Dirac cotangent bundle reduction

Yoshimura, H. and J. E. Marsden

Geom. Mech., 1, (2009), 87-158

Abstract:

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.