By using the critical point theory,the paper gives the results of the existence of the minimal periodic solutions of the symmetrical autonomous natural Hamiltonian systems with the condition that the potential function is one order differentiable.

Using the Nehari argument about the constrained extreme values and the theorem that the functional defined on complete Finsler manifold and satisfying the Palais-Smale condition and having lower bound has a minimal value point,we study the existence of the minimal periodic solutions for nonconvex quadratic and superquadratic second order Hamiltonian systems.

The existence of the minimal periodic solutions to two classes of the nonconvex subquadratic autonomous second order Hamiltonian systems is studied on the basis of the Rabinowitz s saddle point theorem and the Morse index estimation of its critical points.

In this paper,by comparing the Morse indexes of the critical points corresponding to the minimal value and the trivial value of the variational function,the existence of the minimal periodic solutions for a class of nonconvex autonomous second order Hamiltonian systems are discussed.

But how to determine the existence of minimum periodic point on function figure? Here in this paper we translate the complex figure into monotonous continuous broke linegraph.