| Description |
1 online resource (xxii, 629 pages) : illustrations |
| Series |
Oxford graduate texts |
| Contents |
Machine generated contents note: pt. 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 -- pt. 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 |
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Note continued: 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 -- 17.5. Spacetime Diagram -- 17.6. General Fourvectors -- 17.7. Construction of New Fourvectors -- 17.8. Covariant and Contravariant Components -- 17.9. General Lorentz Transformations -- 17.10. Transformation of Components -- 17.11. Examples of Lorentz Transformations -- 17.12. Gradient Fourvector -- 17.13. Manifest Covariance -- 17.14. Formal Covariance -- 17.15. The Lorentz Group -- 17.16. Proper Lorentz Transformations and the Little Group -- 17.17. Parameterization -- 17.18. Fourvector Operators -- 17.19. Fourvector Dyadics -- 17.20. Wedge Products -- 17.21. Scalar, Fourvector, and Operator Fields -- 17.22. Manifestly Covariant Form of Maxwell's Equations -- 17.23. Exercises -- 18. Relativistic Mechanics -- 18.1. Modification of Newton's Laws -- 18.2. The Momentum Fourvector -- 18.3. Fourvector Form of Newton's Second Law -- 18.4. Conservation of Fourvector Momentum -- 18.5. Particles of Zero Mass -- 18.6. Traditional Lagrangian -- 18.7. Traditional Hamiltonian -- 18.8. Invariant Lagrangian -- 18.9. Manifestly Covariant Lagrange Equations -- 18.10. Momentum Fourvectors and Canonical Momenta -- 18.11. Extended Hamiltonian -- 18.12. Invariant Hamiltonian -- 18.13. Manifestly Covariant Hamilton Equations -- 18.14. The Klein-Gordon Equation -- 18.15. The Dirac Equation -- 18.16. The Manifestly Covariant N-Body Problem -- 18.17. Covariant Serret-Frenet Theory -- 18.18. Fermi-Walker Transport -- 18.19. Example of Fermi-Walker Transport -- 18.20. Exercises -- 19. Canonical Transformations -- 19.1. Definition of Canonical Transformations -- 19.2. Example of a Canonical Transformation -- 19.3. Symplectic Coordinates -- 19.4. Symplectic Matrix -- 19.5. Standard Equations in Symplectic Form -- 19.6. Poisson Bracket Condition -- 19.7. Inversion of Canonical Transformations -- 19.8. Direct Condition -- 19.9. Lagrange Bracket Condition -- 19.10. The Canonical Group -- 19.1. 1 Form Invariance of Poisson Brackets -- 19.12. Form Invariance of the Hamilton Equations -- 19.13. Traditional Canonical Transformations -- 19.14. Exercises -- 20. Generating Functions -- 20.1. Proto-Generating Functions -- 20.2. Generating Functions of the F1 Type -- 20.3. Generating Functions of the F2 Type -- 20.4. Examples of Generating Functions -- 20.5. Other Simple Generating Functions -- 20.6. Mixed Generating Functions -- 20.7. Example of a Mixed Generating Function -- 20.8. Finding Simple Generating Functions -- 20.9. Finding Mixed Generating Functions -- 20.10. Finding Mixed Generating Functions-An Example -- 20.11. Traditional Generating Functions -- 20.12. Standard Form of Extended Hamiltonian Recovered -- 20.13. Differential Canonical Transformations -- 20.14. Active Canonical Transformations -- 20.15. Phase-Space Analog of Noether Theorem -- 20.16. Liouville Theorem -- 20.17. Exercises -- 21. Hamilton-Jacobi Theory -- 21.1. Definition of the Action -- 21.2. Momenta from the S1 Function -- 21.3. The S2 Action Function -- 21.4. Example of S1 and S2 Action Functions -- 21.5. The Hamilton-Jacobi Equation -- 21.6. Hamilton's Characteristic Equations -- 21.7. Complete Integrals -- 21.8. Complete Integrals and System Motion -- 21.9. Additive Separation of Variables -- 21.10. General Integrals -- 21.11. Time Independent Hamiltonians -- 21.12. Mono-Energetic Complete Integrals -- 21.13. The Optical Analogy -- 21.14. The Relativistic Hamilton-Jacobi Equation -- 21.15. Schroedinger and Hamilton-Jacobi Equations -- 21.16. The Quantum Cauchy Problem -- 21.17. The Bohm Hidden Variable Model -- 21.18. Feynman Path-Integral Technique -- 21.19. Quantum and Classical Mechanics -- 21.20. Exercises -- 22. Angle-Action Variables -- 22.1. Definition of the Action Variables -- 22.2. Canonical Transformation to Angle-Action Variables -- 22.3. Multiply Periodic Motion -- 22.4. Harmonic Oscillator -- 22.5. Central Force Motion -- 22.6. The Plane Kepler System -- 22.7. Transforming