KAM Theorem and Integrable Systems on b-symplectic Manifolds

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Bagheri, Roxana | 2022

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 56005 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Bahraini, Alireza
Abstract:
The manifolds that will play the central role in this thesis are b-Poisson manifolds. These manifolds are a special sub-class of Poisson manifolds which are in many ways close to being symplectic: For π the Poisson structure dual to a symplectic form, the top wedge π^nnever meets the zero section of Λ^2n TM (non-degeneracy of the symplectic form). We define a b-Poisson manifold to be a Poisson manifold (M,π)such that π^n vanishes transversally to the zero section in Λ^2n TM. A pivotal result about the dynamics of integrable systems is the Liouville-Arnold-Mineur theorem (or action-angle coordinate theorem), which states that the compact common level sets of the integrals f_iare tori that are invariant under the motion of the system and on which the motion is linear. The action-angle coordinates can be computed explicitly by integration. We will investigate this theorem for integrable systems on Poisson Manifolds and b-Symplectic Manifolds in (non)-commutative cases. The main goal of this thesis is to explore integrable systems in the b-setting, including the proof of an action-angle coordinate theorem and a KAM theorem
Keywords:
Manifolds ; Symplectic Geometry ; Integrable Models ; Poisson Manifold ; KAM Theory ; (b)-Symplectic Geometry ; Action-Angle Coordinate Theorem

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