Classical Mechanics

Classical Mechanics
اسم المؤلف
Dr. J.C. Upadhyaya
التاريخ
20 أكتوبر 2022
المشاهدات
343
التقييم
(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#-

  1. 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
  2. Langrangian Dynamics 27—74
    2.1. Introduction 27
    2.2. Basic Concepts 27www.cgaspirants.com
    (x)
    (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
  3. 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)
  4. 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
  5. 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
    (xii)
    Questions 158
    Problems
    — SET-I and SET-II 159 .
    Objective Type Questions 161
    Short Answer Question 162
  6. 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
  7. 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
  8. 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
    (xiii)
    Questions 226
    Problems
    — SET-I and SET-II 226
    Objective Type Questions 230
    Short Answer Questions 231
  9. 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
  10. 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 –
    J1www.cgaspirants.com
    (xiv)
    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
  11. 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
  12. 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
  13. Relativistic Mechanics 367—388/
    13.1. Introduction 367 /
    13.2. Conservation of Momentum at Relativistic Speeds — Variation of Mass /
    with Velocity 367
    / Iwww.cgaspirants.com
    (xv)
    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
  14. 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
  15. 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
    (xvi)
    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
  16. 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
  17. 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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