Communications in Information & Systems

Nonholonomic Systems and Sub-Riemannian Geometry

Ovidiu Calin, Der-Chen Chang, and Stephen S. T. Yau

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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.

Article information

Commun. Inf. Syst., Volume 10, Number 4 (2010), 293-316.

First available in Project Euclid: 24 November 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 93B05: Controllability
Secondary: 49N99: None of the above, but in this section

sub-Riemannian distribution nonholonomic constraint Euler-Lagrange equations


Calin, Ovidiu; Chang, Der-Chen; Yau, Stephen S. T. Nonholonomic Systems and Sub-Riemannian Geometry. Commun. Inf. Syst. 10 (2010), no. 4, 293--316.

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