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Classical Mechanics Second Edition By Tai L. Chow



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Classical Mechanics (Second Edition) written by Tai L. Chow, California State University Stanislaus, Turlock, CA, USA This book presents a reasonably complete account of the theoretical mechanics of particles and systems for physics students at the advanced undergraduate level. It is evolved from a set of lecture notes for a course on the subject, which I have taught at California State University, Stanislaus, for many years. We presume that the student has been exposed to a calculus-based general physics course (from a textbook such as that by Halliday and Resnick) and a course in calculus (including the handling of differentiations of field functions). No prior knowledge of differential equations is required. Differential equations and new mathematical methods are developed in the text as the occasion demands. Vectors are used from the start.
The book has 17 chapters, and with appropriate omission, the essential topics can be covered in a one-semester, four-hour course. We do not make any specific suggestions for a shorter course. We usually vary the topics to suit the ability and mathematical background of the students. We would encourage the more enthusiastic and able students to attempt to master on their own the material not covered in class (for extra credit). A major departure of this book from the conventional approach is the introduction of the Lagrangian and Hamiltonian formulations of mechanics at an early stage. In the conventional approach to the subject, Lagrangian and Hamiltonian formulations are presented near the end of the course, and students rarely develop a reasonable familiarity with these essential methods.


Classical Mechanics (Second Edition) written by Tai L. Chow cover the following topics.


  • 1 Kinematics: Describing the Motion
    1.1 Introduction1
    1.2 Space, Time, and Coordinate Systems
    1.3 Change of Coordinate System (Transformation of Components of a Vector)
    1.4 Displacement Vector
    1.5 Speed and Velocity
    1.6 Acceleration
    1.6.1 Tangential and Normal Acceleration
    1.7 Velocity and Acceleration in Polar Coordinates
    1.7.1 Plane Polar Coordinates (r, θ)
    1.7.2 Cylindrical Coordinates (ρ, ϕ, z)
    1.7.3 Spherical Coordinates (r , θ, ϕ)
    1.8 Angular Velocity and Angular Acceleration
    1.9 Infinitesimal Rotations and the Angular Velocity Vector

  • 2 Newtonian Mechanics
    2.1 The First Law of Motion (Law of Inertia)
    2.1.1 Inertial Frames of Reference
    2.2 The Second Law of Motion; the Equations of Motion
    2.2.1 The Concept of Force
    2.3 The Third Law of Motion
    2.3.1 The Concept of Mass
    2.4 Galilean Transformations and Galilean Invariance
    2.5 Newton’s Laws of Rotational Motion
    2.6 Work, Energy, and Conservation Laws
    2.6.1 Work and Energy
    2.6.2 Conservative Force and Potential Energy
    2.6.3 Conservation of Energy
    2.6.4 Conservation of Momentum
    2.6.5 Conservation of Angular Momentum
    2.7 Systems of Particles
    2.7.1 Center of Mass
    2.7.2 Motion of CM
    2.7.3 Conservation Theorems
    References

  • 3 Integration of Newton’s Equation of Motion
    3.1 Introduction
    3.2 Motion Under Constant Force
    3.3 Force Is a Function of Time
    3.3.1 Impulsive Force and Green’s Function Method66
    3.4 Force Is a Function of Velocity
    3.4.1 Motion in a Uniform Magnetic Field
    3.4.2 Motion in Nearly Uniform Magnetic Field
    3.5 Force Is a Function of Position
    3.5.1 Bounded and Unbounded Motion
    3.5.2 Stable and Unstable Equilibrium
    3.5.3 Critical and Neutral Equilibrium
    3.6 Time-Varying Mass System (Rocket System)

