A2A#1.: Range of applicability; see #2.

A2A#2.: Hamiltonian mechanics generalizes straightforwardly to all classical (and even quantum!) physics — with some notable exceptions: many dissipative systems require adding the dissipative forces “by hand” to the Hamiltonian equations of motion.

To be clear, comparing “classical Newtonian mechanics” and “Hamiltonian mechanics” in the way of this question, I interpreted the former to mean Newton’s original formulation in terms of his three laws of motion. In turn, “Hamiltonian mechanics” (I trust) refers to Hamilton’s system of first order differential equation pairs. Both of these (as well as Lagrangian mechanics) may be derived from a yet more general and overarching (Hamilton’s) principle of least action, for which one needs the “master” function(al) called the action, and which figures prominently in the Feynman-Hibbs path integral formulation of (all) quantum physics — and so is arguably the most general approach to all physics.

You can fully answer this question by reading, for example, Landau and Lifshitz’s book "Mechanics" from the Course of Theoretical Physics. In a nutshell, Hamiltonian mechanics differs from Newtonian mechanics, on which it is based by the fact that it has a clearer structure that allows generalizing to the multidimensional case, finding integrals of motion, introducing new concepts, such as adiabatic invariants, and finally performing transformations from one system of variables to the other which most convenient, for example, an action-angle. Finally, it facilitates the transition to quantum mechanics by transforming the Poisson brackets into a commutator of conjugate quantities.

Hamiltonian mechanics is a more elegant formulation of classical mechanics. It is more elegant in the sense that conserved quantities are more obvious and you have a much larger freedom to choose convenient coordinates. The drawback is that it cannot deal with some systems. When friction is present then it is inconvenient or impossible to use it. It was first developed for celestial mechanics so that wasn’t an issue.

Hamiltonian mechanics really pays off when you study quantum systems. It can help you a tremendous amount in understanding quantum mechanics whereas Newtonian mechanics barely helps at all.

Newtonian mechanics involves causality of the mechanical interactions through forces and their reactive forces. However the Hamiltonian approach evaluates the total energy of the system at any given moment, including the specific components of the system. Sometimes calculating the moment-to-moment dynamics through algebraic or numerical integration using Newton’s method may not only be cumbersome, but also inaccuracies or systematic errors tend to multiply. However, with Hamiltonian mechanics, the energy of the system at any point in time can be assessed, and this balance gives us a reliable overview of the static and dynamic relationships of the components at any given moment, in view of the overall energy.

Lagrangian and Hamiltonian mechanics are mathematically two sides of a single coin, relating the geometry of the space of forms on a manifold to a minimization problem for paths in the tangent bundle obtained from the tangent vector of a curve. It’s basically the Legendre transform. The two functions that transform into each other are referred to as the Hamiltonian and the Lagrangian, respectively.

It all works for rather arbitrary choices of functions, not necessarily having anything to do with physics. And so there is a considerable amount of mathematical machinery available for studying them. The associated dynamics are not dependent on coordinates, which gives you a powerful way to leverage any symmetries.

But as it happens, Newton's equations for conservative forces are the Euler Lagrange equations for the Lagrangian which is kinetic energy minus potential energy, and the symplectic flow equations for the Hamiltonian which is their sum, when expressed in Cartesian coordinates.

Newtonian: everything is written having in mind the position and the velocity of material points. Optimal for any problem which can be easily formulated with the help of these quantities.

Lagrangian: coordinates are completely general. Conservation laws are directly linked with symmetries of the system. Variational description is straightforward.

Hamiltonian: coordinates are united in couples. Excellent for problems with small perturbations, with oscillatory motions, with periodic orbits.

The three approaches to classical mechanics are completely equivalent. Ideally, all three should be mastered. Newtonian approach makes solutions of elementary problems simpler. Lagrangian and Hamilton approaches highlight the relationships and the analogies with optics and quantum mechanics. Lagrangian approach paves the way to field theories. Hamiltonian approach paves the way to statistical mechanics.

These are very good questions.

Let me start with your second question. Hamiltonian mechanics is very useful for summarizing many entire physical theories with one function that represents the total energy of a physical system. The Hamiltonian function H(x, p) (where p[j] is momentum) represents the time evolution operator of a system; this property appears in classical mechanics and quantum mechanics.

• In classical mechanics, dp[j]/dt = - dH/x[j] and dx[j]/dt = dH/dp[j].
• In quantum mechanics, - I hbar d(psi)/dt = H psi, where psi is the quantum wave function.

A theory that is expressed in Hamiltonian form has extra structure that all valid conservative physical theories have; you can’t just throw an ad-hoc term into the field equations unless you add it as a term in the Hamiltonian.

Your second question might be stated as: what is the extra structure that the Hamiltonian form ensures that the theory has? The answer is symplectic structure. The essence of symplectic structure is an antisymmetric 2-form on phase space (x-p space) that maps the pair (d/dx[j], p[j]) to 1, and all other pairs (d/dx[j], p[k]) to zero. In quantum mechanics, (d/dx[j], p[j]) maps to hbar, Planck’s constant. In QM, the symplectic form identifies a minimal area in each of the planes (d/dx[j], p[j]); all other planes orthogonal to these “conjugate pairs” are assigned zero symplectic area. Symplectic area is precisely what is discretized in QM. (It is sometimes claimed that energy and momentum are what is discretized in QM, but that is not so; every bounded classical or quantum system has discrete allowed values of energy and momentum.)

Regarding the difference between Hamiltonian physics, without getting into the “nitty gritty” details of what Hamiltonian physics is, is as follows.

In general, Hamiltonian physics are two different ways of viewing or expressing the exact same set of physical laws. Nearly, anything (if not everything) that is derivable from Hamiltonian mechanics should be derivable from Newtonian mechanics also, except that it may be more difficult to do so without the use of Hamiltonian mechanics.

In general, the one thing that Hamiltonian mechanics adds to Netwon’s laws is the least action principle, which is the concept of that there is some action/energy, which, if minimized (or more precisely extremized or made stationary), will result in the equations of motions.

Interestingly, although this assumption has proven to be essentially correctly experimentally, it was originally based in religion rather than science. It was based on a notion that G-d is perfect and would choose an extremization on which to run the world (as it is more efficient). However, in practice sometimes the extremizations that results in the correct laws of physics are actually a maximization.

Nonetheless, Hamiltonian physics is in fact derived from Newtonian physics. However, Hamiltonian physics attempts to generalize Newtonian physics by asking the question of what action, if minimized, would result in Newtonian mechanics. The generalization makes it easier to apply to figure out the equations of motion in multiple coordinate systems, which makes it easier to find a coordinate system in which the equations of motion are relatively simple to solve. Also Hamiltonian mechanics treats momentum as an essentially independent variable from the position (its conjugate coordinate), whereas in Newtonian mechanics momentum is function of velocity, which is a function of position, and so they are not independent in any sense. Treating the momentum as an independent, but conjugate coordinate to the position further aids in at least some situations in finding the equations of motion for a particular system. Nonetheless, other than providing a more thorough mathematical basis for exploring the laws of physics and a larger tool chest for exploring the laws of physics, it does not in-and-of-itself add any new practical laws of physic beyond the least action principle (which is really more of a new powerful tool for exploring the more practical laws of physics than a new law of physics that cannot be derived from Newtonian physics).

regarding the usefulness of Hamiltonian physics is more powerful in terms of the types of problems that may be solved, relatively easily. Newtonian mechanics requires a careful analysis of the forces, and it may be easy to over look something and set up the equations incorrectly. Whereas, Hamiltonian-Jacobi mechanics, although the math is more elaborate, is more of plug and chug. Also, Hamiltonian mechanics has what can be thought of as the underpinnings of quantum mechanics. In fact, Schrodinger’s equation can be more or less derived from Hamiltonian mechanics, after you throw away the small terms that should not be significant (but without any experimental evidence, there is not way of knowing from Hamiltonian-Jacobi mechanics alone the full significance of the Schrodinger equation). Hamiltonian-Jacobi mechanics provides at least clue of how to quantize new quantities and at least a clue of how to set up Feynman path integrals, for example.

The words Hamiltonian and Lagrangian are more general than the word Newtonian. The words Hamiltonian and Lagrangian actually refer to the type of mathematics needed to solve a problem, not the underlying physics in the problem. You can use Lagrangian and Hamiltonian mathematics to solve problems in Newtonian, relativistic, and quantum physics.

These two adjectives refer to the mathematical representation of a physics theory. Scientists use a Lagrangian representation when they calculate the action integral in a problem. Scientists use a Hamiltonian representation when they calculate the functional dependence of energy on momentum.

Of course, the answers acquired when sticky to a physical hypothesis are mathematically equivalent. But there are ‘latent variables’ associated with each method. Each simplifies a specific problem in a different way.

All 3 are equivalent, but here’s the breakdown in plain English.

Newton’s equations: If you know the position and the derivative at one instant of time, Newton’s equations allow you to construct the whole trajectory, if you know the forces. You need two pieces of information for each coordinate, a position and a velocity.

