Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization

TL;DR

This paper extends Hamilton-Jacobi theory to certain nonholonomic systems by using Hamiltonization, transforming them into Hamiltonian systems, and establishing conditions under which solutions correspond between the two formulations.

Contribution

It provides a geometric framework for Hamiltonization of Chaplygin systems and links solutions of Hamilton-Jacobi equations before and after Hamiltonization.

Findings

Identifies necessary and sufficient conditions for Hamiltonization.

Shows solutions of Hamiltonized systems solve original nonholonomic equations.

Illustrates theory with several concrete examples.

Abstract

We develop Hamilton-Jacobi theory for Chaplygin systems, a certain class of nonholonomic mechanical systems with symmetries, using a technique called Hamiltonization, which transforms nonholonomic systems into Hamiltonian systems. We give a geometric account of the Hamiltonization, identify necessary and sufficient conditions for Hamiltonization, and apply the conventional Hamilton-Jacobi theory to the Hamiltonized systems. We show, under a certain sufficient condition for Hamiltonization, that the solutions to the Hamilton-Jacobi equation associated with the Hamiltonized system also solve the nonholonomic Hamilton-Jacobi equation associated with the original Chaplygin system. The results are illustrated through several examples.