Classical Mechanics

505 Pages·2008·26.47 MB·English

Classical Mechanics
G. ARULDHAS
Formerly
Professor & Head of Physics & Dean Faculty of Science
University of Kerala
New Delhi-110001
2008
CLASSICAL MECHANICS
G. Aruldhas
© 2008 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be
reproduced in any form, by mimeograph or any other means, without permission in writing from the
publisher.
ISBN-978-81-203-3331-4
The export rights of this book are vested solely with the publisher.
Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001
and Printed by Jay Print Pack Private Limited, New Delhi-110015.
To
Myrtle and our children
Vinod & Anitha, Manoj & Bini, Ann & Suresh
Contents

Preface………xi 1. Introduction to Newtonian Mechanics………
1–23
1.1 Frames of Reference……1
Cartesian Co-ordinates (x, y, z)……1
Plane Polar Co-ordinates (r, q)……2
Cylindrical Co-ordinates (r, f, z)……3
Spherical Polar Co-ordinates (r, q, f)……3
1.2 Newton’s Laws of Motion……4
Newton’s First Law of Motion……4
Newton’s Second Law of Motion……4
Newton’s Third Law of Motion……5
1.3 Inertial and Non-inertial Frames……5
1.4 Mechanics of a Particle……6
Conservation of Linear Momentum……6
Angular Momentum and Torque……6
Conservation of Angular Momentum……7
Work Done by a Force……7
Conservative Force……8
Conservation of Energy……8
1.5 Motion under a Constant Force……9
1.6 Motion under a Time-dependent Force……10
1.7 Reflection of Radiowaves from the Ionosphere……10
1.8 Motion under a Velocity Dependent Force……12
1.9 Motion of Charged Particles in Magnetic Fields ……13
Worked Examples……15
Review Questions 22
Problems……22
2. System of Particles………24–38
2.1 Centre of Mass……24
2.2 Conservation of Linear Momentum……25
2.3 Angular Momentum……26
2.4 Conservation of Angular Momentum……27
2.5 Kinetic Energy for a System of Particles……28
2.6 Energy Conservation of a System of Particles……29
2.7 Time Varying Mass Systems—Rockets……31
Worked Examples……34
Review Questions……37
Problems……37
3. Lagrangian Formulation………39–77
3.1 Constraints……39
Holonomic Constraints……39
Non-holonomic Constraints……40
Scleronomous and Rheonomous Constraints……40
3.2 Generalized Co-ordinates……41
Degrees of Freedom……41
Generalized Co-ordinates……41
Configuration Space……42
3.3 Principle of Virtual Work……42
3.4 D’Alembert’s Principle……43
3.5 Lagrange’s Equations……44
3.6 Kinetic Energy in Generalized Co-ordinates……47
3.7 Generalized Momentum……49
3.8 First Integrals of Motion and Cyclic Co-ordinates……49
Cyclic Co-ordinates……50
3.9 Conservation Laws and Symmetry Properties……50
Homogeneity of Space and Conservation of Linear Momentum……50
Isotropy of Space and Conservation of Angular Momentum……51
Homogeneity of Time and Conservation of Energy……53
3.10 Velocity-dependent Potential……54
3.11 Dissipative Force……56
3.12 Newtonian and Lagrangian Formalisms……57
Worked Examples……57
Review Questions……75
Problems……76
4. Variational Principle………78–97
4.1 Hamilton’s Principle……78
4.2 Deduction of Hamilton’s Principle……79
4.3 Lagrange’s Equation from Hamilton’s Principle……81
4.4 Hamilton’s Principle for Non-holonomic Systems……84
Worked Examples……86
Review Questions……96
Problems……96
5. Central Force Motion………98–136
5.1 Reduction to One-body Problem……98
5.2 General Properties of Central Force Motion……100
Angular Momentum……100
Law of Equal Areas……101
5.3 Effective Potential……103
5.4 Classification of Orbits……103
5.5 Motion in a Central Force Field—General Solution……106
Energy Method……106
Lagrangian Analysis……107
5.6 Inverse Square Law Force……107
5.7 Kepler’s Laws……110
5.8 Law of Gravitation from Kepler’s Laws……111
5.9 Satellite Parameters……113
5.10 Communication Satellites……115
5.11 Orbital Transfers……116
5.12 Scattering in a Central Force Field……118
5.13 Scattering Problem in Laboratory Co-ordinates……122
Worked Examples……125
Review Questions……134
Problems……135
6. Hamiltonian Mechanics………137–172
6.1 The Hamiltonian of a System……137
6.2 Hamilton’s Equations of Motion……138
6.3 Hamilton’s Equations from Variational Principle……139
6.4 Integrals of Hamilton’s Equations……140
