def derivatives_in_rectangular_coordinates(): Print_Function() X = (x, y, z) = symbols('x y z') o3d = Ga('e_x e_y e_z', g=[1, 1, 1], coords=X) (ex, ey, ez) = o3d.mv() grad = o3d.grad f = o3d.mv('f', 'scalar', f=True) A = o3d.mv('A', 'vector', f=True) B = o3d.mv('B', 'bivector', f=True) C = o3d.mv('C', 'mv', f=True) print 'f =', f print 'A =', A print 'B =', B print 'C =', C print 'grad*f =', grad * f print 'grad|A =', grad | A print 'grad*A =', grad * A print '-I*(grad^A) =', -o3d.I() * (grad ^ A) print 'grad*B =', grad * B print 'grad^B =', grad ^ B print 'grad|B =', grad | B print 'grad<A =', grad < A print 'grad>A =', grad > A print 'grad<B =', grad < B print 'grad>B =', grad > B print 'grad<C =', grad < C print 'grad>C =', grad > C return
def basic_multivector_operations_3D(): Print_Function() g3d = Ga('e*x|y|z') (ex,ey,ez) = g3d.mv() A = g3d.mv('A','mv') A.Fmt(1,'A') A.Fmt(2,'A') A.Fmt(3,'A') A.even().Fmt(1,'%A_{+}') A.odd().Fmt(1,'%A_{-}') X = g3d.mv('X','vector') Y = g3d.mv('Y','vector') print 'g_{ij} = ',g3d.g X.Fmt(1,'X') Y.Fmt(1,'Y') (X*Y).Fmt(2,'X*Y') (X^Y).Fmt(2,'X^Y') (X|Y).Fmt(2,'X|Y') return
def properties_of_geometric_objects(): Print_Function() global n, nbar g = '# # # 0 0,'+ \ '# # # 0 0,'+ \ '# # # 0 0,'+ \ '0 0 0 0 2,'+ \ '0 0 0 2 0' c3d = Ga('p1 p2 p3 n nbar',g=g) (p1,p2,p3,n,nbar) = c3d.mv() print 'g_{ij} =\n',c3d.g P1 = F(p1) P2 = F(p2) P3 = F(p3) print 'Extracting direction of line from L = P1^P2^n' L = P1^P2^n delta = (L|n)|nbar print '(L|n)|nbar =',delta print 'Extracting plane of circle from C = P1^P2^P3' C = P1^P2^P3 delta = ((C^n)|n)|nbar print '((C^n)|n)|nbar =',delta print '(p2-p1)^(p3-p1) =',(p2-p1)^(p3-p1)
def extracting_vectors_from_conformal_2_blade(): Print_Function() g = '0 -1 #,'+ \ '-1 0 #,'+ \ '# # #' e2b = Ga('P1 P2 a',g=g) (P1,P2,a) = e2b.mv() print 'g_{ij} =\n',e2b.g B = P1^P2 Bsq = B*B print 'B**2 =',Bsq ap = a-(a^B)*B print "a' = a-(a^B)*B =",ap Ap = ap+ap*B Am = ap-ap*B print "A+ = a'+a'*B =",Ap print "A- = a'-a'*B =",Am print '(A+)^2 =',Ap*Ap print '(A-)^2 =',Am*Am aB = a|B print 'a|B =',aB return
def test_reciprocal_frame(): """ Test of formula for general reciprocal frame of three vectors. Let three independent vectors be e1, e2, and e3. The reciprocal vectors E1, E2, and E3 obey the relations: e_i.E_j = delta_ij*(e1^e2^e3)**2 """ g = '1 # #,'+ \ '# 1 #,'+ \ '# # 1' g3dn = Ga('e1 e2 e3',g=g) (e1,e2,e3) = g3dn.mv() E = e1^e2^e3 Esq = (E*E).scalar() Esq_inv = 1 / Esq E1 = (e2^e3)*E E2 = (-1)*(e1^e3)*E E3 = (e1^e2)*E w = (E1|e2) w = w.expand() assert w.scalar() == 0 w = (E1|e3) w = w.expand() assert w.scalar() == 0 w = (E2|e1) w = w.expand() assert w.scalar() == 0 w = (E2|e3) w = w.expand() assert w.scalar() == 0 w = (E3|e1) w = w.expand() assert w.scalar() == 0 w = (E3|e2) w = w.expand() assert w.scalar() == 0 w = (E1|e1) w = (w.expand()).scalar() Esq = expand(Esq) assert simplify(w/Esq) == 1 w = (E2|e2) w = (w.expand()).scalar() assert simplify(w/Esq) == 1 w = (E3|e3) w = (w.expand()).scalar() assert simplify(w/Esq) == 1
def basic_multivector_operations_2D(): Print_Function() g2d = Ga('e*x|y') (ex,ey) = g2d.mv() print 'g_{ij} =',g2d.g X = g2d.mv('X','vector') A = g2d.mv('A','spinor') X.Fmt(1,'X') A.Fmt(1,'A') (X|A).Fmt(2,'X|A') (X<A).Fmt(2,'X<A') (A>X).Fmt(2,'A>X') return
def derivatives_in_spherical_coordinates(): Print_Function() X = (r,th,phi) = symbols('r theta phi') s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True) (er,eth,ephi) = s3d.mv() grad = s3d.grad f = s3d.mv('f','scalar',f=True) A = s3d.mv('A','vector',f=True) B = s3d.mv('B','bivector',f=True) print 'f =',f print 'A =',A print 'B =',B print 'grad*f =',grad*f print 'grad|A =',grad|A print '-I*(grad^A) =',(-s3d.i*(grad^A)).simplify() print 'grad^B =',grad^B
