Lagrangian mechanics on Lie groups: a pedagogical approach

TL;DR

This paper presents a pedagogical approach to formulating classical Lagrangian mechanics on Lie groups, exemplified by rigid body rotation, and discusses generalizations and Hamiltonian aspects.

Contribution

It introduces a straightforward, educational method for Lagrangian mechanics on Lie groups, emphasizing generator matrices and direct derivation of Euler's equations.

Findings

Derivation of Euler's equations from least action on SO(3)

Method generalizes to other Lie groups

Brief discussion of Hamiltonian formulation

Abstract

We describe a new method to formulate classical Lagrangian mechanics on a finite-dimensional Lie group. This new approach is much more pedagogical than many previous treatments of the subject, and it directly introduces students to generator matrices and their usefulness in many manipulations. The example of rigid body rotation, i.e. motion on the Lie group SO(3), is used as an example, and it is shown how to derive Euler's equations directly from the principle of least action. The techniques covered in this paper generalize to other Lie groups in a straightforward manner, which is discussed. We briefly discuss the Hamiltonian formulation of the problem as well.