经典力学

1 Survey of the Elementary Principles 1.1 Mechanics of a Particle 1 1.2 Mechanics of a System of Particles 5 1.3 Constraints 12 1.4 D'Alembert's Principle and Lagrange's Equations 16 1.5 Velocity-Dependent Potentials and the Dissipation Function 22 1.6 Simple Applications of the Lagrangian Formulation 24 2 Variational Principles and I.agrange's Equations 2.1 Hamilton's Principle 34 2.2 Some Techniques of the Calculus of Variations 36 2.3 Derivation of Lagrange's Equations from Hamilton's Principle 44 2.4 Extension of Hamilton's Principle to Nonholonomic Systems 45 2.5 Advantages of a Variational Principle Formulation 51 2.6 Conservation Theorems and Symmetry Properties 54 2.7 Energy Function and the Conservation of Energy 60 3 The Central Force Problem 3.1 Reduction to the Equivalent One-Body Problem 70 3.2 The Equations of Motion and First Integrals 72 3.3 The Equivalent One-Dimensional Problem, and Classification of Orbits 76 3.4 The Virial Theorem 83 3.5 The Differential Equation for the Orbit, and Integrable Power-Law Potentials 86 3.6 Conditions for Closed Orbits (Bertrand's Theorem) 89 3.7 The Kepler Problem: Inverse-Square Law of Force 92 3.8 The Motion in Time in the Kepler Problem 98 3.9 The Laplace-Runge-Lenz Vector 102 3.10 Scattering in a Central Force Field 106 3.11 Transformation of the Scattering Problem to Laboratory Coordinates 114 3.12 The Three-Body Problem 121 4 The Kinematics of Rigid Body Motion 4.1 The Independent Coordinates of a Rigid Body 134 4.2 Orthogonal Transformations 139 4.3 Formal Properties of the Transformation Matrix 144 4.4 The Euler Angles 150 4.5 The Cayley-Klein Parameters and Related Quantities 154 4.6 Euler's Theorem on the Motion of a Rigid Body 155 4.7 Finite Rotations 161 4.8 Infinitesimal Rotations 163 4.9 Rate of Change of a Vector 171 4.10 The Coriolis Effect 174 5 The Rigid Body Equations of Motion 5.1 Angular Momentum and Kinetic Energy of Motion about a Point 184 5.2 Tensors 188 5.3 The Inertia Tensor and the Moment of Inertia 191 5.4 The Eigenvalues of the Inertia Tensor and the Principal Axis Transformation 195 5.5 Solving Rigid Body Problems and the Euler Equations of Motion 198 5.6 Torque-free Motion of a Rigid Body 200 5.7 The Heavy Symmetrical Top with One Point Fixed 208 5.8 Precession of the Equinoxes and of Satellite Orbits 223 5.9 Precession of Systems of Charges in a Magnetic Field 230 6 Oscillations 6.1 Formulation of the Problem 238 6.2 The Eigenvalue Equation and the Principal Axis Transformation 241 6.3 Frequencies of Free Vibration, and Normal Coordinates 250 6.4 Free Vibrations of a Linear Triatomic Molecule 253 6.5 Forced Vibrations and the Effect of Dissipative Forces 259 6.6 Beyond Small Oscillations: The Damped Driven Pendulum and the Josephson Junction 265 7 The Classical Mechanics of the Special Theory of Relativity 7.1 Basic Postulates of the Special Theory 277 7.2 Lorentz Transformations 280 7.3 Velocity Addition and Thomas Precession 282 7.4 Vectors and the Metric Tensor 286 7.5 1-Forms and Tensors 289 7.6 Forces in the Special Theory; Electromagnetism 297 7.7 Relativistic Kinematics of Collisions and Many-Particle Systems 300 7.8 Relativistic Angular Momentum 309 7.9 The Lagrangian Formulation of Relativistic Mechanics 312 7.10 Covariant Lagrangian Formulations 318 7.11 Introduction to the General Theory of Relativity 324 8 The Hamilton Equations of Motion 8.1 Legendre Transformations and the Hamilton Equations of Motion 334 8.2 Cyclic Coordinates and Conservation Theorems 343 8.3 Routh's Procedure 347 8.4 The Hamiltonian Formulation of Relativistic Mechanics 349 8.5 Derivation of Hamilton's Equations from a Variational Principle 353 8.6 The Principle of Least Action 356 9 Canonical Transformations 9.1 The Equations of Canonical Transformation 368 9.2 Examples of Canonical Transformations 375 9.3 The Harmonic Oscillator 377 9.4 The Symplectic Approach to Canonical Transformations 381 9.5 Poisson Brackets and Other Canonical Invariants 388 9.6 Equations of Motion, Infinitesimal Canonical Transformations, and Conservation Theorems in the Poisson Bracket Formulation 396 9.7 The Angular Momentum Poisson Bracket Relations 408 9.8 Symmetry Groups of Mechanical Systems 412 9.9 Liouville's Theorem 419 10 Hamilton-lacobi Theory and Action-Angle Variables 10.1 The Hamilton-Jacobi Equation for Hamilton's Principal Function 430 10.2 The Harmonic Oscillator Problem as an Example of the Hamilton-Jacobi Method 434 10.3 The Hamilton-Jacobi Equation for Hamilton's Characteristic Function 440 10.4 Separation of Variables in the Hamilton-Jacobi Equation 444 10.5 Ignorable Coordinates and the Kepler Problem 445 10.6 Action-angle Variables in Systems of One Degree of Freedom 452 10.7 Action-Angle Variables for Completely Separable Systems 457 10.8 The Kepler Problem in Action-angle Variables 466 11 Classical Chaos 11.1 Periodic Motion 484 11.2 Perturbations and the Kolmogorov-Arnold-Moser Theorem 487 11.3 Attractors 489 11.4 Chaotic Trajectories and Liapunov Exponents 491 11.5 Poincar6 Maps 494 11.6 Hrnon-Heiles Hamiltonian 496 11.7 Bifurcations, Driven-damped Harmonic Oscillator, and Parametric Resonance 505 11.8 The Logistic Equation 509 11.9 Fractals and Dimensionality 516 12 Canonical Perturbation Theory 12.1 Introduction 526 12.2 Time-dependent Perturbation Theory 527 12.3 Illustrations of Time-dependent Perturbation Theory 533 12.4 Time-independent Perturbation Theory 541 12.5 Adiabatic Invariants 549 13 Introduction to the Lagrangian and HamUtonian Formulations for Continuous Systems and Fields 13.1 The Transition from a Discrete to a Continuous System 558 13.2 The Lagrangian Formulation for Continuous Systems 561 13.3 The Stress-energy Tensor and Conservation Theorems 566 13.4 Hamiltonian Formulation 572 13.5 Relativistic Field Theory 577 13.6 Examples of Relativistic Field Theories 583 13.7 Noether's Theorem 589 Appendix A Euler Angles in Alternate Conventions and Cayley-Klein Parameters Appendix B Groups and Algebras Selected Bibliography Author Index Subject Index

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