Lagrangian formulation: Instead of knowing, the initial position and velocity, you are given the location of an object or a particle, whatever is, at 2 different times, and you are asked to fill in the trajectory in between. That corresponds to the actual orbit between those 2 points. This time, the initial velocity will be a part of the solution, you’ll need to aim it in the right way with the right velocity, to get it to go that point in space and time. So the initial velocity will not now be an input, it will be part of the solution. This is the other form of the problems of mechanics. Given an initial and final configuration, find what interpolates between them.

Hamiltonian formulation: A re-writing of the Lagrangian formulation.

In classical physics, particles act like particles, and waves act like waves, and never the twain shall meet.

The classical physics of particles introduces point contact interactions and particle trajectories that follow unique paths. Newton's laws can be used to describe the dynamics of particle motion.

Waves are described using a different theory. In particular, Huygens' principle describes the evolution of a wave, including diffraction phenomena and crucially, interference.

In a material medium, wave mechanics can be reduced to particle interactions, leading one to suspect that the particle picture might be fundamental.

When we add the concept of charges interacting with the electromagnetic field, particles then interact with the field, which mediates interactions with other particles. That's the next level of abstraction.

The electromagnetic field, as described by Maxwell's equations, has wave solutions in the free field devoid of any charges. Here we have moved away from the particle picture completely. The mathematics of wave propagation is now appropriate.

This really was the first stumbling block for classical physics. Wave propagation was assumed to require a medium. This gave rise the the aether theories, which were finally discarded with the development of relativity theory. In other words, the aether was not needed to support wave propagation. What was needed was a new perspective on space-time, known as Lorentz symmetry.

That means that particles and waves were considered to be fundamentally different.

This all changed when so-called fundamental particles were found to exhibit interference phenomena, which is a wave characteristic. This was a direct indication that what were considered to be particles actually exhibited wave-like properties. The degree to which particles act like waves was encapsulated in a new physical constant, called Planck's constant. This is why when you set Planck's constant to zero, you recover classical physics. However, Planck's constant, while small, is not zero. Therefore, one needs to adopt a wave-like picture for particles. In other words, you need to throw out your concept of what a particle is and adopt a new way of thinking. This is how quantum theory was born.

This might still sound rather obscure and a bit silly, but the proof is in the pudding. The particles known as electrons are routinely used for their wave-like properties in electron microscopes. Electron microscopes represent a mature technology widely used around the world to image the finest details down to the atomic scale.

The bottom line is that neither the particle nor wave pictures were adequate for describing all observed phenomena. What was needed was a conceptual readjustment that incorporated both pacticle and wave properties in a single entity from which particle and wave behaviours emerge in the appropriate limits.

OK, let us start with classical physics. Specifically, with the simplest concept in classical physics: a small particle.

Say you have a gun firing small bullets. In front of that gun, there’s a metal plate with two holes. Behind the metal plate, there is a wall. You are spraying the metal plate with your gun. Some of the bullets are blocked. Some make it through the holes. After a while, you will see something like this:

That is to say, on the wall, there will be two compact regions with bullet holes.

Now let us repeat the same experiment with electrons inside a cathode ray tube. Exact same setup, except that the conventional gun is replaced by an electron gun, and in place of the wall we have a fluorescent screen marking impact locations. We fire a bunch of electrons. The result will be something like this:

This result cannot be explained by classical physics. The electrons do not behave like bullets from a gun. Rather, the pattern we get is more akin to an interference pattern characterizing waves. Waves that reach the barrier, pass through the two holes, and then proceeding from these two holes interfere with each other.

This makes no sense from the perspective of classical physics. The math behind this is fairly straightforward but not intuitive. We do not have a brain that is wired to intuit the quantum world. We need to follow through step by step with the math.

The actual math is not terribly hard, but it involves differential operators and complex numbers so I’ll skip the details. The essence: we systematically replace, in a rigorous algebraic process, the positions and momenta (quantities of motion) characterizing the system with “operators”. These operators, acting on the state of the system, yield numbers that we can make sense of as probability densities. The resulting equations no longer describe a specific path for an electron. Rather, they describe all possible paths, with the probability densities obeying a wave equation. It is this wave equation that is responsible for the interference pattern we see with the electrons. But in the end, the probability density tells us the likelihood of the electron impacting at specific locations on the screen. So when the electron arrives (i.e., when its position is measured) it is seen as a particle. But when the electron travels, its travel is governed by the wave-like behavior of that probability density.

Like it or not, this is how the world works, as countless experiments demonstrate. It should remind us that Nature is under no obligation to appeal to our intuition. On the other hand, it’s a testament to human science that we could make sense of all this nonetheless, and today, the quantum world is “ours” to play with, building, among other things, the remarkable technology that allows you to read these very words, wherever you are in the world, seconds after I hit Next and complete this post.

