Routhian reduction for quasi-invariant Lagrangians

TL;DR

This paper generalizes Routhian reduction to quasi-invariant Lagrangians using symplectic reduction, providing a broader framework applicable to systems like a charged particle in a magnetic field.

Contribution

It introduces a new Routhian reduction method for quasi-invariant Lagrangians based on symplectic reduction principles.

Findings

Generalization of Routhian reduction for quasi-invariant Lagrangians

Connection between Routhian reduction and symplectic reduction

Application to charged particle dynamics in magnetic fields

Abstract

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We show how functional Routhian reduction can be seen as a particular instance of reduction of a quasi-invariant Lagrangian, and we exhibit a Routhian reduction procedure for the special case of Lagrangians with quasi-cyclic coordinates. As an application we consider the dynamics of a charged particle in a magnetic field.