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

We consider the question of linear stability of a periodic solution \begin{document}$z(t)$\end{document} with finite spatio-temporal symmetry group of a reversible-equivariant Hamiltonian system obtained as a minimizer of the action functional. Our main theorem states that \begin{document}$z(t)$\end{document} is unstable if a subspace \begin{document}$W$\end{document} associated with the boundary conditions of the minimizing problem is a Lagrangian subspace with no focal points on the time interval defined by the boundary conditions and the second variation restricted to the subspace \begin{document}$W$\end{document} at the minimizer has positive directions. We show that the conditions of our theorem are always met for a class of minimizing periodic orbits with the standard mechanical reversing symmetry. Comparison theorems for Lagrangian subspaces and the use of time-reversing symmetries are essential tools in constructing stable and unstable subspaces for \begin{document}$z(t)$\end{document} . In particular, our results are complementary to the recent paper of Hu and Sun Commun. Math. Phys . 290 , (2009).