Poisson Geometry Of The Dirac Reduction

In Poisson geometry, one of the most important results is the reduction of mechanical systems with symmetry by reducing Poisson structures and symplec-tic structures through the use of the momentum mappings. There are restrictions on what type of systems these techniques can be applied to, e.g., these reduction techniques do not apply to systems that are nonholonomically constrained. Dirac structure were introduced as an attempted to deal with these systems and other systems of differential and algebraic equations. Moreover, the discussion of Dirac structures is more clear and simple after the introducing of the notions of characteristic pairs and dual characteristic pairs. Since a proposed application of Dirac structures is to the reduction of some mechanical systems, the reduction of Dirac structures is considered in detail in this paper, without using the existence of momentum mappings or the introducing of the notion of admissible function, etc..We introduce the notion of characteristic pairs and give the definition of dual characteristic pairs of Dirac structures on Lie bialgebroids. Using the dual characteristic pairs, we give the if and only if conditions for which a maximally isotropic subbundle of the double of a Lie bialgebroid is a Dirac structure. By the characteristic pairs and dual characteristic pairs of Dirac structures, we classify reducible Dirac structures into two classes, i.e., reducible Dirac structures of class I and class II, on the basis of which, we give two kinds of reduction relating Poisson manifolds respectively, without using the existence of momentum mappings. As a result, we give the relationship between the reduced Poisson structures or presymplectic structures with the corresponding structures on the original manifolds. Meanwhile we present some examples and applications.For Jacobi bialgebroids, we introduce the notions of Jacobi-Dirac structures and its characteristic pairs and dual characteristic pairs in the similar ways. Using the dual characteristic pairs, we give the if and only if conditions for which a maximally isotropic subbundle of the double of a Jacobi bialgebroid is a Jacobi-Dirac structure. Also by the characteristic pairs and dual characteristic pairs of Jacobi-Dirac structures, we classify reducible Jacobi-Dirac into two classes, i.e., reducible...