Hamiltonian and symplectic symmetries: An introduction

Pelayo, Álvaro

doi:10.1090/bull/1572

Hamiltonian and symplectic symmetries: An introduction

Author: Álvaro Pelayo
Journal: Bull. Amer. Math. Soc. 54 (2017), 383-436
MSC (2010): Primary 53D20, 53D35, 57R17, 37J35, 57M60, 58D27, 57S25
DOI: https://doi.org/10.1090/bull/1572
Published electronically: March 6, 2017
Full-text PDF Free Access

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Abstract: Classical mechanical systems are modeled by a symplectic manifold $(M,\omega )$ , and their symmetries are encoded in the action of a Lie group on by diffeomorphisms which preserve $\omega$ . These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finite-dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent results on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry.

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