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CLASSICAL MECHANICS BY TAIL L.CHOW PDF FREE DOWNLOAD
CONTENTS
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Chapter 1 Kinematics: Describing the Motion ............................................................................. 1
1.1 Introduction ....................................................................................................... 1
1.2 Space, Time, and Coordinate Systems .............................................................. 1
1.3 Change of Coordinate System (Transformation of Components of a
Vector) ........................................................................................................ 3
1.4 Displacement Vector.......................................................................................... 8
1.5 Speed and Velocity ............................................................................................ 8
1.6 Acceleration..................................................................................................... 10
1.6.1 Tangential and Normal Acceleration.................................................. 11
1.7 Velocity and Acceleration in Polar Coordinates ............................................. 14
1.7.1 Plane Polar Coordinates (r, θ) ............................................................ 14
1.7.2 Cylindrical Coordinates ( ρ, ϕ, z) ........................................................ 15
1.7.3 Spherical Coordinates (r , θ, ϕ) .......................................................... 16
1.8 Angular Velocity and Angular Acceleration................................................... 18
1.9 Infiitesimal Rotations and the Angular Velocity Vector ............................... 19
Chapter 2 Newtonian Mechanics ................................................................................................ 25
2.1 The First Law of Motion (Law of Inertia) ....................................................... 25
2.1.1 Inertial Frames of Reference.............................................................. 26
2.2 The Second Law of Motion; the Equations of Motion.................................... 27
2.2.1 The Concept of Force ......................................................................... 28
2.3 The Third Law of Motion................................................................................ 32
2.3.1 The Concept of Mass.......................................................................... 32
2.4 Galilean Transformations and Galilean Invariance ........................................ 34
2.5 Newton’s Laws of Rotational Motion .............................................................. 36
2.6 Work, Energy, and Conservation Laws ........................................................... 37
2.6.1 Work and Energy................................................................................ 38
2.6.2 Conservative Force and Potential Energy .......................................... 39
2.6.3 Conservation of Energy...................................................................... 40
2.6.4 Conservation of Momentum............................................................... 42
2.6.5 Conservation of Angular Momentum ................................................ 42
2.7 Systems of Particles......................................................................................... 46
2.7.1 Center of Mass.................................................................................... 46
2.7.2 Motion of CM..................................................................................... 48
2.7.3 Conservation Theorems ..................................................................... 49
References .................................................................................................................. 56
Chapter 3 Integration of Newton’s Equation of Motion.............................................................. 57
3.1 Introduction ..................................................................................................... 57
3.2 Motion Under Constant Force ......................................................................... 58
3.3 Force Is a Function of Time ............................................................................ 63
3.3.1 Impulsive Force and Green’s Function Method ................................. 66
3.4 Force Is a Function of Velocity ....................................................................... 67
3.4.1 Motion in a Uniform Magnetic Field ................................................. 71
3.4.2 Motion in Nearly Uniform Magnetic Field ........................................ 73
3.5 Force Is a Function of Position ........................................................................ 74
3.5.1 Bounded and Unbounded Motion ...................................................... 75
3.5.2 Stable and Unstable Equilibrium ....................................................... 76
3.5.3 Critical and Neutral Equilibrium ....................................................... 78
3.6 Time-Varying Mass System (Rocket System) ................................................. 79
Chapter 4 Lagrangian Formulation of Mechanics: Descriptions of Motion in
Confiuration Space ................................................................................................... 85
4.1 Generalized Coordinates and Constraints....................................................... 85
4.1.1 Generalized Coordinates.................................................................... 85
4.1.2 Degrees of Freedom ........................................................................... 85
4.1.3 Confiuration Space ........................................................................... 86
4.1.4 Constraints.......................................................................................... 86
4.1.4.1 Holonomic and Nonholonomic Constraints ....................... 86
4.1.4.2 Scleronomic and Rheonomic Constraints .......................... 88
4.2 Kinetic Energy in Generalized Coordinates ................................................... 88
4.3 Generalized Momentum.................................................................................. 90
4.4 Lagrangian Equations of Motion..................................................................... 91
4.4.1 Hamilton’s Principle........................................................................... 91
4.4.2 Lagrange’s Equations of Motion from Hamilton’s Principle ............. 92
4.5 Nonuniqueness of the Lagrangian................................................................. 102
4.6 Integrals of Motion and Conservation Laws ................................................. 104
4.6.1 Cyclic Coordinates and Conservation Theorems............................. 104
4.6.2 Symmetries and Conservation Laws ................................................ 106
4.6.2.1 Homogeneity of Time and Conservation of Energy ......... 106
4.6.2.2 Spatial Homogeneity and Momentum Conservation........ 107
4.6.2.3 Isotropy of Space and Angular Momentum
Conservation.................................................................. 108
4.6.2.4 Noether’s Theorem ........................................................... 110
4.7 Scale Invariance ............................................................................................ 111
