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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