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Dirac Structures in Lagrangian Mechanics. Part I: Implicit Lagrangian Systems

Yoshimura, Hiroaki and Marsden, Jerrold E. (2006) Dirac Structures in Lagrangian Mechanics. Part I: Implicit Lagrangian Systems. Journal of Geometry and Physics, 57 (1). pp. 133-156. ISSN 0393-0440. doi:10.1016/j.geomphys.2006.02.009.

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This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T^*Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T^*TQ, TT^*Q, and T^*T^*Q. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.

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Additional Information:© 2006 Elsevier Ltd. Received 6 October 2005; revised 21 January 2006; accepted 19 February 2006. Available online 29 March 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. We are very grateful to Tudor Ratiu, Alan Weinstein and Tom Hughes, who kindly provided several very useful remarks. We also thank Jeff Lawson, Shawn Shadden, and Troy Smith for pointing out several improvements.
Funding AgencyGrant Number
Japan Society for the Promotion of Science (JSPS)16560216
NSF- Information Technology Research (ITR)ACI-0204932
Subject Keywords:Dirac structures; Implicit Lagrangian systems; Nonholonomic systems
Issue or Number:1
Record Number:CaltechAUTHORS:20100917-131813412
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20017
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 20:57
Last Modified:08 Nov 2021 23:56

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