Classical Dynamics of Particles and Systems

PrefaceChapter 1. Matrices and Vectors 1.1 Introduction 1.2 The Concept of a Scalar 1.3 Coordinate Transformations 1.4 Properties of Rotation Matrices 1.5 Matrix Operations 1.6 Further Definitions 1.7 Geometrical Significance of Transformation Matrices 1.8 Definitions of a Scalar and a Vector in Terms of Transformation Properties 1.9 Elementary Scalar and Vector Operations 1.10 The Scalar Product of Two Vectors 1.11 The Vector Product of Two Vectors 1.12 Unit Vectors Suggested References ProblemsChapter 2. Vector Calculus 2.1 Introduction 2.2 Differentiation of a Vector with Respect to a Scalar 2.3 Examples of Derivatives —Velocity and Acceleration 2.4 Angular Velocity 2.5 The Gradient Operator 2.6 The Divergence of a Vector 2.7 The Curl of a Vector 2.8 Some Additional Differential Vector Relations 2.9 Integration of Vectors Suggested References ProblemsChapter 3. Fundamentals of Newtonian Mechanics 3.1 Introduction 3.2 Newton's Laws 3.3 Frames of Reference 3.4 The Equation of Motion for a Particle 3.5 Conservation Theorems 3.6 Conservation Theorems for a System of Particles 3.7 Limitations of Newtonian Mechanics Suggested References ProblemsChapter 4. The Special Theory of Relativity 4.1 Introduction 4.2 Galilean Invariance 4.3 The Lorentz Transformation 4.4 Momentum and Energy in Relativity 4.5 Some Consequences of the Lorentz Transformation Suggested References Problems Chapter 5. Gravitational Attraction and Potentials 5.1 Introduction 5.2 The Gravitational Potential 5.3 Lines of Force and Equipotential Surfaces 5.4 The Gravitational Potential of a Spherical Shell 5.5 A Final Comment Suggested References ProblemsChapter 6. Oscillatory Motion 6.1 Introduction 6.2 The Simple Harmonic Oscillator 6.3 Damped Harmonic Motion 6.4 Forcing Functions 6.5 Forced Oscillations 6.6 Phase Diagrams 6.7 The Response of Linear Oscillators to Impulsive Forcing Functions 6.8 Electrical Oscillations 6.9 Harmonic Oscillations in Two Dimensions 6.10 The Use of Complex Notation Suggested References Problems 7Chapter 7. Nonlinear Oscillations 7.1 Oscillations 7.2 Oscillations for General Potential Functions 7.3 Phase Diagrams for Nonlinear Systems 7.4 The Plane Pendulum 7.5 Nonlinear Oscillations in a Symmetric Potential - The Method of Successive Approximations 7.6 Nonlinear Oscillations in an Asymmetric Potential - The Method of Perturbations Suggested References ProblemsChapter 8. Some Methods in the Calculus of Variations 8.1 Introduction 8.2 Statement of the Problem 8.3 Euler's Equation 8.4 The Brachistochrone Problem 8.5 The "Second Form" of Euler's Equation 8.6 Functions with Several Dependent Variables 8.7 The Euler Equations When Auxiliary Conditions Are Imposed 8.8 The δ Notation Suggested References ProblemsChapter 9. Hamilton's Principle — Lagrangian and Hamiltonian Dynamics 9.1 Introduction 9.2 Hamilton's Principle 9.3 Generalized Coordinates 9.4 Lagrange's Equations of Motion in Generalized Coordinates 9.5 Lagrange's Equations with Undetermined Multipliers 9.6 The Equivalence of Lagrange's and Newton's Equations 9.7 The Essence of Lagrangian Dynamics 9.8 A Theorem Concerning the Kinetic Energy 9.9 The Conservation of Energy 9.10 The Conservation of Linear Momentum 9.11 The Conservation of Angular Momentum 9.12 The Canonical Equations of Motion — Hamiltonian Dynamics 9.13 Some Comments Regarding Dynamical Variables and Variational Calculations in Physics 9.14 Phase Space and Liouville's Theorem 9.15 The Virial Theorem 9.16 The Lagrangian