In this paper, the authors investigate the existence and multiplicity of solutions for the following fractional Hamiltonian system: \t\t\t{D∞αt(−∞Dtαu(t))+V(t)u(t)=λu(t)+b(t)|u(t)|q−2u(t)+μh(t),t∈R,u∈Hα(R,RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} { }_{t}D_{\\infty }^{\\alpha } ( {{ }_{-\\infty }D_{t}^{\\alpha }u(t)} )+V(t)u(t)=\\lambda u(t)+b(t) \\vert u(t) \\vert ^{q-2}u(t)+\\mu h(t),\\quad t \\in \\mathbb{R}, \\\\ u\\in H^{\\alpha }(\\mathbb{R},\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where alpha in (1/2,1), q>2, bin C(mathbb{R}, (0,infty )), h in C(mathbb{R}, mathbb{R}^{N}), Nge 1, lambda , mu are parameters, and Vin C(mathbb{R}, mathbb{R}^{Ntimes N}) is a positive definite symmetry matrix for all tin mathbb{R}. By using the variational method and with the help of the Nehari manifold, the existence results of at least one or two nontrivial solutions to the above fractional Hamiltonian system are obtained.