Mathematics

Among finite dimensional Hamiltonian systems, the integrable ones are characterized by the existence of special coordinates (action-angle variables) in which the dynamics is particularly explicit: the angles evolve linearly in time and the actions remain constant for all times.

Nekhoroshev theorem guarantees, under suitable regularity and non-degeneracy hypotheses, that when a small perturbation is added to an integrable Hamiltonian, the action variables are quasi-conserved for exponentially long times.

However, such result strongly depends on the dimension of the system. What happens when dealing with infinite dimensional Hamiltonian systems? After an introduction of the state of the art in the finite dimensional case, the course will explore the partial attempts to answer these questions, presenting some applications of Nekhoroshev theory to quasi-integrable Hamiltonian PDEs.