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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.

J. M. ain't a mathematician
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JLA
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    Have you read http://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
  • @MarkBennet Can you specify what are the initial conditions of this system which does not determine the subsequent motion? – TROLLHUNTER May 17 '12 at 20:59
  • @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? – Mark Bennet May 17 '12 at 21:31
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    Fun fact: the quantum mechanics 3-body problem is solvable! – Alex R. May 17 '12 at 21:53
  • @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. – Willie Wong May 18 '12 at 15:01
  • 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 ... – Willie Wong May 18 '12 at 15:18
  • ...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. – Willie Wong May 18 '12 at 15:22
  • @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). – Mark Bennet May 18 '12 at 15:25
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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 07:08
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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
  • @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? – Ruslan Aug 31 '17 at 09:43
  • @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_version – Alex R. Aug 31 '17 at 16:08
  • @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. – Ruslan Aug 31 '17 at 16:10
  • @Ruslan: It was kind of a joke. – Alex R. Aug 31 '17 at 17:16

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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.

hkr
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