Nonholonomic Hamilton-Jacobi equation and Integrability

TL;DR

This paper extends Hamilton-Jacobi theory to nonholonomic systems, providing an intrinsic proof and a method for integrating equations of motion, including cases where separation of variables is not applicable.

Contribution

It offers a new geometric approach to nonholonomic Hamilton-Jacobi theory, enabling integration of equations of motion similar to unconstrained systems, with broader applicability.

Findings

Intrinsic proof of nonholonomic Hamilton-Jacobi theorem

Method for integrating nonholonomic equations of motion

Application of separation of variables to certain nonholonomic systems

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.