2000/04/24 by Paolo Piccione, Piccione, Paolo, Daniel V. Tausk +1
Mathematics · #37J05 #37J50 #37J60 #53C17 #70H03 #70H20 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Optimization and Control (math.OC) #math.DG #math.OC #msc:37J05 #msc:37J50 #msc:37J60 #msc:53C17 #msc:70H03 #msc:70H20
paper · pdf · doi:10.48550/arxiv.math/0004148
23 pages, LaTeX2e amsart Replacement of May 26th, 2000: expanded Introduction Replacement of September 24th, 2001: shortened version
openalex publication_date 2000/04/24 · arxiv created 2001/09/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ΩP,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 ΩP,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.