اسم المؤلف
Dr. J.C. Upadhyaya
التاريخ
20 أكتوبر 2022
المشاهدات
160
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(لا توجد تقييمات)
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Classical Mechanics
[For M.Sc. (Physics), B.Sc. (Honours), B.E., Net, GATE and Other
Competitive Examinations]
Dr. J.C. Upadhyaya
MSc., PhD., F. Inst. P. (London)
Formerly Director/Professor Incharge,
I Dau Dayal Institute, Dr. 8.R. Ambedkar University, Agra (India),
i Senior Reader in Physics, Agra College, Agra (India)
Contents
Chapter Page N#-
- Introductory Ideas 1—26
(Newtonian Mechanics)
1.1. Introduction 1
1.2. Space and Time (Frame of Reference) 1
T.3. Newton’s Laws of Motion 2 ‘
1.4. Inertial Frames 4
1.5. Gravitational Mass 5
1.6. Mechanics of a Particle : Conservation Laws 6
1.6.1. Conservation of linear momentum 6
1.6.2. Conservation of angular momentum 6
1.6.3. Conservation of energy 7
1.6.3.(a) Work 7
1.6.3.(b) Kinetic energy and work-energy theorem’ 7
1.6.3.(c) Conservative force and potential energy 7
1.6.3.(d) Conservation theorem 8
1.6.3.(e) First integrals of motion 8
1.7. Mechanics of a System of Particles 9
1.7.1. External and internal forces 9
1.7.2. Centre of mass 10
1.7.3. Conservation of linear momentum 10
1.7.4. Centre of mass-frame of reference 11
1.7.5. Conservation of angular momentum 11
1.7.6. Note on consevation throrems of linear and angular
momentum for a system of particles 12
1.7.7. Relation between angular momentum (J) and angular momentum
about center of mass (J x cm’) 12
1.7.8. Conservation of energy 13
1.7.8.(a) Kinetic energy 13
1.7.8.(b) Potential energy 14
1..7.8.(c) Conservation theorem 15
Questions 19′
Problems — SET-I and SET-II 20
Objective Type Questions 25
Short Answer Questions 26 - Langrangian Dynamics 27—74
2.1. Introduction 27
2.2. Basic Concepts 27www.cgaspirants.com
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(1) Coordinate systems 27
(2) Degrees of freedom — Configuration space 28
2.3. Constraints 29
2.3.1. Holonomic constraints 29
2.3.2. Nonholonomic constraints #30
2.3.3. Some more examples of holonomic and non-holonomic constraints 30
2.3.4. Forces of constraints 31
2.3.5. Difficulties introduced by the constraints and their removal 32
2.4. Generalized coordinates 34
2.5. Principle of Virtual Work 35
2.6. D’ Alembert’s principle 36
2.7. Langrange’s Equations from D’Alembert’s principle 38
2.8. Procedure for formation of Langrange’s Equations 41
2.9. Langrange’s Equations in presence of Non-conservative forces 47
2.10. Generalized Potential — Lagrangian for a Charged Particle T
Moving in an Electromagnetic field (Gyroscopic Forces) 49
2.11. Hamilton’s Principle and Langrange’s Equations 51
2.12. Superiority of Lagrangian Mechanics over Newtonian Approach 53
2.13. Guage Invariance of the Lagrangian 53
2.14. Symmetry Properties of Space and Time and Conservation laws 55
2.14. Invariance under Galilean Transformation 57
Questions 64
Problems — SET-I and SET-II 66
Objective Type Questions 73
Short Answer Questions 74 - Hamiltonian Dynamics 75—102
3.1. Introduction 75
3.2. Generalized momentum and cyclic coordinates 75
3.3. Conservation Theorems 77
3.3.1. Conservation of linear momentum 77
3.3.2. Conservation of angular momentum 78
3.3.3. Significance of translation and rotation cyclic coordinates —
symmetry properties 80
3.4. Hamiltonian Function H and Conservation of Energy : Jacobi’s Integral 80
3.5. Hamilton’s Equations 82
3.6.. Hamilton’s Equations in Different Coordinate Systems 84
3.7. Examples in Hamiltonian Dynamics 86
(1) Harmonic oscillator 86
