Twistless Kam Tori, Quasi at Homoclinic Intersections, and Other Cancellations in the Pertur- Bation Series of Certain Completely Integrable Hamiltonian Systems. a Review

Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at xed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the dipohantine tori. We nd in this way a proof of the KAM theorem by direct bounds of the k{th order coeecient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegel's approach, from the linearization of analytic maps to the KAM theory; the convergence radius does not depend, in this case, on the twist strength, which could even vanish ("twistless KAM tori"). The same ideas apply to the case in which the potential couples the pendulum and the rotators: in this case the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"): instead of studying the perturbation theory of the invariant tori we look for the cancellations that must be present because the homoclinic intersections of the whiskers are "quasi at", if the rotation velocity of the quasi periodic motion on the tori is large. We rederive in this way the result that, under suitable conditions, the homoclinic splitting is smaller than any power in the period of the forcing and nd the exact asymptotics in the two dimensional cases (e.g. in the case of a periodically forced pendulum). The technique can be applied to study other quantities: we mention, as another example, the homoclinic scattering phase shifts. We discuss the invariant tori and the splitting of their homoclinic stable and unstable manifolds for a special class of quasi integrable hamiltonian systems. We apply and extend, in the considered class, the important ideas of Melnikov and Eliasson respectively on the theory of low dimensional invariant tori and their manifolds, Me], and on the cancellations behind the convergence of the formal perturbation series for the invariant tori of maximal dimension, E], i.e. for the KAM tori ((K],,A2],,M]). We also point out the analogy of the methods with those used in quantum eld theory, particularly in the renormalization group approaches, G2]. The ideas of Melnikov and Eliasson have been around since quite a while: but it seems that few realized their importance; a …