4.8 Nonconservative Systems and Generalized Potential ................................... 112
4.9 Charged Particle in Electromagnetic Field.................................................... 112
4.10 Forces of Constraint and Lagrange’s Multipliers .......................................... 114
4.11 Lagrangian versus Newtonian Approach to Classical Mechanics ................ 119
Reference.................................................................................................................. 123
Chapter 5 Hamiltonian Formulation of Mechanics: Descriptions of Motion in Phase
Spaces .................................................................................................................. 125
5.1 The Hamiltonian of a Dynamic System........................................................ 125
5.1.1 Phase Space ...................................................................................... 126
5.2 Hamilton’s Equations of Motion ................................................................... 126
5.2.1 Hamilton’s Equations from Lagrange’s Equations........................... 126
5.2.2 Hamilton’s Equations from Hamilton’s Principle ............................ 128
1.1 Introduction ....................................................................................................... 1
1.2 Space, Time, and Coordinate Systems .............................................................. 1
1.3 Change of Coordinate System (Transformation of Components of a
Vector) ........................................................................................................ 3
1.4 Displacement Vector.......................................................................................... 8
1.5 Speed and Velocity ............................................................................................ 8
1.6 Acceleration..................................................................................................... 10
1.6.1 Tangential and Normal Acceleration.................................................. 11
1.7 Velocity and Acceleration in Polar Coordinates ............................................. 14
1.7.1 Plane Polar Coordinates (r, θ) ............................................................ 14
1.7.2 Cylindrical Coordinates ( ρ, ϕ, z) ........................................................ 15
1.7.3 Spherical Coordinates (r , θ, ϕ) .......................................................... 16
1.8 Angular Velocity and Angular Acceleration................................................... 18
1.9 Infiitesimal Rotations and the Angular Velocity Vector ............................... 19
Chapter 2 Newtonian Mechanics ................................................................................................ 25
2.1 The First Law of Motion (Law of Inertia) ....................................................... 25
2.1.1 Inertial Frames of Reference.............................................................. 26
2.2 The Second Law of Motion; the Equations of Motion.................................... 27
2.2.1 The Concept of Force ......................................................................... 28
2.3 The Third Law of Motion................................................................................ 32
2.3.1 The Concept of Mass.......................................................................... 32
2.4 Galilean Transformations and Galilean Invariance ........................................ 34
2.5 Newton’s Laws of Rotational Motion .............................................................. 36
2.6 Work, Energy, and Conservation Laws ........................................................... 37
2.6.1 Work and Energy................................................................................ 38
2.6.2 Conservative Force and Potential Energy .......................................... 39
2.6.3 Conservation of Energy...................................................................... 40
2.6.4 Conservation of Momentum............................................................... 42
2.6.5 Conservation of Angular Momentum ................................................ 42
2.7 Systems of Particles......................................................................................... 46
2.7.1 Center of Mass.................................................................................... 46
2.7.2 Motion of CM..................................................................................... 48
2.7.3 Conservation Theorems ..................................................................... 49
References .................................................................................................................. 56
Chapter 3 Integration of Newton’s Equation of Motion.............................................................. 57
3.1 Introduction ..................................................................................................... 57
3.2 Motion Under Constant Force ......................................................................... 58
3.3 Force Is a Function of Time ............................................................................ 63
3.3.1 Impulsive Force and Green’s Function Method ................................. 66
3.4 Force Is a Function of Velocity ....................................................................... 67
3.4.1 Motion in a Uniform Magnetic Field ................................................. 71
3.4.2 Motion in Nearly Uniform Magnetic Field ........................................ 73
3.5 Force Is a Function of Position ........................................................................ 74
3.5.1 Bounded and Unbounded Motion ...................................................... 75
3.5.2 Stable and Unstable Equilibrium ....................................................... 76
3.5.3 Critical and Neutral Equilibrium ....................................................... 78
3.6 Time-Varying Mass System (Rocket System) ................................................. 79
Chapter 4 Lagrangian Formulation of Mechanics: Descriptions of Motion in
Confiuration Space ................................................................................................... 85
4.1 Generalized Coordinates and Constraints....................................................... 85
4.1.1 Generalized Coordinates.................................................................... 85
4.1.2 Degrees of Freedom ........................................................................... 85
4.1.3 Confiuration Space ........................................................................... 86
4.1.4 Constraints.......................................................................................... 86
4.1.4.1 Holonomic and Nonholonomic Constraints ....................... 86
4.1.4.2 Scleronomic and Rheonomic Constraints .......................... 88
4.2 Kinetic Energy in Generalized Coordinates ................................................... 88
4.3 Generalized Momentum.................................................................................. 90
4.4 Lagrangian Equations of Motion..................................................................... 91
4.4.1 Hamilton’s Principle........................................................................... 91
4.4.2 Lagrange’s Equations of Motion from Hamilton’s Principle ............. 92