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Newtonian mechanics deals with objects that are macroscopic and moving at speeds much smaller that the speed of light in vacuum. In Newtonian mechanics the main principle that is used to predict the motion of an object is : Every object wants to stay in its present state of motion (or rest) and a Force is required to change the momentum (mass times velocity) of such an object. This principle is called Newton's law of motion($\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$) Where $\overrightarrow{F}$ is the Force and $\overrightarrow{p}$ is the momentum of the object. Newtonian mechanics does a good job of describing the motions of most normal sized objects (bricks, cars, buildings etc.).

The Lagrangian and Hamiltonian formulations are not new types of mechanics, they are a new type of math for looking at Newtonian/Classical mechanics. They are collectively called Least action Principles. The Lagrangian is defined as $L=K-U$ and the Hamiltonian is defined as $H=K+U$ where $K$ and $U$ are the Kinetic and Potential energies of a particular particle. Suppose a Particle wants to go from Point A to Point B. There are infinitely many Paths which start at A and end at B. The "Action" is defined as the integral of the Lagrangian or the Hamiltonian over any Path joining A and B. The Least action Principle says that this Action will be smallest for the a particular Path and that's the Path which the particle will actually take to go from A to B. I call it the Lazy-Particle approach to Mechanics. Its as if the Particle tried out all the possible ways of going from A to B and finally chose a Path which takes the least amount of effort.

Relativistic Mechanics is used to describe the motion of objects which are moving at speeds close to the speed of light.(There are actually two different theories of relativity : Special and General. The General theory talks about Gravity.) The central postulate of Special Relativity is that the speed of light is the same for all Inertial frames. In Newtonian mechanics we make a clear distinction between spatial co-ordinates and time but in relativistic mechanics time is not considered to be independent of the frame of reference.(Space and time are collectively called, spacetime :P) This means if two Observers try to describe the same set of events then they might record different times at which the events took place and still be correct in their own frames of reference. All quantities in Newtonian Mechanics get modified accordingly.

Quantum mechanics deals with objects that are very small in size (think electrons, atoms,molecules etc.). It is fundamentally different from Newtonian mechanics because of the Uncertainty principle. According to Quantum mechanics all "objects"(which are localized) are described as "Waves" (Which are non localized ). We can talk about where an object is placed and what is its momentum, but we cannot say the same about a wave which is spread out in space. According to the Born interpretation, this wave tells us about the chances of finding an object at a particular place in space. The Schrodinger Equation which is the quantum analog of Newton's law describes how these Probability waves evolve with time. There's no concept of Forces in Quantum mechanics and all the Observables (things which can be measured about the motion of an object like Energy, Momentum, Angular Momentum etc.) depend on the Probability Wave through Operators (These are basically functions, which take the Probability Wave as the input and spit out the value of the Observable as the output.).

They are all essentially equivalent versions of Newtonian mechanics in the sense that they all give the same answers to mechanics problems. Conceptually, though, they seem worlds apart. Bog standard Newtonian physics starts with a differential equation, Newton’s Second Law. Lagrangian and Hamiltonian mechanics start from a minimum or extremum, from equations derived by finding an extremum of the Lagrangian or Hamiltonian. These equations are often much more convenient for solving some complicated problems and for incorporating symmetries of a physical situation. Moreover, they are generalizable to other physical laws and quantum mechanics.

The easy way is to plug H = p^2/2m + V into the action integral of p dx - H dt and perform a variation: p+sb, x+sw to get the integral of (p+s b)(dx+s dw) - (p+s b)^2/2m dt +V(x+s w)dt . The s derivative at s = 0 is b dx + p dw - p b/m dt + dV/dx w dt.

If p = m dx/dt this becomes p dw - dV/dx w dt “. If ma = - dV/dx then it’s p dw + ma w dt = m v dw + ma w dt = m v dw + m w dv = d (mvw). But that’s 0 because w is 0 at the endpoints.

So solutions to Newton’s laws are solutions. Are there any other solutions?

b dx + p dw - p b/m dt + dV/dx w dt must be 0 for all choices of w and b that vanish at the endpoints.

So first take w =0 in which case we have b (dx -p/m dt) = b (dx/dt - p/m) dt.

Now let b = (dx/dt - p/m) f where f is positive but vanishes at the endpoints. Then the integral of (dx/dt-p/m)^2 f is zero, so dx/dt=p/m. So p=m dx/dt.

So now the action integral reduces to p dw - dV/dx w dt = d (pw) - w dp - dV/dx w dt. But the integral of d (pw) is 0, because w vanishes at the endpoints. So dp/dt + dV/dx =0.’s .

Putting the two together we have ma + dV/dx =0.

