In this paper,periodic solutions for a class of non - autonomous second order Hamiltonian Systems have been studied under the local linking at zero and“nonquadrat- ic”condition,and the proof has been obtained by the local compact condition i.

By using Critical point theory s local linking theorems,the existence positive solution is obtained for a more generally superlinear Dirichlet problem.

With the aids of local linking theorem and mountain pass lemma, existence of periodic solutions of the second order Hamiltonian system under the new superquadratic conditions is obtained.