Numerical Integrators for Mechanical Systems on Lie Groups

TL;DR

This paper develops structure-preserving numerical integrators for mechanical systems on Lie groups, leveraging retraction maps and the trivializability of tangent and cotangent bundles to simplify computations.

Contribution

It introduces a framework for designing integrators specifically for Euler-Poincare and Lie-Poisson equations on Lie groups, enhancing geometric integration methods.

Findings

Simplified integrator construction for Lie group systems

Framework applicable to Euler-Poincare equations

Preserves geometric structure in numerical simulations

Abstract

Retraction maps are known to be the seed for all numerical integrators. These retraction maps-based integrators can be further lifted to tangent and cotangent bundles, giving rise to structure-preserving integrators for mechanical systems. We explore the particular case where the configuration space of our mechanical system is a Lie group with certain symmetries. Here, the integrator simplifies based on the property that the tangent and cotangent bundles of Lie groups are trivializable. Finally, we present a framework for designing numerical integrators for Euler- Poincare and Lie-Poisson type equations.