Nonholonomic Systems and Sub-Riemannian Geometry

Abstract

This paper presents several classical mechanical systems with nonholonomic con- straints from the point of view of sub-Riemannian geometry. For those systems that satisfy the bracket generating condition the system can move continuously between any two given states. How- ever, the paper provides a counterexample to show that the bracket generating condition is not also a sufficient condition for connectivity. All possible motions of the system correspond to curves tan- gent to the distribution defined by the nonholonomic constraints. Among the connecting curves we distinguish an optimal one which minimizes a certain energy induced by a natural sub-Riemannian metric on the non-integrable distribution. The paper discusses several classical problems such as the knife edge, the skater, the rolling disk and the nonholonomic bicycle.