Analytical mechanics for relativity and quantum mechanics
Johns, Oliver DavisAuthor
Oxford : Oxford University Press,
2011.
This work 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.
Bibliographic Level Mode of Issuance: Monograph
Includes bibliographical references and index.
This work 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.
Description based on print version record.
Cover -- Contents -- Dedication -- Prefaces -- Acknowledgments -- PART I: INTRODUCTION: THE TRADITIONAL THEORY -- 1 Basic Dynamics of Point Particles and Collections -- 1.1 Newton's Space and Time -- 1.2 Single Point Particle -- 1.3 Collective Variables -- 1.4 The Law of Momentum for Collections -- 1.5 The Law of Angular Momentum for Collections -- 1.6 "Derivations" of the Axioms -- 1.7 The Work-Energy Theorem for Collections -- 1.8 Potential and Total Energy for Collections -- 1.9 The Center of Mass -- 1.10 Center of Mass and Momentum -- 1.11 Center of Mass and Angular Momentum -- 1.12 Center of Mass and Torque -- 1.13 Change of Angular Momentum -- 1.14 Center of Mass and the Work-Energy Theorems -- 1.15 Center of Mass as a Point Particle -- 1.16 Special Results for Rigid Bodies -- 1.17 Exercises -- 2 Introduction to Lagrangian Mechanics -- 2.1 Configuration Space -- 2.2 Newton's Second Law in Lagrangian Form -- 2.3 A Simple Example -- 2.4 Arbitrary Generalized Coordinates -- 2.5 Generalized Velocities in the q-System -- 2.6 Generalized Forces in the q-System -- 2.7 The Lagrangian Expressed in the q-System -- 2.8 Two Important Identities -- 2.9 Invariance of the Lagrange Equations -- 2.10 Relation Between Any Two Systems -- 2.11 More of the Simple Example -- 2.12 Generalized Momenta in the q-System -- 2.13 Ignorable Coordinates -- 2.14 Some Remarks About Units -- 2.15 The Generalized Energy Function -- 2.16 The Generalized Energy and the Total Energy -- 2.17 Velocity Dependent Potentials -- 2.18 Exercises -- 3 Lagrangian Theory of Constraints -- 3.1 Constraints Defined -- 3.2 Virtual Displacement -- 3.3 Virtual Work -- 3.4 Form of the Forces of Constraint -- 3.5 General Lagrange Equations with Constraints -- 3.6 An Alternate Notation for Holonomic Constraints -- 3.7 Example of the General Method -- 3.8 Reduction of Degrees of Freedom.
3.9 Example of a Reduction -- 3.10 Example of a Simpler Reduction Method -- 3.11 Recovery of the Forces of Constraint -- 3.12 Example of a Recovery -- 3.13 Generalized Energy Theorem with Constraints -- 3.14 Tractable Non-Holonomic Constraints -- 3.15 Exercises -- 4 Introduction to Hamiltonian Mechanics -- 4.1 Phase Space -- 4.2 Hamilton Equations -- 4.3 An Example of the Hamilton Equations -- 4.4 Non-Potential and Constraint Forces -- 4.5 Reduced Hamiltonian -- 4.6 Poisson Brackets -- 4.7 From Lagrangian to Hamiltonian Mechanics -- 4.8 Canonical Transformations -- 4.9 Generating Functions -- 4.10 The Schroedinger Equation -- 4.11 The Ehrenfest Theorem -- 4.12 The Virial Theorem -- 4.13 Exercises -- 5 The Calculus of Variations -- 5.1 Paths in an N-Dimensional Space -- 5.2 Variations of Coordinates -- 5.3 Variations of Functions -- 5.4 Variation of a Line Integral -- 5.5 Finding Extremum Paths -- 5.6 Example of an Extremum Path Calculation -- 5.7 Invariance and Homogeneity -- 5.8 The Brachistochrone Problem -- 5.9 Calculus of Variations with Constraints -- 5.10 An Example with Constraints -- 5.11 Reduction of Degrees of Freedom -- 5.12 Example of a Reduction -- 5.13 Example of a Better Reduction -- 5.14 The Coordinate Parametric Method -- 5.15 Comparison of the Methods -- 5.16 Exercises -- 6 Hamilton's Principle -- 6.1 Hamilton's Principle in Lagrangian Form -- 6.2 Hamilton's Principle with Constraints -- 6.3 Comments on Hamilton's Principle -- 6.4 Phase-Space Hamilton's Principle -- 6.5 Exercises -- 7 Linear Operators and Dyadics -- 7.1 Definition of Operators -- 7.2 Operators and Matrices -- 7.3 Addition and Multiplication -- 7.4 Determinant, Trace, and Inverse -- 7.5 Special Operators -- 7.6 Dyadics -- 7.7 Resolution of Unity -- 7.8 Operators, Components, Matrices, and Dyadics -- 7.9 Complex Vectors and Operators.
