# Dziobek equilibrium configurations on a sphere

## TL;DR

This paper studies equilibrium configurations, especially Dziobek configurations, in the n-body problem on a spherical surface, providing criteria, equations, and derivatives relevant to curved space dynamics.

## Contribution

It introduces a criterion and reduces it to equations for Dziobek equilibrium configurations on a sphere, extending classical n-body problem analysis to curved spaces.

## Key findings

- Derived a criterion for Dziobek equilibrium configurations.
- Reduced the criterion to two sets of equations.
- Calculated the derivative of the Cayley-Menger determinant.

## Abstract

We investigate the n-body problem on a sphere with a general interaction potential that depends on the mutual distances. We focus on the equilibrium configurations, especially on the Dziobek equilibrium configurations, which is an analogy of Dziobek central configurations of the classical n-body problem. We obtain a criterion and then reduce it to two sets of equations. Then we apply these equations to the curved n-body problem in S^3. In the end, we find the derivative of the Cayley-Menger determinant.