Classical Dynamics of Particles and Systems

Preface ......Page v
Contents ......Page xi
1.1 Introduction ......Page 1
1.2 Concept of a Scalar ......Page 2
1.3 Coordinate Transformations ......Page 3
1.4 Properties of Rotation Matrices ......Page 6
1.5 Matrix Operations ......Page 9
1.6 Further Definitions ......Page 12
1.7 Geometrical Significance of Transformation Matrices ......Page 14
1.9 Elementary Scalar and Vector Operations ......Page 20
1.10 Scalar Product of Two Vectors ......Page 21
1.11 Unit Vectors ......Page 23
1.12 Vector Product of Two Vectors ......Page 25
1.13 Differentiation of a Vector with Respect to a Scalar ......Page 29
1.14 Examples of Derivatives—Velocity and Acceleration ......Page 30
1.15 Angular Velocity ......Page 34
1.16 Gradient Operator ......Page 37
1.17 Integration of Vectors ......Page 40
Problems ......Page 43
2.1 Introduction ......Page 48
2.2 Newton's Laws ......Page 49
2.3 Frames of Reference ......Page 53
2.4 The Equation of Motion for a Particle ......Page 55
2.5 Conservation Theorems ......Page 76
2.6 Energy ......Page 82
2.7 Limitations of Newtonian Mechanics ......Page 88
Problems ......Page 90
3.1 Introduction ......Page 99
3.2 Simple Harmonic Oscillator ......Page 100
3.3 Harmonic Oscillations in Two Dimensions ......Page 104
3.4 Phase Diagrams ......Page 106
3.5 Damped Oscillations ......Page 108
3.6 Sinusoidal Driving Forces ......Page 117
3.7 Physical Systems ......Page 123
3.8 Principle of Superposition—^Fourier Series ......Page 126
3.9 The Response of Linear Oscillators to Impulsive Forcing Functions (Optional) ......Page 129
Problems ......Page 138
4.1 Introduction ......Page 144
4.2 Nonlinear Oscillations ......Page 146
4.3 Phase Diagrams for Nonlinear Systems ......Page 150
4.4 Plane Pendulum ......Page 155
4.5 Jumps, Hysteresis, and Phase Lags ......Page 160
4.6 Chaos in a Pendulum ......Page 163
4.7 Mapping ......Page 169
4.8 Chaos Identification ......Page 174
Problems ......Page 178
5.1 Introduction ......Page 182
5.2 Gravitational Potential ......Page 184
5.3 Lines of Force and Equipotential Surfaces ......Page 194
5.4 When Is the Potential Concept Useful ......Page 195
5.5 Ocean Tides ......Page 198
Problems ......Page 204
6.2 Statement of the Problem ......Page 207
6.3 Euler's Equation ......Page 210
6.4 The "Second Form" of the Euler Equation ......Page 216
6.5 Functions with Several Dependent Variables ......Page 218
6.6 Euler Equations When Auxiliary Conditions Are Imposed ......Page 219
6.7 The 8 Notation ......Page 224
Problems ......Page 226
7.1 Introduction ......Page 228
7.2 Hamilton's Principle ......Page 229
7.3 Generalized Coordinates ......Page 233
7.4 Lagrange's Equations of Motion in Generalized Coordinates ......Page 237
7.5 Lagrange's Equations with Undetermined Multipliers ......Page 248
7.6 Equivalence of Lagrange's and Newton's Equations ......Page 254
7.7 Essence of Lagrangian Dynamics ......Page 257
7.8 A Theorem Concerning the Kinetic Energy ......Page 258
7.9 Conservation Theorems Revisited ......Page 260
7.10 Canonical Equations of Motion—Hamiltonian Dynamics ......Page 265
7.11 Some Comments Regarding Dynamical Variables and Variational Calculations in Physics ......Page 272
7.12 Phase Space and Liouville's Theorem (Optional) ......Page 274
7.13 Virial Theorem (Optional) ......Page 277
Problems ......Page 280
8.2 Reduced Mass ......Page 287
8.3 Conservation Theorems—First Integrals of the Motion ......Page 289
8.4 Equations of Motion ......Page 291
8.5 Orbits in a Central Field ......Page 295
8.6 Centrifugal Energy and the Effective Potential ......Page 296
8.7 Planetary Motion—Kepler's Problem ......Page 300
8.8 Orbital Dynamics ......Page 305
8.9 Apsidal Angles and Precession (Optional) ......Page 312
8.10 Stability of Circular Orbits (Optional) ......Page 316
