Analytical mechanics for relativity and quantum mechanics / Oliver Davis Johns.

Format:

Book

Author/Creator:

Johns, Oliver Davis.

Series:

Oxford graduate texts

Language:

English

Subjects (All):

Mechanics, Analytic.

Physical Description:

xx, 597 pages : illustrations ; 25 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2005.

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 introductory graduate level.

A distinguishing feature of the book is its integration of special relativity into the teaching of classical mechanics. After a thorough review of the traditional theory, Part II of the book introduces extended Lagrangian and Hamiltonian methods 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 Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorentz transformation of special relativity.

This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. 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. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarize the student with similar techniques in quantum theory.

Graduate students preparing for research careers will find a graduate mechanics course based on this book to be an essential bridge between their undergraduate training and advanced study in analytical mechanics, relativity, and quantum mechanics.

Contents:

Part I Introduction: The Traditional Theory

1 Basic Dynamics of Point Particles and Collections 3

1.1 Newton's Space and Time 3

1.2 Single Point Particle 5

1.3 Collective Variables 6

1.4 The Law of Momentum for Collections 7

1.5 The Law of Angular Momentum for Collections 8

1.6 "Derivations" of the Axioms 9

1.7 The Work-Energy Theorem for Collections 10

1.8 Potential and Total Energy for Collections 11

1.9 The Center of Mass 11

1.10 Center of Mass and Momentum 13

1.11 Center of Mass and Angular Momentum 14

1.12 Center of Mass and Torque 15

1.13 Change of Angular Momentum 15

1.14 Center of Mass and the Work-Energy Theorems 16

1.15 Center of Mass as a Point Particle 17

1.16 Special Results for Rigid Bodies 17

2 Introduction to Lagrangian Mechanics 24

2.1 Configuration Space 24

2.2 Newton's Second Law in Lagrangian Form 26

2.4 Arbitrary Generalized Coordinates 27

2.5 Generalized Velocities in the q-System 29

2.6 Generalized Forces in the q-System 29

2.7 The Lagrangian Expressed in the q-System 30

2.8 Two Important Identities 31

2.9 Invariance of the Lagrange Equations 32

2.10 Relation Between Any Two Systems 33

2.11 More of the Simple Example 34

2.12 Generalized Momenta in the q-System 35

2.13 Ignorable Coordinates 35

2.14 Some Remarks About Units 36

2.15 The Generalized Energy Function 36

2.16 The Generalized Energy and the Total Energy 37

2.17 Velocity Dependent Potentials 38

3 Lagrangian Theory of Constraints 46

3.1 Constraints Defined 46

3.2 Virtual Displacement 47

3.3 Virtual Work 48

3.4 Form of the Forces of Constraint 50

3.5 General Lagrange Equations with Constraints 52

3.6 An Alternate Notation for Holonomic Constraints 53

3.7 Example of the General Method 54

3.8 Reduction of Degrees of Freedom 54

3.9 Example of a Reduction 57

3.10 Example of a Simpler Reduction Method 58

3.11 Recovery of the Forces of Constraint 59

3.12 Example of a Recovery 60

3.13 Generalized Energy Theorem with Constraints 61

3.14 Tractable Non-Holonomic Constraints 63

4 Introduction to Hamiltonian Mechanics 71

4.1 Phase Space 71

4.2 Hamilton Equations 74

