After locating the libration points, the next step is to search for an orbit. That is, we try to find a set of initial conditions (x,y,z,Vx,Vy,Vz) that will allow the spacecraft to enter into orbit around the desired libration point.
Here is the strategy:
The analytical approximation that we use in our project is given in detail in the paper Periodic Orbits About the L1 and L2 Collinear Points in the Circular Restricted Three Body Problem (1978) by D.L. Richardson.
One can now discuss the remaining part of the process, which is using the approximation
to discover a set of initial conditions for an actual orbit. The mathematics of this step
is discussed in the link below.
Computing the actual orbits and their stable and unstable manifolds