Momentum and energy preserving integrators for nonholonomic dynamics

TL;DR

This paper introduces a geometric integrator for nonholonomic mechanical systems that preserves momentum, constraints, and in some cases energy, without requiring discretization of the constraints.

Contribution

It presents a novel integrator that maintains key geometric properties of nonholonomic systems, including momentum and energy preservation, without discretizing the constraints.

Findings

Preserves discrete nonholonomic momentum map

Maintains nonholonomic constraints in average

Energy-preserving in certain cases

Abstract

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and a (generally nonintegrable) distribution in TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study in particular the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.