Practical aspects of a total low energy transfer to the Moon have been seen in missions like GENESIS, which uses Weak stability Boundary legs of Earth and Sun to reach ESL-2. This four body model transfers justifies that:
It is possible to use the unstable manifolds of the planar Lyapunov periodic orbits about the Sun-Earth L2 point to provide a low energy transfer from the Earth to the stable manifolds of planar Lyapunov periodic orbits around the Earth-Moon L2 point.
Thus transferring the spacecraft around EML2 which
act as separatrices in the energy manifold of the flow through the equilibrium point, provide the dynamical channels in phase space that enable ballistic captures of the spacecraft by the Moon.
Such Transfers have theoretically been proved to have 25-40% delta-v savings.
Now, instead of going 1.5 million miles away, can we use just the Earth-Moon system? Dynamical nature of CRTBP in Earth-Moon system suggests that certain states in phase space if attained, can lead spacecraft to asymptotically approach L-points into periodic/quasiperiodic orbits (In case of Earth-Moon system, lets just have Jacobi Energy constant, just enough to open up zero-velocity surface at both EML-1 and EML-2)
Also, Homo/Heteroclinic connection between Lyapunov orbits between two L-points, like utilized in ARTEMIS allows us to traverse space in a way shown here:
Can we in intermediate stage of such trajectory execute a decay manoeuvrer to get captured around the Moon in some way (because I'm guessing, we cannot get ballistically captured by the Moon from EML-1) ? What margin of delta-v would be needed in such case?
Alternatively, is there a possibility of ballistic capture from a periodic orbit around EML-2, spacecraft being transferred to EML-2 from EML-1 as in the figure ?
Quoted Text reference: Low Energy Transfer to the Moon, W.S.Koon