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

Abstract

In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighbourhood of a given I0 for nearly integrable and analytic Hamiltonian system H(I,θ)=H0(I)+ϵH1(I,θ) 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 only on initial I0.