{-# OPTIONS -Wall #-}
{-# LANGUAGE MultiParamTypeClasses #-}

{- | 
Module      :  LPFP.MultipleObjects
Copyright   :  (c) Scott N. Walck 2023
License     :  BSD3 (see LICENSE)
Maintainer  :  Scott N. Walck <walck@lvc.edu>
Stability   :  stable

Code from chapter 19 of the book Learn Physics with Functional Programming
-}

module LPFP.MultipleObjects where

import LPFP.SimpleVec
    ( Vec, R, (^+^), (^-^), (*^), (^*), (^/), zeroV, magnitude )
import LPFP.Mechanics1D
    ( RealVectorSpace(..), Diff(..), NumericalMethod, Mass, TimeStep, euler )
import LPFP.Mechanics3D
    ( OneBodyForce, ParticleState(..), DParticleState(..), HasTime(..)
    , defaultParticleState, newtonSecondPS )

type TwoBodyForce
    =  ParticleState  -- force is produced BY particle with this state
    -> ParticleState  -- force acts ON particle with this state
    -> ForceVector

type ForceVector = Vec

oneFromTwo :: ParticleState  -- state of particle PRODUCING the force
           -> TwoBodyForce
           -> OneBodyForce
oneFromTwo stBy f = f stBy

gravityMagnitude :: Mass -> Mass -> R -> R
gravityMagnitude m1 m2 r = let gg = 6.67408e-11  -- N m^2 / kg^2
                           in gg * m1 * m2 / r**2

universalGravity :: TwoBodyForce
universalGravity st1 st2
    = let gg = 6.67408e-11  -- N m^2 / kg^2
          m1 = mass st1
          m2 = mass st2
          r1 = posVec st1
          r2 = posVec st2
          r21 = r2 ^-^ r1
      in (-gg) *^ m1 *^ m2 *^ r21 ^/ magnitude r21 ** 3

constantRepulsiveForceWrong :: ForceVector -> TwoBodyForce
constantRepulsiveForceWrong force = \_ _ -> force

constantRepulsiveForce :: R -> TwoBodyForce
constantRepulsiveForce force st1 st2
    = let r1 = posVec st1
          r2 = posVec st2
          r21 = r2 ^-^ r1
      in force *^ r21 ^/ magnitude r21

linearSpring :: R  -- spring constant
             -> R  -- equilibrium length
             -> TwoBodyForce
linearSpring k re st1 st2
    = let r1 = posVec st1
          r2 = posVec st2
          r21 = r2 ^-^ r1
          r21mag = magnitude r21
      in (-k) *^ (r21mag - re) *^ r21 ^/ r21mag

-- | Force provided by a spring that is fixed at one end.
fixedLinearSpring :: R -> R -> Vec -> OneBodyForce
fixedLinearSpring k re r1
    = oneFromTwo (defaultParticleState { posVec = r1 }) (linearSpring k re)

centralForce :: (R -> R) -> TwoBodyForce
centralForce f st1 st2
    = let r1 = posVec st1
          r2 = posVec st2
          r21 = r2 ^-^ r1
          r21mag = magnitude r21
      in f r21mag *^ r21 ^/ r21mag

linearSpringCentral :: R  -- spring constant
                    -> R  -- equilibrium length
                    -> TwoBodyForce
linearSpringCentral k re = centralForce (\r -> -k * (r - re))

billiardForce :: R  -- spring constant
              -> R  -- threshold center separation
              -> TwoBodyForce
billiardForce k re
    = centralForce $ \r -> if r >= re
                           then 0
                           else (-k * (r - re))

data Force = ExternalForce Int OneBodyForce
           | InternalForce Int Int TwoBodyForce

data MultiParticleState
    = MPS { particleStates :: [ParticleState] } deriving Show

instance HasTime MultiParticleState where
    timeOf (MPS sts) = time (sts !! 0)

data DMultiParticleState = DMPS [DParticleState] deriving Show

newtonSecondMPS :: [Force]
                -> MultiParticleState -> DMultiParticleState  -- a diff eqn

newtonSecondMPS fs mpst@(MPS sts)
    = let deriv (n,st) = newtonSecondPS (forcesOn n mpst fs) st
      in DMPS $ map deriv (zip [0..] sts)

forcesOn :: Int -> MultiParticleState -> [Force] -> [OneBodyForce]
forcesOn n mpst = map (forceOn n mpst)

forceOn :: Int -> MultiParticleState -> Force -> OneBodyForce
forceOn n _         (ExternalForce n0 fOneBody)
    | n == n0    = fOneBody
    | otherwise  = const zeroV
forceOn n (MPS sts) (InternalForce n0 n1 fTwoBody)
    | n == n0    = oneFromTwo (sts !! n1) fTwoBody  -- n1 acts on n0
    | n == n1    = oneFromTwo (sts !! n0) fTwoBody  -- n0 acts on n1
    | otherwise  = const zeroV

instance RealVectorSpace DMultiParticleState where
    DMPS dsts1 +++ DMPS dsts2 = DMPS $ zipWith (+++) dsts1 dsts2
    scale w (DMPS dsts) = DMPS $ map (scale w) dsts

instance Diff MultiParticleState DMultiParticleState where
    shift dt (DMPS dsts) (MPS sts) = MPS $ zipWith (shift dt) dsts sts

eulerCromerMPS :: TimeStep        -- dt for stepping
               -> NumericalMethod MultiParticleState DMultiParticleState
eulerCromerMPS dt deriv mpst0
    = let mpst1 = euler dt deriv mpst0
          sts0 = particleStates mpst0
          sts1 = particleStates mpst1
          -- now update positions
          in MPS $ [ st1 { posVec = posVec st0 ^+^ velocity st1 ^* dt }
                         | (st0,st1) <- zip sts0 sts1 ]

updateMPS :: NumericalMethod MultiParticleState DMultiParticleState
          -> [Force]
          -> MultiParticleState -> MultiParticleState
updateMPS method = method . newtonSecondMPS

statesMPS :: NumericalMethod MultiParticleState DMultiParticleState
          -> [Force]
          -> MultiParticleState -> [MultiParticleState]
statesMPS method = iterate . method . newtonSecondMPS

speed :: ParticleState -> R
speed st = undefined st

universalGravity' :: TwoBodyForce
universalGravity' (ParticleState m1 _ _ r1 _) (ParticleState m2 _ _ r2 _)
    = undefined m1 r1 m2 r2

universalGravityCentral :: TwoBodyForce
universalGravityCentral = undefined

lennardJones :: R  -- dissociation energy
             -> R  -- equilibrium length
             -> TwoBodyForce
lennardJones de re = centralForce $ \r -> undefined de re r

systemKE :: MultiParticleState -> R
systemKE mpst = undefined mpst

forcesOn' :: Int -> MultiParticleState -> [Force] -> [OneBodyForce]
forcesOn' n mpst fs = externalForcesOn n fs ++ internalForcesOn n mpst fs

externalForcesOn :: Int -> [Force] -> [OneBodyForce]
externalForcesOn n fs = undefined n fs

internalForcesOn :: Int -> MultiParticleState -> [Force] -> [OneBodyForce]
internalForcesOn n (MPS sts) fs
    = [oneFromTwo (sts !! n1) f | InternalForce n0 n1 f <- fs, n == n0] ++
      [oneFromTwo (sts !! n0) f | InternalForce n0 n1 f <- fs, n == n1]