  • 4 Lagrangian Formulation of Mechanics: Descriptions of Motion in Configuration Space
    4.1 Generalized Coordinates and Constraints
    4.1.1 Generalized Coordinates
    4.1.2 Degrees of Freedom
    4.1.3 Configuration Space.
    4.1.4 Constraints
    4.1.4.1 Holonomic and Nonholonomic Constraints
    4.1.4.2 Scleronomic and Rheonomic Constraints.
    4.2 Kinetic Energy in Generalized Coordinates
    4.3 Generalized Momentum
    4.4 Lagrangian Equations of Motion
    4.4.1 Hamilton’s Principle
    4.4.2 Lagrange’s Equations of Motion from Hamilton’s Principle
    4.5 Nonuniqueness of the Lagrangian
    4.6 Integrals of Motion and Conservation Laws
    4.6.1 Cyclic Coordinates and Conservation Theorems
    4.6.2 Symmetries and Conservation Laws
    4.6.2.1 Homogeneity of Time and Conservation of Energy
    4.6.2.2 Spatial Homogeneity and Momentum Conservation
    4.6.2.3 Isotropy of Space and Angular Momentum Conservation
    4.6.2.4 Noether’s Theorem
    4.7 Scale Invariance
    4.8 Nonconservative Systems and Generalized Potential
    4.9 Charged Particle in Electromagnetic Field
    4.10 Forces of Constraint and Lagrange’s Multipliers
    4.11 Lagrangian versus Newtonian Approach to Classical Mechanics
    Reference

  • 5 Hamiltonian Formulation of Mechanics: Descriptions of Motion in Phase Spaces
    5.1 The Hamiltonian of a Dynamic System
    5.1.1 Phase Space
    5.2 Hamilton’s Equations of Motion
    5.2.1 Hamilton’s Equations from Lagrange’s Equations
    5.2.2 Hamilton’s Equations from Hamilton’s Principle
    5.3 Integrals of Motion and Conservation Theorems
    5.3.1 Energy Integrals
    5.3.2 Cyclic Coordinates and Integrals of Motion
    5.3.3 Conservation Theorems of Momentum and Angular Momentum
    5.4 Canonical Transformations.
    5.5 Poisson Brackets
    5.5.1 Fundamental Properties of Poisson Brackets
    5.5.2 Fundamental Poisson Brackets
    5.5.3 Poisson Brackets and Integrals of Motion
    5.5.4 Equations of Motion in Poisson Bracket Form
    5.5.5 Canonical Invariance of Poisson Brackets
    5.6 Poisson Brackets and Quantum Mechanics
    5.7 Phase Space and Liouville’s Theorem
    5.8 Time Reversal in Mechanics (Optional)
    5.9 Passage from Hamiltonian to Lagrangian
    References

  • 6 Motion Under a Central Force
    6.1 Two-Body Problem and Reduced Mass
    6.2 General Properties of Central Force Motion
    6.3 Effective Potential and Classification of Orbits
    6.4 General Solutions of Central Force Problem
    6.4.1 Energy Method
    6.4.2 Lagrangian Analysis
    6.5 Inverse Square Law of Force.
    6.6 Kepler’s Three Laws of Planetary Motion
    6.7 Applications of Central Force Motion
    6.7.1 Satellites and Spacecraft..
    6.7.2 Communication Satellites
    6.7.3 Flyby Missions to Outer Planets
    6.8 Newton’s Law of Gravity from Kepler’s Laws
    6.9 Stability of Circular Orbits (Optional)
    6.10 Apsides and Advance of Perihelion (Optional)
    6.10.1 Advance of Perihelion and Inverse-Square Force
    6.10.2 Method of Perturbation Expansion
    6.11 Laplace–Runge–Lenz Vector and the Kepler Orbit (Optional)
    References.

