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

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.