# Relative Equilibria of the Curved N-Body Problem

• Florin Diacu
Book

Part of the Atlantis Series in Dynamical Systems book series (ASDS, volume 1)

1. Front Matter
Pages i-xiv
2. Florin Diacu
Pages 1-10
3. ### Background and Equations of Motion

1. Front Matter
Pages 11-11
2. Florin Diacu
Pages 13-24
3. Florin Diacu
Pages 25-42
4. ### Isometries and Relative Equilibria

1. Front Matter
Pages 43-43
2. Florin Diacu
Pages 45-52
3. Florin Diacu
Pages 53-59
4. Florin Diacu
Pages 61-64
5. ### Criteria and Qualitative Behavior

1. Front Matter
Pages 65-65
2. Florin Diacu
Pages 67-77
3. Florin Diacu
Pages 79-87
6. ### Examples

1. Front Matter
Pages 89-89
2. Florin Diacu
Pages 91-97
3. Florin Diacu
Pages 99-108
4. Florin Diacu
Pages 109-112
7. ### The 2-dimensional case

1. Front Matter
Pages 113-113
2. Florin Diacu
Pages 115-119
3. Florin Diacu
Pages 121-130
4. Florin Diacu
Pages 131-134
8. Back Matter
Pages 135-143

### Introduction

The guiding light of this monograph is a question easy to understand but difficult to answer: {What is the shape of the universe? In other words, how do we measure the shortest distance between two points of the physical space? Should we follow a straight line, as on a flat table, fly along a circle, as between Paris and New York, or take some other path, and if so, what would that path look like? If you accept that the model proposed here, which assumes a gravitational law extended to a universe of constant curvature, is a good approximation of the physical reality (and I will later outline a few arguments in this direction), then we can answer the above question for distances comparable to those of our solar system. More precisely, this monograph provides a mathematical proof that, for distances of the order of 10 AU, space is Euclidean. This result is, of course, not surprising for such small cosmic scales. Physicists take the flatness of space for granted in regions of that size. But it is good to finally have a mathematical confirmation in this sense. Our main goals, however, are mathematical. We will shed some light on the dynamics of N point masses that move in spaces of non-zero constant curvature according to an attraction law that naturally extends classical Newtonian gravitation beyond the flat (Euclidean) space. This extension is given by the cotangent potential, proposed by the German mathematician Ernest Schering in 1870. He was the first to obtain this analytic expression of a law suggested decades earlier for a 2-body problem in hyperbolic space by Janos Bolyai and, independently, by Nikolai Lobachevsky. As Newton's idea of gravitation was to introduce a force inversely proportional to the area of a sphere the same radius as the Euclidean distance between the bodies, Bolyai and Lobachevsky thought of a similar definition using the hyperbolic distance in hyperbolic space. The recent generalization we gave to the cotangent potential to any number N of bodies, led to the discovery of some interesting properties. This new research reveals certain connections among at least five branches of mathematics: classical dynamics, non-Euclidean geometry, geometric topology, Lie groups, and the theory of polytopes.

### Keywords

Celestial mechanics Dynamical systems N-body problem Spaces of constant curvature non-Euclidean geometry

#### Authors and affiliations

• Florin Diacu
• 1
1. 1., Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada

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