4.3 An Example of the Hamilton Equations 76

4.4 Non-Potential and Constraint Forces 77

4.5 Reduced Hamiltonian 78

4.6 Poisson Brackets 80

4.7 The Schroedinger Equation 82

4.8 The Ehrenfest Theorem 83

5 The Calculus of Variations 88

5.1 Paths in an N-Dimensional Space 89

5.2 Variations of Coordinates 90

5.3 Variations of Functions 91

5.4 Variation of a Line Integral 92

5.5 Finding Extremum Paths 94

5.6 Example of an Extremum Path Calculation 95

5.7 Invariance and Homogeneity 98

5.8 The Brachistochrone Problem 100

5.9 Calculus of Variations with Constraints 102

5.10 An Example with Constraints 105

5.11 Reduction of Degrees of Freedom 106

5.12 Example of a Reduction 107

5.13 Example of a Better Reduction 108

5.14 The Coordinate Parametric Method 108

5.15 Comparison of the Methods 111

6 Hamilton's Principle 117

6.1 Hamilton's Principle in Lagrangian Form 117

6.2 Hamilton's Principle with Constraints 118

6.3 Comments on Hamilton's Principle 119

6.4 Phase-Space Hamilton's Principle 120

7 Linear Operators and Dyadics 123

7.1 Definition of Operators 123

7.2 Operators and Matrices 125

7.3 Addition and Multiplication 127

7.4 Determinant, Trace, and Inverse 127

7.5 Special Operators 129

7.6 Dyadics 130

7.7 Resolution of Unity 133

7.8 Operators, Components, Matrices, and Dyadics 133

7.9 Complex Vectors and Operators 134

7.10 Real and Complex Inner Products 136

7.11 Eigenvectors and Eigenvalues 136

7.12 Eigenvectors of Real Symmetric Operator 137

7.13 Eigenvectors of Real Anti-Symmetric Operator 137

7.14 Normal Operators 139

7.15 Determinant and Trace of Normal Operator 141

7.16 Eigen-Dyadic Expansion of Normal Operator 142

7.17 Functions of Normal Operators 143

7.18 The Exponential Function 145

7.19 The Dirac Notation 146

8 Kinematics of Rotation 152

8.1 Characterization of Rigid Bodies 152

8.2 The Center of Mass of a Rigid Body 153

8.3 General Definition of Rotation Operator 155

8.4 Rotation Matrices 157

8.5 Some Properties of Rotation Operators 158

8.6 Proper and Improper Rotation Operators 158

8.7 The Rotation Group 160

8.8 Kinematics of a Rigid Body 161

8.9 Rotation Operators and Rigid Bodies 163

8.10 Differentiation of a Rotation Operator 164

8.11 Meaning of the Angular Velocity Vector 166

8.12 Velocities of the Masses of a Rigid Body 168

8.13 Savio's Theorem 169

8.14 Infinitesimal Rotation 170

8.15 Addition of Angular Velocities 171

8.16 Fundamental Generators of Rotations 172

8.17 Rotation with a Fixed Axis 174

8.18 Expansion of Fixed-Axis Rotation 176

8.19 Eigenvectors of the Fixed-Axis Rotation Operator 178

8.20 The Euler Theorem 179

8.21 Rotation of Operators 181

8.22 Rotation of the Fundamental Generators 181

8.23 Rotation of a Fixed-Axis Rotation 182

8.24 Parameterization of Rotation Operators 183

8.25 Differentiation of Parameterized Operator 184

8.26 Euler Angles 185

8.27 Fixed-Axis Rotation from Euler Angles 188

8.28 Time Derivative of a Product 189

8.29 Angular Velocity from Euler Angles 190

8.30 Active and Passive Rotations 191

8.31 Passive Transformation of Vector Components 192

8.32 Passive Transformation of Matrix Elements 193

8.33 The Body Derivative 194

8.34 Passive Rotations and Rigid Bodies 195

8.35 Passive Use of Euler Angles 196

9 Rotational Dynamics 202

9.1 Basic Facts of Rigid-Body Motion 202

9.2 The Inertia Operator and the Spin 203

9.3 The Inertia Dyadic 204

