Stochastic Chaplygin systems

TL;DR

This paper extends stochastic Hamiltonian reduction techniques to certain non-holonomic systems, demonstrating that the drift in a stochastically perturbed Chaplygin ball relates to the gradient of its preserved measure density.

Contribution

It introduces a stochastic reduction framework for non-holonomic systems with symmetries, specifically applying it to the Chaplygin ball.

Findings

The drift of the stochastic Chaplygin ball is a gradient of the preserved measure density.

The approach generalizes stochastic Hamiltonian reduction to non-holonomic systems.

The non-holonomic connection plays a key role in the analysis.

Abstract

We mimic the stochastic Hamiltonian reduction of Lazaro-Cami and Ortega [17, 18] for the case of certain non-holonomic systems with symmetries. Using the non-holonomic connection it is shown that the drift of the stochastically perturbed $n$-dimensional Chaplygin ball is a certain gradient of the density of the preserved measure of the deterministic system.