### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 019, 26 pages      arXiv:1511.00061      https://doi.org/10.3842/SIGMA.2017.019
Contribution to the Special Issue “Gone Fishing”

### Lagrangian Mechanics and Reduction on Fibered Manifolds

Songhao Li, Ari Stern and Xiang Tang
Department of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis MO 63130-4899, USA

Received October 05, 2016, in final form March 13, 2017; Published online March 22, 2017

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.

Key words: Lagrangian mechanics; reduction; fibered manifolds; Lie algebroids; Lie groupoids.

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