Abstract: By making use of the least action principle and the saddle point theorem in critical point theory.The existence of  periodic solutions is studied for some non-autonomous second order Hamiltonian systems in the cases of and linear growth the  sublinear growth.Some new solvability conditions are obtained.

Key words: non-autonomous second order Hamiltonian systems, the least action principle, saddle point theorem, sublinear growth, linear growth, periodic solutions