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

Abstract

<p style='text-indent:20px;'>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 construct a family of maps, labeled by an arbitrary <inline-formula><tex-math id="M1">\begin{document}$ \ell \in \mathbb{N} $\end{document}</tex-math></inline-formula> indicating the desired order of accuracy, and prove that our method is structure preserving in the sense that the discrete flow is interpolated to order <inline-formula><tex-math id="M2">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> by the flow of a continuous system possessing the same structure as the vector field that is being discretized. In particular, our discretization preserves a smooth measure on the phase space to the arbitrary order <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>. We present applications to a remarkable class of nonholonomic mechanical systems that allow Hamiltonization. To our best knowledge, these results provide the first instance of a measure preserving discretization (to arbitrary order) of measure preserving nonholonomic systems.</p>