On the stability of some properly-degenerate Hamiltonian systems with two degrees of freedom

摘要: Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the "intermediate system" depend explicitly upon angle--variable conjugated to non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples Celestial Mechanics, are investigated. Under suitable "convexity" assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, action variables stay forever close to their initial values. In "non convex" cases, stability holds up a small set where, in principle, might (in exponentially long times) drift away from its value quantity independent perturbation. Proofs based "blow up" (complex) analysis near separatrices, KAM techniques and conservation arguments.