def test_coordinate_vars():
"""Tests the coordinate variables functionality"""
assert CoordinateSym('Ax', A, 0) == A[0]
assert CoordinateSym('Ax', A, 1) == A[1]
assert CoordinateSym('Ax', A, 2) == A[2]
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_time_derivative():
#The use of time_derivative for calculations pertaining to scalar
#fields has been tested in test_coordinate_vars in test_essential.py
A = ReferenceFrame('A')
q = dynamicsymbols('q')
qd = dynamicsymbols('q', 1)
B = A.orientnew('B', 'Axis', [q, A.z])
d = A.x | A.x
assert time_derivative(d, B) == (-qd) * (A.y | A.x) + \
(-qd) * (A.x | A.y)
d1 = A.x | B.y
assert time_derivative(d1, A) == - qd*(A.x|B.x)
assert time_derivative(d1, B) == - qd*(A.y|B.y)
d2 = A.x | B.x
assert time_derivative(d2, A) == qd*(A.x|B.y)
assert time_derivative(d2, B) == - qd*(A.y|B.x)
d3 = A.x | B.z
assert time_derivative(d3, A) == 0
assert time_derivative(d3, B) == - qd*(A.y|B.z)
q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4')
q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1)
q1dd, q2dd, q3dd, q4dd = dynamicsymbols('q1 q2 q3 q4', 2)
C = B.orientnew('C', 'Axis', [q4, B.x])
v1 = q1 * A.z
v2 = q2*A.x + q3*B.y
v3 = q1*A.x + q2*A.y + q3*A.z
assert time_derivative(B.x, C) == 0
assert time_derivative(B.y, C) == - q4d*B.z
assert time_derivative(B.z, C) == q4d*B.y
assert time_derivative(v1, B) == q1d*A.z
assert time_derivative(v1, C) == - q1*sin(q)*q4d*A.x + \
q1*cos(q)*q4d*A.y + q1d*A.z
assert time_derivative(v2, A) == q2d*A.x - q3*qd*B.x + q3d*B.y
assert time_derivative(v2, C) == q2d*A.x - q2*qd*A.y + \
q2*sin(q)*q4d*A.z + q3d*B.y - q3*q4d*B.z
assert time_derivative(v3, B) == (q2*qd + q1d)*A.x + \
(-q1*qd + q2d)*A.y + q3d*A.z
assert time_derivative(d, C) == - qd*(A.y|A.x) + \
sin(q)*q4d*(A.z|A.x) - qd*(A.x|A.y) + sin(q)*q4d*(A.x|A.z)