Classical Dynamics of Particles and Systems (eBook)

Jerry B. Marion (Autor)

eBook Download: PDF

2013 | 1. Auflage
592 Seiten
Elsevier Science (Verlag)
978-1-4832-7281-8 (ISBN)

Infos

Classical Dynamics of Particles and Systems presents a modern and reasonably complete account of the classical mechanics of particles, systems of particles, and rigid bodies for physics students at the advanced undergraduate level. The book aims to present a modern treatment of classical mechanical systems in such a way that the transition to the quantum theory of physics can be made with the least possible difficulty; to acquaint the student with new mathematical techniques and provide sufficient practice in solving problems; and to impart to the student some degree of sophistication in handling both the formalism of the theory and the operational technique of problem solving.
Vector methods are developed in the first two chapters and are used throughout the book. Other chapters cover the fundamentals of Newtonian mechanics, the special theory of relativity, gravitational attraction and potentials, oscillatory motion, Lagrangian and Hamiltonian dynamics, central-force motion, two-particle collisions, and the wave equation.

Inhalt

Front Cover 1
Classical Dynamics of Particles and Systems 4
Copyright Page 5
Table of Contents 10
Preface 6
Chapter 1. Matrices and Vectors 18
1.1 Introduction 18
1.2 The Concept of a Scalar 19
1.3 Coordinate Transformations 20
1.4 Properties of Rotation Matrices 23
1.5 Matrix Operations 26
1.6 Further Definitions 28
1.7 Geometrical Significance of Transformation Matrices 30
1.8 Definitions of a Scalar and a Vector in Terms of Transformation Properties 37
1.9 Elementary Scalar and Vector Operations 37
1.10 The Scalar Product of Two Vectors 38
1.11 The Vector Product of Two Vectors 41
1.12 Unit Vectors 44
Suggested References 46
Problems 46
Chapter 2. Vector Calculus 49
2.1 Introduction 49
2.2 Differentiation of a Vector with Respect to a Scalar 50
2.3 Examples of Derivatives —Velocity and Acceleration 51
2.4 Angular Velocity 55
2.5 The Gradient Operator 58
2.6 The Divergence of a Vector 62
2.7 The Curl of a Vector 65
2.8 Some Additional Differential Vector Relations 67
2.9 Integration of Vectors 68
Suggested References 70
Problems 70
Chapter 3. Fundamentals of Newtonian Mechanics 73
3.1 I ntroduction 73
3.2 Newton's Laws 74
3.3 Frames of Reference 78
3.4 The Equation of Motion for a Particle 81
3.5 Conservation Theorems 91
3.6 Conservation Theorems for a System of Particl 96
3.7 Limitations of Newtonian Mechanics 105
Suggested References 107
Problems 107
Chapter 4. The Special Theory of Relativity 112
4.1 I ntroduction 112
4.2 Galilean Invariance 113
4.3 The Lorentz Transformation 115
4.4 Momentum and Energy in Relativity 119
4.5 Some Consequences of the Lorentz Transformation 124
Suggested References 128
Problems 129
Chapter 5. Gravitational Attraction and Potentials 132
5.1 Introduction 132
5.2 The Gravitational Potential 133
5.3 Lines of Force and Equipotential Surfaces 137
5.4 The Gravitational Potential of a Spherical Shell 139
5.5 A Final Comment 142
Suggested References 143
Problems 143
Chapter 6. Oscillatory Motion 145
6.1 I ntroduction 145
6.2 The Simple Harmonic Oscillator 147
6.3 Damped Harmonic Motion 149
6.4 Forcing Functions 153
6.5 Forced Oscillations 155
6.6 Phase Diagrams 160
6.7 The Response of Linear Oscillators to Impulsive Forcing Functions 
165 
6.8 Electrical Oscillations 173
6.9 Harmonic Oscillations in Two Dimensions 175
6.10 The Use of Complex Notation 178
Suggested References 180
Problems 181
Chapter 7. Nonlinear Oscillations 186
7.1 Oscillations 186
7.2 Oscillations for General Potential Functions 187
7.3 Phase Diagrams for Nonlinear Systems 192
7.4 The Plane Pendulum 196
7.5 Nonlinear Oscillations in a Symmetric Potential —The Method of Successive Approximations 202
7.6 Nonlinear Oscillations in an Asymmetric Potential —The Method of Perturbations 207
Suggested References 210
Problem 210
Chapter 8. Some Methods in the Calculus of Variations 213
8.1 Introduction 213
8.2 Statement of the Problem 214
8.3 Euler's Equation 218
8.4 The Brachistochrone Problem 219
8.5 The "Second Form" of Euler's Equation 221
8.6 Functions with Several Dependent Variables 223
8.7 The Euler Equations When Auxiliary Conditions Are I mposed 224
8.8 The d Notation 227
Suggested References 228
Problems 229
Chapter 9. Hamilton's Principle — Lagrangian and Hamiltonian Dynamics 231
9.1 Introduction 231
9.2 Hamilton's Principle 233
9.3 Generalized Coordinates 236
9.4 Lagrange's Equations of Motion in Generalized Coordinates 239
9.5 Lagrange's Equations with Undetermined Multipliers 243
