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We prove a nonholonomic version of the classical Mauper\-tuis-Jacobiprinciple which transforms an autonomous mechanical nonholonomic problem,determined by a kinetic minus potential energy and a distribution, in a kineticnonholonomic problem over a fixed level set of the Lagrangian energy. To provethis result we introduce an appropriate contact bundle structure clarifying thegeometric equivalence between both problems.  By using the nonholonomic Maupertuis-Jacobi principle, we prove that theregular solutions of a mechanical nonholonomic problem starting from a fixedpoint and in the same level set of the Lagrangian energy are reparametrizationsof geodesics for a family of Riemannian metrics defined on the image of thenonholonomic exponential map. In particular, these trajectories minimizeRiemannian length.

Contact bundle formulation of nonholonomic Maupertuis-Jacobi principle and a length minimizing property of nonholonomic dynamics