if and only if the two potentials F and G are functionally dependent. MORERA 4 obtained differential equations to be satisfied by the Monge potentials of a screw field. In geometrical terms, these equations assert that for c to be a screw field the surfaces F=const and G=const, which by Eq. (35.1) are always vector sheets of curlc, must simultaneously be vector sheets of c. Formally, c · grad F = (grad H + F grad G) · grad F = 0, } c · grad G = (gradH + Fgrad G) · gradG = o. (3 5 .4) Conversely, if Eq. (35.4) hold and if F and G be functionally independent, then both c and curl c are perpendicular to grad F and to grad G; hence they are parallel, so c is a screw field. 36. STOKEs' potentials. In a finite region v, an arbitrary vector field c has a representation of the form 5 c = - grad S + curl v (36.1) and hence is the sum of a lamellar and a solenoidal field. Functions S and v satisfying Eq. (36.1) are called, respectively, a scalar potential and a vector potential of c; together, they are called the Stokes' potentials. An infinite nurober of potentials correspond to a given field c. Let c be piecewise differentiable in a finite region v, bounded by 6; within v, Iet c be continuous except upon a surface 6 ', on each side of which it has finite Iimits; then a pair of potentials for c 1 E.g. VAN DANTZIG [1934,1, p. 646]. 2 Cf. HADAMARD [1903, 2, p. 80]. 3 This result was implied, but not stated explicitly, by MoNGE [1787, 1, §§XVI-XVIII, XX] and PFAFF [1818, 1, § 4]. The above derivation is due to HANKEL [1861, 1, § 11]. 4 [1889, 2]. See also BJ0RGUM [1951, 4, § 5.1]. STOKES [1851, 2, Part I, Sect. 1, §§ 3 -8]. Proofs are given in works on vector analysis, e.g. PHILLIPS [1933. 1, § 83]. Four stronger decomposition theorems are given by ·WEYL [1940, 2]. Cf. also BLUMENTHAL [1905, 1]. Sect. 37. Definitionsand conditions of reality. 829 is given by S = ~1-J 4n divc_dv d + ~1-J 4n da· d [c] __ - __ 4n 1 ___ ,f.. 'j" da:.!!_ d ' l .. 6' 6 V = _1_! curl_!!_ dv + 1J_da x ~] __ 1_ ,f.. !-_a x~_ 4n d 4n d 4n 'j" d ". .. " (36.2) where d is the distance from the point of integration to the point where S and v are being calculated, and where the bold-face bracket denotes the fump of c across 11', (36-3) c+ and c- being the limiting values of c on the two sides of 11' and the sense of da being fixed appropriately in terms of the choice of signs + and - for the two sides of 6'. In the case of a region extending to infinity in all directions, if c = ö (p-2) the formulae (36.2) still hold, providing the integrals over 6 be omitted. For suitably selected potentials, e.g., those given by Eqs. (36.2), the representation (36.1) is valid globally1 and v is solenoidal. We always assume the potentials are so selected. Since curlv is solenoidal, we may replace it in Eq. (36.1) by an expression of the form (32.2), so obtaining a local representation c = - grad S + gradF X grad G. VI. Tensors of order two. a) Proper numbers and vectors. (36.4) 37. Definitionsand conditions of reality. A proper number a of a second order tensor a is a root of the equation (37.1) Since in n dimensions Eq. (37.1) is a polynomial equation of degree n in a, there are always n proper numbers, which need not be real or distinct. The left and right proper vectors of a corresponding to the proper number a are, respectively, the non-zero vectors m and q, in general complex, such that The directions of the vectors are the principal directions of a. A sulficient condition that alt proper numbers of a be real is that 2 { b is symmetric and positive definite 3 ,} a = b-1 • c, where c is symmetric. Inded, if a be so expressible, its proper numbers are given by 1 HADAMARD [1903, 2, p. 80]. (37.2) (37-3) 2 An alternative statement is: For some choice of the metric tensor (viz. gkm=bkm), a is symmetric. In general, such a metric is not Euclidean. 3 Throughout this work we write "the tensor b is positive definite" in place of "the components bkm are cocfficients of a positive definite quadratic form". 8)0 J. L. ERICKSEN: Tensor Fields. Sect. 37. the assertion then follows because each root a of an equation ofthistype is real 1 • Furtherroore, if the roots of Eq. (37.4) are a1 , ... , a,., there exist reallinearly independent vectors q1 , •.• , q,. such that (37.5) whence follows -1 a"',.q~ = b"'' c,,.q~ = abq';, (3 7.6) so qb is a right proper vector of a corresponding to the proper nurober ab. Defining the vector mb by ~,.=b,.. qb, froro Eq. (37.5) we get -1 a", ~" = b"• c., bku qb = a4 b,. qg =ab mb,, (3 7.7) so mb is a left proper vector of a corresponding to the proper nurober ab. Were the above condition also necessary, then all proper nurobers' being real would ensure the existence of n linearly independent right proper vectors. A counter-exarople is shown by the roatrix 0 0 0 1 0 0 ' 0 0 1 (3 7.8) which has the real proper nurobers 0, 0, 1, but only two linearly independent right proper vectors 2, e.g. (0, 1, 0) and (0, 0, 1). A necessary condition follows: If the proper numbers a1 , ... , a,. of a are all real, and if there exist corresponding right proper vectors q1 , ... , q,. forming a linearly independent set 3, then Eq. (37.3) holds. To show this, we first note that without loss of generality the qb roay be assuroed real. Since they are linearly independent, there exists a unique real reciprocal set me such that If we set .. me,.q~ = t5eb• L: mb,.q~ = t5k. b=l .. b,.,- L: ~k ~, b=I then b is a syroroetric positive definite tensor. Moreover, -1 (37.9) (37.10) ~" = b,.,q{,, q~ = b"'~,. (37.11) Since the q4 are right proper vectors of a, froro Eqs. (37.2) 2 , (37.9) 2 , and (37.11) 2 it follows that a~ = aP, t5k = at b=1 ~ mbk ql, = b=1 f. ab mb,. q€ l .. -1 -1 = L: mbk ab bf>' mb, = bf>' c,,., b=1 (37.12) 1 See e.g. CoURANT and HILBERT [1931, 2, p. 32). The proofthat the roots of Eq. (31.4)3 are real is implicit in the work of CAUCHY [1828, 1], [1829, 1], who treated explicitly the case b,.,= d,.,, thereby showing that the proper numbers of a symmetric tensor are always real. HERMITE [1855, 1] extended CAUCHY's result to complex matrices, showing that the proper numbers of a Hermitian matrix are always real. · 2 TRUESDELL [1954, 2, § 22) attributes this example to WHAPLES and gives an example of a matrix having all its proper numbers real and all its proper vector proportional to a single vector. Those familiar with the theory of elementary divisorswill see easily that the number of linearly independent proper vectors equals the number of elementary divisors. 