def test_coordinate_vars():
"""Tests the coordinate variables functionality"""
A = ReferenceFrame('A')
assert CoordinateSym('Ax', A, 0) == A[0]
assert CoordinateSym('Ax', A, 1) == A[1]
assert CoordinateSym('Ax', A, 2) == A[2]
raises(ValueError, lambda: CoordinateSym('Ax', A, 3))
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
assert isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym)
assert A.variable_map(A) == {A[0]: A[0], A[1]: A[1], A[2]: A[2]}
assert A[0].frame == A
B = A.orientnew('B', 'Axis', [q, A.z])
assert B.variable_map(A) == {
B[2]: A[2],
B[1]: -A[0] * sin(q) + A[1] * cos(q),
B[0]: A[0] * cos(q) + A[1] * sin(q)
}
assert A.variable_map(B) == {
A[0]: B[0] * cos(q) - B[1] * sin(q),
A[1]: B[0] * sin(q) + B[1] * cos(q),
A[2]: B[2]
}
assert time_derivative(B[0], A) == -A[0] * sin(q) * qd + A[1] * cos(q) * qd
assert time_derivative(B[1], A) == -A[0] * cos(q) * qd - A[1] * sin(q) * qd
assert time_derivative(B[2], A) == 0
assert express(B[0], A, variables=True) == A[0] * cos(q) + A[1] * sin(q)
assert express(B[1], A, variables=True) == -A[0] * sin(q) + A[1] * cos(q)
assert express(B[2], A, variables=True) == A[2]
assert time_derivative(A[0] * A.x + A[1] * A.y + A[2] * A.z,
B) == A[1] * qd * A.x - A[0] * qd * A.y
assert time_derivative(B[0] * B.x + B[1] * B.y + B[2] * B.z,
A) == -B[1] * qd * B.x + B[0] * qd * B.y
assert express(B[0]*B[1]*B[2], A, variables=True) == \
A[2]*(-A[0]*sin(q) + A[1]*cos(q))*(A[0]*cos(q) + A[1]*sin(q))
assert (time_derivative(B[0] * B[1] * B[2], A) -
(A[2] * (-A[0]**2 * cos(2 * q) - 2 * A[0] * A[1] * sin(2 * q) +
A[1]**2 * cos(2 * q)) * qd)).trigsimp() == 0
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == \
(B[0]*cos(q) - B[1]*sin(q))*A.x + (B[0]*sin(q) + \
B[1]*cos(q))*A.y + B[2]*A.z
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A, variables=True) == \
A[0]*A.x + A[1]*A.y + A[2]*A.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == \
(A[0]*cos(q) + A[1]*sin(q))*B.x + \
(-A[0]*sin(q) + A[1]*cos(q))*B.y + A[2]*B.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B, variables=True) == \
B[0]*B.x + B[1]*B.y + B[2]*B.z
N = B.orientnew('N', 'Axis', [-q, B.z])
assert N.variable_map(A) == {N[0]: A[0], N[2]: A[2], N[1]: A[1]}
C = A.orientnew('C', 'Axis', [q, A.x + A.y + A.z])
mapping = A.variable_map(C)
assert mapping[A[0]] == 2*C[0]*cos(q)/3 + C[0]/3 - 2*C[1]*sin(q + pi/6)/3 +\
C[1]/3 - 2*C[2]*cos(q + pi/3)/3 + C[2]/3
assert mapping[A[1]] == -2*C[0]*cos(q + pi/3)/3 + \
C[0]/3 + 2*C[1]*cos(q)/3 + C[1]/3 - 2*C[2]*sin(q + pi/6)/3 + C[2]/3
assert mapping[A[2]] == -2*C[0]*sin(q + pi/6)/3 + C[0]/3 - \
2*C[1]*cos(q + pi/3)/3 + C[1]/3 + 2*C[2]*cos(q)/3 + C[2]/3
def g_DF(self):
#gDF = TM_ds * b_hex(self.a, self.c) * self.g
gC = b_hex(self.a, self.c) * self.g # still matrix form
gCF = gC[0] * self.CF.x + gC[1] * self.CF.y + gC[
2] * self.CF.z # vector explicitely defined
gDF = express(gCF, self.DF)
return gDF
def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.
Parameters
==========
otherframe : ReferenceFrame
The frame for this Vector to be described in
variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector, dynamicsymbols
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z
"""
from sympy.physics.vector import express
return express(self, otherframe, variables=variables)
def gradient(scalar, frame):
"""
Returns the vector gradient of a scalar field computed wrt the
coordinate symbols of the given frame.
Parameters
==========
scalar : sympifiable
The scalar field to take the gradient of
frame : ReferenceFrame
The frame to calculate the gradient in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import gradient
>>> R = ReferenceFrame('R')
>>> s1 = R[0]*R[1]*R[2]
>>> gradient(s1, R)
R_y*R_z*R.x + R_x*R_z*R.y + R_x*R_y*R.z
>>> s2 = 5*R[0]**2*R[2]
>>> gradient(s2, R)
10*R_x*R_z*R.x + 5*R_x**2*R.z
"""
_check_frame(frame)
outvec = Vector(0)
scalar = express(scalar, frame, variables=True)
for i, x in enumerate(frame):
outvec += diff(scalar, frame[i]) * x
return outvec
def gradient(scalar, frame):
"""
Returns the vector gradient of a scalar field computed wrt the
coordinate symbols of the given frame.
