def simple_root(self, i):
"""The ith simple root of F_4
Every lie algebra has a unique root system.
Given a root system Q, there is a subset of the
roots such that an element of Q is called a
simple root if it cannot be written as the sum
of two elements in Q. If we let D denote the
set of simple roots, then it is clear that every
element of Q can be written as a linear combination
of elements of D with all coefficients non-negative.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("F4")
>>> c.simple_root(3)
[0, 0, 0, 1]
"""
if i < 3:
return self.basic_root(i-1, i)
if i == 3:
root = [0]*4
root[3] = 1
return root
if i == 4:
root = [Rational(-1, 2)]*4
return root
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols('t r m g v')
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols('q1:7')
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols('u:6')
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols('u1:7')
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame('N') # Inertial frame
NO = Point('NO') # Inertial origin
A = N.orientnew('A', 'Axis', [q1, N.z]) # Yaw intermediate frame
B = A.orientnew('B', 'Axis', [q2, A.x]) # Lean intermediate frame
C = B.orientnew('C', 'Axis', [q3, B.y]) # Disc fixed frame
CO = NO.locatenew('CO', q4*N.x + q5*N.y + q6*N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1*B.x + u2*B.y + u3*B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4*C.x + u5*C.y + u6*C.z)
# Disc Ground Contact Point
P = CO.locatenew('P', r*B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix([dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B] +
[dot(v_co_n_qd - CO.vel(N), uv) for uv in N])
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(N, C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m*g*A.z
# Create inertia dyadic of disc C about point CO
I = (m * r**2) / 4
J = (m * r**2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody('Disc', CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(N, [q1, q2, q3, q4, q5], [u1, u2, u3], kd_eqs=kindiffs,
q_dependent=[q6], configuration_constraints=f_c,
u_dependent=[u4, u5, u6], velocity_constraints=f_v)
with warns_deprecated_sympy():
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t)
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qd:
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix([0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r*cos(q2)}
u_op = {u1: 0,
u2: sin(q2)*q1d + q3d,
u3: cos(q2)*q1d,
u4: -r*(sin(q2)*q1d + q3d)*cos(q3),
u5: 0,
u6: -r*(sin(q2)*q1d + q3d)*sin(q3)}
qd_op = {q2d: 0,
q4d: -r*(sin(q2)*q1d + q3d)*cos(q1),
q5d: -r*(sin(q2)*q1d + q3d)*sin(q1),
q6d: 0}
ud_op = {u1d: 4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5,
u2d: 0,
u3d: 0,
u4d: r*(sin(q2)*sin(q3)*q1d*q3d + sin(q3)*q3d**2),
u5d: r*(4*g*sin(q2)/(5*r) + sin(2*q2)*q1d**2/2 + 6*cos(q2)*q1d*q3d/5),
u6d: -r*(sin(q2)*cos(q3)*q1d*q3d + cos(q3)*q3d**2)}
A, B = linearizer.linearize(op_point=[q_op, u_op, qd_op, ud_op], A_and_B=True, simplify=True)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix([[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1)*q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1)*q3d, 0, 0, 0, 0, cos(q1), -sin(q1), 0],
[0, Rational(4, 5), 0, 0, 0, 0, 0, 6*q3d/5],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -2*q3d, 0, 0]])
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert A.subs(upright_nominal).subs(q3d, 1/sqrt(3)).eigenvals() == {0: 8}
def positive_roots(self):
"""
This method generates all the positive roots of
A_n. This is half of all of the roots of E_n;
by multiplying all the positive roots by -1 we
get the negative roots.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("A3")
>>> c.positive_roots()
{1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0],
5: [0, 1, 0, -1], 6: [0, 0, 1, -1]}
"""
n = self.n
if n == 6:
posroots = {}
k = 0
for i in range(n - 1):
for j in range(i + 1, n - 1):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
root = [
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(-1, 2),
Rational(-1, 2),
Rational(1, 2)
]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
if (a + b + c + d + e) % 2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
posroots[k] = root
return posroots
if n == 7:
posroots = {}
k = 0
for i in range(n - 1):
for j in range(i + 1, n - 1):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
k += 1
posroots[k] = [0, 0, 0, 0, 0, 1, 1, 0]
root = [
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(-1, 2),
Rational(-1, 2),
Rational(1, 2)
]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
for f in range(0, 2):
if (a + b + c + d + e + f) % 2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
if f == 1:
root[5] = Rational(1, 2)
posroots[k] = root
return posroots
if n == 8:
posroots = {}
k = 0
for i in range(n):
for j in range(i + 1, n):
k += 1
root = self.basic_root(i, j)
posroots[k] = root
k += 1
root = self.basic_root(i, j)
root[i] = 1
posroots[k] = root
root = [
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(-1, 2),
Rational(-1, 2),
Rational(1, 2)
]
for a in range(0, 2):
for b in range(0, 2):
for c in range(0, 2):
for d in range(0, 2):
for e in range(0, 2):
for f in range(0, 2):
for g in range(0, 2):
if (a + b + c + d + e + f +
g) % 2 == 0:
k += 1
if a == 1:
root[0] = Rational(-1, 2)
if b == 1:
root[1] = Rational(-1, 2)
if c == 1:
root[2] = Rational(-1, 2)
if d == 1:
root[3] = Rational(-1, 2)
if e == 1:
root[4] = Rational(-1, 2)
if f == 1:
root[5] = Rational(1, 2)
if g == 1:
root[6] = Rational(1, 2)
posroots[k] = root
return posroots
def positive_roots(self):
"""Generate all the positive roots of A_n
This is half of all of the roots of F_4; by multiplying all the
positive roots by -1 we get the negative roots.