9.4 Kinetic Energy of a Rigid Body 205

9.5 Meaning of the Inertia Operator 205

9.6 Principal Axes 206

9.7 Guessing the Principal Axes 208

9.8 Time Evolution of the Spin 210

9.9 Torque-Free Motion of a Symmetric Body 211

9.10 Euler Angles of the Torque-Free Motion 215

9.11 Body with One Point Fixed 217

9.12 Preserving the Principal Axes 220

9.13 Time Evolution with One Point Fixed 221

9.14 Body with One Point Fixed, Alternate Derivation 221

9.15 Work-Energy Theorems 222

9.16 Rotation with a Fixed Axis 223

9.17 The Symmetric Top with One Point Fixed 224

9.18 The Initially Clamped Symmetric Top 229

9.19 Approximate Treatment of the Symmetric Top 230

9.20 Inertial Forces 231

9.21 Laboratory on the Surface of the Earth 234

9.22 Coriolis Force Calculations 236

9.23 The Magnetic - Coriolis Analogy 237

10 Small Vibrations About Equilibrium 246

10.1 Equilibrium Defined 246

10.2 Finding Equilibrium Points 247

10.3 Small Coordinates 248

10.4 Normal Modes 249

10.5 Generalized Eigenvalue Problem 250

10.6 Stability 252

10.7 Initial Conditions 252

10.8 The Energy of Small Vibrations 253

10.9 Single Mode Excitations 254

10.11 Zero-Frequency Modes 260

Part II Mechanics with Time as a Coordinate

11 Lagrangian Mechanics with Time as a Coordinate 267

11.1 Time as a Coordinate 268

11.2 A Change of Notation 268

11.3 Extended Lagrangian 269

11.4 Extended Momenta 270

11.5 Extended Lagrange Equations 272

11.7 Invariance Under Change of Parameter 275

11.8 Change of Generalized Coordinates 276

11.9 Redundancy of the Extended Lagrange Equations 277

11.10 Forces of Constraint 278

11.11 Reduced Lagrangians with Time as a Coordinate 281

12 Hamiltonian Mechanics with Time as a Coordinate 285

12.1 Extended Phase Space 285

12.2 Dependency Relation 285

12.3 Only One Dependency Relation 286

12.4 From Traditional to Extended Hamiltonian Mechanics 288

12.5 Equivalence to Traditional Hamilton Equations 290

12.6 Example of Extended Hamilton Equations 291

12.7 Equivalent Extended Hamiltonians 292

12.8 Alternate Hamiltonians 293

12.9 Alternate Traditional Hamiltonians 295

12.10 Not a Legendre Transformation 295

12.11 Dirac's Theory of Phase-Space Constraints 296

12.12 Poisson Brackets with Time as a Coordinate 298

12.13 Poisson Brackets and Quantum Commutators 300

13 Hamilton's Principle and Noether's Theorem 305

13.1 Extended Hamilton's Principle 305

13.2 Noether's Theorem 307

13.3 Examples of Noether's Theorem 308

13.4 Hamilton's Principle in an Extended Phase Space 310

14 Relativity and Spacetime 313

14.1 Galilean Relativity 313

14.2 Conflict with the Aether 315

14.3 Einsteinian Relativity 316

14.4 What Is a Coordinate System? 318

14.5 A Survey of Spacetime 319

14.6 The Lorentz Transformation 331

14.7 The Principle of Relativity 337

14.8 Lorentzian Relativity 339

14.9 Mechanism and Relativity 340

15 Fourvectors and Operators 343

15.1 Fourvectors 343

15.2 Inner Product 346

15.3 Choice of Metric 347

15.4 Relativistic Interval 347

15.5 Spacetime Diagram 349

15.6 General Fourvectors 350

15.7 Construction of New Fourvectors 351

15.8 Covariant and Contravariant Components 352

15.9 General Lorentz Transformations 355

15.10 Transformation of Components 356

15.11 Examples of Lorentz Transformations 358

15.12 Gradient Fourvector 360

15.13 Manifest Covariance 361

15.14 Formal Covariance 362

15.15 The Lorentz Group 362

15.16 Proper Lorentz Transformations and the Little Group 364

