Structure preserving discretization of time-reparametrized Hamiltonian systems with application to nonholonomic mechanics

We propose a discretization of vector fields that are Hamiltonian up to multiplication by a positive function on the phase space that may be interpreted as a time reparametrization. We prove that our method is structure preserving in the sense that the discrete flow is interpolated to arbitrary order by the flow of a continuous system possessing the same structure. In particular, our discretization preserves a smooth measure on the phase space to arbitrary order. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first occurrence in the literature of a measure preserving discretization of measure preserving nonholonomic systems.