4.5 Nonuniqueness of the Lagrangian................................................................. 102
4.6 Integrals of Motion and Conservation Laws ................................................. 104
4.6.1 Cyclic Coordinates and Conservation Theorems............................. 104
4.6.2 Symmetries and Conservation Laws ................................................ 106
4.6.2.1 Homogeneity of Time and Conservation of Energy ......... 106
4.6.2.2 Spatial Homogeneity and Momentum Conservation........ 107
4.6.2.3 Isotropy of Space and Angular Momentum
Conservation.................................................................. 108
4.6.2.4 Noether’s Theorem ........................................................... 110
4.7 Scale Invariance ............................................................................................ 111
4.8 Nonconservative Systems and Generalized Potential ................................... 112
4.9 Charged Particle in Electromagnetic Field.................................................... 112
4.10 Forces of Constraint and Lagrange’s Multipliers .......................................... 114
4.11 Lagrangian versus Newtonian Approach to Classical Mechanics ................ 119
Reference.................................................................................................................. 123
Chapter 5 Hamiltonian Formulation of Mechanics: Descriptions of Motion in Phase
Spaces .................................................................................................................. 125
5.1 The Hamiltonian of a Dynamic System........................................................ 125
5.1.1 Phase Space ...................................................................................... 126
5.2 Hamilton’s Equations of Motion ................................................................... 126
5.2.1 Hamilton’s Equations from Lagrange’s Equations........................... 126
5.2.2 Hamilton’s Equations from Hamilton’s Principle ............................ 128
5.3 Integrals of Motion and Conservation Theorems.......................................... 132
5.3.1 Energy Integrals ............................................................................... 132
5.3.2 Cyclic Coordinates and Integrals of Motion .................................... 132
5.3.3 Conservation Theorems of Momentum and Angular
Momentum ................................................................................ 133
5.4 Canonical Transformations ........................................................................... 135
5.5 Poisson Brackets ............................................................................................ 140
5.5.1 Fundamental Properties of Poisson Brackets................................... 141
5.5.2 Fundamental Poisson Brackets......................................................... 141
5.5.3 Poisson Brackets and Integrals of Motion........................................ 141
5.5.4Equations of Motion in Poisson Bracket Form ................................ 144
5.5.5 Canonical Invariance of Poisson Brackets ....................................... 144
5.6 Poisson Brackets and Quantum Mechanics .................................................. 145
5.7 Phase Space and Liouville’s Theorem........................................................... 147
5.8 Time Reversal in Mechanics (Optional)........................................................ 150
5.9 Passage from Hamiltonian to Lagrangian..................................................... 151
References ................................................................................................................ 154
Chapter 6 Motion Under a Central Force.................................................................................. 155
6.1 Two-Body Problem and Reduced Mass ........................................................ 155
6.2 General Properties of Central Force Motion................................................. 157
6.3 Effective Potential and Classifiation of Orbits ............................................ 159
6.4 General Solutions of Central Force Problem................................................. 163
6.4.1 Energy Method................................................................................. 163
6.4.2 Lagrangian Analysis ........................................................................ 164
6.5 Inverse Square Law of Force......................................................................... 167
6.6 Kepler’s Three Laws of Planetary Motion .................................................... 172
6.7 Applications of Central Force Motion........................................................... 174
6.7.1 Satellites and Spacecraft .................................................................. 174
6.7.2 Communication Satellites ................................................................ 178
6.7.3 Flyby Missions to Outer Planets ...................................................... 179
6.8 Newton’s Law of Gravity from Kepler’s Laws .............................................. 182
6.9 Stability of Circular Orbits (Optional) .......................................................... 183
6.10 Apsides and Advance of Perihelion (Optional) ............................................. 188
6.10.1 Advance of Perihelion and Inverse-Square Force............................ 189
6.10.2 Method of Perturbation Expansion .................................................. 190
6.11 Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional) .................... 192
References ................................................................................................................ 198
Chapter 7 Harmonic Oscillator................................................................................................. 199
7.1 Simple Harmonic Oscillator.......................................................................... 199
7.1.1 Motion of Mass m on the End of a Spring ....................................... 199
7.1.2 The Bob of Simple Pendulum Swinging through a Small Arc ........ 200
7.1.3 Solution of Equation of Motion of SHM.......................................... 201
7.1.4 Kinetic, Potential, Total, and Average Energies of Harmonic
Oscillator.......................................................................................... 203
7.2 Adiabatic Invariants and Quantum Condition............................................... 206
7.3 Damped Harmonic Oscillator ....................................................................... 209
7.4 Phase Diagram for Damped Oscillator ......................................................... 218
7.5 Relaxation Time Phenomena......................................................................... 220
7.6 Forced Oscillations without Damping........................................................... 220
7.6.1 Periodic Driving Force..................................................................... 221
7.6.2 Arbitrary Driving Forces ................................................................. 223
7.7 Forced Oscillations with Damping................................................................ 225
7.7.1 Resonance......................................................................................... 227
7.7.2 Power Absorption............................................................................. 231
