Recent Progress in Classical

The goal of this lecture is to review several developments in classical mechanics that have taken place in the last years, that will t in the time of the talk and that I have become aware of. Unfortunately, the latter is a constraint more severe than what I would like and I apologize to the authors and the audience for many things that have been left out. In particular, I have left out topics such as \twist mappings", or \geometric phases" and \quantum chaos" that are generating a great deal ofactivity in the literature. One of the central problems of mechanics has been to integrate Hamilton's equations of motion, or at least decide if such an integration is impossible. The most geometrically natural notion of integrability is that the system should have as many conserved quantities with vanishing Poisson brackets as degrees of freedom. It has long been known that the fact that a system is integrable severely restricts the topology of the phase space and of the energy surface. For example, if the system admits action-angle variables | which is strictly stronger than being integrable in the above sense | the phase space should be R n T n so that the topology is determined. The rst foothold in the geometric theory of integrable systems is the Liouville/Arnol'd theorem that says that the system can be decomposed in pieces R n i T m i , but the n i ; m i can change. For example in the Kepler system we have bounded trajectories along ellipses and unbounded ones along hyperbolas. These two sets get \glued" in the intermediate set of parabolic trajectories. An important realization was that the gluing of the diierent pieces has to be done in very precise ways so as to preserve the very rigid structure imposed by integrability. Hence, the phase space of an integrable system can be considered as pieces of R n i T m i glued in very precise ways. This turns out to impose severe restrictions on the phase spaces and energy surfaces of integrable systems. (Notice that having standard pieces that get glued in well deened ways according to the singular submanifolds of a vector eld is somewhat reminiscent of Morse Theory.) This beautiful theory, which one may start to learn in the paper Fo] written by the indefatigable leader of the method, has many spin-oos. Since one …