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

Let (M,g)(M,g) be a closed connected Riemannian manifold, L:TM→RL:TM→R be a Tonelli Lagrangian. Given two non-empty closed submanifolds Q0,Q1⊆MQ0,Q1⊆M and a real number k, we study the existence of Euler–Lagrange orbits with energy k   connecting Q0Q0 to Q1Q1 and satisfying suitable boundary conditions, known as 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.