Three-Body Problem - how to find the Figure-eight solution?

Question

Suppose the coordinates of the Earth and the Moon are fixed and let $(u,v)$ be the coordinates of the satellite. I'm looking for the numerical solution of the three-body problem: $$u'' = 2v + u - \frac{c_1(u+c_2)}{((u+c_2)^2 + v^2)^\frac{1}{2}} - \frac{c_2(u-c_1)}{((u-c_1)^2 + v^2)^\frac{1}{2}}$$ $$v'' = -2u + v - \frac{c_1v}{((u+c_2)^2 + v^2)^\frac{1}{2}} - \frac{c_2v}{((u-c_1)^2 + v^2)^\frac{1}{2}}$$ where $c_1 \in (0,1)$ and $c_2 = 1 - c_2.$
Initial conditions are $[u,u',v,v']' = [1,0,0,-1]'.$
I need to find the constants such that the solution becomes a figure 8. I'm not sure if I can vary only $c_1$ or also the initial conditions, but any approach or idea would be helpful.
I've read about similar problems but this one is very specific and very simplified and I can't find anything that helps.
Thanks in advance!

Answer 1

I would start your simulation on the line through the Earth and Moon, on the outside of the Earth. The nice thing about that start is you know $u'=0$, so the motion is fully in the $y$ direction. You now have a two parameter question instead of four (starting position and velocity in one vs. two dimensions). You need the total energy to be negative so the orbit is bound. If you compute the apogee of the orbit ignoring the moon, it has to be at least close to the gravitational balance point between the Earth and Moon. This gives you a range of starting velocities depending on the starting altitude. I am guessing that if you start too slow, the spacecraft will just orbit the earth with the ellipse distorted a bit by the Moon's gravity. Somewhat faster will get your figure 8. Faster yet and you will do one orbit around the two bodies. Finally you will escape.