Energy Integral……140
Integrals Associated with Cyclic Co-ordinates……141
6.5 Canonical Transformations……142
6.6 Poisson Brackets……146
Fundamental Poisson Brackets……146
Fundamental Properties of Poisson Brackets……147
Equations of Motion in Poisson Bracket Form……148
6.7 Poisson Bracket and Integrals of Motion……149
6.8 The Canonical Invariance of Poisson Bracket……150
6.9 Lagrange Brackets……151
6.10 D-Variation……152
6.11 The Principle of Least Action……153
Different Forms of Least Action Principle……155
6.12 Poisson Brackets and Quantum Mechanics……157
Worked Examples……158
Review Questions……170
Problems……171
7. Hamilton-Jacobi Theory………173–195
7.1 Hamilton–Jacobi Equation……173
Physical Significance of S……175
7.2 Hamilton’s Characteristic Function……175
7.3 Harmonic Oscillator in The H-J Method……177
7.4 Separation of Variables in The H-J Equation……179
7.5 Central Force Problem in Plane Polar Co-ordinates……181
7.6 Action-Angle Variables……182
7.7 Harmonic Oscillator in Action-Angle Variables……184
7.8 Kepler Problem in Action-Angle Variables……185
7.9 Road to Quantization……188
Worked Examples……189
Review Questions……194
Problems……195
8. The Motion of Rigid Bodies………196–230
8.1 Introduction……196
8.2 Angular Momentum……197
8.3 Kinetic Energy……199
8.4 Inertia Tensor……200
8.5 Principal Axes……201
8.6 Euler’s Angles……203
8.7 Infinitesimal Rotations……207
8.8 Rate of Change of a Vector……208
8.9 Coriolis Force……209
8.10 Euler’s Equations of Motion……211
8.11 Force-free Motion of a Symmetrical Top……212
8.12 Heavy Symmetric Top with One Point Fixed……215
Worked Examples……220
Review Questions……229
Problems……230
9. Theory of Small Oscillations………231–251
9.1 Equilibrium and Potential Energy……231
9.2 Theory of Small Oscillations……232
9.3 Normal Modes……235
9.4 Two Coupled Pendula ……237
Resonant Frequencies……237
Normal Modes……238
9.5 Longitudinal Vibrations of CO Molecule……241
2
Normal Frequencies……241
Normal Modes……242
Normal Co-ordinates……243
Worked Examples……244
Review Questions……250
Problems……251
10. Special Theory of Relativity………252–300
10.1 Galilean Transformation……252
10.2 Electromagnetism and Galilean Transformation……254
10.3 Michelson–Morley Experiment……255
The Interferometer……255
The Experiment……256
10.4 The Postulates of Special Theory of Relativity……258
10.5 Lorentz Transformation……258
10.6 Velocity Transformation……261
10.7 Length Contraction……262
10.8 Time Dilation……263
10.9 Simultaneity……264
10.10 Mass in Relativity……264
10.11 Mass and Energy……266
10.12 Relativistic Lagrangian of a Particle……268
10.13 Relativistic Hamiltonian of a Particle……269
10.14 Space-Time Diagram……270
10.15 Geometrical Interpretation of Lorentz Transformation……272
10.16 Principle of Covariance……273
10.17 Four-Vectors in Mechanics……274
Position Four-Vector……275
Four-Velocity……275
Momentum Four-Vector……276
Four-Force……277
Four-Acceleration……278
10.18 Charge Current Four-Vector……278
10.19 Invariance of Maxwell’s Equations……279
Maxwell’s Equations……279
Vector and Scalar Potentials……279
Gauge Transformations……280
Four-Vector Potential……281
10.20 Electromagnetic Field Tensor……282
10.21 General Theory of Relativity……283
Principle of Equivalence……284
Bending of Light in a Gravitational Field……284
Precession of the Perihelion of Planetary Orbits……285
Space Curvature……285
Gravitational Red Shift……286
Worked Examples……287
Review Questions……297
Problems……298
11. Introduction to Nonlinear Dynamics………301–323
11.1 Linear and Nonlinear Systems……301
11.2 Integration of Linear Equation: Quadrature Method……302
11.3 Integration of Nonlinear Second Order Equation……304
11.4 The Pendulum Equation……305
11.5 Phase Plane Analysis of Dynamical Systems……308
Phase Curve of Simple Harmonic Oscillator……308
Phase Curve of Damped Oscillator……309
11.6 Phase Portrait of the Pendulum……310
11.7 Matching of Phase Curve with Potential V(x)……311
Simple Harmonic Oscillator……312
Simple Pendulum……312
11.8 Linear Stability Analysis……313
Stability Matrix 313
Classification of Fixed Points……314
11.9 Fixed Point Analysis of a Damped Oscillator……316