Gallavotti Giovanni
Twistless KAM tori, quasi flat homoclinic intersections, and other
cancellations in the perturbation series of certain completely
integrable hamiltonian systems. A review.
(217K, TeX)
ABSTRACT. THIS IS A CORRECTED VERSION OF A PREVIOUSLY POSTED PAPER.
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 dipohantine 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 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 ({\it "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 ({\it
"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 {\it "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 ({\it 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 {\it homoclinic scattering phase
shifts}.