Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

Details Lagrangian and Hamiltonian Formalism for Constrained Variational Problems Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

2000-04-24

arxiv.org/abs/math/0004148v3

Mathematics

Optimization and Control

23 pages, LaTeX2e amsart Replacement of May 26th, 2000: expanded Introduction Replacement of September 24th, 2001: shortened v

Scientific paper

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 $TM^*$ 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 $\mathcal L$.

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