def rounding_numerical_components(): Print_Function() o3d = Ga('e_x e_y e_z',g=[1,1,1]) (ex,ey,ez) = o3d.mv() X = 1.2*ex+2.34*ey+0.555*ez Y = 0.333*ex+4*ey+5.3*ez print 'X =',X print 'Nga(X,2) =',Nga(X,2) print 'X*Y =',X*Y print 'Nga(X*Y,2) =',Nga(X*Y,2) return
def basic_multivector_operations_2D_orthogonal(): Print_Function() o2d = Ga('e*x|y',g=[1,1]) (ex,ey) = o2d.mv() print 'g_{ii} =',o2d.g X = o2d.mv('X','vector') A = o2d.mv('A','spinor') X.Fmt(1,'X') A.Fmt(1,'A') (X*A).Fmt(2,'X*A') (X|A).Fmt(2,'X|A') (X<A).Fmt(2,'X<A') (X>A).Fmt(2,'X>A') (A*X).Fmt(2,'A*X') (A|X).Fmt(2,'A|X') (A<X).Fmt(2,'A<X') (A>X).Fmt(2,'A>X') return
class LieAlgebra(object): def __init__(self,n): self.n = n e = '' ebar = '' for i in range(1,n+1): e += ' e_' + str(i) if GaLatexPrinter.latex_flg: ebar += r' \bar{e}_' + str(i) else: ebar += ' ebar_' + str(i) g = n * [1] + n * [-1] basis = e[1:] + ebar self.Ga = Ga(basis, g=g) self.basis = self.Ga.mv() self.e = self.basis[:n] self.ebar = self.basis[n:] self.w = [] self.wstar = [] for i in range(n): self.w.append(self.e[i] + self.ebar[i]) self.wstar.append(self.e[i] - self.ebar[i]) self.Nu_bais = self.w + self.wstar self.Eij = [] self.Fij = [] self.Ki = [] for i in range(n): self.Ki.append(self.e[i] * self.ebar[i]) print r'%F_{'+str(i)+'} =',self.Ki[-1] * self.Ki[-1].rev() for j in range(i): self.Eij.append(self.e[i] * self.e[j] - self.ebar[i] * self.ebar[j]) self.Fij.append(self.e[i] * self.ebar[j] - self.ebar[i] * self.e[j]) print r'%E_{'+str(i)+str(j)+'} =',self.Eij[-1] * self.Eij[-1].rev() print r'%F_{'+str(i)+str(j)+'} =',self.Fij[-1] * self.Fij[-1].rev() print 'K_{i} =',self.Ki print 'E_{ij} =',self.Eij print 'F_{ij} =',self.Fij E = self.Eij[0]/2 for i in range(2*n): print E E *= self.Eij[0]/2
def check_generalized_BAC_CAB_formulas(): Print_Function() g4d = Ga('a b c d') (a,b,c,d) = g4d.mv() print 'g_{ij} =',g4d.g print '\\bm{a|(b*c)} =',a|(b*c) print '\\bm{a|(b^c)} =',a|(b^c) print '\\bm{a|(b^c^d)} =',a|(b^c^d) print '\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a)) print '\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b) print '\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) print '\\bm{(a^b)|(c^d)} =',(a^b)|(c^d) print '\\bm{((a^b)|c)|d} =',((a^b)|c)|d print '\\bm{(a^b)\\times (c^d)} =',Com(a^b,c^d) return
def LieBasis(n): g = n * [1] + n * [-1] basis = '' for i in range(1,n+1): basis += 'e_' +str(i) + ' ' for i in range(1,n+1): basis += r'\bar{e}_' +str(i) + ' ' basis = basis[:-1] LieGA = Ga(basis,g=g) bases = LieGA.mv() e = bases[:n] ebar = bases[n:] print e print ebar print LieGA.g E = [] F = [] K = [] indexes = [] for i in range(n): K.append(e[i]*ebar[i]) for j in range(n): if i < j: indexes.append((i+1,j+1)) E.append(e[i]*e[j]-ebar[i]*ebar[j]) F.append(e[i]*ebar[j]-ebar[i]*e[j]) print indexes print 'E =',E print 'F =',F print 'K =',K,'\n' for k in range(len(indexes)): k_i = indexes[k][0] k_j = indexes[k][1] for l in range(len(indexes)): l_i = indexes[l][0] l_j = indexes[l][1] print 'E_'+str(k_i)+str(k_j)+' x F_'+str(l_i)+str(l_j)+' = '+str(Com(E[k],F[l])) return
def conformal_representations_of_circles_lines_spheres_and_planes(): global n,nbar Print_Function() g = '1 0 0 0 0,0 1 0 0 0,0 0 1 0 0,0 0 0 0 2,0 0 0 2 0' cnfml3d = Ga('e_1 e_2 e_3 n nbar',g=g) (e1,e2,e3,n,nbar) = cnfml3d.mv() print 'g_{ij} =\n',cnfml3d.g e = n+nbar #conformal representation of points A = make_vector(e1,ga=cnfml3d) # point a = (1,0,0) A = F(a) B = make_vector(e2,ga=cnfml3d) # point b = (0,1,0) B = F(b) C = make_vector(-e1,ga=cnfml3d) # point c = (-1,0,0) C = F(c) D = make_vector(e3,ga=cnfml3d) # point d = (0,0,1) D = F(d) X = make_vector('x',3,ga=cnfml3d) print 'F(a) =',A print 'F(b) =',B print 'F(c) =',C print 'F(d) =',D print 'F(x) =',X print 'a = e1, b = e2, c = -e1, and d = e3' print 'A = F(a) = 1/2*(a*a*n+2*a-nbar), etc.' print 'Circle through a, b, and c' print 'Circle: A^B^C^X = 0 =',(A^B^C^X) print 'Line through a and b' print 'Line : A^B^n^X = 0 =',(A^B^n^X) print 'Sphere through a, b, c, and d' print 'Sphere: A^B^C^D^X = 0 =',(((A^B)^C)^D)^X print 'Plane through a, b, and d' print 'Plane : A^B^n^D^X = 0 =',(A^B^n^D^X) L = (A^B^e)^X L.Fmt(3,'Hyperbolic Circle: (A^B^e)^X = 0 =') return
