2022/12/31 by Eva Miranda, Miranda, Eva, Arnau Planas +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Nonlinear Waves and Solitons #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2301.00266
This monograph explores classification and perturbation problems for integrable systems on a class of Poisson manifolds called bm-Poisson manifolds. Even if the class of bm-Poisson manifolds is not ample enough to represent general Poisson manifolds, this investigation can be seen as a first step for the study of perturbation theory for general Poisson manifolds. We prove an action-angle coordinate and a KAM theorem for bm-Poisson manifolds which improves the one obtained for b-Poisson manifolds for m=1 in [KMS16]. As an outcome of this result together with the extension of the desingularization techniques of Guillemin-Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds. We also obtain a new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In several problems in celestial mechanics, this distinguished hypersurface can be the line at infinity or can represent the collision set.