(2) Motion of a particle in a central force field 87
(3) Charged particle moving in an electomagnetic field 88
(4) Compound pendulam 89
(5) Two dimensional harmonic oscillator 89,
3.8. Routhain94
Questions 96
^Problems — SET-I and SET-I 97www.cgaspirants.com
Objective Type Questions 100
Short Answer Questions 101
(xi) - Two-Body Central Force Problem 103—137
4.1. Reduction of Two – Body Central Force Problem to
the Equivalent One-Body Problem 103
4.2. Central Force and Motion in a Plane 106
4.3. Equations of Motion under Central Force and.First Integrals 107
4.4. Differential Equation for an Orbit 108
‘ 4.5. Inverse Square Law of Force 109
4.6. Kepler’s Laws of Plantery Motion and their Deduction 110
4.6.1. Deduction of the Kepler’s first law 110
4.6.2. Deduction of the Kepler’s second law 112
4.6.3. Deduction of the Kepler’s third law (Period of Motiom in an Elliptical Orbit) 112
4.7. Stability of Orbit under Central Force 114
4.8. Artificial Statellites 121
4.9. Virial Theorem 124
4.10. Scattering in a Central Force Field — Scattering cross-section,
Scattering angle, Impact parameter 125
4.11. Rutherford Scattering Cross-section 127
Questions
Problems — SET-I and SET-II 132
Objective Type Questions 136
Short Answer Questions 137 - Variational Principles 138—162
5.1. Introduction 138
5.2. The Calculus of Variations and Euler-Lagrange’s Equations 138
5.3. Deduction of Hamilton’s principle from D’Alembert’s principle 146
5.4. Modified Hamilton’s principle 147
5.5. Deduction of Hamiltion’s Equations from Modified
Hamiltion’s Principle (or Variational Principle) 147
5.6. Deduction of Lagrange’s Equations from Variational Principle for
Non-conservative Systems (Holohomic Constraints) 148
5.7. Langrange’s Equations of Motion for Non-holonomic Systems .
(Lagrange’s Method of Undetermined Multipliers) 149
5.8. Physical Significance of Langrange’s Multipliers \ 151
5.9. Examples of Lagrange’s Method of Undetermined Multipliers 151
(1) Rolling hoop on an inlined plane 151
(2) Simple pendulum 152
5.10. A-Variation 1^3’ ‘
5.11. Principle of,Least Action 154
5.12. Other Forms.oUPrinciple of least Action 156
(1). For a conservation system 1 56
(2) Jacobi’s form of the principle of least action 157
(3) Principle of least action in terms of arc length of the particle trajectory 157www.cgaspirants.com
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Questions 158
Problems
— SET-I and SET-II 159 .
Objective Type Questions 161
Short Answer Question 162 - Canonical Transformations 163—178
6.1. Canonical Transformations 163
6.2. Legendre Transformations 163
6.3. Generating Functions 164
6.4. Procedure for Application of Canonical Transformations 167
6.5. Condition for Canonical Transformations 167
6.6. Bilinear Invariant Condition 170
6.7. Intergal Invariant of Poincare 171
6.8. Infinitesimal Contact Transformations 173
Questions 174
Problems — SET-I and SET-II 174
Objective Type Questions 177
Short Answer Questions 178 - Brackets and Liouville’s Theorem 179—196
7.1. Introduction 179
7.2. Poission’s Brackets 179
7.3. Lagrange Brackets 182
7.4. Relation Between Lagrange and Poisson Brackets 182
7.5. Angular Momentum and Poisson Brackets 182
7.6. Invariance of Poisson Bracket with respect to Canonical Transformations 183
7.7. Phase Space 189
7.8. Liouville’s Theorem 190
Questions 174
Problems — SET-I and SET-II 194
Objective Type Questions 195
Short Answer Questions 196 - Hamilton-Jacobi Theory and Transition to Quantum Mechanics 197—231
8.1. Introduction 197
8.2. The Hamilton-Jacobi Equation 197
8.3. . Solution of Harmonic Oscillatior Problem by Hamilton-Jacobi Method 199
