Celestial Encounters

by Diacu, Florin; Holmes, Philip

• ISBN13: 9780691027432
• ISBN10: 0691027439
• Format: Hardcover
• Copyright: 1996-10-01
• Publisher: Princeton Univ Pr

Summary

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 Poincare's work, Florin Diacu and Philip Holmes trace the history of attempts to solve the problems of celestial mechanics first posed in Isaac Newton's Principia in 1686. In describing how mathematical rigor was brought to bear on one of our oldest fascinations - the motions of the heavens - they introduce the people whose ideas led to the flourishing field now called nonlinear dynamics.

Table of Contents

Preface and Acknowledgments	xi
A Note to the Reader	xvii
A Great Discovery And a Mistake
A Walk in Paris	4	(3)
Newton's Insight	7	(2)
A Language for the Laws of Nature	9	(4)
Models of Reality	13	(5)
Manifold Worlds	18	(2)
The n-Body Problem	20	(3)
King Oscar's Prize	23	(4)
Poincare's Achievement	27	(3)
Les methodes nouvelles	30	(1)
Fixed Points*	31	(3)
First Returns*	34	(3)
A Glimpse of Chaos*	37	(5)
Pandora's Box	42	(2)
Poincare's Mistake	44	(4)
A Surprising Discovery	48	(3)
Symbolic Dynamics
A Fixed Point Begins a Career	51	(4)
On the Beach at Rio	55	(5)
Smale's Horseshoe*	60	(5)
Shifts on Symbols*	65	(3)
Symbols for Chaos*	68	(5)
Oscillations and Revolutions	73	(5)
A New Science?	78	(4)
Collisions and Other Singularities
A Singular Man	82	(6)
Collision or Blowup	88	(3)
Computer Games	91	(5)
How to Catch a Rabbit	96	(5)
A Measure of Success	101	(3)
Regularizing Collisions	104	(2)
Celestial Billiards	106	(6)
Encounters at a Conference	112	(1)
From Four to Five Bodies	113	(3)
The End of a Century's Quest	116	(6)
A Symmetric Digression	122	(1)
An Idea at Dinner	123	(5)
Stability
A Longing for Order	128	(3)
The Marquis and the Emperor	131	(4)
Music of the Spheres	135	(4)
Eternal Return	139	(4)
Perturbing the World	143	(4)
How Stable is Stable?	147	(1)
The Qualitative Age	147	(6)
Linearization and Its Limits	153	(4)
The Stability of Models	157	(2)
Planets in Balance	159	(6)
KAM Theory
Simplify and Solve	165	(5)
Quasi-periodic Motions*	170	(4)
Perturbing the Tori*	174	(3)
Letters, a Lost Solution, and Politics	177	(5)
Worrying at the Proof	182	(5)
Twist Maps*	187	(3)
A Gifted Student	190	(3)
Chaos Diffuses	193	(8)
Epilogue	201	(2)
Notes	203	(10)
Bibliography	213	(12)
Index	225