On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions

Abstract

Let $(M,g)$ be a closed Riemannian manifold, $L:TM\to \mathbb{R}$ be a Tonelli Lagrangian. Given two closed submanifolds $Q_0$ and $Q_1$ of $M$ and a real number $k$, we study the existence of Euler-Lagrange orbits with energy $k$ connecting $Q_0$ to $Q_1$ and satisfying the conormal boundary conditions. We introduce the Mañé critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.

arXiv commentIn this revised version some sharper examples have been added in Section 6 and some gaps have been eliminated. To appear in Journal of Functional Analysis