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

In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighbourhood of given I0 for nearly integrable and analytic Hamiltonian system H(I,θ)=H0(I)+ϵH1(I,θ) degree freedom short range. That to say, we will give an extension original Kolmogorov theorem infinite-dimensional case The proof based on approximation finite-dimensional improved machinery which works normal form depending only initial I0.