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Symplectic connections and the linearisation of Hamiltonian systems

Published online by Cambridge University Press:  14 November 2011

J. E. Marsden ,
T. Ratiu  and
G. Raugel
J. E. Marsden
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A. and Cornell University, Ithaca, NY 14853-7901, U.S.A.
T. Ratiu
Affiliation:
Department of Mathematics, University of California, Santa Cruz, CA 95064, and MSRI, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A.
G. Raugel
Affiliation:
Laboratoire d'Analyse Numérique (Unité Associée au CNRS D760) Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France
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Synopsis

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 329 - 380
Copyright
Copyright © Royal Society of Edinburgh 1991

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