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The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system $$\left\{\begin{array}{c}\ddot{u}\left(t\right)=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\hfill \\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0.\hfill \end{array}\right.$$ Some new existence theorems are obtained by the least action principle.

The existence of positive periodic solutions for a kind of Rayleigh equation with a deviating argument $$x^{\text{'}\text{'}}\left(t\right)+f\left(x^{\text{'}}\left(t\right)\right)+g(t,x(t-\tau \left(t\right)))=p\left(t\right)$$ is studied. Using the coincidence degree theory, some sufficient conditions on the existence of positive periodic solutions are obtained.