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Poisson reduction for nonholonomic mechanical systems with symmetry

Koon, Wang Sang and Marsden, Jerrold E. (1998) Poisson reduction for nonholonomic mechanical systems with symmetry. Reports on Mathematical Physics, 42 . pp. 101-134. ISSN 0034-4877. http://resolver.caltech.edu/CaltechAUTHORS:20100820-093224864

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Abstract

This paper continues the work of Koon and Marsden [1997b] that began the comparison of the Hamiltonian and Lagrangian formulations of nonholonomic systems. Because of the necessary replacement of conservation laws with the momentum equation, it is natural to let the value of momentum be a variable and for this reason it is natural to take a Poisson viewpoint. Some of this theory has been started in van der Schaft and Maschke [1994]. We build on their work, further develop the theory of nonholonomic Poisson reduction, and tie this theory to other work in the area. We use this reduction procedure to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic momentum equation and the reduced Lagrange d’Alembert equations in Hamiltonian form. We also show that these equations are equivalent to those given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden and Murray [1996]. Because of the results of Koon and Marsden [1997b], this is also equivalent to the results of Bates and Sniatycki [1993], obtained by nonholonomic symplectic reduction. Two interesting complications make this effort especially interesting. First of all, as we have mentioned, symmetry need not lead to conservation laws but rather to a momentum equation. Second, the natural Poisson bracket fails to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum that vanishes when the Jacobi identity holds), or equivalently, the Schouten bracket, is an interesting expression involving the curvature of the underlying distribution describing the nonholonomic constraints. The Poisson reduction results in this paper are important for the future development of the stability theory for nonholonomic mechanical systems with symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they should be useful for the development of the powerful block diagonalization properties of the energy-momentum method developed by Simo, Lewis and Marsden [1991].


Item Type:Article
Additional Information:© 1998, The papers are being reviewed individually. Research partially supported by the DOE contract DE-FG0395-ER25251.
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Department of EnergyDE-FG0395-ER25251
Record Number:CaltechAUTHORS:20100820-093224864
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100820-093224864
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19548
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:20 Aug 2010 20:18
Last Modified:26 Dec 2012 12:20

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