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

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... 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.