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 in the problem, the mass of the perturbing
body for instance, and for = 0 the system becomes completely integrable.
One then tries to show that its periodic solutions will subsist for -# 0
small enough. Poincare also introduced global methods, relying on the
topological properties of the flow, and the fact that it preserves the
2-form L =l dPi 1\ dqi' The most celebrated result he obtained in this
direction is his last geometric theorem, which states that an
area-preserving map of the annulus which rotates the inner circle and
the outer circle in opposite directions must have two fixed points. And
now another ancient theme appear: the least action principle. It states
that the periodic solutions of a Hamiltonian system are extremals of a
suitable integral over closed curves. In other words, the problem is
variational. This fact was known to Fermat, and Maupertuis put it in the
Hamiltonian formalism. In spite of its great aesthetic appeal, the least
action principle has had little impact in Hamiltonian mechanics. There
is, of course, one exception, Emmy Noether's theorem, which relates
integrals ofthe motion to symmetries of the equations. But until
recently, no periodic solution had ever been found by variational
methods.