On the geometry of Lagrangian mechanics with non-holonomic constraints

Abstract

As is well known, Lagrangian mechanics have been entirely geometrized in terms of symplectic geometry. On the other hand, the geometrization of non-holonomic mechanics has been less developed. However, due to the interest aroused by non-holonomic geometry, many papers have been devoted to this subject. In this article we generalize the construction of a connection whose geodesics are the trajectories of a system, obtained by Vershik and Feddeef in the case where the Lagrangian is quadratic and the constraints are linear on the velocities. Using the algebraic formalism of the connections theory introduced by the first author, we carry out the construction in the general case of an arbitrary mechanical system (i.e. of a manifold with a convex Lagrangian not necessarily homogeneous) with ideal non-holonomic constraints. Moreover, we prove something stronger than the result of Vershik and Feddeev: our connection has not only the above-mentioned property for the geodesics, but it preserves the Hamiltonian by parallel transport. This connection is then a generalization of the Levi-Civita connection for the Riemannian manifolds for which the metric (i.e. the kinetic energy) is preserved by parallel transport.