Reduction of Dirac structures and the Hamilton-Pontryagin principle

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. s 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 g * with its Lie-Poisson structure. However, the basic step of reducing Hamiltonʹs phase space principle already shows that it is important to use g ⊕ g * for the reduced space, rather than just g * . 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 μ ∈ g * , produces a Dirac structure on g ⊕ g * . This geometry then simultaneously supports both Euler-Poincaré and Lie-Poisson reduction. 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.