# Orbits in the problem of two fixed centers on the sphere

## TL;DR

This paper establishes a mathematical isomorphism between the two fixed center problem on a sphere and its planar counterpart, enabling analysis of spherical orbits through elliptic integrals and functions.

## Contribution

It introduces a novel isomorphism via gnomonic projections that links spherical and planar two fixed center problems, facilitating bifurcation and dynamical analysis.

## Key findings

- Bifurcation diagrams of the spherical problem are derived from planar systems.
- Spherical orbits are expressed using Jacobi elliptic functions.
- Quadratures are transformed into elliptic integrals for analysis.

## Abstract

A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts the original quadratures into elliptic integrals and allows the bifurcation diagram of the spherical problem to be analyzed in terms of the corresponding ones of the planar systems. The dynamics along the orbits in the different regimes for the problem in $S^2$ is expressed in terms of Jacobi elliptic functions.