Variational integrators in discrete vakonomic mechanics

We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of "admissible section" and "admissible infinitesimal variation" of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections that satisfy a certain regularity condition, we prove that critical sections are extremals of a variational problem without constraints canonically associated to the initial system (Lagrange multiplier rule). We introduce a notion of "constrained variational integrator", which is characterized by a Cartan equation that ensures its simplecticity. Moreover, under certain regularity conditions we prove that these integrators can be locally constructed from a generating function of the second kind in the sense of symplectic geometry. Finally, the theory is illustrated with two elementary examples: an isoperimetric problem and an optimal control problem.