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

TL;DR

This paper investigates the existence of Euler-Lagrange orbits with specific energy levels connecting submanifolds under conormal boundary conditions on a Riemannian manifold, introducing a critical value concept and providing sharp results.

Contribution

It introduces the Ma e critical value for this problem and establishes existence results for different energy regimes, with counterexamples confirming their sharpness.

Findings

Existence of Euler-Lagrange orbits depends on the energy level relative to the Ma e critical value.

Results are sharp, with counterexamples demonstrating the limits of the theorems.

The study extends understanding of boundary value problems in Lagrangian dynamics.

Abstract

Let $(M,g)$ be a closed Riemannian manifold, $L: TM\rightarrow \mathbb R$ 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 Ma\~n\'e 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.