r"""vegie

Module for vector algebra
(C) Alexander Heger, 2023

This module was wrtting to do some calcualtions for
quadrupole-quarupole interaction(s) of multi-star/planet systems.

Select features

     - transform flipsco/contravariance of valence slots (all by default)

     - transpose re-orders valnce slots.  The last two by default


Example:

from vegie import *
i,j,k,l,m,n = indices('i,j,k,l,m,n')
T = Variable('T','--++')
V = Variable('v','+')
U = Variable('u','+')
t = T(i,j,k,l)
u = U(-i)
v = V(+j)
s = Tensor('s')



f = ((3 * u(-i)*u(+j) - u(-k)*u(+k)*Delta(-i,+j)) * (3 * v(-i)*v(+j) - u(-l)*u(+l)*Delta(-i,+j)).T)

f // u
f // v


from vegie import *
i, j = indices('i, j')
P, Q = variables('$!P++, $!Q--')
r, s, t, v = variables('r+', 's+', 't+', 'v+')

### paper

# two permanent - A3
Phi = -3/2 * Q(i,j) * r(i) * r(j) / r**5
G = -Phi // r(i) // r(j) | S
U = -1/2 * G * P(i,j) | S
F = -U // r(i) | S
T = F @ r(-i) | S

# two tidal - A4.1
Qpq = (r(i)*r(j) - 1/3 * r**2 * Delta(+i,+j)) / r**5
Qqp = (r(i)*r(j) - 1/3 * r**2 * Delta(+i,+j)) / r**5
Phi = -3/2*Qpq*r(-r)*r(-j) / r**5 | S
G =  -Phi // r(i) // r(j) | S
U = -1/2 * G * Qpq | S
F = - U // r(i) | S
T = F @ r(-i) | S

# permanent in tidal - A4.2
Phi = -3/2 / r**6
G =  -Phi // r(i) // r(j) | S
U = -1/2 * G * P(i,j) | S
F = - U // r(i) | S
T = F @ r(-i) | S

# tidal in permament - A4.3
Phi = -3/2 * Q(i,j) * r(i) * r(j) / r**5
G = -Phi // r(i) // r(j) | S
Qqp = (r(i)*r(j)-1/3 * r**2 * Delta(+i,+j)) / r**5
U = -1/2 * G * Qqp | S
F = -U // r(i) | S
T = F @ r(-i) | S

# spin-spin - A5.1
Wp, Wq = variables('Wp+, Wq+')
Wq.set_latex('\Omega','\mathrm{q}')
Wp.set_latex('\Omega','\mathrm{p}')

Qp = (Wp(i) * Wp(j) - 1/3 * Wp**2 * Delta(+i, +j))
Phi = -3/2 * ((r(i)*Wq(-i))**2 - 1/3 * r**2 * Wq**2) / r**5 | S
G = -Phi // r(i) // r(j) | S
U = -1/2*G1 * Qp | S
F = -U // r(i) | S
T = F @ r(-i) | S

G | extract(3/2*Norm(r)**(-9)) | rename('k+') | jtex
G | extract(3/2*Norm(r)**(-9)) | jvec(doc=True)
U | extract(3/4/r**9) | jvec(doc=True)
F | extract(15/4/r**11) | jvec(doc=True)
T | extract(15/2/r**11) | jvec(doc=True)

# ALTERNATE (using repleace) - spin-spin - A5.1
Qp = (Wp(i) * Wp(j) - 1/3 * Wp**2 * Delta(+i, +j))
Qq = (Wq(-i) * Wq(-j) - 1/3 * Wq**2 * Delta(-i, -j))

Phi = -3/2 * Q(i,j) * r(i) * r(j) / r**5
G = -Phi // r(i) // r(j) | S
U = -1/2 * G * P(i,j) | S
F = -U // r(i) | S
T = F @ r(-i) | S

G1 = G | replace(Q, Qq) | S
U1 = U | replace(Q, Qq) | replace(P, Qp) | S
F1 = F | replace(Q, Qq) | replace(P, Qp) | S
T1 = T | replace(Q, Qq) | replace(P, Qp) | S


# spin (P) in tidal field (Q) - A5.2
Wp = variables('Wp+')
Wp.set_latex('\Omega','\mathrm{p}')

Qp = (Wp(+i) * Wp(+j) - 1/3 * Wp**2 * Delta(+i, +j))

Phi = -3/2 / r**6
G = -Phi // r(i) // r(j) | S
U = -1/2 * G * Qp | S
F = -U // r(i) | S
T = F @ r(-i) | S

G | extract(9/r**10) | rename('k+') | jtex
G | extract(9/r**10) | jvec(doc=True)
U | extract(12/r**10) | jvec(doc=True)
F | extract(24/r**12) | jvec(doc=True)
T | extract(72/r**10) | jvec(doc=True)



% =======================================================================

TODO - add simplifyer that extracts values from inside powers

TODO - automatically extract maximum negative powers of norms, common integer factors and fractions

TODO - *** automatic simplification ***

TODO - add more algebra for Variable objects (non-commutative), inclduing number-typed objects

     - double contraction should likely be of that kind

     - have special tensors to host these, similar to Norm

     - there could be a special FundamentalNorm (or Scalar) for them

     - possibly a separate Norm for non-fundamental scalars

     - Norm is just a fancy way to take square root of contraction, same as contraction power 1/2.

TODO - group by common scalar groups if they are long

     - add function to count complexity/length

TODO - group Norms to beginning and combine with values for LaTeX output of Dot

TODO - proper re-design of __str__ and __repr__

TODO - VariableDoubleContraction similar to VariableNorm

TODO - Apply assumptions such as two vectors being parallel or pendicular

TODO - add doube contaction for complex tensors for vector notation

TODO - expand even powers of compound (Sum) Norm / VariableNorm

TODO - expand positive integer powers of Power of scalrs containing Sum


"""

from .command import *
from .index import indices
from .variable import variables
from .norm import Norm
from .special import Delta, Epsilon