2nd Edition

Classical Mechanics

By Tai L. Chow Copyright 2013
    639 Pages 318 B/W Illustrations
    by CRC Press

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    Classical Mechanics, Second Edition presents a complete account of the classical mechanics of particles and systems for physics students at the advanced undergraduate level. The book evolved from a set of lecture notes for a course on the subject taught by the author at California State University, Stanislaus, for many years. It assumes the reader has been exposed to a course in calculus and a calculus-based general physics course. However, no prior knowledge of differential equations is required. Differential equations and new mathematical methods are developed in the text as the occasion demands.

    The book begins by describing fundamental concepts, such as velocity and acceleration, upon which subsequent chapters build. The second edition has been updated with two new sections added to the chapter on Hamiltonian formulations, and the chapter on collisions and scattering has been rewritten. The book also contains three new chapters covering Newtonian gravity, the Hamilton-Jacobi theory of dynamics, and an introduction to Lagrangian and Hamiltonian formulations for continuous systems and classical fields. To help students develop more familiarity with Lagrangian and Hamiltonian formulations, these essential methods are introduced relatively early in the text.

    The topics discussed emphasize a modern perspective, with special note given to concepts that were instrumental in the development of modern physics, for example, the relationship between symmetries and the laws of conservation. Applications to other branches of physics are also included wherever possible. The author provides detailed mathematical manipulations, while limiting the inclusion of the more lengthy and tedious ones. Each chapter contains homework problems of varying degrees of difficulty to enhance understanding of the material in the text. This edition also contains four new appendices on D'Alembert's principle and Lagrange's equations, derivation of Hamilton’s principle, Noether’s theorem, and conic sections.

    Kinematics: Describing the Motion
    Space, Time, and Coordinate Systems
    Change of Coordinate System (Transformation of Components of a Vector)
    Displacement Vector
    Speed and Velocity
    Velocity and Acceleration in Polar Coordinates
    Angular Velocity and Angular Acceleration
    Infinitesimal Rotations and the Angular Velocity Vector

    Newtonian Mechanics
    The First Law of Motion (Law of Inertia)
    The Second Law of Motion; the Equations of Motion
    The Third Law of Motion
    Galilean Transformations and Galilean Invariance
    Newton’s Laws of Rotational Motion
    Work, Energy, and Conservation Laws
    Systems of Particles

    Integration of Newton’s Equation of Motion
    Motion Under Constant Force
    Force Is a Function of Time
    Force Is a Function of Velocity
    Force Is a Function of Position
    Time-Varying Mass System (Rocket System)

    Lagrangian Formulation of Mechanics: Descriptions of Motion in Configuration Space
    Generalized Coordinates and Constraints
    Kinetic Energy in Generalized Coordinates
    Generalized Momentum
    Lagrangian Equations of Motion
    Nonuniqueness of the Lagrangian
    Integrals of Motion and Conservation Laws
    Scale Invariance
    Nonconservative Systems and Generalized Potential
    Charged Particle in Electromagnetic Field
    Forces of Constraint and Lagrange’s Multipliers
    Lagrangian versus Newtonian Approach to Classical Mechanics

    Hamiltonian Formulation of Mechanics: Descriptions of Motion in PhaseSpaces
    The Hamiltonian of a Dynamic System
    Hamilton’s Equations of Motion
    Integrals of Motion and Conservation Theorems
    Canonical Transformations
    Poisson Brackets
    Poisson Brackets and Quantum Mechanics
    Phase Space and Liouville’s Theorem
    Time Reversal in Mechanics (Optional)
    Passage from Hamiltonian to Lagrangian

    Motion Under a Central Force
    Two-Body Problem and Reduced Mass
    General Properties of Central Force Motion
    Effective Potential and Classification of Orbits
    General Solutions of Central Force Problem
    Inverse Square Law of Force
    Kepler’s Three Laws of Planetary Motion
    Applications of Central Force Motion
    Newton’s Law of Gravity from Kepler’s Laws
    Stability of Circular Orbits (Optional)
    Apsides and Advance of Perihelion (Optional)
    Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional)

