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Classical Mechanics (Third Edtion) by Herbert Goldstein

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Classical Mechanics (Third Edtion) written by Herbert GoldsteinColumbia University and Charles P. PooleUniversity of South Carolina and John L. SafkoUniversity of South Carolina Several of the new features and approaches in this third edition had been mentioned as possibilities in the preface to the second edition, such as properties of group theory, tensors in non-Euclidean spaces, and “new mathematics” of theoretical physics such as manifolds. The reference to “One area omitted that deserves special attention—nonlinear oscillation and associated stability questions” now constitutes the subject matter of our new Chapter 11 “Classical Chaos.” We debated whether to place this new chapter after Perturbation theory where it fits more logically, or before Perturbation theory where it is more likely to be covered in class, and we chose the latter. The referees who reviewed our manuscript were evenly divided on this question.
The mathematical level of the present edition is about the same as that of the first two editions. Some of the mathematical physics, such as the discussions of hermitean and unitary matrices, was omitted because it pertains much more to quantum mechanics than it does to classical mechanics, and little used notations like dyadics were curtailed. Space devoted to power law potentials, Cayley-Klein parameters, Routh’s procedure, time independent perturbation theory, and the stress-energy tensor was reduced. In some cases reference was made to the second edition for more details. The problems at the end of the chapters were divided into “derivations” and “exercises,” and some new ones were added.

Classical Mechanics (Third Edtion) written by Herbert Goldstein, Charles P. Pooleand John L. Safko cover the following topics.

  • 1. Survey of the Elementary Principles
    1.1 Mechanics of a Particle
    1.2 Mechanics of a System of Particles
    1.3 Constraints
    1.4 D’Alembert’s Principle and Lagrange’s Equations
    1.5 Velocity-Dependent Potentials and the Dissipation Function
    1.6 Simple Applications of the Lagrangian Formulation

  • 2. Variational Principles and Lagrange’s Equations
    2.1 Hamilton’s Principle
    2.2 Some Techniques of the Calculus of Variations
    2.3 Derivation of Lagrange’s Equations from Hamilton’s Principle
    2.4 Extending Hamilton’s Principle to Systems with Constraints
    2.5 Advantages of a Variational Principle Formulation
    2.6 Conservation Theorems and Symmetry Properties
    2.7 Energy Function and the Conservation of Energy

  • 3. The Central Force Problem
    3.1 Reduction to the Equivalent One-Body Problem
    3.2 The Equations of Motion and First Integrals
    3.3 The Equivalent One-Dimensional Problem, and Classification of Orbits
    3.4 The Virial Theorem
    3.5 The Differential Equation for the Orbit, and Integrable Power-Law Potentials
    3.6 Conditions for Closed Orbits (Bertrand’s Theorem)
    3.7 The Kepler Problem: Inverse-Square Law of Force
    3.8 The Motion in Time in the Kepler Problem
    3.9 The Laplace–Runge–Lenz Vector
    3.10 Scattering in a Central Force Field
    3.11 Transformation of the Scattering Problem to Laboratory Coordinates
    3.12 The Three-Body Problem

  • 4. The Kinematics of Rigid Body Motion
    4.1 The Independent Coordinates of a Rigid Body
    4.2 Orthogonal Transformations
    4.3 Formal Properties of the Transformation Matrix
    4.4 The Euler Angles
    4.5 The Cayley–Klein Parameters and Related Quantities
    4.6 Euler’s Theorem on the Motion of a Rigid Body
    4.7 Finite Rotations
    4.8 Infinitesimal Rotations
    4.9 Rate of Change of a Vector
    4.10 The Coriolis Effect

  • 5. The Rigid Body Equations of Motion
    5.1 Angular Momentum and Kinetic Energy of Motion about a Point
    5.2 Tensors
    5.3 The Inertia Tensor and the Moment of Inertia
    5.4 The Eigenvalues of the Inertia Tensor and the Principal Axis Transformation
    5.5 Solving Rigid Body Problems and the Euler Equations of Motion
    5.6 Torque-Free Motion of a Rigid Body
    5.7 The Heavy Symmetrical Top with One Point Fixed
    5.8 Precession of the Equinoxes and of Satellite Orbits
    5.9 Precession of Systems of Charges in a Magnetic Field

