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Dirac cotangent bundle reduction

Yoshimura, Hiroaki and Marsden, Jerrold E. (2009) Dirac cotangent bundle reduction. Journal of Geometric Mechanics, 1 (1). pp. 87-158. ISSN 1941-4889.

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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.
Funding AgencyGrant Number
Waseda UniversityUNSPECIFIED
JSPS grant-in-aid for Scientific Research20560229
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
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15570
Deposited By: Jason Perez
Deposited On:22 Sep 2009 17:39
Last Modified:26 Dec 2012 11:18

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