Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

2000 Mathematics Subject Classification. Primary 37J40, 70H08.We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order nondegenerate conditions of Bruno-Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.The first author is partially supported by NSFC Grant 10601019 and Chinese Postdoctoral Science Foundation. The second author is partially supported by NSFC Grant 10531050, National 973 project of China 2006CD805903, SRFDP Grant 20040183030, and 985 Program of Jilin University. The third author is partially supported by NSF grant DMS0708331, NSFC Grant 10428101, and a Changjiang Scholarship from Jilin University