Classical mechanics

Cover......Page 1
Classical Mechanics......Page 6
Copyright......Page 7
Contents......Page 8
Preface......Page 14
1.1 Classical Mechanics......Page 20
1.2 Space and Time......Page 21
1.3 Mass and Force......Page 26
1.4 Newton\'s First and Second Laws; Inertial Frames......Page 30
1.5 The Third Law and Conservation of Momentum......Page 34
1.6 Newton\'s Second Law in Cartesian Coordinates......Page 40
1.7 Two-Dimensional Polar Coordinates......Page 43
Principal Definitions and Equations of Chapter 1......Page 50
Problems for Chapter 1......Page 51
2.1 Air Resistance......Page 60
2.2 Linear Air Resistance......Page 63
2.3 Trajectory and Range in a Linear Medium......Page 71
2.4 Quadratic Air Resistance......Page 74
2.5 Motion of a Charge in a Uniform Magnetic Field......Page 82
2.6 Complex Exponentials......Page 85
2.7 Solution for the Charge in a B Field......Page 87
Principal Definitions and Equations of Chapter 2......Page 88
Problems for Chapter 2......Page 89
3.1 Conservation of Momentum......Page 100
3.2 Rockets......Page 102
3.3 The Center of Mass......Page 104
3.4 Angular Momentum for a Single Particle......Page 107
3.5 Angular Momentum for Several Particles......Page 110
Principal Definitions and Equations of Chapter 3......Page 115
Problems for Chapter 3......Page 116
4.1 Kinetic Energy and Work......Page 122
4.2 Potential Energy and Conservative Forces......Page 126
4.3 Force as the Gradient of Potential Energy......Page 133
4.4 The Second Condition that F be Conservative......Page 135
4.5 Time-Dependent Potential Energy......Page 138
4.6 Energy for Linear One-Dimensional Systems......Page 140
4.7 Curvilinear One-Dimensional Systems......Page 146
4.8 Central Forces......Page 150
4.9 Energy of Interaction of Two Particles......Page 155
4.10 The Energy of a Multiparticle System......Page 161
Principal Definitions and Equations of Chapter 4......Page 165
Problems for Chapter 4......Page 167
5.1 Hooke\'s Law......Page 178
5.2 Simple Harmonic Motion......Page 180
5.3 Two-Dimensional Oscillators......Page 187
5.4 Damped Oscillations......Page 190
5.5 Driven Damped Oscillations......Page 196
5.6 Resonance......Page 204
5.7 Fourier Series *......Page 209
5.8 Fourier Series Solution for the Driven Oscillator *......Page 214
5.9 The RMS Displacement; Parseval\'s Theorem *......Page 220
Principal Definitions and Equations of Chapter 5......Page 222
Problems for Chapter 5......Page 224
CHAPTER 6 Calculus of Variations......Page 232
6.1 Two Examples......Page 233
6.2 The Euler-Lagrange Equation......Page 235
6.3 Applications of the Euler-Lagrange Equation......Page 238
6.4 More than Two Variables......Page 243
Problems for Chapter 6......Page 247
CHAPTER 7 Lagrange\'s Equations......Page 254
7.1 Lagrange\'s Equations for Unconstrained Motion......Page 255
7.2 Constrained Systems; an Example......Page 262
7.3 Constrained Systems in General......Page 264
7.4 Proof of Lagrange\'s Equations with Constraints......Page 267
7.5 Examples of Lagrange\'s Equations......Page 271
7.6 Generalized Momenta and Ignorable Coordinates......Page 283
7.7 Conclusion......Page 284
7.8 More about Conservation Laws *......Page 285
7.9 Lagrange\'s Equations for Magnetic Forces *......Page 289
7.10 Lagrange Multipliers and Constraint Forces *......Page 292
Principal Definitions and Equations of Chapter 7......Page 297
Problems for Chapter 7......Page 298
8.1 The Problem......Page 310
8.2 CM and Relative Coordinates; Reduced Mass......Page 312
8.3 The Equations of Motion......Page 314
8.4 The Equivalent One-Dimensional Problem 3QO 8.5 The Equation of the Orbit......Page 322
8.6 The Kepler Orbits......Page 325
8.7 The Unbounded Kepler Orbits......Page 330
8.8 Changes of Orbit......Page 332
Principal Definitions and Equations of Chapter 8......Page 336
Problems for Chapter 8......Page 337
9.1 Acceleration without Rotation......Page 344
9.2 The Tides......Page 347
9.3 The Angular Velocity Vector......Page 353
9.4 Time Derivatives in a Rotating Frame......Page 356
9.5 Newton\'s Second Law in a Rotating Frame......Page 359
9.6 The Centrifugal Force......Page 361
9.7 The Coriolis Force......Page 365
9.8 Free Fall and the Coriolis Force......Page 368
9.9 The Foucault Pendulum......Page 371
9.10 Coriolis Force and Coriolis Acceleration......Page 375
Principal Definitions and Equations of Chapter 9......Page 376
Problems for Chapter 9......Page 377
10.1 Properties of the Center of Mass......Page 384
10.2 Rotation about a Fixed Axis......Page 389
10.3 Rotation about Any Axis; the Inertia Tensor......Page 395
