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arXiv · 0906.3357

Nonholonomic Hamilton-Jacobi equation and Integrability

Tomoki Ohsawa·Anthony M. Bloch
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Abstract

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 Leon, and Martin de Diego 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.

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Tomoki Ohsawa, Anthony M. Bloch. 2009-06-18. Nonholonomic Hamilton-Jacobi equation and Integrability. https://doi.org/10.3934/jgm.2009.1.461

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