Instability for periodic solutions with spatio-temporal symmetries in Hamiltonian systems

We consider the question of stability/instability of a periodic solution z(t) of an equivariant Hamiltonian system obtained by a minimizing variational principle. We suppose that z(t) lies in a constant energy surface, and that the symmetry group of the solution is finite and does not contain time reversing symmetries. Our main theorem states that z(t) is unstable if the second variation at the minimizer has positive directions and the Lagrange plane W associated with the symmetric minimizing problem has no focal points in the fundamental time domain, determined by the boundary conditions. Examples from Celestial Mechanics are presented.