  We prove a nonholonomic version of the classical Mauper\-tuis-Jacobi
principle which transforms an autonomous mechanical nonholonomic problem,
determined by a kinetic minus potential energy and a distribution, in a kinetic
nonholonomic problem over a fixed level set of the Lagrangian energy. To prove
this result we introduce an appropriate contact bundle structure clarifying the
geometric equivalence between both problems.
  By using the nonholonomic Maupertuis-Jacobi principle, 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.