def check_generalized_BAC_CAB_formulas(): Print_Function() g5d = Ga('a b c d e') (a, b, c, d, e) = g5d.mv() print 'g_{ij} =\n', g5d.g print 'a|(b*c) =', a | (b * c) print 'a|(b^c) =', a | (b ^ c) print 'a|(b^c^d) =', a | (b ^ c ^ d) print 'a|(b^c)+c|(a^b)+b|(c^a) =', (a | ( b ^ c)) + (c | (a ^ b)) + (b | (c ^ a)) print 'a*(b^c)-b*(a^c)+c*(a^b) =',a*(b^c)-b*(a^c)+c*(a^b) print 'a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) print '(a^b)|(c^d) =',(a^b)|(c^d) print '((a^b)|c)|d =',((a^b)|c)|d print '(a^b)x(c^d) =',Com(a^b,c^d) print '(a|(b^c))|(d^e) =',(a|(b^c))|(d^e) return
def basic_multivector_operations(): Print_Function() g3d = Ga('e*x|y|z') (ex, ey, ez) = g3d.mv() A = g3d.mv('A', 'mv') A.Fmt(1, 'A') A.Fmt(2, 'A') A.Fmt(3, 'A') X = g3d.mv('X', 'vector') Y = g3d.mv('Y', 'vector') print 'g_{ij} =\n', g3d.g X.Fmt(1, 'X') Y.Fmt(1, 'Y') (X * Y).Fmt(2, 'X*Y') (X ^ Y).Fmt(2, 'X^Y') (X | Y).Fmt(2, 'X|Y') g2d = Ga('e*x|y') (ex, ey) = g2d.mv() print 'g_{ij} =\n', g2d.g X = g2d.mv('X', 'vector') A = g2d.mv('A', 'spinor') X.Fmt(1, 'X') A.Fmt(1, 'A') (X | A).Fmt(2, 'X|A') (X < A).Fmt(2, 'X<A') (A > X).Fmt(2, 'A>X') o2d = Ga('e*x|y', g=[1, 1]) (ex, ey) = o2d.mv() print 'g_{ii} =\n', o2d.g X = o2d.mv('X', 'vector') A = o2d.mv('A', 'spinor') X.Fmt(1, 'X') A.Fmt(1, 'A') (X * A).Fmt(2, 'X*A') (X | A).Fmt(2, 'X|A') (X < A).Fmt(2, 'X<A') (X > A).Fmt(2, 'X>A') (A * X).Fmt(2, 'A*X') (A | X).Fmt(2, 'A|X') (A < X).Fmt(2, 'A<X') (A > X).Fmt(2, 'A>X') return
def reciprocal_frame_test(): Print_Function() g = '1 # #,'+ \ '# 1 #,'+ \ '# # 1' g3dn = Ga('e1 e2 e3',g=g) (e1,e2,e3) = g3dn.mv() print 'g_{ij} =\n',g3dn.g E = e1^e2^e3 Esq = (E*E).scalar() print 'E =',E print 'E**2 =',Esq Esq_inv = 1 / Esq E1 = (e2^e3)*E E2 = (-1)*(e1^e3)*E E3 = (e1^e2)*E print 'E1 = (e2^e3)*E =',E1 print 'E2 =-(e1^e3)*E =',E2 print 'E3 = (e1^e2)*E =',E3 w = (E1|e2) w = w.expand() print 'E1|e2 =',w w = (E1|e3) w = w.expand() print 'E1|e3 =',w w = (E2|e1) w = w.expand() print 'E2|e1 =',w w = (E2|e3) w = w.expand() print 'E2|e3 =',w w = (E3|e1) w = w.expand() print 'E3|e1 =',w w = (E3|e2) w = w.expand() print 'E3|e2 =',w w = (E1|e1) w = (w.expand()).scalar() Esq = expand(Esq) print '(E1|e1)/E**2 =',simplify(w/Esq) w = (E2|e2) w = (w.expand()).scalar() print '(E2|e2)/E**2 =',simplify(w/Esq) w = (E3|e3) w = (w.expand()).scalar() print '(E3|e3)/E**2 =',simplify(w/Esq) return
#ALGEBRA & DEFINITIONS ######################################################################## #Clifford(1,4) #Flat space, no metric, just signature #All constants = 1 metric=[1 ,-1 ,-1 ,-1 ,-1] #Dimensions variables = (t, x, y, z, w) = symbols('t x y z w', real=True) myBasis='gamma_t gamma_x gamma_y gamma_z gamma_w' #Algebra sp5d = Ga(myBasis, g=metric, coords=variables,norm=True) (gamma_t, gamma_x, gamma_y, gamma_z, gamma_w) = sp5d.mv() (grad, rgrad) = sp5d.grads() #Imaginary unit imag=gamma_w imag.texLabel='i' #Associative Hyperbolic Quaternions ihquat=gamma_t jhquat=gamma_t*gamma_x*gamma_y*gamma_z*gamma_w khquat=gamma_x*gamma_y*gamma_z*gamma_w ihquat.texLabel='\\mathbf{i}' jhquat.texLabel='\\mathbf{j}' khquat.texLabel='\\mathbf{k}' #Quaternions iquat=gamma_y*gamma_z jquat=gamma_z*gamma_x
from sympy import symbols, sin, latex, diff, Function, expand from sympy.galgebra.ga import Ga from sympy.galgebra.lt import Mlt from sympy.galgebra.printer import Eprint, Format, xpdf Format() #Define spherical coordinate system in 3-d coords = (r, th, phi) = symbols('r,theta,phi', real=True) sp3d = Ga('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords) (er, eth, ephi) = sp3d.mv() #Define coordinates for 2-d (u,v) and 1-d (s) manifolds u,v,s,alpha = symbols('u v s alpha',real=True) sub_coords = (u,v) smap = [1, u, v] # Coordinate map for sphere of r = 1 in 3-d print r'(u,v)\rightarrow (r,\theta,\phi) = ',smap #Define unit sphere manifold sph2d = sp3d.sm(smap,sub_coords) print '#Unit Sphere Manifold:'
