Relative equilibria in curved restricted 4-and 5-body problems

Abstract

Abstract We consider a 5-body problem on 2-dimensional surfaces of constant curvature κ, with four of the masses arranged at the vertices of a square and the fifth mass at the north pole of the sphere. The five-body set up is discussed for κ > 0 and for κ < 0. When the curvature is positive, it is shown that relative equilibria exists when the four masses at the vertices of the square are either equal or two of them are infinitesimal such that it doesn’t effect the motion of the remaining three masses. However with two pairs of masses at the vertices of the square, no relative equilibria exists. In the hyperbolic case, κ < 0, there exist two values for the angular velocity which produce negative elliptic relative equilibria when the masses at the vertices of the square are equal. We also show that the solutions with non-equal masses do not exist in H2.