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I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of elementary functions. Is this correct? I get that for each linearly independent constant of motion you can reduce the degree of freedom by one, but I don't understand why the theorem

Theorem (First integrals of the n-body problem) The only linearly independent integrals of the $n$-body problem, which are algebraic with respect to $q$, $p$ and $t$ are the $10$ described above. (http://en.wikipedia.org/wiki/N-body_problem#Three-body_problem)

implies that there is no analytic solution (I think this is synonymous with closed-form solution, and solution in terms of elementary functions). I've been trying to think about it, but I can't reason it, and apparently integrability implies no chaos, which I can't see either.

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Have you read en.wikipedia.org/wiki/Integrable_system ? –  lhf May 17 '12 at 20:21
    
I had understood that a perfectly elastic simultaneous collision between three bodies could not be solved - momentum and energy are conserved, but this does not determine the subsequent motion. –  Mark Bennet May 17 '12 at 20:47
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Fun fact: the quantum mechanics 3-body problem is solvable! –  Alex R. May 17 '12 at 21:53
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"Integrable by quadrature", which is the classical (Liouville) notion of integrability, does not mean "integrable in terms of elementary functions". It means that you can in principle write down the solution, provided that you are able to compute all antiderivatives and inverses of functions that you happen to come across along the way. But unfortunately not all antiderivatives and inverses of elementary functions are elementary, as you probably know... –  Hans Lundmark May 28 '12 at 7:08
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All of which leads to the curious, and unanswered question, is the solar system stable? –  thisfeller Jul 16 '12 at 19:04
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1 Answer

2-Body Problems also exist which have no specific solution such that there is a range of solutions for a given physical condition. This means solvability is not based on the number of bodies but the state and representation of space.

Indian Journal of Science and Technology published a physical proof called, “Binary Precession Solutions based on Synchronized Field Couplings”

http://www.indjst.org/index.php/indjst/article/view/30008/25962

In this research, a generalized wave function with classical characteristics was isolated within the motion of binary stars. The wave function provided the first tool for cracking the complex motion of DI Herculis and other binary stars that had several measured precession solutions.

http://xxx.lanl.gov/pdf/1111.3328v2.pdf

In this research, published about a year after the Indian Journal of Science and Technology publication, mathematicians from Imperial College London produced a proof for the physical existence of wave functions. The research was published in Nature Magazine.

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This does not answer the question. This answer and these citations concern quantum mechanics, not celestial mechanics. –  robjohn Feb 26 '13 at 7:54
    
Binary stars are celestial and the first paper describes the mechanics of those systems. The second paper supports one conclusion reached in the first paper. The whole bases of the 3-Body problem is centered around the number of bodies in the system and it is clear the bases has more to do with the representation of space then number of bodies. The proof is simply, we have 2-Body problems with unpredictable motion as it relates to Newtonian mechanics and General Relativity. These are fact based citations not just conjecture based citations. –  Jamahl Peavey Feb 26 '13 at 18:41
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Looking at the beginning of the first paper, I missed that it was regarding celestial mechanics; however, it does involve GR. classical-mechanics specifically refers to Newtonian-only mechanics. So, although this is closer than I previously thought, and your paper looks quite interesting, I don't think this answer really addresses the question. –  robjohn Feb 26 '13 at 19:12
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Why isn't the 3-Body problem solvable? Implies that 2-Body systems are solvable. I prove that most 2-Body systems are unsolvable such that solvability as it relates to exactness cannot be achieved when the motion exist when a statistical form. The classical wave function produced natural solutions. GR, Newtonian mechanics and MOND failed to calculate precessions for systems such as DI Herculis and others. Ed Guinan spent over 30 years studying the DI Herculis system and why it did not conform to GR. The Classical Wave Function calculated both measurements so we know it is correct. –  Jamahl Peavey Feb 27 '13 at 14:55
    
The classical Wave Function analysis meant Ed Guinan's original measurement was correct as well as the current measurement. An MIT group published in Nature claiming to solve the problem in 2009. This is Ed Guinan’s response. adsabs.harvard.edu/abs/2010AAS...21541934Z Notice the last line in the abstract and comments related to the MIT group’s work. It is experimental proof the system is changing between both measurements. The system’s motion is not exact but exist within the statistical form of the classical wave function. –  Jamahl Peavey Feb 27 '13 at 14:56
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