Classical Mechanics

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Chapter 1: The Foundations of Motion and Computation
	1.1 THE WORLD OF PHYSICS
	1.2 THE BASICS OF CLASSICAL MECHANICS
		1.2.1 The Basic Descriptors of Motion
			1.2.1.1 Position and Displacement
			1.2.1.2 Velocity
			1.2.1.3 Acceleration
		1.2.2 Mass and Force
			1.2.2.1 Mass
			1.2.2.2 Force
	1.3 NEWTON’S LAWS OF MOTION
		1.3.1 Newton’s First Law
		1.3.2 Newton’s second law
		1.3.3 Newton’s third law
	1.4 REFERENCE FRAMES
	1.5 COMPUTATION IN PHYSICS
		1.5.1 The Use of Computation in Physics
		1.5.2 Different Computational Tools
		1.5.3 Some Warnings
	1.6 CLASSICAL MECHANICS IN THE MODERN WORLD
	1.7 CHAPTER SUMMARY
	1.8 END-OF-CHAPTER PROBLEMS
Chapter 2: Single-Particle Motion in One Dimension
	2.1 EQUATIONS OF MOTION
	2.2 ORDINARY DIFFERENTIAL EQUATIONS
	2.3 CONSTANT FORCES
	2.4 TIME-DEPENDENT FORCES
	2.5 AIR RESISTANCE AND VELOCITY-DEPENDENT FORCES
	2.6 POSITION-DEPENDENT FORCES
	2.7 NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS
	2.8 CHAPTER SUMMARY
	2.9 END-OF-CHAPTER PROBLEMS
Chapter 3: Motion in Two and Three Dimensions
	3.1 POSITION, VELOCITY, AND ACCELERATION IN CARTESIAN COORDI-NATE SYSTEMS
	3.2 VECTOR PRODUCTS
		3.2.1 The Dot Product
		3.2.2 The Cross Product
	3.3 POSITION, VELOCITY, AND ACCELERATION IN NON-CARTESIAN COORDINATE SYSTEMS
		3.3.1 Polar Coordinates
		3.3.2 Position, Velocity, and Acceleration in Cylindrical Coordinates
		3.3.3 Position, Velocity, and Acceleration in Spherical Coordinates
	3.4 THE GRADIENT, DIVERGENCE, AND CURL
		3.4.1 The Gradient
		3.4.2 The Divergence
		3.4.3 The Curl
		3.4.4 Second Derivatives with the Del Operator
	3.5 CHAPTER SUMMARY
	3.6 END-OF-CHAPTER PROBLEMS
Chapter 4: Momentum, Angular Momentum, and Multiparticle Systems
	4.1 CONSERVATION OF MOMENTUM AND NEWTON’S THIRD LAW
	4.2 ROCKETS
	4.3 CENTER OF MASS
	4.4 NUMERICAL INTEGRATION AND THE CENTER OF MASS
		4.4.1 Trapezoidal Rule
		4.4.2 Simpson’s Rule
	4.5 MOMENTUM OF A SYSTEM OF MULTIPLE PARTICLES
	4.6 ANGULAR MOMENTUM OF A SINGLE PARTICLE
	4.7 ANGULAR MOMENTUM OF MULTIPLE PARTICLES
	4.8 CHAPTER SUMMARY
	4.9 END-OF-CHAPTER PROBLEMS
Chapter 5: Energy
	5.1 WORK AND ENERGY IN ONE-DIMENSIONAL SYSTEMS
	5.2 POTENTIAL ENERGY AND EQUILIBRIUM POINTS IN ONE-DIMENSIONAL SYSTEMS
	5.3 WORK AND LINE INTEGRALS
	5.4 THE WORK-KINETIC ENERGY THEOREM, REVISITED
	5.5 CONSERVATIVE FORCES AND POTENTIAL ENERGY
	5.6 ENERGY AND MULTIPARTICLE SYSTEMS
	5.7 CHAPTER SUMMARY
	5.8 END-OF-CHAPTER PROBLEMS
Chapter 6: Harmonic Oscillations
	6.1 DIFFERENTIAL EQUATIONS
	6.2 THE SIMPLE HARMONIC OSCILLATOR
		6.2.1 The Equation of Motion of the Simple Harmonic Oscillator
