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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Hamiltonian and symplectic symmetries: An introduction
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by Álvaro Pelayo PDF
Bull. Amer. Math. Soc. 54 (2017), 383-436 Request permission


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
  • MR Author ID: 731609
  • Email:
  • 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)
  • © Copyright 2017 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 54 (2017), 383-436
  • MSC (2010): Primary 53D20, 53D35, 57R17, 37J35, 57M60, 58D27, 57S25
  • DOI:
  • MathSciNet review: 3662913