On the existence of orbits satisfying periodic or conormal boundary conditions for Euler-Lagrange flows

2015/11/24 by Asselle, Luca · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1511.07612

Abstract

Let (M,g) be a closed Riemannian manifold and L:TM→ \mathbb R be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the Euler-Lagrange flow associated with L satisfying suitable boundary conditions. We first look for orbits connecting two given closed submanifolds of M satisfying the conormal boundary conditions: We introduce the Mañé critical value that is relevant for the problem and prove existence results for supercritical and subcritical energies; we also complement these with counterexamples, thus showing the sharpness of our results. We then move to the problem of finding periodic orbits: We provide an existence result of periodic orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and a multiplicity result in case the configuration space is the 2-torus.

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