Innovated　by　the　nonlinear　modes　concept　in　the　vibrational　dynamics,　the　vertical　periodic　orbits　around　the　triangular　libration　points　are　revisited　for　the　Circular　Restricted　Three-body　Problem.　The.　zeta-component　motion　is　treated　as　the　dominant　motion　and　the　xi　and　eta-component　motions　are　treated　as　the　slave　motions.　The　slave　motions　are　in　nature　related　to　the　dominant　motion　through　the　approximate　nonlinear　polynomial　expansions　with　respect　to　the　zeta-position　and　zeta-velocity　during　the　one　of　the　periodic　orbital　motions.　By　employing　the　relations　among　the　three　directions,　the　three-dimensional　system　can　be　transferred　into　one-dimensional　problem.　Then　the　approximate　three-dimensional　vertical　periodic　solution　can　be　analytically　obtained　by　solving　the　dominant　motion　only　on　zeta-direction.　To　demonstrate　the　effectiveness　of　the　proposed　method,　an　accuracy　study　was　carried　out　to　validate　the　polynomial　expansion　(PE)　method.　As　one　of　the　applications,　the　invariant　nonlinear　relations　in　polynomial　expansion　form　are　used　as　constraints　to　obtain　numerical　solutions　by　differential　correction.　The　nonlinear　relations　among　the　directions　provide　an　alternative　point　of　view　to　explore　the　overall　dynamics　of　periodic　orbits　around　libration　points　with　general　rules.