Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction Book

Introduction x

Chapter 1: FEYNMAN MEETS FERMAT 1

1.1 The Physicist as Mathematician 1

1.2 Fermat's Last Theorem 2

1.3 "Proof" by Probability 3

1.4 Feynman's Double Integral 6

1.5 Things to come 10

1.6 Challenge Problems 11

1.7 Notes and References 13

Chapter 2: Just for Fun: Two Quick Number-Crunching Problems 16

2.1 Number-Crunching in the Past 16

2.2 A Modern Number-Cruncher 20

2.3 Challenge Problem 25

2.4 Notes and References 25

Chapter 3: Computers and Mathematical Physics 27

3.1 When Theory Isn't Available 27

3.2 The Monte Carlo Technique 28

3.3 The Hot Plate Problem 34

3.4 Solving the Hot Plate Problem with Analysis 38

3.5 Solving the Hot Plate Problem by Iteration 44

3.6 Solving the Hot Plate Problem with the Monte Carlo Technique 50

3.7 ENIAC and MANIAC-I: the Electronic Computer Arrives 55

3.8 The Fermi-Pasta-Ulam Computer Experiment 58

3.9 Challenge Problems 73

3.10 Notes and References 74

Chapter 4: The Astonishing Problem of the Hanging Masses 82

4.1 Springs and Harmonic Motion 82

4.2 A Curious Oscillator 87

4.3 Phase-Plane Portraits 96

4.4 Another (Even More?) Curious Oscillator 99

4.5 Hanging Masses 104

4.6 Two Hanging Masses and the Laplace Transform 108

4.7 Hanging Masses and MATLAB 113

4.8 Challenge Problems 124

4.9 Notes and References 124

Chapter 5: The Three-Body Problem and Computers 131

5.1 Newton's Theory of Gravity 131

5.2 Newton's Two-Body Solution 139

5.3 Euler's Restricted Three-Body Problem 147

5.4 Binary Stars 155

5.5 Euler's Problem in Rotating Coordinates 166

5.6 Poincar? and the King Oscar II Competition 177

5.7 Computers and the Pythagorean Three-Body Problem 184

5.8 Two Very Weird Three-Body Orbits 195

5.9 Challenge Problems 205

5.10 Notes and References 207

Chapter 6: Electrical Circuit Analysis and Computers 218

6.1 Electronics Captures a Teenage Mind 218

6.2 My First Project 220

6.3 "Building" Circuits on a Computer 230

6.4 Frequency Response by Computer Analysis 234

6.5 Differential Amplifiers and Electronic Circuit Magic 249

6.6 More Circuit Magic: The Inductor Problem 260

6.7 Closing the Loop: Sinusoidal and Relaxation Oscillators by Computer 272

6.8 Challenge Problems 278

6.9 Notes and References 281

Chapter 7: The Leapfrog Problem 288

7.1 The Origin of the Leapfrog Problem 288

7.2 Simulating the Leapfrog Problem 290

7.3 Challenge Problems 296

7.4 Notes and References 296

Chapter 8: Science Fiction: When Computers Become Like Us 297

8.1 The Literature of the Imagination 297

8.2 Science Fiction "Spoofs" 300

8.3 What If Newton Had Owned a Calculator? 305

8.4 A Final Tale: the Artificially Intelligent Computer 314

8.5 Notes and References 324

Chapter 9: A Cautionary Epilogue 328

9.1 The Limits of Computation 328

9.2 The Halting Problem 330

9.3 Notes and References 333

Appendix

(FPU Computer Experiment MATLAB Code) 335

Solutions to the Challenge Problems 337

Acknowledgments 371

Index 373

Also by Paul J. Nahin 377