7.10 Real and Complex Inner Products -- 7.11 Eigenvectors and Eigenvalues -- 7.12 Eigenvectors of Real Symmetric Operator -- 7.13 Eigenvectors of Real Anti-Symmetric Operator -- 7.14 Normal Operators -- 7.15 Determinant and Trace of Normal Operator -- 7.16 Eigen-Dyadic Expansion of Normal Operator -- 7.17 Functions of Normal Operators -- 7.18 The Exponential Function -- 7.19 The Dirac Notation -- 7.20 Exercises -- 8 Kinematics of Rotation -- 8.1 Characterization of Rigid Bodies -- 8.2 The Center of Mass of a Rigid Body -- 8.3 General Definition of Rotation Operator -- 8.4 Rotation Matrices -- 8.5 Some Properties of Rotation Operators -- 8.6 Proper and Improper Rotation Operators -- 8.7 The Rotation Group -- 8.8 Kinematics of a Rigid Body -- 8.9 Rotation Operators and Rigid Bodies -- 8.10 Differentiation of a Rotation Operator -- 8.11 Meaning of the Angular Velocity Vector -- 8.12 Velocities of the Masses of a Rigid Body -- 8.13 Savio's Theorem -- 8.14 Infinitesimal Rotation -- 8.15 Addition of Angular Velocities -- 8.16 Fundamental Generators of Rotations -- 8.17 Rotation with a Fixed Axis -- 8.18 Expansion of Fixed-Axis Rotation -- 8.19 Eigenvectors of the Fixed-Axis Rotation Operator -- 8.20 The Euler Theorem -- 8.21 Rotation of Operators -- 8.22 Rotation of the Fundamental Generators -- 8.23 Rotation of a Fixed-Axis Rotation -- 8.24 Parameterization of Rotation Operators -- 8.25 Differentiation of Parameterized Operator -- 8.26 Euler Angles -- 8.27 Fixed-Axis Rotation from Euler Angles -- 8.28 Time Derivative of a Product -- 8.29 Angular Velocity from Euler Angles -- 8.30 Active and Passive Rotations -- 8.31 Passive Transformation of Vector Components -- 8.32 Passive Transformation of Matrix Elements -- 8.33 The Body Derivative -- 8.34 Passive Rotations and Rigid Bodies -- 8.35 Passive Use of Euler Angles -- 8.36 Exercises -- 9 Rotational Dynamics.