Parameters
==========
scalar : sympifiable
The scalar field to take the gradient of
frame : ReferenceFrame
The frame to calculate the gradient in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import gradient
>>> R = ReferenceFrame('R')
>>> s1 = R[0]*R[1]*R[2]
>>> gradient(s1, R)
R_y*R_z*R.x + R_x*R_z*R.y + R_x*R_y*R.z
>>> s2 = 5*R[0]**2*R[2]
>>> gradient(s2, R)
10*R_x*R_z*R.x + 5*R_x**2*R.z
"""
_check_frame(frame)
outvec = Vector(0)
scalar = express(scalar, frame, variables=True)
for i, x in enumerate(frame):
outvec += diff(scalar, frame[i]) * x
return outvec
def ghex_DF(self):
gC = b_hex(self.a,
self.c) * self.g # g in real space Cartesian coordinates
gCF = gC[0] * self.CF.x + gC[1] * self.CF.y + gC[
2] * self.CF.z # vector explicitely defined
gDF = express(gCF, self.DF)
return gDF
def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.
Parameters
==========
otherframe : ReferenceFrame
The frame for this Vector to be described in
variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Vector, dynamicsymbols
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z
"""
from sympy.physics.vector import express
return express(self, otherframe, variables=variables)
def b_SF(self):
bDF = self.b * self.DF.z
bSF = express(bDF, self.SF)
x_b = bSF.dot(self.SF.x)
y_b = bSF.dot(self.SF.y)
z_b = bSF.dot(self.SF.z)
return [x_b, y_b, z_b]
def Beta_hex_RincF(self):
''' numerical function of above for plotting
Return result in incident beam frame'''
betaRincF = express(self.Beta_hex_DF(), self.RincF, variables=True)
betaRincF_n = betaRincF.subs([(self.RincF[0], x), (self.RincF[1], y),
(self.RincF[2], z)],
simultaneous=True)
return betaRincF_n
def rinc_SF(self):
'''
the incident beam is along the lab cartesian direction (0,0,-1)
return in vector form in sample frame
'''
rinc_LF = -1. * self.LF.z
rinc_SF = express(rinc_LF, self.SF)
return rinc_SF
def Beta_hex_RincF_n(self, xN, yN, zN):
''' numerical function of above for plotting
Return result in incident beam frame'''
betaRincF = express(self.Beta_hex_DF(), self.RincF, variables=True)
betaRincF_n = betaRincF.subs([(self.RincF[0], x), (self.RincF[1], y),
(self.RincF[2], z)],
simultaneous=True)
np_function = lambdify((x, y, z), betaRincF_n, "numpy")
return np_function(xN, yN, zN)
def Beta_cubic_SF_n(self, xN, yN, zN):
''' numerical function of above for plotting
Return result in sample frame'''
betaSF = express(self.Beta_cubic_DF(), self.SF, variables=True)
betaSF_n = betaSF.subs([(self.SF[0], x), (self.SF[1], y),
(self.SF[2], z)],
simultaneous=True)
np_function = lambdify((x, y, z), betaSF_n, "numpy")
return np_function(xN, yN, zN)
def gcubic_SF(self):
gC = b_cubic(self.a) * self.g # g in real space Cartesian coordinates
gCF = gC[0] * self.CF.x + gC[1] * self.CF.y + gC[
2] * self.CF.z # vector explicitely defined
gSF = express(gCF, self.SF)
x_g = gSF.dot(self.SF.x)
y_g = gSF.dot(self.SF.y)
z_g = gSF.dot(self.SF.z)
return [x_g, y_g, z_g]
def b_SF(self):
# b = a * [100] in displacement frame
#bDF = self.b * self.DF.y # vector explicitely defined
bDF = self.b * self.DF.x
bSF = express(bDF, self.SF)
x_b = bSF.dot(self.SF.x)
y_b = bSF.dot(self.SF.y)
z_b = bSF.dot(self.SF.z)
return [x_b, y_b, z_b]
def test_coordinate_vars():
"""Tests the coordinate variables functionality"""
A = ReferenceFrame('A')
assert CoordinateSym('Ax', A, 0) == A[0]
assert CoordinateSym('Ax', A, 1) == A[1]
assert CoordinateSym('Ax', A, 2) == A[2]
raises(ValueError, lambda: CoordinateSym('Ax', A, 3))
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
assert isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym) and \
isinstance(A[0], CoordinateSym)