from sympy.galgebra.printer import Format, xpdf from sympy.galgebra.ga import Ga Format() g3d = Ga('e*x|y|z') A = g3d.mv('A','mv') print r'\bm{A} =',A A.Fmt(2,r'\bm{A}') A.Fmt(3,r'\bm{A}') xpdf(paper='letter')
def main(): Format() (g3d,ex,ey,ez) = Ga.build('e*x|y|z') A = g3d.mv('A','mv') print r'\bm{A} =',A A.Fmt(2,r'\bm{A}') A.Fmt(3,r'\bm{A}') X = (x,y,z) = symbols('x y z',real=True) o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X) (ex,ey,ez) = o3d.mv() f = o3d.mv('f','scalar',f=True) A = o3d.mv('A','vector',f=True) B = o3d.mv('B','bivector',f=True) print r'\bm{A} =',A print r'\bm{B} =',B print 'grad*f =',o3d.grad*f print r'grad|\bm{A} =',o3d.grad|A print r'grad*\bm{A} =',o3d.grad*A print r'-I*(grad^\bm{A}) =',-o3d.i*(o3d.grad^A) print r'grad*\bm{B} =',o3d.grad*B print r'grad^\bm{B} =',o3d.grad^B print r'grad|\bm{B} =',o3d.grad|B g4d = Ga('a b c d') (a,b,c,d) = g4d.mv() print 'g_{ij} =',g4d.g print '\\bm{a|(b*c)} =',a|(b*c) print '\\bm{a|(b^c)} =',a|(b^c) print '\\bm{a|(b^c^d)} =',a|(b^c^d) print '\\bm{a|(b^c)+c|(a^b)+b|(c^a)} =',(a|(b^c))+(c|(a^b))+(b|(c^a)) print '\\bm{a*(b^c)-b*(a^c)+c*(a^b)} =',a*(b^c)-b*(a^c)+c*(a^b) print '\\bm{a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c)} =',a*(b^c^d)-b*(a^c^d)+c*(a^b^d)-d*(a^b^c) print '\\bm{(a^b)|(c^d)} =',(a^b)|(c^d) print '\\bm{((a^b)|c)|d} =',((a^b)|c)|d print '\\bm{(a^b)\\times (c^d)} =',Com(a^b,c^d) g = '1 # #,'+ \ '# 1 #,'+ \ '# # 1' ng3d = Ga('e1 e2 e3',g=g) (e1,e2,e3) = ng3d.mv() E = e1^e2^e3 Esq = (E*E).scalar() print 'E =',E print '%E^{2} =',Esq Esq_inv = 1/Esq E1 = (e2^e3)*E E2 = (-1)*(e1^e3)*E E3 = (e1^e2)*E print 'E1 = (e2^e3)*E =',E1 print 'E2 =-(e1^e3)*E =',E2 print 'E3 = (e1^e2)*E =',E3 print 'E1|e2 =',(E1|e2).expand() print 'E1|e3 =',(E1|e3).expand() print 'E2|e1 =',(E2|e1).expand() print 'E2|e3 =',(E2|e3).expand() print 'E3|e1 =',(E3|e1).expand() print 'E3|e2 =',(E3|e2).expand() w = ((E1|e1).expand()).scalar() Esq = expand(Esq) print '%(E1\\cdot e1)/E^{2} =',simplify(w/Esq) w = ((E2|e2).expand()).scalar() print '%(E2\\cdot e2)/E^{2} =',simplify(w/Esq) w = ((E3|e3).expand()).scalar() print '%(E3\\cdot e3)/E^{2} =',simplify(w/Esq) X = (r,th,phi) = symbols('r theta phi') s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True) (er,eth,ephi) = s3d.mv() f = s3d.mv('f','scalar',f=True) A = s3d.mv('A','vector',f=True) B = s3d.mv('B','bivector',f=True) print 'A =',A print 'B =',B print 'grad*f =',s3d.grad*f print 'grad|A =',s3d.grad|A print '-I*(grad^A) =',-s3d.i*(s3d.grad^A) print 'grad^B =',s3d.grad^B coords = symbols('t x y z') m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords) (g0,g1,g2,g3) = m4d.mv() I = m4d.i B = m4d.mv('B','vector',f=True) E = m4d.mv('E','vector',f=True) B.set_coef(1,0,0) E.set_coef(1,0,0) B *= g0 E *= g0 J = m4d.mv('J','vector',f=True) F = E+I*B print 'B = \\bm{B\\gamma_{t}} =',B print 'E = \\bm{E\\gamma_{t}} =',E print 'F = E+IB =',F print 'J =',J gradF = m4d.grad*F gradF.Fmt(3,'grad*F') print 'grad*F = J' (gradF.get_grade(1)-J).Fmt(3,'%\\grade{\\nabla F}_{1} -J = 0') (gradF.get_grade(3)).Fmt(3,'%\\grade{\\nabla F}_{3} = 0') (alpha,beta,gamma) = symbols('alpha beta gamma') (x,t,xp,tp) = symbols("x t x' t'") m2d = Ga('gamma*t|x',g=[1,-1]) (g0,g1) = m2d.mv() R = cosh(alpha/2)+sinh(alpha/2)*(g0^g1) X = t*g0+x*g1 Xp = tp*g0+xp*g1 print 'R =',R print r"#%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} = t'\bm{\gamma'_{t}}+x'\bm{\gamma'_{x}} = R\lp t'\bm{\gamma_{t}}+x'\bm{\gamma_{x}}\rp R^{\dagger}" Xpp = R*Xp*R.rev() Xpp = Xpp.collect() Xpp = Xpp.trigsimp() print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp Xpp = Xpp.subs({sinh(alpha):gamma*beta,cosh(alpha):gamma}) print