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Variational Principles for Lie—Poisson and Hamilton—Poincaré Equations

Cendra, Hernán and Marsden, Jerrold E. and Pekarsky, Sergey and Ratiu, Tudor S. (2003) Variational Principles for Lie—Poisson and Hamilton—Poincaré Equations. Moscow Mathematical Journal, 3 (3). pp. 833-867. ISSN 1609-3321. http://resolver.caltech.edu/CaltechAUTHORS:20100727-093528997

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Abstract

As is well-known, there is a variational principle for the Euler—Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton's principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie—Poisson equations on g^*, the dual of g, and also to generalize this construction. The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q→Q/G becomes a principal bundle. Starting with a Lagrangian system on TQ invariant under the tangent lifted action of G, the reduced equations on (TQ)/G, appropriately identified, are the Lagrange—Poincaré equations. Similarly, if we start with a Hamiltonian system on T^*Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T^*Q)/G are called the Hamilton—Poincaré equations. Amongst our new results, we derive a variational structure for the Hamilton—Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors.


Item Type:Article
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http://www.ams.org/distribution/mmj/vol3-3-2003/cendra-etal.pdfAuthorUNSPECIFIED
Additional Information:© 2003 Independent University of Moscow. Dedicated to Vladimir Arnold on his 65th birthday. Received December 24, 2002; in revised form July 24, 2003. H.C. supported in part by the EPFL. J. E. M. and S.P. supported in part by NSF grant DMS-0204474 and a Max Planck Research Award. T. S. R. supported in part by the European Commission and the Swiss Federal Government (Research Training Network Mechanics and Symmetry in Europe (MASIE)), by Swiss National Science Foundation and by Humboldt Foundation.
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Funding AgencyGrant Number
Ecole Polytechnique Fédérale de Lausanne (French: Swiss Federal Institute of Technology, Lausanne, Switzerland) (EPFL. J. E. M)UNSPECIFIED
NSFDMS-0204474
European CommissionUNSPECIFIED
Swiss National Science Foundation and by Humboldt FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20100727-093528997
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ID Code:19198
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:27 Jul 2010 20:40
Last Modified:26 Dec 2012 12:16

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