Persistence Of Invariant Tori In Generalized Hamiltonian Systems Under Weak Non-resonant Condition

In this paper,we adopt the method of rational approximation to prove the persistence of invariant tori under Brjuno-Russmann condition.The Brjuno-Riissmann non-resonant condition is|<k,?>|??/?(|K|),0?k?Zn where ?>0,Russmann approximation function ?:[1,+?)?[1,+?)is a continuous increasing unbounded function and satisfies ?(1)= 1,The method of rational approximation avoids the problem of small denominator,when solving the Homological function equation,and making the KAM iteration converge at the rate of qn?(0<q<1),rather than at the speed of the super-exponential function ??n(1<?<2).