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Variational principles for Lie-Poisson and Hamilton-Poincaré equations

Hernán, Cendra 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.

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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.

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Additional Information:© 2003 Independent University of Moscow. 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.
Funding AgencyGrant Number
École Polytechnique Fédérale de Lausanne (EPFL)UNSPECIFIED
Max Planck Research AwardUNSPECIFIED
European CommissionUNSPECIFIED
Swiss Federal GovernmentUNSPECIFIED
Swiss National Science FoundationUNSPECIFIED
Humboldt FoundationUNSPECIFIED
Subject Keywords:Geometric mechanics, Euler—Lagrange, Lagrangian reduction, Euler—Poincaré, Lagrange—Poincaré, Hamilton—Poincaré
Classification Code:2000 Mathematics Subject Classification: 37J15, 70H25
Record Number:CaltechAUTHORS:20101004-155049051
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20286
Deposited By: Tony Diaz
Deposited On:16 Nov 2010 23:33
Last Modified:26 Dec 2012 12:29

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