Relative equilibria of four identical satellites

Abstract

We consider the Newtonian 5-body problem in the plane, where four bodies have the same mass m , which is small compared with the mass M of the remaining body. We consider the (normalized) relative equilibria in this system and follow them to the limit when m / M →0. In some cases, two small bodies will coalesce at the limit. We call the other equilibria the relative equilibria of four separate identical satellites. We prove rigorously that there are only three such equilibria, all already known after the numerical researches by H. Salo and C. F. Yoder. Our main contribution is to prove that any equilibrium configuration possesses a symmetry, a statement indicated by J. Llibre as the missing key to proving that there is no other equilibrium.