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Rotators interacting with a pendulum via small, velocity independent,potentials are considered. If the interaction potential does not depend on thependulum position then the pendulum and the rotators are decoupled and we studythe invariant tori of the rotators system at fixed rotation numbers: we exhibitcancellations, to all orders of perturbation theory, that allow proving thestability and analyticity of the dipohantine tori. We find in this way a proofof the KAM theorem by direct bounds of the $k$--th order coefficient of theperturbation expansion of the parametric equations of the tori in terms oftheir average anomalies: this extends Siegel's approach, from the linearizationof analytic maps to the KAM theory; the convergence radius does not depend, inthis case, on the twist strength, which could even vanish ({\it "twistless KAMtori"}). The same ideas apply to the case in which the potential couples thependulum and the rotators: in this case the invariant tori with diophantinerotation numbers are unstable and have stable and unstable manifolds ({\it"whiskers"}): instead of studying the perturbation theory of the invariant toriwe look for the cancellations that must be present because the homoclinicintersections of the whiskers are {\it "quasi flat"}, if the rotation velocityof the quasi periodic motion on the tori is large. We rederive in this way theresult that, under suitable conditions, the homoclinic splitting is smallerthan any power in the period of the forcing and find the exact asymptotics inthe two dimensional cases ({\it e.g.} in the case of a periodically forcedpendulum). The technique can be applied to study other quantities: we mention,as another example, the {\it homoclinic scattering phase shifts}.}Comment: 46 pages, Plain Tex, generates four figures named f1.ps,f2.ps,  f3.ps,f4.ps. This paper replaces a preceding version which contained an error  at the last paragraph of section 6, invalidating section 7 (but not the rest  of the paper). The error is corrected here. If you already printed the  previous paper only p.1,3, p.29 and section 7 with the appendices 3,4 need to  be reprinted (ie: p. 30,31,32 and 4

reviews in mathematical physics

TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION SERIES OF CERTAIN COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS: A REVIEW