Numerical Simulation of Nonholonomic Dynamics

Abstract

We study the numerical integration of nonholonomic problems. The problems are formulated using Lagrangian and Hamiltonian mechanics. We review briefly the theoretical concepts used in geometric mechanics. We reconstruct two nonholonomic variational integrators from the monograph of Monforte. We also construct two one-step integrators based on a combination of the continuous Legendre transform and the discrete Legendre transform from an article by Marsden and West. Inintially these integrators display promising behavior, but they turn out to be unstable. The variational integrators are compared with a classical Runge-Kutta method. We compare the methods on three nonholonomic systems: The nonholonomic particle from the monograph of Monforte, the nonholonomic system of particles from an article by McLachlan and Perlmutter, and a variation of the Chaplygin sleigh from Bloch.