Routh Reduction by Stages

TL;DR

This paper extends Routh reduction to a staged Lagrangian framework, introducing magnetic Lagrangian systems and demonstrating their closure under reduction, thus preserving the Lagrangian structure of dynamics.

Contribution

It develops a staged Routh reduction method for Lagrangian systems, introducing magnetic Lagrangian systems and relating sequential reductions to full symmetry reduction.

Findings

Magnetic Lagrangian systems are closed under Routh reduction.

A transformation relates sequential Routh reductions to full symmetry reduction.

The approach preserves the Lagrangian nature of the dynamics.

Abstract

This paper deals with the Lagrangian analogue of symplectic or point reduction by stages. We develop Routh reduction as a reduction technique that preserves the Lagrangian nature of the dynamics. To do so we heavily rely on the relation between Routh reduction and cotangent symplectic reduction. The main results in this paper are: (i) we develop a class of so called magnetic Lagrangian systems and this class has the property that it is closed under Routh reduction; (ii) we construct a transformation relating the magnetic Lagrangian system obtained after two subsequent Routh reductions and the magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full symmetry group.