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 pointlike particles, of unspecified but nonzero mass, gravitating together but confined to a plane? Can the general threebody 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 
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 246  
June 25th, 2018, 05:42 PM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 556 Thanks: 321 Math Focus: Dynamical systems, analytic function theory, numerics  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 (noncollision) 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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bodies, path, plane 
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