On the geometry of nonholonomic mechanical systems with vertical distribution

Abstract

Let (Q,g) be the configuration space of a nonholonomic mechanical system, where g is a Riemannian metric on Q. Suppose the horizontal distribution D on Q admits a vertical distribution D¯, that is D¯ is an integrable complementary (not necessarily orthogonal) distribution to D in TQ. We prove the existence and uniqueness of a linear connection on (Q,g) subject to some conditions on its torsion and on the covariant derivative of g. Then we show that the solutions of the Lagrange-d’Alembert equations are the geodesics of ∇ and vice versa. All the local components of the torsion and curvature tensor fields of ∇ with respect to an adapted frame field are determined. Finally, two examples are given to illustrate the theory we develop in the paper.