Analytical Mechanics for Relativity And Quantum Mechanics

by Johns, Oliver

• ISBN13: 9780198567264
• ISBN10: 019856726X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-09-01
• Publisher: Oxford University Press

Summary

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. It is intended for use at the graduate level. A distinguishing feature of the book is its integration of special relativity into the teaching of classical mechanics. Extended Lagrangian and Hamiltonian methods are introduced that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Lagrangians and Hamiltonians, canonical transformations, and the Hamilton-Jacobi equation are developed using this extended theory. This permits the Lorentz transformation of special relativity to become a canonical transformation. This is also a book for those who study analytical mechanics as a preliminary to a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text, and classical mechanics itself is presented in a way that will aid the reader in the study of quantum theory. A chapter is devoted to linear vector operators and dyadics, including a comparison to the bra-ket notation of quantum mechanics. Rotations are presented using an operator formalism similar to that used in quantum theory, and the definition of the Euler angles follows the quantum mechanical convention. The extended Hamiltonian theory with time as a coordinate is compared to Dirac's formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. The book provides a necessary bridge to carry graduate students from their previous undergraduate classical mechanics courses to the future study of advanced relativity and quantum theory. Several of the current fundamental problems in theoretical physics---the development of quantum information technology, and the problem of quantizing the gravitational field, to name two---require a rethinking of the quantum-classical connection. This text is intended to encourage the retention or restoration of introductory graduate analytical mechanics courses. It is written for the intellectually curious graduate student, and the teacher who values mathematical precision in addition to accessibility.

Author Biography

Oliver Davis Johns is Professor of Physics at San Francisco State University.

Table of Contents

Dedication	p. v
Preface	p. vii
Acknowledgments	p. ix
Introduction: The Traditional Theory
Basic Dynamics of Point Particles and Collections	p. 3
Newton's Space and Time	p. 3
Single Point Particle	p. 5
Collective Variables	p. 6
The Law of Momentum for Collections	p. 7
The Law of Angular Momentum for Collections	p. 8
"Derivations" of the Axioms	p. 9
The Work-Energy Theorem for Collections	p. 10
Potential and Total Energy for Collections	p. 11
The Center of Mass	p. 11
Center of Mass and Momentum	p. 13
Center of Mass and Angular Momentum	p. 14
Center of Mass and Torque	p. 15
Change of Angular Momentum	p. 15
Center of Mass and the Work-Energy Theorems	p. 16
Center of Mass as a Point Particle	p. 17
Special Results for Rigid Bodies	p. 17
Exercises	p. 18
Introduction to Lagrangian Mechanics	p. 24
Configuration Space	p. 24
Newton's Second Law in Lagrangian Form	p. 26
A Simple Example	p. 27
Arbitrary Generalized Coordinates	p. 27
Generalized Velocities in the q-System	p. 29
Generalized Forces in the q-System	p. 29
The Lagrangian Expressed in the q-System	p. 30
Two Important Identities	p. 31
Invariance of the Lagrange Equations	p. 32
Relation Between Any Two Systems	p. 33