7.8 Oscillator Under Arbitrary Periodic Force.................................................... 235
7.8.1 Fourier’s Series Solution .................................................................. 236
7.9 Vibration Isolation ......................................................................................... 239
7.10 Parametric Excitation .................................................................................... 241
Chapter 8 Coupled Oscillations and Normal Coordinates........................................................ 249
8.1 Coupled Pendulum......................................................................................... 249
8.1.1 Normal Coordinates ......................................................................... 251
8.2 Coupled Oscillators and Normal Modes: General Analytic Approach ........... 254
8.2.1 The Equation of Motion of a Coupled System................................. 254
8.2.2 Normal Modes of Oscillation........................................................... 255
8.2.3 Orthogonality of Eigenvectors ......................................................... 257
8.2.4 Normal Coordinates ......................................................................... 259
8.3 Forced Oscillations of Coupled Oscillators................................................... 264
8.4 Coupled Electric Circuits .............................................................................. 266
Chapter 9 Nonlinear Oscillations ............................................................................................. 273
9.1 Qualitative Analysis: Energy and Phase Diagrams....................................... 274
9.2 Elliptical Integrals and Nonlinear Oscillations............................................. 280
9.3 Fourier Series Expansions ............................................................................. 283
9.3.1 Symmetrical Potential: V(x) = V(−x)................................................. 284
9.3.2 Asymmetrical Potential: V(−x) = −V(x) ............................................ 287
9.4 The Method of Perturbation .......................................................................... 288
9.4.1 Bogoliuboff–Kryloff Procedure and Removal of Secular Terms.........292
9.5 Ritz Method................................................................................................... 295
9.6 Method of Successive Approximation........................................................... 297
9.7 Multiple Solutions and Jumps........................................................................ 299
9.8 Chaotic Oscillations ...................................................................................... 301
9.8.1 Some Helpful Tools for an Understanding of Chaos........................ 301
9.8.2 Conditions for Chaos........................................................................ 306
9.8.3 Routes to Chaos................................................................................ 307
9.8.4 Lyapunov Exponentials .................................................................... 308
References ................................................................................................................ 312
Chapter 10 Collisions and Scatterings ........................................................................................ 313
10.1 Direct Impact of Two Particles...................................................................... 313
10.2 Scattering Cross Sections and Rutherford Scattering ................................... 318
10.2.1 Scattering Cross Sections................................................................. 319
10.2.2 Rutherford’s α-Particle Scattering Experiment................................ 320
10.2.3Cross Section Is Lorentz Invariant................................................... 324
10.3 Laboratory and Center-of-Mass Frames of Reference .................................. 324
10.4 Nuclear Sizes ................................................................................................. 328
10.5 Small-Angle Scattering (Optional)................................................................ 329
References ................................................................................................................ 336
Chapter 11 Motion in Non-Inertial Systems............................................................................... 337
11.1 Accelerated Translational Coordinate System .............................................. 337
11.2 Dynamics in Rotating Coordinate System .................................................... 341
11.2.1 Centrifugal Force ............................................................................. 345
11.2.2 The Coriolis Force............................................................................ 349
11.2.2.1 Trade Winds and Circulation of Ocean Currents ............. 351
11.2.2.2 Weather Systems............................................................... 352
11.2.2.3 Hurricanes ........................................................................ 354
11.2.2.4 Bathtub Vortex and Earth Rotation .................................. 354
11.3 Motion of Particle Near the Surface of the Earth ......................................... 355
11.4 Foucault Pendulum ........................................................................................ 361
11.5 Larmor’s Theorem......................................................................................... 364
11.6 Classical Zeeman Effect................................................................................ 365
11.7 Principle of Equivalence................................................................................ 368
11.7.1 Principle of Equivalence and Gravitational Red Shift..................... 369
Chapter 12 Motion of Rigid Bodies ............................................................................................ 377
12.1 Independent Coordinates of Rigid Body ....................................................... 378
12.2 Eulerian Angles ............................................................................................. 379
12.3 Rate of Change of Vector .............................................................................. 382
12.4 Rotational Kinetic Energy and Angular Momentum.................................... 384
12.5 Inertia Tensor................................................................................................. 394
12.5.1 Diagonalization of a Symmetric Tensor........................................... 396
12.5.2 Moments and Products of Inertia..................................................... 397
12.5.3 Parallel-Axis Theorem ..................................................................... 398
12.5.4 Moments of Inertia about an Arbitrary Axis ................................... 401
12.5.5 Principal Axes of Inertia .................................................................. 403
12.6 Euler’s Equations of Motion.......................................................................... 407
12.7 Motion of a Torque-Free Symmetrical Top................................................... 409
12.8 Motion of Heavy Symmetrical Top with One Point Fixed............................ 414
12.8.1 Precession without Nutation............................................................. 417