3 Sufficient for this is that the proper numbers all be distinct. Sect. 37. where we have set Definitions and conditions of reality. .. c,,. =Lab mb, mbk· b=l Since c,,.=c11 ,, Eq. (37.12) is the desired result. 831 (37.13) The tensors b and c are not uniquely determined by a. Given any admissible set of proper vectors qb, we can obtain an infinite number of different admissible sets by multiplying each vector qb by an arbitrary non-zero scalar factor. As is easily seen from Eqs. (37.8), (37.10), and (37.13), different sets generally will determine different tensors b and c. It follows from Eqs. (37-9) and (37.11) that b as defined by Eqs. (37.10) satisfies (37-5), whatever be the choice of the qb. From Eqs. (37.11) and (37-7) we conclude that the reciprocal set ofvectors me, defined by the condition (37-9), are in fact left proper vectors corresponding to the proper numbers ae. The intermediate formula (37.12)a is particularly important in that it expresses the tensor a uniquely in terms of its proper numbers, supposed real, and a set of its independent right proper vectors, supposed n in number, and left proper vectors so chosen as to form a reciprocal set. The invariance of this representation under possible different choices of the qe is immediate from Eq. (37-9). It can occur that the vectors qe are mutually orthogonal. If they are normalized so as tobe unit vectors, Eq. (37.10) implies that bkm=gkm• whence we conclude that a =a'. That is, if all proper numbers of a are real, and if a has n linearly independent mutually orthogonal proper vectors, then a is symmetric; this theorem is due in principle to KELVIN and TAIT1• The example (37.8), interpreted as being referred to reetangular Cartesian co-ordinates, shows that the reality of all proper numbers and orthogonality of a maximal linearly independent set of right proper vectors is insufficient to imply a =a'. There must exist n orthogonal proper vectors, which is not the case for the example. Aside from the result of KELVIN and TAIT, there has been no need to introduce the metric tensor if a is taken as a mixed tensor, b and c as covariant tensors. For any field a(:r) such that Eq. (37-3) holds, it is possible to select a coordinate system such that at a given point P, the right proper vectors have the components <5~. Indeed, let the qb (:r) constitute any linearly independent set of right proper vectors, in any Co-ordinate system; then we need only select :r* = :r* (:r) such that at P we have 8x*" qk (:r*) = --qm (:r) = 15" b 8xm b b· From Eq. (37-9) 2 it follows that me,.(:r*) =t:5ek at P. Hence From Eqs. (37.12) 2, (37.14), and (37.15), at P we derive k * _ 8 x* k 8 :>:5 u ) _ ~ 8 x* " 8 :>:5 a ,(:r ) - 8x" ax*' a .(:r - LJ 8x" 8x*' mbs qb u ab, s l b=1 " =Lab 6~t:5br· b=1 (37.14) (37.15) (37.16) This equation asserts that in the co-ordinate system :r*, the matrixlla",ll assumes diagonal form with entries ab. Conversely, if there is a real, non-singular 1 [1867, 1, § 183]. 832 J. L. ERICKSEN: Tensor Fields. Sect. 38. co-ordinate transformation which diagonalizes II a" mll at a point, then a = b-1 · c, " where band c are the tensors having at P the components !5,.m and 2; ab !5b,. !5bm• b=1 respectively, in the system in which II a" mll is diagonal. Such a co-ordinate system, hereafter called a principal Co-ordinate system, need not be orthogonal, which means that lla"mll and lla,.mll need not be diagonalized by the transformation. Combining these results, we have the following characterization: A necessary and sulficient condition that a = b-1 · c, where b and c are symmetric and b is positive definite, is that at an arbitrarily selected point P, there exist a real nonsingular co-ordinate transformation which diagonalizes the matrix II a" mll· In the langnage of matrices, this result may be restated as follows: A necessary and sutficient condition that a = b-1 • c, where b and c are symmetric and b is positive definite, is that a be ·similar to a diagonal matrix modulo the real matrices, i.e. that a =I· d ·l-1, where I is real and non-singular, d real and diagonal. A restaterneut of the theorems of CAUCHY and KELVIN and TAIT is: I may be taken as orthogonal, i.e.f' = 1-1, if and only if a = a'. In a principal co-ordinate system we have m 0 ak m + { m} { k m) { d} { } ak ,P = oxP kp ak - am unsumme ; 37.17 when k = m, the second term vanishes. In general, this formula holds only at the point where the principal co-ordinate system is defined, and only for the diagonal mixed components do covariant derivatives reduce to ordinary partial derivatives. Even if lla"mll and llakmll also be diagonal matrices, we have a"",p=l= oa""foxP and akk,P =!= oakkfoxP, in general. When the principal CO-ordinate system is rea~ and orthogonal, Eq. {37.17) may be read in terms of physical components: a- a)' (37.18} OSp OS_. where we have used Eq. (25.6). For the case of most interest here, namely n =3, we may make use of the obvious but useful fact that the number of real proper vectors is fixed by the sign of the discriminant of the left member of Eq. (37.1}. We may use also the fact that the proper numbers of a are all real if and only if the same be true of a+m 1, where m is any scalar, as follows immediately from Eq. (37.1}. If a admits the decomposition {37-3), we have a+m1 = b-1 • (c+mb) as a corresponding decomposition for a +m1. FROBENIUS 1 showed that a has a proper number which is real, positive, simple, and greater in absolute value than any other proper number provided, for some choice of Coordinates, the numbers akm be all positive. BANG 2 noted that in three dimensions the proper numbers of a are all real if, in some co-ordinate system, a 12 a 23 a31 = a\ a32 a13 and a12 a 23/ a 13 , a23 a31f a21 , and a\ a12f a32 are of the same sign. 38. Principal and related invariants. The K-th principal invariant I~KJ of a second order tensor a is the K-th elementary symmetric function of the proper numbers of a, (38.1) so that (38.2) 1 [1908, 1]. 