LPFPCore

Code from the book Learn Physics with Functional Programming

(Approximations to) Real numbers

type R = Double

An approximation to a real number.

type Time = R

Time is a real number.

Vectors

data Vec

A type for three-dimensional vectors.

Instances

Show Vec

Defined in LPFPCore.SimpleVec

Methods

showsPrec :: Int -> Vec -> ShowS

show :: Vec -> String

showList :: [Vec] -> ShowS

Eq Vec

Defined in LPFPCore.SimpleVec

Methods

(==) :: Vec -> Vec -> Bool

(/=) :: Vec -> Vec -> Bool

type PosVec = Vec

The position of a particle can be represented as a vector.

type Velocity = Vec

Velocity is a vector.

type Acceleration = Vec

Acceleration is a vector.

vec

Arguments

:: R

x component

-> R

y component

-> R

z component

-> Vec

Form a vector by giving its x, y, and z components.

(^+^) :: Vec -> Vec -> Vec infixl 6

Vector addition.

(^-^) :: Vec -> Vec -> Vec infixl 6

Vector subtraction.

(*^) :: R -> Vec -> Vec infixr 7

Scalar multiplication of a number and a vector.

(^*) :: Vec -> R -> Vec infixl 7

Scalar multiplication of a vector and a number.

(^/) :: Vec -> R -> Vec infixr 7

Division of a vector by a number.

(<.>) :: Vec -> Vec -> R infixr 7

Dot product of two vectors.

(><) :: Vec -> Vec -> Vec infixl 7

Cross product of two vectors.

magnitude :: Vec -> R

Magnitude of a vector.

zeroV :: Vec

The zero vector.

negateV :: Vec -> Vec

Negate a vector.

sumV :: [Vec] -> Vec

Add a list of vectors.

xComp :: Vec -> R

x component of a vector

yComp :: Vec -> R

y component of a vector

zComp :: Vec -> R

z component of a vector

iHat :: Vec

A unit vector in the x direction.

jHat :: Vec

A unit vector in the y direction.

kHat :: Vec

A unit vector in the z direction.

positionCV :: PosVec -> Velocity -> Time -> PosVec

Given initial position and a constant velocity, return a position function.

velocityCA :: Velocity -> Acceleration -> Time -> Velocity

Given initial velocity and a constant acceleration, return a velocity function.

positionCA :: PosVec -> Velocity -> Acceleration -> Time -> PosVec

Given initial position, initial velocity, and a constant acceleration, return a position function.

aParallel :: Vec -> Vec -> Vec

Given a nonzero velocity and an acceleration, return the component of acceleration parallel to the velocity.

aPerp :: Vec -> Vec -> Vec

Given a nonzero velocity and an acceleration, return the component of acceleration perpendicular to the velocity.

speedRateChange :: Vec -> Vec -> R

Given velocity and acceleration, return the rate at which speed is changing.

Calculus

type Derivative = (R -> R) -> R -> R

A derivative takes a real-valued function of a real variable (often time) as input, and produces a real-valued function of a real variable as output.

type VecDerivative = (R -> Vec) -> R -> Vec

A vector derivative takes a vector-valued function of a real variable (usually time) as input, and produces a vector-valued function of a real variable as output.

derivative :: R -> Derivative

Given a step size, calculate the derivative of a real-valued function of a real variable (often time).

vecDerivative :: R -> VecDerivative

Given a step size, calculate the vector derivative of a vector-valued function of a real variable (usually time).

integral :: R -> (R -> R) -> R -> R -> R

Given a step size, a function, a lower limit, and an upper limit, return the definite integral of the function.

antiDerivative :: R -> R -> (R -> R) -> R -> R

Given a step size, a y-intercept, and a function, return a function with the given y-intercept whose derivative is the given function.

velFromPos

Arguments

:: R

dt

-> (Time -> PosVec)

position function

-> Time -> Velocity

velocity function

Given a time step and a position function, return a velocity function.

accFromVel :: R -> (Time -> Velocity) -> Time -> Acceleration

Given a time step and a velocity function, return an acceleration function.

