Table of Contents
Preface xiii
Chapter 1 The Foundations of Motion and Computation 1
1.1 The World Of Physics 1
1.2 The Basics Of Classical Mechanics 3
1.2.1 The Basic Descriptors of Motion 3
1.2.1.1 Position and Displacement 3
1.2.1.2 Velocity 4
1.2.1.3 Acceleration 5
1.2.2 Mass and Force 5
1.2.2.1 Mass 6
1.2.2.2 Force 6
1.3 Newton's Laws Of Motion 7
1.3.1 Newton's First Law 7
1.3.2 Newton's second law 8
1.3.3 Newton's third law 10
1.4 Reference Frames 11
1.5 Computation In Physics 13
1.5.1 The Use of Computation in Physics 14
1.5.2 Different Computational Tools 20
1.5.3 Some Warnings 22
1.6 Classical Mechanics In The Modern World 22
1.7 Chapter Summary 23
1.8 End-Of-Chapter Problems 24
Chapter 2 Single-Particle Motion in One Dimension 29
2.1 Equations Of Motion 29
2.2 Ordinary Differential Equations 30
2.3 Constant Forces 32
2.4 Time-Dependent Forces 36
2.5 Air Resistance And Velocity-Dependent Forces 39
2.6 Position-Dependent Forces 45
2.7 Numerical Solutions Of Differential Equations 48
2.8 Chapter Summary 54
2.9 End-Of-Chapter Problems 55
Chapter 3 Motion in Two and Three Dimensions 61
3.1 Position, Velocity, And Acceleration In Cartesian Coordinate Systems 61
3.2 Vector Products 69
3.2.1 The Dot Product 69
3.2.2 The Cross Product 71
3.3 Position, Velocity, And Acceleration In Non-Cartesian Coordinate Systems 75
3.3.1 Polar Coordinates 75
3.3.2 Position, Velocity, and Acceleration in Cylindrical Coordinates 82
3.3.3 Position, Velocity, and Acceleration in Spherical Coordinates 84
3.4 The Gradient, Divergence, And Curl 86
3.4.1 The Gradient 86
3.4.2 The Divergence 92
3.4.3 The Curl 94
3.4.4 Second Derivatives with the Del Operator 96
3.5 Chapter Summary 97
3.6 End-Of-Chapter Problems 99
Chapter 4 Momentum, Angular Momentum, and Multiparticle Systems 107
4.1 Conservation Of Momentum And Newton's Third Law 107
4.2 Rockets 111
4.3 Center Of Mass 113
4.4 Numerical Integration And The Center Of Mass 118
4.4.1 Trapezoidal Rule 118
4.4.2 Simpson's Rule 119
4.5 Momentum Of A System Of Multiple Particles 123
4.6 Angular Momentum Of A Single Particle 125
4.7 Angular Momentum Of Multiple Particles 126
4.8 Chapter Summary 129
4.9 End-Of-Chapter Problems 131
Chapter 5 Energy 135
5.1 Work And Energy In One-Dimensional Systems 135
5.2 Potential Energy And Equilibrium Points In One-Dimensional Systems 139
5.3 Work And Line Integrals 146
5.4 The Work-Kinetic Energy Theorem, Revisited 149
5.5 Conservative Forces And Potential Energy 150
5.6 ENERGY AND MULTIPARTICLE SYSTEMS 154
5.7 CHAPTER SUMMARY 156
5.8 END-OF-CHAPTER PROBLEMS 157
Chapter 6 Harmonic Oscillations 163
6.1 Differential Equations 163
6.2 THE SIMPLE HARMONIC OSCILLATOR 164
6.2.1 The Equation of Motion of the Simple Harmonic Oscillator 165
6.2.2 Potential and Kinetic Energy in Simple Harmonic Motion 167
6.2.3 The Simple Plane Pendulum as an Example of a Harmonic Oscillator 168
6.3 Numerical Solutions Using The Euler Method For Harmonic Oscillations 170
6.4 Damped Harmonic Oscillator 172
6.4.1 Overdamped Oscillations 173
6.4.2 Underdamped Oscillation 174
6.4.3 Critically Damped Oscillations 176
6.5 Energy In Damped Harmonic Motion 177
6.6 Forced Harmonic Oscillator 179
6.7 Energy Resonance And The Quality Factor For Driven Oscillations 185
6.8 Electrical Circuits 188
6.9 Principle Of Superposition And Fourier Series 191
6.9.1 The Principle of Superposition 191
6.9.2 Fourier Series 192
6.9.3 Example of Superposition Principle and Fourier Series 195
6.10 Phase Space 198
6.11 Chapter Summary 200
6.12 End-Of-Chapter Problems 202
Chapter 7 The Calculus of Variations 209
7.1 The Motivation For Learning The Calculus Of Variations 209
7.2 The Shortest Distance Between Two Points-Setting Up The Calculus Of Variations 210
7.3 The First Form Of The Euler Equation 212
7.4 The Second Form Of The Euler Equation 215
7.5 Some Examples Of Problems Solved Using The Calculus Of Variations 216
7.5.1 The Brachistochrone Problem 216
7.5.2 Geodesies 219
7.5.3 Minimum Surface of Revolution 220
7.6 Multiple Dependent Variables 223
7.7 Chapter Summary 225
7.8 End-Of-Chapter Problems 225
Chapter 8 Lagrangian and Hamiltonian Dynamics 229
8.1 An Introduction To The Lagrangian 230