r'%\f{\sinh}{\alpha} = \gamma\beta' print r'%\f{\cosh}{\alpha} = \gamma' print r"%t\bm{\gamma_{t}}+x\bm{\gamma_{x}} =",Xpp.collect() coords = symbols('t x y z') m4d = Ga('gamma*t|x|y|z',g=[1,-1,-1,-1],coords=coords) (g0,g1,g2,g3) = m4d.mv() I = m4d.i (m,e) = symbols('m e') psi = m4d.mv('psi','spinor',f=True) A = m4d.mv('A','vector',f=True) sig_z = g3*g0 print '\\bm{A} =',A print '\\bm{\\psi} =',psi dirac_eq = (m4d.grad*psi)*I*sig_z-e*A*psi-m*psi*g0 dirac_eq.simplify() dirac_eq.Fmt(3,r'\nabla \bm{\psi} I \sigma_{z}-e\bm{A}\bm{\psi}-m\bm{\psi}\gamma_{t} = 0') xpdf() return
from sympy import symbols, sin from sympy.galgebra.printer import Format, xpdf from sympy.galgebra.ga import Ga Format() print r'#\newline Normalized Spherical Coordinates:' sph_coords = (r, th, phi) = symbols('r theta phi', real=True) sp3d = Ga('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords, norm=True) f = sp3d.mv('f', 'scalar', f=True) F = sp3d.mv('F', 'vector', f=True) lap = sp3d.grad*sp3d.grad print 'g =', sp3d.g print r'%\nabla =', sp3d.grad print r'%\nabla^{2} =', lap print r'%\lp\nabla^{2}\rp f =', lap*f print r'%\nabla\cdot\lp\nabla f\rp =', sp3d.grad | (sp3d.grad * f) print 'F =', F print r'%\nabla\cdot F =', sp3d.grad | F print '#Unnormalized Spherical Coordinates:' sp3du = Ga('e', g=[1, r**2, r**2 * sin(th)**2], coords=sph_coords) f = sp3du.mv('f', 'scalar', f=True) F = sp3du.mv('F', 'vector', f=True) lap = sp3du.grad*sp3du.grad print 'g =', sp3du.g print r'%\nabla =', sp3du.grad print r'%\nabla^{2} =', lap
from sympy import symbols, sin from sympy.galgebra.printer import Format, xpdf from sympy.galgebra.ga import Ga Format() X = (r,th,phi) = symbols('r theta phi') s3d = Ga('e_r e_theta e_phi',g=[1,r**2,r**2*sin(th)**2],coords=X,norm=True) (er,eth,ephi) = s3d.mv() grad = s3d.grad f = s3d.mv('f','scalar',f=True) A = s3d.mv('A','vector',f=True) B = s3d.mv('B','bivector',f=True) print 'f =',f print 'A =',A print 'B =',B print 'grad*f =',grad*f print 'grad|A =',grad|A print '-I*(grad^A) =',(-s3d.i*(grad^A)).simplify() print 'grad^B =',grad^B xpdf(paper='letter')
def test_submanifolds(): #Define spherical coordinate system in 3-d coords = (r, th, phi) = symbols('r,theta,phi', real=True) sp3d = Ga('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords) (er, eth, ephi) = sp3d.mv() #Define coordinates for 2-d (u,v) and 1-d (s) manifolds u,v,s,alpha = symbols('u v s alpha',real=True) sub_coords = (u,v) smap = [1, u, v] # Coordinate map for sphere of r = 1 in 3-d #Define unit sphere manifold sph2d = sp3d.sm(smap,sub_coords) (eu,ev) = sph2d.mv() #Define vector and vector field on unit sphere tangent space a = sph2d.mv('a','vector') b = sph2d.mv('b','vector') c = sph2d.mv('c','vector') f = sph2d.mv('f','vector',f=True) #Define directional derivative in direction a for unit sphere manifold dd = a|sph2d.grad assert str(dd) == 'a__u*D{u} + a__v*D{v}' assert str(dd * eu) == 'a__v*e_v/tan(u)' assert str(dd * ev) == '-a__v*sin(2*u)*e_u/2 + a__u*e_v/tan(u)' assert str(dd * f) == '(a__u*D{u}f__u - a__v*f__v*sin(2*u)/2 + a__v*D{v}f__u)*e_u + (a__u*f__v/tan(u) + a__u*D{u}f__v + a__v*f__u/tan(u) + a__v*D{v}f__v)*e_v' V = Mlt('V',sph2d,nargs=1,fct=True) T = Mlt('T',sph2d,nargs=2,fct=True) assert str(T.contract(1,2)) == 'a_1__u**2*D{u}^2T_uu + a_1__u**2*D{v}^2T_uu/sin(u)**2 + a_1__u*a_1__v*D{u}^2T_uv + a_1__u*a_1__v*D{u}^2T_vu + a_1__u*a_1__v*D{v}^2T_uv/sin(u)**2 + a_1__u*a_1__v*D{v}^2T_vu/sin(u)**2 + a_1__v**2*D{u}^2T_vv + a_1__v**2*D{v}^2T_vv/sin(u)**2' #Tensor Evaluation assert str(T(a,b)) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv' assert str(T(a,b+c).expand()) == 'a__u*b__u*T_uu + a__u*b__v*T_uv + a__u*c__u*T_uu + a__u*c__v*T_uv + a__v*b__u*T_vu + a__v*b__v*T_vv + a__v*c__u*T_vu + a__v*c__v*T_vv' assert str(T(a,alpha*b)) == 'a__u*alpha*b__u*T_uu + a__u*alpha*b__v*T_uv + a__v*alpha*b__u*T_vu + a__v*alpha*b__v*T_vv' #Geometric Derivative With Respect To Slot assert str(T.pdiff(1)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2' assert str(T.pdiff(2)) == '(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv)*e_u + (a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)*e_v/sin(u)**2' #Covariant Derivatives assert str(V.cderiv()) == 'a_2__u*(a_1__u*D{u}V_u + a_1__v*D{u}V_v) + a_2__v*(a_1__u*D{v}V_u + a_1__v*D{v}V_v)' DT = T.cderiv() assert str(DT) == 'a_3__u*(a_1__u*a_2__u*D{u}T_uu + a_1__u*a_2__v*D{u}T_uv + a_1__v*a_2__u*D{u}T_vu + a_1__v*a_2__v*D{u}T_vv) + a_3__v*(a_1__u*a_2__u*D{v}T_uu + a_1__u*a_2__v*D{v}T_uv + a_1__v*a_2__u*D{v}T_vu + a_1__v*a_2__v*D{v}T_vv)' #Define curve on unit sphere manifold us = Function('u__s')(s) vs = Function('v__s')(s) #Define 1-d submanifold on unit shpere manifold crv1d = sph2d.sm([us,vs],[s]) (es,) = crv1d.mv() #1-D Manifold On Unit Sphere: assert str(crv1d.grad) == 'e_s*1/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)*D{s}' #Define scalar and vector fields on 1-d manifold tangent space g = crv1d.mv('g','scalar',f=True) h = crv1d.mv('h','vector',f=True) assert str(crv1d.grad * g) == 'D{s}g*e_s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)' assert str(crv1d.grad | h) == 'D{s}h__s + h__s*sin(u__s)**2*D{s}v__s*D{s}^2v__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2) + h__s*sin(2*u__s)*D{s}u__s*D{s}v__s**2/(2*(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)) + h__s*D{s}u__s*D{s}^2u__s/(sin(u__s)**2*D{s}v__s**2 + D{s}u__s**2)' return
from sympy import expand, simplify from sympy.galgebra.ga import Ga from sympy.galgebra.printer import Format, xpdf Format() g = "1 # #," + "# 1 #," + "# # 1" ng3d = Ga("e1 e2 e3", g=g) (e1, e2, e3) = ng3d.mv() print "g_{ij} =", ng3d.g E = e1 ^ e2 ^ e3 Esq = (E * E).scalar() print "E =", E print "%E^{2} =", Esq Esq_inv = 1 / Esq E1 = (e2 ^ e3) * E E2 = (-1) * (e1 ^ e3) * E E3 = (e1 ^ e2) * E print "E1 = (e2^e3)*E =", E1 print "E2 =-(e1^e3)*E =", E2 print "E3 = (e1^e2)*E =", E3 w = E1 | e2 w = w.expand() print "E1|e2 =", w
return [a1,a2] def norm(X): Y=sqrt((X*X).scalar()) return Y Get_Program(True) Eprint() g='1 0 0 0, \ 0 1 0 0, \ 0 0 0 2, \ 0 0 2 0' c2d = Ga('e_1 e_2 n \\bar{n}',g=g) (e1,e2,n,nbar) = c2d.mv() global n,nbar,I def F(x): global n,nbar Fx = ((x*x)*n+2*x-nbar) / 2 return(Fx) e = (n+nbar)/2 ebar = n - e I=e1*e2*e*ebar def intersect_lines(L1,L2): global I '''
def noneuclidian_distance_calculation(): from sympy import solve,sqrt Print_Function() g = '0 # #,# 0 #,# # 1' necl = Ga('X Y e',g=g) (X,Y,e) = necl.mv() print 'g_{ij} =',necl.g print '(X^Y)**2 =',(X^Y)*(X^Y) L = X^Y^e B = (L*e).expand().blade_rep() # D&L 10.152 print 'B =',B Bsq = B*B print 'B**2 =',Bsq.obj Bsq = Bsq.scalar() print '#L = X^Y^e is a non-euclidian line' print 'B = L*e =',B BeBr =B*e*B.rev() print 'B*e*B.rev() =',BeBr print 'B**2 =',B*B print 'L**2 =',L*L # D&L 10.153 (s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha') XdotY = necl.g[0,1] Xdote = necl.g[0,2] Ydote = necl.g[1,2] Bhat = Binv*B # D&L 10.154 R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2) print 's = sinh(alpha/2) and c = cosh(alpha/2)' print 'exp(alpha*B/(2*|B|)) =',R Z = R*X*R.rev() # D&L 10.155 Z.obj = expand(Z.obj) Z.obj = Z.obj.collect([Binv,s,c,XdotY]) Z.Fmt(3,'R*X*R.rev()') W = Z|Y # Extract scalar part of multivector # From this point forward all calculations are with sympy scalars print 'Objective is to determine value of C = cosh(alpha) such that W = 0' W = W.scalar() print 'Z|Y =',W W = expand(W) W = simplify(W) W = W.collect([s*Binv]) M = 1/Bsq W = W.subs(Binv**2,M) W = simplify(W) Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote) W = W.collect([Binv*c*s,XdotY]) #Double angle substitutions W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2)) W = W.subs(2*c*s,S) W = W.subs(c**2,(C+1)/2) W = W.subs(s**2,(C-1)/2) W = simplify(W) W = W.subs(1/Binv,Bmag) W = expand(W) print 'S = sinh(alpha) and C = cosh(alpha)' print 'W =',W Wd = collect(W,[C,S],exact=True,evaluate=False) print 'Wd =', Wd Wd_1 = Wd[one] Wd_C = Wd[C] Wd_S = Wd[S] print 'Scalar Coefficient =',Wd_1 print 'Cosh Coefficient =',Wd_C print 'Sinh Coefficient =',Wd_S print '|B| =',Bmag Wd_1 = Wd_1.subs(Bmag,1/Binv) Wd_C = Wd_C.subs(Bmag,1/Binv) Wd_S = Wd_S.subs(Bmag,1/Binv) lhs = Wd_1+Wd_C*C rhs = -Wd_S*S lhs = lhs**2 rhs = rhs**2 W = expand(lhs-rhs) W = expand(W.subs(1/Binv**2,Bmag**2)) W = expand(W.subs(S**2,C**2-1)) W = W.collect([C,C**2],evaluate=False) a = simplify(W[C**2]) b = simplify(W[C]) c = simplify(W[one]) print 'Require a*C**2+b*C+c = 0' print 'a =',a print 'b =',b print 'c =',c x = Symbol('x') C = solve(a*x**2+b*x+c,x)[0] print 'cosh(alpha) = C = -b/(2*a) =',expand(simplify(expand(C))) return
def main(): Print_Function() (x, y, z) = xyz = symbols('x,y,z',real=True) (o3d, ex, ey, ez) = Ga.build('e_x e_y e_z', g=[1, 1, 1], coords=xyz) grad = o3d.grad (u, v) = uv = symbols('u,v',real=True) (g2d, eu, ev) = Ga.build('e_u e_v', coords=uv) grad_uv = g2d.grad v_xyz = o3d.mv('v','vector') A_xyz = o3d.mv('A','vector',f=True) A_uv = g2d.mv('A','vector',f=True) print '#3d orthogonal ($A$ is vector function)' print 'A =', A_xyz print '%A^{2} =', A_xyz * A_xyz print 'grad|A =', grad | A_xyz print 'grad*A =', grad * A_xyz print 'v|(grad*A) =',v_xyz|(grad*A_xyz) print '#2d general ($A$ is vector function)' print 'A =', A_uv print '%A^{2} =', A_uv * A_uv print 'grad|A =', grad_uv | A_uv print 'grad*A =', grad_uv * A_uv A = o3d.lt('A') print '#3d orthogonal ($A,\\;B$ are linear transformations)' print 'A =', A print '\\f{\\det}{A} =', A.det() print '\\overline{A} =', A.adj() print '\\f{\\Tr}{A} =', A.tr() print '\\f{A}{e_x^e_y} =', A(ex^ey) print '\\f{A}{e_x}^\\f{A}{e_y} =', A(ex)^A(ey) B = o3d.lt('B') print 'A + B =', A + B print 'AB =', A * B print 'A - B =', A - B print '#2d general ($A,\\;B$ are linear transformations)' A2d = g2d.lt('A') print 'A =', A2d print '\\f{\\det}{A} =', A2d.det() print '\\overline{A} =', A2d.adj() print '\\f{\\Tr}{A} =', A2d.tr() print '\\f{A}{e_u^e_v} =', A2d(eu^ev) print '\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev) B2d = g2d.lt('B') print 'B =', B2d print 'A + B =', A2d + B2d print 'AB =', A2d * B2d print 'A - B =', A2d - B2d a = g2d.mv('a','vector') b = g2d.mv('b','vector') print r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify() m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True)) T = m4d.lt('T') print 'g =', m4d.g print r'\underline{T} =',T print r'\overline{T} =',T.adj() print r'\f{\det}{\underline{T}} =',T.det() print r'\f{\mbox{tr}}{\underline{T}} =',T.tr() a = m4d.mv('a','vector') b = m4d.mv('b','vector') print r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify() coords = (r, th, phi) = symbols('r,theta,phi', real=True) (sp3d, er, eth, ephi) = Ga.build('e_r e_th e_ph', g=[1, r**2, r**2*sin(th)**2], coords=coords) grad = sp3d.grad sm_coords = (u, v) = symbols('u,v', real=True) smap = [1, u, v] # Coordinate map for sphere of r = 1 sph2d = sp3d.sm(smap,sm_coords,norm=True) (eu, ev) = sph2d.mv() grad_uv = sph2d.grad F = sph2d.mv('F','vector',f=True) f = sph2d.mv('f','scalar',f=True) print 'f =',f print 'grad*f =',grad_uv * f print 'F =',F print 'grad*F =',grad_uv * F tp = (th,phi) = symbols('theta,phi',real=True) smap = [sin(th)*cos(phi),sin(th)*sin(phi),cos(th)] sph2dr = o3d.sm(smap,tp,norm=True) (eth, ephi) = sph2dr.mv() grad_tp = sph2dr.grad F = sph2dr.mv('F','vector',f=True) f = sph2dr.mv('f','scalar',f=True) print 'f =',f print 'grad*f =',grad_tp * f print 'F =',F print 'grad*F =',grad_tp * F return
