Routh's procedure for non-Abelian symmetry groups

TL;DR

This paper generalizes Routh's reduction method to systems with non-Abelian symmetry groups, introducing a new analysis technique using quasi-velocities to handle more complex symmetries.

Contribution

It extends Routh's procedure to non-Abelian symmetry groups and develops a novel quasi-velocity based analysis method for such systems.

Findings

Successfully generalized Routh's reduction for non-Abelian groups

Introduced a new quasi-velocity based analytical framework

Provided a method for reconstructing full solutions from reduced equations

Abstract

We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum in velocity phase space. We present a new method of analysis based on the use of quasi-velocities. We discuss the reconstruction of solutions of the full Euler-Lagrange equations from those of the reduced equations.