Differential equations

type UpdateFunction s = s -> s

An update function takes a state as input and returns an updated state as output.

type DifferentialEquation s ds = s -> ds

A differential equation takes a state as input and returns as output the rate at which the state is changing.

type NumericalMethod s ds = DifferentialEquation s ds -> UpdateFunction s

A numerical method turns a differential equation into a state-update function.

class RealVectorSpace ds where

A real vector space allows vector addition and scalar multiplication by reals.

Methods

(+++) :: ds -> ds -> ds

scale :: R -> ds -> ds

Instances

RealVectorSpace DParticleFieldState

Defined in LPFPCore.Lorentz

Methods

(+++) :: DParticleFieldState -> DParticleFieldState -> DParticleFieldState

scale :: R -> DParticleFieldState -> DParticleFieldState

RealVectorSpace DParticleState

Defined in LPFPCore.Mechanics3D

Methods

(+++) :: DParticleState -> DParticleState -> DParticleState

scale :: R -> DParticleState -> DParticleState

RealVectorSpace DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

(+++) :: DMultiParticleState -> DMultiParticleState -> DMultiParticleState

scale :: R -> DMultiParticleState -> DMultiParticleState

RealVectorSpace (R, R)

Defined in LPFPCore.Mechanics1D

Methods

(+++) :: (R, R) -> (R, R) -> (R, R)

scale :: R -> (R, R) -> (R, R)

RealVectorSpace (R, R, R)

A triple of real numbers is a real vector space.

Defined in LPFPCore.Mechanics1D

Methods

(+++) :: (R, R, R) -> (R, R, R) -> (R, R, R)

scale :: R -> (R, R, R) -> (R, R, R)

class RealVectorSpace ds => Diff s ds where

A type class that expresses a relationship between a state space and a time-derivative-state space.

Methods

shift :: R -> ds -> s -> s

Instances

Diff ParticleFieldState DParticleFieldState

Defined in LPFPCore.Lorentz

Methods

shift :: R -> DParticleFieldState -> ParticleFieldState -> ParticleFieldState

Diff ParticleState DParticleState

Defined in LPFPCore.Mechanics3D

Methods

shift :: R -> DParticleState -> ParticleState -> ParticleState

Diff MultiParticleState DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

shift :: R -> DMultiParticleState -> MultiParticleState -> MultiParticleState

Diff State1D (R, R, R)

A triple of real numbers can serve as the time derivative of a State1D.

Defined in LPFPCore.Mechanics1D

Methods

shift :: R -> (R, R, R) -> State1D -> State1D

Diff (Time, Velocity) (R, R)

Defined in LPFPCore.Mechanics1D

Methods

shift :: R -> (R, R) -> (Time, Velocity) -> (Time, Velocity)

solver :: NumericalMethod s ds -> DifferentialEquation s ds -> s -> [s]

Given a numerical method, a differential equation, and an initial state, return a list of states.

euler :: Diff s ds => R -> (s -> ds) -> s -> s

Given a step size, return the numerical method that uses the Euler method with that step size.

rungeKutta4 :: Diff s ds => R -> (s -> ds) -> s -> s

Given a step size, return the numerical method that uses the 4th order Runge Kutta method with that step size.

3D Mechanics

Single particle state

data ParticleState

Data type for the state of a single particle in three-dimensional space.

Constructors

ParticleState

Fields

• mass :: R
• charge :: R
• time :: R
• posVec :: Vec
• velocity :: Vec

Instances

HasTime ParticleState

Defined in LPFPCore.Mechanics3D

Methods

timeOf :: ParticleState -> Time

Show ParticleState

Defined in LPFPCore.Mechanics3D

Methods

showsPrec :: Int -> ParticleState -> ShowS

show :: ParticleState -> String

showList :: [ParticleState] -> ShowS

Diff ParticleState DParticleState

Defined in LPFPCore.Mechanics3D

Methods

shift :: R -> DParticleState -> ParticleState -> ParticleState

data DParticleState

Data type for the time-derivative of a particle state.

