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December  2009, 1(4): 461-481. doi: 10.3934/jgm.2009.1.461

Nonholonomic Hamilton-Jacobi equation and integrability

Tomoki Ohsawa 1, and  Anthony M. Bloch 2,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043, United States
2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States

Received  June 2009 Revised  January 2010 Published  January 2010

We discuss an extension of the Hamilton-Jacobi theory to nonholonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholonomic analogue of the Hamilton-Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton-Jacobi theory for unconstrained systems. The proof also helps us identify a geometric meaning of the conditions on the solutions of the Hamilton-Jacobi equation that arise from nonholonomic constraints. The major advantage of our result is that it provides us with a method of integrating the equations of motion just as the unconstrained Hamilton-Jacobi theory does. In particular, we build on the work by Iglesias-Ponte, de Léon, and Martín de Diego [15] so that the conventional method of separation of variables applies to some nonholonomic mechanical systems. We also show a way to apply our result to systems to which separation of variables does not apply.
Keywords: Hamilton-Jacobi equation, Nonholonomic mechanics, Integrable systems..
Mathematics Subject Classification: Primary: 70F25, 70H06, 70H20; Secondary: 37N0.
Citation: Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
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