Reduction and relative equilibria for the two-body problem on spaces of constant curvature

@article{Borisov2018ReductionAR,
  title={Reduction and relative equilibria for the two-body problem on spaces of constant curvature},
  author={Alexey Vladimirovich Borisov and Luis C. Garc{\'i}a-Naranjo and Ivan S. Mamaev and James Montaldi},
  journal={Celestial Mechanics and Dynamical Astronomy},
  year={2018},
  volume={130},
  pages={1-36}
}
We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each $$q>0$$q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of… 
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