Constructors

DParticleState

Fields

• dmdt :: R
• dqdt :: R
• dtdt :: R
• drdt :: Vec
• dvdt :: Vec

Instances

RealVectorSpace DParticleState

Defined in LPFPCore.Mechanics3D

Methods

(+++) :: DParticleState -> DParticleState -> DParticleState

scale :: R -> DParticleState -> DParticleState

Show DParticleState

Defined in LPFPCore.Mechanics3D

Methods

showsPrec :: Int -> DParticleState -> ShowS

show :: DParticleState -> String

showList :: [DParticleState] -> ShowS

Diff ParticleState DParticleState

Defined in LPFPCore.Mechanics3D

Methods

shift :: R -> DParticleState -> ParticleState -> ParticleState

class HasTime s where

Methods

timeOf :: s -> Time

Instances

HasTime ParticleFieldState

Defined in LPFPCore.Lorentz

Methods

timeOf :: ParticleFieldState -> Time

HasTime ParticleState

Defined in LPFPCore.Mechanics3D

Methods

timeOf :: ParticleState -> Time

HasTime MultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

timeOf :: MultiParticleState -> Time

defaultParticleState :: ParticleState

A default particle state.

newtonSecondPS

Arguments

:: [OneBodyForce]

-> ParticleState

-> DParticleState

a differential equation

Given a list of forces, return a differential equation based on Newton's second law.

relativityPS :: [OneBodyForce] -> ParticleState -> DParticleState

Given a list of forces, return a differential equation based on the theory of special relativity.

eulerCromerPS :: TimeStep -> NumericalMethod ParticleState DParticleState

Euler-Cromer method for the ParticleState data type.

statesPS

Arguments

:: NumericalMethod ParticleState DParticleState

numerical method

-> [OneBodyForce]

list of force funcs

-> ParticleState

-> [ParticleState]

evolver

Given a numerical method, a list of one-body forces, and an initial state, return a list of states describing how the particle evolves in time.

updatePS :: NumericalMethod ParticleState DParticleState -> [OneBodyForce] -> ParticleState -> ParticleState

Given a numerical method and a list of one-body forces, return a state-update function.

One-body forces

type OneBodyForce = ParticleState -> Vec

Data type for a one-body force.

earthSurfaceGravity :: OneBodyForce

The force of gravity near Earth's surface. The z direction is toward the sky. Assumes SI units.

sunGravity :: OneBodyForce

The force of the Sun's gravity on an object. The origin is at center of the Sun. Assumes SI units.

airResistance

Arguments

:: R

drag coefficient

-> R

air density

-> R

cross-sectional area of object

-> OneBodyForce

The force of air resistance on an object.

windForce

Arguments

:: Vec

wind velocity

-> R

drag coefficient

-> R

air density

-> R

cross-sectional area of object

-> OneBodyForce

The force of wind on an object.

uniformLorentzForce

Arguments

:: Vec

E

-> Vec

B

-> OneBodyForce

The force of uniform electric and magnetic fields on an object.

fixedLinearSpring :: R -> R -> Vec -> OneBodyForce

Force provided by a spring that is fixed at one end.

Interacting particles

data Force

Constructors

ExternalForce Int OneBodyForce

InternalForce Int Int TwoBodyForce

data MultiParticleState

Constructors

MPS

Fields

• particleStates :: [ParticleState]

Instances

HasTime MultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

timeOf :: MultiParticleState -> Time

Show MultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

showsPrec :: Int -> MultiParticleState -> ShowS

show :: MultiParticleState -> String

showList :: [MultiParticleState] -> ShowS

Diff MultiParticleState DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

shift :: R -> DMultiParticleState -> MultiParticleState -> MultiParticleState

data DMultiParticleState

Constructors

DMPS [DParticleState]