Now you go back and consider more generally the variational problem for the integral of p dx -H dt, where the same arguments will yield Hamilton’s equations.

Let’s start with some historical notes and definitions.

Lagrangian mechanics was named after Joseph-Louis Lagrange (1736–1813), Hamiltonian mechanics was named after William Rowan Hamilton (1805–1865). They can be regarded as reformulations of classical or Newtonian mechanics, and as branches of Analytical mechanics :

Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries.

Lagrange elaborated his formulation of mechanics in a treatise entitled Mécanique analytique, where he used the Principle of Virtual Work with d’Alembert’s principle, dealt with statics, hydrostatics, and dynamics, and introduced the Lagrange multiplier method to treat constraints.

Lagrange relied mostly on algebraic methods and stated in the Preface to his book:

No figures will be found in this work. The methods I present require neither constructions nor geometrical or mechanical arguments, but solely algebraic operations subject to a regular and uniform procedure. Those who appreciate mathematical analysis will see with pleasure mechanics becoming a new branch of it and hence, will recognize that I have enlarged its domain.

Hamilton exposed his ideas about a reformulation of classical and Lagrangian mechanics in 1834, in a paper entitled On a General Method in Dynamics. He introduced the paper by writing:

The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem.

Here are some details about Lagrangian and Hamiltonian mechanics and the difference or relation between them. To avoid confusion, one has to to take into account that different books or textbooks about this topic use different (but coherent) notations or symbols.

Lagrangian and Hamiltonian mechanics are also related to the branch of mathematical analysis known as the calculus of variations, which studies how to minimize and maximize functionals, and deals with extremal functions that make the functional attain a maximum or minimum value – or stationary functions – i,e. those where the rate of change of the functional is zero.

General coordinates $q_1,q_2,\dots q_n$ may represent angles, distances, or quantities related to them.

Let

$r_{\mu }=x_{\mu }i+y_{\mu }j+z_{\mu }k$

be the position vector of a given particle in a coordinate system.

The relation between the position coordinates and the generalized coordinates can be written as ($t$ is the time):

$r_{\mu }=r_{\mu }(q_1,q_2,\dots q_n,t)$

The total kinetic energy of a given system is:

$T=\frac{1}{2}\sum _{\mu =1}^N{m}_{\mu }{\dot{r}}_{\mu }^2=\frac{1}{2}\sum _{\mu =1}^N{m}_{\mu }{v}_{\mu }^2$

$W$ being the total work done on a system of particles $F_{\mu }$ which act on the $\mu th$ particle:

$dW=\sum _{\alpha =1}^n{\chi }_{\alpha }d{q}_{\alpha }$

The quantity

${\chi }_{\alpha }=\sum _{\mu =1}^N{F}_{\mu }\cdot \frac{\partial r_{\mu }}{\partial q_{\alpha }}$

is called the generalized force related to the generalized coordinate $q_{\alpha }.$

It can be shown that the generalized force is related to the kinetic energy by the equations:

$\frac{d}{dt}\left(\frac{\partial T}{\partial {\dot{q}}_{\alpha }}\right)-\frac{\partial T}{\partial q_{\alpha }}={\chi }_{\alpha }$

When the system is conservative, and the forces can be derived from a potential or potential energy $V$, the equations above can be expressed as:

$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_{\alpha }}\right)-\frac{\partial L}{\partial q_{\alpha }}=0\left(1\right)$

where $L=T-V$ is the Lagrangian function, or the Lagrangian, of the system.

For the simple case of one dimension, the equations $\left(1\right)$ , called Lagrange’s equations or the Euler-Lagrange equations, can be written in the form:

$\frac{d}{dt}\left(\frac{\partial L}{\partial v}\right)-\frac{\partial L}{\partial x}=0,$

where the velocity $v$ is the time derivative of the position: $v=\dot{x}.$

So in classical mechanics, for a system of $N$ particles the Lagrangian can be written as:

$L=\frac{1}{2}\sum _{\mu =1}^N{m}_{\mu }{\dot{r}}_{\mu }^2-V(r_1,...,r_N)$

For the case of one dimension and one particle (or point mass), the Lagrangian can be expressed as:

$L=T-V=\frac{1}{2}m{\dot{x}}^2-V\left(x\right).$

The generalized momentum related to the generalized coordinate $q_{\alpha }$ (sometimes called the conjugate momentum) is defined as:

$p_{\alpha }(q_{\alpha },{\dot{q}}_{\alpha },t)=\frac{\partial L}{\partial {\dot{q}}_{\alpha }}$

Now we can cross to Hamiltonian mechanics. The Hamiltonian (function) $H$ is defined in terms of the Lagrangian $L$ as ($H$ represents the Legendre transformation of $L$):

