FM 93-4 mp_arc 93-17?
Author: Giovanni Gallavotti
Title: Twistless KAM tori, quasi flat homoclinic
intersections, and other cancellations in the perturbation
series of certain completely integrable hamiltonian
systems. A review.
Keywords: KAM theorem, invariant tori, classical
mechanics, perturbation theory, chaos
Abstract: 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
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, whichcould 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}.
Fisica, Universita' di Roma La Sapienza,
P.le Moro 2, 00185, Roma, Italia.
e-mail giovanni@ipparco.roma1.infn.it
tel. 6-49914370, fax 6-4957697