def normalize_metric(self):
if self.de is None:
return
renorm = []
# Generate mapping for renormalizing reciprocal basis vectors
for ib in self.n_range: # e^{ib} --> e^{ib}/|e_{ib}|
renorm.append((self.r_symbols[ib], self.r_symbols[ib] / self.e_norm[ib]))
# Normalize derivatives of basis vectors
for x_i in self.n_range:
for jb in self.n_range:
self.de[x_i][jb] = Simp.apply((((self.de[x_i][jb].subs(renorm)
- diff(self.e_norm[jb], self.coords[x_i]) *
self.basis[jb]) / self.e_norm[jb])))
if self.debug:
for x_i in self.n_range:
for jb in self.n_range:
print '\partial_{' + str(self.coords[x_i]) + '}\hat{e}_{' + str(self.coords[jb]) + '} =', self.de[x_i][jb]
# Normalize metric tensor
for ib in self.n_range:
for jb in self.n_range:
self.g[ib, jb] = Simp.apply(self.g[ib, jb] / (self.e_norm[ib] * self.e_norm[jb]))
if self.debug:
printer.oprint('e^{i}->e^{i}/|e_{i}|', renorm)
printer.oprint('renorm(g)', self.g)
return
def normalize_metric(self):
if self.de is None:
return
renorm = []
# Generate mapping for renormalizing reciprocal basis vectors
for ib in self.n_range: # e^{ib} --> e^{ib}/|e_{ib}|
renorm.append((self.r_symbols[ib], self.r_symbols[ib] / self.e_norm[ib]))
# Normalize derivatives of basis vectors
for x_i in self.n_range:
for jb in self.n_range:
self.de[x_i][jb] = Simp.apply((((self.de[x_i][jb].subs(renorm)
- diff(self.e_norm[jb], self.coords[x_i]) *
self.basis[jb]) / self.e_norm[jb])))
if self.debug:
for x_i in self.n_range:
for jb in self.n_range:
print '\partial_{' + str(self.coords[x_i]) + '}\hat{e}_{' + str(self.coords[jb]) + '} =', self.de[x_i][jb]
# Normalize metric tensor
for ib in self.n_range:
for jb in self.n_range:
self.g[ib, jb] = Simp.apply(self.g[ib, jb] / (self.e_norm[ib] * self.e_norm[jb]))
if self.debug:
printer.oprint('e^{i}->e^{i}/|e_{i}|', renorm)
printer.oprint('renorm(g)', self.g)
return
def derivatives_of_basis(
self): # Derivatives of basis vectors from Christ-Awful symbols
dg = [] # dg[i][j][k] = \partial_{x_{k}}g_{ij}
for i in self.n_range:
dg_row = []
for j in self.n_range:
dg_row.append(
[diff(self.g[i, j], coord) for coord in self.coords])
dg.append(dg_row)
# See if metric is flat
self.connect_flg = False
for i in self.n_range:
for j in self.n_range:
for k in self.n_range:
if dg[i][j][k] != 0:
self.connect_flg = True
break
if not self.connect_flg:
self.de = None
return
n_range = range(len(self.basis))
de = [] # de[i][j] = \partial_{x_{i}}e^{x_{j}}
# Christoffel symbols of the first kind, \Gamma_{ijk}
for i in n_range:
de_row = []
for j in n_range:
Gamma = []
for k in n_range:
gamma = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
Gamma.append(Simp.apply(gamma))
de_row.append(
sum([
gamma * base
for (gamma, base) in zip(Gamma, self.r_symbols)
]))
de.append(de_row)
if self.debug:
printer.oprint('D_{i}e^{j}', de)
self.de = de
return
def derivatives_of_basis(self): # Derivatives of basis vectors from Christ-Awful symbols
dg = [] # dg[i][j][k] = \partial_{x_{k}}g_{ij}
for i in self.n_range:
dg_row = []
for j in self.n_range:
dg_row.append([diff(self.g[i, j], coord) for coord in self.coords])
dg.append(dg_row)
# See if metric is flat
self.connect_flg = False
for i in self.n_range:
for j in self.n_range:
for k in self.n_range:
if dg[i][j][k] != 0:
self.connect_flg = True
break
if not self.connect_flg:
self.de = None
return
n_range = range(len(self.basis))
de = [] # de[i][j] = \partial_{x_{i}}e^{x_{j}}
# Christoffel symbols of the first kind, \Gamma_{ijk}
for i in n_range:
de_row = []
for j in n_range:
Gamma = []
for k in n_range:
gamma = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
Gamma.append(Simp.apply(gamma))
de_row.append(sum([gamma * base for (gamma, base) in zip(Gamma, self.r_symbols)]))
de.append(de_row)
if self.debug:
printer.oprint('D_{i}e^{j}', de)
self.de = de
return
def main():
enhance_print()
coords = symbols('x y z')
(ex,ey,ez,grad) = MV.setup('ex ey ez',metric='[1,1,1]',coords=coords)
mfvar = (u,v) = symbols('u v')
eu = ex+ey
ev = ex-ey
(eu_r,ev_r) = ReciprocalFrame([eu,ev])
oprint('Frame',(eu,ev),'Reciprocal Frame',(eu_r,ev_r))
print 'eu.eu_r =',eu|eu_r
print 'eu.ev_r =',eu|ev_r
print 'ev.eu_r =',ev|eu_r
print 'ev.ev_r =',ev|ev_r
eu = ex+ey+ez
ev = ex-ey
(eu_r,ev_r) = ReciprocalFrame([eu,ev])
oprint('Frame',(eu,ev),'Reciprocal Frame',(eu_r,ev_r))
print 'eu.eu_r =',eu|eu_r
print 'eu.ev_r =',eu|ev_r
print 'ev.eu_r =',ev|eu_r
print 'ev.ev_r =',ev|ev_r
print 'eu =',eu
print 'ev =',ev
def_prec(locals())
print GAeval('eu^ev|ex',True)
print GAeval('eu^ev|ex*eu',True)
return
def main():
enhance_print()
coords = symbols('x y z')
(ex, ey, ez, grad) = MV.setup('ex ey ez', metric='[1,1,1]', coords=coords)
mfvar = (u, v) = symbols('u v')
eu = ex + ey
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
eu = ex + ey + ez
ev = ex - ey
(eu_r, ev_r) = ReciprocalFrame([eu, ev])
oprint('Frame', (eu, ev), 'Reciprocal Frame', (eu_r, ev_r))
print('eu.eu_r =', eu | eu_r)
print('eu.ev_r =', eu | ev_r)
print('ev.eu_r =', ev | eu_r)
print('ev.ev_r =', ev | ev_r)
print('eu =', eu)
print('ev =', ev)
def_prec(locals())
print(GAeval('eu^ev|ex', True))
print(GAeval('eu^ev|ex*eu', True))
return
def main():
Eprint()
coords = symbols('x y z')
(o3d,ex,ey,ez) = Ga.build('ex ey ez',g=[1,1,1],coords=coords)
mfvar = (u,v) = symbols('u v')
eu = ex+ey
ev = ex-ey
(eu_r,ev_r) = o3d.ReciprocalFrame([eu,ev])
oprint('Frame',(eu,ev),'Reciprocal Frame',(eu_r,ev_r))
print 'eu.eu_r =',eu|eu_r
print 'eu.ev_r =',eu|ev_r
print 'ev.eu_r =',ev|eu_r
print 'ev.ev_r =',ev|ev_r
eu = ex+ey+ez
ev = ex-ey
(eu_r,ev_r) = o3d.ReciprocalFrame([eu,ev])
oprint('Frame',(eu,ev),'Reciprocal Frame',(eu_r,ev_r))
print 'eu.eu_r =',eu|eu_r
print 'eu.ev_r =',eu|ev_r
print 'ev.eu_r =',ev|eu_r
print 'ev.ev_r =',ev|ev_r
print 'eu =',eu
print 'ev =',ev
def_prec(locals())
print GAeval('eu^ev|ex',True)
print GAeval('eu^ev|ex*eu',True)
return
def __init__(self, basis, **kwargs):
kwargs = test_init_slots(Metric.init_slots, **kwargs)
self.name = 'GA' + str(Metric.count)
Metric.count += 1
if not isinstance(basis, basestring):
raise TypeError('"' + str(basis) + '" must be string')
X = kwargs['X'] # Vector manifold
g = kwargs['g'] # Explicit metric or base metric for vector manifold
debug = kwargs['debug']
coords = kwargs['coords'] # Manifold coordinates (sympy symbols)
norm = kwargs['norm'] # Normalize basis vectors
self.sig = kwargs['sig'] # Hint for metric signature
"""
String for symbolic metric determinant. If self.gsym = 'g'
then det(g) is sympy scalar function of coordinates with
name 'det(g)'. Useful for complex non-orthogonal coordinate
systems or for calculations with general metric.
