Reduction of Dirac Structures and the Hamilton-Pontryagin Principle

Yoshimura H., and J. E. Marsden

Reports on Mathematical Physics, 60, (2007), 381-426

Abstract:

This paper develops a reduction theory for Dirac structures that includes in a unified way, reduction of both Lagrangian and Hamiltonian systems. It includes the reduction of variational principles and in particular, the Hamilton-Pontryagin variational principle. It also includes reduction theory for implicit Lagrangian systems that could be degenerate and have constraints.

In this paper we focus on the special case in which the configuration manifold is a Lie group G. In our earlier papers we established the link between the Hamilton-Pontryagin principle and Dirac structures. We begin the paper with the reduction of this principle. The traditional view of Poisson reduction in this case is to reduce T ^* G with its natural Poisson structure to $\mathfrak{g}^\ast$ with its Lie-Poisson structure. However, the basic step of reducing Hamilton's phase space principle already shows that it is important to use $\mathfrak{g} \oplus \mathfrak{g}^\ast$ for the reduced space, rather than just $\mathfrak{g}^\ast$. In this way, our construction includes both Euler-Poincaré as well as Lie-Poisson reduction. The geometry behind this procedure, which we call Lie-Dirac reduction starts with the standard (i.e., canonical) Dirac structure on T ^* G (which can be viewed either symplectically or from the Poisson viewpoint) and for each $\mu$ $\in$ $\mathfrak{g}^\ast$, produces a Dirac structure on $\mathfrak{g} \oplus \mathfrak{g}^\ast$. This geometry then simultaneously supports both Euler-Poincaré and Lie-Poisson reduction.

In the last part of the paper, we include nonholonomic constraints, and illustrate this construction with Suslov systems in nonholonomic mechanics, both from the Euler-Poincaré and Lie-Poisson viewpoints.