to Plane Delaunay Variables -- 22.8. The Bohr Model -- 22.9. The Old Quantum Theory -- 22.10. Inclined Orbits -- 22.11. Old and New Quantum Theories -- 22.12. Exercises -- pt. III MATHEMATICAL APPENDICES -- A. Vector Fundamentals -- A.1. Properties of Vectors -- A.2. Dot Product -- A.3. Cross Product -- A.4. Linearity -- A.5. Cartesian Basis -- A.6. The Position Vector -- A.7. Fields -- A.8. Polar Coordinates -- A.9. The Algebra of Sums -- A.10. Miscellaneous Vector Formulae -- A.11. Gradient Vector Operator -- A.12. The Serret-Frenet Formulae -- B. Matrices and Determinants -- B.1. Definition of Matrices -- B.2. Transposed Matrix -- B.3. Column Matrices and Column Vectors -- B.4. Square, Symmetric, and Hermitian Matrices -- B.5. Algebra of Matrices: Addition -- B.6. Algebra of Matrices: Multiplication -- B.7. Diagonal and Unit Matrices -- B.8. Trace of a Square Matrix -- B.9. Differentiation of Matrices -- B.10. Determinants of Square Matrices -- B.11. Properties of Determinants -- B.12. Cofactors -- B.13. Expansion of a Determinant by Cofactors -- B.14. Inverses of Nonsingular Matrices -- B.15. Partitioned Matrices -- B.16. Cramer's Rule -- B.17. Minors and Rank -- B.18. Linear Independence -- B.19. Homogeneous Linear Equations -- B.20. Inner Products of Column Vectors -- B.21. Complex Inner Products -- B.22. Orthogonal and Unitary Matrices -- B.23. Eigenvalues and Eigenvectors of Matrices -- B.24. Eigenvectors of Real Symmetric Matrix -- B.25. Eigenvectors of Complex Hermitian Matrix -- B.26. Normal Matrices -- B.27. Properties of Normal Matrices -- B.28. Functions of Normal Matrices -- C. Eigenvalue Problem with General Metric -- C.1. Positive-Definite Matrices -- C.2. Generalization of the Real Inner Product -- C.3. The Generalized Eigenvalue Problem -- C.4. Finding Eigenvectors in the Generalized Problem -- C.5. Uses of the Generalized Eigenvectors -- D. The Calculus of Many Variables -- D.1. Basic Properties of Functions -- D.2. Regions of Definition of Functions -- D.3. Continuity of Functions -- D.4. Compound Functions -- D.5. The Same Function in Different Coordinates -- D.6. Partial Derivatives -- D.7. Continuously Differentiable Functions -- D.8. Order of Differentiation -- D.9. Chain Rule -- D.10. Mean Values -- D.11. Orders of Smallness -- D.12. Differentials -- D.13. Differential of a Function of Several Variables -- D.14. Differentials and the Chain Rule -- D.15. Differentials of Second and Higher Orders -- D.16. Taylor Series -- D.17. Higher-Order Differential as a Difference -- D.18. Differential Expressions -- D.19. Line Integral of a Differential Expression -- D.20. Perfect Differentials -- D.21. Perfect Differential and Path Independence -- D.22. Jacobians -- D.23. Global Inverse Function Theorem -- D.24. Local Inverse Function Theorem -- D.25. Derivatives of the Inverse Functions -- D.26. Implicit Function Theorem -- D.27. Derivatives of Implicit Functions -- D.28. Functional Independence -- D.29. Dependency Relations -- D.30. Legendre Transformations -- D.31. Homogeneous Functions -- D.32. Derivatives of Homogeneous Functions -- D.33. Stationary Points -- D.34. Lagrange Multipliers -- D.35. Geometry of the Lagrange Multiplier Theorem -- D.36. Coupled Differential Equations -- D.37. Surfaces and Envelopes -- E. Geometry of Phase Space -- E.1. Abstract Vector Space -- E.2. Subspaces -- E.3. Linear Operators -- E.4. Vectors in Phase Space -- E.5. Canonical Transformations in Phase Space -- E.6. Orthogonal Subspaces -- E.7. A Special Canonical Transformation -- E.8. Special Self-Orthogonal Subspaces -- E.9. Arnold's Theorem -- E.10. Existence of a Mixed Generating Function |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 619-621) and index |
| Notes |
Print version record |
| Subject |
Mechanics, Analytic.
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Quantum theory.
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Mechanics, Analytic
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Quantum theory
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| Form |
Electronic book
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| ISBN |
0191775169 |
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9780191775161 |
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