A relative morse index for the symplectic action

The notion of a Morse index of a function on a finite‐dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.