assert A.variable_map(A) == {A[0]:A[0], A[1]:A[1], A[2]:A[2]}
assert A[0].frame == A
B = A.orientnew('B', 'Axis', [q, A.z])
assert B.variable_map(A) == {B[2]: A[2], B[1]: -A[0]*sin(q) + A[1]*cos(q),
B[0]: A[0]*cos(q) + A[1]*sin(q)}
assert A.variable_map(B) == {A[0]: B[0]*cos(q) - B[1]*sin(q),
A[1]: B[0]*sin(q) + B[1]*cos(q), A[2]: B[2]}
assert time_derivative(B[0], A) == -A[0]*sin(q)*qd + A[1]*cos(q)*qd
assert time_derivative(B[1], A) == -A[0]*cos(q)*qd - A[1]*sin(q)*qd
assert time_derivative(B[2], A) == 0
assert express(B[0], A, variables=True) == A[0]*cos(q) + A[1]*sin(q)
assert express(B[1], A, variables=True) == -A[0]*sin(q) + A[1]*cos(q)
assert express(B[2], A, variables=True) == A[2]
assert time_derivative(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == A[1]*qd*A.x - A[0]*qd*A.y
assert time_derivative(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == - B[1]*qd*B.x + B[0]*qd*B.y
assert express(B[0]*B[1]*B[2], A, variables=True) == \
A[2]*(-A[0]*sin(q) + A[1]*cos(q))*(A[0]*cos(q) + A[1]*sin(q))
assert (time_derivative(B[0]*B[1]*B[2], A) -
(A[2]*(-A[0]**2*cos(2*q) -
2*A[0]*A[1]*sin(2*q) +
A[1]**2*cos(2*q))*qd)).trigsimp() == 0
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == \
(B[0]*cos(q) - B[1]*sin(q))*A.x + (B[0]*sin(q) + \
B[1]*cos(q))*A.y + B[2]*A.z
assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A, variables=True) == \
A[0]*A.x + A[1]*A.y + A[2]*A.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == \
(A[0]*cos(q) + A[1]*sin(q))*B.x + \
(-A[0]*sin(q) + A[1]*cos(q))*B.y + A[2]*B.z
assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B, variables=True) == \
B[0]*B.x + B[1]*B.y + B[2]*B.z
N = B.orientnew('N', 'Axis', [-q, B.z])
assert N.variable_map(A) == {N[0]: A[0], N[2]: A[2], N[1]: A[1]}
C = A.orientnew('C', 'Axis', [q, A.x + A.y + A.z])
mapping = A.variable_map(C)
assert mapping[A[0]] == 2*C[0]*cos(q)/3 + C[0]/3 - 2*C[1]*sin(q + pi/6)/3 +\
C[1]/3 - 2*C[2]*cos(q + pi/3)/3 + C[2]/3
assert mapping[A[1]] == -2*C[0]*cos(q + pi/3)/3 + \
C[0]/3 + 2*C[1]*cos(q)/3 + C[1]/3 - 2*C[2]*sin(q + pi/6)/3 + C[2]/3
assert mapping[A[2]] == -2*C[0]*sin(q + pi/6)/3 + C[0]/3 - \
2*C[1]*cos(q + pi/3)/3 + C[1]/3 + 2*C[2]*cos(q)/3 + C[2]/3
def scalar_potential(field, frame):
"""
Returns the scalar potential function of a field in a given frame
(without the added integration constant).
Parameters
==========
field : Vector
The vector field whose scalar potential function is to be
calculated
frame : ReferenceFrame
The frame to do the calculation in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import scalar_potential, gradient
>>> R = ReferenceFrame('R')
>>> scalar_potential(R.z, R) == R[2]
True
>>> scalar_field = 2*R[0]**2*R[1]*R[2]
>>> grad_field = gradient(scalar_field, R)
>>> scalar_potential(grad_field, R)
2*R_x**2*R_y*R_z
"""
#Check whether field is conservative
if not is_conservative(field):
raise ValueError("Field is not conservative")
if field == Vector(0):
return S(0)
#Express the field exntirely in frame
#Subsitute coordinate variables also
_check_frame(frame)
field = express(field, frame, variables=True)
#Make a list of dimensions of the frame
dimensions = [x for x in frame]
#Calculate scalar potential function
temp_function = integrate(field.dot(dimensions[0]), frame[0])
for i, dim in enumerate(dimensions[1:]):
partial_diff = diff(temp_function, frame[i + 1])
partial_diff = field.dot(dim) - partial_diff
temp_function += integrate(partial_diff, frame[i + 1])
return temp_function
def scalar_potential(field, frame):
"""
Returns the scalar potential function of a field in a given frame
(without the added integration constant).