$H=\sum _{\alpha =1}^n{p}_{\alpha }{\dot{q}}_{\alpha }-L$

The equations of motion of a given system in terms of the Hamiltonian are expressed in the symmetrical form:

${\dot{p}}_{\alpha }=-\frac{\partial H}{\partial q_{\alpha }},{\dot{q}}_{\alpha }=\frac{\partial H}{\partial p_{\alpha }}$

For a conservative system, the Hamiltonian represents the total energy of this system:

$H=T+V.$

As a simple application of the relations above, for one dimension and for a single particle, remembering that $p=m\dot{x}=mv$, and using the ordinary definition of kinetic energy and potential energy, we have:

$H=p\dot{q}-L=\dot{x}\frac{\partial L}{\partial \dot{x}}-L=p\dot{x}-T+V=p\dot{x}-\frac{m}{2}{\dot{x}}^2+V$

$H=\frac{p^2}{m}-\frac{p^2}{2m}+V=\frac{p^2}{2m}+V=\frac{1}{2}m{v}^2+mgx$

Hamiltonian mechanics provides symmetry between $p_{\alpha }$ and $q_{\alpha }$, the momentum and position coordinates. A space of $2n$ dimensions where a point is represented by $2n$ coordinates

$(p_1,\dots ,p_n,q_1,\dots ,q_n)$

is called a $2n$ dimensional phase space, or a $pq$ phase space.

According to Hamilton’s principle of least action, the motion of a conservative system from time $t_1$ to time $t_2$ is described by the integral

$S={\int }_{t_1}^{t_2}L(q_{\alpha },\dot{q_{\alpha }},t)dt.$

This integral is called the action integral, is a functional of the generalized coordinates defining the configuration of the system, and has an extreme value.

The action integral is an extremum:

$\delta S=\delta {\int }_{t_1}^{t_2}L(q_{\alpha },\dot{q_{\alpha }},t)dt=0$

This condition is true or verified if the Euler-Lagrange equations of motion $\left(1\right)$ are satisfied.

Lagrangian and Hamiltonian mechanics and formulations have many applications and uses in various branches or sub-fields of physics.

As an example, the Hamiltonian operator (for a single particle) in quantum mechanics can be expressed as:

$\begin{array}{cc}\hat{H}& =\hat{T}+\hat{V}\\ & =\frac{\hat{p}\cdot \hat{p}}{2m}+V(r,t)\\ & =-\frac{{\hslash }^2}{2m}{\nabla }^2+V(r,t)\end{array}$

This Hamiltonian is used in the expression of the time-dependent Schrödinger equation :

$i\hslash \frac{\partial }{\partial t}\Psi (r,t)=\hat{H}\Psi (r,t)$

The time-independent Schrödinger equation is expressed in terms of the Hamiltonian as:

$\hat{H}\Psi =E\Psi$

The Lagrangian is used in branches of physics such as electromagnetism, General Relativity and classical field theory. For example, in classical electrodynamics, the Lagrangian density can be expressed as:

$L=-\frac{1}{4{\mu }_0}{F}^{\alpha \beta }{F}_{\alpha \beta }-A_{\alpha }{J}^{\alpha }.$

$F_{\alpha \beta }$ is the (covariant) electromagnetic field tensor, $A_{\alpha }$ is the four-potential, and $J^{\alpha }$ is the four-current. ${\mu }_0$ is the magnetic permeability of vacuum.

The Lagrangian density for quantum electrodynamics is:

$L_{QED}=i\hslash c\overline{\psi }D/\psi -m{c}^2\overline{\psi }\psi -\frac{1}{4{\mu }_0}{F}_{\mu \nu }{F}^{\mu \nu },$

where $F^{\mu \nu }$ is the electromagnetic tensor, $D$ is the gauge covariant derivative, and $D/$ is Feynman notation for ${\gamma }^{\sigma }{D}_{\sigma }$ with $D_{\sigma }={\partial }_{\sigma }-ie{A}_{\sigma }$ where $A_{\sigma }$ is the electromagnetic four-potential.

For more info, see also the following relevant links:

Generalized coordinates - Wikipedia

Canonical transformation - Wikipedia

Hamiltonian (quantum mechanics) - Wikipedia

Lagrangian (field theory) - Wikipedia

Action (physics) - Wikipedia

Classical Mechanics/Lagrangian

What is the difference between a Lagrangian and a Hamiltonian?