"""
self.gsym = kwargs['gsym']
self.debug = debug
self.is_ortho = False # Is basis othogonal
self.coords = coords # Manifold coordinates
if self.coords is None:
self.connect_flg = False
else:
self.connect_flg = True # Connection needed for postion dependent metric
self.norm = norm # True to normalize basis vectors
self.detg = None # Determinant of g
self.g_adj = None # Adjugate of g
self.g_inv = None # Inverse of g
# Generate list of basis vectors and reciprocal basis vectors
# as non-commutative symbols
if ' ' in basis or ',' in basis or '*' in basis: # bases defined by substrings separated by spaces or commas
self.basis = symbols_list(basis)
self.r_symbols = symbols_list(basis, sub=False)
else:
if coords is not None: # basis defined by root string with symbol list as indices
self.basis = symbols_list(basis, coords)
self.r_symbols = symbols_list(basis, coords, sub=False)
self.coords = coords
if self.debug:
printer.oprint('x^{i}', self.coords)
else:
raise ValueError('for basis "' + basis + '" coords must be entered')
if self.debug:
printer.oprint('e_{i}', self.basis, 'e^{i}', self.r_symbols)
self.n = len(self.basis)
self.n_range = range(self.n)
# Generate metric as list of lists of symbols, rationals, or functions of coordinates
if g is None:
if X is None: # default metric from dot product of basis as symbols
self.g = self.metric_symbols_list()
else: # Vector manifold
if coords is None:
raise ValueError('For metric derived from vector field ' +
' coordinates must be defined.')
else: # Vector manifold defined by vector field
dX = []
for coord in coords: # Get basis vectors by differentiating vector field
dX.append([diff(x, coord) for x in X])
g_tmp = []
for dx1 in dX:
g_row = []
for dx2 in dX:
dx1_dot_dx2 = trigsimp(Metric.dot_orthogonal(dx1, dx2, g))
g_row.append(dx1_dot_dx2)
g_tmp.append(g_row)
self.g = Matrix(g_tmp)
if self.debug:
printer.oprint('X_{i}', X, 'D_{i}X_{j}', dX)
else: # metric is symbolic or list of lists of functions of coordinates
if isinstance(g, basestring): # metric elements are symbols or constants
if g == 'g': # general symbolic metric tensor (g_ij functions of position)
g_lst = []
g_inv_lst = []
for coord1 in self.coords:
i1 = str(coord2)
tmp = []
tmp_inv = []
for coord2 in self.coords:
i2 = str(coord2)
tmp.append(Function('g_'+i1+'_'+i2)(*self.coords))
tmp_inv.append(Function('g__'+i1+'__'+i2)(*self.coords))
g_lst.append(tmp)
g_inv_lst.append(tmp_inv)
self.g = Matrix(g_lst)
self.g_inv = Matrix(g_inv_lst)
else: # specific symbolic metric tensor (g_ij are symbolic or numerical constants)
self.g = self.metric_symbols_list(g) # construct symbolic metric from string and basis
else: # metric is given as list of function or list of lists of function or matrix of functions
if isinstance(g, Matrix):
self.g = g
else:
if isinstance(g[0], list):
self.g = Matrix(g)
else:
m = eye(len(g))
for i in range(len(g)):
m[i, i] = g[i]
self.g = m
self.g_raw = copy.deepcopy(self.g) # save original metric tensor for use with submanifolds
if self.debug:
printer.oprint('g', self.g)
# Determine if metric is orthogonal
self.is_ortho = True
for i in self.n_range:
for j in self.n_range:
if i < j:
if self.g[i, j] != 0:
self.is_ortho = False
break
self.g_is_numeric = True
for i in self.n_range:
for j in self.n_range:
if i < j:
if not self.g[i, j].is_number:
self.g_is_numeric = False
break
if self.coords is not None:
self.derivatives_of_basis() # calculate derivatives of basis
if self.norm: # normalize basis, metric, and derivatives of normalized basis
self.e_norm = []
for i in self.n_range:
self.e_norm.append(square_root_of_expr(self.g[i, i]))
if debug:
printer.oprint('|e_{i}|', self.e_norm)
else:
self.e_norm = None
if self.norm:
if self.is_ortho:
self.normalize_metric()
else:
raise ValueError('!!!!Basis normalization only implemented for orthogonal basis!!!!')
