# How can you construct the periodic orbits around libration points L1 and L2 in the circular restricted three-body problem and prove their existence?

Following up on a previous question about the classification of periodic orbits, how can they be constructed, especially the planar Lyapunov family, around libration points $$L_1$$ and $$L_2$$? And how can we prove their existence around those points?

Most papers I have found focus mainly on Halo orbits (see example) but I haven't found any that focus on the planar Lyapunov family.

• A very simplistic argument is that if a starting state that obeys Lyapunov stability has no vertical component, it must necessarily stay planar by conservation of momentum. It's hence just the "easy" task of proving convergence, with no further component analysis required. Apr 6, 2020 at 20:28
• The whole zoo of closed periodic orbits in the CR3BP are discussed in this answer and this and this may also be helpful. The first link refers to this paper which is not paywalled here.
– uhoh
Apr 6, 2020 at 21:01
• here is an earlier work by the same authors.
– uhoh
Apr 6, 2020 at 21:03
• @uhoh Thank you for sharing the papers. I went through them and they are very useful. But my question was more related to computation and construction of these orbits numerically.
– John
Apr 8, 2020 at 9:39
• @John ya I understand and that's exactly why those are in a comment and not an answer. It's possible they will also be helpful to someone else writing an answer. This is a hard problem and it may take some time before you get an answer. This answer doesn't answer your question either, but it may also be helpful.
– uhoh
Apr 8, 2020 at 9:43