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.
Now my question is: What is this proof? Can anybody explain it? Do we have to use the method of "Proof by contradiction"?
And also another question: What are the approximate methods used to solve this problem? Numerical analytical methods? Perturbation theory?