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Good evening everyone, since Poincaré demonstrated that there are no exact solutions to the Three-body problem, many scientists have developed techniques to approximate these solutions.

The Newtonian approach has been well studied so far, and if I understand correctly the relativistic approach is not really the best because General Relativity is not an N-body theory but rather a field theory.

I read that the numerical simulations of this problem are very accurate nowadays but that many of the solutions found are very unstable in reality.

  • Can we consider today that this problem has been sufficiently explored ? and should we spend more time on the N-body problem ?
  • Or should we continue to deepen our knowledge of this problem ?
  • If so, what are the aspects of this problem that are unexplored or not well understood?

I hope my questions are not too vague. If that's the case, let me know and I'll rephrase them.

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    $\begingroup$ "the relativistic approach is not really the best because General Relativity is not an N-body theory but rather a field theory" I don't agree—GR is equally well-suited to study the N-body problem. Erik Poisson's book "Gravity: Newtonian, Post-Newtonian, Relativistic" explains in great detail how this is done. Apart from that, I don't find your comparison between Newtonian mechanics and GR very apt, for two reasons: 1) Newtonian mechanics can also be seen as a field theory, by virtue of Poisson's law $\Delta \phi = -4\pi\rho$. … $\endgroup$ – balu Aug 18 '18 at 11:05
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    $\begingroup$ …On the other hand, GR can also be cast into a formulation that resembles Newtonian gravity more closely. (In the sense that purely relativistic effects can be treated as correctional forces in Newton's law.) See again Poisson's book. 2) Things always become messy once you consider complicated matter distributions/several bodies, but this is equally true in Newtonian mechanics, not just in GR, so I wouldn't say one is better suited to treat the N-body problem than the other. $\endgroup$ – balu Aug 18 '18 at 11:08
  • $\begingroup$ @balu : you are right, I don't know a lot about GR, this is from an answer that I read on another post. You're the second person in the same week that advised me to read this book (physics.stackexchange.com/questions/422613/…). Thank for your answer, I am just starting to discover this field and I hope to learn more from the book you recommended. $\endgroup$ – Loïc Poncin Aug 23 '18 at 12:04
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Sure, many papers have been put out on the three-body problem in recent years, like this one and this one. It's a nice example of a chaotic system that's still reasonably tractable (as chaotic systems go), so there's probably still stuff to be learned from it. Just in past few decades, people have found exact solutions to the few-body problem with really counterintuitive behavior - for example, a five-body system where one particle's velocity diverges to infinity in finite time without colliding with any of the other particles.

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