		6.2.2 Potential and Kinetic Energy in Simple Harmonic Motion
		6.2.3 The Simple Plane Pendulum as an Example of a Harmonic Oscillator
	6.3 NUMERICAL SOLUTIONS USING THE EULER METHOD FOR HARMONIC OSCILLATIONS
	6.4 DAMPED HARMONIC OSCILLATOR
		6.4.1 Overdamped Oscillations
		6.4.2 Underdamped Oscillation
		6.4.3 Critically Damped Oscillations
	6.5 ENERGY IN DAMPED HARMONIC MOTION
	6.6 FORCED HARMONIC OSCILLATOR
	6.7 ENERGY RESONANCE AND THE QUALITY FACTOR FOR DRIVEN OSCILLATIONS
	6.8 ELECTRICAL CIRCUITS
	6.9 PRINCIPLE OF SUPERPOSITION AND FOURIER SERIES
		6.9.1 The Principle of Superposition
		6.9.2 Fourier Series
		6.9.3 Example of Superposition Principle and Fourier Series
	6.10 PHASE SPACE
	6.11 CHAPTER SUMMARY
	6.12 END-OF-CHAPTER PROBLEMS
Chapter 7: The Calculus of Variations
	7.1 THE MOTIVATION FOR LEARNING THE CALCULUS OF VARIATIONS
	7.2 THE SHORTEST DISTANCE BETWEEN TWO POINTS—SETTING UP THE CALCULUS OF VARIATIONS
	7.3 THE FIRST FORM OF THE EULER EQUATION
	7.4 THE SECOND FORM OF THE EULER EQUATION
	7.5 SOME EXAMPLES OF PROBLEMS SOLVED USING THE CALCULUS OF VARIATIONS
		7.5.1 The Brachistochrone Problem
		7.5.2 Geodesics
		7.5.3 Minimum Surface of Revolution
	7.6 MULTIPLE DEPENDENT VARIABLES
	7.7 CHAPTER SUMMARY
	7.8 END-OF-CHAPTER PROBLEMS
Chapter 8: Lagrangian and Hamiltonian Dynamics
	8.1 AN INTRODUCTION TO THE LAGRANGIAN
	8.2 GENERALIZED COORDINATES AND DEGREES OF FREEDOM
	8.3 HAMILTON’S PRINCIPLE
	8.4 SOME EXAMPLES OF LAGRANGIAN DYNAMICS
	8.5 NUMERICAL SOLUTIONS TO ODE’S USING THE FOURTH-ORDER RUNGE-KUTTA METHOD
	8.6 CONSTRAINT FORCES AND LAGRANGE’S EQUATION WITH UNDE-TERMINED MULTIPLIERS
	8.7 CONSERVATION THEOREMS AND THE LAGRANGIAN
		8.7.1 Conservation of Momentum
		8.7.2 Conservation of Energy
	8.8 HAMILTONIAN DYNAMICS
	8.9 ADDITIONAL EXPLORATIONS INTO THE HAMILTONIAN
	8.10 CHAPTER SUMMARY
	8.11 END-OF-CHAPTER PROBLEMS
Chapter 9: Central Forces and Planetary Motion
	9.1 CENTRAL FORCES
		9.1.1 Central Forces and the Conservation of Energy
		9.1.2 Central Forces and the Conservation of Angular Momentum
	9.2 THE TWO-BODY PROBLEM
	9.3 EQUATIONS OF MOTION FOR THE TWO-BODY PROBLEM
	9.4 PLANETARY MOTION AND KEPLER’S FIRST LAW
	9.5 ORBITS IN A CENTRAL FORCE FIELD
	9.6 KEPLER’S LAWS OF PLANETARY MOTION
		9.6.1 Kepler’s First Law
		9.6.2 Kepler’s Second Law
		9.6.3 Kepler’s Third Law
	9.7 THE PLANAR CIRCULAR RESTRICTED THREE-BODY PROBLEM
	9.8 CHAPTER SUMMARY
	9.9 END-OF-CHAPTER PROBLEMS
Chapter 10: Motion in Noninertial Reference Frames
	10.1 MOTION IN A NONROTATING ACCELERATING REFERENCE FRAME
	10.2 ANGULAR VELOCITY AS A VECTOR
	10.3 TIME DERIVATIVES OF VECTORS IN ROTATING COORDINATE FRAMES
