Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

Piccione, Paolo; Tausk, Daniel V.

Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

Authors

Description

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 $\mathcal D$ 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 version