9.6 The Equivalence of Lagrange's and Newton's Equations 246
9.7 The Essence of Lagrangian Dynamics 249
9.8 A Theorem Concerning the Kinetic Energy 251
9.9 The Conservation of Energy 253
9.10 The Conservation of Linear Momentum 255
9.11 The Conservation of Angular Momentum 256
9.12 The Canonical Equations of Motion —Hamiltonian Dynamics 259
9.13 Some Comments Regarding Dynamical Variables and Variational Calculations in Physics 266
9.14 Phase Space and Liouville's Theorem 269
9.15 The Virial Theorem 273
9.16 The Lagrangian Function in Special Relativity 274
Suggested References 276
Problems 277
Chapter 10. Central-Force Motion 284
10.1 Introduction 284
10.2 The Reduced Mass 285
10.3 Conservation Theorems—First Integralsof the Motion 286
10.4 Equations of Motion 288
10.5 Orbits in a Central Field 291
10.6 Centrifugal Energy and the Effective Potential 292
10.7 Planetary Motion—Kepler's Problem 295
10.8 Kepler's Equation 299
10.9 Approximate Solution of Kepler's Equation 305
10.10 Apsidal Angles and Precession 306
10.11 Stability of Circular Orbits 312
10.12 The Problem of Three Bodies 319
Suggested References 326
Problems 326
Chapter 11. Kinematics of Two-Particle Collisions 332
11.1 Introduction 332
11.2 Elastic Collisions —Center-of-Mass and Laboratory Coordinate Systems 333
11.3 Kinematics of Elastic Collisions 340
11.4 Cross Sections 344
11.5 The Rutherford Scattering Formula 349
11.6 The Total Cross Section 351
11.7 Relativistic Kinematics 352
Suggested References 355
Problems 356
Chapter 12. Motion in a Noninertial Reference Frame 359
12.1 Introduction 359
12.2 Rotating Coordinate Systems 360
12.3 The Coriolis Force 362
12.4 Motion Relative to the Earth 365
Suggested References 375
Problems 376
Chapter 13. Dynamics of Rigid Bodies 378
13.1 Introduction 378
13.2 The Inertia Tensor 380
13.3 Angular Momentum 386
13.4 Principal Axes of Inertia 389
13.5 Moments of Inertia for Different Body Coordinate Systems 395
13.6 Further Properties of the Inertia Tensor 399
13.7 The Eulerian Angles 408
13.8 Euler's Equations for a Rigid Body 411
13.9 Force-Free Motion of a Symmetrical Top 414
13.10 The Motion of a Symmetrical Top with One Point Fixed 417
13.11 The Stability of Rigid-Body Rotations 424
Suggested References 427
Problems 427
Chapter 14. Systems with Many Degrees of Freedom — Small Oscillations and Normal Coordinates 433
14.1 I ntroduction 433
14.2 Two Coupled Harmonic Oscillators 434
14.3 The General Problem of Coupled Oscillations 438
14.4 The Orthogonality of the Eigenvectors 444
14.5 Normal Coordinates 447
14.6 Two Linearly Coupled Plane Pendula 452
14.7 Three Linearly Coupled Plane Pendula —An Example of Degeneracy 455
14.8 The Loaded String 458
14.9 The Continuous String as a Limiting Caseof the Loaded String 467
14.10 The Wave Equation 470
14.11 The Nonuniform String —Orthogonal Functions and Perturbation Theory 471
14.12 Fourier Analysis 480
Suggested References 487
Problems 488
Chapter 15. The Wave Equation in One Dimension 495
15.1 Introduction 495
15.2 Separation of the Wave Equation 496
15.3 Phase Velocity, Dispersion, and Attenuation 503
15.4 Electrical Analogies — Filtering Networks 509
15.5 Group Velocity and Wave Packets 512
15.6 Fourier Integral Representation of Wave Packets 516
15.7 Energy Propagation in the Loaded String 522
15.8 Further Comments Regarding Phase and Group Velocities 526
15.9 Reflected and Transmitted Waves 528
15.10 Damped Plane Waves 531
Suggested References 533
Problems 534
Solutions, Hints, and References for Selected Problems 539
Appendix A. Taylor's Theorem 544
Exercises 546
Appendix B. Complex Numbers 548
B.1 Complex Numbers 548
B.2 Geometrical Representation of Complex Numbers 549
B.3 Trigonometric Functions of Complex Variables 550
B.4 Hyperbolic Functions 551
Exercises 552
Appendix C. Ordinary Differential Equations of Second Order 554
C.1 Linear Homogeneous Equations 554
C.2 Linear Inhomogeneous Equations 559
Exercises 561
Appendix D. Useful Formulas 563
D.1 Binomial Expansion 563
D.2 Trigonometric Relations 564
D.3 Trigonometric Series 565
D.4 Exponential and Logarithmic Series 565
D.5 Hyperbolic Functions 566
Appendix E. Useful Integrals 567
E.1 Algebraic Functions 567
E.2 Trigonometric Functions 568
E.3 Gamma Functions 569
E.4 Elliptic Integrals 570
Appendix F. Differential Relations in Curvilinear Coordinate Systems 571
F.1 Cylindrical Coordinates 571
F.2 Spherical Coordinates 573
Appendix G. A "Proof" of the Relation 575
SELECTED REFERENCES 577
BIBLIOGRAPHY 578

Produkt-Details

• Erscheint lt. Verlag: 22.10.2013
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik
• ISBN-10: 1-4832-7281-8 / 1483272818
• ISBN-13: 978-1-4832-7281-8 / 9781483272818