    Harmonic Oscillator
    Simple Harmonic Oscillator
    Adiabatic Invariants and Quantum Condition
    Damped Harmonic Oscillator
    Phase Diagram for Damped Oscillator
    Relaxation Time Phenomena
    Forced Oscillations without Damping
    Forced Oscillations with Damping
    Oscillator Under Arbitrary Periodic Force
    Vibration Isolation
    Parametric Excitation

    Coupled Oscillations and Normal Coordinates
    Coupled Pendulum
    Coupled Oscillators and Normal Modes: General Analytic Approach
    Forced Oscillations of Coupled Oscillators
    Coupled Electric Circuits

    Nonlinear Oscillations
    Qualitative Analysis: Energy and Phase Diagrams
    Elliptical Integrals and Nonlinear Oscillations
    Fourier Series Expansions
    The Method of Perturbation
    Ritz Method
    Method of Successive Approximation
    Multiple Solutions and Jumps
    Chaotic Oscillations

    Collisions and Scatterings
    Direct Impact of Two Particles
    Scattering Cross Sections and Rutherford Scattering
    Laboratory and Center-of-Mass Frames of Reference
    Nuclear Sizes
    Small-Angle Scattering (Optional)

    Motion in Non-Inertial Systems
    Accelerated Translational Coordinate System
    Dynamics in Rotating Coordinate System
    Motion of Particle Near the Surface of the Earth
    Foucault Pendulum
    Larmor’s Theorem
    Classical Zeeman Effect
    Principle of Equivalence

    Motion of Rigid Bodies
    Independent Coordinates of Rigid Body
    Eulerian Angles
    Rate of Change of Vector
    Rotational Kinetic Energy and Angular Momentum
    Inertia Tensor
    Euler’s Equations of Motion
    Motion of a Torque-Free Symmetrical Top
    Motion of Heavy Symmetrical Top with One Point Fixed
    Stability of Rotational Motion

    Theory of Special Relativity
    Historical Origin of Special Theory of Relativity
    Michelson–Morley Experiment
    Postulates of Special Theory of Relativity
    Lorentz Transformations
    Doppler Effect
    Relativistic Space–Time (Minkowski Space)
    Equivalence of Mass and Energy
    Conservation Laws of Energy and Momentum
    Generalization of Newton’s Equation of Motion
    Relativistic Lagrangian and Hamiltonian Functions
    Relativistic Kinematics of Collisions
    Collision Threshold Energies

    Newtonian Gravity and Newtonian Cosmology
    Newton’s Law of Gravity
    Gravitational Field and Gravitational Potential
    Gravitational Field Equations: Poisson’s and Laplace’s Equations
    Gravitational Field and Potential of Extended Body
    General Theory of Relativity: Relativistic Theory of Gravitation
    Introduction to Cosmology
    Brief History of Cosmological Ideas
    Discovery of Expansion of the Universe, Hubble’s Law
    Big Bang
    Formulating Dynamical Models of the Universe
    Cosmological Red Shift and Hubble Constant H
    Critical Mass Density and Future of the Universe
    Microwave Background Radiation
    Dark Matter

    Hamilton–Jacobi Theory of Dynamics
    Canonical Transformation and H-J Equation
    Action and Angle Variables
    Infinitesimal Canonical Transformations and Time Development Operator
    H-J Theory and Wave Mechanics

    Introduction to Lagrangian and Hamiltonian Formulations for Continuous Systems and Classical Fields
    Vibration of Loaded String
    Vibrating Strings and the Wave Equation
    Continuous Systems and Classical Fields
    Scalar and Vector of Fields

    Appendix 1: Vector Analysis and Ordinary Differential Equations
    Appendix 2: D’Alembert’s Principle and Lagrange’s Equations
    Appendix 3: Derivation of Hamilton’s Principle from D’Alembert’s Principle
    Appendix 4: Noether’s Theorem
    Appendix 5: Conic Sections, Ellipse, Parabola, and Hyperbola



    Dr. Tai Chow was born and raised in China. He received the Bachelor of Science degree in physics from National Taiwan University, a master’s degree in physics from Case Western Reserve University in Cleveland, and a Ph.D. degree in physics from the University of Rochester in New York. Since 1970, Dr. Chow has been in the Department of Physics at California State University, Stanislaus, and served as the department chairman for 18 years. He has published more than 40 articles in physics and astrophysics journals and is the author of four textbooks.