Learn Physics with Functional Programming: A Hands-on Guide to Exploring Physics with Haskell

Title Page
Copyright Page
Dedication
About the Author
About the Technical Reviewer
Brief Contents
Contents in Detail
ACKNOWLEDGMENTS
INTRODUCTION
	Who This Book Is For
	Why Functional Programming, and Why Haskell?
	About This Book
PART I A HASKELL PRIMER FOR PHYSICISTS
1 CALCULATING WITH HASKELL
	A Kinematics Problem
	The Interactive Compiler
	Numeric Functions
	Operators
		Precedence and Associativity
		The Application Operator
	Functions with Two Arguments
	Numbers in Haskell
		Negative Numbers in Haskell
		Decimal Numbers in Haskell
		Exponential Notation
	Approximate Calculation
	Errors
	Getting Help and Quitting
	More Information
	Summary
	Exercises
2 WRITING BASIC FUNCTIONS
	Constants, Functions, and Types
	How We Talk About Functions
	Anonymous Functions
	Composing Functions
	Variable Not in Scope Error
	Summary
	Exercises
3 TYPES AND ENTITIES
	Basic Types
		The Boolean Type
		The Character Type
		The String Type
		Numeric Types
	Function Types
	Summary
	Exercises
4 DESCRIBING MOTION
	Position and Velocity on an Air Track
	Types for Physical Quantities
	Introducing Derivatives
	Derivatives in Haskell
	Modeling the Car’s Position and Velocity
	Modeling Acceleration
	Approximate Algorithms and Finite Precision
	Summary
	Exercises
5 WORKING WITH LISTS
	List Basics
		Selecting an Element from a List
		Concatenating Lists
		Arithmetic Sequences
		List Types
		Functions for Lists of Numbers
		When Not to Use a List
	Type Variables
	Type Conversion
	The Length of Lists
	A String Is a List of Characters
	List Comprehensions
	Infinite Lists
	List Constructors and Pattern Matching
	Summary
	Exercises
6 HIGHER-ORDER FUNCTIONS
	How to Think About Functions with Parameters
	Mapping a Function Over a List
	Iteration and Recursion
	Anonymous Higher-Order Functions
	Operators as Higher-Order Functions
	Combinators
	Predicate-Based Higher-Order Functions
	Numerical Integration
		Introducing Integrators
		Digital Integration
		Implementing Antiderivatives
	Summary
	Exercises
7 GRAPHING FUNCTIONS
	Using Library Modules
		Standard Library Modules
		Other Library Modules
	Plotting
		Function Only
		Function and Module
		Function, Module, and Plot Definition
	Summary
	Exercises
8 TYPE CLASSES
	Type Classes and Numbers
	Type Classes from the Prelude
		The Eq Type Class
		The Show Type Class
		The Num Type Class
		The Integral Type Class
		The Ord Type Class
		The Fractional Type Class
		The Floating Type Class
	Exponentiation and Type Classes
	Sections
	Example of Type Classes and Plotting
	Summary
	Exercises
9 TUPLES AND TYPE CONSTRUCTORS
	Pairs
	Currying a Function of Two Variables
	Triples
	Comparing Lists and Tuples
	Maybe Types
	Lists of Pairs
	Tuples and List Comprehensions
	Type Constructors and Kinds
	Numerical Integration Redux
	Summary
	Exercises
10 DESCRIBING MOTION IN THREE DIMENSIONS
	Three-Dimensional Vectors
		Coordinate-Free Vectors
		Geometric Definition of Vector Addition
		Geometric Definition of Scaling a Vector
		Geometric Definition of Vector Subtraction
		Geometric Definition of Dot Product
		Geometric Definition of Cross Product
		Derivative of a Vector-Valued Function
	Coordinate Systems
		Vector Addition with Coordinate Components
		Vector Scaling with Coordinate Components
		Vector Subtraction with Coordinate Components
		Dot Product with Coordinate Components
		Cross Product with Coordinate Components
		Derivative with Coordinate Components
	Kinematics in 3D
		Defining Position, Velocity, and Acceleration
		Two Components of Acceleration
		Projectile Motion
	Making Your Own Data Type
		Single Data Constructor
		Multiple Data Constructors
	Defining a New Data Type for 3D Vectors
		Possible Implementations
		Data Type Definition for Vec
		Vec Functions
	Summary
	Exercises
11 CREATING GRAPHS
	Title and Axis Labels
	Other Labels
	Plotting Data
	Multiple Curves on One Set of Axes
	Controlling the Plot Ranges
	Making a Key
	Summary
	Exercises
12 CREATING STAND-ALONE PROGRAMS
	Using GHC to Make a Stand-Alone Program
		Hello, World!
