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...

A geometric derivation of numerical integrators for nonholonomic systems and optimal control problems is obtained. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems and optimal control problems.

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian Ld : Q × Q → R, where Q is the configuration manifold, and a (generally nonintegrable) distribution D ⊂ TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the...

320 NOTICES OF THE AMS VOLUME 52, NUMBER 3 Introduction Nonholonomic systems are, roughly speaking, mechanical systems with constraints on their velocity that are not derivable from position constraints. They arise, for instance, in mechanical systems that have rolling contact (for example, the rolling of wheels without slipping) or certain kinds of sliding contact (such as the sliding of skate...

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 theor...

This paper is concerned with the stabilization of nonholonomic systems in portcontrolled Hamiltonian formulae based on time-varying generalized canonical transformations. A special class of time-varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity. Utilizing these...

We study a discrete analog of the Lagrange-d’Alembert principle of nonhonolomic mechanics and give conditions for it to define a map and to be reversible. In specific cases it can generate linearly implicit, semi-implicit, or implicit numerical integrators for nonholonomic systems which, in several examples, exhibit superior preservation of the dynamics. We also study discrete nonholonomic syst...