# American Institute of Mathematical Sciences

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

## Nonholonomic Hamilton-Jacobi equation and integrability

 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.
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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