Giovanni Gallavotti

Twistless KAM Tori

A selfcontained proof of the KAM theorem in the Thirring model is discussed.

TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION 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 fixed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the diophantine tori. We find in this way a proof of the KAM theorem by direct bounds of the k-th order coefficient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegels 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 flat, 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 find 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.

Smooth prime integrals for quasi-integrable Hamiltonian systems

SummaryA Hamiltonian withN degrees of freedom, analytic perturbation of a canonically integrable strictly nonisochronous analytic Hamiltonian, is considered. We show the existence ofN functions on phase space and of classC∞ which are prime integrals for the perturbed motions on a suitable region whose Lebesgue measure tends to fill locally the phase space as the perturbation’s magnitude approaches zero. An application to the perturbations of isochronous nonresonant linear oscillators is given.RiassuntoSi considera un sistema hamiltoniano aN gradi di libertà, perturbazione analitica di un sistema analiticamente e canonicamente integrabile e strettamente non isocrono. Si mostra l’esistenza diN funzioni definite sullo spazio delle fasi e ivi di classeC∞ che sono integrali primi per il moto perturbato su opportune regioni la cui misura di Lebesgue tende a riempire localmente lo spazio delle fasi al tendere a zero della perturbazione. S’illustra un’applicazione alle perturbazioni di oscillatori isocroni non risonanti.

A rigorous implementation of the Jeans - Landau - Teller approximation for adiabatic invariants

Rigorous bounds on the rate of energy exchanges between vibrational and translational degrees of freedom are established in simple classical models of diatomic molecules. The results are in agreement with an elementary approximation introduced by Landau and Teller. The method is perturbative theory `beyond all orders, with diagrammatic techniques (tree expansions) to organize and manipulate terms, and to look for compensations, like in recent studies on KAM theorem homoclinic splitting.