More of the Simple Example	p. 34
Generalized Momenta in the q-System	p. 35
Ignorable Coordinates	p. 35
Some Remarks About Units	p. 36
The Generalized Energy Function	p. 36
The Generalized Energy and the Total Energy	p. 37
Velocity Dependent Potentials	p. 38
Exercises	p. 41
Lagrangian Theory of Constraints	p. 46
Constraints Defined	p. 46
Virtual Displacement	p. 47
Virtual Work	p. 48
Form of the Forces of Constraint	p. 50
General Lagrange Equations with Constraints	p. 52
An Alternate Notation for Holonomic Constraints	p. 53
Example of the General Method	p. 54
Reduction of Degrees of Freedom	p. 54
Example of a Reduction	p. 57
Example of a Simpler Reduction Method	p. 58
Recovery of the Forces of Constraint	p. 59
Example of a Recovery	p. 60
Generalized Energy Theorem with Constraints	p. 61
Tractable Non-Holonomic Constraints	p. 63
Exercises	p. 64
Introduction to Hamiltonian Mechanics	p. 71
Phase Space	p. 71
Hamilton Equations	p. 74
An Example of the Hamilton Equations	p. 76
Non-Potential and Constraint Forces	p. 77
Reduced Hamiltonian	p. 78
Poisson Brackets	p. 80
The Schroedinger Equation	p. 82
The Ehrenfest Theorem	p. 83
Exercises	p. 84
The Calculus of Variations	p. 88
Paths in an N-Dimensional Space	p. 89
Variations of Coordinates	p. 90
Variations of Functions	p. 91
Variation of a Line Integral	p. 92
Finding Extremum Paths	p. 94
Example of an Extremum Path Calculation	p. 95
Invariance and Homogeneity	p. 98
The Brachistochrone Problem	p. 100
Calculus of Variations with Constraints	p. 102
An Example with Constraints	p. 105
Reduction of Degrees of Freedom	p. 106
Example of a Reduction	p. 107
Example of a Better Reduction	p. 108
The Coordinate Parametric Method	p. 108
Comparison of the Methods	p. 111
Exercises	p. 113
Hamilton's Principle	p. 117
Hamilton's Principle in Lagrangian Form	p. 117
Hamilton's Principle with Constraints	p. 118
Comments on Hamilton's Principle	p. 119
Phase-Space Hamilton's Principle	p. 120
Exercises	p. 122
Linear Operators and Dyadics	p. 123
Definition of Operators	p. 123
Operators and Matrices	p. 125
Addition and Multiplication	p. 127
Determinant, Trace, and Inverse	p. 127
Special Operators	p. 129
Dyadics	p. 130
Resolution of Unity	p. 133
Operators, Components, Matrices, and Dyadics	p. 133
Complex Vectors and Operators	p. 134
Real and Complex Inner Products	p. 136
Eigenvectors and Eigenvalues	p. 136
Eigenvectors of Real Symmetric Operator	p. 137
Eigenvectors of Real Anti-Symmetric Operator	p. 137
Normal Operators	p. 139
Determinant and Trace of Normal Operator	p. 141
Eigen-Dyadic Expansion of Normal Operator	p. 142
Functions of Normal Operators	p. 143
The Exponential Function	p. 145
The Dirac Notation	p. 146
Exercises	p. 147
Kinematics of Rotation	p. 152
Characterization of Rigid Bodies	p. 152
The Center of Mass of a Rigid Body	p. 153
General Definition of Rotation Operator	p. 155
Rotation Matrices	p. 157
Some Properties of Rotation Operators	p. 158
Proper and Improper Rotation Operators	p. 158
The Rotation Group	p. 160
Kinematics of a Rigid Body	p. 161
Rotation Operators and Rigid Bodies	p. 163
Differentiation of a Rotation Operator	p. 164
Meaning of the Angular Velocity Vector	p. 166
Velocities of the Masses of a Rigid Body	p. 168
Savio's Theorem	p. 169
Infinitesimal Rotation	p. 170
Addition of Angular Velocities	p. 171
Fundamental Generators of Rotations	p. 172
Rotation with a Fixed Axis	p. 174
Expansion of Fixed-Axis Rotation	p. 176
Eigenvectors of the Fixed-Axis Rotation Operator	p. 178
The Euler Theorem	p. 179
Rotation of Operators	p. 181
