Numerical Integrators for Mechanical Systems on Lie Groups

TL;DR

This work develops a unified framework for numerical integration of mechanical systems on Lie groups using retraction and discretization maps together with trivializations of tangent, cotangent, and higher-order bundles. It provides trivialized Tulczyjew triple maps and explicit discretization schemes for Euler–Poincaré, Lie–Poisson, and Euler–Arnold dynamics. An illustrative Lie–Poisson example demonstrates the concrete steps and update rules. The framework enables efficient, geometry-preserving simulations for rigid-body-like systems and other mechanics on Lie groups.

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.