2 [1893, 3]. MuiR [1896, 2] extended BANG's result to n dimensions. Sect. 38. Principal and related invariants. 833 Therefore the principal invariants determine the proper numbers uniquely up to order. When the proper numbers are real, unique determination is effected by the order convention ~~ a2 ;;::: • • • ~ a,., so that, lor values ol the J~K) such that all roots ol Eq. (38.2) are real, any single-valued lunction ol proper numbers equals a single-valued lunction ol principal invariants. Comparison of Eq. (38.2) with Eq. (37.1) shows that J~KJ is the sum of the principal K-rowed minors of the matrix lla"mll: I (K)=_1_s.•l···'K am1 amx a K! umJ ..• mg r1 • • • •x • (38.3) F or a real tensor a such that there is a real non-singular co-ordinate translormation which diagonalizes lla"mll, any single-valued absolute scalar invariant l(a) under arbitrary co-ordinate translormations is expressible as a single-valued lunction ol principal invariants1• For proof, we evaluate I in a principal co-ordinate system; by Eq. (37.16), I is a function of the ab, which are real; the assertion follows by the italicized statement in the paragraph preceding. Q.E.D. This result does not extend to an arbitrary second order tensor a, it being necessary to adjoin to the principal invariants WEvR's 2 characteristics of a to obtain a complete set. If a can be diagonalized, the WEYR characteristics of a are uniquely determined by the principal invariants. It is true that if f is a scalar polynomial in the a"m• then it is always expressible as a polynomial in the principal invariants 3 • The WEYR characteristics are not such polynomials; in fact they are not even continuous functions of a at a = o. Of more interest in mechanics are corresponding results for scalar invariants of both akm and Ckm under arbitrary co-ordinate transformations, or of a8 m under orthogonal transformations. The K-th moment I~K) of a is the sum of K-th powers of the proper numbers of a: .. - 1cK)- ~ (a )K _ am1 ams amx a = LJ b - ms m1 · · · m1 · (38.4) b=l If we set (38.5) .. where the sum runs over all sets of n non-negative integers Mb such that ~ Mb= L, .. b=l ~ b Mb = K, then an expression for the moments I~K) in terms of the principal b=l invariants J~Kl is" K J(K) =K "_L u(K,L) a L.J L Yla • L~l (38.6) Except where otherwise noted, we henceforth assume n = 3 and write 5 Ia, Ha, lila, la, IIa, lila in place of J~>, II!:>, I!:>, I~>, 1!:1, 1!:1, respectively. In this notation, a is a proper number of a if and only if a3 - Ia a2 + IIa a - lila = 0; in particular, a = 1 is a proper number if and only if la - Ha + lila = 1 . (38.7) (38.8) 1 For the symmetric case, RANKINE [1856, 2, § 3] refers to this result as a discovery of CAYLEY. 2 [1885, 1] and [1890, 1]. See also MAcDuFFEE [1933, 2, § 40]. 3 For n = 3 this follows immediately from a result due to WEITZENBÖCK [1923, I, pp. 65-66]. See also TRUESDELL [1952, 2, p. 132], [1953. 4, p. 594]. 4 BURNSIDE and PANTON [1901, 5, § 159] attribute this result to WARING. 5 In a space whose co-ordinates are Ia, lla, IIIa, BoRDONI [1955. 4] discusses the surface where the discriminant of (38.7) is constant. Handbuch der Physik, Bd. III/1. 53 834 J. L. ERICKSEN: Tensor Fields. From Eq. (38.6) we have Ha = I! - 2 Ha, 2 IIa = I! - IIa, IIIa = I! - 3 Ia IIa + 3 IIIa, ) IIIa =t I!--! Ia IIa+ i IIIa. The inverse of a, which exists when IIIa =j= 0, is given explicitly by IIIa a-1 = a 2 - la a + IIa 1. The octahedral invariant! Ua is defined by Ua- L [j (ab- ac)J2 = i (Ila- IIa),) b ßak;;;- = 3 a r a k > (38.15) (3 8.16) 1 The invariance and significance of Ua were known to MAXWELL in 1856 [1937. 10, pp. 32-38]; it was introduced by v. MISES [1913, 2, § 1]. The name "octahedral invariant" is usually attached to (2/V3) ut on the basis of a geometrical interpretation given by NADAI and LODE [1933, 3, § II], [1937, 2, pp. 206-207]. 2 KLEITZ [1873. 3, § 23] was the first to make an explicit study of the deviator. 3 More generally, for any a that can be diagonalized by real transformations. 4 LIPSCHITZ [ 187 5. 1]; HAMEL [ 1936, 1, § 1]. 5 For the symmetric case, these results are given by MuRNAGHAN [1937. 3, § 3], SIGNORINI [1943,1, § 17), and REINER [1945, 3, §4}. Sect, 39. Inequalities. 835 where Eq. (38.16) 5 , which follows from Eq. (38.16) 4 by Eq. (38.10), holds only when u-1 exists. Also1, when u-1 exists, we have da= -a · da-1 · a and hence (38.17) 39. Inequalities 2• CAUCHY 3 was first to establish bounds for the proper numbers of matrices. Let the proper numbers of a real symmetric three-dimensional matrix be ordered so that ~ ~ a3 • Then, if a be referred to a reetangular Cartesian co-ordinate system, CAUCHY's results are that a1 is never less than, while a3 is never greater than any one of the six quantities (k =f= m) (3 9.1) and that, for each k and m, a2 lies between the two numbers given by the expressions (39.1). Of course, one may interpret the components in (39.1) as physical components of a in an arbitrary orthogonal co-ordinate system. Let a be any second order tensor, real or complex, in n dimensions, set b == -! (a + a'), c =-! (a- a'), where the bar denotes the complex conjugate; select any proper number a of a, and write it in the form a =P +iq, where p and q are real. Refer a to any reetangular Cartesian co-ordinate system and let A, B, and C be the maxima of the absolute values of the components of a, b, and c, respectively. We then have HrRSCH's inequalities 4 [a[ ~nA, IPI ~nB, [q[ ~nC. (39.2) One corollary is HERMITE's theorem 5 : A sufficient condition that the proper numbers of a alt be real is that a be Hermitian, i.e. that c =0. Another is WEIERSTRAss' theorem 6 : A sufficient condition that the proper numbers of a alt be pure imaginary is that a be skew-Hermitian, i.e., that b =0. HrRSCH7 showed also that when b is real, q~ [!(n-1)]!C, and that, if the proper numbers of b, which in this case HERMITE's theorem shows to be real, be ordered so that ~ • • · ~ b,., then b 1 ~ p ~ b,.. When the proper numbers of the Hermitian tensor d = a . a' are similarly ordered, BROW~E's inequality 8 asserts that any proper number a of a satisfies dl~ a a~ d,.. We again restriet attention to the case when n = 3 and a is real. From Eq. (38.15) follows (39-3) the equality holding if and only if a be symmetric. Further inequalities follow from the observation that since complex proper numbers appear in conjugate pairs, all symmetric functions of them are real, so that an inequality may be inferred from every identity in which the square of such a function occurs. From (38.15) we thus obtain (39.4) 1 SrGNORINI [1943, 1, § 17]. 