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Hamiltonian and symplectic symmetries: An introduction


Author: Álvaro Pelayo
Journal: Bull. Amer. Math. Soc.
MSC (2010): Primary 53D20, 53D35, 57R17, 37J35, 57M60, 58D27, 57S25
DOI: https://doi.org/10.1090/bull/1572
Published electronically: March 6, 2017
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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 $ G$ on $ M$ 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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Additional Information

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