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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 http://en.wikipedia.org/wiki/Integrable_system ?2012-05-17
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    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.2012-05-17
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    @MarkBennet Can you specify what are the initial conditions of this system which does not determine the subsequent motion?2012-05-17
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    @Xnyyrznaa: I think the idea was that, given an energy E, you could fire three point particles of from the origin in a variety of ways having zero total momentum: particles of equal mass at equal speeds and at $120^o$ angles will illustrate. Reverse one of these motions. How can you tell how the particles emerge?2012-05-17
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    Fun fact: the quantum mechanics 3-body problem *is* solvable!2012-05-17
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    @Mark: while that is true, the set of initial conditions which leads to a three-body simultaneous collision has (at least morally) measure zero, which I don't think is completely the resolution to the question asked.2012-05-18
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    Roughly speaking, the idea is counting the number of degrees of freedom. For each particle you have 6 degrees of freedom (3 for position and 3 for momentum). At three or more particles you have at least 18 degrees and only 10 integrals, so you don't have closed form solutions. How about 2 particles you ask? As it turns out you can cheat a bit there: by working in the centre of mass reference frame, you kill 6 integrals (0 center of mass and 0 linear momentum). However, conservation of momentum means that the two body evolution is only two dimensional! So for the two body problem you really ...2012-05-18
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    ...only have 4 degrees of freedom, which is just enough to kill using the remaining integrals of motion (conservation of angular momentum and conservation of energy). For three and more bodies you don't have such nice reductions.2012-05-18
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    @WillieWong Indeed - I put it in a comment for that reason. It was a one-liner which I remember from a lecture once. There is an article in the Princeton Companion to Mathematics which suggests that Sundman produced a series solution for cases with non-zero angular momentum (which avoids triple collisions).2012-05-18
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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...2012-05-28
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    All of which leads to the curious, and unanswered question, is the solar system stable?2012-07-16
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    @AlexR. in what sense is it solvable? Does it admit a closed form solution or is the solution representable in terms of e.g. quadratures? And, to be precise, by the quantum 3-body problem problem do you mean initial-value problem for TDSE or boundary-value problem for TISE?2017-08-31
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    @Ruslan: the energies of the particles can be derived as generalizations of Lambert's W Function. See here for example: https://en.m.wikipedia.org/wiki/Euler%27s_three-body_problem#Quantum_mechanical_version2017-08-31
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    @AlexR. it's not the _true_ 3-body problem: there the problem is simplified to that of motion of electron (1 body!) in the field of two fixed nuclei.2017-08-31
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    @Ruslan: It was kind of a joke.2017-08-31

2 Answers 2

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For the classical 3-body problem, the obstacle to a solution is, as you said, integrability. This is also sometimes called separability, and when it fails, it means that there does not exist a manifold in phase space such that on that manifold, the equations for the independent degrees of freedom of the equation are separated into independent equations. This is in turn related to being able to interchange mixed partial derivatives as you mention for the Poisson brackets, because if the equations separate, derivatives (and therefore integrals) can be performed in any order.

The relationship between this and chaos is that non-integrable systems are generically chaotic -- meaning "usually" or "observably" chaotic, the obstacle to separating the degrees of freedom being that there are intersecting stable and unstable manifolds of hyperbolic periodic points which cause the solutions to fold endlessly in phase space. "Generic" has a definition here, it means true on a countable interesection of open dense sets -- in other words, for every solution, there is an open subset of solutions arbitrarily close which have this property.

Hope this helps. There is a completely worked out solution for what is called the "restricted 3-body problem" (3 body problem in which one of the bodies has no mass) in Jurgen Moser's Stable and Random Motions in Dynamical Systems, which shows that even in this case, the motion of the massless body is chaotic for most initial conditions.

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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.2013-02-26
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    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.2013-02-26
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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. [tag: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.2013-02-26
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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.2013-02-27
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    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. http://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.2013-02-27
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    Furthermore, DI Herculis’ was once postulated to be a 3-Body elliptical problem as a solution to its complex motion. http://adsabs.harvard.edu/abs/1991ApJ...375..314K No evidence ever identified a third-body. Observational evidence does show DI Herculis is a detached binary system and the classical wave functional analysis models the system this way such that only the primary mass contributes to the complex statistical motion of the system. The solution is the most accurate for both measurements not excluding the MIT group’s evaluation. http://adsabs.harvard.edu/abs/2010arXiv1002.2949C2013-03-01
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    In conclusion, GR and Special Relativity’s beta structure in the Lorentz Factor is how coupling is defined in the Classical Wave Function. Einstein’s gravitational lensing for light around the Sun as well as Mercury’s precession was calculated via this synthesis proving Einstein’s Space-Time and Wave Functions are the same phenomena.2013-03-02