Celestial Encounters: The Origins of Chaos and Stability

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Princeton University Press, 1996 - Mathematics - 233 pages
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Celestial Encounters is for anyone who has ever wondered about the foundations of chaos. In 1888, the 34-year-old Henri Poincar 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, Poincar found a serious mistake in his work. While correcting it, he discovered the phenomenon of chaos.

Starting with the story of Poincar '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.

In presenting the modern theory of dynamical systems, the models underlying much of modern science are described pictorially, using the geometrical language invented by Poincar . More generally, the authors reflect on mathematical creativity and the roles that chance encounters, politics, and circumstance play in it.

 

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Contents

A Great DiscoveryAnd a Mistake
3
A WALK in PARIS
4
NEWTONS INSIGHT
7
A LANGUAGE for the LAWS of NATURE
9
MODELS of REALITY
13
MANIFOLD WORLDS
18
THE nBODY PROBLEM
20
KING OSCARS PRIZE
23
ENCOUNTERS at a CONFERENCE
112
FROM FOUR to FIVE BODIES
113
THE END of a CENTURYS QUEST
116
A SYMMETRIC DIGRESSION
122
AN IDEA at DINNER
123
Stability
127
A LONGING for ORDER
128
THE MARQUIS and the EMPEROR
131

POINCARES ACHIEVEMENT
27
LES MÉTHODES NOUVELLES
30
FIXED POINTS
31
FIRST RETURNS
34
A GLIMPSE OF CHAOS
37
PANDORAS BOX
42
POINCARES MISTAKE
44
A SURPRISING DISCOVERY
48
Symbolic Dynamics
51
ON THE BEACH AT RIO
55
SMALES HORSESHOE
60
SHIFTS on SYMBOLS
65
SYMBOLS for CHAOS
68
OSCILLATIONS and REVOLUTIONS
73
A NEW SCIENCE?
78
A SINGULAR MAN
82
COLLISION or BLOWUP
88
COMPUTER GAMES
91
HOW to CATCH a RABBIT
96
A MEASURE of SUCCESS
101
REGULARIZING COLLISIONS
104
CELESTIAL BILLIARDS
106
MUSIC of the SPHERES
135
ETERNAL RETURN
139
PERTURBING the WORLD
143
How STABLE is STABLE?
147
LINEARIZATION and ITS LIMITS
153
THE STABILITY of MODELS
157
PLANETS in BALANCE
159
KAM Theory
164
SIMPLIFY and SOLVE
165
QUASIPERIODIC MOTIONS
170
PERTURBING the TORI
174
LETTERS a LOST SOLUTION and POLITICS
177
WORRYING at the PROOF
182
TWIST MAPS
187
A GIFTED STUDENT
190
CHAOS DIFFUSES
193
EPILOGUE
201
Notes
203
Bibliography
213
Index
225
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About the author (1996)

Florin Diacu is professor of mathematics and former director of the Pacific Institute for the Mathematical Sciences at the University of Victoria in Canada. He is the coauthor of "Celestial Encounters: The Origins of Chaos and Stability" and the coeditor of "Classical and Celestial Mechanics: The Recife Lectures" (both Princeton).

Philip Holmes is Eugene Higgins Professor of Mechanical and Aerospace Engineering and Professor of Applied and Computational Mathematics, Princeton University. He works on nonlinear dynamics and differential equations.

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