Geometric Integrators for Nonholonomic Systems on Lie Groups

TL;DR

This paper develops a framework for structure-preserving numerical integrators for nonholonomic systems on Lie groups, utilizing retraction maps to handle manifold constraints while respecting system symmetries.

Contribution

It introduces a general geometric integrator construction for nonholonomic Lie group systems using retraction maps, extending existing methods to constrained dynamics.

Findings

The integrator preserves nonholonomic constraints at each step.

Application to the Suslov problem demonstrates the method's effectiveness.

Framework generalizes to various Lie group-based nonholonomic systems.

Abstract

We present a general framework for constructing structure-preserving numerical integrators for nonholonomically constrained mechanical systems evolving on Lie groups using retraction maps. Retraction maps generalize the exponential map and provide a convenient tool for performing numerical integration on manifolds. In nonholonomic mechanics, the constraints restrict the dynamics to a nonintegrable distribution rather than the entire tangent bundle. Using the Hamel formulation, the equations of motion can be expressed in local coordinates adapted to this constraint distribution. We then specialize the framework to the case of Lie groups, where both the dynamics and the constraints exhibit symmetries, allowing a simplified formulation of the numerical scheme. The resulting integrator respects the constraint distribution and enforces the nonholonomic constraints at each discrete time step. The approach is illustrated using the Suslov problem.