Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

Abstract

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

Keywords

• M R 2 %22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> conformal sphere <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3444"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="double-struck">M</mml:mi> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>
• the two-body problem
• relative equilibria
• antipodal points