Examples
========
>>> from sympy.liealgebras.cartan_type import CartanType
>>> c = CartanType("A3")
>>> c.positive_roots()
{1: [1, -1, 0, 0], 2: [1, 0, -1, 0], 3: [1, 0, 0, -1], 4: [0, 1, -1, 0],
5: [0, 1, 0, -1], 6: [0, 0, 1, -1]}
"""
n = self.n
posroots = {}
k = 0
for i in range(0, n-1):
for j in range(i+1, n):
k += 1
posroots[k] = self.basic_root(i, j)
k += 1
root = self.basic_root(i, j)
root[j] = 1
posroots[k] = root
for i in range(0, n):
k += 1
root = [0]*n
root[i] = 1
posroots[k] = root
k += 1
root = [Rational(1, 2)]*n
posroots[k] = root
for i in range(1, 4):
k += 1
root = [Rational(1, 2)]*n
root[i] = Rational(-1, 2)
posroots[k] = root
posroots[k+1] = [Rational(1, 2), Rational(1, 2), Rational(-1, 2), Rational(-1, 2)]
posroots[k+2] = [Rational(1, 2), Rational(-1, 2), Rational(1, 2), Rational(-1, 2)]
posroots[k+3] = [Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(1, 2)]
posroots[k+4] = [Rational(1, 2), Rational(-1, 2), Rational(-1, 2), Rational(-1, 2)]
return posroots
def matrix_form(self, weylelt):
"""
This method takes input from the user in the form of products of the
generating reflections, and returns the matrix corresponding to the
element of the Weyl group. Since each element of the Weyl group is
a reflection of some type, there is a corresponding matrix representation.
This method uses the standard representation for all the generating
reflections.
Examples
========
>>> from sympy.liealgebras.weyl_group import WeylGroup
>>> f = WeylGroup("F4")
>>> f.matrix_form('r2*r3')
Matrix([
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, -1],
[0, 0, 1, 0]])
"""
elts = list(weylelt)
reflections = elts[1::3]
n = self.cartan_type.rank()
if self.cartan_type.series == "A":
matrixform = eye(n + 1)
for elt in reflections:
a = int(elt)
mat = eye(n + 1)
mat[a - 1, a - 1] = 0
mat[a - 1, a] = 1
mat[a, a - 1] = 1
mat[a, a] = 0
matrixform *= mat
return matrixform
if self.cartan_type.series == "D":
matrixform = eye(n)
for elt in reflections:
a = int(elt)
mat = eye(n)
if a < n:
mat[a - 1, a - 1] = 0
mat[a - 1, a] = 1
mat[a, a - 1] = 1
mat[a, a] = 0
matrixform *= mat
else:
mat[n - 2, n - 1] = -1
mat[n - 2, n - 2] = 0
mat[n - 1, n - 2] = -1
mat[n - 1, n - 1] = 0
matrixform *= mat
return matrixform
if self.cartan_type.series == "G":
matrixform = eye(3)
for elt in reflections:
a = int(elt)
if a == 1:
gen1 = Matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])
matrixform *= gen1
else:
gen2 = Matrix(
[
[Rational(2, 3), Rational(2, 3), Rational(-1, 3)],
[Rational(2, 3), Rational(-1, 3), Rational(2, 3)],
[Rational(-1, 3), Rational(2, 3), Rational(2, 3)],
]
)
matrixform *= gen2
return matrixform
if self.cartan_type.series == "F":
matrixform = eye(4)
for elt in reflections:
a = int(elt)
if a == 1:
mat = Matrix(
[[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]
)
matrixform *= mat
elif a == 2:
mat = Matrix(
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]
)
matrixform *= mat
elif a == 3:
mat = Matrix(
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1]]
)
matrixform *= mat
else:
mat = Matrix(
[
[
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
Rational(1, 2),
],
[
Rational(1, 2),
Rational(1, 2),
Rational(-1, 2),
Rational(-1, 2),
],
[
Rational(1, 2),
Rational(-1, 2),
Rational(1, 2),
Rational(-1, 2),
],
[
Rational(1, 2),
Rational(-1, 2),
Rational(-1, 2),
Rational(1, 2),
],
]
)
matrixform *= mat
return matrixform
if self.cartan_type.series == "E":
matrixform = eye(8)
for elt in reflections:
a = int(elt)
if a == 1:
mat = Matrix(
[
[
Rational(3, 4),
Rational(1, 4),
Rational(1, 4),
Rational(1, 4),
Rational(1, 4),
Rational(1, 4),
Rational(1, 4),
Rational(-1, 4),
],
[
Rational(1, 4),
Rational(3, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(1, 4),
Rational(-1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(3, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(3, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(3, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(3, 4),
Rational(-1, 4),
Rational(1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-3, 4),
Rational(1, 4),
],
[
Rational(1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(-1, 4),
Rational(3, 4),
],
]
)
matrixform *= mat
elif a == 2:
mat = eye(8)
mat[0, 0] = 0
mat[0, 1] = -1
mat[1, 0] = -1
mat[1, 1] = 0
matrixform *= mat
else:
mat = eye(8)
mat[a - 3, a - 3] = 0
mat[a - 3, a - 2] = 1
mat[a - 2, a - 3] = 1
mat[a - 2, a - 2] = 0
matrixform *= mat
return matrixform
if self.cartan_type.series == "B" or self.cartan_type.series == "C":
matrixform = eye(n)
for elt in reflections:
a = int(elt)
mat = eye(n)
if a == 1:
mat[0, 0] = -1
matrixform *= mat
else:
mat[a - 2, a - 2] = 0
mat[a - 2, a - 1] = 1
mat[a - 1, a - 2] = 1
mat[a - 1, a - 1] = 0
matrixform *= mat
return matrixform