Resonance conjecture via weak KAM theory

Abstract

Poincaré established the problem how much of the stability mechanism of integrable Hamiltonian systems can persist under small perturbations, which he called “the fundamental problem of dynamics”. This paper deals with the fundamental problem in general resonant case. We give a weak KAM type result that for each y in the g (with rank m0)-resonant surface, the nearly integrable Hamiltonian system has at least m0+1 weak KAM solutions associated with relative equilibria.