  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.