8.4. Hamilton-Jacobi Equation: Hamilton’s Characteristic Function—
Conservative Systems 201
8.5. Kepler’s Problem : Solution by Hamilton’s-Jacobi Method 204
8.6. Action and Angle Variables 207
8.7. Problem of Harmonic Oscillator using Action-Angle Variables
(Deduction of Frequency of Motion) 209
8.8. Action-Angle Variables in General Case 210
8.9. Hamilton-Jacobi Equation-Geometrical Optics and Wave Mechanics
(Transition from Classical to Quantum Mechanics) 214www.cgaspirants.com
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Questions 226
Problems
— SET-I and SET-II 226
Objective Type Questions 230
Short Answer Questions 231 - Small Oscillations and Normal Modes (Coupled-Oscillators) 232—272
9.1. Introduction 232
9.2. Potential Energy and Equilibrium—One Dimensional Oscillator 232
9.2.1. Stable, Unstable and Neutral Equilibrium 233
9.2.2. One-dimensional oscillator 233-
9.3. Two Coupled Oscillators 235
9.3.1. Solution of the differential equations 236
9.3.2. Normal coordinates and normal, modes 237
9.3.3. Kinetic and potential energies in normal coordinates 239
9.4. General Theory of Small Oscillations 240
9.4.1. Secular equation and eigenvalue equation 241
9.4.2. Solution of the eigenvalue equation 242
9.4.3. Small oscillations in normal coordinates 243
9.5. Examples of Two Coupled Oscillators 246
(1) Two coupled pendulums 246
(2) Double pendulum 250
9.6.’ Vibrations of a Linear Triatomic Molecule
9.7. Transverse Oscillations of N-coupled Masses on an Elastic String :
Many Coupled Oscillators 256
9.8. Transition from Discrete to a Continuous System : Waves on a String 264 .
Questions 266
j Problems — SET-I and SET-II 267
Objective Type Questions 270
Short Answer Questions 272 - Dynamics of a Rigid Body 273—319
10.1. Generalized Coordinates of a Rigid Body 273
10.2. Body and Space Reference Systems 274
j 10.3. Euler’s Angles 276
’ 10.4. Infinitesimal Rotations as Vectors — Angular Velocity 280
10.5. Components of Angular Velocity 280
10.6. Angular Momentum and Inertia Tensor 282
10.7. Principle Axes-Principle Moments of Inertia 284
10.8. Rotational Kinetic Energy of a Rigid Body 285
10..9. Symmetric Bodies 287
10.10. Moments of Interia for Different Body Systems 287
10.11. Euler’s Equations of Motion for a Rigid Body 289
. 10.12. Torque-Freq .Motion of a Rigid Body 291
I 10.13. Force-free Motion of a Symmetrical Top 295
I 10.14. Motion of a. Heavy Symmetrical Top 298
| 10.15.FastTop303
L –
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10.16. Sleeping Top 306
10.17. Gyroscope 307
Questions 314
Problems — SET-I and SET-II 315
Objective Type Questions 318
Short Answer Question 319 - Noninertial and Rotating Coordinate Systems 32,0—333
11.1. Noninertial Frames of Reference 320
11.2. Fictitious or Pseudo Force 320
11.3. Centrifugal Force 322
11.4. Uniformly Rotating Frames 323
11.5. Free Fall of a Body on Earth’s Surface 325
11.6, Foucault’s Pendulum 327
Questions 329
Problems
— SET-I and SET-II 330
Objective Type Questions 3-32
Short Answer Questions 333 - Special Theory of Relativity-Lorentz Transformations 334—366
12.1. Introduction 334
12.2. Galilean Transformations 334
12.3. Principle of Relativity 336
12.4. Transformation of Force from One Inertial System to Another 336
12.5. Covariance of the physical Laws 337
12.6. Principle of Relativity and Speed of Light 337
12.7. The Michelson-Morley Experiments 339
12.8. Ether Hypothesis 341
12.9. Postulates of Special Theory of Relativity 342
12.10. Lorentz Transformations 342
12.11. Consequences of Lorentz Transformations 345
(1) Length contraction 345
