Routh reduction for singular Lagrangians

TL;DR

This paper develops a generalized Routh reduction method for singular Lagrangian systems with symmetry, avoiding regularity conditions and preserving the Euler-Lagrange structure, with a presymplectic framework and detailed equations.

Contribution

It introduces a regularity-agnostic Routh reduction procedure that maintains the Euler-Lagrange form and provides a presymplectic framework for singular Lagrangian systems.

Findings

Routh reduction applicable without regularity conditions

Presymplectic framework for reduced systems

Detailed interpretation of reduced Euler-Lagrange equations

Abstract

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian $L$ or the momentum map $J_L$ are required apart from the momentum being a regular value of $J_L$. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler-Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler-Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.