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

Rubén Darío Ortiz Ortiz (), Ana Magnolia Marín Ramírez and Ismael Oviedo de Julián
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Rubén Darío Ortiz Ortiz: Grupo Ondas, Departamento de Matemáticas, Universidad de Cartagena, Sede San Pablo, Cartagena de Indias 130001, Colombia
Ana Magnolia Marín Ramírez: Grupo Ondas, Departamento de Matemáticas, Universidad de Cartagena, Sede San Pablo, Cartagena de Indias 130001, Colombia
Ismael Oviedo de Julián: Unidad Azcapotzalco, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Cd. de México 02128, Mexico

Mathematics, 2024, vol. 12, issue 7, 1-17

Abstract: We consider the two- and n -body problems on the two-dimensional conformal sphere M R 2 , 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 M R 2 . 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: conformal sphere M R 2; the two-body problem; relative equilibria; antipodal points (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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