(2) Simultaneity 345
(3) Time dilation 346
(4) Addition of velocities 349
12.12. Aberration of Light from Stars 353
12.13 Relativistic Doppler’s Effect 355
Questions 359
Problems — SET-I and SET-II 360
Objective Type Questions 365
Short Answer Questions 365 - Relativistic Mechanics 367—388/
13.1. Introduction 367 /
13.2. Conservation of Momentum at Relativistic Speeds — Variation of Mass /
with Velocity 367
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13.3. Relativistic Energy — Mass-Energy Relation (E = me2} 370
13.4. Examples of Mass — Energy Conversion 371
13.5. Relation between Momentum and Energy and Conversation Laws 372
.1-3.6. Transformation of Momentum and Energy 373
l-3?7. Praticles with Zero Rest Mass 374
13.8. Force in Relativistic Mechanics 374
13.9. Lorentz Transformation for Force 375
13.10. Equilibrium of Right-angled Lever 375
13.11. The Lagrangian and Hamiltonian of a Particle in Relativistic Mechanics 380
13.12. Relativistic Lagrangian and Hamiltonian of a Charged Particle in an
Electromagnetic Field-Velocity Dependent Potential 382
Questions 383
Problems — SET-I and SET-II 383
Objective Type Questions 387
Short Answer Questions 387 - Four Dimensional Formulation — Minkowski Space 389—419
14.1. Introduction 389
14.2. Minkowski Space and Lorentz Transformations 389
14.3. World Point and World Line 392
14.4. Space-time Invertvals 392
14.5. Four-vectors 396
14.6. Examples of Four-vectors 398
(1) Position four-vector 398
(2) Four velocity or velocity four-vector 398
(3) Momentum four-vector 399
(4) Acceleration four-vector 399
(5) Four-force Minkowski force 400
14.7. Consevation of Four-momentum — application of Four-vectors 403
14.8. Covariant Formulation of Lagrangian and Hamiltonian 407
14.9. Geometrial Interpretation of Lorentz Transformations: Minkowski
Diagrams 411
14.10. Geometrical Representation of Simultaneity, Length Contraction and
Time Dilation 414
Questions 416
Problems – SET-I and SET-II 417
Objective Type Questions 418
Short Answer Questions 418 - Convariant Formulation of Electrodynamics 420—437
15.1. D’Alembertian Operator’tF420
15.2. Maxwell’s Feild Equations 421
15.3. Maxwell’s Equations in terms of Electromagnetic Potentials A and (|) 423
15.4. Current Four-Vector 425
15.5. Transformation of Electromagnetic Potentials A and (|>
(Four-vector Potential) 425www.cgaspirants.com
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15.6. Covariance of Maxwell’s Field Equations in Terms of Four-Vectors 426
15.7. The Electromagnetic Field Tensor 427
15.8. Lorentz Transformations of Electric and Magnetic Fields
15.9. Covariant Form of Maxwell’s Fields Equations in terms of Electromagnetic Field Tensor
15.10. Lorentz Force on Charged Particle 432
1 5:.l I . Lorentz Force in Covariant Form 433
Questions 434
Problems — SET-I and SET-II 435
Objective Type Questions 435
Short Answer Questions 436 - Nonlinear Dynamics and Chaos 438—475
16.1. Indroduction 438
16.2. Nonlinear Differential Equations 438
16.3. Phase Trajectories-Singular Points (Topological Methods) 439
16.4. Phase Trajectories of Linear Systems 440
16.5. Phase Trajectories of Non linear Systems 444
16.6. Limit Cycles-Attractors 451
16.7. N-Torus 453
16.8. Chaos 455
16.9. Logistic Map 455
16.10. Strange Attractor 462
16.11. Sensitivity to Initial Conditions and Parameters — Lyapunov Exponent 463
16.12. Poincare Sections 464
16.13. Driven Damped Harmonic Oscillator 464
16.14. Fractals 466 ’
16.15. Integrable Hamiltonian and Invariant Tori 469 - 1 6. KAM Theorem 470
Questions 471
Problems 472
Objective Type Questions 473
