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. https://resolver.caltech.edu/CaltechAUTHORS:20100903-092444291
![]() |
PDF
- Published Version
Restricted to Repository administrators only See Usage Policy. 161kB |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20100903-092444291
Abstract
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.
Item Type: | Article | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Related URLs: |
| |||||||||
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. | |||||||||
Funders: |
| |||||||||
Subject Keywords: | Euler–Poincaré; Symplectic; Poisson | |||||||||
Issue or Number: | 1-2 | |||||||||
Classification Code: | MSC: 70H35; 70E15; 58F | |||||||||
DOI: | 10.1016/S0393-0440(00)00018-8 | |||||||||
Record Number: | CaltechAUTHORS:20100903-092444291 | |||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20100903-092444291 | |||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||
ID Code: | 19776 | |||||||||
Collection: | CaltechAUTHORS | |||||||||
Deposited By: | Ruth Sustaita | |||||||||
Deposited On: | 04 Sep 2010 03:38 | |||||||||
Last Modified: | 08 Nov 2021 23:55 |
Repository Staff Only: item control page