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 June 24th, 2018, 02:09 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 394 Thanks: 27 Math Focus: Number theory Path of 3 bodies in a plane... Can an exact solution be found for the trajectories of three point-like particles, of unspecified but nonzero mass, gravitating together but confined to a plane? Can the general three-body problem be solved exactly for any interaction; e.g., by any central force, where one body is a massive horizon?
June 25th, 2018, 02:01 AM   #2
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Quote:
 Originally Posted by Loren Can an exact solution be found for the trajectories of three point-like particles, of unspecified but nonzero mass, gravitating together but confined to a plane?
Not in general, although very specific cases can be known.

June 25th, 2018, 05:42 PM   #3
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Quote:
 Originally Posted by Micrm@ss Not in general, although very specific cases can be known.
This isn't really true. This is commonly referred to as the "3 body problem" in general or often the "circular restricted" 3 body problem. The vector field associated with it is analytic and Hamiltonian for all values of the masses so it admits a perfectly good solution for any (non-collision) initial conditions and these solutions are perfectly easy to compute.

Perhaps the misunderstanding is due to the fact that the 3 body problem is NOT integrable? This isn't true for the 2 body (Kepler) problem which is always integrable. However, this result doesn't mean the 3 body problem has no solutions or that they can't be computed. It says the solutions aren't "nice" (the solution curves aren't restricted to invariant tori).

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