Persistence Of Invariant Torus In Hamiltonian Systems With Two-degree Of Freedom

This paper consider the persistence of invariant tori for small perturbation of integrable Hamiltonian systems with two freedom degree. By using the special property of two freedom and an improved KAM iteration, without any non-degeneracy conditions we prove the exitence of invariant tori for nearly integrable Hamiltonian systems with two freedom degree. The main difficulty is caused by the little shift of frequency in KAM iteration, and we must have that the non-degenercy condition appear as soon and it will be kept for ever in KAM iteration. So that the usual KAM method can be available. If the non-degeneracy conditions never ocurr, then the KAM iteration is also convergent. Therefor, we must develope some new technology to overcome these difficulties.