2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59065We 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 $\mathcal D$ on $M$; such curves are called horizontal. We study the manifold structure of the set $Ω_{P,Q}(M,\mathcal D)$ of horizontal curves that join two submanifolds $P$ and $Q$ of $M$. We consider an action functional $\mathcal L$ defined on $Ω_{P,Q}(M,\mathcal D)$ associated to a time-dependent Lagrangian defined on $\mathcal D$. 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 {\em Legendre transform} for maps on vector bundles. We prove that the solutions of the Hamilton equations of $H$ are precisely the critical points of $\mathcal L$.23 pages, LaTeX2e amsart Replacement of May 26th, 2000: expanded Introduction Replacement of September 24th, 2001: shortened versionOptimization and ControlDifferential Geometry37J05; 37J50; 37J60; 53C17; 70H03; 70H20text