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Symmetry reduction of discrete Lagrangian mechanics on Lie groups

Marsden, Jerrold E. and Pekarsky, Sergey and Shkoller, Steve (2000) Symmetry reduction of discrete Lagrangian mechanics on Lie groups. Journal of Geometry and Physics, 36 (1-2). pp. 140-151. ISSN 0393-0440. doi:10.1016/S0393-0440(00)00018-8.

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For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie–Poisson structure on the dual of the Lie algebra g^* by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ωL on G×G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton–Jacobi equation are made. The rigid body is discussed as an example.

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Additional Information:© 2000 Elsevier Science. Received 14 October 1999; revised 7 February 2000. Available online 25 September 2000. February 1999; current version October 26, 2000. The authors would like to thank Alan Weinstein for pointing out the connections with the general theory of dynamics on groupoids and algebroids. SP and SS would also like to thank the Center for Nonlinear Science in Los Alamos for providing a valuable setting for part of this work. SS was partially supported by NSF-KDI grant ATM-98-73133, and JEM and SP were partially supported by NSF-KDI grant ATM-98-73133 and the Air Force Office of Scientific Research.
Funding AgencyGrant Number
NSF-Knowledge and Distributed Intelligence (KDI)ATM-98-73133
NSF-Knowledge and Distributed Intelligence (KDI)ATM-98-73133
Air Force Oce of Scientific ResearchUNSPECIFIED
Subject Keywords:Euler–Poincaré; Symplectic; Poisson
Issue or Number:1-2
Classification Code:MSC: 70H35; 70E15; 58F
Record Number:CaltechAUTHORS:20100903-092444291
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19776
Deposited By: Ruth Sustaita
Deposited On:04 Sep 2010 03:38
Last Modified:08 Nov 2021 23:55

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