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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

##### MSC:
 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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