Multiple periodic solutions of some forced hamiltonian systems and the generalized saddle point theorem

In this paper we prove the existence of geometrically distinct periodic solutions of $$J\dot{u} + \nabla H(t,u)=0$$ Where $H(t, x)$ is periodic with respect to $t, x_1,\cdots, x_p$ and goes to zero uniformly with respect to $(t, x_1,\cdots,x_p)$ when ($x_{p+1},\cdots, x_{2N})$ goes to infinity.