12.8.2 Precession with Nutation.................................................................. 419
12.9 Stability of Rotational Motion....................................................................... 420
References ................................................................................................................ 425
Chapter 13 Theory of Special Relativity .................................................................................... 427
13.1 Historical Origin of Special Theory of Relativity......................................... 427
13.2 Michelson–Morley Experiment..................................................................... 430
13.3 Postulates of Special Theory of Relativity .................................................... 433
13.3.1 Time Is Not Absolute ....................................................................... 434
13.4 Lorentz Transformations ............................................................................... 434
13.4.1 Relativity of Simultaneity, Causality................................................ 437
13.4.2 Time Dilation, Relativity of Co-Locality......................................... 438
13.4.3 Length Contraction........................................................................... 439
13.4.4 Visual Apparent Shape of Rapidly Moving Object.......................... 441
13.4.5 Relativistic Velocity Addition .......................................................... 441
13.5 Doppler Effect ............................................................................................... 445
13.6 Relativistic Space–Time (Minkowski Space)................................................ 446
13.6.1 Four-Velocity and Four-Acceleration ............................................... 449
13.6.2 Four-Energy and Four-Momentum Vectors ..................................... 450
13.6.3 Particles of Zero Rest Mass.............................................................. 452
13.7 Equivalence of Mass and Energy .................................................................. 453
13.8 Conservation Laws of Energy and Momentum............................................. 459
13.9 Generalization of Newton’s Equation of Motion........................................... 459
13.9.1 Force Transformation ....................................................................... 461
13.10 Relativistic Lagrangian and Hamiltonian Functions..................................... 463
13.11 Relativistic Kinematics of Collisions ............................................................ 467
13.12 Collision Threshold Energies ........................................................................ 470
References ................................................................................................................ 474
Chapter 14 Newtonian Gravity and Newtonian Cosmology ...................................................... 475
14.1 Newton’s Law of Gravity............................................................................... 475
14.2 Gravitational Field and Gravitational Potential............................................. 477
14.3 Gravitational Field Equations: Poisson’s and Laplace’s Equations ............... 479
14.4 Gravitational Field and Potential of Extended Body..................................... 480
14.5 Tides .............................................................................................................. 481
14.6 General Theory of Relativity: Relativistic Theory of Gravitation ................ 487
14.6.1 Gravitational Shift of Spectral Lines (Gravitational Red Shift) ...........488
14.6.2 Bending of Light Beam .................................................................... 489
14.7 Introduction to Cosmology............................................................................ 491
14.8 Brief History of Cosmological Ideas ............................................................. 492
14.8.1 Newton and Infiite Universe .......................................................... 493
14.8.2 Newton’s Law of Gravity Predicts Nonstationary Universe ............ 493
14.8.3 An Infiite Steady Universe Is an Empty Universe ......................... 495
14.8.4 Olbers’ Paradox ................................................................................ 496
14.9 Discovery of Expansion of the Universe, Hubble’s Law ............................... 497
14.10 Big Bang ........................................................................................................ 499
14.10.1 Age of the Universe.......................................................................... 499
14.11 Formulating Dynamical Models of the Universe.......................................... 499
14.12 Cosmological Red Shift and Hubble Constant H.......................................... 503
14.13 Critical Mass Density and Future of the Universe ........................................ 504
14.13.1 Density Parameter Ω ........................................................................ 505
14.13.2 Deceleration Parameter q0................................................................ 505
14.13.3 An Accelerating Universe?............................................................... 507
14.14 Microwave Background Radiation ................................................................ 507
14.15 Dark Matter ................................................................................................... 511
Reference.................................................................................................................. 514
Chapter 15 Hamilton–Jacobi Theory of Dynamics .................................................................... 515
15.1 Canonical Transformation and H-J Equation................................................ 515
15.2 Action and Angle Variables........................................................................... 522
15.3 Infiitesimal Canonical Transformations and Time Development
Operator .................................................................................................... 527
15.4 H-J Theory and Wave Mechanics.................................................................. 530
Reference.................................................................................................................. 533
Chapter 16 Introduction to Lagrangian and Hamiltonian Formulations for Continuous
Systems and Classical Fields.................................................................................... 535
16.1 Vibration of Loaded String............................................................................ 535
16.2 Vibrating Strings and the Wave Equation ..................................................... 541
16.2.1 Wave Equation.................................................................................. 541
16.2.2 Separation of Variables .................................................................... 543
16.2.3 Wave Number and Phase Velocity ................................................... 543
16.2.4 Group Velocity and Wave Packets ................................................... 544
16.3 Continuous Systems and Classical Fields...................................................... 547
16.3.1 Lagrangian Formulation................................................................... 547
