On the geometry of Lagrangian mechanics with non-holonomic constraints

As is well known, Lagrangian mechanics have been entirely geometrized in te rms of symplectic geometry. On the other hand, the geometrization of non-ho lonomic mechanics has been less developed. However, due to the interest aro used by non-holonomic geometry, many papers have been devoted to this subje ct. In this article we generalize the construction of a connection whose ge odesics are the trajectories of a system, obtained by Vershik and Feddeef i n 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 gener al case of an arbitrary mechanical system (i.e. of a manifold with a convex Lagrangian not necessarily homogeneous) with ideal non-holonomic constrain ts. Moreover, we prove something stronger than the result of Vershik and Fe ddeev: our connection has not only the above-mentioned property for the geo desics, but it preserves the Hamiltonian by parallel transport. This connec tion is then a generalization of the Levi-Civita connection for the Riemann ian manifolds for which the metric (i.e. the kinetic energy) is preserved b y parallel transport. (C) 1999 Elsevier Science B.V. All rights reserved.