Analytical Mechanics for Relativity and Quantum Mechanics

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1 Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS

2 CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION: THE TRADITIONAL THEORY 1 Basic Dynamics of Point Particles and Collections Newton's Space and Time Single Point Particle Collective Variables The Law of Momentum for Collections The Law of Angular Momentum for Collections "Derivations" of the Axioms The Work-Energy Theorem for Collections Potential and Total Energy for Collections The Center of Mass Center of Mass and Momentum Center of Mass and Angular Momentum Center of Mass and Torque Change of Angular Momentum Center of Mass and the Work-Energy Theorems Center of Mass as a Point Particle Special Results for Rigid Bodies Exercises 18 2 Introduction to Lagrangian Mechanics Configuration Space Newton's Second Law in Lagrangian Form, A Simple Example Arbitrary Generalized Coordinates Generalized Velocities in the q-system Generalized Forces in the q-system The Lagrangian Expressed in the q-system Two Important Identities Invariance of the Lagrange Equations Relation Between Any Two Systems More of the Simple Example Generalized Momenta in the q-system Ignorable Coordinates Some Remarks About Units 36 XI

3 xii CONTENTS 2.15 The Generalized Energy Function The Generalized Energy and the Total Energy Velocity Dependent Potentials Exercises 41 3 Lagrangian Theory of Constraints Constraints Defined Virtual Displacement Virtual Work Form of the Forces of Constraint General Lagrange Equations with Constraints An Alternate Notation for Holonomic Constraints Example of the General Method Reduction of Degrees of Freedom Example of a Reduction Example of a Simpler Reduction Method Recovery of the Forces of Constraint Example of a Recovery Generalized Energy Theorem with Constraints Tractable Non-Holonomic Constraints Exercises 64 4 Introduction to Hamiltonian Mechanics Phase Space Hamilton Equations An Example of the Hamilton Equations Non-Potential and Constraint Forces ' Reduced Hamiltonian Poisson Brackets The Schroedinger Equation The Ehrenfest Theorem Exercises 84 5 The Calculus of Variations Paths in an A?-Dimensional Space Variations of Coordinates ' Variations of Functions Variation of a Line Integral Finding Extremum Paths Example of an Extremum Path Calculation Invariance and Homogeneity The Brachistochrone Problem Calculus of Variations with Constraints An Example with Constraints Reduction of Degrees of Freedom Example of'a Reduction Example of a Better Reduction The Coordinate Parametric Method 108

4 CONTENTS xiii 5.15 Comparison of the Methods Exercises Hamilton's Principle Hamilton's Principle in Lagrangian Form Hamilton's Principle with Constraints Comments on Hamilton's Principle Phase-Space Hamilton's Principle Exercises Linear Operators and Dyadics Definition of Operators Operators and Matrices Addition and Multiplication Determinant, Trace, and Inverse Special Operators Dyadics Resolution of Unity Operators, Components, Matrices, and Dyadics Complex Vectors and Operators Real and Complex Inner Products Eigenvectors and Eigenvalues Eigenvectors of Real Symmetric Operator Eigenvectors of Real Anti-Symmetric Operator Normal Operators Determinant and Trace of Normal Operator Eigen-Dyadic Expansion of Normal Operator Functions of Normal Operators The Exponential Function The Dirac Notation Exercises Kinematics of Rotation Characterization of Rigid Bodies The Center of Mass of a Rigid Body General Definition of Rotation Operator Rotation Matrices Some Properties of Rotation Operators Proper and Improper Rotation Operators The Rotation Group Kinematics of a Rigid Body Rotation Operators and Rigid Bodies Differentiation of a Rotation Operator Meaning of the Angular Velocity Vector Velocities of the Masses of a Rigid Body Savio's Tr/eorem Infinitesimal Rotation Addition of Angular Velocities 171

5 xiv CONTENTS 8.16 Fundamental Generators of Rotations Rotation with a Fixed Axis Expansion of Fixed-Axis Rotation Eigenvectors of the Fixed-Axis Rotation Operator The Euler Theorem Rotation of Operators Rotation of the Fundamental Generators Rotation of a Fixed-Axis Rotation Parameterization of Rotation Operators Differentiation of Parameterized Operator Euler Angles Fixed-Axis Rotation from Euler Angles Time Derivative of a Product Angular Velocity from Euler Angles Active and Passive Rotations Passive Transformation of Vector Components Passive Transformation of Matrix Elements The Body Derivative Passive Rotations and Rigid Bodies Passive Use of Euler Angles Exercises Rotational Dynamics Basic Facts of Rigid-Body Motion The Inertia Operator and the Spin The Inertia Dyadic Kinetic Energy of a Rigid Body Meaning of the Inertia Operator Principal Axes Guessing the Principal Axes Time Evolution of the Spin Torque-Free Motion of a Symmetric Body Euler Angles of the Torque-Free Motion Body with One Point Fixed Preserving the Principal Axes * Time Evolution with One Point Fixed Body with One Point Fixed, Alternate Derivation Work-Energy Theorems Rotation with a Fixed Axis The Symmetric Top with One Point Fixed The Initially Clamped Symmetric Top Approximate Treatment of the Symmetric Top Inertial Forces Laboratory on the Surface of the Earth Coriolis Force Calculations The Magnetic - Coriolis Analogy Exercises 239

