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The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems

Koon, Wang Sang and Marsden, Jerrold E. (1997) The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems. Reports on Mathematical Physics, 40 (1). pp. 21-62. ISSN 0034-4877. doi:10.1016/S0034-4877(97)85617-0.

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This paper compares the Hamiltonian approach to systems with nonholonomic constraints (see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke [1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996] and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the approaches and each has its own advantages; some structures have been discovered on one side and their analogues on the other side are interesting to clarify. For example, the momentum equation and the reconstruction equation were first found on the Lagrangian side and are useful for the control theory of these systems, while the failure of the reduced two form to be closed (i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the Hamiltonian side. Clarifying the relation between these approaches is important for the future development of the control theory and stability and bifurcation theory for such systems. In addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum phase) control system.

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Additional Information:© 1997 Published by Elsevier Science Ltd. Received 26 November 1996; revised 24 April 1997. Available online 11 June 1999. April 28, 1997. Research partially supported by the DOE contract DE-FG0395-ER25251
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Department of Energy (DOE)DE-FG0395-ER25251
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Record Number:CaltechAUTHORS:20100820-092142844
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19546
Deposited By: Ruth Sustaita
Deposited On:20 Aug 2010 20:13
Last Modified:08 Nov 2021 23:53

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