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arXiv · 1511.07612

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

Luca Asselle
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Abstract

Let $(M,g)$ be a closed Riemannian manifold and $L:TM\rightarrow \mathbb R$ 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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Luca Asselle. 2015-11-24. On the existence of orbits satisfying periodic or conormal boundary conditions for Euler-Lagrange flows. https://arxiv.org/abs/1511.07612

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