Discretization of Dirac systems and port-Hamiltonian systems: the role of the constraint algorithm

We study the discretization of (almost-)Dirac structures using the notion of retraction and discretization maps on manifolds. Additionally, we apply the proposed discretization techniques to obtain numerical integrators for port-Hamiltonian systems and we discuss how to merge the discretization procedure and the constraint algorithm associated to systems of implicit differential equations. After fixing a time step, discretization maps are used to discretize the configuration manifold and we obtain different geometric integrators for the given implicit differential equations. As a result we propose a new method that exactly preserves the integrable part of those equations.