A symplectic map and its application to the persistence of lower dimensional invariant tori

We give a nonlinear symplectic coordinator transformation, which can move the normal frequencies of the lower dimensional torus up to <k,w> where w is the frequency vector of the torus. That means the normal frequencies with a difference <k,w> may be regarded as the same. As an application, we derive a persistence result on lower dimensional tori of nearly integrable Hamiltonian systems when the second Melnikov’s condition is partially violated.