Discrete Dirac Structures and Implicit Discrete Lagrangian and Hamiltonian Systems

We present discrete analogues of Dirac structures and the Tulczyjew’s triple by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete analogues of implicit Lagrangian and Hamiltonian systems. In particular, this yields implicit nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. 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.