Hamilton's principle and the splitting of periodic orbits

It is shown that bifurcation of periodic orbits, and orbit splitting in general, are closely related to the development of new saddle-point directions of the Lagrangian integral I1. Critical parameter values for orbit splitting can be found by looking at zeros of the second variation of I1 over a suitably chosen function space of periodic functions. I1 is a minimum for unstable periodic orbits and KAM trajectories. The rapid convergence of this method for finding bifurcations is illustrated with the example of a particle in a standing wave field