In this work we investigate the following fractional Hamiltonian systems tDα∞ (-∞Dαt u(t))+L(t)u(t) = ∇W(t,u(t)), where α ∈ (1/2,1), L ∈C(ℝ,ℝn2 ) is a positive definite symmetric matrix,W(t,u) =a(t)V(t) with a ∈C(ℝ,ℝ+) and V ∈C1(ℝn,ℝ). By using the Mountain pass theorem and assuming that there exist M > 0 such that (L(t)u,u) ≥ M|u|2 for all (t,u) ∈ ℝ×ℝn and V satisfies the global Ambrosetti-Rabinowitz condition and other suitable conditions, we prove that the above mentioned equation at least has one nontrivial weak solution.