Copyright	(c) Scott N. Walck 2014
License	BSD3 (see LICENSE)
Maintainer	Scott N. Walck <walck@lvc.edu>
Stability	experimental
Safe Haskell	Trustworthy
Language	Haskell98

Physics.Learn.Mechanics

Contents

Simple one-particle state
One-particle state
Two-particle state
Many-particle state

Description

Newton's second law and all that

Synopsis

type TheTime = Double
type TimeStep = Double
type Velocity = Vec
type SimpleState = (TheTime, Position, Velocity)
type SimpleAccelerationFunction = SimpleState -> Vec
simpleStateDeriv :: SimpleAccelerationFunction -> DifferentialEquation SimpleState
simpleRungeKuttaStep :: SimpleAccelerationFunction -> TimeStep -> SimpleState -> SimpleState
data St = St {
- position :: Position
- velocity :: Velocity
}
data DSt = DSt Vec Vec
type OneParticleSystemState = (TheTime, St)
type OneParticleAccelerationFunction = OneParticleSystemState -> Vec
oneParticleStateDeriv :: OneParticleAccelerationFunction -> DifferentialEquation OneParticleSystemState
oneParticleRungeKuttaStep :: OneParticleAccelerationFunction -> TimeStep -> OneParticleSystemState -> OneParticleSystemState
oneParticleRungeKuttaSolution :: OneParticleAccelerationFunction -> TimeStep -> OneParticleSystemState -> [OneParticleSystemState]
type TwoParticleSystemState = (TheTime, St, St)
type TwoParticleAccelerationFunction = TwoParticleSystemState -> (Vec, Vec)
twoParticleStateDeriv :: TwoParticleAccelerationFunction -> DifferentialEquation TwoParticleSystemState
twoParticleRungeKuttaStep :: TwoParticleAccelerationFunction -> TimeStep -> TwoParticleSystemState -> TwoParticleSystemState
type ManyParticleSystemState = (TheTime, [St])
type ManyParticleAccelerationFunction = ManyParticleSystemState -> [Vec]
manyParticleStateDeriv :: ManyParticleAccelerationFunction -> DifferentialEquation ManyParticleSystemState
manyParticleRungeKuttaStep :: ManyParticleAccelerationFunction -> TimeStep -> ManyParticleSystemState -> ManyParticleSystemState

Documentation

type TheTime = Double #

Time (in s).

type TimeStep = Double #

A time step (in s).

type Velocity = Vec #

Velocity of a particle (in m/s).

Simple one-particle state

type SimpleState = (TheTime, Position, Velocity) #

A simple one-particle state, to get started quickly with mechanics of one particle.

type SimpleAccelerationFunction = SimpleState -> Vec #

An acceleration function gives the particle's acceleration as a function of the particle's state. The specification of this function is what makes one single-particle mechanics problem different from another. In order to write this function, add all of the forces that act on the particle, and divide this net force by the particle's mass. (Newton's second law).

simpleStateDeriv #

Arguments

:: SimpleAccelerationFunction	acceleration function for the particle
-> DifferentialEquation SimpleState	differential equation

Time derivative of state for a single particle with a constant mass.

simpleRungeKuttaStep #

Arguments

:: SimpleAccelerationFunction	acceleration function for the particle
-> TimeStep	time step
-> SimpleState	initial state
-> SimpleState	state after one time step

Single Runge-Kutta step

One-particle state

data St #

The state of a single particle is given by the position of the particle and the velocity of the particle.

Constructors

St
Fields position :: Position velocity :: Velocity

Instances

Show St #
Instance details Defined in Physics.Learn.Mechanics Methods showsPrec :: Int -> St -> ShowS # show :: St -> String # showList :: [St] -> ShowS #
StateSpace St #
Instance details Defined in Physics.Learn.Mechanics Associated Types type Diff St :: * # Methods (.-.) :: St -> St -> Diff St # (.+^) :: St -> Diff St -> St #
type Diff St #
Instance details Defined in Physics.Learn.Mechanics type Diff St = DSt

data DSt #

The associated vector space for the state of a single particle.

Constructors

DSt Vec Vec

Instances

Show DSt #
Instance details Defined in Physics.Learn.Mechanics Methods showsPrec :: Int -> DSt -> ShowS # show :: DSt -> String # showList :: [DSt] -> ShowS #
VectorSpace DSt #
Instance details Defined in Physics.Learn.Mechanics Associated Types type Scalar DSt :: * # Methods (*^) :: Scalar DSt -> DSt -> DSt #
AdditiveGroup DSt #
Instance details Defined in Physics.Learn.Mechanics Methods zeroV :: DSt # (^+^) :: DSt -> DSt -> DSt # negateV :: DSt -> DSt # (^-^) :: DSt -> DSt -> DSt #
type Scalar DSt #
Instance details Defined in Physics.Learn.Mechanics type Scalar DSt = Double

type OneParticleSystemState = (TheTime, St) #

The state of a system of one particle is given by the current time, the position of the particle, and the velocity of the particle. Including time in the state like this allows us to have time-dependent forces.

type OneParticleAccelerationFunction = OneParticleSystemState -> Vec #

An acceleration function gives the particle's acceleration as a function of the particle's state.

oneParticleStateDeriv #

Arguments

:: OneParticleAccelerationFunction	acceleration function for the particle
-> DifferentialEquation OneParticleSystemState	differential equation

Time derivative of state for a single particle with a constant mass.

oneParticleRungeKuttaStep #

Arguments

:: OneParticleAccelerationFunction	acceleration function for the particle
-> TimeStep	time step
-> OneParticleSystemState	initial state
-> OneParticleSystemState	state after one time step

Single Runge-Kutta step

oneParticleRungeKuttaSolution #

Arguments

:: OneParticleAccelerationFunction	acceleration function for the particle
-> TimeStep	time step
-> OneParticleSystemState	initial state
-> [OneParticleSystemState]	state after one time step

List of system states

Two-particle state

type TwoParticleSystemState = (TheTime, St, St) #

The state of a system of two particles is given by the current time, the position and velocity of particle 1, and the position and velocity of particle 2.

type TwoParticleAccelerationFunction = TwoParticleSystemState -> (Vec, Vec) #

An acceleration function gives a pair of accelerations (one for particle 1, one for particle 2) as a function of the system's state.

twoParticleStateDeriv #

Arguments

:: TwoParticleAccelerationFunction	acceleration function for two particles
-> DifferentialEquation TwoParticleSystemState	differential equation

Time derivative of state for two particles with constant mass.

twoParticleRungeKuttaStep #

Arguments

:: TwoParticleAccelerationFunction	acceleration function
-> TimeStep	time step
-> TwoParticleSystemState	initial state
-> TwoParticleSystemState	state after one time step

Single Runge-Kutta step for two-particle system

Many-particle state

type ManyParticleSystemState = (TheTime, [St]) #

The state of a system of many particles is given by the current time and a list of one-particle states.

type ManyParticleAccelerationFunction = ManyParticleSystemState -> [Vec] #

An acceleration function gives a list of accelerations (one for each particle) as a function of the system's state.

manyParticleStateDeriv #

Arguments

:: ManyParticleAccelerationFunction	acceleration function for many particles
-> DifferentialEquation ManyParticleSystemState	differential equation

Time derivative of state for many particles with constant mass.

manyParticleRungeKuttaStep #

Arguments

:: ManyParticleAccelerationFunction	acceleration function
-> TimeStep	time step
-> ManyParticleSystemState	initial state
-> ManyParticleSystemState	state after one time step

Single Runge-Kutta step for many-particle system