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.Communicated by

Stochastic disks that roll

“…Collectively, rolling particles have different phase behaviours than those that slide [27]. Yet despite their intriguing dynamics, rolling has been considered in stochastic settings only for simple systems such as a rolling ball or sled [28][29][30], or as a noisy relaxation of the rolling 2 constraint itself [31]. This paper studies a natural model of stochastic, rolling particles, with the aim of determining how rolling could affect quantities that are macroscopically measurable.…”

Geometric Analysis of Noisy Perturbations to Nonholonomic Constraints

“…On the other hand, the energy is in general not conserved in the case of affine, or inhomogeneous, noisy constraints. Furthermore, in [4], we have proved that the expectation value for the energy can either increase indefinitely, or remain finite, depending on the type of noise used in the equations. These results are derived using analytical studies stemming from the cases of reduced dynamics, such as rolling of a ball along a one-dimensional line.…”

Dynamics of non-holonomic systems with stochastic transport

“…may also be interested in combining stochastic extensions of non-holonomic rolling conditions as in [6] with the ST noise. Making this combination represents another interesting and challenging problem which should be treatable by our methods.…”

“…A different case of stochasticity arising in non-holonomic systems was considered by Gay-Balmaz & Putkaradze [6], where no random forces were acting on the system itself, but the constraint was stochastic. A physical realization of such system would be the motion of a deterministic rolling ball on a rough plane, or experiencing a random slippage at the contact point.…”

“…This result contrasts with [5], where no integrals of motion were found to be preserved for general stochastic forces. The preservation of integrals of motion described in [6] was due to the nature of the forces introduced by the stochastic constraints.…”