Lagrangian Mechanics and Reduction on Fibered Manifolds

TL;DR

This paper generalizes Lagrangian mechanics to fibered manifolds, introduces a reduction theory for symmetries via Lie groupoid actions, and extends classical reduction methods within this new framework.

Contribution

It develops a unified fibered manifold framework for Lagrangian mechanics and reduction, including new results for Lie algebroids and Hamilton-Pontryagin principles.

Findings

Unified formulation of Lagrangian mechanics on fibered manifolds

New coordinate-free equations of motion for Lie algebroids

Extension of reduction theories to fibered and Lie algebroid settings

Abstract

This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the Hamilton-Pontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.