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How to make formulation of the three-body problem if we suppose that the motion is periodic and angular momentum is zero for the example of three equal masses.

Another question for understanding can be, as we know that the problem of two bodies has solution, how to know when the orbit has shape of ellipse and when it is a circular shape?

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I'm very ignorant about this, but I think the problem is largely that the solutions are not even approximately periodic. I once wrote a crude a computer sim (to play with it together with my son), and we saw strange things happening: a planet and two stars will occasionally result in the planet swapping from one star to the other after an irregular number of periods, or be slingshot out to the emptiness! Or something like that? – Jyrki Lahtonen Mar 15 '13 at 18:57
@Jyrki thank you very much for your attention, but can you help me how to simulate three body problem? – Pipe Mar 15 '13 at 19:02
My sim was very crude. I discretized time: run the clock a fixed time interval at a time assuming constant acceleration for the duration of that small interval. Not accurate (because the behavior is probably chaotic), but let's you see those strange things. I guess pde-people and numerical analysts will know of techniques of improving the accuracy of such a sim. – Jyrki Lahtonen Mar 15 '13 at 19:06
@Pipe Most easy way is to use Lagrangian dynamics – Occupy Gezi Mar 15 '13 at 21:34
@ Anil I am not sure what you mean. – Pipe Mar 16 '13 at 0:16

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