  Firstly,we generalize the classical Palais-Smale-Cerami condition for $C^1$
functional to the local Lipschitz case,then generalize the famous
Benci-Rabinowitz's and Rabinowitz's Saddle Point Theorems with classical
Cerami-Palais-Smale condition to the local Lipschitz functional, then we apply
these Theorems to study the existence of new periodic solutions for second
order Hamiltonian systems with local Lipschitz potentials which are weaker than
Rabinowitz's original conditions .The key point of our proof is proving
Cerami-Palais-Smale condition for local Lipschitz case,which is difficult since
no smooth and symmetry for the potential.
