Melnikov method and transversal homoclinic points in the restricted three-body problem

In this paper we show, by Melnikov method, the existence of the transversal homoclinic orbits in the circular restricted three-body problem for all but some finite number of values of the mass ratio of the two primaries. This implies the existence of a family of oscillatory and capture motion. This also shows the non-existence of any real analytic integral in the circular restricted three-body problem besides the well-known Jacobi integral for all but possibly finite number of values of the mass ratio of the two primaries. This extends a classical theorem of Poincaré [10]. Because the resulting singularities in our equation are degenerate, a stable manifold theorem of McGehee [7] is used.