		A Program That Imports Modules
	Using Cabal to Make a Stand-Alone Program
	Using Stack to Make a Stand-Alone Program
	Summary
	Exercises
13 CREATING 2D AND 3D ANIMATIONS
	2D Animation
		Displaying a 2D Picture
		Making a 2D Animation
		Making a 2D Simulation
	3D Animation
		Displaying a 3D Picture
		Making a 3D Animation
		Making a 3D Simulation
	Summary
	Exercises
PART II EXPRESSING NEWTONIAN MECHANICS AND SOLVING PROBLEMS
14 NEWTON’S SECOND LAW AND DIFFERENTIAL EQUATIONS
	Newton’s First Law
	Newton’s Second Law in One Dimension
	Second Law with Constant Forces
	Second Law with Forces That Depend Only on Time
	Air Resistance
	Second Law with Forces That Depend Only on Velocity
		Euler Method by Hand
		Euler Method in Haskell
	The State of a Physical System
	Second Law with Forces That Depend on Time and Velocity
		Method 1: Produce a List of States
		Method 2: Produce a Velocity Function
	Example: Pedaling and Coasting with Air Resistance
		Euler Method by Hand
		Method 1: Produce a List of States
		Method 2: Produce a Velocity Function
	Summary
	Exercises
15 MECHANICS IN ONE DIMENSION
	Introductory Code
	Forces That Depend on Time, Position, and Velocity
		A General Strategy for Solving Mechanics Problems
		Solving with Euler’s Method
		Producing a List of States
		Position and Velocity Functions
	A Damped Harmonic Oscillator
		Euler Method by Hand
		Method 1: Producing a List of States
		Method 2: Producing Position and Velocity Functions
	Euler-Cromer Method
	Solving Differential Equations
		Generalizing the State Space
		Type Classes for State Spaces
		One More Numerical Method
		Comparison of Numerical Methods
	Summary
	Exercises
16 MECHANICS IN THREE DIMENSIONS
	Introductory Code
	Newton’s Second Law in Three Dimensions
	The State of One Particle
	Solving Newton’s Second Law
	One-Body Forces
		Earth Surface Gravity
		Gravity Produced by the Sun
		Air Resistance
		Wind Force
		Force from Uniform Electric and Magnetic Fields
	State Update for One Particle
	Preparing for Animation
		Two Helpful Animation Functions
		How the Functions Work
	Summary
	Exercises
17 SATELLITE, PROJECTILE, AND PROTON MOTION
	Satellite Motion
		State-Update Function
		Initial State
		Time-Scale Factor
		Animation Rate
		Display Function
	Projectile Motion with Air Resistance
		Calculating a Trajectory
		Finding the Angle for Maximum Range
		2D Animation
		3D Animation
	Proton in a Magnetic Field
	Summary
	Exercises
18 A VERY SHORT PRIMER ON RELATIVITY
	A Little Theory
	A Replacement for Newton’s Second Law
	Response to a Constant Force
	Proton in a Magnetic Field
	Summary
	Exercises
19 INTERACTING PARTICLES
	Newton’s Third Law
	Two-Body Forces
		Universal Gravity
		Constant Repulsive Force
		Linear Spring
		Central Force
		Elastic Billiard Interaction
	Internal and External Forces
	The State of a Multi-Particle System
	State Update for Multiple Particles
		Implementing Newton’s Second Law
		Numerical Methods for Multiple Particles
		Composite Functions
	Summary
	Exercises
20 SPRINGS, BILLIARD BALLS, AND A GUITAR STRING
	Introductory Code
	Two Masses and Two Springs
		Forces
		Animation Functions
		Stand-Alone Animation Program
		Using Mechanical Energy as a Guide to Numerical Accuracy
	A Collision
		Data Representations
		Spring Constant and Time Step
		Momentum and Energy Conservation
		Numerical Issues
		Animated Results
	Wave on a Guitar String
		Forces
		State-Update Function
