Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

Abstract

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $\gamma$ 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 ${\Omega }_{P,Q}(M,\mathcal{D})$ of horizontal curves that join two submanifolds $P$ and $Q$ of $M$. We consider an action functional $\mathcal{L}$ defined on ${\Omega }_{P,Q}(M,\mathcal{D})$ 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 $T{M}^{\ast }$ 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 version