Routh reduction and the class of magnetic Lagrangian systems

TL;DR

This paper introduces a new transformation concept for magnetic Lagrangian systems obtained via Routh reduction, showing equivalence of different reduction approaches under certain conditions, advancing the understanding of symmetry reduction in Lagrangian mechanics.

Contribution

It presents a novel transformation applicable to magnetic Lagrangian systems after Routh reduction, linking reductions by different group structures.

Findings

Reduction with respect to a semi-direct product group is equivalent to reduction with respect to an Abelian normal subgroup under certain conditions.

The new transformation simplifies the analysis of magnetic Lagrangian systems post-reduction.

Results relate to the broader theory of Routh reduction by stages.

Abstract

In this paper, some new aspects related to Routh reduction of Lagrangian systems with symmetry are discussed. The main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after Routh reduction of Lagrangian systems with symmetry, so-called magnetic Lagrangian systems. We use these transformations in order to show that, under suitable conditions, the reduction with respect to a (full) semi-direct product group is equivalent to the reduction with respect to an Abelian normal subgroup. The results in this paper are closely related to the more general theory of Routh reduction by stages.