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Jun 5, 2018 at 23:30 comment added Richard Montgomery This is a nice problem Joseph. I do not think the Koon-Marsden work will help. The system you speak of is called the ``N-center problem''. The fact that you do not want to collide with the centers themselves adds topology in that is not there without them. For the planar version with periodic solutions related to braid types, see: arxiv:1201.0280.
Jun 3, 2018 at 20:24 comment added Piyush Grover A typical case here is to think of a planet-moon environment, where there is a dominant large body, and N-1 smaller "large" bodies (aka moons). Then one can build arbitrarily ordered "tours" of the moons, given certain conditions are met, i.e. it is possible to find a trajectory that goes around moon1 five times, then moon3 two times, and moon2 three times. See here: cds.caltech.edu/~koon/presentations/barcelona_june.pdf
Jun 3, 2018 at 20:21 comment added Piyush Grover @AlexandreEremenko The results hold for 3D case as well. The application to OP's problem requires the regions need to be defined properly. However, the intuition here is that as N, the number of bodies, increase, more of the phase space region becomes chaotic (as N->infinity, we get ergodicity). Hence, there are infinite number of periodic orbits, and they are dense in the chaotic region of the phase space. So orbits can be found arbitrarily close to any "itinerary" that one can write.
Jun 2, 2018 at 21:59 comment added Joseph O'Rourke It does seems this is a suggestion for a possible existence proof, but far from a constructive proof, which would be most satisfying.
Jun 2, 2018 at 13:54 comment added Alexandre Eremenko The paper you cite deals with PLANAR 3-body problem. In the original problem with knot, what "regions around each body" mean? There is only one region.
Jun 2, 2018 at 2:36 history edited Piyush Grover CC BY-SA 4.0
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Jun 2, 2018 at 1:18 comment added Joseph O'Rourke Koon, Wang Sang, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross. "Dynamical systems, the three-body problem and space mission design." (2008). World Scientific link.
Jun 1, 2018 at 23:51 history answered Piyush Grover CC BY-SA 4.0