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Provides us to know how to measure the shortest distance between two points of the physical space. This book offers a mathematical proof that, for distances of the order of 10 AU, space is Euclidean.

Celestial Encounters is for anyone who has ever wondered about the foundations of chaos. In 1888, the 34-year-old Henri Poincare submitted a paper that was to change the course of science, but not before it underwent significant changes itself. "The Three-Body Problem and the Equations of Dynamics" won a prize sponsored by King Oscar II of Sweden and Norway and the journal Acta Mathematica, but after accepting the prize, Poincare found a serious mistake in his work. While correcting it, he discovered the phenomenon of chaos.

Starting with the story of...

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...

The author considers the 3 -dimensional gravitational n -body problem, n 2 , in spaces of constant Gaussian curvature K ' 0 , i.e. on spheres S 3 ?' , for ?>0 , and on hyperbolic manifolds H 3 ?', for ?<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 S 3 ? and H 3 ? . Then he proves...