On Noisy Extensions of Nonholonomic Constraints

Abstract

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange–d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with “rolling ball type” constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.