Parameters
==========
field : Vector
The vector field whose scalar potential function is to be
calculated
frame : ReferenceFrame
The frame to do the calculation in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import scalar_potential, gradient
>>> R = ReferenceFrame('R')
>>> scalar_potential(R.z, R) == R[2]
True
>>> scalar_field = 2*R[0]**2*R[1]*R[2]
>>> grad_field = gradient(scalar_field, R)
>>> scalar_potential(grad_field, R)
2*R_x**2*R_y*R_z
"""
#Check whether field is conservative
if not is_conservative(field):
raise ValueError("Field is not conservative")
if field == Vector(0):
return S(0)
#Express the field exntirely in frame
#Subsitute coordinate variables also
_check_frame(frame)
field = express(field, frame, variables=True)
#Make a list of dimensions of the frame
dimensions = [x for x in frame]
#Calculate scalar potential function
temp_function = integrate(field.dot(dimensions[0]), frame[0])
for i, dim in enumerate(dimensions[1:]):
partial_diff = diff(temp_function, frame[i + 1])
partial_diff = field.dot(dim) - partial_diff
temp_function += integrate(partial_diff, frame[i + 1])
return temp_function
def curl(vect, frame):
"""
Returns the curl of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the curl in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import curl
>>> R = ReferenceFrame('R')
>>> v1 = R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z
>>> curl(v1, R)
0
>>> v2 = R[0]*R[1]*R[2]*R.x
>>> curl(v2, R)
R_x*R_y*R.y - R_x*R_z*R.z
"""
_check_vector(vect)
if vect == 0:
return Vector(0)
vect = express(vect, frame, variables=True)
#A mechanical approach to avoid looping overheads
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
outvec = Vector(0)
outvec += (diff(vectz, frame[1]) - diff(vecty, frame[2])) * frame.x
outvec += (diff(vectx, frame[2]) - diff(vectz, frame[0])) * frame.y
outvec += (diff(vecty, frame[0]) - diff(vectx, frame[1])) * frame.z
return outvec
def curl(vect, frame):
"""
Returns the curl of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the curl in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import curl
>>> R = ReferenceFrame('R')
>>> v1 = R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z
>>> curl(v1, R)
0
>>> v2 = R[0]*R[1]*R[2]*R.x
>>> curl(v2, R)
R_x*R_y*R.y - R_x*R_z*R.z
"""
_check_vector(vect)
if vect == 0:
return Vector(0)
vect = express(vect, frame, variables=True)
#A mechanical approach to avoid looping overheads
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
outvec = Vector(0)
outvec += (diff(vectz, frame[1]) - diff(vecty, frame[2])) * frame.x
outvec += (diff(vectx, frame[2]) - diff(vectz, frame[0])) * frame.y
outvec += (diff(vecty, frame[0]) - diff(vectx, frame[1])) * frame.z
return outvec
def divergence(vect, frame):
"""
Returns the divergence of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the divergence in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import divergence
>>> R = ReferenceFrame('R')
>>> v1 = R[0]*R[1]*R[2] * (R.x+R.y+R.z)
>>> divergence(v1, R)
R_x*R_y + R_x*R_z + R_y*R_z
>>> v2 = 2*R[1]*R[2]*R.y
>>> divergence(v2, R)
2*R_z
"""
_check_vector(vect)
if vect == 0:
return S(0)
vect = express(vect, frame, variables=True)
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
out = S(0)
out += diff(vectx, frame[0])
out += diff(vecty, frame[1])
out += diff(vectz, frame[2])
return out
def divergence(vect, frame):
"""
Returns the divergence of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the divergence in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import divergence
>>> R = ReferenceFrame('R')
>>> v1 = R[0]*R[1]*R[2] * (R.x+R.y+R.z)
>>> divergence(v1, R)
R_x*R_y + R_x*R_z + R_y*R_z
>>> v2 = 2*R[1]*R[2]*R.y
>>> divergence(v2, R)
2*R_z
"""
_check_vector(vect)
if vect == 0:
return S(0)
vect = express(vect, frame, variables=True)
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
out = S(0)
out += diff(vectx, frame[0])
out += diff(vecty, frame[1])
out += diff(vectz, frame[2])
return out
def test_coordinate_vars():
"""Tests the coordinate variables functionality"""
A = ReferenceFrame("A")
assert CoordinateSym("Ax", A, 0) == A[0]
assert CoordinateSym("Ax", A, 1) == A[1]
assert CoordinateSym("Ax", A, 2) == A[2]
raises(ValueError, lambda: CoordinateSym("Ax", A, 3))
q = dynamicsymbols("q")
qd = dynamicsymbols("q", 1)
assert (isinstance(A[0], CoordinateSym)
