# Relative equilibria for the positive curved $n-$body problem

## TL;DR

This paper investigates the existence of relative equilibria in the positive curved n-body problem, explicitly characterizing initial positions and mass parameters for 5 and 7 bodies on curved surfaces.

## Contribution

It provides explicit conditions and demonstrates that the parameter sets leading to relative equilibria have positive measure for 5 and 7 bodies.

## Key findings

- Explicit initial positions for relative equilibria in 5 and 7 body cases.
- Mass parameters form infinite sets for given initial positions.
- Parameter sets have positive Lebesgue measure.

## Abstract

We consider the $n$ body problem defined on surfaces of constant positive curvature. For the 5 and 7 body problem in a collinear symmetric configuration we obtain initial positions which lead to relative equilibria. We give explicitly the values of masses in terms of the initial positions. For positions for which relative equilibria exist, there are infinitely many values of the masses that generate such solutions. For the 5 and 7 body problem, the set of parameters (masses and positions) leading to relative equilibria has positive Lebesgue measure.