10.4 Principal Axes of Inertia......Page 404
10.5 Finding the Principal Axes; Eigenvalue Equations......Page 406
10.6 Precession of a Top due to a Weak Torque......Page 409
10.7 Euler\'s Equations......Page 411
10.8 Euler\'s Equations with Zero Torque......Page 414
10.9 Euler Angles *......Page 418
10.10 Motion of a Spinning Top *......Page 420
Principal Definitions and Equations of Chapter 10......Page 424
Problems for Chapter 10......Page 425
11.1 Two Masses and Three Springs......Page 434
11.2 Identical Springs and Equal Masses......Page 438
11.3 Two Weakly Coupled Oscillators......Page 443
11.4 Lagrangian Approach: The Double Pendulum......Page 447
11.5 The General Case......Page 453
11.6 Three Coupled Pendulums......Page 458
11. 7 Normal Coordinates *......Page 461
Principal Definitions and Equations of Chapter 11......Page 464
Problems for Chapter 11......Page 465
CHAPTER 12 Nonlinear Mechanics and Chaos......Page 474
12.1 Linearity and Nonlinearity......Page 475
12.2 The Driven Damped Pendulum DDP......Page 479
12.3 Some Expected Features of the DDP......Page 480
12.4 The DDP: Approach to Chaos......Page 484
12.5 Chaos and Sensitivity to Initial Conditions......Page 493
12.6 Bifurcation Diagrams......Page 500
12.7 State-Space Orbits......Page 504
12.8 Poincare Sections......Page 512
12.9 The Logistic Map......Page 515
Principal Definitions and Equations of Chapter 12......Page 530
Problems for Chapter 12......Page 531
CHAPTER 13 Hamiltonian Mechanics......Page 538
13.1 The Basic Variables......Page 539
13.2 Hamilton\'s Equations for One-Dimensional Systems......Page 541
13.3 Hamilton\'s Equations in Several Dimensions......Page 545
13.4 Ignorable Coordinates......Page 552
13.5 Lagrange\'s Equations vs. Hamilton\'s Equations......Page 553
13.6 Phase-Space Orbits......Page 555
13.7 Liouville\'s Theorem *......Page 560
Problems for Chapter 13......Page 567
CHAPTER 14 Collision Theory......Page 574
14.1 The Scattering Angle and Impact Parameter......Page 575
14.2 The Collision Cross Section......Page 577
14.3 Generalizations of the Cross Section......Page 580
14.4 The Differential Scattering Cross Section......Page 585
14.5 Calculating the Differential Cross Section......Page 589
14.6 Rutherford Scattering......Page 591
14.7 Cross Sections in Various Frames *......Page 596
14.8 Relation of the CM and Lab Scattering Angles *......Page 599
Principal Definitions and Equations of Chapter 14......Page 603
Problems for Chapter 14......Page 604
CHAPTER 15 Special Relativity......Page 612
15.2 Galilean Relativity......Page 613
15.3 The Postulates of Special Relativity......Page 618
15.4 The Relativity of Time; Time Dilation......Page 620
15.5 Length Contraction......Page 625
15.6 The Lorentz Transformation......Page 627
15.7 The Relativistic Velocity-Addition Formula......Page 632
15.8 Four-Dimensional Space-Time; Four-Vectors......Page 634
15.9 The Invariant Scalar Product......Page 640
15.10 The Light Cone......Page 642
15.11 The Quotient Rule and Doppler Effect......Page 647
15.12 Mass, Four-Velocity, and Four-Momentum......Page 650
15.13 Energy, the Fourth Component of Momentum......Page 655
15.14 Collisions......Page 661
15.15 Force in Relativity......Page 666
15.16 Massless Particles; the Photon......Page 669
15.17 Tensors *......Page 673
15.18 Electrodynamics and Relativity......Page 677
Principal Definitions and Equations of Chapter 15......Page 681
Problems for Chapter 15......Page 683
CHAPTER 16 Continuum Mechanics......Page 698
16.1 Transverse Motion of a Taut String......Page 699
16.2 The Wave Equation......Page 702
16.3 Boundary Conditions; Waves on a Finite String *......Page 705
16.4 The Three-Dimensional Wave Equation......Page 711
16.5 Volume and Surface Forces......Page 714
16.6 Stress and Strain: The Elastic Moduli......Page 718
16.7 The Stress Tensor......Page 721
16.8 The Strain Tensor for a Solid......Page 726
16.9 Relation between Stress and Strain: Hooke\'s Law......Page 732
16.10 The Equation of Motion for an Elastic Solid......Page 735
16.11 Longitudinal and Transverse Waves in a Solid......Page 738
16.12 Fluids: Description of the Motion *......Page 740
16.13 Waves in a Fluid *......Page 744
Principal Definitions and Equations of Chapter 16......Page 747
Problems for Chapter 16......Page 749
A.1 Diagonalizing a Single Matrix......Page 756
A.2 Simultaneous Diagonalization of Two Matrices......Page 760
Further Reading......Page 764
Answers for Odd-Numbered Problems......Page 766
Index......Page 794
Endpapers......Page 805