On the Morse index of Lagrangian systems Original Research Article

We give a new proof of the identity between the Morse index of a periodic solution of a Lagrangian system, positive definite in the velocities, and its Maslov index. Furthermore, we give an interpretation of the Maslov index when the Lagrangian system is not positive definite in the velocities. Our approach consists in relating the Morse index of the solution to a relative Morse index obtained passing to the Hamiltonian formulation. The main concepts used are the notion of commensurable subspaces, relative dimension, Fredholm pairs and relative index of strongly indefinite bilinear forms.