6 CONTENTS xv 10 Small Vibrations About Equilibrium Equilibrium Defined Finding Equilibrium Points Small Coordinates Normal Modes Generalized Eigenvalue Problem Stability Initial Conditions The Energy of Small Vibrations Single Mode Excitations A Simple Example Zero-Frequency Modes Exercises 261 PART II MECHANICS WITH TIME AS A COORDINATE 11 Lagrangian Mechanics with Time as a Coordinate Time as a Coordinate A Change of Notation Extended Lagrangian Extended Momenta Extended Lagrange Equations A Simple Example Invariance-Under Change of Parameter Change of Generalized Coordinates Redundancy of the Extended Lagrange Equations Forces of Constraint Reduced Lagrangians with Time as a Coordinate Exercises Hamiltonian Mechanics with Time as a Coordinate Extended Phase Space Dependency Relation Only One Dependency Relation From Traditional to Extended Hamiltonian Mechanics Equivalence to Traditional Hamilton Equations Example of Extended Hamilton Equations Equivalent Extended Hamiltonians Alternate Hamiltonians ' Alternate Traditional Hamiltonians Not a Legendre Transformation Dirac's Theory of Phase-Space Constraints Poisson Brackets with Time as a Coordinate Poisson Brackets and Quantum Commutators Exercises Hamilton's Principle and Noether's Theorem Extended Hamilton's Principle 305

7 xvi CONTENTS 13.2 Noether's Theorem Examples of Noether's Theorem Hamilton's Principle in an Extended Phase Space Exercises Relativity and Spacetime Galilean Relativity Conflict with the Aether Einsteinian Relativity What Is a Coordinate System? A Survey of Spacetime The Lorentz Transformation The Principle of Relativity Lorentzian Relativity Mechanism and Relativity Exercises Fourvectors and Operators Fourvectors Inner Product Choice of Metric Relativistic Interval Spacetime Diagram General Fourvectors Construction of New Fourvectors Covariant and Contravariant Components General Lorentz Transformations Transformation of Components Examples of Lorentz Transformations Gradient Fourvector Manifest Covariance Formal Covariance The Lorentz Group Proper Lorentz Transformations and the Little Group Parameterization Fourvector Operators ' Fourvector Dyadics Wedge Products Scalar, Fourvector, and Operator Fields Manifestly Covariant Form of Maxwell's Equations Exercises Relativistic Mechanics Modification of Newton's Laws The Momentum Fourvector Fourvector Form of Newton's Second Law Conservation of Fourvector Momentum Particles of Zero Mass 380

8 CONTENTS xvii 16.6 Traditional Lagrangian Traditional Hamiltonian Invariant Lagrangian Manifestly Covariant Lagrange Equations Momentum Fourvectors and Canonical Momenta Extended Hamiltonian Invariant Hamiltonian Manifestly Covariant Hamilton Equations The Klein-Gordon Equation The Dirac Equation The Manifestly Covariant iv-body Problem Covariant Serret-Frenet Theory Fermi-Walker Transport Example of Fermi-Walker Transport Exercises Canonical Transformations Definition of Canonical Transformations Example of a Canonical Transformation Symplectic Coordinates Symplectic Matrix Standard Equations in Symplectic Form Poisson Bracket Condition Inversion of Canonical Transformations Direct Condition Lagrange Bracket Condition the Canonical Group Form Invariance of Poisson Brackets Form Invariance of the Hamilton Equations Traditional Canonical Transformations Exercises Generating Functions Proto-Generating Functions Generating Functions of the Ft Type Generating Functions of the F 2 Type Examples of Generating Functions Other Simple Generating Functions Mixed Generating Functions Example of a Mixed Generating Function Finding Simple Generating Functions Finding Mixed Generating Functions Finding Mixed Generating Functions An Example Traditional Generating Functions Standard Form of Extended Hamiltonian Recovered Differential Canonical Transformations Active Canonical Transformations Phase-Space Analog of Noether Theorem 454

9 xviii CONTENTS Liouville Theorem Exercises Hamilton-Jacobi Theory Definition of the Action Momenta from the 5i Action Function The S2 Action Function Example of S\ and 52 Action Functions The Hamilton-Jacobi Equation Hamilton's Characteristic Equations Complete Integrals Separation of Variables Canonical Transformations General Integrals Mono-Energetic Integrals The Optical Analogy The Relativistic Hamilton-Jacobi Equation Schroedinger and Hamilton-Jacobi Equations The Quantum Cauchy Problem The Bohm Hidden Variable Model Feynman Path-Integral Technique Quantum and Classical Mechanics Exercises 489 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.I 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

10 CONTENTS xix B.8 Trace of a Square Matrix 513 B.9 Differentiation of Matrices 513 B.10 Determinants of Square Matrices 513 B.ll 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.I5 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.I 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.I 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.ll 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.I6 Taylor Series ' 547 D.I7 Higher-Order Differential as a Difference 548 D.18 Differential Expressions 548

11 xx CONTENTS 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.I 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 References 588 Index 591

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