Instances

RealVectorSpace DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

(+++) :: DMultiParticleState -> DMultiParticleState -> DMultiParticleState

scale :: R -> DMultiParticleState -> DMultiParticleState

Show DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

showsPrec :: Int -> DMultiParticleState -> ShowS

show :: DMultiParticleState -> String

showList :: [DMultiParticleState] -> ShowS

Diff MultiParticleState DMultiParticleState

Defined in LPFPCore.MultipleObjects

Methods

shift :: R -> DMultiParticleState -> MultiParticleState -> MultiParticleState

Two-body forces

type TwoBodyForce = ParticleState -> ParticleState -> ForceVector

universalGravity :: TwoBodyForce

linearSpring :: R -> R -> TwoBodyForce

centralForce :: (R -> R) -> TwoBodyForce

billiardForce :: R -> R -> TwoBodyForce

newtonSecondMPS :: [Force] -> MultiParticleState -> DMultiParticleState

eulerCromerMPS :: TimeStep -> NumericalMethod MultiParticleState DMultiParticleState

updateMPS :: NumericalMethod MultiParticleState DMultiParticleState -> [Force] -> MultiParticleState -> MultiParticleState

statesMPS :: NumericalMethod MultiParticleState DMultiParticleState -> [Force] -> MultiParticleState -> [MultiParticleState]

data Justification

Constructors

LJ

RJ

Instances

Show Justification

Defined in LPFPCore.MOExamples

Methods

showsPrec :: Int -> Justification -> ShowS

show :: Justification -> String

showList :: [Justification] -> ShowS

data Table a

Constructors

Table Justification [[a]]

Instances

Show a => Show (Table a)

Defined in LPFPCore.MOExamples

Methods

showsPrec :: Int -> Table a -> ShowS

show :: Table a -> String

showList :: [Table a] -> ShowS

kineticEnergy :: ParticleState -> R

systemKE :: MultiParticleState -> R

momentum :: ParticleState -> Vec

systemP :: MultiParticleState -> Vec

linearSpringPE :: R -> R -> ParticleState -> ParticleState -> R

earthSurfaceGravityPE :: ParticleState -> R

tenths :: R -> Float

sigFigs :: Int -> R -> Float

Electricity

elementaryCharge :: Charge

coulombForce :: TwoBodyForce

Coordinate Systems

data Position

Instances

Show Position

Defined in LPFPCore.CoordinateSystems

Methods

showsPrec :: Int -> Position -> ShowS

show :: Position -> String

showList :: [Position] -> ShowS

type Displacement = Vec

type ScalarField = Position -> R

type VectorField = Position -> Vec

type CoordinateSystem = (R, R, R) -> Position

cartesian :: CoordinateSystem

cylindrical :: CoordinateSystem

spherical :: CoordinateSystem

cart :: R -> R -> R -> Position

cyl :: R -> R -> R -> Position

sph :: R -> R -> R -> Position

cartesianCoordinates :: Position -> (R, R, R)

cylindricalCoordinates :: Position -> (R, R, R)

sphericalCoordinates :: Position -> (R, R, R)

displacement :: Position -> Position -> Displacement

shiftPosition :: Displacement -> Position -> Position

rHat :: VectorField

thetaHat :: VectorField

phiHat :: VectorField

sHat :: VectorField

xHat :: VectorField

yHat :: VectorField

zHat :: VectorField

origin :: Position

xSF :: ScalarField

ySF :: ScalarField

rSF :: ScalarField

rVF :: VectorField

fst3 :: (a, b, c) -> a

snd3 :: (a, b, c) -> b

thd3 :: (a, b, c) -> c

addScalarFields :: [ScalarField] -> ScalarField

addVectorFields :: [VectorField] -> VectorField

sfTable :: ((R, R) -> Position) -> [R] -> [R] -> ScalarField -> Table Int

Geometry

data Curve

Constructors

Curve

Fields

• curveFunc :: R -> Position
• startingCurveParam :: R
• endingCurveParam :: R

unitCircle :: Curve

straightLine :: Position -> Position -> Curve

data Surface

Constructors

Surface

Fields

• surfaceFunc :: (R, R) -> Position
• lowerLimit :: R
• upperLimit :: R
• lowerCurve :: R -> R
• upperCurve :: R -> R