	10.4 NEWTON’S SECOND LAW IN A ROTATING FRAME
		10.4.1 The Centrifugal Force
		10.4.2 The Coriolis Force
	10.5 FOUCAULT PENDULUM
	10.6 PROJECTILE MOTION IN A NONINERTIAL FRAME
	10.7 CHAPTER SUMMARY
	10.8 END-OF-CHAPTER PROBLEMS
Chapter 11: Rigid Body Motion
	11.1 ROTATIONAL MOTION OF PARTICLES AROUND A FIXED AXIS
	11.2 REVIEW OF ROTATIONAL PROPERTIES FOR A SYSTEM OF PARTICLES
		11.2.1 The Center of Mass
		11.2.2 Momentum of a System of Particles
		11.2.3 Angular Momentum of a System of Particles
		11.2.4 Work and Kinetic Energy for a System of Particles
	11.3 THE MOMENT OF INERTIA TENSOR
	11.4 KINETIC ENERGY AND THE INERTIA TENSOR
	11.5 INERTIA TENSOR IN DIFFERENT COORDINATE SYSTEMS—THE PARALLEL AXIS THEOREM
	11.6 PRINCIPAL AXES OF ROTATION
	11.7 PRECESSION OF A SYMMETRIC SPINNING TOP WITH ONE POINT FIXED AND EXPERIENCING A WEAK TORQUE
	11.8 RIGID BODY MOTION IN THREE DIMENSIONS AND EULER’S EQUATIONS
	11.9 THE FORCE-FREE SYMMETRIC TOP
	11.10 CHAPTER SUMMARY
	11.11 END-OF-CHAPTER PROBLEMS
Chapter 12: Coupled Oscillations
	12.1 COUPLED OSCILLATIONS OF A TWO-MASS THREE-SPRING SYSTEM
		12.1.1 The Equations of Motion—Numerical Solution
		12.1.2 Equal Masses and Identical Springs: The Normal Modes
		12.1.3 The General Case: Linear Combination of Normal Modes
	12.2 NORMAL MODE ANALYSIS OF THE TWO-MASS THREE-SPRING SYSTEM
		12.2.1 Equal Masses and Identical Springs—Analytical Solution
		12.2.2 Solving the Two-Mass and Three-Spring System as an Eigenvalue Problem
	12.3 THE DOUBLE PENDULUM
		12.3.1 The Lagrangian and Equations of Motion—Numerical Solutions
		12.3.2 Identical Masses and Lengths—Analytical Solutions
		12.3.3 The Double Pendulum as an Eigenvector/Eigenvalue Problem
	12.4 GENERAL THEORY OF SMALL OSCILLATIONS AND NORMAL COOR-DINATES
		12.4.1 The Lagrangian for Small Oscillations Around an Equilibrium Position
		12.4.2 The Equations of Motion for Small Oscillations Around an Equilibrium Point
		12.4.3 Normal Coordinates
	12.5 CHAPTER SUMMARY
	12.6 END-OF-CHAPTER PROBLEMS
Chapter 13: Nonlinear Systems
	13.1 LINEAR VS. NONLINEAR SYSTEMS
	13.2 THE DAMPED HARMONIC OSCILLATOR, REVISITED
	13.3 FIXED POINTS AND PHASE PORTRAITS
		13.3.1 The Simple Plane Pendulum, Revisited
		13.3.2 The Double-Well Potential, Revisited
		13.3.3 Damped Double-Well
		13.3.4 Bifurcations of Fixed Points
	13.4 LIMIT CYCLES
		13.4.1 The Duffing Equation
		13.4.2 Limit Cycles and Period Doubling Bifurcations
	13.5 CHAOS
		13.5.1 Chaos and Initial Conditions
		13.5.2 Lyapunov Exponents
	13.6 A FINAL WORD ON NONLINEAR SYSTEMS
	13.7 CHAPTER SUMMARY
	13.8 END-OF-CHAPTER PROBLEMS
Bibliography
Index