Equal masses Eulerian relative equilibria on a rotating meridian of S^2

Abstract

Relative equilibria on a rotating meridian on S 2 in equal-mass threebody problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria.Almost all isosceles triangles, including equilateral, can form a relative equilibrium, except for the two equal arc angles θ = π/2. For θ ∈ (0, 2π/3) \ {π/2}, the mid mass must be on the rotation axis, in our case, at the north or south pole of S 2 .For θ ∈ (2π/3, π), the mid mass must be on the equator. For θ = 2π/3, we obtain the equilateral triangle, where the position of the masses is arbitrary.When the largest arc angle a is in a ∈ (π/2, a c ), with a c = 1.8124..., two scalene configurations exist for given a .

Relative equilibria of mechanical systems with rotational symmetry

“…In section 4 we apply theorem 1 to obtain a complete and self-contained classification of the RE for the equal-mass spherical three-body problem. We would like to point out that this classification is not new, having recently been established in a series of preprints by Fujiwara and Pérez-Chavela [5][6][7][8][9] and the article [10]. In the final section we show how the energy-momentum method of [16] for assessing stability fits into our formalism.…”