Variational structure of the zones of instability

Abstract

We give a variational characterization of the regions of stability and instability for planar Hamiltonian systems which are periodic in the time dependence, and convex in the momentum variables. In particular, we give necessary and sufficient conditions for stability, in terms of the Morse index of the variational problem with periodic or anti-periodic boundary conditions. We also show that this index coincides with the Maslov index of a curve of one-dimensional lines in the space $\mathbb{R}^2$. This result generalizes a result of Poincaré (1892) on unstable closed geodesics on orientable surfaces which minimize the arc length functional, and also gives a new test for stability based on the variational description of parametric resonance equivalent to that given by Liapunov (1892).