Linear instability of periodic orbits of free period Lagrangian systems

Minimizing closed geodesics on surfaces are linearly unstable. By starting with this classical Poincaré’s instability result, in the present paper we prove a result that allows to deduce the linear instability of periodic solutions of autonomous Lagrangian systems admitting an orbit cylinder (condition which is satisfied for instance if the periodic orbit is transversally non-degenerate) in terms of the parity properties of a suitable quantity which is obtained by adding the dimension of the configuration space to a suitably defined spectral index. Such a spectral index coincides with the Morse index of the periodic orbit seen as a critical point of the free period action functional in the case the Lagrangian is Tonelli, namely fibrewise strictly convex and superlinear, and it encodes the functional and symplectic properties of the problem. The main result of the paper is a generalization of the celebrated Poincaré ’s instability result for closed geodesics on surfaces and at the same time extends to the autonomous case several previous results which have been proved by the authors (as well as by others) in the case of non-autonomous Lagrangian systems.