Rotation of the Fundamental Generators	p. 181
Rotation of a Fixed-Axis Rotation	p. 182
Parameterization of Rotation Operators	p. 183
Differentiation of Parameterized Operator	p. 184
Euler Angles	p. 185
Fixed-Axis Rotation from Euler Angles	p. 188
Time Derivative of a Product	p. 189
Angular Velocity from Euler Angles	p. 190
Active and Passive Rotations	p. 191
Passive Transformation of Vector Components	p. 192
Passive Transformation of Matrix Elements	p. 193
The Body Derivative	p. 194
Passive Rotations and Rigid Bodies	p. 195
Passive Use of Euler Angles	p. 196
Exercises	p. 198
Rotational Dynamics	p. 202
Basic Facts of Rigid-Body Motion	p. 202
The Inertia Operator and the Spin	p. 203
The Inertia Dyadic	p. 204
Kinetic Energy of a Rigid Body	p. 205
Meaning of the Inertia Operator	p. 205
Principal Axes	p. 206
Guessing the Principal Axes	p. 208
Time Evolution of the Spin	p. 210
Torque-Free Motion of a Symmetric Body	p. 211
Euler Angles of the Torque-Free Motion	p. 215
Body with One Point Fixed	p. 217
Preserving the Principal Axes	p. 220
Time Evolution with One Point Fixed	p. 221
Body with One Point Fixed, Alternate Derivation	p. 221
Work-Energy Theorems	p. 222
Rotation with a Fixed Axis	p. 223
The Symmetric Top with One Point Fixed	p. 224
The Initially Clamped Symmetric Top	p. 229
Approximate Treatment of the Symmetric Top	p. 230
Inertial Forces	p. 231
Laboratory on the Surface of the Earth	p. 234
Coriolis Force Calculations	p. 236
The Magnetic - Coriolis Analogy	p. 237
Exercises	p. 239
Small Vibrations About Equilibrium	p. 246
Equilibrium Defined	p. 246
Finding Equilibrium Points	p. 247
Small Coordinates	p. 248
Normal Modes	p. 249
Generalized Eigenvalue Problem	p. 250
Stability	p. 252
Initial Conditions	p. 252
The Energy of Small Vibrations	p. 253
Single Mode Excitations	p. 254
A Simple Example	p. 255
Zero-Frequency Modes	p. 260
Exercises	p. 261
Mechanics with Time as a Coordinate
Lagrangian Mechanics with Time as a Coordinate	p. 267
Time as a Coordinate	p. 268
A Change of Notation	p. 268
Extended Lagrangian	p. 269
Extended Momenta	p. 270
Extended Lagrange Equations	p. 272
A Simple Example	p. 273
Invariance Under Change of Parameter	p. 275
Change of Generalized Coordinates	p. 276
Redundancy of the Extended Lagrange Equations	p. 277
Forces of Constraint	p. 278
Reduced Lagrangians with Time as a Coordinate	p. 281
Exercises	p. 282
Hamiltonian Mechanics with Time as a Coordinate	p. 285
Extended Phase Space	p. 285
Dependency Relation	p. 285
Only One Dependency Relation	p. 286
From Traditional to Extended Hamiltonian Mechanics	p. 288
Equivalence to Traditional Hamilton Equations	p. 290
Example of Extended Hamilton Equations	p. 291
Equivalent Extended Hamiltonians	p. 292
Alternate Hamiltonians	p. 293
Alternate Traditional Hamiltonians	p. 295
Not a Legendre Transformation	p. 295
Dirac's Theory of Phase-Space Constraints	p. 296
Poisson Brackets with Time as a Coordinate	p. 298
Poisson Brackets and Quantum Commutators	p. 300
Exercises	p. 302
Hamilton's Principle and Noether's Theorem	p. 305
Extended Hamilton's Principle	p. 305
Noether's Theorem	p. 307
Examples of Noether's Theorem	p. 308
Hamilton's Principle in an Extended Phase Space	p. 310
Exercises	p. 312
Relativity and Spacetime	p. 313
Galilean Relativity	p. 313
Conflict with the Aether	p. 315
Einsteinian Relativity	p. 316
What Is a Coordinate System?	p. 318
A Survey of Spacetime	p. 319
The Lorentz Transformation	p. 331
The Principle of Relativity	p. 337
Lorentzian Relativity	p. 339
