Analytic approach on invariant manifold of unstable equilibrium of restricted n+1 body problem

This paper focusses on the analytical solutions of invariant manifold of unstable Lagrange equilibrium points of restricted N+1 body problem. Because the linear behavior of these equilibrium points is type center x center x saddle. Therefore, there exist stable manifold and unstable manifold, which are respectively tangent to the stable subspaces and unstable subspaces. We can use the improved center manifold theory to obtain Taylor series expansion of neighborhood of these points, these algebraic equation is independent of time. We obtain the geometric structure of restricted N+1 body problem. Compared with numerical integration, the results of three order expansion are consistent with the numerical results in the small neighborhood of libration point, and this approach will avoid explicit numerical differentiation. Restricted three body problem of Earth-moon is studied in detail.