and isinstance(A[0], CoordinateSym)
and isinstance(A[0], CoordinateSym))
assert A.variable_map(A) == {A[0]: A[0], A[1]: A[1], A[2]: A[2]}
assert A[0].frame == A
B = A.orientnew("B", "Axis", [q, A.z])
assert B.variable_map(A) == {
B[2]: A[2],
B[1]: -A[0] * sin(q) + A[1] * cos(q),
B[0]: A[0] * cos(q) + A[1] * sin(q),
}
assert A.variable_map(B) == {
A[0]: B[0] * cos(q) - B[1] * sin(q),
A[1]: B[0] * sin(q) + B[1] * cos(q),
A[2]: B[2],
}
assert time_derivative(B[0], A) == -A[0] * sin(q) * qd + A[1] * cos(q) * qd
assert time_derivative(B[1], A) == -A[0] * cos(q) * qd - A[1] * sin(q) * qd
assert time_derivative(B[2], A) == 0
assert express(B[0], A, variables=True) == A[0] * cos(q) + A[1] * sin(q)
assert express(B[1], A, variables=True) == -A[0] * sin(q) + A[1] * cos(q)
assert express(B[2], A, variables=True) == A[2]
assert (time_derivative(A[0] * A.x + A[1] * A.y + A[2] * A.z,
B) == A[1] * qd * A.x - A[0] * qd * A.y)
assert (time_derivative(B[0] * B.x + B[1] * B.y + B[2] * B.z,
A) == -B[1] * qd * B.x + B[0] * qd * B.y)
assert express(B[0] * B[1] * B[2], A, variables=True) == A[2] * (
-A[0] * sin(q) + A[1] * cos(q)) * (A[0] * cos(q) + A[1] * sin(q))
assert (time_derivative(B[0] * B[1] * B[2], A) -
(A[2] * (-A[0]**2 * cos(2 * q) - 2 * A[0] * A[1] * sin(2 * q) +
A[1]**2 * cos(2 * q)) * qd)).trigsimp() == 0
assert (express(B[0] * B.x + B[1] * B.y + B[2] * B.z,
A) == (B[0] * cos(q) - B[1] * sin(q)) * A.x +
(B[0] * sin(q) + B[1] * cos(q)) * A.y + B[2] * A.z)
assert (express(B[0] * B.x + B[1] * B.y + B[2] * B.z, A,
variables=True) == A[0] * A.x + A[1] * A.y + A[2] * A.z)
assert (express(A[0] * A.x + A[1] * A.y + A[2] * A.z,
B) == (A[0] * cos(q) + A[1] * sin(q)) * B.x +
(-A[0] * sin(q) + A[1] * cos(q)) * B.y + A[2] * B.z)
assert (express(A[0] * A.x + A[1] * A.y + A[2] * A.z, B,
variables=True) == B[0] * B.x + B[1] * B.y + B[2] * B.z)
N = B.orientnew("N", "Axis", [-q, B.z])
assert N.variable_map(A) == {N[0]: A[0], N[2]: A[2], N[1]: A[1]}
C = A.orientnew("C", "Axis", [q, A.x + A.y + A.z])
mapping = A.variable_map(C)
assert (mapping[A[0]] == 2 * C[0] * cos(q) / 3 + C[0] / 3 -
2 * C[1] * sin(q + pi / 6) / 3 + C[1] / 3 -
2 * C[2] * cos(q + pi / 3) / 3 + C[2] / 3)
assert (mapping[A[1]] == -2 * C[0] * cos(q + pi / 3) / 3 + C[0] / 3 +
2 * C[1] * cos(q) / 3 + C[1] / 3 - 2 * C[2] * sin(q + pi / 6) / 3 +
C[2] / 3)
assert (mapping[A[2]] == -2 * C[0] * sin(q + pi / 6) / 3 + C[0] / 3 -
2 * C[1] * cos(q + pi / 3) / 3 + C[1] / 3 + 2 * C[2] * cos(q) / 3 +
C[2] / 3)
def test_express():
assert express(Vector(0), N) == Vector(0)
assert express(S(0), N) == S(0)
assert express(A.x, C) == cos(q3) * C.x + sin(q3) * C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
assert express(A.x, N) == cos(q1) * N.x + sin(q1) * N.y
assert express(A.y, N) == -sin(q1) * N.x + cos(q1) * N.y
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == cos(q2) * B.y - sin(q2) * B.z
assert express(A.z, B) == sin(q2) * B.y + cos(q2) * B.z
assert express(A.x, C) == cos(q3) * C.x + sin(q3) * C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
# Check to make sure UnitVectors get converted properly
assert express(N.x, N) == N.x
assert express(N.y, N) == N.y
assert express(N.z, N) == N.z
assert express(N.x, A) == (cos(q1) * A.x - sin(q1) * A.y)
assert express(N.y, A) == (sin(q1) * A.x + cos(q1) * A.y)
assert express(N.z, A) == A.z
assert express(N.x, B) == (cos(q1) * B.x - sin(q1) * cos(q2) * B.y +
sin(q1) * sin(q2) * B.z)
assert express(N.y, B) == (sin(q1) * B.x + cos(q1) * cos(q2) * B.y -
sin(q2) * cos(q1) * B.z)
assert express(N.z, B) == (sin(q2) * B.y + cos(q2) * B.z)
assert express(
N.x, C) == ((cos(q1) * cos(q3) - sin(q1) * sin(q2) * sin(q3)) * C.x -
sin(q1) * cos(q2) * C.y +
(sin(q3) * cos(q1) + sin(q1) * sin(q2) * cos(q3)) * C.z)
assert express(
N.y, C) == ((sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1)) * C.x +
cos(q1) * cos(q2) * C.y +
(sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)) * C.z)
assert express(N.z, C) == (-sin(q3) * cos(q2) * C.x + sin(q2) * C.y +
cos(q2) * cos(q3) * C.z)
assert express(A.x, N) == (cos(q1) * N.x + sin(q1) * N.y)
assert express(A.y, N) == (-sin(q1) * N.x + cos(q1) * N.y)
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == (cos(q2) * B.y - sin(q2) * B.z)
assert express(A.z, B) == (sin(q2) * B.y + cos(q2) * B.z)
assert express(A.x, C) == (cos(q3) * C.x + sin(q3) * C.z)