Function in Special Relativity Suggested References ProblemsChapter 10. Central-Force Motion 10.1 Introduction 10.2 The Reduced Mass 10.3 Conservation Theorems — First Integrals of the Motion 10.4 Equations of Motion 10.5 Orbits in a Central Field 10.6 Centrifugal Energy and the Effective Potential 10.7 Planetary Motion-Kepler's Problem 10.8 Kepler's Equation 10.9 Approximate Solution of Kepler's Equation 10.10 Apsidal Angles and Precession 10.11 Stability of Circular Orbits 10.12 The Problem of Three Bodies Suggested References ProblemsChapter 11. Kinematics of Two-Particle Collisions 11.1 Introduction 11.2 Elastic Collisions —Center-of-Mass and Laboratory Coordinate Systems 11.3 Kinematics of Elastic Collisions 11.4 Cross Sections 11.5 The Rutherford Scattering Formula 11.6 The Total Cross Section 11.7 Relativistic Kinematics Suggested References ProblemsChapter 12. Motion in a Noninertial Reference Frame 12.1 Introduction 12.2 Rotating Coordinate Systems 12.3 The Coriolis Force 12.4 Motion Relative to the Earth Suggested References ProblemsChapter 13. Dynamics of Rigid Bodies 13.1 Introduction 13.2 The Inertia Tensor 13.3 Angular Momentum 13.4 Principal Axes of Inertia 13.5 Moments of Inertia for Different Body Coordinate Systems 13.6 Further Properties of the Inertia Tensor 13.7 The Eulerian Angles 13.8 Euler's Equations for a Rigid Body 13.9 Force-Free Motion of a Symmetrical Top 13.10 The Motion of a Symmetrical Top with One Point Fixed 13.11 The Stability of Rigid-Body Rotations Suggested References ProblemsChapter 14. Systems with Many Degrees of Freedom — Small Oscillations and Normal Coordinates 14.1 Introduction 14.2 Two Coupled Harmonic Oscillators 14.3 The General Problem of Coupled Oscillations 14.4 The Orthogonality of the Eigenvectors 14.5 Normal Coordinates 14.6 Two Linearly Coupled Plane Pendula 14.7 Three Linearly Coupled Plane Pendula — An Example of Degeneracy 14.8 The Loaded String 14.9 The Continuous String as a Limiting Case of the Loaded String 14.10 The Wave Equation 14.11 The Nonuniform String - Orthogonal Functions and Perturbation Theory 14.12 Fourier Analysis Suggested References ProblemsChapter 15. The Wave Equation in One Dimension 15.1 Introduction 15.2 Separation of the Wave Equation 15.3 Phase Velocity, Dispersion, and Attenuation 15.4 Electrical Analogies — Filtering Networks 15.5 Group Velocity and Wave Packets 15.6 Fourier Integral Representation of Wave Packets 15.7 Energy Propagation in the Loaded String 15.8 Further Comments Regarding Phase and Group Velocities 15.9 Reflected and Transmitted Waves 15.10 Damped Plane Waves Suggested References ProblemsSolutions, Hints, and References for Selected ProblemsAppendix A. Taylor's Theorem ExercisesAppendix B. Complex Numbers B.1 Complex Numbers B.2 Geometrical Representation of Complex Numbers B.3 Trigonometric Functions of Complex Variables B.4 Hyperbolic Functions ExercisesAppendix C. Ordinary Differential Equations of Second Order C.1 Linear Homogeneous Equations C.2 Linear Inhomogeneous Equations ExercisesAppendix D. Useful Formulas D.1 Binomial Expansion D.2 Trigonometric Relations D.3 Trigonometric Series D.4 Exponential and Logarithmic Series D.5 Hyperbolic FunctionsAppendix E. Useful Integrals E.1 Algebraic Functions E.2 Trigonometric Functions E.3 Gamma Functions E.4 Elliptic IntegralsAppendix F. Differential Relations in Curvilinear Coordinate Systems F.1 Cylindrical Coordinates F.2 Spherical CoordinatesAppendix G. A Proof of the Relation Σµ χ 2µ = Σµ χ' 2µSelected ReferencesBibliography