Communications in Information and Systems

Volume 10 (2010)

Number 4

Nonholonomic Systems and Sub-Riemannian Geometry

Pages: 293 – 316



Ovidiu Calin

Der-Chen Chang

Stephen S. T. Yau


This paper presents several classical mechanical systems with nonholonomic constraints 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. However, 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 tangent 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.


sub-Riemannian distribution; nonholonomic constraint; Euler-Lagrange equations

2010 Mathematics Subject Classification

Primary 93B05. Secondary 49N99.

Full Text (PDF format)