if not self.g_is_numeric:
self.signature()
# Sign of square of pseudo scalar
self.e_sq_sgn = '+'
if ((self.n*(self.n-1))/2+self.sig[1])%2 == 1:
self.e_sq_sgn = '-'
if self.debug:
print 'signature =', self.sig
def Christoffel_symbols(self,mode=1):
"""
mode = 1 Christoffel symbols of the first kind
mode = 2 Christoffel symbols of the second kind
"""
if mode == 1:
dg = [] # dg[i][j][k] = \partial_{x_{k}}g_{ij}
for i in self.n_range:
dg_row = []
for j in self.n_range:
dg_row.append([diff(self.g[i, j], coord) for coord in self.coords])
dg.append(dg_row)
# See if connection is zero
self.connect_flg = False
for i in self.n_range:
for j in self.n_range:
for k in self.n_range:
if dg[i][j][k] != 0:
self.connect_flg = True
break
if not self.connect_flg:
self.de = None
return
n_range = range(len(self.basis))
dG = [] # dG[i][j][k] = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
# Christoffel symbols of the first kind, \Gamma_{ijk}
# \partial_{x^{i}}e_{j} = \Gamma_{ijk}e^{k}
for i in n_range:
dG_j = []
for j in n_range:
dG_k = []
for k in n_range:
gamma = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
dG_k.append(Simp.apply(gamma))
dG_j.append(dG_k)
dG.append(dG_j)
if self.debug:
printer.oprint('Gamma_{ijk}', dG)
return dG
elif mode == 2:
Gamma1 = self.Christoffel_symbols(mode=1)
if self.connect_flg: # Connection is not zero
self.inverse_metric()
# Christoffel symbols of the second kind, \Gamma_{ij}^{k} = \Gamma_{ijl}g^{lk}
# \partial_{x^{i}}e_{j} = \Gamma_{ij}^{k}e_{k}
Gamma = []
for Gi in Gamma1:
Gamma_i = []
for Gij in Gi:
Gamma_ij = []
Gamma_ijk = S(0)
l = 0
for Gijl in Gij:
Gamma_ijk += Gijl * g(l,)
else:
return
else:
raise ValueError('In Christoffle_symobols mode = ' +str(mode) +' is not allowed\n')
def __init__(self, basis, **kwargs):
kwargs = test_init_slots(Metric.init_slots, **kwargs)
self.name = 'GA' + str(Metric.count)
Metric.count += 1
if not isinstance(basis, basestring):
raise TypeError('"' + str(basis) + '" must be string')
X = kwargs['X'] # Vector manifold
g = kwargs['g'] # Explicit metric or base metric for vector manifold
debug = kwargs['debug']
coords = kwargs['coords'] # Manifold coordinates (sympy symbols)
norm = kwargs['norm'] # Normalize basis vectors
self.sig = kwargs['sig'] # Hint for metric signature
"""
String for symbolic metric determinant. If self.gsym = 'g'
then det(g) is sympy scalar function of coordinates with
name 'det(g)'. Useful for complex non-orthogonal coordinate
systems or for calculations with general metric.
"""
self.gsym = kwargs['gsym']
self.debug = debug
self.is_ortho = False # Is basis othogonal
self.coords = coords # Manifold coordinates
if self.coords is None:
self.connect_flg = False
else:
self.connect_flg = True # Connection needed for postion dependent metric
self.norm = norm # True to normalize basis vectors
self.detg = None # Determinant of g
self.g_adj = None # Adjugate of g
self.g_inv = None # Inverse of g
# Generate list of basis vectors and reciprocal basis vectors
# as non-commutative symbols
if ' ' in basis or ',' in basis or '*' in basis: # bases defined by substrings separated by spaces or commas
self.basis = symbols_list(basis)
self.r_symbols = symbols_list(basis, sub=False)
else:
if coords is not None: # basis defined by root string with symbol list as indices
self.basis = symbols_list(basis, coords)
self.r_symbols = symbols_list(basis, coords, sub=False)
self.coords = coords
if self.debug:
printer.oprint('x^{i}', self.coords)
else:
raise ValueError('for basis "' + basis + '" coords must be entered')
if self.debug:
printer.oprint('e_{i}', self.basis, 'e^{i}', self.r_symbols)
self.n = len(self.basis)
self.n_range = range(self.n)
# Generate metric as list of lists of symbols, rationals, or functions of coordinates
if g is None:
if X is None: # default metric from dot product of basis as symbols
self.g = self.metric_symbols_list()
else: # Vector manifold
if coords is None:
raise ValueError('For metric derived from vector field ' +
' coordinates must be defined.')
