Yoshimura, Hiroaki and Marsden, Jerrold E.
(2009)
*Dirac cotangent bundle reduction.*
Journal of Geometric Mechanics, 1
(1).
pp. 87-158.
ISSN 1941-4889.
http://resolver.caltech.edu/CaltechAUTHORS:20090903-112420240

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## Abstract

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie--Dirac reduction . This procedure simultaneously includes Lagrangian, Hamiltonian, and a variational view of reduction. The goal of the present paper is to generalize Lie-Dirac reduction to the case of a general configuration manifold; we refer to this as Dirac cotangent bundle reduction. This reduction procedure encompasses, in particular, a reduction theory for Hamiltonian as well as implicit Lagrangian systems, including the case of degenerate Lagrangians. First of all, we establish a reduction theory starting with the Hamilton-Pontryagin variational principle, which enables one to formulate an implicit analogue of the Lagrange-Poincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure . Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called Lagrange-Poincaré-Dirac reduction . This procedure naturally induces the horizontal and vertical implicit Lagrange-Poincaré equations , which are consistent with those derived from the reduced Hamilton-Pontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, Hamilton-Poincaré-Dirac reduction for the horizontal and vertical Hamilton-Poincaré equations . We illustrate the reduction procedures by an example of a satellite with a rotor. The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages . This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.

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Additional Information: | © AIMS 2009. Received: August 2008; Revised: March 2009; Published: April 2009 We are very grateful to Tudor Ratiu and Hern´an Cendra for providing helpful remarks and suggestions. We also thank the reviewers for their helpful suggestions and comments. 2000 Mathematics Subject Classification. Primary: 70H03, 70H05, 70H30; Secondary: 53D20. | ||||||||||

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Subject Keywords: | Dirac cotangent bundle reduction; gauged Dirac structures; Lagrange-Poincaré-Dirac reduction; Hamilton-Poincaré-Dirac reduction; implicit Lagrange-Poincaré equations; Hamilton-Poincaré equations | ||||||||||

Record Number: | CaltechAUTHORS:20090903-112420240 | ||||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20090903-112420240 | ||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||

ID Code: | 15570 | ||||||||||

Collection: | CaltechAUTHORS | ||||||||||

Deposited By: | Jason Perez | ||||||||||

Deposited On: | 22 Sep 2009 17:39 | ||||||||||

Last Modified: | 26 Dec 2012 11:18 |

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