Perturbations of the spherical pendulum and Abelian integrals.

Integrable Hamiltonian Systems were the main subject of classical mechanics in the last Century. Most of the interesting examples of such Systems — the geodesic flow on the ellipsoid, K. Neumann's System, the integrable cases of the rigid body problem, etc. — were solved by Inversion of Abelian integrals. Recently the list of interesting Systems was considerably enlargened in connection with the so called inverse scattering method (see [8] for some discussions and references). On the other hand after the works of Bruns and Poincare it is considered that most of the Hamiltonian Systems are not integrable — they do not possess enough conserved quantities and, s a consequence, cannot be solved explicitely. The most powerful approach to non-integrable Systems is perturbation theory and especially KAM-theory. Abelian integrals play a considerable role here, too.