Equivalence between Hamiltonian and Lagrangian Mechanics

http://www.dzre.com/alex/P441/lectures/lec_18.pdf [Examples in Lagrangian mechanics]

There are a number of ways in which the Hamiltonian formulation is important in fluid mechanics. I am not sure if you are very familiar with the Hamiltonian approach to fluid mechanics, but if not this article is my favorite reference:
http://www.ph.utexas.edu/~morrison/98RMP_morrison.pdf
It was written by Phil Morrison, and he is responsible for much of the work in this field. Hamiltonian fluid mechanics is likely different than the Hamiltonian mechanics which you are familiar with, in particular it occurs on Poisson manifolds (which are sort of like stacks of symplectic manifolds). Hamiltonian descriptions of fluids are very beautiful because there is an extremely nice geometric picture which you can read about in the link if you are not already familiar with it. Unfortunately, as will be mentioned later, the picture is not really mathematically rigorous.

The primary reason to use the Hamiltonian formulation to describe fluid theories is because different Hamiltonian systems share many similarities. Fluid theories are infinite dimensional, and there isn't really a rigorous mathematical theory of infinite-dimensional Hamiltonian systems like one would find in the finite-dimensional case. In the finite-dimensional case there is a pretty geometric picture that is true rigorously (see the following book by Arnold for a decent exposition: http://books.google.com/books/about/Mathematical_methods_of_classical_mechan.html?id=Pd8-s6rOt_cC)
because there are universal existence theorems for ODEs, in infinite-dimensions this is much more difficult and one has to do things in a case by case basis (PDEs are much harder).

This means it is difficult to get theorems that apply to all Hamiltonian fluid theories. Despite these difficulties with rigorous mathematics, many rigorous results from finite-dimensional Hamiltonian systems seem to have analogies in infinite-dimensional Hamiltonian systems, and these ideas are crucial for understanding the behavior of these systems, and are a source of ideas for theorems to prove about specific systems. This is true both within fluid mechanics and outside of it. For example, integrability, chaos, energy principles for stability, geometry of phase space, the KAM theorem (nonlinear Landau damping, part of the citation for the latest fields medal, relied on this), and my personal favorite (because half my thesis was based on it) Krein's theorem, all generalize in various ways to infinite-dimensional Hamiltonian systems. These ideas are extraordinarily useful when you look at linearized fluid equations (linearizing maintains the Hamiltonian character of the equations, solving a problem of this type was the second half of my thesis). Fluid mechanics is one of the most difficult subjects in classical physics and it is important to use all the intuition that we can get for it.

The second reason to use the Hamiltonian formalism is that it prevents you from making mistakes when deriving reduced descriptions of various phenomenon occurring in fluids. Within fluid mechanics there is an entire zoo of approximate models for different things. These range from finite-dimensional models for fluid turbulence to infinite-dimensional theories that are reductions or simplifications of other systems to asymptotic theories. It is easy to screw up these models by adding dissipation unintentionally. For example, in my opinion, if you derive some reduced, it should become Hamiltonian when the viscosity terms are set to zero. If this is not the case I would say that the model is likely unphysical. This is actually something that happens frequently in this field.

In a similar vein, when analyzing a newly derived fluid equation, it is important to find all of the conserved quantities. These are most easily calculated using the Hamiltonian formulation of your equation (assuming it has no dissipation). In the fluid case the invariants can be subtle, almost all Hamiltonian fluid theories are "non-canonical Hamiltonian systems with Lie-Poisson brackets". Systems of this type share a number of universal features (Point 1 helping us out here), in particular the existence of invariants due to the geometric structure of phase space. These are called Casimir invariants, and always accompany Hamiltonian fluid theories. It is essential to find all of these invariants to analyze such a theory.

There are also a number of ideas for improving treatment of fluid theories that come purely from Hamiltonian systems theory. These theories have not been around that long, and much of the potential is unrealized. When in possession of a theory with a lot of invariants, the natural thing to do is to rewrite the theory in a space where the invariants have been projected out. It is possible to do this for Hamiltonian fluid theories, but no one has analyzed the equations that have resulted from this reduction. These equations are called 'leaf equations'. There are also a number of interesting ideas for using Lie-groups to perform reductions of complicated systems with continuous symmetries. See the beautiful work of Predrag Cvitanovic et. al.http://www.cns.gatech.edu/~predrag/papers/pubs.html which was inspired by Hamiltonian methods.

Sometimes it is important to look at individual particle orbits. This is important in mixing problems and also in plasma physics when the goal is to understand confinement. In these cases the finite-dimensional Hamiltonian systems theory is essential, as one must determine the structure of the magnetic field and the resulting orbits using ideas related to KAM theory and chaos.

