Nonholonomic Hamilton–jacobi Equation and Integrability

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 IglesiasPonte, de Léon, and Mart́ın 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.