In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter e in the problem, the mass of the perturbing body for instance, and for e = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for e -# 0 small enough. Poincare...