I really hope this answer is helpful. If you think anything should be added to it or needs clarifying please do not hesitate to ask. I can provide very detailed references or descriptions and will do so as requested (but maybe with a large lag time). This field is very close to mine and is an area which I actively follow. Also I realize I didn't explain much about what the Hamiltonian description of a fluid is like, but if I did the answer who be way too long. If I think of a short and sweet description I will post it.

Because the most common way to derive the mathematical formalism of quantum mechanics is via a multiple-paths version of Hamilton’s Principle. Hamilton’s principle states that the generalised coordinates of the Lagrangian describe the trajectory a particle will follow if the action functional, that is, the time integral of the Lagrangian, is stationary.

In classical mechanics, there is only one possible trajectory the particle can follow because there is only one state for the system under which the action is stationary. In quantum mechanics, however, the action for several possible trajectories is calculated and weighted with a probability that the particle will go down that specific path. In the aggregate, we end up with a spectrum of possible states the system can find itself in and the probability of each state manifesting; this spectrum is known as the wave function.

Under the de Broglie interpretation of quantum mechanics, what we conclude is that, in reality, nature does allow for several possible states to coexist for one same system, but only in very low energy regimes. As we “zoom out” from the scale at which quantum effects are usually observed, these multiple states blend together into one “single” state, which is why only one possible trajectory is considered in classical mechanics, meant to deal with higher energy regimes.

Hamiltonian mechanics reveals the total energy of a system. Whereas Newtonian mechanics reveals how that energy creates movement. The conversion is best used with an example.

fig 1. Particle accelerating to the ground due to gravity.

The particle has a mass m,

the gravitational acceleration is g = +9.8m/s²

and is accelerating in the negative direction with a force F = −mg

The Hamiltonian is written as:

H = KE + PE (kenetic energy + potential energy)

where the kenetic energy is calculated as 1/2 mv², and potential energy as m·g from a height X. Now the equation reads:

H = 1/2 mv² + mgx

the kenetic energy is often written in the form of momentum. Where momentum is calculated as p = mass × velocity. And squaring to aid substitution p² = m²v². Now p² is sub'd into KE

H = 1/2 · m²/m · v² + mgx

H = 1/2m · p² + mgx

Now take the partial derivative of the Hamiltonian with respect to X, then with respect to momentum

∂H/∂x = mg

∂H/∂p = p/m

though as p = mv. We can sub mv in and simplify

∂H/∂p = mv/m = v

the mass of the particle falling multiplied by the time derivative of the partial of the Hamiltonian with respect to momentum + the partial of the Hamiltonian with respect to its position x = 0

m·d/dt (∂H/∂p) + ∂H/∂x = 0

when the time derivative of the Hamiltonian with respect to momentum is calculated it equals acceleration

d/dt (∂H/∂p) = a

which can be sub'd and simplified

ma + mg = 0

factor out the mass m, leaving

a + g = 0

and solving for acceleration

a = −g

So the acceleration of the particle with mass m is the gravitational value in the downward direction. As shown in fig 1.

Then inserting the particle’s mass back into the acceleration equation which reveals the force F acting on the particle we get:

ma = −mg

F = ma

Force = mass times acceleration.

Hamiltonian mechanics takes the total energy of any system examined, converts it to momentum, which in turn reveals the equations of motion.

If you are looking for a model that describes in an abstract and general way the development of what we call a physical event, you will find that Hamiltonian theory is that model. That's why it's a bad idea to call that theory mechanics. Obviously you can use Hamiltonian theory to formulate mechanical phenomena. You can also use it to formulate phenomena specific to fields, including the electromagnetic field. You can even use it to formulate computer phenomena. The Hamiltonian theory is the general mathematical model of what science understands by an event, be it physical, computer science or whatever.

Well, Newtonian gravity emerges from Einsteinian gravity, in suitable limits. Most of the time Newtonian gravity is completely adequate. For example, you can understand the solar system (99.9% at least) using it, you can plan space missions using it, etc. If you look around enough, though, you can find situations where it’s experimentally clear that the answer Newtonian gravity gives is wrong. We began to notice in the late 1800’s that it made incorrect predictions about the orbit of Mercury - our observational data had gotten good enough that this became clear.

Einstein’s theory, on the other hand, gets the orbit of Mercury right, and in fact gets everything we have ever checked right. We believe currently it’s correct for all situation where we can ignore quantum effects on the gravitational front. We don’t have any “replacement theory” that will cover those quantum gravity cases.

So, I think the thing to tell you is that yes, in fact it is possible to show that Einstein’s theory is superior to Newtons, but in most practical cases it doesn’t matter which one you use and Newton’s is a lot easier to do computationally. In most cases, the practical difference is that you’ll have to work a lot harder to do it Einstein’s way and you’ll get the same answer in the end. Most cases - but not all.

Stay safe and well!

Kip

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