Equilibrium points in restricted problems on S2 and H2

Abstract

In this paper, we consider the restricted four-body problem on S2 and the restricted three-body problem on H2. In the first case, the primary particles are considered to be rotating around the vertical axis. In the latter case, the primaries move at a hyperbolic relative equilibrium. If the primary particles locate at height z0 = 0 and they form an isosceles triangle on S2, then the number of symmetric equilibrium points of the restricted four-body problem depends on the angular velocity, the configuration, and the masses of the primaries. For this case, the number of equilibrium points is 0, 2, 4, 6, or 8. If the primaries are at height z0 ∈ (0, 1), then the primaries form an equilateral triangle and the number of symmetric equilibrium points is 8, 11, 14, 17, or 20. The number depends on the position of the primaries, and it is higher than the previous case due to the higher number of symmetry axes of the triangle formed by the primaries. On the other hand, for the restricted three-body problem on H2, the number of equilibrium points is 3, considering that the primaries move at a hyperbolic relative equilibrium. We analyzed the general (non-symmetric) case, showing the locations of the equilibrium points for the given positions of the primaries. We also studied the stability of these equilibrium points.