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Symmetries of Hamiltonian Systems on Symplectic and Poisson Manifolds

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Similarity and Symmetry Methods

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 73))

Abstract

This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions in differential geometry. Full proofs of many results are provided.

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Notes

  1. 1.

    In plane Euclidean geometry, the power of a point \(O\) with respect to a circle \(\mathcal C\) is the real number \(\overrightarrow{OA}.\overrightarrow{OB}\), where \(A\) and \(B\) are the two intersection points of \(\mathcal C\) with a straight line \(\mathcal D\) through \(O\). That number does not depend on \(\mathcal D\) and is equal to \(\Vert \overrightarrow{OC}\Vert ^2-{\mathcal R}^2\), where \(C\) is the centre and \(\mathcal R\) the radius of \(\mathcal C\).

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Acknowledgments

I address my thanks to Ivaïlo Mladenov for his kind invitation to present lectures at the International Conference on Geometry, Integrability and Quantization held in Varna in June 2013. For their moral support and many stimulating scientific discussions, I thank my colleagues and friends Alain Albouy, Marc Chaperon, Alain Chenciner, Maylis Irigoyen, Jean-Pierre Marco, Laurent Lazzarini, Fani Petalidou, Géry de Saxcé, Wlodzimierz Tulczyjew, Paweł Urbański, Claude Vallée and Alan Weinstein.

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  1. Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, Paris, France

    Charles-Michel Marle

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  1. Charles-Michel Marle
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Correspondence to Charles-Michel Marle .

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  1. LEMTA—Université de Lorraine, Vandoeuvre, France

    Jean-François Ganghoffer

  2. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivaïlo Mladenov

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Marle, CM. (2014). Symmetries of Hamiltonian Systems on Symplectic and Poisson Manifolds. In: Ganghoffer, JF., Mladenov, I. (eds) Similarity and Symmetry Methods. Lecture Notes in Applied and Computational Mechanics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-08296-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-08296-7_4

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