A KAM Theorem for a Class of Nearly Integrable Symplectic Mappings

2015 ◽  
Vol 29 (1) ◽  
pp. 131-154 ◽  
Author(s):  
Xuezhu Lu ◽  
Jia Li ◽  
Junxiang Xu
Keyword(s):  
Kam Theorem ◽  
Symplectic Mappings

scholarly journals A KAM Theorem for Lower Dimensional Elliptic Invariant Tori of Nearly Integrable Symplectic Mappings

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Shunjun Jiang
Keyword(s):  
Generating Functions ◽  
Invariant Tori ◽  
Kam Theorem ◽  
Symplectic Mappings ◽  
Lower Dimensional

This paper develops a new KAM theorem for a class of lower dimensional elliptic invariant tori of nearly integrable symplectic mappings with generating functions but without assuming any nondegenerate condition.

scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

A New KAM Theorem for the Hyperbolic Lower Dimensional Tori in Reversible Systems

2015 ◽  
Vol 143 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Xiaocai Wang ◽  
Junxiang Xu ◽  
Dongfeng Zhang
Keyword(s):  
Reversible Systems ◽  
Kam Theorem ◽  
Lower Dimensional Tori ◽  
Lower Dimensional

The KAM Theorem

2020 ◽  
pp. 197-204
Author(s):  
Walter Dittrich ◽  
Martin Reuter
Keyword(s):  
Kam Theorem

scholarly journals The KAM theorem and renormalization group

2009 ◽  
Vol 29 (2) ◽  
pp. 419-431 ◽  
Author(s):  
E. DE SIMONE ◽  
A. KUPIAINEN
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Renormalization Group ◽  
Elementary Proof ◽  
Renormalization Group Equation ◽  
Quadratic Nonlinearity ◽  
Picard Iteration ◽  
Group Equation ◽  
Quantum Field ◽  
Kam Theorem

AbstractWe give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a certain PDE with quadratic nonlinearity, the so-called Polchinski renormalization group equation studied in quantum field theory.

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