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

Abstract

Properly degenerate nearly--integrable Hamiltonian systems with twodegrees of freedomsuch that the 'intermediate system' depend explicitly upon the angle--variableconjugated to the non--degenerate action--variable are considered and, inparticular,model problems motivated by classical examples of Celestial Mechanics, areinvestigated. Under suitable 'convexity' assumptions on theintermediate Hamiltonian,it is proved that, in every energy surface, the action variables stayforever close totheir initial values. In 'non convex' cases, stability holds up to a small set where,in principle, the degenerate action--variable might (in exponentially long times)drift away from its initial value by a quantity independent of the perturbation.Proofs are based on a 'blow up' (complex) analysis near separatrices, KAMtechniques and energy conservation arguments.