On the persistence of lower dimensional invariant tori under quasi-periodic perturbations
A class of perturbed Hamiltonian systems is studied which can roughly be characterised as follows. The integrable, unperturbed Hamiltonian gives rise to invariant tori of dimension lower than the number of degrees of freedom and these are not contained in a resonant higher-dimensional torus. The (analytic) perturbation is time-dependent, but quasi-periodic. The issue, of course, is the study of survival of the lower dimensional tori under suitable Diophantine-type conditions. Two ways of estimating the measure of destroyed tori are presented. The first one corresponds to the study of persistence of a single torus, when the size of the perturbation is used as a parameter. In the second approach, with fixed size of the perturbation parameter, existence of invariant tori is proved for every vector of frequencies close to the unperturbed one, except for a set of frequencies of measure exponentially small with respect to the distance to the unperturbed torus. One of the extra assumptions is that the linear variational equations around the initial torus can be reduced to equations with constant coefficients (as in Floquet's theorem when the torus is a periodic orbit). A couple of applications are discussed to specific situations in the restricted three-body problem of celestial mechanics. As is customary with KAM-theory, the proofs are quite technical, but the first part of the paper extensively discusses the general ideas.
scientific article; zbMATH DE number 1129284