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 I 0 for nearly integrable and analytic Hamiltonian system H ( I , θ ) = H 0 ( I ) + ϵ H 1 ( 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 I 0 .