15.17 Parameterization 364

15.18 Fourvector Operators 366

15.19 Fourvector Dyadics 367

15.20 Wedge Products 368

15.21 Scalar, Fourvector, and Operator Fields 369

15.22 Manifestly Covariant Form of Maxwell's Equations 370

16 Relativistic Mechanics 376

16.1 Modification of Newton's Laws 376

16.2 The Momentum Fourvector 378

16.3 Fourvector Form of Newton's Second Law 378

16.4 Conservation of Fourvector Momentum 380

16.5 Particles of Zero Mass 380

16.6 Traditional Lagrangian 381

16.7 Traditional Hamiltonian 383

16.8 Invariant Lagrangian 383

16.9 Manifestly Covariant Lagrange Equations 384

16.10 Momentum Fourvectors and Canonical Momenta 385

16.11 Extended Hamiltonian 386

16.12 Invariant Hamiltonian 387

16.13 Manifestly Covariant Hamilton Equations 388

16.14 The Klein-Gordon Equation 389

16.15 The Dirac Equation 390

16.16 The Manifestly Covariant N-Body Problem 392

16.17 Covariant Serret-Frenet Theory 399

16.18 Fermi-Walker Transport 401

16.19 Example of Fermi-Walker Transport 403

17 Canonical Transformations 411

17.1 Definition of Canonical Transformations 411

17.2 Example of a Canonical Transformation 412

17.3 Symplectic Coordinates 412

17.4 Symplectic Matrix 416

17.5 Standard Equations in Symplectic Form 417

17.6 Poisson Bracket Condition 418

17.7 Inversion of Canonical Transformations 419

17.8 Direct Condition 420

17.9 Lagrange Bracket Condition 422

17.10 The Canonical Group 423

17.11 Form Invariance of Poisson Brackets 424

17.12 Form Invariance of the Hamilton Equations 426

17.13 Traditional Canonical Transformations 428

18 Generating Functions 434

18.1 Proto-Generating Functions 434

18.2 Generating Functions of the F[subscript 1] Type 436

18.3 Generating Functions of the F[subscript 2] Type 438

18.4 Examples of Generating Functions 439

18.5 Other Simple Generating Functions 441

18.6 Mixed Generating Functions 442

18.7 Example of a Mixed Generating Function 444

18.8 Finding Simple Generating Functions 445

18.9 Finding Mixed Generating Functions 446

18.10 Finding Mixed Generating Functions-An Example 448

18.11 Traditional Generating Functions 449

18.12 Standard Form of Extended Hamiltonian Recovered 451

18.13 Differential Canonical Transformations 452

18.14 Active Canonical Transformations 453

18.15 Phase-Space Analog of Noether Theorem 454

18.16 Liouville Theorem 455

19 Hamilton-Jacobi Theory 461

19.1 Definition of the Action 461

19.2 Momenta from the S[subscript 1] Action Function 462

19.3 The S[subscript 2] Action Function 464

19.4 Example of S[subscript 1] and S[subscript 2] Action Functions 465

19.5 The Hamilton-Jacobi Equation 466

19.6 Hamilton's Characteristic Equations 467

19.7 Complete Integrals 469

19.8 Separation of Variables 472

19.9 Canonical Transformations 473

19.10 General Integrals 475

19.11 Mono-Energetic Integrals 480

19.12 The Optical Analogy 482

19.13 The Relativistic Hamilton-Jacobi Equation 483

19.14 Schroedinger and Hamilton-Jacobi Equations 483

19.15 The Quantum Cauchy Problem 485

19.16 The Bohm Hidden Variable Model 486

19.17 Feynman Path-Integral Technique 487

19.18 Quantum and Classical Mechanics 488

Part III Mathematical Appendices

A Vector Fundamentals 495

A.1 Properties of Vectors 495

A.2 Dot Product 495

A.3 Cross Product 496

A.4 Linearity 496

A.5 Cartesian Basis 497

A.6 The Position Vector 498

A.7 Fields 499

A.8 Polar Coordinates 499