2 Further discussion of results in this section is given by MAcDUFFEE [1933, 2, § 18]. 3 [1828, J], [1830, 1, Th. 1]; in [1829, 1, Th. 1] he extended the result to n dimensions. 4 [1901, 2]. 5 [1855. 1]. For real a, this theorem was proved earlier by CAUCHY [1828, 1], [1829, 1]. s [1879. 2]. For real a, the theoremwas proved earlier by CLEBSCH [1863, 2]. 7 [1901, 2]. For real a, the results are due to BENDIXSON [1901, 3]. 8 [1928, 1, Th. V]. i3* J. L. ERICKSEN: Tensor Fields. Sect. 39. with equality holdingifand only if a be symmetric. From Eqs. (38.9h and (38.11) it follows that II0 ~- 2Il0 , From Eq. (38.13) we get II,a ~Ha. In all cases of Ineqs. (39.5) and (39.6), equality holds if and only if Ia =0. (3 9. 5) (39.6) When we add the condition that all proper numbers of a be real, as is the case when a is symmetric, from the fact that then IIa ~ 0 with equality holding if and only if a =0, a further sequence of inequalities may be inferred. From Eqs. (38.9) and (38.13) we thus get 3 IIa ~ I! ~ 2 IIa, (39.7) where equality holds on the left if and only if a is spherical; on the right, if and only if a=O. From Eq. (38.11) follows (39.8) where equality holdingifand only if a =0. Also from Eq. (38.11), since Ua ~ 0, we derive I!~ 3 Ha, with equality holding if and only if a is spherical. When all proper numbers are non-negative_, we have obviously1 , Ia ~ o, IIa ~ o, I! ~ 27 IIIa ~ o. (39.9) (39.10) For a real symmetric tensor referred to reetangular Cartesian Co-ordinates, from Eq. (38.4) 2 we see that (akm)2 ~ Ha • There is also HADAMARD's inequality 2 (39.11) (39.12) If !Ia! ~ K and lila!~ K, Eq. (38.9h and Ineq. (39.11) imply that for a symmetric tensor we have (39.13) In Ineqs. (39.11) to (39.13), akm can be interpreted as a physical component of a in any orthogonal co-ordinate system. In any co-ordinate system, we have (39-14) where M is the maximum of the absolute values of the mixed components of a. Results of WEDDERBURN 3 imply for any two symmetric tensors a and b the inequalities (39.15) 1 The third inequality, for the symmetric case, was given in more complicated form by SIGNORINI [1949, 7, Chap. II, § 5]. 2 [1893. 2], ScHUR [1909, 2] gives an extension of this inequality as well as several others. Cf. also MuiR [1930, 2, Chap. I(a)]. 3 [1925,E2]. Sects. 40, 41. Real powers of positive tensors. 837 b) Powers and matrix polynomials. 40. Integral powers. In this Part, the dimension of the underlying space is arbitrary. When K is an integer, the K-th power of a tensor a is defined inductively by1 (40.1) where K is restricted to be positive when a has no inverse. It follows from Eq. (37.2) that (40.2) hence the K-th power of any proper number of a is a proper number of aK, and every right (left) proper vector of a is a right (left) proper vector of aK. To obtain a sharper result, we note the following identities, valid for any scalar m and any tensor a: when K> 0, aK- m 1 = (a-1 - m1 1) ... (a-1 - mK 1) when K < 0, (40.3) (40.4) where m1 , ... , mK are the I Kl complex I Kl-th roots of m. Taking the determinant of both sides of Eqs. (40.3) and (40.4) shows that m isaproper number of aK if and only ij at least one IKI-th root of m isaproper number 2 of a when K >O, or of a-1 when K < 0. The proper numbers of a-1 are the reciprocals 3 of those of a. It can occur for some K that there exist proper vectors of aK which are not proper vectors of a. E.g., if II aKmll = II ~ _ ~ II, then a 2 = 1, so that an arbitrary vector is a proper vector of a 2, while the only right proper vectors of a are (b, 0) and (0, c). By the Hamilton-Cayley theorem, we may replace (ab)K by aK in Eq. (38.2), obtaining (40.5) This formula makes it possible to write any power of a as a linear combination of 1, a, ... , an-I, with scalar coefficients that are polynomials in the principal invariants ij the power is positive, rational functions if it is negative. For positive K, RANUM 4 gave the formula N an+K = L: H~K,L) a{L)' L~1 where H~K,L) is defined by Eq. (38.5), and where n-L a{L)= L (-1t-Q+l I~n-Q)aQ+L-1. Q~O (40.6) (40.7) 41. Real powers of positive tensors. We call a tensor akm positive when its proper numbers arereal and positive and it possesses n linearly independentproper vectors. In particular, a positive definite (symmetric) tensor is positive. For a positive tensor we have the representation (37.12h: n a's = L ab q(, mbs' b~I 1 STICKELBURGER [1881, 4] was first to define general powers of matrices. 2 BoRCHARDT [1846, 1], [1847, 1], when K>o. (41.1) 3 This was known to SPOTTISWOODE [1856, 1]; the result then follows for all K=J= o. 4 [1911, 1]. 838 J. L. ERICKSEN: Tensor Fields. Sect. 42. where ab> 0 and where the qb and mb are reciprocal sets of right and left proper vectors. For any real K, the K-th power of a positive tensor a is defined to be K the positive tensor aK whose components a~ are given by (41.2) wherein (ab)K is the positive real K-th power of ab. That is, aK is the unique tensor having the sameproper vectors as a and having as its proper numbers the positive K-th powers of the proper numbers of a. This definition is equivalent to Eq. ( 40.1) when both are applicable. The usuallaws of exponents apply to aK as defined by Eq. (41.2). For example, by Eq. (37.9) 1 we obtain n n L;(ab)K+Lqf,mbs= L (ao)Kq{,mbu(a,)Lq~mcs, (41.3) b=l ~c=l so that aK+L =aK. aL. For a non-negative tensor, i.e., a tensor having n linearly independent proper vectors and real proper numbers which are positive or zero1, Eq. (41.2) serves to define a unique K-th power when K~ 0. If we accept the possibility that aK may be complex and multivalued, by using different determinations of the K-th powers of the several proper numbers in Eq. (41.2) we may define various K-th powers of any matrix having a representation of the form (41.1). Such a definition is unsatisfactory in two respects. First, once one admits the possibility that aK be multivalued, it seems preferable that al/M should represent any solution x of xM=a (41.4) when M is an integer, whereas the definition just mentioned excludes some solutions. Second, there is little or no motivation for excluding tensors not representable in the form ( 41.1). A more satisfactory definition of aK is easily obtained by regarding it as a complex multivalued function of a in the sense of CrPOLLA2 . We omit the details, as the definitions already given are adequate for this appendix. Several writers, beginning with CA YLEY 3, have studied the solution of Eq. (41.4). 