Jacobi metrics, contact bundle formulation of mechanical nonholonomic systems and a length minimizing property of nonholonomic dynamics

The trajectories of a mechanical nonholonomic system with fixed energy are the trajectories of a kinetic nonholonomic system with energy 1. The constraint distribution of both systems is just the same and the kinetic energy of the second system is associated with the corresponding Jacobi metric Bakša [1975] and Koiller [1992]. In this paper, we prove this result using an appropriate contact bundle structure clarifying the geometric equivalence between both problems. Then, as a consequence, we prove that the regular solutions of a mechanical nonholonomic problem starting from a fixed point and in the same level set of the Lagrangian energy are reparametrizations of geodesics for a family of Riemannian metrics defined on the image of the nonholonomic exponential map. In particular, these trajectories minimize Riemannian length.