Morse index theorem for heteroclinic orbits of Lagrangian systems

The classical Morse Index Theorem plays a central role in Lagrangian dynamics and differential geometry. Although many generalization of this result are well-known, in the case of orbits of Lagrangian systems with self-adjoint boundary conditions parametrized by a finite length interval, essentially no results are known in the case of either heteroclinic or half-clinic orbits of Lagrangian systems. The main goal of this paper is to fill up this gap by providing a new version of the Morse index theorem for heteroclinic, homoclinic and half-clinic orbits of a Lagrangian system.