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The classical Newtonian three-body problem remains unsolved today and had been the subject of many investigations by the best minds of the $18^{th}$ and $19^{th}$ centuries. Though the simpler version, the restricted three-body problem has a solution, the original problem is a much complicated nonlinear problem.

Newton himself speculated that an exact solution to the problem "exceeds, if I am not mistaken, the force of any human mind," whereas Hilbert, in his celebrated Paris address of 1900, put the problem in a category similar to Fermat’s last theorem.

I am writing this question to get an expert answer on whether the problem is difficult to solve or is impossible to solve. Also Wikipedia mentions that no general closed-form solution exists for this problem due to the chaotic behavior of the system for most initial conditions.

If the problem is solvable, are there any recent advancements in finding solutions for this problem?

Richard
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    Even the restricted three-body problem is chaotic (as famously discovered by Poincaré), so what do you mean when you say that it has a solution? By the way, here is a nice video with a talk by Rick Moeckel about this. – Hans Lundmark Dec 05 '19 at 09:15
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    Even if finding the general solution is hopeless, there are fairly recent advances about the existence of particular kinds of solutions. You could search for “N-body choreographies” or “off to infinity in finite time”, for example. – Hans Lundmark Dec 05 '19 at 09:17

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Very recent advance:

The 'Three-Body Problem' Has Perplexed Astronomers Since Newton Formulated It. A.I. Just Cracked It in Under a Second.

Quote:

The mind-bending calculations required to predict how three heavenly bodies orbit each other have baffled physicists since the time of Sir Isaac Newton. Now artificial intelligence (A.I.) has shown that it can solve the problem in a fraction of the time required by previous approaches.

Newton was the first to formulate the problem in the 17th century, but finding a simple way to solve it has proved incredibly difficult. The gravitational interactions between three celestial objects like planets, stars and moons result in a chaotic system — one that is complex and highly sensitive to the starting positions of each body.

Current approaches to solving these problems involve using software that can take weeks or even months to complete calculations. So researchers decided to see if a neural network — a type of pattern recognizing A.I. that loosely mimics how the brain works — could do better.

The algorithm they built provided accurate solutions up to 100 million times faster than the most advanced software program, known as Brutus. That could prove invaluable to astronomers trying to understand things like the behavior of star clusters and the broader evolution of the universe, said Chris Foley, a biostatistician at the University of Cambridge and co-author of a paper to the arXiv database, which has yet to be peer-reviewed.

Community
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Martín-Blas Pérez Pinilla
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    After reading the article in the link, it seems AI works when you fit it with stuff similar to something it has seen before. The is no indication it will continue to work for more complicated input. It is an advance but still not there (though probably good enough for another grant proposal) – achille hui Dec 05 '19 at 09:00
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    I don't think a popular science article (very often wrong or misleading, especially on hyped up subjects like machine learning/AI) that contains essentially no mathematics is a useful answer on this site. – Mees de Vries Dec 05 '19 at 09:20
  • @MeesdeVries, OK about the popular science press. Is simply the first hit in the Google search of "three body AI". The most relevant bit is the reference to the arXiv article: https://arxiv.org/abs/1910.07291. – Martín-Blas Pérez Pinilla Dec 05 '19 at 09:40
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    This does not answer the question, which asked for closed-form, symbolic solutions. The ANN produces just another type of numerical approximation, and from the images in the article, with visible errors, that is, with relative errors larger than 1%, which is quite a reduction in quality from the training data set that was computed with a relative tolerance of $10^{-11}$. "Faster" is here only a distinction in quantity, not in quality. – Lutz Lehmann Dec 05 '19 at 16:43
  • speaking of AI and orbital mechanics: How might one approach using AI (convolutional neural network) to predict collisions in orbit? I wonder if you can add an answer there as well? – uhoh Jul 01 '20 at 03:44