  • 7 Harmonic Oscillator
    7.1 Simple Harmonic Oscillator
    7.1.1 Motion of Mass m on the End of a Spring
    7.1.2 The Bob of Simple Pendulum Swinging through a Small Arc
    7.1.3 Solution of Equation of Motion of SHM
    7.1.4 Kinetic, Potential, Total, and Average Energies of Harmonic Oscillator
    7.2 Adiabatic Invariants and Quantum Condition
    7.3 Damped Harmonic Oscillator
    7.4 Phase Diagram for Damped Oscillator
    7.5 Relaxation Time Phenomena
    7.6 Forced Oscillations without Damping
    7.6.1 Periodic Driving Force
    7.6.2 Arbitrary Driving Forces
    7.7 Forced Oscillations with Damping
    7.7.1 Resonance
    7.7.2 Power Absorption
    7.8 Oscillator Under Arbitrary Periodic Force
    7.8.1 Fourier’s Series Solution
    7.9 Vibration Isolation
    7.10 Parametric Excitation

  • 8 Coupled Oscillations and Normal Coordinates
    8.1 Coupled Pendulum
    8.1.1 Normal Coordinates
    8.2 Coupled Oscillators and Normal Modes: General Analytic Approach
    8.2.1 The Equation of Motion of a Coupled System
    8.2.2 Normal Modes of Oscillation
    8.2.3 Orthogonality of Eigenvectors
    8.2.4 Normal Coordinates
    8.3 Forced Oscillations of Coupled Oscillators
    8.4 Coupled Electric Circuits

  • 9 Nonlinear Oscillations
    9.1 Qualitative Analysis: Energy and Phase Diagrams
    9.2 Elliptical Integrals and Nonlinear Oscillations
    9.3 Fourier Series Expansions
    9.3.1 Symmetrical Potential: V(x) = V(−x)
    9.3.2 Asymmetrical Potential: V(−x) = −V(x)
    9.4 The Method of Perturbation
    9.4.1 Bogoliuboff–Kryloff Procedure and Removal of Secular Terms292
    9.6 Method of Successive Approximation
    9.7 Multiple Solutions and Jumps
    9.8 Chaotic Oscillations.
    9.8.1 Some Helpful Tools for an Understanding of Chaos
    9.8.2 Conditions for Chaos
    9.8.3 Routes to Chaos
    9.8.4 Lyapunov Exponentials
    References.312

  • 10 Collisions and Scatterings
    10.1 Direct Impact of Two Particles
    10.2 Scattering Cross Sections and Rutherford Scattering
    10.2.1 Scattering Cross Sections
    10.2.2 Rutherford’s α-Particle Scattering Experiment.
    10.2.3 Cross Section Is Lorentz Invariant
    10.3 Laboratory and Center-of-Mass Frames of Reference
    10.4 Nuclear Sizes
    10.5 Small-Angle Scattering (Optional)
    References

  • 11 Motion in Non-Inertial Systems
    11.1 Accelerated Translational Coordinate System
    11.2 Dynamics in Rotating Coordinate System
    11.2.1 Centrifugal Force
    11.2.2 The Coriolis Force
    11.2.2.1 Trade Winds and Circulation of Ocean Currents
    11.2.2.2 Weather Systems
    11.2.2.3 Hurricanes
    11.2.2.4 Bathtub Vortex and Earth Rotation
    11.3 Motion of Particle Near the Surface of the Earth
    11.4 Foucault Pendulum.
    11.5 Larmor’s Theorem
    11.6 Classical Zeeman Effect
    11.7 Principle of Equivalence
    11.7.1 Principle of Equivalence and Gravitational Red Shift

  • 12 Motion of Rigid Bodies
    12.1 Independent Coordinates of Rigid Body
    12.2 Eulerian Angles
    12.3 Rate of Change of Vector
    12.4 Rotational Kinetic Energy and Angular Momentum
    12.5 Inertia Tensor
    12.5.1 Diagonalization of a Symmetric Tensor
    12.5.2 Moments and Products of Inertia
    12.5.3 Parallel-Axis Theorem
    12.5.4 Moments of Inertia about an Arbitrary Axis
    12.5.5 Principal Axes of Inertia.
    12.6 Euler’s Equations of Motion
    12.7 Motion of a Torque-Free Symmetrical Top
    12.8 Motion of Heavy Symmetrical Top with One Point Fixed
    12.8.1 Precession without Nutation
    12.8.2 Precession with Nutation
    12.9 Stability of Rotational Motion.
    References