42. Matrix functions. A matrix b given by (42.1) where the cGi are scalar constants, is a matrix polynomial in the variable matrix a. Matrix polynomials may be set into one-to-one correspondence with the polynomials in a scalar variable x which are defined by the same set of constants, and the algebra of matrix polynomials is isomorphic to the algebra of polynomials in a scalar indeterminate. Similarly, an infinite set of constants c<» defines a formal matrix power series in a, corresponding to the formal scalar power series b (x) determined by the same coefficients. The matrix power series thus obtained converges if and only if every proper number of a lies inside or on the circle of convergence of the scalar series 1 For real tensors, this is a slight generalization of the usual definition of non-negative tensors or linear transformations. See e.g. HALMOS [1942, J, §56]. 2 [1932, 2]. See also MAcDUFFEE [1933, 2, §SO]. 3 [1858, 2], [1872, 1]. WEITZENBÖCK [1932, 3] gave a method for determining all solutions. AuTONNE [1902, 1], [1903, 1] showed that in the complex field, if a is nonnegative and Hermitian, then there is a unique solution which is non-negative and Hermitian. A similar analysis shows that if a is non-negative according to the definition given above, then there is a unique solution which is non-negative. For further references, see MAcDuFFEE [1933. 2, §§ 48, SO] and WEDDERBURN [1934. 2, p. 171]. Sect. 42. Matrix functions. 839 b (x) and, for every proper number a of multiplicity m, the power series for the formal m -1st derivative b(m-1) (a) converges 1• Each proper number of a matrix polynomial b (a) is a function of a single proper number of a, as follows from a more general and more explicit theorem of FROBENIUS: Let r(x, ... , y) =P(x, ... , y)fq(x, ... , y) be a rational function of the scalar indeterminates x, ... , y, p and q being polynomials; let a, ... , b be commutative matrices such that the matrix q (a, ... , b) is non-singular; then the proper numbers ab, ... , bn of a, ... , b can be ordered so that r (a1 , ... , b1), ... , r(a,., ... , b,.) are the proper numbers of r=p(a, ... , b) · q-1 (a, ... , b) =q-1 · p, the ordering being the same for all rational /unctions 2• Loosely related to this is SYLVESTER's assertion that, whether or not a and b commute, the proper numbers of a · b and b · a coincide 3• RANUM's equation (40.6) serves to reduce any matrix polynomial or power series to the form (42.2) where the coefficients d(J) are, respectively, polynomials or power series 4 in the principal invariants of a. There are many functions representable in this form that are neither matrix polynomials nor matrixpower series: for example, b (a) = n1l 1. Any function of the form (42.2) satisfies b (I· a · 1-1) = 1 · b (a) · r, (42.3) where I is an arbitrary non-singular matrix. Conversely, if each component of b be a polynomial in the components of a and if b (a) satisfy Eq. (42.3) for arbitrary non-singular I, then b is representable 6 in the form (42.2) with coefficients d(J) which are polynomials in the principal invariants of a. DIRAC6 proposed Eq. (42.3), with a, b, and I interpreted as elements of any algebra 7, as a part of the definition of b's being a function of a. He noted that Eq. (42.3) implies that I commutes with b whenever I commutes with a. TURNBULL and AITKEN8 showed that if a and b are complex n x n matrices and if b commutes with every complex matrix that commutes with a, then b is expressible as a linear combination of 1, a, ... , a"- 1. An analogaus result for matrices over the real field is readily established. Consequently Eq. (42.3) implies that b is a linear combination of 1, a, ... , a"- 1 with coefficients depending on a in such a way as to be scalars under arbitrary symmetry transformations. Under reasonably general conditions, these scalar coefficients are expressible as functions of the principal invariants of a, as follows from the theorems given at the beginning of Sect. 38. 1 HENSEL [1926, 1]. WEYR [1887, 2] had previously treated the case where no proper number of a lies on the circle of convergence. PHILLIPS [1919, 2] gave sufficient conditions for the convergence of a matrixpower series in any finite set of commuting matrices. 2 According to MACDUFFER [1933, 2, p. 23], BROMWICH [1901, 4] noted that this theorem may fail to hold when r (x, ... , y) is .not rational. We find no explicit statement to this effect in BROMWICH's paper, though it may follow from results given in his § 3. 3 SYLVESTER [ 1883, 2] stated this without proof. MACDUFFER [ 1933, 2, p. 23] gives an elegant proof. 4 Expressions of the type (42.2) are sometimes called "polynomials in a" even when the d!Jj arenot polynomials in a. See e.g. MAcDUFFER [1933. 2, Th. 15.3] 5 For the case n = 3, this follows immediately from a theorem of WEITZENBÖCK [ 1923, 1, pp. 65-66]. The result for arbitrary n can be established similarly. 6 [1926, 2]. 7 The elements of some, but not all algebras are representable by square matrices of finite order. 8 [1932, 5, p. 150]. 