assert express(A.y, C) == (sin(q2) * sin(q3) * C.x + cos(q2) * C.y -
sin(q2) * cos(q3) * C.z)
assert express(A.z, C) == (-sin(q3) * cos(q2) * C.x + sin(q2) * C.y +
cos(q2) * cos(q3) * C.z)
assert express(B.x, N) == (cos(q1) * N.x + sin(q1) * N.y)
assert express(B.y, N) == (-sin(q1) * cos(q2) * N.x +
cos(q1) * cos(q2) * N.y + sin(q2) * N.z)
assert express(B.z, N) == (sin(q1) * sin(q2) * N.x -
sin(q2) * cos(q1) * N.y + cos(q2) * N.z)
assert express(B.x, A) == A.x
assert express(B.y, A) == (cos(q2) * A.y + sin(q2) * A.z)
assert express(B.z, A) == (-sin(q2) * A.y + cos(q2) * A.z)
assert express(B.x, B) == B.x
assert express(B.y, B) == B.y
assert express(B.z, B) == B.z
assert express(B.x, C) == (cos(q3) * C.x + sin(q3) * C.z)
assert express(B.y, C) == C.y
assert express(B.z, C) == (-sin(q3) * C.x + cos(q3) * C.z)
assert express(
C.x, N) == ((cos(q1) * cos(q3) - sin(q1) * sin(q2) * sin(q3)) * N.x +
(sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1)) * N.y -
sin(q3) * cos(q2) * N.z)
assert express(C.y, N) == (-sin(q1) * cos(q2) * N.x +
cos(q1) * cos(q2) * N.y + sin(q2) * N.z)
assert express(
C.z, N) == ((sin(q3) * cos(q1) + sin(q1) * sin(q2) * cos(q3)) * N.x +
(sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)) * N.y +
cos(q2) * cos(q3) * N.z)
assert express(C.x, A) == (cos(q3) * A.x + sin(q2) * sin(q3) * A.y -
sin(q3) * cos(q2) * A.z)
assert express(C.y, A) == (cos(q2) * A.y + sin(q2) * A.z)
assert express(C.z, A) == (sin(q3) * A.x - sin(q2) * cos(q3) * A.y +
cos(q2) * cos(q3) * A.z)
assert express(C.x, B) == (cos(q3) * B.x - sin(q3) * B.z)
assert express(C.y, B) == B.y
assert express(C.z, B) == (sin(q3) * B.x + cos(q3) * B.z)
assert express(C.x, C) == C.x
assert express(C.y, C) == C.y
assert express(C.z, C) == C.z == (C.z)
# Check to make sure Vectors get converted back to UnitVectors
assert N.x == express((cos(q1) * A.x - sin(q1) * A.y), N)
assert N.y == express((sin(q1) * A.x + cos(q1) * A.y), N)
assert N.x == express(
(cos(q1) * B.x - sin(q1) * cos(q2) * B.y + sin(q1) * sin(q2) * B.z), N)
assert N.y == express(
(sin(q1) * B.x + cos(q1) * cos(q2) * B.y - sin(q2) * cos(q1) * B.z), N)
assert N.z == express((sin(q2) * B.y + cos(q2) * B.z), N)
"""
These don't really test our code, they instead test the auto simplification
(or lack thereof) of SymPy.
assert N.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N)
assert N.y == express((
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N)
assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), N)
"""
assert A.x == express((cos(q1) * N.x + sin(q1) * N.y), A)
assert A.y == express((-sin(q1) * N.x + cos(q1) * N.y), A)
assert A.y == express((cos(q2) * B.y - sin(q2) * B.z), A)
assert A.z == express((sin(q2) * B.y + cos(q2) * B.z), A)
assert A.x == express((cos(q3) * C.x + sin(q3) * C.z), A)
# Tripsimp messes up here too.
#print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
# sin(q2)*cos(q3)*C.z), A)
assert A.y == express(
(sin(q2) * sin(q3) * C.x + cos(q2) * C.y - sin(q2) * cos(q3) * C.z), A)
assert A.z == express(
(-sin(q3) * cos(q2) * C.x + sin(q2) * C.y + cos(q2) * cos(q3) * C.z),
A)
assert B.x == express((cos(q1) * N.x + sin(q1) * N.y), B)
assert B.y == express(
(-sin(q1) * cos(q2) * N.x + cos(q1) * cos(q2) * N.y + sin(q2) * N.z),
B)
assert B.z == express(
(sin(q1) * sin(q2) * N.x - sin(q2) * cos(q1) * N.y + cos(q2) * N.z), B)
assert B.y == express((cos(q2) * A.y + sin(q2) * A.z), B)
assert B.z == express((-sin(q2) * A.y + cos(q2) * A.z), B)
assert B.x == express((cos(q3) * C.x + sin(q3) * C.z), B)
assert B.z == express((-sin(q3) * C.x + cos(q3) * C.z), B)
"""
assert C.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z), C)
assert C.y == express((
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C)
assert C.z == express((
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z), C)
"""
assert C.x == express(
(cos(q3) * A.x + sin(q2) * sin(q3) * A.y - sin(q3) * cos(q2) * A.z), C)
assert C.y == express((cos(q2) * A.y + sin(q2) * A.z), C)
assert C.z == express(
(sin(q3) * A.x - sin(q2) * cos(q3) * A.y + cos(q2) * cos(q3) * A.z), C)
assert C.x == express((cos(q3) * B.x - sin(q3) * B.z), C)
assert C.z == express((sin(q3) * B.x + cos(q3) * B.z), C)
def scalar_potential_difference(field, frame, point1, point2, origin):
"""
Returns the scalar potential difference between two points in a
certain frame, wrt a given field.
