Lagrangian mechanics on Lie groups: a pedagogical approach

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.