Collisions, Rings, and Other Newtonian $N$-Body Problems

This book is directed toward readers who are interested in learning about the Newtonian $N$-body problem, as well as toward students and experts in this area who are interested in new expositions of past results, previously unpublished research conclusions, and new research problems. As many readers will have no previous knowledge about this fascinating area, each chapter starts with introductory material that is motivated by unanswered research questions, includes some history with an occasional anecdote, provides discussions intended to develop intuition, introduces new technical approaches that answer open questions, and raises unsolved research problems. The first chapter, for instance, starts with simple explanations of the apparent 'looping' orbit of Mars and the unexpected 'Sunrise, Sunset' behavior as viewed from Mercury, to lead up to the unexplained and weird dynamics exhibited by Saturn's F-ring. The second chapter, which introduces a way to decompose the velocity of the system, is motivated by a seemingly easy but unanswered conjecture involving the dynamics of the system when the system's diameter is a constant. The third chapter, which describes questions about the structure of the rings of Saturn, introduces new and surprisingly simple ways to find configurations of the particles that are 'central' to any discussion of the $N$-body problem, or even about those expanding cracks in a car's windshield. The fourth chapter analyzes collisions, while the last chapter discusses the likelihood of collisions and other events.