If a scalar field is provided, its values at the two points are
considered. If a conservative vector field is provided, the values
of its scalar potential function at the two points are used.
Returns (potential at position 2) - (potential at position 1)
Parameters
==========
field : Vector/sympyfiable
The field to calculate wrt
frame : ReferenceFrame
The frame to do the calculations in
point1 : Point
The initial Point in given frame
position2 : Point
The second Point in the given frame
origin : Point
The Point to use as reference point for position vector
calculation
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import scalar_potential_difference
>>> R = ReferenceFrame('R')
>>> O = Point('O')
>>> P = O.locatenew('P', R[0]*R.x + R[1]*R.y + R[2]*R.z)
>>> vectfield = 4*R[0]*R[1]*R.x + 2*R[0]**2*R.y
>>> scalar_potential_difference(vectfield, R, O, P, O)
2*R_x**2*R_y
>>> Q = O.locatenew('O', 3*R.x + R.y + 2*R.z)
>>> scalar_potential_difference(vectfield, R, P, Q, O)
-2*R_x**2*R_y + 18
"""
_check_frame(frame)
if isinstance(field, Vector):
#Get the scalar potential function
scalar_fn = scalar_potential(field, frame)
else:
#Field is a scalar
scalar_fn = field
#Express positions in required frame
position1 = express(point1.pos_from(origin), frame, variables=True)
position2 = express(point2.pos_from(origin), frame, variables=True)
#Get the two positions as substitution dicts for coordinate variables
subs_dict1 = {}
subs_dict2 = {}
for i, x in enumerate(frame):
subs_dict1[frame[i]] = x.dot(position1)
subs_dict2[frame[i]] = x.dot(position2)
return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
def test_express():
assert express(Vector(0), N) == Vector(0)
assert express(S(0), N) == S(0)
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
assert express(A.x, N) == cos(q1)*N.x + sin(q1)*N.y
assert express(A.y, N) == -sin(q1)*N.x + cos(q1)*N.y
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == cos(q2)*B.y - sin(q2)*B.z
assert express(A.z, B) == sin(q2)*B.y + cos(q2)*B.z
assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z
assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \
sin(q2)*cos(q3)*C.z
assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \
cos(q2)*cos(q3)*C.z
# Check to make sure UnitVectors get converted properly
assert express(N.x, N) == N.x
assert express(N.y, N) == N.y
assert express(N.z, N) == N.z
assert express(N.x, A) == (cos(q1)*A.x - sin(q1)*A.y)
assert express(N.y, A) == (sin(q1)*A.x + cos(q1)*A.y)
assert express(N.z, A) == A.z
assert express(N.x, B) == (cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z)
assert express(N.y, B) == (sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z)
assert express(N.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(N.x, C) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.z)
assert express(N.y, C) == (
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z)
assert express(N.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(A.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(A.y, N) == (-sin(q1)*N.x + cos(q1)*N.y)
assert express(A.z, N) == N.z
assert express(A.x, A) == A.x
assert express(A.y, A) == A.y
assert express(A.z, A) == A.z
assert express(A.x, B) == B.x
assert express(A.y, B) == (cos(q2)*B.y - sin(q2)*B.z)
assert express(A.z, B) == (sin(q2)*B.y + cos(q2)*B.z)
assert express(A.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(A.y, C) == (sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z)
assert express(A.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z)
assert express(B.x, N) == (cos(q1)*N.x + sin(q1)*N.y)
assert express(B.y, N) == (-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(B.z, N) == (sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z)
assert express(B.x, A) == A.x
assert express(B.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(B.z, A) == (-sin(q2)*A.y + cos(q2)*A.z)
assert express(B.x, B) == B.x
assert express(B.y, B) == B.y
assert express(B.z, B) == B.z
assert express(B.x, C) == (cos(q3)*C.x + sin(q3)*C.z)
assert express(B.y, C) == C.y
assert express(B.z, C) == (-sin(q3)*C.x + cos(q3)*C.z)
assert express(C.x, N) == (
(cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z)
assert express(C.y, N) == (
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z)
assert express(C.z, N) == (
(sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z)
assert express(C.x, A) == (cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z)
assert express(C.y, A) == (cos(q2)*A.y + sin(q2)*A.z)
assert express(C.z, A) == (sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z)
assert express(C.x, B) == (cos(q3)*B.x - sin(q3)*B.z)
assert express(C.y, B) == B.y
assert express(C.z, B) == (sin(q3)*B.x + cos(q3)*B.z)
assert express(C.x, C) == C.x
assert express(C.y, C) == C.y
assert express(C.z, C) == C.z == (C.z)
# Check to make sure Vectors get converted back to UnitVectors
assert N.x == express((cos(q1)*A.x - sin(q1)*A.y), N)
assert N.y == express((sin(q1)*A.x + cos(q1)*A.y), N)
assert N.x == express((cos(q1)*B.x - sin(q1)*cos(q2)*B.y +
sin(q1)*sin(q2)*B.z), N)
assert N.y == express((sin(q1)*B.x + cos(q1)*cos(q2)*B.y -
sin(q2)*cos(q1)*B.z), N)
assert N.z == express((sin(q2)*B.y + cos(q2)*B.z), N)
"""
These don't really test our code, they instead test the auto simplification
(or lack thereof) of SymPy.