9.1 Basic Facts of Rigid-Body Motion -- 9.2 The Inertia Operator and the Spin -- 9.3 The Inertia Dyadic -- 9.4 Kinetic Energy of a Rigid Body -- 9.5 Meaning of the Inertia Operator -- 9.6 Principal Axes -- 9.7 Guessing the Principal Axes -- 9.8 Time Evolution of the Spin -- 9.9 Torque-Free Motion of a Symmetric Body -- 9.10 Euler Angles of the Torque-Free Motion -- 9.11 Body with One Point Fixed -- 9.12 Preserving the Principal Axes -- 9.13 Time Evolution with One Point Fixed -- 9.14 Body with One Point Fixed, Alternate Derivation -- 9.15 Work-Energy Theorems -- 9.16 Rotation with a Fixed Axis -- 9.17 The Symmetric Top with One Point Fixed -- 9.18 The Initially Clamped Symmetric Top -- 9.19 Approximate Treatment of the Symmetric Top -- 9.20 Inertial Forces -- 9.21 Laboratory on the Surface of the Earth -- 9.22 Coriolis Force Calculations -- 9.23 The Magnetic - Coriolis Analogy -- 9.24 Exercises -- 10 Small Vibrations About Equilibrium -- 10.1 Equilibrium Defined -- 10.2 Finding Equilibrium Points -- 10.3 Small Coordinates -- 10.4 Normal Modes -- 10.5 Generalized Eigenvalue Problem -- 10.6 Stability -- 10.7 Initial Conditions -- 10.8 The Energy of Small Vibrations -- 10.9 Single Mode Excitations -- 10.10 A Simple Example -- 10.11 Zero-Frequency Modes -- 10.12 Exercises -- 11 Central Force Motion -- 11.1 Formulation of the Problem -- 11.2 Kepler's Law of Areas -- 11.3 Orbital Motion -- 11.4 General Features of the Motion -- 11.5 Inverse Square Force: the Kepler Problem -- 11.6 Details of the Kepler Orbits -- 11.7 Kepler's Third Law -- 11.8 The Eccentricity Vector -- 11.9 Periodic Orbits -- 11.10 The Isotropic Harmonic Oscillator -- 11.11 Exercises -- 12 Scattering -- 12.1 Cross Sections -- 12.2 Differential Cross Sections -- 12.3 Scattering by Hard Spheres -- 12.4 Scattering by an Inverse-Square Central Force.
12.5 Scattering by General Central Forces -- 12.6 Exercises -- PART II: MECHANICS WITH TIME AS A COORDINATE -- 13 Lagrangian Mechanics with Time as a Coordinate -- 13.1 Time as a Coordinate -- 13.2 A Change of Notation -- 13.3 Extended Lagrangian -- 13.4 Extended Momenta -- 13.5 Extended Lagrange Equations -- 13.6 A Simple Example -- 13.7 Invariance Under Change of Parameter -- 13.8 Change of Generalized Coordinates -- 13.9 Redundancy of the Extended Lagrange Equations -- 13.10 Forces of Constraint -- 13.11 Reduced Lagrangians with Time as a Coordinate -- 13.12 Exercises -- 14 Hamiltonian Mechanics with Time as a Coordinate -- 14.1 Extended Phase Space -- 14.2 Dependency Relation -- 14.3 Only One Dependency Relation -- 14.4 From Traditional to Extended Hamiltonian Mechanics -- 14.5 Equivalence to Traditional Hamilton Equations -- 14.6 Example of Extended Hamilton Equations -- 14.7 Equivalent Extended Hamiltonians -- 14.8 Alternate Hamiltonians -- 14.9 Alternate Traditional Hamiltonians -- 14.10 Not a Legendre Transformation -- 14.11 Dirac's Theory of Phase-Space Constraints -- 14.12 Poisson Brackets with Time as a Coordinate -- 14.13 Poisson Brackets and Quantum Commutators -- 14.14 Exercises -- 15 Hamilton's Principle and Noether's Theorem -- 15.1 Extended Hamilton's Principle -- 15.2 Noether's Theorem -- 15.3 Examples of Noether's Theorem -- 15.4 Hamilton's Principle in an Extended Phase Space -- 15.5 Exercises -- 16 Relativity and Spacetime -- 16.1 Galilean Relativity -- 16.2 Conflict with the Aether -- 16.3 Einsteinian Relativity -- 16.4 What Is a Coordinate System? -- 16.5 A Survey of Spacetime -- 16.6 The Lorentz Transformation -- 16.7 The Principle of Relativity -- 16.8 Lorentzian Relativity -- 16.9 Mechanism and Relativity -- 16.10 Exercises -- 17 Fourvectors and Operators -- 17.1 Fourvectors -- 17.2 Inner Product -- 17.3 Choice of Metric.
17.4 Relativistic Interval.
Mechanics, Analytic
Quantum theory
Engineering & Applied Sciences
Applied Mathematics