A.9 The Algebra of Sums 502

A.10 Miscellaneous Vector Formulae 502

A.11 Gradient Vector Operator 504

A.12 The Serret-Frenet Formulae 505

B Matrices and Determinants 508

B.1 Definition of Matrices 508

B.2 Transposed Matrix 508

B.3 Column Matrices and Column Vectors 509

B.4 Square, Symmetric, and Hermitian Matrices 509

B.5 Algebra of Matrices: Addition 510

B.6 Algebra of Matrices: Multiplication 511

B.7 Diagonal and Unit Matrices 512

B.8 Trace of a Square Matrix 513

B.9 Differentiation of Matrices 513

B.10 Determinants of Square Matrices 513

B.11 Properties of Determinants 514

B.12 Cofactors 515

B.13 Expansion of a Determinant by Cofactors 515

B.14 Inverses of Nonsingular Matrices 516

B.15 Partitioned Matrices 517

B.16 Cramer's Rule 518

B.17 Minors and Rank 519

B.18 Linear Independence 520

B.19 Homogeneous Linear Equations 520

B.20 Inner Products of Column Vectors 521

B.21 Complex Inner Products 523

B.22 Orthogonal and Unitary Matrices 523

B.23 Eigenvalues and Eigenvectors of Matrices 524

B.24 Eigenvectors of Real Symmetric Matrix 525

B.25 Eigenvectors of Complex Hermitian Matrix 528

B.26 Normal Matrices 528

B.27 Properties of Normal Matrices 530

B.28 Functions of Normal Matrices 533

C Eigenvalue Problem with General Metric 534

C.1 Positive-Definite Matrices 534

C.2 Generalization of the Real Inner Product 535

C.3 The Generalized Eigenvalue Problem 536

C.4 Finding Eigenvectors in the Generalized Problem 537

C.5 Uses of the Generalized Eigenvectors 538

D The Calculus of Many Variables 540

D.1 Basic Properties of Functions 540

D.2 Regions of Definition of Functions 540

D.3 Continuity of Functions 541

D.4 Compound Functions 541

D.5 The Same Function in Different Coordinates 541

D.6 Partial Derivatives 542

D.7 Continuously Differentiable Functions 543

D.8 Order of Differentiation 543

D.9 Chain Rule 543

D.10 Mean Values 544

D.11 Orders of Smallness 544

D.12 Differentials 545

D.13 Differential of a Function of Several Variables 545

D.14 Differentials and the Chain Rule 546

D.15 Differentials of Second and Higher Orders 546

D.16 Taylor Series 547

D.17 Higher-Order Differential as a Difference 548

D.18 Differential Expressions 548

D.19 Line Integral of a Differential Expression 550

D.20 Perfect Differentials 550

D.21 Perfect Differential and Path Independence 552

D.22 Jacobians 553

D.23 Global Inverse Function Theorem 556

D.24 Local Inverse Function Theorem 559

D.25 Derivatives of the Inverse Functions 560

D.26 Implicit Function Theorem 561

D.27 Derivatives of Implicit Functions 561

D.28 Functional Independence 562

D.29 Dependency Relations 563

D.30 Legendre Transformations 563

D.31 Homogeneous Functions 565

D.32 Derivatives of Homogeneous Functions 565

D.33 Stationary Points 566

D.34 Lagrange Multipliers 566

D.35 Geometry of the Lagrange Multiplier Theorem 569

D.36 Coupled Differential Equations 570

D.37 Surfaces and Envelopes 572

E Geometry of Phase Space 575

E.1 Abstract Vector Space 575

E.2 Subspaces 577

E.3 Linear Operators 578

E.4 Vectors in Phase Space 580

E.5 Canonical Transformations in Phase Space 581

E.6 Orthogonal Subspaces 582

E.7 A Special Canonical Transformation 582

E.8 Special Self-Orthogonal Subspaces 583

E.9 Arnold's Theorem 585

E.10 Existence of a Mixed Generating Function 586.

Notes:

Includes bibliographical references (pages 588-590) and index.

ISBN:

019856726X

OCLC:

60512562