Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems

Hiroaki Yoshimura; Jerrold E. Marsden

doi:10.1007/978-3-540-73890-9_18

Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems

Hiroaki Yoshimura^*, Jerrold E. Marsden

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

11 Citations (Scopus)

Abstract

This paper begins by recalling how a constraint distribution on a configuration manifold induces a Dirac structure together with an implicit Lagrangian system, a construction that is valid even for degenerate Lagrangians. In such degenerate cases, it is shown in this paper that an implicit Hamiltonian system can be constructed by using a generalized Legendre transformation, where the primary constraints are incorporated into a generalized Hamiltonian on the Pontryagin bundle. Some examples of degenerate Lagrangians for L-C circuits, nonholonomic systems, and point vortices illustrate the theory.

Original language	English
Title of host publication	Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 - Proceedings from the 3rd IFAC Workshop
Publisher	Springer Verlag
Pages	233-247
Number of pages	15
ISBN (Print)	3540738894, 9783540738893
DOIs	https://doi.org/10.1007/978-3-540-73890-9_18
Publication status	Published - 2007
Event	3rd IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control, LHMNLC 2006 - Nagoya, Japan Duration: 2006 Jul 19 → 2006 Jul 21

Publication series

Name	Lecture Notes in Control and Information Sciences
Volume	366 LNCIS
ISSN (Print)	0170-8643

Conference

Conference	3rd IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control, LHMNLC 2006
Country/Territory	Japan
City	Nagoya
Period	06/7/19 → 06/7/21

ASJC Scopus subject areas

Library and Information Sciences

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10.1007/978-3-540-73890-9_18

Cite this

Yoshimura, H., & Marsden, J. E. (2007). Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems. In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 - Proceedings from the 3rd IFAC Workshop (pp. 233-247). (Lecture Notes in Control and Information Sciences; Vol. 366 LNCIS). Springer Verlag. https://doi.org/10.1007/978-3-540-73890-9_18

Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems. / Yoshimura, Hiroaki; Marsden, Jerrold E.
Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 - Proceedings from the 3rd IFAC Workshop. Springer Verlag, 2007. p. 233-247 (Lecture Notes in Control and Information Sciences; Vol. 366 LNCIS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Yoshimura, H & Marsden, JE 2007, Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems. in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 - Proceedings from the 3rd IFAC Workshop. Lecture Notes in Control and Information Sciences, vol. 366 LNCIS, Springer Verlag, pp. 233-247, 3rd IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control, LHMNLC 2006, Nagoya, Japan, 06/7/19. https://doi.org/10.1007/978-3-540-73890-9_18

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AB - This paper begins by recalling how a constraint distribution on a configuration manifold induces a Dirac structure together with an implicit Lagrangian system, a construction that is valid even for degenerate Lagrangians. In such degenerate cases, it is shown in this paper that an implicit Hamiltonian system can be constructed by using a generalized Legendre transformation, where the primary constraints are incorporated into a generalized Hamiltonian on the Pontryagin bundle. Some examples of degenerate Lagrangians for L-C circuits, nonholonomic systems, and point vortices illustrate the theory.

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Dirac structures and the legendre transformation for implicit lagrangian and hamiltonian systems

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