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

MathSciNet review: 3662913

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Abstract: Classical mechanical systems are modeled by a symplectic manifold , and their symmetries are encoded in the action of a Lie group on by diffeomorphisms which preserve . 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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Álvaro Pelayo

Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive $#$0112, La Jolla, California 92093-0112

Email: alpelayo@math.ucsd.edu



DOI: https://doi.org/10.1090/bull/1572

Received by editor(s): October 14, 2016

Published electronically: March 6, 2017

Additional Notes: The author is supported by NSF CAREER Grant DMS-1518420.

Dedicated: In memory of Professor J.J. Duistermaat (1942–2010)

Article copyright: © Copyright 2017 American Mathematical Society