840 J. L. ERICKSEN: Tensor Fields. Sect. 43. Various writers1 have considered the problern of inverting a matrix polynomial to obtain a in terms of b. Theinverse a(b), when it exists 2, is in general complex and multivalued. It can be shown that for arbitrary non-singular I we have {42.4) this being interpreted in the sense that for given I each value of a (b) is equal to some value of 1-1 • [a(l · b · l-1)] ·I· Contrary to what might be expected from the results of WEITZENBÖCK and of TuRNBULL and AITKEN concerning Eq. (42.3). in general there are values of a(b) which arenot expressible as linear combinations of 1, ... , bn--1 ; values, that is which cannot be regarded as matrix polynomials or power series. Recognizing this, algebraists have attempted to devise more general definitions of matrix functions 3 to include such possibilities, still insisting upon a correspondence between matrix functions and functions of a single scalar variable. According to these definitions, a matrix function x(a) always satisfies Eq. (42.3). this being interpreted as was Eq. (42.4) in cases where b (a) is multivalued. Such definitions are not easily generalized to the case of matrices depending on several matrices. The reader who has even slight familiarity with recent development in continuum mechanics will see that here we have come up against a central problern in modern theories of materials. We now formulate lunctional delinitions which seem particularly weil suited for application in classical mechanics, if perhaps not so well for other fields: 1. If each component of the matrix b be a function of the components of the matrices a, ... , c, we say that b is a matrix function of a, ... , c. 2. A matrix function b (a, ... , c) such that b(o · a · o-1, ... , o · c · o-1) =o · b(a, ... , c) · o-1 (42.5) for all orthogonal matrices o, i.e., for all o such that o-1 = o', {42.6) is an isotropic 4 matrix function of a, ... , c. 3. If each component of the matrix function b (a, ... , c) be a polynomial in the components of a, ... , c, then b (a, ... , c) is a polynomial matrix function of a, ... , c. 4. An isotropic polynomial matrix function is one satisfying both 2 and 3. c) Decompositions. 43. Invariant decompositions. The group of transformations of reetangular Cartesian co-ordinate systems decomposes the linear vector space of second order tensors into three linear subspaces, the tensors comprising these spaces being, respectively, the spherical tensors, the symmetric traceless tensors, and the skewsymmetric tensors. Any orthogonal transformation maps each subspace onto itself; the only tensor common to two of them is the zero tensor; none possesses a proper linear subspace mapped into itself by every orthogonal transformation. In modern terminology, these three are the irreducible invariant subspaces of 1 Cf. MAcDuFFEE [1933, 2, §§ 47, 48] for references. RuTHERFORD [1932, 4] gives certain rather explicit solutions for a fairly general class of equations. MAcDuFFEE [1933, 2, p. 94] indicates how all solutions can be obtained in the case when b is spherical. 2 WEITZENBÖCK [ 1932, 3, p. 161] gives a simple example of a case where no inverse exists for the equation b =a2 • 3 Cf. MAcDuFFEE [1933, 2, §SO] for references. 4 The terminology agrees with that used by RIVLIN and ERICKSEN [1954, 5, § 21]. Sect. 43. Invariant decompositions. 841 the space of tensors or order two with respect to the orthogonal group1• Explicitly, we have the decomposition - -1 J(1) + [ - -1 J(1) ] + a,.".- n a g,.". a(km) n a gkm a[km] • (43 .1) which is, in the sense indicated above, maximal. An arbitrary matrix a can be written in infinitely many ways as the product of two symmetric matrices, one of which is non-singular2. This can be formulated in the following two ways: a" ". = b"' c,m, a"". = d"' e,m, b[km]- c -0 det b""'=l= 0, -[km]- • d[km] = e[km] = 0, det e,..,. =!= 0. (43.2) (43-3) As is clear from the results at the beginning of Sect. 37, b cannot always be chosen tobe positive definite; the same applies to e. Any non-singular matrix a may be written in the forms a = s . o = o . s*, (43.4) where o is orthogonal, and s and s* are symmetric and positive definite 3 ; o, s, and s* are uniquely determined. We now give a proof ofthispolar decompositio'n theorem4• Since a · a' is a positive definite 5 symmetric tensor, by the results of Sect. 41, it has a unique positive square root s; since a is non-singular, so is s. Therefore we ma y set s= (a·a')i, o = s-1 - a. (43-5) Now s-1 • a. a'. s-1= 1, or (s-1 • a) · (s-1 • a)' = 1; therefore o is orthogonal. From Eq. (43.5) 2 follows Eq. (43.4)1 • In the same way we obtain a decomposition a = o* · s*, with s* = (a'. a)!, o* _ a-s-1 • (43.6) 1 The corresponding decomposition of the space of tensors of any given order is derived and discussed in some detail by WEYL [1946, 3, Chap. VB], references to relevant literature being given on pp. 310-311. 2 FROBENIUS [1910, 1]. Voss [1878, 1, p. 343] previously established this for nonsingular a. HILTON [1914, 1] characterized the matrices which can be written as the product of two skew-symmetric matrices or as the product of a symmetric and a skew-symmetric matrix. 3 We need to interpret this theoremalso in terms of lineartransformations; equivalently, any second order tensor with non-vanishing determinant may be expressed as the product of a unique orthogonal tensor by a unique positive definite symmetric tensor. An orthogonal matrix was defined by Eq. (42.6). An orthogonal tensor ok". is defined by the property that under the transformation vk = okpvP, w"' = o"'pwP, for arbitrary vectors v and w, the inner product g,.".vkwm is invariant. That is, g,.".v"w"' = g,.". okp o"'qvPwq = g,.".v"w"'. (A) In order for this relation to hold for arbitrary v and w, o must satisfy (B) alternatively, o,."'o"p = IJp · In a reetangular Cartesian system, the matrix ok". is an orthogonal matrix. In general co-ordinates, (B) asserts that (g · o)' = (o · g-1tl, where g = II g,.".IJ. ' An equivalent algebraic statement and all the essential ideas for an algebraic proof were given by FINGER [1892, 4, Eq. (25)]; the first algebraic proof, by E. and F. CosSERAT [1896. 3, § 6] (cf. also BURGATTI [1914, 3], SIGNORINI [1930, 3]). The ideas of FINGERand the CosSERATS were put into matrix notation and extended to complex matrices by AuTONNE [1902, T9, Lemma II], hisform of the proof being that given above. If a is singular, decompositions a = s · o = o* · s* exist but are not unique; cf. HALM OS [ 1942, 1, § 67]. 5 If we set w := v · a, then, since a is non-singular, w = 0 if and only if v = o. Since v · (a · a') · v =W • 1v, it follows that a · a' is positive definite. 