Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization

This document is a brief overview of the Hamilton-Jacobi theory of Chaplygin systems based on [1]. In this paper, after reducing Chaplygin systems, Ohsawa et al. use a technique that they call Chaplygin Hamiltonization to turn the reduced Chaplygin systems into Hamiltonian systems. This method was first introduced in a paper by Chaplygin in 1911 where he reduced some nonholonomic systems by the action of Rk, for some k, and turned the corresponding dynamical equations into the Hamilton’s equations. In [1] Ohsawa et al. take another step forward and formulate the conventional Hamilton-Jacobi equation, which they name Chaplygin Hamilton-Jacobi equation, in order to integrate Chaplygin systems. They also establish the link between this approach and the direct approach of extending Hamilton-Jacobi equation to the nonholonomic systems [2], which is called nonholonomic Hamilton-Jacobi equation. Consider a conserved nonholonomic system with the constant energy E, the configuration manifold Q equipped with a non-involutive distribution D ⊂ TQ (defined by the nonholonomic constraints), and a Hamiltonian H : T ∗Q → R, the corresponding nonholonomic Hamilton-Jacobi equation can be written as