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Let's consider the set of ODEs of the famous three-body problem. My question is about the statement "there is no closed-form solution of the three-body problem".

Background - The three-body motion equations can be written in Hamiltonian form as a set of 18 first-order differential equations. Besides the 10 integrals (the 6 integrals of the motion of the centre of mass, the 3 integrals of angular momentum and the integral of energy), Poincaré and Bruns proved there are no others. Together with the “elimination of the time” and the “elimination of the nodes” (or SO2 symmetry), the original 18 equations can be reduced to 6... but they can't be reduced further. Finally, Sundman proved that a series of power expansion solutions to the three-body problem exists. However, it is often stated that (e.g. here) that the problem has no closed-form solution: it can't be expressed analytically in terms of a finite number of certain "well-known" functions.

Question: How to prove the closed-solution statement? Is there any any reference to a rigorous proof? Or is it a motivated and reasonable conjecture?

Quillo
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AInseven
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    That's a bit too much of a request, to expect us, based on a six-part question, to prove them for you?? – amWhy Jan 07 '23 at 19:53
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    @amWhy the problem is when they reduced to 6, there is no closed form solution. So, if we know the 6 equations, maybe it will help to understand why. – AInseven Jan 07 '23 at 20:08
  • "But, Sundman had proved there exist a series of power expansion solutions to the three-body problem." Can you cite the source of your question? – amWhy Jan 07 '23 at 20:12
  • @amWhy I read his original paper.He proved this by Cauchy's theorem.The theorem is on page 113. He proved the original 18 equations had series of power expansion solutions by using some transformation.You can also see them on the last 5 page of the paper. – AInseven Jan 07 '23 at 20:40
  • Thanks for the link, Alnseven. It would help if you included that link in your question. Thanks for your assistance. – amWhy Jan 07 '23 at 20:42
  • Related and interesting question but lacking technical answers: https://math.stackexchange.com/q/146457/532409 – Quillo Apr 10 '24 at 13:34
  • @AmWhy: The (5) Lagrange Points are positions where the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. This general “Three-Body-Problem” was addressed by Lagrange in his prize-winning paper (Essai sur le Probleme des Trois Corps, 1772)... Wiki – Narasimham Apr 10 '24 at 13:37
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    It is not clear why this question is "not focused" and can not be reopened. – Quillo Apr 10 '24 at 17:25

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A nice historical overview of the three-body problem is given in [1] on pp. 726. Reading quickly it does not seem the case that the non-existence of a general closed form solution is mathematically established. It is rather the case that

despite the discovery of the particular solutions and a century of unrelenting work on the problem, the mathematicians of the nineteenth century were unable to find a general solution. Indeed, the problem was considered so hard that in 1890 Poincare was led to declare that he thought it impossible without the discovery of some significant new mathematics.

T. Gowers, J. Barrow-Green, I. Leader (eds.) The Princeton Companion to Mathematics. 2008.

Kurt G.
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    I also doubt if there is a proof.But I find this"...This means that in general there is no closed form solution ..."in here – AInseven Jan 07 '23 at 20:58
  • You could be more specific when you quote from a 40 page paper. Which page? Having said that: I doubt that June Barrow-Green states here anything different than what she wrote above. The non-existence result that you want to proove is too grand to be buried in side remarks. Imho this is millenium price/fields medal stuff. – Kurt G. Jan 08 '23 at 07:31
  • Sorry about the page. It's on page 13, but you can directly search "This means that in general" by Ctrl+F after you click the link. And another one also said about this.And someone here tried to find a proof, something about differential Galois group which I don't know. But as you said, it shouldn't be buried in side remarks. – AInseven Jan 08 '23 at 14:49