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أخواني في الله أحضرت لكم كتاب 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)
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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 16. 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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