  • 6. Oscillations
    6.1 Formulation of the Problem
    6.2 The Eigenvalue Equation and the Principal Axis Transformation
    6.3 Frequencies of Free Vibration, and Normal Coordinates
    6.4 Free Vibrations of a Linear Triatomic Molecule
    6.5 Forced Vibrations and the Effect of Dissipative Forces
    6.6 Beyond Small Oscillations: The Damped Driven Pendulum and the Josephson Junction

  • 7. The Classical Mechanics of the Special Theory of Relativity
    7.1 Basic Postulates of the Special Theory
    7.2 Lorentz Transformations
    7.3 Velocity Addition and Thomas Precession
    7.4 Vectors and the Metric Tensor
    7.5 1-Forms and Tensors
    7.6 Forces in the Special Theory; Electromagnetism
    7.7 Relativistic Kinematics of Collisions and Many-Particle Systems
    7.8 Relativistic Angular Momentum
    7.9 The Lagrangian Formulation of Relativistic Mechanics
    7.10 Covariant Lagrangian Formulations
    7.11 Introduction to the General Theory of Relativity

  • 8. The Hamilton Equations of Motion
    8.1 Legendre Transformations and the Hamilton Equations of Motion
    8.2 Cyclic Coordinates and Conservation Theorems
    8.3 Routh’s Procedure
    8.4 The Hamiltonian Formulation of Relativistic Mechanics
    8.5 Derivation of Hamilton’s Equations from a Variational Principle
    8.6 The Principle of Least Action

  • 9. Canonical Transformations
    9.1 The Equations of Canonical Transformation
    9.2 Examples of Canonical Transformations
    9.3 The Harmonic Oscillator
    9.5 Poisson Brackets and Other Canonical Invariants
    9.6 Equations of Motion, Infinitesimal Canonical Transformations, and Conservation Theorems in the Poisson Bracket Formulation
    9.7 The Angular Momentum Poisson Bracket Relations
    9.8 Symmetry Groups of Mechanical Systems
    9.9 Liouville’s Theorem

  • 10. Hamilton–Jacobi Theory and Action-Angle Variables
    10.1 The Hamilton–Jacobi Equation for Hamilton’s Principal Function
    10.2 The Harmonic Oscillator Problem as an Example of the Hamilton–Jacobi Method
    10.3 The Hamilton–Jacobi Equation for Hamilton’s Characteristic Function
    10.4 Separation of Variables in the Hamilton–Jacobi Equation
    10.5 Ignorable Coordinates and the Kepler Problem
    10.6 Action-Angle Variables in Systems of One Degree of Freedom
    10.7 Action-Angle Variables for Completely Separable Systems
    10.8 The Kepler Problem in Action-Angle Variables

  • 11. Classical Chaos
    11.1 Periodic Motion
    11.2 Perturbations and the Kolmogorov–Arnold–Moser Theorem
    11.3 Attractors
    11.4 Chaotic Trajectories and Liapunov Exponents
    11.5 Poincar´e Maps
    11.6 H´enon–Heiles Hamiltonian
    11.7 Bifurcations, Driven-Damped Harmonic Oscillator, and Parametric Resonance
    11.8 The Logistic Equation
    11.9 Fractals and Dimensionality

  • 12. Canonical Perturbation Theory
    12.1 Introduction 526
    12.2 Time-Dependent Perturbation Theory
    12.4 Time-Independent Perturbation Theory
    12.5 Adiabatic Invariants

  • 13. Introduction to the Lagrangian and Hamiltonian Formulations for Continuous Systems and Fields
    13.1 The Transition from a Discrete to a Continuous System
    13.2 The Lagrangian Formulation for Continuous Systems
    13.3 The Stress-Energy Tensor and Conservation Theorems
    13.4 Hamiltonian Formulation
    13.5 Relativistic Field Theory
    13.6 Examples of Relativistic Field Theories
    13.7 Noether’s Theorem
    Exercises 598

  • Appendix
    A Euler Angles in Alternate Conventions and Cayley–Klein Parameters
    B Groups and Algebras
    C Solutions to Select Exercises

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