# Relative Equilibria in the 3-Dimensional Curved \(n\)-Body Problem

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*Florin Diacu*

The author considers the \(3\)-dimensional gravitational \(n\)-body problem, \(n\ge 2\), in spaces of constant Gaussian curvature \(\kappa\ne 0\), i.e. on spheres \({\mathbb S}_\kappa^3\), for \(\kappa>0\), and on hyperbolic manifolds \({\mathbb H}_\kappa^3\), for \(\kappa<0\). His goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. He also briefly discusses the issue of singularities in order to avoid impossible configurations. He derives the equations of motion and defines six classes of relative equilibria, which follow naturally from the geometric properties of \({\mathbb S}_\kappa^3\) and \({\mathbb H}_\kappa^3\). Then he proves several criteria, each expressing the conditions for the existence of a certain class of relative equilibria, some of which have a simple rotation, whereas others perform a double rotation, and he describes their qualitative behaviour.

#### Table of Contents

# Table of Contents

## Relative Equilibria in the 3-Dimensional Curved $n$-Body Problem

- Chapter 1. Introduction 18 free
- Chapter 2. BACKGROUND AND EQUATIONS OF MOTION 916
- Chapter 3. ISOMETRIES AND RELATIVE EQUILIBRIA 2734
- Chapter 4. CRITERIA AND QUALITATIVE BEHAVIOUR 4148
- Chapter 5. EXAMPLES 5764
- 16. Examples of ๐ -positive elliptic relative equilibria 5764
- 17. Examples of ๐ -positive elliptic-elliptic relative equilibria 6269
- 18. Examples of ๐ -negative elliptic relative equilibria 7077
- 19. Examples of ๐ -negative hyperbolic relative equilibria 7178
- 20. Examples of ๐ -negative elliptic-hyperbolic relative equilibria 7279

- Chapter 6. CONCLUSIONS 7380
- Bibliography 7784