Geometric analysis of noisy perturbations to nonholonomic constraints

TL;DR

This paper introduces stochastic extensions to nonholonomic constraints in mechanical systems, focusing on invariant systems and analyzing the preservation of integrals of motion under noise perturbations.

Contribution

It develops a stochastic Lagrange-d'Alembert framework for nonholonomic systems and explores stochastic deformations of the Suslov problem.

Findings

Stochastic extensions preserve some integrals of motion.

Analysis of stochastic perturbations on nonholonomic systems.

Application to systems on rotation and Euclidean groups.

Abstract

We propose two types of stochastic extensions of nonholonomic constraints for mechanical systems. Our approach relies on a stochastic extension of the Lagrange-d'Alembert framework. We consider in details the case of invariant nonholonomic systems on the group of rotations and on the special Euclidean group. Based on this, we then develop two types of stochastic deformations of the Suslov problem and study the possibility of extending to the stochastic case the preservation of some of its integrals of motion such as the Kharlamova or Clebsch-Tisserand integrals.