16.3.2 Hamiltonian Formulation................................................................. 550
16.3.3 Conservation Laws ........................................................................... 552
16.4 Scalar and Vector of Fields............................................................................ 553
16.4.1 Scalar Fields ..................................................................................... 553
16.4.2 Vector Fields..................................................................................... 554
Appendix 1: Vector Analysis and Ordinary Differential Equations....................................... 557
Appendix 2: D’Alembert’s Principle and Lagrange’s Equations ............................................ 587
Appendix 3: Derivation of Hamilton’s Principle from D’Alembert’s Principle..................... 595
Appendix 4: Noether’s Theorem................................................................................................. 599
Appendix 5: Conic Sections, Ellipse, Parabola, and Hyperbola ............................................. 605
5.3.1 Energy Integrals ............................................................................... 132
5.3.2 Cyclic Coordinates and Integrals of Motion .................................... 132
5.3.3 Conservation Theorems of Momentum and Angular
Momentum ................................................................................ 133
5.4 Canonical Transformations ........................................................................... 135
5.5 Poisson Brackets ............................................................................................ 140
5.5.1 Fundamental Properties of Poisson Brackets................................... 141
5.5.2 Fundamental Poisson Brackets......................................................... 141
5.5.3 Poisson Brackets and Integrals of Motion........................................ 141
5.5.4Equations of Motion in Poisson Bracket Form ................................ 144
5.5.5 Canonical Invariance of Poisson Brackets ....................................... 144
5.6 Poisson Brackets and Quantum Mechanics .................................................. 145
5.7 Phase Space and Liouville’s Theorem........................................................... 147
5.8 Time Reversal in Mechanics (Optional)........................................................ 150
5.9 Passage from Hamiltonian to Lagrangian..................................................... 151
References ................................................................................................................ 154
Chapter 6 Motion Under a Central Force.................................................................................. 155
6.1 Two-Body Problem and Reduced Mass ........................................................ 155
6.2 General Properties of Central Force Motion................................................. 157
6.3 Effective Potential and Classifiation of Orbits ............................................ 159
6.4 General Solutions of Central Force Problem................................................. 163
6.4.1 Energy Method................................................................................. 163
6.4.2 Lagrangian Analysis ........................................................................ 164
6.5 Inverse Square Law of Force......................................................................... 167
6.6 Kepler’s Three Laws of Planetary Motion .................................................... 172
6.7 Applications of Central Force Motion........................................................... 174
6.7.1 Satellites and Spacecraft .................................................................. 174
6.7.2 Communication Satellites ................................................................ 178
6.7.3 Flyby Missions to Outer Planets ...................................................... 179
6.8 Newton’s Law of Gravity from Kepler’s Laws .............................................. 182
6.9 Stability of Circular Orbits (Optional) .......................................................... 183
6.10 Apsides and Advance of Perihelion (Optional) ............................................. 188
6.10.1 Advance of Perihelion and Inverse-Square Force............................ 189
6.10.2 Method of Perturbation Expansion .................................................. 190
6.11 Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional) .................... 192
References ................................................................................................................ 198
Chapter 7 Harmonic Oscillator................................................................................................. 199
7.1 Simple Harmonic Oscillator.......................................................................... 199
7.1.1 Motion of Mass m on the End of a Spring ....................................... 199
7.1.2 The Bob of Simple Pendulum Swinging through a Small Arc ........ 200
7.1.3 Solution of Equation of Motion of SHM.......................................... 201
7.1.4 Kinetic, Potential, Total, and Average Energies of Harmonic
Oscillator.......................................................................................... 203
7.2 Adiabatic Invariants and Quantum Condition............................................... 206
7.3 Damped Harmonic Oscillator ....................................................................... 209
7.4 Phase Diagram for Damped Oscillator ......................................................... 218
7.5 Relaxation Time Phenomena......................................................................... 220
7.6 Forced Oscillations without Damping........................................................... 220
7.6.1 Periodic Driving Force..................................................................... 221
7.6.2 Arbitrary Driving Forces ................................................................. 223
7.7 Forced Oscillations with Damping................................................................ 225
7.7.1 Resonance......................................................................................... 227
7.7.2 Power Absorption............................................................................. 231
7.8 Oscillator Under Arbitrary Periodic Force.................................................... 235
7.8.1 Fourier’s Series Solution .................................................................. 236
7.9 Vibration Isolation ......................................................................................... 239
7.10 Parametric Excitation .................................................................................... 241
Chapter 8 Coupled Oscillations and Normal Coordinates........................................................ 249
8.1 Coupled Pendulum......................................................................................... 249
8.1.1 Normal Coordinates ......................................................................... 251
8.2 Coupled Oscillators and Normal Modes: General Analytic Approach ........... 254
8.2.1 The Equation of Motion of a Coupled System................................. 254
8.2.2 Normal Modes of Oscillation........................................................... 255
