Classical Mechanics
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Book Description
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 calculusbased 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 HamiltonJacobi 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.
Table of Contents
Kinematics: Describing the Motion
Introduction
Space, Time, and Coordinate Systems
Change of Coordinate System (Transformation of Components of a Vector)
Displacement Vector
Speed and Velocity
Acceleration
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
References
Integration of Newton’s Equation of Motion
Introduction
Motion Under Constant Force
Force Is a Function of Time
Force Is a Function of Velocity
Force Is a Function of Position
TimeVarying 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
Reference
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
References
Motion Under a Central Force
TwoBody 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)
References
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
References
Collisions and Scatterings
Direct Impact of Two Particles
Scattering Cross Sections and Rutherford Scattering
Laboratory and CenterofMass Frames of Reference
Nuclear Sizes
SmallAngle Scattering (Optional)
References
Motion in NonInertial 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 TorqueFree Symmetrical Top
Motion of Heavy Symmetrical Top with One Point Fixed
Stability of Rotational Motion
References
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
References
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
Tides
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
Reference
Hamilton–Jacobi Theory of Dynamics
Canonical Transformation and HJ Equation
Action and Angle Variables
Infinitesimal Canonical Transformations and Time Development Operator
HJ Theory and Wave Mechanics
Reference
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
Index
Author(s)
Biography
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.
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