else: # Vector manifold defined by vector field
dX = []
for coord in coords: # Get basis vectors by differentiating vector field
dX.append([diff(x, coord) for x in X])
g_tmp = []
for dx1 in dX:
g_row = []
for dx2 in dX:
dx1_dot_dx2 = trigsimp(Metric.dot_orthogonal(dx1, dx2, g))
g_row.append(dx1_dot_dx2)
g_tmp.append(g_row)
self.g = Matrix(g_tmp)
if self.debug:
printer.oprint('X_{i}', X, 'D_{i}X_{j}', dX)
else: # metric is symbolic or list of lists of functions of coordinates
if isinstance(g, basestring): # metric elements are symbols or constants
if g == 'g': # general symbolic metric tensor (g_ij functions of position)
g_lst = []
g_inv_lst = []
for coord1 in self.coords:
i1 = str(coord2)
tmp = []
tmp_inv = []
for coord2 in self.coords:
i2 = str(coord2)
tmp.append(Function('g_'+i1+'_'+i2)(*self.coords))
tmp_inv.append(Function('g__'+i1+'__'+i2)(*self.coords))
g_lst.append(tmp)
g_inv_lst.append(tmp_inv)
self.g = Matrix(g_lst)
self.g_inv = Matrix(g_inv_lst)
else: # specific symbolic metric tensor (g_ij are symbolic or numerical constants)
self.g = self.metric_symbols_list(g) # construct symbolic metric from string and basis
else: # metric is given as list of function or list of lists of function or matrix of functions
if isinstance(g, Matrix):
self.g = g
else:
if isinstance(g[0], list):
self.g = Matrix(g)
else:
m = eye(len(g))
for i in range(len(g)):
m[i, i] = g[i]
self.g = m
self.g_raw = copy.deepcopy(self.g) # save original metric tensor for use with submanifolds
if self.debug:
printer.oprint('g', self.g)
# Determine if metric is orthogonal
self.is_ortho = True
for i in self.n_range:
for j in self.n_range:
if i < j:
if self.g[i, j] != 0:
self.is_ortho = False
break
if self.coords is not None:
self.derivatives_of_basis() # calculate derivatives of basis
if self.norm: # normalize basis, metric, and derivatives of normalized basis
self.e_norm = []
for i in self.n_range:
self.e_norm.append(square_root_of_expr(self.g[i, i]))
if debug:
printer.oprint('|e_{i}|', self.e_norm)
else:
self.e_norm = None
if self.norm:
if self.is_ortho:
self.normalize_metric()
else:
raise ValueError('!!!!Basis normalization only implemented for orthogonal basis!!!!')
self.signature()
# Sign of square of pseudo scalar
self.e_sq_sgn = '+'
if ((self.n*(self.n-1))/2+self.sig[1])%2 == 1:
self.e_sq_sgn = '-'
if self.debug:
print 'signature =', self.sig
def Christoffel_symbols(self,mode=1):
"""
mode = 1 Christoffel symbols of the first kind
mode = 2 Christoffel symbols of the second kind
"""
if mode == 1:
dg = [] # dg[i][j][k] = \partial_{x_{k}}g_{ij}
for i in self.n_range:
dg_row = []
for j in self.n_range:
dg_row.append([diff(self.g[i, j], coord) for coord in self.coords])
dg.append(dg_row)
# See if connection is zero
self.connect_flg = False
for i in self.n_range:
for j in self.n_range:
for k in self.n_range:
if dg[i][j][k] != 0:
self.connect_flg = True
break
if not self.connect_flg:
self.de = None
return
n_range = range(len(self.basis))
dG = [] # dG[i][j][k] = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
# Christoffel symbols of the first kind, \Gamma_{ijk}
# \partial_{x^{i}}e_{j} = \Gamma_{ijk}e^{k}
for i in n_range:
dG_j = []
for j in n_range:
dG_k = []
for k in n_range:
gamma = half * (dg[j][k][i] + dg[i][k][j] - dg[i][j][k])
dG_k.append(Simp.apply(gamma))
dG_j.append(dG_k)
dG.append(dG_j)
if self.debug:
printer.oprint('Gamma_{ijk}', dG)
return dG
elif mode == 2:
Gamma1 = self.Christoffel_symbols(mode=1)
if self.connect_flg: # Connection is not zero
self.inverse_metric()
# Christoffel symbols of the second kind, \Gamma_{ij}^{k} = \Gamma_{ijl}g^{lk}
# \partial_{x^{i}}e_{j} = \Gamma_{ij}^{k}e_{k}
Gamma = []
for Gi in Gamma1:
Gamma_i = []
for Gij in Gi:
Gamma_ij = []
Gamma_ijk = S(0)
l = 0
for Gijl in Gij:
Gamma_ijk += Gijl * g(l,)
else:
return
else:
raise ValueError('In Christoffle_symobols mode = ' +str(mode) +' is not allowed\n')