On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

Fernando Jiménez; Jürgen Scheurle

doi:10.3934/jgm.2015.7.43

Article Contents

2015, Volume 7, Issue 1: 43-80. Doi: 10.3934/jgm.2015.7.43

This issue Previous Article Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations Next Article Higher-order variational calculus on Lie algebroids

On the discretization of nonholonomic dynamics in $\mathbb{R}^n$

Fernando Jiménez^1, and
Jürgen Scheurle^1,

1.
Zentrum Mathematik der Technische Universität München, D-85747 Garching bei Munchen

Received: June 30, 2014

Revised: January 31, 2015

Published: February 2015

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Abstract

In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion to a set of ordinary differential equations on the underlying distribution manifold $D$. Moreover, we show that any $D-$preserving discretization may be understood as being generated by the exact evolution map of a time-periodic non-autonomous perturbation of the original continuous-time nonholonomic system. By means of discretizing the corresponding Lagrange-d'Alembert principle, we construct geometric integrators for the original nonholonomic system. We give precise conditions under which these integrators generate a discrete flow preserving the distribution $D$. Also, we derive corresponding consistency estimates. Finally, we carefully treat the example of the nonholonomic particle, showing how to discretize the equations of motion in a reasonable way, particularly regarding the nonholonomic constraints.

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Mathematics Subject Classification: 34C15, 37J15, 37N05, 65P10, 70F25.

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