 Understanding of the geometric shape of the universe
 New mathematical results
 Bridges laid between several fields of mathematics
 Opening of new directions of research
 Clearly exposed material
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 About this book

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 nonzero 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 2body 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, nonEuclidean geometry, geometric topology, Lie groups, and the theory of polytopes.
 Reviews

From the reviews:
“The book is divided into 5 parts with several chapters and sections in each. … The book is clear, well written, interesting and easy to read. … The book is an invitation to more research on this topic and it is a nice source of new problems, particularly for people working in celestial mechanics, dynamical systems, numerical analysis and geometric mechanics. I enjoyed reading it.” (Ernesto PérezChavela, Mathematical Reviews, June, 2013)
 Table of contents (14 chapters)


Introduction
Pages 110

Preliminary Developments
Pages 1324

Equations of motion
Pages 2542

Isometric Rotations
Pages 4552

Relative Equilibria (RE)
Pages 5359

Table of contents (14 chapters)
 Download Preface 1 PDF (1.6 MB)
 Download Sample pages 2 PDF (1.6 MB)
 Download Table of contents PDF (1.4 MB)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Relative Equilibria of the Curved NBody Problem
 Authors

 Florin Diacu
 Series Title
 Atlantis Studies in Dynamical Systems
 Series Volume
 1
 Copyright
 2012
 Publisher
 Atlantis Press
 Copyright Holder
 Florin Diacu
 eBook ISBN
 9789491216688
 DOI
 10.2991/9789491216688
 Hardcover ISBN
 9789491216671
 Softcover ISBN
 9789462390386
 Edition Number
 1
 Number of Pages
 XIV, 146
 Topics