On the existence of periodic solutions to second order Hamiltonian systems

In this paper, the existence of periodic solutions to the second order Hamil-tonian systems is investigated. By introducing a new growth condition which gener-alizes the Ambrosetti-Rabinowitz condition, we prove a existence result of nontrivial T-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the po-tential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions.