Dynamics of non-holonomic systems with stochastic transport

TL;DR

This paper develops a variational framework for modeling stochastic transport in nonholonomic dynamical systems, demonstrated through a stochastic rolling ball example, preserving key mathematical properties.

Contribution

It introduces a stochastic variational approach using Hamilton-Pontryagin principles for nonholonomic systems, extending deterministic methods to include uncertainty.

Findings

The stochastic approach preserves conserved quantities analytically.

Numerical simulations show the impact of stochasticity on system behavior.

The method applies broadly to stochastic, nonholonomically constrained systems.

Abstract

This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under nonholonomic constraints. For this purpose, we derive, analyze and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton-Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange-d'Alembert principle. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, nonholonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle. Keywords: Nonholonomic constraints, Stochastic dynamics, Transport noise.