Is the three-body problem still unsolvable even when the three bodies are located in the same plane?
The three body problem, in which the three bodies are not in the same plane, is so incredibly difficult that it is widely belived to have no solution.
You can't have three bodies 'not in the same plane'. They define their own plane.
but their movement need not be coplanar. the plane defined by the instantaneous position of the 3 bodies will be movin' all over the place.
duh! I was not thinking - apologies.
I wasn't thinking either actually, but yeah, what I meant is: can the 3-body problem be solved if the movement of the three bodies is coplanar? (like, when the plane defined by the three bodies isn't rotating)
You might try looking at
for some general info on the three body problem.
If you don't mind a series solution that takes 10^8000000 terms to converge :-), there is a solution to the restricted 3-body problem. The restricted three body problem is the coplanar three body problem when one of the masses is small and a circular orbit for the two "large" masses around their common COM.
The 3B problem is only solvable for some special cases.
For instance, what I think is called the "Lagrange position" with another planet directly opposite the Earth on the other side of the Sun.
Of course, numerical solution is always possible (often using perturbation theory),
and has been used for centuries to predict eclipses.
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