8.2.3 Orthogonality of Eigenvectors ......................................................... 257
8.2.4 Normal Coordinates ......................................................................... 259
8.3 Forced Oscillations of Coupled Oscillators................................................... 264
8.4 Coupled Electric Circuits .............................................................................. 266
Chapter 9 Nonlinear Oscillations ............................................................................................. 273
9.1 Qualitative Analysis: Energy and Phase Diagrams....................................... 274
9.2 Elliptical Integrals and Nonlinear Oscillations............................................. 280
9.3 Fourier Series Expansions ............................................................................. 283
9.3.1 Symmetrical Potential: V(x) = V(−x)................................................. 284
9.3.2 Asymmetrical Potential: V(−x) = −V(x) ............................................ 287
9.4 The Method of Perturbation .......................................................................... 288
9.4.1 Bogoliuboff–Kryloff Procedure and Removal of Secular Terms.........292
9.5 Ritz Method................................................................................................... 295
9.6 Method of Successive Approximation........................................................... 297
9.7 Multiple Solutions and Jumps........................................................................ 299
9.8 Chaotic Oscillations ...................................................................................... 301
9.8.1 Some Helpful Tools for an Understanding of Chaos........................ 301
9.8.2 Conditions for Chaos........................................................................ 306
9.8.3 Routes to Chaos................................................................................ 307
9.8.4 Lyapunov Exponentials .................................................................... 308
References ................................................................................................................ 312
Chapter 10 Collisions and Scatterings ........................................................................................ 313
10.1 Direct Impact of Two Particles...................................................................... 313
10.2 Scattering Cross Sections and Rutherford Scattering ................................... 318
10.2.1 Scattering Cross Sections................................................................. 319
10.2.2 Rutherford’s α-Particle Scattering Experiment................................ 320
10.2.3Cross Section Is Lorentz Invariant................................................... 324
10.3 Laboratory and Center-of-Mass Frames of Reference .................................. 324
10.4 Nuclear Sizes ................................................................................................. 328
10.5 Small-Angle Scattering (Optional)................................................................ 329
References ................................................................................................................ 336
Chapter 11 Motion in Non-Inertial Systems............................................................................... 337
11.1 Accelerated Translational Coordinate System .............................................. 337
11.2 Dynamics in Rotating Coordinate System .................................................... 341
11.2.1 Centrifugal Force ............................................................................. 345
11.2.2 The Coriolis Force............................................................................ 349
11.2.2.1 Trade Winds and Circulation of Ocean Currents ............. 351
11.2.2.2 Weather Systems............................................................... 352
11.2.2.3 Hurricanes ........................................................................ 354
11.2.2.4 Bathtub Vortex and Earth Rotation .................................. 354
11.3 Motion of Particle Near the Surface of the Earth ......................................... 355
11.4 Foucault Pendulum ........................................................................................ 361
11.5 Larmor’s Theorem......................................................................................... 364
11.6 Classical Zeeman Effect................................................................................ 365
11.7 Principle of Equivalence................................................................................ 368
11.7.1 Principle of Equivalence and Gravitational Red Shift..................... 369
Chapter 12 Motion of Rigid Bodies ............................................................................................ 377
12.1 Independent Coordinates of Rigid Body ....................................................... 378
12.2 Eulerian Angles ............................................................................................. 379
12.3 Rate of Change of Vector .............................................................................. 382
12.4 Rotational Kinetic Energy and Angular Momentum.................................... 384
12.5 Inertia Tensor................................................................................................. 394
12.5.1 Diagonalization of a Symmetric Tensor........................................... 396
12.5.2 Moments and Products of Inertia..................................................... 397
12.5.3 Parallel-Axis Theorem ..................................................................... 398
12.5.4 Moments of Inertia about an Arbitrary Axis ................................... 401
12.5.5 Principal Axes of Inertia .................................................................. 403
12.6 Euler’s Equations of Motion.......................................................................... 407
12.7 Motion of a Torque-Free Symmetrical Top................................................... 409
12.8 Motion of Heavy Symmetrical Top with One Point Fixed............................ 414
12.8.1 Precession without Nutation............................................................. 417
12.8.2 Precession with Nutation.................................................................. 419
12.9 Stability of Rotational Motion....................................................................... 420
References ................................................................................................................ 425
Chapter 13 Theory of Special Relativity .................................................................................... 427
13.1 Historical Origin of Special Theory of Relativity......................................... 427
13.2 Michelson–Morley Experiment..................................................................... 430
13.3 Postulates of Special Theory of Relativity .................................................... 433
13.3.1 Time Is Not Absolute ....................................................................... 434
13.4 Lorentz Transformations ............................................................................... 434
13.4.1 Relativity of Simultaneity, Causality................................................ 437