Mechanism and Relativity	p. 340
Exercises	p. 341
Fourvectors and Operators	p. 343
Fourvectors	p. 343
Inner Product	p. 346
Choice of Metric	p. 347
Relativistic Interval	p. 347
Spacetime Diagram	p. 349
General Fourvectors	p. 350
Construction of New Fourvectors	p. 351
Covariant and Contravariant Components	p. 352
General Lorentz Transformations	p. 355
Transformation of Components	p. 356
Examples of Lorentz Transformations	p. 358
Gradient Fourvector	p. 360
Manifest Covariance	p. 361
Formal Covariance	p. 362
The Lorentz Group	p. 362
Proper Lorentz Transformations and the Little Group	p. 364
Parameterization	p. 364
Fourvector Operators	p. 366
Fourvector Dyadics	p. 367
Wedge Products	p. 368
Scalar, Fourvector, and Operator Fields	p. 369
Manifestly Covariant Form of Maxwell's Equations	p. 370
Exercises	p. 373
Relativistic Mechanics	p. 376
Modification of Newton's Laws	p. 376
The Momentum Fourvector	p. 378
Fourvector Form of Newton's Second Law	p. 378
Conservation of Fourvector Momentum	p. 380
Particles of Zero Mass	p. 380
Traditional Lagrangian	p. 381
Traditional Hamiltonian	p. 383
Invariant Lagrangian	p. 383
Manifestly Covariant Lagrange Equations	p. 384
Momentum Fourvectors and Canonical Momenta	p. 385
Extended Hamiltonian	p. 386
Invariant Hamiltonian	p. 387
Manifestly Covariant Hamilton Equations	p. 388
The Klein-Gordon Equation	p. 389
The Dirac Equation	p. 390
The Manifestly Covariant N-Body Problem	p. 392
Covariant Serret-Frenet Theory	p. 399
Fermi-Walker Transport	p. 401
Example of Fermi-Walker Transport	p. 403
Exercises	p. 405
Canonical Transformations	p. 411
Definition of Canonical Transformations	p. 411
Example of a Canonical Transformation	p. 412
Symplectic Coordinates	p. 412
Symplectic Matrix	p. 416
Standard Equations in Symplectic Form	p. 417
Poisson Bracket Condition	p. 418
Inversion of Canonical Transformations	p. 419
Direct Condition	p. 420
Lagrange Bracket Condition	p. 422
The Canonical Group	p. 423
Form Invariance of Poisson Brackets	p. 424
Form Invariance of the Hamilton Equations	p. 426
Traditional Canonical Transformations	p. 428
Exercises	p. 430
Generating Functions	p. 434
Proto-Generating Functions	p. 434
Generating Functions of the F[subscript 1] Type	p. 436
Generating Functions of the F[subscript 2] Type	p. 438
Examples of Generating Functions	p. 439
Other Simple Generating Functions	p. 441
Mixed Generating Functions	p. 442
Example of a Mixed Generating Function	p. 444
Finding Simple Generating Functions	p. 445
Finding Mixed Generating Functions	p. 446
Finding Mixed Generating Functions-An Example	p. 448
Traditional Generating Functions	p. 449
Standard Form of Extended Hamiltonian Recovered	p. 451
Differential Canonical Transformations	p. 452
Active Canonical Transformations	p. 453
Phase-Space Analog of Noether Theorem	p. 454
Liouville Theorem	p. 455
Exercises	p. 456
Hamilton-Jacobi Theory	p. 461
Definition of the Action	p. 461
Momenta from the S[subscript 1] Action Function	p. 462
The S[subscript 2] Action Function	p. 464
Example of S[subscript 1] and S[subscript 2] Action Functions	p. 465
The Hamilton-Jacobi Equation	p. 466
Hamilton's Characteristic Equations	p. 467
Complete Integrals	p. 469
Separation of Variables	p. 472
Canonical Transformations	p. 473
General Integrals	p. 475
Mono-Energetic Integrals	p. 480
The Optical Analogy	p. 482
The Relativistic Hamilton-Jacobi Equation	p. 483
Schroedinger and Hamilton-Jacobi Equations	p. 483
The Quantum Cauchy Problem	p. 485
The Bohm Hidden Variable Model	p. 486
