Electronic ISBN:  9781470414832 
Product Code:  MEMO/228/1071.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 228; 2014; 80 ppMSC: Primary 70; Secondary 34; 37;
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

Chapters

1. Introduction

2. BACKGROUND AND EQUATIONS OF MOTION

3. ISOMETRIES AND RELATIVE EQUILIBRIA

4. CRITERIA AND QUALITATIVE BEHAVIOUR

5. EXAMPLES

6. CONCLUSIONS


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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.

Chapters

1. Introduction

2. BACKGROUND AND EQUATIONS OF MOTION

3. ISOMETRIES AND RELATIVE EQUILIBRIA

4. CRITERIA AND QUALITATIVE BEHAVIOUR

5. EXAMPLES

6. CONCLUSIONS