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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 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.

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+1 for asking about an actual proof –  David Z Nov 23 '10 at 8:11
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If such proof exist, why are there still people trying to solve 3 body motion under gravity? –  hwlau Nov 23 '10 at 10:30
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@hwlau, what do you mean by "solve" here? Perturbation series? Or analytical solution? Or? –  Graviton Nov 23 '10 at 15:07
    
Since our computer will never be too fast for such problem, improve the efficiency of numerical calculation is always valuable. Even we know that pi is an irrational number, we are still trying to give a longer expression. –  gerry Nov 23 '10 at 15:07
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@gerry: No one looks for more digitsof pi 'cause they need more digits. It's a burning in exercise for large computers or a stunt for publicity. –  dmckee Nov 23 '10 at 15:57

2 Answers 2

up vote 29 down vote accepted

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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@Ngu Soon Hui: The question asked for a proof of the impossibleness of solving the n-body problem, but the problem is actually solvable. –  reallygoodname Nov 24 '10 at 10:08
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@reallygoodname, you have such a proof that there is an analytical solution for all $N$ ? I don't deny that this problem is in principle solvable by applying perturbation techniques or numerical simulations, but the existence ( or non-existence) of an analytical solution is what is required. –  Graviton Nov 24 '10 at 11:18
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@Ngu: strange definition of analyticity. –  Cedric H. Nov 24 '10 at 12:31
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The proof is in the Wang paper (adsabs.harvard.edu/abs/1991CeMDA..50...73W), and is not actually that complicated. The series can be made to be arbitrarily accurate, by increasing the number of terms in the expansion, though the time they are valid for may be finite as there are non-regularizable singularities for n > 3. As for solutions which are exact for all time, there are some very specific configurations (highly symmetric) for which people have found, i think i remember seeing a 12 body one. These solutions are not stable though. –  reallygoodname Nov 24 '10 at 12:32
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That is a bit ambiguious. It solves the problem for all n, and it solves the 0 angular momentum case for the 3-body problem. The 0 case was the only excluded case for Sundman's 1912 solution, which was the first to solve the 3-body problem. Wang's work is an extension of the methods used by Sundman. –  reallygoodname Nov 24 '10 at 13:00

One easy way to see this is that the N-body problem can be used, with appropriate potentials, to simulate a classical computer, so that as N becomes large, any algorithm which predicts the future behavior at arbitrarily long times has to be at least as computationally complex as a general cN-bit computer (where c is the number of bits you can usefully code per-particle) . Summation of convergent infinite series also simulates a computer, so that's not a useful interpretation of the word "solve". But any good definition of saying "solve" should mean that you reduced the computational complexity of predicting the future behavior from the present state, which can't be done for a general purpose computer.

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I agree. I'd only add that we know that the problem is chaotic for $n > 2$. What this means in the context of a series solution is that if you change the number of bits used to represent the data, the answer you get differs exponentially from that with the original number of bits. –  Paul J. Gans Dec 23 '12 at 0:50
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@PaulJ.Gans: The problem has chaotic sectors for n>2, but these don't fill the majority of the phase space. The KAM regions are larger, at least until you get to some dozens of particles, and then the generic behavior tends to become ergodic. This is something people notice when doing simulations. –  Ron Maimon Jan 4 '13 at 2:49

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