13.4.2 Time Dilation, Relativity of Co-Locality......................................... 438
13.4.3 Length Contraction........................................................................... 439
13.4.4 Visual Apparent Shape of Rapidly Moving Object.......................... 441
13.4.5 Relativistic Velocity Addition .......................................................... 441
13.5 Doppler Effect ............................................................................................... 445
13.6 Relativistic Space–Time (Minkowski Space)................................................ 446
13.6.1 Four-Velocity and Four-Acceleration ............................................... 449
13.6.2 Four-Energy and Four-Momentum Vectors ..................................... 450
13.6.3 Particles of Zero Rest Mass.............................................................. 452
13.7 Equivalence of Mass and Energy .................................................................. 453
13.8 Conservation Laws of Energy and Momentum............................................. 459
13.9 Generalization of Newton’s Equation of Motion........................................... 459
13.9.1 Force Transformation ....................................................................... 461
13.10 Relativistic Lagrangian and Hamiltonian Functions..................................... 463
13.11 Relativistic Kinematics of Collisions ............................................................ 467
13.12 Collision Threshold Energies ........................................................................ 470
References ................................................................................................................ 474
Chapter 14 Newtonian Gravity and Newtonian Cosmology ...................................................... 475
14.1 Newton’s Law of Gravity............................................................................... 475
14.2 Gravitational Field and Gravitational Potential............................................. 477
14.3 Gravitational Field Equations: Poisson’s and Laplace’s Equations ............... 479
14.4 Gravitational Field and Potential of Extended Body..................................... 480
14.5 Tides .............................................................................................................. 481
14.6 General Theory of Relativity: Relativistic Theory of Gravitation ................ 487
14.6.1 Gravitational Shift of Spectral Lines (Gravitational Red Shift) ...........488
14.6.2 Bending of Light Beam .................................................................... 489
14.7 Introduction to Cosmology............................................................................ 491
14.8 Brief History of Cosmological Ideas ............................................................. 492
14.8.1 Newton and Infiite Universe .......................................................... 493
14.8.2 Newton’s Law of Gravity Predicts Nonstationary Universe ............ 493
14.8.3 An Infiite Steady Universe Is an Empty Universe ......................... 495
14.8.4 Olbers’ Paradox ................................................................................ 496
14.9 Discovery of Expansion of the Universe, Hubble’s Law ............................... 497
14.10 Big Bang ........................................................................................................ 499
14.10.1 Age of the Universe.......................................................................... 499
14.11 Formulating Dynamical Models of the Universe.......................................... 499
14.12 Cosmological Red Shift and Hubble Constant H.......................................... 503
14.13 Critical Mass Density and Future of the Universe ........................................ 504
14.13.1 Density Parameter Ω ........................................................................ 505
14.13.2 Deceleration Parameter q0................................................................ 505
14.13.3 An Accelerating Universe?............................................................... 507
14.14 Microwave Background Radiation ................................................................ 507
14.15 Dark Matter ................................................................................................... 511
Reference.................................................................................................................. 514
Chapter 15 Hamilton–Jacobi Theory of Dynamics .................................................................... 515
15.1 Canonical Transformation and H-J Equation................................................ 515
15.2 Action and Angle Variables........................................................................... 522
15.3 Infiitesimal Canonical Transformations and Time Development
Operator .................................................................................................... 527
15.4 H-J Theory and Wave Mechanics.................................................................. 530
Reference.................................................................................................................. 533
Chapter 16 Introduction to Lagrangian and Hamiltonian Formulations for Continuous
Systems and Classical Fields.................................................................................... 535
16.1 Vibration of Loaded String............................................................................ 535
16.2 Vibrating Strings and the Wave Equation ..................................................... 541
16.2.1 Wave Equation.................................................................................. 541
16.2.2 Separation of Variables .................................................................... 543
16.2.3 Wave Number and Phase Velocity ................................................... 543
16.2.4 Group Velocity and Wave Packets ................................................... 544
16.3 Continuous Systems and Classical Fields...................................................... 547
16.3.1 Lagrangian Formulation................................................................... 547
16.3.2 Hamiltonian Formulation................................................................. 550
16.3.3 Conservation Laws ........................................................................... 552
16.4 Scalar and Vector of Fields............................................................................ 553
16.4.1 Scalar Fields ..................................................................................... 553
16.4.2 Vector Fields..................................................................................... 554
Appendix 1: Vector Analysis and Ordinary Differential Equations....................................... 557
Appendix 2: D’Alembert’s Principle and Lagrange’s Equations ............................................ 587
Appendix 3: Derivation of Hamilton’s Principle from D’Alembert’s Principle..................... 595
Appendix 4: Noether’s Theorem................................................................................................. 599
Appendix 5: Conic Sections, Ellipse, Parabola, and Hyperbola ............................................. 605
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