		Initial State
		Stand-Alone Program
		Asynchronous Animation
	Summary
	Exercises
PART III EXPRESSING ELECTROMAGNETIC THEORY AND SOLVING PROBLEMS
21 ELECTRICITY
	Electric Charge
	Coulomb’s Law
	Two Charges Interacting
		Looking at Extremes
		Modeling the Situation in Haskell
	Summary
	Exercises
22 COORDINATE SYSTEMS AND FIELDS
	Polar Coordinates
	Cylindrical Coordinates
	Spherical Coordinates
	Introductory Code
	A Type for Position
		Defining the New Type
		Making a Position
		Using a Position
	Displacement
	The Scalar Field
	The Vector Field
	Functions for Visualizing Scalar Fields
		3D Visualization
		2D Visualization
	Functions for Visualizing Vector Fields
		3D Visualization
		2D Visualization
		Gradient Visualization
	Summary
	Exercises
23 CURVES, SURFACES, AND VOLUMES
	Introductory Code
	Curves
		Parameterizing Curves
		Examples of Curves
	Surfaces
		Parameterizing Surfaces
		Examples of Surfaces
		Orientation
	Volumes
	Summary
	Exercises
24 ELECTRIC CHARGE
	Charge Distributions
	Introductory Code
	A Type for Charge Distribution
	Examples of Charge Distributions
	Total Charge
		Total Charge of a Line Charge
		Total Charge of a Surface Charge
		Total Charge of a Volume Charge
		Calculating Total Charge in Haskell
	Electric Dipole Moment
	Summary
	Exercises
25 ELECTRIC FIELD
	What Is an Electric Field?
	Introductory Code
	Charge Creates an Electric Field
		Electric Field Created by a Point Charge
		Electric Field Created by Multiple Charges
		Electric Field Created by a Line Charge
		Electric Field Created by a Surface Charge
		Electric Field Created by a Volume Charge
	Scalar Integrals
		Scalar Line Integral
		Scalar Surface Integral
		Scalar Volume Integral
	Approximating Curves, Surfaces, and Volumes
		Approximating a Curve
		Approximating a Surface
		Approximating a Volume
	Summary
	Exercises
26 ELECTRIC CURRENT
	Current Distributions
	Introductory Code
	A Type for Current Distribution
	Examples of Current Distributions
	Conservation of Charge and Constraints on Steady Current Distributions
	Magnetic Dipole Moment
	Summary
	Exercises
27 MAGNETIC FIELD
	A Simple Magnetic Effect
	Introductory Code
	Current Creates Magnetic Field
		Magnetic Field Created by a Line Current
		Magnetic Field Created by a Surface Current
		Magnetic Field Created by a Volume Current
	Summary
	Exercises
28 THE LORENTZ FORCE LAW
	Introductory Code
	Statics and Dynamics
	State of One Particle and Fields
	Lorentz Force Law
	Do We Really Need an Electric Field?
	State Update
	Animating a Particle in Electric and Magnetic Fields
		Uniform Fields
		Classical Hydrogen
	Summary
	Exercises
29 THE MAXWELL EQUATIONS
	Introductory Code
	The Maxwell Equations
		Relationships Between Electricity and Magnetism
		Connection to Coulomb’s Law and Biot-Savart Law
		State Update
		Spatial Derivatives and the Curl
		A Naive Method
	The FDTD Method
		The Yee Cell
		A Type for State
		FDTD and the Curl
		State Update
	Animation
		Current Density
		Grid Boundary
		Display Function
		Two Helping Functions
		Main Program
	Summary
	Exercises
APPENDIX: INSTALLING HASKELL
	Installing GHC
	Installing a Text Editor
	Installing Gnuplot
	Installing Haskell Library Packages
		Using Cabal
		Using Stack
	Installing Gloss
	Installing Diagrams
	Setting Up Your Coding Environment
		What We Want in a Coding Environment
		All Code in One Directory
		One Way to Use Stack
	Summary
BIBLIOGRAPHY
INDEX