Lagrangian Mechanics and Reduction on Fibered Manifolds
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, coordinatefree formulation of the equations of motion. Finally, we extend the foregoing to include fibered and Lie algebroid generalizations of the HamiltonPontryagin principle of Yoshimura and Marsden, along with the associated reduction theory.
 Publication:

SIGMA
 Pub Date:
 March 2017
 DOI:
 10.3842/SIGMA.2017.019
 arXiv:
 arXiv:1511.00061
 Bibcode:
 2017SIGMA..13..019L
 Keywords:

 Lagrangian mechanics;
 reduction;
 fibered manifolds;
 Lie algebroids;
 Lie groupoids;
 Mathematics  Dynamical Systems;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 SIGMA 13 (2017), 019, 26 pages