unitSphere :: Surface

centeredSphere :: R -> Surface

sphere :: R -> Position -> Surface

northernHemisphere :: Surface

disk :: R -> Surface

shiftSurface :: Vec -> Surface -> Surface

data Volume

Constructors

Volume

Fields

• volumeFunc :: (R, R, R) -> Position
• loLimit :: R
• upLimit :: R
• loCurve :: R -> R
• upCurve :: R -> R
• loSurf :: R -> R -> R
• upSurf :: R -> R -> R

unitBall :: Volume

centeredBall :: R -> Volume

northernHalfBall :: Volume

centeredCylinder :: R -> R -> Volume

Electromagnetic Theory

Charge

type Charge = R

data ChargeDistribution

Constructors

PointCharge Charge Position

LineCharge ScalarField Curve

SurfaceCharge ScalarField Surface

VolumeCharge ScalarField Volume

MultipleCharges [ChargeDistribution]

totalCharge :: ChargeDistribution -> Charge

electricDipoleMoment :: ChargeDistribution -> Vec

Electric Field

epsilon0 :: R

cSI :: R

mu0 :: R

eField :: ChargeDistribution -> VectorField

type ScalarLineIntegral = ScalarField -> Curve -> R

type ScalarSurfaceIntegral = ScalarField -> Surface -> R

type ScalarVolumeIntegral = ScalarField -> Volume -> R

type VectorLineIntegral = VectorField -> Curve -> Vec

type VectorSurfaceIntegral = VectorField -> Surface -> Vec

type VectorVolumeIntegral = VectorField -> Volume -> Vec

type CurveApprox = Curve -> [(Position, Vec)]

type SurfaceApprox = Surface -> [(Position, Vec)]

type VolumeApprox = Volume -> [(Position, R)]

scalarLineIntegral :: CurveApprox -> ScalarField -> Curve -> R

scalarSurfaceIntegral :: SurfaceApprox -> ScalarField -> Surface -> R

scalarVolumeIntegral :: VolumeApprox -> ScalarField -> Volume -> R

vectorLineIntegral :: CurveApprox -> VectorField -> Curve -> Vec

vectorSurfaceIntegral :: SurfaceApprox -> VectorField -> Surface -> Vec

vectorVolumeIntegral :: VolumeApprox -> VectorField -> Volume -> Vec

dottedLineIntegral :: CurveApprox -> VectorField -> Curve -> R

dottedSurfaceIntegral :: SurfaceApprox -> VectorField -> Surface -> R

curveSample :: Int -> Curve -> [(Position, Vec)]

surfaceSample :: Int -> Surface -> [(Position, Vec)]

volumeSample :: Int -> Volume -> [(Position, R)]

type Field a = Position -> a

Current

type Current = R

data CurrentDistribution

Constructors

LineCurrent Current Curve

SurfaceCurrent VectorField Surface

VolumeCurrent VectorField Volume

MultipleCurrents [CurrentDistribution]

crossedLineIntegral :: CurveApprox -> VectorField -> Curve -> Vec

totalCurrent :: VectorField -> Surface -> Current

magneticDipoleMoment :: CurrentDistribution -> Vec

Magnetic Field

bField :: CurrentDistribution -> VectorField

Lorentz Force Law

lorentzForce :: ParticleFieldState -> Vec

newtonSecondPFS :: ParticleFieldState -> DParticleFieldState

defaultPFS :: ParticleFieldState

Maxwell Equations

directionalDerivative :: Vec -> ScalarField -> ScalarField

curl :: R -> VectorField -> VectorField

type FieldState = (R, VectorField, VectorField)