Estimates in the kolmogorov theorem on conservation of conditionally periodic motions

Abstract

A Hamiltonian system which differs from the integrable system by a small perturbation is considered. According to the Kolmogorov theorem /1–3/ the majority of invariant tori present in the unperturbed system do not decompose under a perturbation, and few of them become deformed. The following estimates are obtained below: for the perturbation ε under the usual conditions of nondegeneracy, the measure of the set of the decomposing tori and the deformation of the remaining tori are both estimate from above by the quantities of the order of √ε, and these estimates cannot be improved. The proof follows that /2,3/ of the Kolmogorov theorem with the intermediate estimates obtained more accurately. Similar estimates were obtained in the Moser theorem concerning the invariant curves of the mapping of a plane onto itself in /4/, and for the mapping in the multidimensional case, in /5/.