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I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential topology, general relativity, Hamilton–Jacobi equation, and see why in integrable systems the motion takes place on an N torus and in general fills the surface. Also, know that even the reduced three-body problem could have a chaotic, highly irregular dynamical evolution.

Referring to what is reported in the news you can see that specifically one of her discoveries show that even the potentially chaotic dynamic of the three-body problem follows some deeply geometric laws. What we consider unpredictable, in the interaction of Sun, Moon, and Earth, is still to some extent predictable.

Is it possible for you to explain this restriction on phase space using what I now know and understand, I mean without moduli spaces and ergodic theory?

In addition I greatly appreciate it if you please suggest good and accessible resources for a beginner in moduli spaces and ergodic theory assuming understanding of the essential topology used in general relativity.

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    $\begingroup$ I think one should point out on a physics thread, that mathematical research into regular solutions of Hamiltonian problems is very different from actual physical problems. The solar system is very, very far from a three or n-body problem. Not only does one have to take the masses of hundreds of bodies into account, those bodies are also not point-like, which means that cusp scenarios become non-elastic impacts, and there are important relativistic corrections, solar wind pressure, and forces from e.g. mass loss of these bodies that are very important for the long term dynamics. $\endgroup$ – CuriousOne Aug 14 '14 at 20:14
  • $\begingroup$ @Victor, I would consider cross-posting this on MathOverflow, or possibly on Math.SE. You can look in their chatrooms for the moderators and see what they think. (Of course, if you cross-post, clearly label both posts as such.) $\endgroup$ – Emilio Pisanty Aug 18 '14 at 13:36
  • $\begingroup$ @EmilioPisanty thanks for the suggestion here is what I got from there link $\endgroup$ – user56963 Aug 19 '14 at 15:01
  • $\begingroup$ Well, it looks like the comments on MO give you your answer. $\endgroup$ – Emilio Pisanty Aug 19 '14 at 16:31
  • $\begingroup$ The MO post is removed. Can anyone post the summary of the discussion here? $\endgroup$ – stochastic Jan 4 '18 at 16:19

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