842 J. L. ERICKSEN: Tensor Fields. Sect. 44. We shall show presently that both decompositions of a are unique. Since a = o . (o-1 . s . o), it is a consequence of this uniqueness that s* = o-1 · s · o, o*=o. (43 .7) To prove uniqueness1, we note that s · o = s' · o' implies o = s-1 · s' · o' and o' =o-1 = (o')-1. s'. s-1, or o = s · (s')-1 · o'. Hence (s-1 · s'- s · (s')-1) · o' =0, whence follows s-1 • s' =S. (s')-1, or s 2 = (s') 2 ; therefore, since s and s' are positive definite, we derive s = s', and hence o = o'. Q.E.D. By this uniqueness, it follows also that s and o commute if and only if s = s*; that is, if and only if a. a' =a'. a, in which case a is called normal 2• The dass of normal matrices includes all symmetric, skew-symmetric, or orthogonal matrices. lf a is non-singular, there exists a non-singular matrix b such that3 a = b2 and, if a is also symmetric, a matrix c such that 4 a = c'·c. (43.8) (43.9) In Eqs. (43.8) and (43-9), b and c in general are complex and are not uniquely determined. 44. Certain canonical forms. For any second order tensor a, there exists a co-ordinate transformation, which may be chosen to be unitary, reducing a at a given point to superdiagonal form 5 : a1 1 a12 ... a\ JJakmll = 0 a22 ... (44.1) 0 0 ... ann When a is real and all its proper numbers are real, this transformation is real and may be chosen tobe orthogonal. When there are complex proper numbers, the transformation is necessarily complex. In three dimensions a real tensor field has either three real proper numbers or one real, two complex conjugate. In the latter case, there isareal transformation, which may be taken to be orthogonal, reducing a to the form 6 a b c llakmll = - b d e (44.2) o o t When Eq. (44.1) holds in reetangular Cartesian co-ordinates, a is normal (Sect. 43) if and only if the matrix ( 44.1) is in fact diagonal. Hence a tensor all of whose proper numbers are real is normal if and only if it is symmetric. A tensor of the form (44.2) is normal if and only if c = e = 0, a = d. It is known7 that a real symmetric traceless matrix a can be transformed by orthogonal transformations to a system in which all diagonal components of a 1 MURNAGHAN and WINTNER [1931, 5]. 2 MURNAGHAN and WINTNER [1931, 4], [1931, 5]. 3 Cf. MACDUFFEE [1933, 2, §§ 35, 48]. 4 This follows immediately from the fami!iar result, used by LAGRANGE [1759, 1], that a quadratic form can be reduced to a sum of squares by linear transformations. 5 SCHUR [1909, 2]. 6 MURNAGHAN and WINTNER [1931, 4]. There is a generalization to n dimensions. 7 LovE [1906, 1, § 16] stated this without proof. WHAPLES has shown us a proof and an extension of the theorem to matrices defined over essentially arbitrary fields. Sect. 44. Certain canonical forms. 843 vanish. Now consider the matrix (43.1); in every co-ordinate system the diagonal components of 1 are all equal, those of the skew-symmetric part i (a- a') vanish, and the remaining tensor in the matrix (43.1) is symmetric and traceless, whence it follows by the preceding theorem that in a suitably chosenreetangular Cartesian co-ordinate system, all diagonal components of a are equal. When n = 3, we have a b c a = da e I g a (44.3) When the proper numbers of a are all distinct, 11 akmll can be transformed by real, but not necessarily orthogonal transformations to the form 1 0 0 (44.4) 0 0 ... 1 0 where cK= (-1)K+l J~l. When a has multipleproper numbers it can be reduced to a direct sum of matrices of the type (44.4). Denote by b the matrix on the right of Eq. (44.4). Then c-1 · b · c is diagonal 2, where C= (al)"-1 (a2)"-1 (a1)n-2 (a2)"-2 (44.5) This provides another proof of the result, noted in Sect. 3 7, tha t II ak m II can be red uced to diagonal form by real transformations if its proper numbers be real and distinct. Any matrix can be reduced by a complex transformation to a direct sum of matrices of the form 3 a 1 0 0 0 a 1 0 (44.6) a 1 0 0 0 a If all the proper numbers are real, the transformation may be taken as real also. NoLL has informed us that any real matrix can be transformed by a real transformation to a direct sum of matrices of the two forms (44.6) and 0 1 0 . ~I a b 1 0 0 0 1 0 0 a b 0 0 0 0 0 0 0 (44.7) 0 a b 1 0 0 0 1 0 • a b When allproper numbers are distinct, the forms (44.6) and (44.7) reduce to llall and ~~~ ~~~· respectively. This furnishes still another proof that a matrix with real and distinct proper numbers can be diagonalized by a real transformation. 1 Cf. MAcDuFFEE [1933, 2, § 39] for references and discussion. 2 SCHUR [1909, 3]. 3 JORDAN [1!170, 4, p. 114]. 844 J. L. ERICKSEN: Tensor Fields. Sects. 45. 46. d) Normaland shear components. 45. Definitions. Except where otherwise noted, the second order tensors considered in the remainder of this chapter are assumed to be symmetric, there then being no distinction between right and left proper vectors. The scalar akm vk vm, where v is any unit vector, is called the normal component of a for the direction v. The scalar akmukvm, where u and v are unit vectors, is the shear component of a for the directions u and v, the normal components being included as a special case. When u and v are perpendicular unit vectors, we call akmukvm the corresponding orthogonal shear component. In orthogonal co-ordinates, the physical component a(kk) is the normal component of a for the direction of the tangent to the k-th Co-ordinate curve, while a(km), k=f=m, is the shear component for the directions of the tangents to the k-th and m-th co-ordinate curves. In passages where a fixed orthogonal coordinate system has been laid down, the phrases "normal components" and "shear components ", with no further qualification, mean the components a(kk) and a(km), k =f= m, respectively. Because of the existence of the canonical form exemplified by Eq. ( 44.3), it is always possible to refer a to an orthogonal co-ordinate system in which the normal components of a at a given point are alt equal. They can all be made to vanish if and only if a is traceless. In order for the normal components to be zero in all orthogonal co-ordinate system, it is necessary and sufficient that a = o. Since by assumption a is symmetric, there exist orthogonal co-ordinate systems in which llall is diagonal at a given point; that is, it is always possible to refer a to an orthogonal co-ordinate system in which at a given point all the shear components are zero. These Co-ordinate systems are principal co-ordinate systems as defined in Sect. 37. In order for the shear components to vanish in allorthogonal co-ordinate systems, it is necessary and sufficient that a be spherical; for a spherical tensor, the shear component for the directions u and v is zero if and only if u and v be orthogonal. The proper numbers of a, which are real, we order as follows: (45.1) 46. Extremal properties. Setting C- akmukvm, (46.1) for a fixed tensor a, we consider the problern of finding the extremes of the scalar C when the vectors u and v are varied subject to certain constraints. Problem 1. Let the constraints be u = v, uk uk = 1, so that C is the normal component of a for the direction u. Referring a to a reetangular Cartesian principal co-ordinate system, we then have C = a1 (u1) 2 + · · · +an (un) 2 , (u1) 2 + · · · + (un) 2 = 1, (46.2) and from Eq. (45.1) it follows that a1=a1{ (u1) 2+ · · · + (un) 2} ~aJ(u + · · · +an(un) ~a,.