Dirac cotangent bundle reduction (original) (raw)

2009, The Journal of Geometric Mechanics

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie-Dirac reduction. This procedure simultaneously includes Lagrangian, Hamiltonian, and a variational view of reduction. The goal of the present paper is to generalize Lie-Dirac reduction to the case of a general configuration manifold; we refer to this as Dirac cotangent bundle reduction. This reduction procedure encompasses, in particular, a reduction theory for Hamiltonian as well as implicit Lagrangian systems, including the case of degenerate Lagrangians. First of all, we establish a reduction theory starting with the Hamilton-Pontryagin variational principle, which enables one to formulate an implicit analogue of the Lagrange-Poincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure. Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called Lagrange-Poincaré-Dirac reduction. This procedure naturally induces the horizontal and vertical implicit Lagrange-Poincaré equations, which are consistent with those derived from the reduced Hamilton-Pontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, Hamilton-Poincaré-Dirac reduction for the horizontal and vertical Hamilton-Poincaré equations. We illustrate the reduction procedures by an example of a satellite with a rotor. The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages. This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.