**CBMS Regional Conference Series in Mathematics**

Volume: 104;
2005;
235 pp;
Softcover

MSC: Primary 70;

**Print ISBN: 978-0-8218-3250-9
Product Code: CBMS/104**

List Price: $52.00

Individual Price: $41.60

**Electronic ISBN: 978-1-4704-2464-0
Product Code: CBMS/104.E**

List Price: $49.00

Individual Price: $39.20

#### Supplemental Materials

# Collisions, Rings, and Other Newtonian \(N\)-Body Problems

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*Donald G. Saari*

A co-publication of the AMS and CBMS

Written by well-known expert Donald Saari, this book is directed toward readers
who want to learn about the Newtonian \(N\)-body problem. It is also intended for
students and experts who are interested in new expositions of past results in
this area, previously unpublished research conclusions, and new research
problems.

Professor Saari has written the book for a broad audience, including readers
with no previous knowledge about this fascinating area. He begins each chapter
with introductory material motivated by unanswered research questions. He then
includes some history, discussions intended to develop intuition, descriptions
of open problems, and examples taken from real problems in astronomy.

The first chapter starts with simple explanations of the apparent "looping"
orbit of Mars and the unexpected "Sunrise, Sunset" behavior as viewed from
Mercury and then leads 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. The fourth chapter analyzes collisions, and the last chapter
discusses the likelihood of collisions and other events.

The book is suitable for graduate students and researchers interested in
celestial mechanics.

#### Readership

Graduate students and research mathematicians interested in celestial mechanics.

#### Reviews & Endorsements

The book can be useful for readers who are interested in learning celestial mechanics and particularly the Newtonian N-body problem as well as for students, postgraduate students and experts in this area who are interested in new expositions of past results, previously unpublished research conclusions, and new research problems.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents vii8
- Preface v6 free
- 1 Introduction 112 free
- 2 Central configurations 3142
- 2.1 Equations of motion and integrals 3243
- 2.2 Central Configurations 3445
- 2.3 A conjecture and a velocity decomposition 4758
- 2.4 More conjectures 6576
- 2.5 Jacobi coordinates help "see" the dynamics 6980
- 2.5.1 Velocity decomposition and a basis 7182
- 2.5.2 Describing P" with the basis 7485
- 2.5.3 "Seeing" the gradient of U 7687
- 2.5.4 An illustrating example 7788
- 2.5.5 Finding central and other configurations 7990
- 2.5.6 Equations of motion for constant I 8091
- 2.5.7 Basis for the coplanar N-body problem 8192

- 3 Finding Central Configurations 8394
- 4 Collisions — both real and imaginary 137148
- 4.1 One body problem 138149
- 4.2 Sundman and the three-body problem 147158
- 4.3 Generalized Weierstrass-Sundman theorem 150161
- 4.3.1 A simple case-the central force problem 151162
- 4.3.2 Larger p-values and "Black Holes" 151162
- 4.3.3 Lagrange-Jacobi equation 153164
- 4.3.4 Proof of the Weierstrass-Sundman Theorem 154165
- 4.3.5 Bounded above means bounded below 159170
- 4.3.6 Problems 161172
- 4.3.7 An interesting historical footnote 161172

- 4.4 Singularities-an overview 162173
- 4.5 Rate of approach of collisions 172183
- 4.6 Sharper asymptotic results 184195
- 4.7 Spin, or no spin? 185196
- 4.8 Manifolds defined by collisions 191202
- 4.9 Proof of the slowly varying assertion 195206

- 5 How likely is it? 207218
- Bibliography 223234
- Index 232243
- Back Cover Back Cover1250