{ (u1) 2+ · · · + (un) 2}=a,., (46.3) so that ~C~a,.. To see when equality can hold on the left, we set a1=C in Eq. (46.2h and use Eq. (46.2) 2 , so obtaining 0 = a1 {(u1) 2 - 1} + a2 (u2) 2+ · ·· + a"(u,.) 2 } = (a2 - a1) (u2)2 + ... + (a,.- a1) (u,.)2, (46.4) Sect. 46. Extremal properties. 845 whence it follows by Eq. (45.1) that u11 =0 when k > 1 unless a11 =a1 • Therefore, letting K be the largest integer for which aK =a1 , so that a1 =a1 when I::;;;; K, we conclude that uK+1 = · · · =u,. =0, whence it follows by direct calculation that a~um = a1 u11 • Thus the normal component of a is greatest for the directions which are principal directions corresponding to the greatest proper number, and the value of the greatest normal component is the greatest proper number. Sirnilary, C takes on its minimum value a,. if and only if u be a proper vector of a corresponding to the proper nurober a,.. The remaining proper numbers of a are extremal values of C; if b> 1, ab is the maximum value of C when u ranges over the vectors which satisfy the constraints and are perpendicular to b - 1 mutually orthogonal proper vectors of a corresponding to the proper numbers1 a1 , ... , ab-l· In general C takes on these intermediate values for infinitely many vectors u which do not satisfy the orthogonality conditions and are not proper vectors of a. Problem 2. Let the constraints be u,. u" = v11 v" = 1, so that C is the shear component for the directions u and v. Extremal conditions for C are (46.5) where a and bare multipliers. Since u"oCfou"=v"oCfov"=C. we conclude that a = b; hence (46.6) lf a has two proper numbers ± a that are numerically equal but opposite in sign, Eq. (46.6) can be satisfied by choosing u and v to be non-parallel vectors such that u +v and u- v are proper vectors corresponding to a and - a. This includes the limiting case where zero is a multiple proper nurober of a. Otherwise, we must have u ± v = 0; the case u = v is that considered in Problem 1, while the case u = -v is similar. We have shown that extrema of shear components are always to be found among those for the coincident directions, which are the normal components, and among those for opposite directions; for additional extrema of the shear components to exist, it is necessary and sulficient that a have a pair of proper numbers ± a. Problem 3. In Problem 2, we add the further constraint u · v = 0, thus confining attention to orthogonal shear components. Clearly, the largest (smallest) value of C attainable with this additional constraint is never greater (less) than that attainable with the constraints of Problem 2. Extremal conditions are (46.7) where a, b, and c are multipliers. It follows that a = c and the corresponding extremal value of C is a. Therefore a,.". (um+ v"') = (a + b) (u11 + v,.), a,.m (um- v"') = (b- a) (u11 - v11). (46.8) The constraints require that the vectors u ± v be non-null, whence it follows that these are proper vectors corresponding to the proper numbers (b ±a) of a. This result is the theorem of Coulomb and Hopkins 2 : The maximum and minimum 1 See, e.g., COURANT and HILBERT [1931, 2, pp. 20-23]. CAUCHY [1829, 1], [1830, 1, Chap. 11] noted that the proper numbers of a are the (real) extremal values of akm X uk u"'fli, u'. · 2 The theoremwas derived in the plane case by CouLOMB [1 i76, 1, §VIII]; in the general case, by HoPKINS [1847, 2, §§ 4, 5]. 846 J. L. ERICKSEN: Tensor Fields. Sect. 46. orthogonal shear components of a are given by max C = t (a1 - an), min C = t (an- a1) (46.9) respectively, these being takenon if and only if V2u=m1 +mn, V2v =m1 -m,., where m 1 and mn areorthogonal unit proper vectors of a corresponding to the greatest and least proper numbers a1 and an. When a1 =an, a is spherical, so that all orthogonal shear components vanish, as follows also from (46.9). The remaining extremal values of C, as given by (46.8), are f(a0 -ae), b=J=e. Each of these is a minimax unless a0 - ae = ± (a 1 - an)· The octahedral invariant l..la, defined by Eq. ( 46.11), is thus the sum of the squares of the extremal orthogonal shear components of a. From Eq. (46.9) we observe that f (an- a1) ;;;;: akm uk v"' (u, u' V5 vs)-§ ;;;;: t (a 1 - an) for all non-null orthogonal vectors u and v. (46.10) In a reetangular Cartesian co-ordinate system such that at a given point n- 2 axes are principal axes of a and the remaining two are directions satisfying Eq. (46.7), a assumes the form d 0 0 llakmll = 0 b a 0 a b where d isadiagonal (n-2)X(n-2) matrix 1 . (46.11) Problem 4. We now take v as a fixed unit vector and vary u subject to the constraints uk uk = 1, uk vk = 0. Thus we seek the greatest shear components among all directions perpendicular to a given direction. As conditions that C be extremal, we obtain ak".v"'=auk+bvk. (46.12) Therefore b = ak". vk v"'; that is, bis the normal component of a for the direction v, and a, which satisfies (46.13) is the extremal value of C. When v is a proper vector of a, then C = 0 for all u satisfying the constraints. This is no more than the statement that the shear component for a principal direction and any direction orthogonal to it is zero. Otherwise Eq. (46.13) determine a unique value for a 2, and therefore Eq. (46.12) determines u up to sign. It can be shown that2 a 2 - '\' (a - a )2 cos2 {} cos2 {} - L... b e b e• (46.14) b