Title

Persistence and (in)stability of invariant tori in action-angle Hamilton equations

Abstract

In the latest 19th century, King Oscar II from Sweden proposed the following contest for all scientists from all over the world: solving the N-body problem.
In his "Sur le probléme des trois corps et les équations de la dynamique", Poincaré wrote a perturbative expansions for periodic solutions of the 3-body problem close to those of 2-body problem. However, this series blows up because of resonances! It seems that we are destined to collapse to the Sun...
In 1954, however, Kolmogorov showed that in fact, there are many other solutions, which are quasi-periodic for the N-body problem.
This fact is better known as KAM Theorem: geometrically speaking, quasi-periodic solutions (or KAM Tori) are "preserved" under suitable perturbations of integrable Hamiltonian systems.
Are (quasi-)periodic solutions stable? In 1977, Nekhoroshev showed that this is true for exponentially long times. After such times, however, Arnol'd shows, with an example of 1962, that the actions could increase because of the eventual "large" number of degrees of freedom. This phenomenon is better known as Arnol'd Diffusion, which is conjectured in 1989 to be true for all action-angle Hamiltonian systems with many degrees of freedom.
I will try to give such ideas and intuitions on Hamiltonian ODEs and, if time allows, I will show how people try to "extend" them on Hamiltonian PDEs.

Giuseppe La Scala is a guest PhD student from Scuola Superiore Meridionale.