Discrete Dirac Structures and Variational Discrete Dirac Mechanics

Dr. Melvin Leok, Department of Mathematics, Purdue University

We construct discrete Dirac structures by considering the geometry of symplectic maps and their associated generating functions, in a manner analogous to the construction of continuous Dirac structures in terms of the geometry of symplectic vector fields and their associated Hamiltonians. We demonstrate that this framework provides a means of deriving implicit discrete Lagrangian and Hamiltonian systems, while incorporating discrete Dirac constraints. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators.

We also introduce a discrete Hamilton--Pontryagin variational principle on the discrete Pontryagin bundle, which provides an alternative derivation of the same set of integration algorithms. In so doing, we explicitly characterize the discrete Dirac structures that are preserved by Hamilton--Pontryagin integrators. In addition to providing a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of Dirac mechanics, it provides a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

The commutative diagrams which arise in the development of continuous Dirac structures are related to Tulczyjew's triples in classical mechanics, which have generalizations to Lie algebroids and classical field theories. This provides a natural framework for extending Dirac mechanics to Lie algebroids, groupoids, and multisymplectic field theories. The generalization to algebroids and groupoids would in turn provide an alternative approach to the theory of Dirac reduction by stages.