I was playing a game on my smartphone whose goal is to draw certain orbit in presence of certain central gravitational potential. I noticed that when there are two center of force is possible to have a eight(or lemniscate)-shaped orbit. I would like to find analytical solution to that problem in the most symmetric situation: I have a potential such as
$$ V=-\frac{1}{d_1}-\frac{1}{d_2} $$
where
$d_1=\sqrt{(x-1)^2 + y^2}$ and $d_2=\sqrt{(x+1)^2+y^2}$.
I would like to write down the equation of orbit in the having as initial conditions: $x(0)=0$, $y(0)=0$, $x'(0)=v\cos{θ}$, $y'(0)=v\sin{θ}$.
Of course in euclidean coordinates the square roots makes the problem difficult (Mathematica doesn't succede in solving them analytically). I was thinking what is the most clever system of coordinates but I'm a bit doubtful.
Do you know any book or paper that solves this problem that seems very simple?
