Lagrangian Mechanics and Variational Integrators on Two-Spheres

TL;DR

This paper develops global Euler-Lagrange equations and variational integrators for mechanical systems on two-spheres, avoiding local parameterization issues and preserving geometric properties.

Contribution

It introduces a geometric framework for deriving global equations and integrators on two-spheres, improving upon local parameterization methods.

Findings

Global equations are more compact than angle-based equations.

Variational integrators preserve symplecticity and momentum.

Computational experiments demonstrate effectiveness on mechanical systems.

Abstract

Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton's principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems.