On the stability of tetrahedral relative equilibria in the positively   curved 4-body problem
  We consider the motion of point masses given by a natural extension of Newtonian gravitation to spaces of constant positive curvature. Our goal is to explore the spectral stability of tetrahedral orbits of the corresponding 4-body problem in the 2-dimensional case, a situation that can be reduced to studying the motion of the bodies on the unit sphere. We first perform some extensive and highly precise numerical experiments to find the likely regions of stability and instability, relative to the values of the masses and to the latitude of the position of three equal masses. Then we support the numerical evidence with rigorous analytic proofs in the vicinity of some limit cases in which certain masses are either very large or negligible, or the latitude is close to zero. 
Florin Diacu, Regina Martinez, Ernesto Perez-Chavela, and Carles Simo
math.DS math-ph math.MP