Feynman Path-Integral Technique	p. 487
Quantum and Classical Mechanics	p. 488
Exercises	p. 489
Mathematical Appendices
Vector Fundamentals	p. 495
Properties of Vectors	p. 495
Dot Product	p. 495
Cross Product	p. 496
Linearity	p. 496
Cartesian Basis	p. 497
The Position Vector	p. 498
Fields	p. 499
Polar Coordinates	p. 499
The Algebra of Sums	p. 502
Miscellaneous Vector Formulae	p. 502
Gradient Vector Operator	p. 504
The Serret-Frenet Formulae	p. 505
Matrices and Determinants	p. 508
Definition of Matrices	p. 508
Transposed Matrix	p. 508
Column Matrices and Column Vectors	p. 509
Square, Symmetric, and Hermitian Matrices	p. 509
Algebra of Matrices: Addition	p. 510
Algebra of Matrices: Multiplication	p. 511
Diagonal and Unit Matrices	p. 512
Trace of a Square Matrix	p. 513
Differentiation of Matrices	p. 513
Determinants of Square Matrices	p. 513
Properties of Determinants	p. 514
Cofactors	p. 515
Expansion of a Determinant by Cofactors	p. 515
Inverses of Nonsingular Matrices	p. 516
Partitioned Matrices	p. 517
Cramer's Rule	p. 518
Minors and Rank	p. 519
Linear Independence	p. 520
Homogeneous Linear Equations	p. 520
Inner Products of Column Vectors	p. 521
Complex Inner Products	p. 523
Orthogonal and Unitary Matrices	p. 523
Eigenvalues and Eigenvectors of Matrices	p. 524
Eigenvectors of Real Symmetric Matrix	p. 525
Eigenvectors of Complex Hermitian Matrix	p. 528
Normal Matrices	p. 528
Properties of Normal Matrices	p. 530
Functions of Normal Matrices	p. 533
Eigenvalue Problem with General Metric	p. 534
Positive-Definite Matrices	p. 534
Generalization of the Real Inner Product	p. 535
The Generalized Eigenvalue Problem	p. 536
Finding Eigenvectors in the Generalized Problem	p. 537
Uses of the Generalized Eigenvectors	p. 538
The Calculus of Many Variables	p. 540
Basic Properties of Functions	p. 540
Regions of Definition of Functions	p. 540
Continuity of Functions	p. 541
Compound Functions	p. 541
The Same Function in Different Coordinates	p. 541
Partial Derivatives	p. 542
Continuously Differentiable Functions	p. 543
Order of Differentiation	p. 543
Chain Rule	p. 543
Mean Values	p. 544
Orders of Smallness	p. 544
Differentials	p. 545
Differential of a Function of Several Variables	p. 545
Differentials and the Chain Rule	p. 546
Differentials of Second and Higher Orders	p. 546
Taylor Series	p. 547
Higher-Order Differential as a Difference	p. 548
Differential Expressions	p. 548
Line Integral of a Differential Expression	p. 550
Perfect Differentials	p. 550
Perfect Differential and Path Independence	p. 552
Jacobians	p. 553
Global Inverse Function Theorem	p. 556
Local Inverse Function Theorem	p. 559
Derivatives of the Inverse Functions	p. 560
Implicit Function Theorem	p. 561
Derivatives of Implicit Functions	p. 561
Functional Independence	p. 562
Dependency Relations	p. 563
Legendre Transformations	p. 563
Homogeneous Functions	p. 565
Derivatives of Homogeneous Functions	p. 565
Stationary Points	p. 566
Lagrange Multipliers	p. 566
Geometry of the Lagrange Multiplier Theorem	p. 569
Coupled Differential Equations	p. 570
Surfaces and Envelopes	p. 572
Geometry of Phase Space	p. 575
Abstract Vector Space	p. 575
Subspaces	p. 577
Linear Operators	p. 578
Vectors in Phase Space	p. 580
Canonical Transformations in Phase Space	p. 581
Orthogonal Subspaces	p. 582
A Special Canonical Transformation	p. 582
Special Self-Orthogonal Subspaces	p. 583
Arnold's Theorem	p. 585
Existence of a Mixed Generating Function	p. 586
References	p. 588
Index	p. 591
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