Mathematics

The KAM theory deals with small perturbations of completely integrable Hamiltonian systems. The phase space of an integrable Hamiltonian system is completely foliated by invariant tori on which the flow is periodic or quasi-periodic, being conjugated to a linear flow. The goal of the KAM theory is to prove that, under a sufficiently small Hamiltonian perturbation, most of these invariant tori survive, being just slightly deformed. The problem takes the form of an equation in some functional space, that cannot be solved with the standard implicit function theorem. After a short survey of the main properties of Hamiltonian systems, we shall introduce an iterative method to deal with a family of perturbative equations in functional spaces, and apply it to the problem of the invariant tori.