On the stability of tetrahedral relative equilibria in the positively curved
4-body problem

Florin Diacu
Pacific Institute for the Mathematical Sciences and
Department of Mathematics and Statistics, University of Victoria
Victoria, Canada

Regina Mart\'{\i}nez
Departament de Matem\`atiques, Universitat Aut\`onoma de Barcelona
Bellatera, Barcelona, Spain

Ernesto P\'erez-Chavela
Departamento de Matem\'aticas, Universidad Aut\'onoma Metropolitana
Iztapalapa, Mexico, D.F., Mexico

Carles Sim\'o
Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona
Barcelona, Spain

Abstract
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.