# Regularization of the restricted $(n+1)$-body problem on curved spaces

## TL;DR

This paper studies the regularization of singularities in the restricted $(n+1)$-body problem on curved spaces, demonstrating how to remove collision singularities via coordinate transformations.

## Contribution

It introduces methods to regularize binary collision singularities in the curved space restricted $(n+1)$-body problem using Levi-Civita and Birkhoff transformations.

## Key findings

- Binary collision singularities can be regularized in curved spaces.
- Regular polygon configurations of primaries are considered.
- Coordinate transformations effectively remove singularities.

## Abstract

We consider $(n+1)$ bodies moving under their mutual gravitational attraction in spaces with constant Gaussian curvature $\kappa$. In this system, $n$ primary bodies with equal masses form a relative equilibrium solution with a regular polygon configuration, and the remaining body of negligible mass does not affect the motion of the others. We show that the singularity due to binary collision between the negligible mass and the primaries can be regularized local and globally through suitable changes of coordinates (Levi-Civita and Birkhoff type transformations).