Problems ......Page 323
9.1 Introduction ......Page 328
9.2 Center of Mass ......Page 329
9.3 Linear Momentum of the System ......Page 331
9.4 Angular Momentum of the System ......Page 336
9.5 Energy of the System ......Page 339
9.6 Elastic Collisions of Two Particles ......Page 345
9.7 Kinematics of Elastic Collisions ......Page 352
9.8 Inelastic Collisions ......Page 358
9.9 Scattering Cross Sections ......Page 363
9.10 Rutherford Scattering Formula ......Page 369
9.11 Rocket Motion ......Page 371
Problems ......Page 378
10.1 Introduction ......Page 387
10.2 Rotating Coordinate Systems ......Page 388
10.3 Centrifugal and Coriolis Forces ......Page 391
10.4 Motion Relative to the Earth ......Page 395
Problems ......Page 408
11.1 Introduction ......Page 411
11.2 Simple Planar Motion ......Page 412
11.3 Inertia Tensor ......Page 415
11.4 Angular Momentum ......Page 419
11.5 Principal Axes of Inertia ......Page 424
11.6 Moments of Inertia for Different Body Coordinate Systems ......Page 428
11.7 Further Properties of the Inertia Tensor ......Page 433
11.8 Eulerian Angles ......Page 440
11.9 Euler's Equations for a Rigid Body ......Page 444
11.10 Force-Free Motion of a Symmetric Top ......Page 448
11.11 Motion of a Symmetric Top with One Point Fixed ......Page 454
11.12 Stability of Rigid-Body Rotations ......Page 460
Problems ......Page 463
12.1 Introduction ......Page 468
12.2 Two Coupled Harmonic Oscillators ......Page 469
12.3 Weak Coupling ......Page 473
12.4 General Problem of Coupled Oscillations ......Page 475
12.5 Orthogonality of the Eigenvectors (Optional) ......Page 481
12.6 Normal Coordinates ......Page 483
12.7 Molecular Vibrations ......Page 490
12.8 Three Linearly Coupled Plane Pendula—an Example of Degeneracy ......Page 495
12.9 The Loaded String ......Page 498
Problems ......Page 507
13.1 Introduction ......Page 512
13.2 Continuous String as a Limiting Case of the Loaded String ......Page 513
13.3 Energy of a Vibrating String ......Page 516
13.4 Wave Equation ......Page 520
13.5 Forced and Damped Motion ......Page 522
13.6 General Solutions of the Wave Equation ......Page 524
13.7 Separation of the Wave Equation ......Page 527
13.8 Phase Velocity, Dispersion, and Attenuation ......Page 533
13.9 Group Velocity and Wave Packets ......Page 538
Problems ......Page 542
14.1 Introduction ......Page 546
14.2 Galilean Invariance ......Page 547
14.3 Lorentz Transformation ......Page 548
14.4 Experimental Verification of the Special Theory ......Page 555
14.5 Relativistic Doppler Effect ......Page 558
14.6 Twin Paradox ......Page 561
14.7 Relativistic Momentum ......Page 562
14.8 Energy ......Page 566
14.9 Spacetime and Four-Vectors ......Page 569
14.10 Lagrangian Function in Special Relativity ......Page 578
14.11 Relativistic Kinematics ......Page 579
Problems ......Page 583
A. Taylor's Theorem ......Page 589
Problems ......Page 593
B.1 Elliptic Integrals of the First Kind ......Page 594
B.3 Elliptic Integrals of the Third Kind ......Page 595
Problems ......Page 598
C.1 Linear Homogeneous Equations ......Page 599
C.2 Linear Inhomogeneous Equations ......Page 603
Problems ......Page 606
D.1 Binomial Expansion ......Page 608
D.2 Trigonometric Relations ......Page 609
D.4 Exponential and Logarithmic Series ......Page 610
D.6 Hyperbolic Functions ......Page 611
Problems ......Page 612
E.1 Algebraic Functions ......Page 613
E.2 Trigonometric Functions ......Page 614
E.3 Gamma Functions ......Page 615
F.2 Cylindrical Coordinates ......Page 617
F.3 Spherical Coordinates ......Page 619
G. A "Proof" of the Relation Σ_μ x_μ^2 = Σ_μ x'_μ^2 ......Page 621
H. Numerical Solution for Example 2.7 ......Page 623
Selected References ......Page 626
Bibliography ......Page 628
Answers to Even-Numbered Problems ......Page 633
Index ......Page 643