  • 13 Theory of Special Relativity
    13.1 Historical Origin of Special Theory of Relativity
    13.2 Michelson–Morley Experiment
    13.3 Postulates of Special Theory of Relativity
    13.3.1 Time Is Not Absolute
    13.4 Lorentz Transformations
    13.4.1 Relativity of Simultaneity, Causality
    13.4.2 Time Dilation, Relativity of Co-Locality
    13.4.3 Length Contraction
    13.4.4 Visual Apparent Shape of Rapidly Moving Object
    13.4.5 Relativistic Velocity Addition
    13.5 Doppler Effect
    13.6 Relativistic Space–Time (Minkowski Space)
    13.6.1 Four-Velocity and Four-Acceleration.
    13.6.2 Four-Energy and Four-Momentum Vectors.
    13.6.3 Particles of Zero Rest Mass
    13.7 Equivalence of Mass and Energy.
    13.8 Conservation Laws of Energy and Momentum
    13.9 Generalization of Newton’s Equation of Motion
    13.9.1 Force Transformation.
    13.10 Relativistic Lagrangian and Hamiltonian Functions.
    13.11 Relativistic Kinematics of Collisions
    13.12 Collision Threshold Energies
    References.

  • 14 Newtonian Gravity and Newtonian Cosmology
    14.1 Newton’s Law of Gravity.
    14.2 Gravitational Field and Gravitational Potential.
    14.3 Gravitational Field Equations: Poisson’s and Laplace’s Equations
    14.4 Gravitational Field and Potential of Extended Body.
    14.5 Tides
    14.6 General Theory of Relativity: Relativistic Theory of Gravitation
    14.6.1 Gravitational Shift of Spectral Lines (Gravitational Red Shift)
    14.6.2 Bending of Light Beam
    14.7 Introduction to Cosmology
    14.8 Brief History of Cosmological Ideas
    14.8.1 Newton and Infinite Universe
    14.8.2 Newton’s Law of Gravity Predicts Nonstationary Universe
    14.8.3 An Infinite Steady Universe Is an Empty Universe
    14.8.4 Olbers’ Paradox
    14.9 Discovery of Expansion of the Universe, Hubble’s Law
    14.10 Big Bang.
    14.10.1 Age of the Universe
    14.11 Formulating Dynamical Models of the Universe
    14.12 Cosmological Red Shift and Hubble Constant H
    14.13 Critical Mass Density and Future of the Universe
    14.13.1 Density Parameter Ω
    14.13.2 Deceleration Parameter q
    14.13.3 An Accelerating Universe?
    14.14 Microwave Background Radiation
    14.15 Dark Matter
    Reference.

  • 15 Hamilton–Jacobi Theory of Dynamics
    15.1 Canonical Transformation and H-J Equation
    15.2 Action and Angle Variables
    15.3 Infinitesimal Canonical Transformations and Time Development Operator527
    15.4 H-J Theory and Wave Mechanics
    Reference.

  • 16 Introduction to Lagrangian and Hamiltonian Formulations for Continuous Systems and Classical Fields
    16.1 Vibration of Loaded String
    16.2 Vibrating Strings and the Wave Equation
    16.2.1 Wave Equation
    16.2.2 Separation of Variables
    16.2.3 Wave Number and Phase Velocity
    16.2.4 Group Velocity and Wave Packets
    16.3 Continuous Systems and Classical Fields
    16.3.1 Lagrangian Formulation
    16.3.2 Hamiltonian Formulation
    16.3.3 Conservation Laws
    16.4 Scalar and Vector of Fields
    16.4.1 Scalar Fields
    16.4.2 Vector Fields

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

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