Classical Mechanics, 3rd Edition

The third edition of Classical Mechanics presents a complete account of the classical mechanics of particles and systems for physics students at the advanced undergraduate level. The textbook evolved from a set of lecture notes for a course on the subject taught for many years by the author at California State University, Stanislaus. 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 with fundamental concepts, such as velocity and acceleration. Vectors are used from the start. Unlike conventional textbooks, this textbook introduces Lagrangian and Hamiltonian formulations of mechanics early on, enabling students to develop confidence in these essential methods. Special note is given to concepts instrumental in the development of modern physics, including the relationship between symmetries and the laws of conservation. Applications to other branches of physics are also included wherever possible. Generalized coordinates are covered to enable discussion of Lagrangian dynamics and Hamiltonian dynamics, which have important applications in quantum mechanics, statistical mechanics, celestial mechanics, and electrodynamics. Key features: • Lengthier sections in early chapters have been rewritten as short and easy-to-understand sections. • Each chapter contains homework problems of varying degrees of difficulty to enhance understanding of the material in the text. • Detailed but not overlong mathematical manipulations are provided. • The essential topics can be covered in a one-semester, four-hour course. Cover Half Title Title Copyright Contents Preface Author 1 Kinematics—Describing the Motion 1.1 Introduction 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 (r̄, 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 2.4.1 Is Space Absolute or Relative? 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 the Center of Mass 2.7.3 Conservation Theorems 3 Integration of Newton’s Equation of Motion 3.1 Introduction 3.2 Motion under a Constant Force 3.3 Force Is a Function of Time 3.3.1 Impulsive Force and the Green’s Function Method 3.4 Force Is a Function of Velocity 3.4.1 Motion in a Uniform Magnetic Field 3.4.2 Motion in a 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 The Lagrangian Formulation of Mechanics: Description 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 4.2 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 Non-Uniqueness of the Lagrangian 4.6 Integrals of the Motion and Conservation Laws 4.6.1 Cyclic Coordinates and Conservation Theorems 4.6.2 Symmetries and Conservation Laws 4.7 Scale Invariance 4.8 Nonconservative Systems and Generalized Potential 4.9 Charged Particle in an Electromagnetic Field 4.10 Forces of Constraint and Lagrange’s Multipliers 4.11 The Lagrangian Versus the Newtonian Approach to Classical Mechanics 5 The Hamiltonian Formulation of Mechanics: Description of Motion in Phase Space 5.1 The Hamiltonian of a Dynamical 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 the 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 The 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 The 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 The Passage from the Hamiltoniian to the Lagrangian 6 Motion Under a Central Force 6.1 The Two-Body Problem and the Reduced Mass 6.2 General Properties of Central Force Motion 6.3 Effective Potential and Classification of Orbits 6.4 General Solutions of the Central Force Problem 6.4.1 The 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 the Outer Planets 6.8 Newton’s Law of Gravity from Kepler’s Laws 6.9 Stability of Circular Orbits (Optional) 6.10 The Apsides and the Advance of Perihelion (Optional) 6.10.1 Advance of the Perihelion and the Inverse-Square Force 6.10.2 Method of the Perturbation Expansion 6.11 The Laplace-Runge-Lenz Vector and the Kepler Orbit (Optional) 7 The Harmonic Oscillator 7.1 The Simple Harmonic Oscillator 7.1.1 The Motion of Mass m on the End of a Spring 7.1.2 The Bob of a Simple Pendulum Swinging through a Small Arc 7.1.3 Solution of the Equation of Motion of the SHM 7.1.4 Kinetic, Potential, Total, and Average Energies of the Harmonic Oscillator 7.2 Adiabatic Invariants and Quantum Condition 7.3 The 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 The Orthogonality of the 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 The Bogoliuboff-Kryloff Procedure and the Removal of Secular Terms 9.5 The Ritz Method 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 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 The Laboratory and Center-of-Mass Frames of Reference 10.4 Nuclear Sizes 10.5 Small-Angle Scattering (Optional) 11 Motion in Non-Inertial Systems 11.1 Accelerated Translational Coordinate System 11.2 Dynamics in Rotating Coordinate System 11.2.1 The Centrifugal Force 11.2.2 The Coriolis Force 11.3 Motion of a Particle Near the Surface of the Earth 11.4 The Foucault Pendulum 11.5 Larmor’s Theorem 11.6 The Classical Zeeman Effect 11.7 The Principle of Equivalence 11.7.1 The Principle of Equivalence and the Gravitational Red Shift 12 The Motion of Rigid Bodies 12.1 The Independent Coordinates of a Rigid Body 12.2 The Eulerian Angles 12.3 Rate of Change of a Vector 12.4 Rotational Kinetic Energy and Angular Momentum 12.5 The 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 The Motion of a 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 13 Newtonian Gravity and Newtonian Cosmology 13.1 Newton’s Law of Gravity 13.2 Gravitational Field and Gravitational Potential 13.3 Gravitational Field Equations: Poisson’s and Laplace’s Equations 13.4 Gravitational Field and Potential of an Extended Body 13.5 The Tides 13.6 The General Theory of Relativity: Relativistic Theory of Gravitation 13.6.1 Gravitational Shift of Spectral Lines (Gravitational Redshift) 13.6.2 The Bending of a Light Beam 13.7 Introduction to Newtonian Cosmology 13.8 A Brief History of Cosmological Ideas 13.8.1 Newton and Infinite Universe 13.8.2 Newton’s Law of Gravity Predicts a Non-Stationary Universe 13.8.3 An Infinite Steady Universe Is an Empty Universe 13.8.4 Olbers’s Paradox 13.9 Discovery of the Expansion of the Universe, Hubble’s Law 13.10 The Big Bang 13.10.1 Age of the Universe 13.11 Formulating Dynamical Models of the Universe 13.12 Cosmological Redshift and the Hubble Constant H 13.13 The Critical Mass Density and the Future of the Universe 13.13.1 The Density Parameter Ω 13.13.2 The Deceleration Parameter q0 13.13.3 An Accelerating Universe? 13.14 The Microwave Background Radiation 13.15 Dark Matter 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