Short Answer Questions 4^4
Index 477—480www.cgaspirants.com
Index
Aberration of light from stars 353
Accelerated frames 320
Acceleration in rotating frame 324
Action 3, 51,154,207
Action and angle variables 207, 210
Angle variable 208
Angular velocity 280, 282
Aperiodic motion 442
Areal velocity 18, 107
Attroctors 451
Atwood machine 16,43
Bifurcations458
Bilinear invariant condition 170
Brackets 179
-Poisson’s 179
-Lagrange’s 182
Brachristochrone problem 143
Body angle 278
-coordinate system 274
Calculus of variation 198
Canonical momentum 75
Canonical transformations 163
Central force 18, 47, 108
Centre of mass 10
-frameof reference 11
Centrifugal force 322
Chaos 455,460
Conjugate momentum 75
Compound pendulum 44, 89
Configuration space 28
Contact transformations 163
Conservation laws 6
Conservation theorem 77
-oflinear momentum6,55,77
-of momentumat relativisticspeeds 421
-of angular momentum6, 56, 78
of energy6, 80, 81
Conservative force 7
Constraints 29
-holonomic 29
- nonholonomic 30
- rheonomous 30
-scleronomous 30
Coordinate systems 2, 27
-cartesian 2, 27
-cylindrical 27
-spherical 28
Coriolis force 325
Coupled oscillators (two) 235
-pendulums 246
-many 256
Covariance of Maxwell’s field equations in terms of
four vectors 426
1
-in termsof electromagnetic field tensor 430
Covariance of physical laws 337
Covariant formulation ofelectrodynamics 420
Covariant formulation of Lagrangian and Hamiltonian
407
Cycliccoordinates 76, 83
D’alembert’s principle 36
De Broglie relation 217
Degrees of freedom 28
5-variation 153
A- variation 153
Differential equation for an orbit 108
Differential scattering cross-section 126
Dispersion curve 261
Dispersion relation 261
Double pendulum 250
Dynamics of a rigid body 273
Eigenvalue equation 242
Eigenvectors 243
Einstein’s mass-energy relation 371
Eikonel216www.cgaspirants.com
478 Classical Mechanics j
Electromagnetic field tensor 427 Gravitational mass 5
Equation of continuity 422 Gyroscope 307
Equilibrium of right-angled level 375 Hamilton-Jacobiequation 197
Ether hypothesis 341 – time-independent 202
Euler-Lagrange’s equations 138, 139 – geometrical optics and wave te chanics 214
Euler’s angles 276 Hamilton’s equations 91 ‘
Euler’s equations of motion for a rigid body 289 – canonical equations 92 |
Fermat’s principle 157,218 -from modified Hamilton’s principle 147 j
Fictitious force 320 Hamiltonian82
First integrals 77 – in relative mechanics 380 j
Force 2 Hamilton’s characteristic function 201, 202 j
-in relativistic mechanics 374 Hamilton’s principal function 52, 198 i
Force-free motion of a symmetrical top 295 51, 140
Foucault’s pendulum 372 -extended 148 I
Four dimensional formulation 327 -fromB’Alembert’s principle 146 j
-modified 147 ,
Four space 389
Four vector 396 Ignorablecoordinates 76 !
-acceleration 399 Impact parameter 127 :
Inertia 2 |
-application 403
-current 424 -ellipsoid 292 j
-moment of 283 1
-force 400 1
-tensor 282
-momentum399
Inertial frames 4
-potential 425 Inertial mass 2 j
-scalar product 397 Infinitesimal contact transformations 173 J
-velocity 398
Integral invariance of Poincare 171 1
Frame of reference 1
Invariable plane 293
-inertial 4
Invariant tori 469 i
-noninertial 320
Inversesquare law of force 109 !
Galilean invariance hypothesis 336 Jacobi’s indenlity 187 ;
Galilean transformations 334 KAM theorem 470
Galilean law of addition of velocities 335 Kepler’s laws 110
Gauge transformation 422 Kepler’s problem-solution by Hamilton-Jacobi
Generalized-coordinates 29, 34 method 204
-force 38 – in action-angle variables 211 j
-momentum75 Kinetic energy 7 !
