On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range

TL;DR

This paper extends the Kolmogorov theorem to infinite-dimensional Hamiltonian systems with short-range interactions, proving the existence of invariant tori near a given state under small perturbations.

Contribution

It provides an extension of the classical Kolmogorov theorem to infinite-dimensional, short-range Hamiltonian systems, with a novel proof based on finite-dimensional approximation and enhanced KAM techniques.

Findings

Existence of infinite KAM tori near a prescribed state in short-range Hamiltonian systems.

Extension of Kolmogorov theorem to infinite degrees of freedom.

Development of an improved KAM machinery for normal form dependence.

Abstract

In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,\theta) = H_{0}(I)+ \epsilon H_{1}(I,\theta)$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending on initial $ I^{0}$.