Continuation of Periodic Orbits in Symmetric Hamiltonian Systems

The idea of everything returning eventually to its point of departure has a strong hold on humanity, with many historical, philosophical and religious implications. Classical examples are the need to construct a calendar and the subsequent search for orbits in the solar system in which the planets follow a closed track and repeat their history over and over again.

Nature, at its most basic level, has decided to be Hamiltonian; non- Hamiltonian systems come up in physics only as phenomenological models for the more complicated underlying processes. However, Hamiltonian systems are nongeneric dynamical systems with remarkable properties, in particular with respect to periodic orbits. The role of periodic solutions in Hamiltonian systems and their importance in modern physics was first recognized by Poincaré [26]. Today periodic orbits are at the basis of both classical and quantum mechanics [13]. Poincaré conjectured that periodic orbits, that is, solutions that return to their initial conditions after some finite time, are densely distributed among all possible bounded classical trajectories; and he suggested that the study of periodic orbits would provide the clue to the overall behavior of any mechanical system. Quoting the original work [26]:

• “It seems at first that the existence of periodic solutions could not be of any practical interest whatsoever. Indeed, the probability is zero for the initial condition to correspond precisely to those of a periodic solution. But it may happen that they differ by very little. […] Here is a fact which I have not been able to demonstrate rigorously, but which nevertheless seems very plausible to me. Given equations of the Hamiltonian form and any particular solution of these equations, we can always find a periodic solution (whose period may admittedly be very long) such that the difference between the two solutions is as small as we wish during as long a time as we wish. Besides this, what renders these periodic solutions so precious is that they are, so to speak, the only opening through which we may try to penetrate into the fortress which has the reputation of being impregnable.”