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Symplectic Reduction for Semidirect Products and Central Extensions

Marsden, Jerrold E. and Misiołek, Gerard and Perlmutter, Matthew and Ratiu, Tudor S. (1998) Symplectic Reduction for Semidirect Products and Central Extensions. Differential Geometry and its Applications, 9 (1-2). pp. 173-212. ISSN 0926-2245.

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This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980's by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein, 1974) and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misiołek, Perlmutter and Ratiu, 1998). The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott-Virasoro group and the KdV equation.

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Additional Information:© 1998 Published by Elsevier Science. Received 14 January 1998; revised 20 April 1998. Communicated by M. Gotay Available online 17 September 1998. March, 1994; this version May 24, 1998, to appear in Differential Geometry and its Applications. We thank Hernan Cendra, Mark Gotay and the referee for several helpful suggestions. We also thank the Erwin Schrödinger Institute for Mathematical Physics for providing the opportunity to meet in pleasant surroundings during 1994 at which time some of the ideas in the paper were first worked out. The research of JEM was partially supported by NSF grant DMS 96–33161, that of GM by the faculty leave program at Notre Dame, that of MP by by DOE contract DE–FG0395–ER25251 and that of TSR by NSF Grant DMS-9503273, DOE contract DE-FG03-95ER25245-A000, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.
Funding AgencyGrant Number
NSFDMS 96–33161
Department of Energy (DOE)DE–FG0395–ER25251
Department of Energy (DOE)DE-FG03-95ER25245-A000
Erwin Schrödinger InstituteUNSPECIFIED
Miller Institute for Basic Research in ScienceUNSPECIFIED
Subject Keywords:Symplectic reduction; Hamiltonian systems; reduction by stages; semidirect products; group extensions
Classification Code:MSC:37J15; 53D20
Record Number:CaltechAUTHORS:20100831-143247267
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19747
Deposited By: Ruth Sustaita
Deposited On:01 Sep 2010 18:26
Last Modified:08 May 2015 17:47

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