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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 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.

Qmechanic
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Graviton
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    If such proof exist, why are there still people trying to solve 3 body motion under gravity? – unsym 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
  • @hwlau: People are strange? =) – Jens Nov 23 '10 at 15:10
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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 --- ex-moderator kitten Nov 23 '10 at 15:57
  • @Ngu: There is some form of 'solution' for the three body problem. However, is there a 'solution' that converge faster than the using simulation for the same time period? – unsym Nov 24 '10 at 02:55
  • From the practical point of view, I want a 'solution' that converge at the same rate for the $t = 10, 10^2, 10^3, ...$. Perturbation series could be a good 'solution' if you can find good perturbation function. – unsym Nov 24 '10 at 02:56
  • @hwlau, I don't know, which is why I'm asking the question here. – Graviton Nov 24 '10 at 03:25
  • @hwlau: It is solvable in certain special cases. For instance, it's trivial to solve for N=1, and Newton solved the N=2 case for gravity. There are special cases for N=3 that are solvable. The impossibility of a general solution refers to a solution that would cover all cases for every N. –  Aug 09 '11 at 14:26

2 Answers2

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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.

Art Brown
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reallygoodname
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  • @reallygoodname, although this receives some upvotes but I don't see how it actually answers the question. – Graviton Nov 24 '10 at 03:24
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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 Soon Hui, what exactly do you mean by analytic? – reallygoodname Nov 24 '10 at 11:53
  • A formula; it can contain lots of terms but it mustn't sum to infinity – Graviton Nov 24 '10 at 11:56
  • Also, if there is a proof showing that for an infinite series that solve the $N$-body problem and that series must always converge, that's also an answer I want. – Graviton Nov 24 '10 at 12:00
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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 (http://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
  • @reallygoodname, I afraid your paper only addresses the case where the angular momentum is zero, this is actually a special case of the $N$ body problem; see the last sentence of the abstract. And from what I read, someone ( not sure whether it's Poincaré or who) actually proved that there could be no such analytical solution. – Graviton Nov 24 '10 at 12:53
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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
  • upvoted for citing June Barrow-Green's book on Poincare – Geremia Dec 16 '14 at 06:19
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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.

Ron Maimon
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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 00:50
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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 04 '13 at 02:49