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Dirac structures in Lagrangian mechanics. Part II: Variational structures

Yoshimura, Hiroaki and Marsden, Jerrold E. (2006) Dirac structures in Lagrangian mechanics. Part II: Variational structures. Journal of Geometry and Physics, 57 (1). pp. 209-250. ISSN 0393-0440. doi:10.1016/j.geomphys.2006.02.012.

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Part I of this paper introduced the notion of implicit Lagrangian systems and their geometric structure was explored in the context of Dirac structures. In this part, we develop the variational structure of implicit Lagrangian systems. Specifically, we show that the implicit Euler–Lagrange equations can be formulated using an extended variational principle of Hamilton called the Hamilton–Pontryagin principle. This variational formulation incorporates, in a natural way, the generalized Legendre transformation, which enables one to treat degenerate Lagrangian systems. The definition of this generalized Legendre transformation makes use of natural maps between iterated tangent and cotangent spaces. Then, we develop an extension of the classical Lagrange–d’Alembert principle called the Lagrange–d’Alembert–Pontryagin principle for implicit Lagrangian systems with constraints and external forces. A particularly interesting case is that of nonholonomic mechanical systems that can have both constraints and external forces. In addition, we define a constrained Dirac structure on the constraint momentum space, namely the image of the Legendre transformation (which, in the degenerate case, need not equal the whole cotangent bundle). We construct an implicit constrained Lagrangian system associated with this constrained Dirac structure by making use of an Ehresmann connection. Two examples, namely a vertical rolling disk on a plane and an L–C circuit are given to illustrate the results.

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Additional Information:© 2006 Elsevier Ltd. Received 6 October 2005; revised 17 February 2006; accepted 26 February 2006. Available online 17 April 2006. This paper was written during a visit of this author during 2002-2003 in the Department of Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA. Research partially supported by JSPS Grant 16560216. Research partially supported by NSF-ITR Grant ACI-0204932.
Funding AgencyGrant Number
Japan Society for the Promotion of Science (JSPS)16560216
NSF - Information Technology Research (ITR)ACI-0204932
Subject Keywords:The Hamilton–Pontryagin principle; Degenerate Lagrangians; Constrained Dirac structures; L–C circuits
Issue or Number:1
Record Number:CaltechAUTHORS:20100917-130658487
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20016
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 20:38
Last Modified:08 Nov 2021 23:56

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