Resonance conjecture via weak KAM theory

Poincare 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. (c) 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

This paper addresses the fundamental problem of dynamics, namely, how much of the stability mechanism of integrable Hamiltonian systems can persist under small perturbations, as established by Poincare. We provide a weak KAM type result, showing that for each y on the g (with rank m0)-resonant surface, the nearly integrable Hamiltonian system has at least m0 + 1 weak KAM solutions associated with relative equilibria.