  Inspired by the classical Poincar\'e criterion about the instability of
orientation preserving minimizing closed geodesics on surfaces, we investigate
the relation intertwining the instability and the variational properties of
periodic solutions of a non-autonomous Lagrangian on a finite dimensional
Riemannian manifold.
  We establish a general criterion for a priori detecting the linear
instability of a periodic orbit on a Riemannian manifold for a (maybe not
Legendre convex) non-autonomous Lagrangian simply by looking at the parity of
the spectral index, which is the right substitute of the Morse index in the
framework of strongly indefinite variational problems and defined in terms of
the spectral flow of a path of Fredholm quadratic forms on a Hilbert bundle.
