On the variational mechanics with non-linear constraints

This paper concerns a geometric formulation of the so-called variational mechanics for mechanical systems with non-linear constraints. Given a smooth Lagrangian L on the tangent bundle of the configuration space M of the constrained mechanical system, its variational trajectories are defined, through a generalization of Hamilton's principle of stationary action, as extremals of the smooth Lagrangian functional gamma --> integral L(<(gamma)over dot>) defined on a convenient Banach manifold of curves compatible with the constraint manifold C subset of TM. In the particular case of a Lagrangian given by the positive definite quadratic form induced by a metric tensor on M, this amounts to a generalization of sub-Riemannian geometry. Among the main results, it is proven that, under a regularity condition on the Lagrangian L, the normal extremals of the Lagrangian functional are given by the projections on M of a Hamiltonian vector field defined on the generalized mixed bundle W. (C) 2003 Elsevier SAS. All rights reserved.