8.2 Generalized Coordinates And Degrees Of Freedom 231
8.3 Hamilton's Principle 233
8.4 Some Examples Of Lagrangian Dynamics 235
8.5 Numerical Solutions To Ode's Using The Fourth-Order Runge-Kutta Method 243
8.6 Constraint Forces And Lagrange's Equation With Undetermined Multipliers 248
8.7 Conservation Theorems And The Lagrangian 254
8.7.1 Conservation of Momentum 254
8.7.2 Conservation of Energy 256
8.8 Hamiltonian Dynamics 258
8.9 Additional Explorations Into The Hamiltonian 263
8.10 Chapter Summary 266
8.11 End-Of-Chapter Problems 266
Chapter 9 Central Forces and Planetary Motion 273
9.1 Central Forces 273
9.1.1 Central Forces and the Conservation of Energy 274
9.1.2 Central Forces and the Conservation of Angular Momentum 275
9.2 The Two-Body Problem 276
9.3 Equations Of Motion For The Two-Body Problem 279
9.4 Planetary Motion And Kepler's First Law 284
9.5 Orbits In A Central Force Field 285
9.6 Kepler's Laws Of Planetary Motion 287
9.6.1 Kepler's First Law 288
9.6.2 Kepler's Second Law 292
9.6.3 Kepler's Third Law 295
9.7 The Planar Circular Restricted Three-Body Problem 297
9.8 Chapter Summary 302
9.9 End-Of-Chapter Problems 304
Chapter 10 Motion in Noninertial Reference Frames 311
10.1 Motion In A Nonrotating Accelerating Reference Frame 311
10.2 Angular Velocity As A Vector 313
10.3 Time Derivatives Of Vectors In Rotating Coordinate Frames 316
10.4 Newton's Second Law In A Rotating Frame 318
10.4.1 The Centrifugal Force 320
10.4.2 The Coriolis Force 323
10.5 Foucault Pendulum 326
10.6 Projectile Motion In A Noninertial Frame 329
10.7 Chapter Summary 331
10.8 End-Of-Chapter Problems 331
Chapter 11 Rigid Body Motion 335
11.1 Rotational Motion Of Particles Around A Fixed Axis 335
11.2 Review Of Rotational Properties For A System Of Particles 338
11.2.1 The Center of Mass 339
11.2.2 Momentum of a System of Particles 340
11.2.3 Angular Momentum of a System of Particles 340
11.2.4 Work and Kinetic Energy for a System of Particles 341
11.3 The Moment Of Inertia Tensor 341
11.4 Kinetic Energy And The Inertia Tensor 346
11.5 Inertia Tensor In Different Coordinate Systems-The Parallel Axis Theorem 348
11.6 Principal Axes Of Rotation 351
11.7 Precession Of A Symmetric Spinning Top With One Point Fixed And Experiencing A Weak Torque 355
11.8 Rigid Body Motion In Three Dimensions And Euler's Equations 357
11.9 The Force-Free Symmetric Top 359
11.10 Chapter Summary 361
11.11 End-Of-Chapter Problems 363
Chapter 12 Coupled Oscillations 373
12.1 Coupled Oscillations Of A Two-Mass Three-Spring System 373
12.1.1 The Equations of Motion-Numerical Solution 373
12.1.2 Equal Masses and Identical Springs: The Normal Modes 375
12.1.3 The General Case: Linear Combination of Normal Modes 378
12.2 Normal Mode Analysis Of The Two-Mass Three-Spring System 381
12.2.1 Equal Masses and Identical Springs-Analytical Solution 381
12.2.2 Solving the Two-Mass and Three-Spring System as an Eigenvalue Problem 384
12.3 The Double Pendulum 387
12.3.1 The Lagrangian and Equations of Motion-Numerical Solutions 387
12.3.2 Identical Masses and Lengths-Analytical Solutions 389
12.3.3 The Double Pendulum as an Eigenvector/Eigenvalue Problem 391
12.4 General Theory Of Small Oscillations And Normal Coordinates 392
12.4.1 The Lagrangian for Small Oscillations Around an Equilibrium Position 392
12.4.2 The Equations of Motion for Small Oscillations Around an Equilibrium Point 394
12.4.3 Normal Coordinates 396
12.5 Chapter Summary 397
12.6 End-Of-Chapter Problems 399
Chapter 13 A Nonlinear Systems 407
13.1 Linear Vs. Nonlinear Systems 407
13.2 The Damped Harmonic Oscillator, Revisited 409
13.3 Fixed Points And Phase Portraits 412
13.3.1 The Simple Plane Pendulum, Revisited 421
13.3.2 The Double-Well Potential, Revisited 423
13.3.3 Damped Double-Well 424
13.3.4 Bifurcations of Fixed Points 429
13.4 Limit Cycles 430
13.4.1 The Duffing Equation 430
13.4.2 Limit Cycles and Period Doubling Bifurcations 431
13.5 Chaos 434
13.5.1 Chaos and Initial Conditions 435
13.5.2 Lyapunov Exponents 437
13.6 A Final Word On Nonlinear Systems 437
13.7 Chapter Summary 438
13.8 End-Of-Chapter Problems 439
Bibliography 445
Index 447
