Symmetry and Reduction in Implicit Generalized Hamiltonian Systems

In this paper we study the notion of symmetry for implicit generalized Hamiltonian systems, which are Hamiltonian systems with respect to a generalized Dirac structure. We investigate the reduction of these systems admitting a symmetry Lie group with corresponding conserved quantities. Main features in this approach concern the projection and restriction of Dirac structures, generalizing the corresponding theory for symplectic forms and Poisson brackets. The results are applied to the theory of symmetries and reduction in nonholonomically constrained mechanical systems. The main result extends the reduction theory for explicit Hamiltonian systems and constrained mechanical systems to a general unified reduction theory for implicit generalized Hamiltonian systems.