assert N.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x -
sin(q1)*cos(q2)*C.y +
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N)
assert N.y == express((
(sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x +
cos(q1)*cos(q2)*C.y +
(sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N)
assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), N)
"""
assert A.x == express((cos(q1)*N.x + sin(q1)*N.y), A)
assert A.y == express((-sin(q1)*N.x + cos(q1)*N.y), A)
assert A.y == express((cos(q2)*B.y - sin(q2)*B.z), A)
assert A.z == express((sin(q2)*B.y + cos(q2)*B.z), A)
assert A.x == express((cos(q3)*C.x + sin(q3)*C.z), A)
# Tripsimp messes up here too.
#print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
# sin(q2)*cos(q3)*C.z), A)
assert A.y == express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y -
sin(q2)*cos(q3)*C.z), A)
assert A.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y +
cos(q2)*cos(q3)*C.z), A)
assert B.x == express((cos(q1)*N.x + sin(q1)*N.y), B)
assert B.y == express((-sin(q1)*cos(q2)*N.x +
cos(q1)*cos(q2)*N.y + sin(q2)*N.z), B)
assert B.z == express((sin(q1)*sin(q2)*N.x -
sin(q2)*cos(q1)*N.y + cos(q2)*N.z), B)
assert B.y == express((cos(q2)*A.y + sin(q2)*A.z), B)
assert B.z == express((-sin(q2)*A.y + cos(q2)*A.z), B)
assert B.x == express((cos(q3)*C.x + sin(q3)*C.z), B)
assert B.z == express((-sin(q3)*C.x + cos(q3)*C.z), B)
"""
assert C.x == express((
(cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x +
(sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y -
sin(q3)*cos(q2)*N.z), C)
assert C.y == express((
-sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C)
assert C.z == express((
(sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x +
(sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y +
cos(q2)*cos(q3)*N.z), C)
"""
assert C.x == express((cos(q3)*A.x + sin(q2)*sin(q3)*A.y -
sin(q3)*cos(q2)*A.z), C)
assert C.y == express((cos(q2)*A.y + sin(q2)*A.z), C)
assert C.z == express((sin(q3)*A.x - sin(q2)*cos(q3)*A.y +
cos(q2)*cos(q3)*A.z), C)
assert C.x == express((cos(q3)*B.x - sin(q3)*B.z), C)
assert C.z == express((sin(q3)*B.x + cos(q3)*B.z), C)
def scalar_potential_difference(field, frame, point1, point2, origin):
"""
Returns the scalar potential difference between two points in a
certain frame, wrt a given field.
If a scalar field is provided, its values at the two points are
considered. If a conservative vector field is provided, the values
of its scalar potential function at the two points are used.
Returns (potential at position 2) - (potential at position 1)
Parameters
==========
field : Vector/sympyfiable
The field to calculate wrt
frame : ReferenceFrame
The frame to do the calculations in
point1 : Point
The initial Point in given frame
position2 : Point
The second Point in the given frame
origin : Point
The Point to use as reference point for position vector
calculation
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import scalar_potential_difference
>>> R = ReferenceFrame('R')
>>> O = Point('O')
>>> P = O.locatenew('P', R[0]*R.x + R[1]*R.y + R[2]*R.z)
>>> vectfield = 4*R[0]*R[1]*R.x + 2*R[0]**2*R.y
>>> scalar_potential_difference(vectfield, R, O, P, O)
2*R_x**2*R_y
>>> Q = O.locatenew('O', 3*R.x + R.y + 2*R.z)
>>> scalar_potential_difference(vectfield, R, P, Q, O)
-2*R_x**2*R_y + 18
"""
_check_frame(frame)
if isinstance(field, Vector):
#Get the scalar potential function
scalar_fn = scalar_potential(field, frame)
else:
#Field is a scalar
scalar_fn = field
#Express positions in required frame
position1 = express(point1.pos_from(origin), frame, variables=True)
position2 = express(point2.pos_from(origin), frame, variables=True)
#Get the two positions as substitution dicts for coordinate variables
subs_dict1 = {}
subs_dict2 = {}
for i, x in enumerate(frame):
subs_dict1[frame[i]] = x.dot(position1)
subs_dict2[frame[i]] = x.dot(position2)
return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
