One can obtain the solution to a $2$-Body problem analytically. However, I understand that obtaining a general solution to a $N$-body problem is impossible.
Is there a proof somewhere that shows this possibility/impossibility?
Edit: I am looking to prove or disprove the below statement:
there exists a power series that that solve this problem, for all the terms in the series and the summation of the series must converge.
Answer
While the N-body Problem is chaotic, a convergent expansion exists. The 3-Body expansion was found by Sundman in 1912, and the full N-body problem in 1991 by Wang.
However, These expansions are pretty much useless for real problems( millions of terms are required for even short times); you're much better off with a numerical integration.
The history of the 3-Body problem is in itself pretty interesting stuff. Check out June Barrow-Green's book which include a pretty good analysis of all the relevant physics, along with a ripping tale.
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