In astronomy, the 3-body question is how to calculate exactly the motion of three bodies under the influence of gravity. It is believed that no general solution exists because the situation is chaotic, although Lagrange is famous for coming up with five special cases where the three bodies retain their relative positions.
However I believe I’ve heard somewhere that solutions have been found for a few other special cases besides Lagrange’s original five. Can anyone give me a site/cite for any of these? I tried searching for “three body problem”, but all this gives me are sites devoted to the problem in classical physics of simultaneous elastic collisions between three particles. I need the astronomer’s three-body problem.
There’s also the one where two large equal bodies are orbiting each other in a circle, and a third smaller body is bouncing up and down along the line that goes through the center of that circle.
For analytic solutions, there’s precious few. However, if you just want a non-chaotic numerical solution, there’s also plenty if the distances are large enough and the masses are different enough. For instance, we can very precisely solve for the motions of nine planets and over fifty moons in our Solar System, because the Sun is big enough to dominate all the planets, and with one exception, the moons are all dominated by their respective parent planets.
By the way, the sites you found actually are relevant: Long-range gravitational interactions are just a special case of collisions. They might not be the easiest thing to work from, though.