Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves γ in a differentiable manifold M that are everywhere tangent to a smooth distribution D on M ; such curves are called horizontal. We study the manifold structure of the set ΩP,Q(M,D) of horizontal curves that join two submanifolds P and Q of M . We consider an action functional L defined on ΩP,Q(M,D) associated to a time-dependent Lagrangian defined on D. If the Lagrangian satisfies a suitable hyper-regularity assumption, it is shown how to construct an associated degenerate Hamiltonian H on TM using a general notion of Legendre transform for maps on vector bundles. We prove that the solutions of the Hamilton equations of H are precisely the critical points of L.