# Proof of unsolvability of $n$-body problem for $n\geqslant 3$ in general

We know there are general solutions for 1-body problem and 2-body problem and also we know in some "special cases" there are some possible solutions for $n$-body problem for $n \geqslant 3$ and there might not be a general solution to this problem. But I heard it could be possible to prove there is no general solution in terms of elementary functions for $n$-body problem in general.

1. Now my question is: What is this proof? Can anybody explain it? Do we have to use the method of "Proof by contradiction"?

2. And also another question: What are the approximate methods used to solve this problem? Numerical analytical methods? Perturbation theory?

• Have you read the Wikipedia article that describes this? If not, this would have made valuable preliminary reading. In particular it mentions Sundman's theorm that tells us there is a general solution for the three body problem, though admittedly one that is not useful in practice. Nov 17, 2016 at 16:47
• Nov 17, 2016 at 17:00
• Please be more precise what you mean by "there is no general solution". No solution that we can write down in closed form (also define what a closed form is)? No solution at all for some initial conditions? Nov 26, 2016 at 15:29
• @ACuriousMind I meant proving there is no solution for $n\geqslant 3$ for the equations of motion to be stated in terms of elementary functions Nov 26, 2016 at 15:34
• Related math.SE post (on the topic of proofs of non-existence of elementary solutions): math.stackexchange.com/q/439248/143136, Nov 26, 2016 at 15:46