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On elliptic lower dimensional tori in Hamiltonian systems. (English) Zbl 0662.58037
This paper presents a perturbation theory of KAM-type for finite dimensional, elliptic invariant tori in finite or infinite dimensional Hamiltonian systems. In the model problem, the unperturbed integrable Hamiltonian is \[ N=\sum^{n}_{i=1}\omega_ iy_ i+\sum^{m}_{j=1}\Omega_ j(u^ 2_ j+v^ 2_ j) \] where \(2\leq n<\infty\) and \(1\leq m\leq \infty\), and the underlying phase space is \({\mathbb{T}}^ n\times {\mathbb{R}}^ n\times {\mathbb{R}}^ m\times {\mathbb{R}}^ m\) with coordinates (x,y,u,v). The \(\omega_ i\), \(\Omega_ j\) are real frequencies, and the equations of motion are \(\dot x=\omega\), \(\dot y=0\), \(\dot u=\Omega v\), \(\dot v=-\Omega u\) in usual vector notation. In this setting the frequencies \(\omega\) on the torus \({\mathbb{T}}^ n\) are considered as parameters, while the frequencies \(\Omega\) of the elliptic fixed point \((u,v)=(0,0)\) are usually functions of \(\omega\).
The result is that such a configuration persists under small real analytic perturbations of the Hamiltonian N. The perturbation has to be small in a special weighted norm, and the dependence of \(\Omega\) on \(\omega\) must not be too degenerate to avoid certain low order resonances.
Reviewer: J.Pöschel

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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