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Reduction of Dirac Structures and the Hamilton-Pontryagin Principle

Yoshimura, Hiroaki and Marsden, Jerrold E. (2007) Reduction of Dirac Structures and the Hamilton-Pontryagin Principle. Reports on Mathematical Physics, 60 (3). pp. 381-426. ISSN 0034-4877. http://resolver.caltech.edu/CaltechAUTHORS:20100917-141535653

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Abstract

This paper develops a reduction theory for Dirac structures that includes, in a unified way, reduction of both Lagrangian and Hamiltonian systems. It includes the reduction of variational principles and in particular, the Hamilton-Pontryagin variational principle. It also includes reduction theory for implicit Lagrangian systems that could be degenerate and have constraints. In this paper we focus on the special case in which the configuration manifold is a Lie group G. In our earlier papers we established the link between the Hamilton-Pontryagin principle and Dirac structures. We begin the paper with the reduction of this principle. The traditional view of Poisson reduction in this case is to reduce T^*G with its natural Poisson structure to g^* with its Lie-Poisson structure. However, the basic step of reducing Hamilton's phase space principle already shows that it is important to use g^* ⊕ g^* for the reduced space, rather than just g^*. In this way, our construction includes both Euler-Poincaré as well as Lie-Poisson reduction. The geometry behind this procedure, which we call Lie-Dirac reduction starts with the standard (i.e., canonical) Dirac structure on T^*G (which can be viewed either symplectically or from the Poisson viewpoint) and for each µ є g^*, produces a Dirac structure on g^* ⊕ g^* . This geometry then simultaneously supports both Euler-Poincaré and Lie-Poisson reduction. In the last part of the paper, we include nonholonomic constraints, and illustrate this construction with Suslov systems in nonholonomic mechanics, both from the Euler-Poincaré and Lie-Poisson viewpoints.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1016/S0034-4877(08)00004-9DOIUNSPECIFIED
Additional Information:© 2007 Published by Elsevier Ltd. Received 19 February 2007; revised 17 September 2007. Available online 20 May 2008. Research partially supported by JSPS Grant 16560216. Research partially supported by NSF-ITR Grant ACI-0204932. We are very grateful to Tudor Ratiu, Alan Weinstein and Troy Smith for providing useful remarks and suggestions. We also thank the reviewers for their helpful suggestions and comments.
Funders:
Funding AgencyGrant Number
Japan Society for the Promotion of Science (JSPS) 16560216
NSF-ITRACI-0204932
Subject Keywords:reduced Hamilton-Pontryagin principle, Lie-Dirac reduction, implicit Lagrangian systems, Suslov problems
Record Number:CaltechAUTHORS:20100917-141535653
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100917-141535653
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20022
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 21:50
Last Modified:26 Dec 2012 12:26

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