def noneuclidian_distance_calculation(): Print_Function() from sympy import solve,sqrt g = '0 # #,# 0 #,# # 1' nel = Ga('X Y e',g=g) (X,Y,e) = nel.mv() print 'g_{ij} =',nel.g print '%(X\\W Y)^{2} =',(X^Y)*(X^Y) L = X^Y^e B = L*e # D&L 10.152 Bsq = (B*B).scalar() print '#%L = X\\W Y\\W e \\text{ is a non-euclidian line}' print 'B = L*e =',B BeBr =B*e*B.rev() print '%BeB^{\\dagger} =',BeBr print '%B^{2} =',B*B print '%L^{2} =',L*L # D&L 10.153 (s,c,Binv,M,S,C,alpha) = symbols('s c (1/B) M S C alpha') XdotY = nel.g[0,1] Xdote = nel.g[0,2] Ydote = nel.g[1,2] Bhat = Binv*B # D&L 10.154 R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2) print '#%s = \\f{\\sinh}{\\alpha/2} \\text{ and } c = \\f{\\cosh}{\\alpha/2}' print '%e^{\\alpha B/{2\\abs{B}}} =',R Z = R*X*R.rev() # D&L 10.155 Z.obj = expand(Z.obj) Z.obj = Z.obj.collect([Binv,s,c,XdotY]) Z.Fmt(3,'%RXR^{\\dagger}') W = Z|Y # Extract scalar part of multivector # From this point forward all calculations are with sympy scalars #print '#Objective is to determine value of C = cosh(alpha) such that W = 0' W = W.scalar() print '%W = Z\\cdot Y =',W W = expand(W) W = simplify(W) W = W.collect([s*Binv]) M = 1/Bsq W = W.subs(Binv**2,M) W = simplify(W) Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote) W = W.collect([Binv*c*s,XdotY]) #Double angle substitutions W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2)) W = W.subs(2*c*s,S) W = W.subs(c**2,(C+1)/2) W = W.subs(s**2,(C-1)/2) W = simplify(W) W = W.subs(1/Binv,Bmag) W = expand(W) print '#%S = \\f{\\sinh}{\\alpha} \\text{ and } C = \\f{\\cosh}{\\alpha}' print 'W =',W Wd = collect(W,[C,S],exact=True,evaluate=False) Wd_1 = Wd[one] Wd_C = Wd[C] Wd_S = Wd[S] print '%\\text{Scalar Coefficient} =',Wd_1 print '%\\text{Cosh Coefficient} =',Wd_C print '%\\text{Sinh Coefficient} =',Wd_S print '%\\abs{B} =',Bmag Wd_1 = Wd_1.subs(Bmag,1/Binv) Wd_C = Wd_C.subs(Bmag,1/Binv) Wd_S = Wd_S.subs(Bmag,1/Binv) lhs = Wd_1+Wd_C*C rhs = -Wd_S*S lhs = lhs**2 rhs = rhs**2 W = expand(lhs-rhs) W = expand(W.subs(1/Binv**2,Bmag**2)) W = expand(W.subs(S**2,C**2-1)) W = W.collect([C,C**2],evaluate=False) a = simplify(W[C**2]) b = simplify(W[C]) c = simplify(W[one]) print '#%\\text{Require } aC^{2}+bC+c = 0' print 'a =',a print 'b =',b print 'c =',c x = Symbol('x') C = solve(a*x**2+b*x+c,x)[0] print '%b^{2}-4ac =',simplify(b**2-4*a*c) print '%\\f{\\cosh}{\\alpha} = C = -b/(2a) =',expand(simplify(expand(C))) return
print '#2d general ($A,\\;B$ are linear transformations)' A2d = g2d.lt('A') print 'A =', A2d print '\\f{\\det}{A} =', A2d.det() #A2d.adj().Fmt(4,'\\overline{A}') print '\\f{\\Tr}{A} =', A2d.tr() print '\\f{A}{e_u^e_v} =', A2d(eu^ev) print '\\f{A}{e_u}^\\f{A}{e_v} =', A2d(eu)^A2d(ev) B2d = g2d.lt('B') print 'B =', B2d print 'A + B =', A2d + B2d print 'AB =', A2d * B2d print 'A - B =', A2d - B2d a = g2d.mv('a','vector') b = g2d.mv('b','vector') print r'a|\f{\overline{A}}{b}-b|\f{\underline{A}}{a} =',((a|A2d.adj()(b))-(b|A2d(a))).simplify() print '#4d Minkowski spaqce (Space Time)' m4d = Ga('e_t e_x e_y e_z', g=[1, -1, -1, -1],coords=symbols('t,x,y,z',real=True)) T = m4d.lt('T') print 'g =', m4d.g print r'\underline{T} =',T print r'\overline{T} =',T.adj() #m4d.mv(T.det()).Fmt(4,r'\f{\det}{\underline{T}}') print r'\f{\mbox{tr}}{\underline{T}} =',T.tr() a = m4d.mv('a','vector') b = m4d.mv('b','vector') print r'a|\f{\overline{T}}{b}-b|\f{\underline{T}}{a} =',((a|T.adj()(b))-(b|T(a))).simplify() xpdf(paper='landscape')
from sympy.galgebra.printer import Format, xpdf from sympy.galgebra.ga import Ga from sympy import symbols Format() X = (x,y,z) = symbols('x y z') o3d = Ga('e_x e_y e_z',g=[1,1,1],coords=X) f = o3d.mv('f','scalar',f=True) A = o3d.mv('A','vector',f=True) B = o3d.mv('B','bivector',f=True) print r'\bm{A} =',A print r'\bm{B} =',B print 'grad*f =',o3d.grad*f print r'grad|\bm{A} =',o3d.grad|A (o3d.grad*A).Fmt(2,r'grad*\bm{A}') print r'-I*(grad^\bm{A}) =',-o3d.mv_I*(o3d.grad^A) (o3d.grad*B).Fmt(2,r'grad*\bm{B}') print r'grad^\bm{B} =',o3d.grad^B print r'grad|\bm{B} =',o3d.grad|B xpdf(paper='letter')