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

TL;DR

This paper extends the Maupertuis-Jacobi principle to nonholonomic systems, showing that solutions are length-minimizing geodesics in a Riemannian metric, with a geometric framework based on contact bundles.

Contribution

It introduces a nonholonomic Maupertuis-Jacobi principle and a contact bundle structure, establishing a geometric equivalence and length-minimizing properties of solutions.

Findings

Solutions are reparametrizations of geodesics in a Riemannian metric.

Trajectories minimize Riemannian length.

Provides a geometric framework for nonholonomic mechanics.

Abstract

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.