-potential 49 – in generalized coordinates 41 !
-velocities 38 Lagrange brackets 182 1
Generating functions 164 Lagrange’s equations 40, 4 1 I
Geometrical interpretation of Lorentz transformations -for L-C circuit 46 ‘
411
-from D’Alembert’sprinciple38
Geometrical representation of simultaneity 414 – from variational principle for non-conservative ।
-length contradiction 415 system148 1
-time dilation 415 -from Hamilton’s principle 51 1www.cgaspirants.com
Index 479
obi - in presence of non-conservative forces 47
-of motion for nonholonomicsystems 149 - for a charged particle moving in an electro¬
magnetic field 49
Lagrange’s method of undetermined multipliers 149
Lagrangian 40
-dynamics 27
-inrelativistic mechanics380
Lagendre transformations 163
Length contraction 345
Light cone 396
Ligt like interval 395
Limit cycles 451
Line of nodes 278
Liouville’s theorem 190
Logistic map 455
Lorentz condition 423
Lorentz force on charged particle 432
Lorentz force in covariant form 433
Lorentz transformations 342, 391
-offorce375
-ofelectricand magnetic fields429
Lyapunov exponent 463
Maxwells field equations 421
-interms of electromagnetic potentials Aand
423
-covariance 426
Mechanics of a particle 6
Mechanics of a system of particle 9
Michelson-Morleyexperiment 339
Minkowski diagrams 411
Minkowski space 389
Modes 232
-normal 232
Modified Hamilton’s principle 147
Moment oflnertia 283
Motion under central force 107
itive
N-couples masses 256
Newton’s equation from Lagrange’s equations 41
Newton’s laws of motion 2
Noninertial frames 320
Nonlineardifferential equations 439
Nonlinear systems 444
Normal coordinates 23.8, 245
-frequency 232, 237
-modes 238,245
-mode frequency 245
N-torus453
Nutational angle 278
Phase integral 208
Phase trajectories 439
Phase space 171, 189
Phase velocity 215
Poincaresections 464
Poisson brackets 179
-and quantum mechanics 219
-fundamental 181
Positronium 1 05
Potential energy 7
-and equilibrium232
-curve 233
Principal axes 284
Principal moments of inertia 2’84
Precessional angle 277
Principle of least action 154
-Jacobi’s form 157
Principle of relativity 337
Principle of virtual work 35
Rayleigh’s dissipation function 48
Reduced mass 104
Reduction oftwo-body problem to one-body problem
103
Redudant coordinates 29
Relativistic Doppler’s sffect 355
Relativistic energy 370
Relativistic Hamiltonian of a charged particle 382
Relativistic kineticenergy 370
Relativistic Lagrangian of a charged particle 382
Relativistic law of addition of velocities 349
Rigid body 273
Rotating frames 323
Routherford scattering cross-section 127
Routhian 94
Scattering angle 127 - cross-section 126
- in a central force field 125
- in a repulsive force field 128www.cgaspirants.com
480
Schrodinger equation 217,218
Simple pendulum 42, 90
Simultaneity 345
Singular points 439
Small oscillations 232, 240
Sommerfield-Wilson rule of quantization 219
Space-time continuum 389
Space-time intervals 392
Space and time 1
Special theory of relativity 334, 342
Spherical pendulum 63
Stability of orbit under central force 114
Stable, unstable and neutral equilibrium 233
Strange attractor 462
Superfluous coordinates 29
Symmetrical top 295
-heavy298
-sleeping306
Theorem ofparallel axes 288
Time dilation 346
Classical Mechanics
Torque-free motion of a rigid body 291
Transformations of four vector potential 425
Transformation of force 336, 375
Triatomic molecule (vibrations of linear) 252
Twin paradox 348
Two-body central force problem103
Variation of mass with velocity367, 369
Variational principles 138
Velocity dependent potential 49
Vibrations of continuous string 266
Virial theorem 124
-ofClausius 125
Virtual work 35
Work 7
Work-energy theorem 7
Worldline 392
World point 392
World region 395
World space 385
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