def get_leg_position(front=True, left=True):
anchor = Translate(body_width / 2, body_width / 2, 0)
if not left:
anchor = ReflectX() @ anchor
if not front:
anchor = ReflectY() @ anchor
atoms = sp.symbols(r'w_b, w_r, w_h, u_b, u_r, u_h')
rot_from_pivot, rot_y_from_z, rot_d_from_z, u_a, u_r, u_h = atoms
chain = get_active_kinematic_chain(rot_from_pivot, rot_y_from_z,
rot_d_from_z)
Leg_Bottom = anchor @ chain[-1] @ origin
foot_pos = []
f_a = get_body_linkage_constraint(rot_from_pivot, u_a)
f_t = get_leg_top_constraint(rot_y_from_z, u_r)
f_b = get_leg_bottom_constraint(rot_d_from_z, u_h)
for i in range(0, 3):
foot_pos.append(
filter_tiny_numbers(sp.expand_trig(sp.simplify(Leg_Bottom[i]))))
constraints = [
2 * sp.expand_trig(sp.simplify(sp.expand(f))) for f in (f_a, f_t, f_b)
]
return atoms, foot_pos, constraints
def _get_Ylm(self, l, m):
"""
Compute an expression for spherical harmonic of order (l,m)
in terms of Cartesian unit vectors, :math:`\hat{z}`
and :math:`\hat{x} + i \hat{y}`
Parameters
----------
l : int
the degree of the harmonic
m : int
the order of the harmonic; |m| < l
Returns
-------
expr :
a sympy expression that corresponds to the
requested Ylm
References
----------
https://en.wikipedia.org/wiki/Spherical_harmonics
"""
import sympy as sp
# the relevant cartesian and spherical symbols
x, y, z, r = sp.symbols('x y z r', real=True, positive=True)
xhat, yhat, zhat = sp.symbols('xhat yhat zhat',
real=True,
positive=True)
xpyhat = sp.Symbol('xpyhat', complex=True)
phi, theta = sp.symbols('phi theta')
defs = [(sp.sin(phi), y / sp.sqrt(x**2 + y**2)),
(sp.cos(phi), x / sp.sqrt(x**2 + y**2)),
(sp.cos(theta), z / sp.sqrt(x**2 + y**2 + z**2))]
# the cos(theta) dependence encoded by the associated Legendre poly
expr = sp.assoc_legendre(l, m, sp.cos(theta))
# the exp(i*m*phi) dependence
expr *= sp.expand_trig(sp.cos(
m * phi)) + sp.I * sp.expand_trig(sp.sin(m * phi))
# simplifying optimizations
expr = sp.together(expr.subs(defs)).subs(x**2 + y**2 + z**2, r**2)
expr = expr.expand().subs([(x / r, xhat), (y / r, yhat),
(z / r, zhat)])
expr = expr.factor().factor(extension=[sp.I]).subs(
xhat + sp.I * yhat, xpyhat)
expr = expr.subs(xhat**2 + yhat**2, 1 - zhat**2).factor()
# and finally add the normalization
amp = sp.sqrt((2 * l + 1) / (4 * numpy.pi) * sp.factorial(l - m) /
sp.factorial(l + m))
expr *= amp
return expr
def substitute_trig(l1x,l1y,l2x,l2y,l1,l2):
phi1 = Symbol('phi1')
phi2 = Symbol('phi2')
cos2t12 = sympy.cos(2*(phi1-phi2))
sin2t12 = sympy.sin(2*(phi1-phi2))
simpcos = sympy.expand_trig(cos2t12)
simpsin = sympy.expand_trig(sin2t12)
cos2t12 = sympy.expand(sympy.simplify(simpcos.subs([(sympy.cos(phi1),l1x/l1),(sympy.cos(phi2),l2x/l2),
(sympy.sin(phi1),l1y/l1),(sympy.sin(phi2),l2y/l2)])))
sin2t12 = sympy.expand(sympy.simplify(simpsin.subs([(sympy.cos(phi1),l1x/l1),(sympy.cos(phi2),l2x/l2),
(sympy.sin(phi1),l1y/l1),(sympy.sin(phi2),l2y/l2)])))
return cos2t12,sin2t12
def _get_Ylm(self, l, m):
"""
Compute an expression for spherical harmonic of order (l,m)
in terms of Cartesian unit vectors, :math:`\hat{z}`
and :math:`\hat{x} + i \hat{y}`
Parameters
----------
l : int
the degree of the harmonic
m : int
the order of the harmonic; |m| < l
Returns
-------
expr :
a sympy expression that corresponds to the
requested Ylm
References
----------
https://en.wikipedia.org/wiki/Spherical_harmonics
"""
import sympy as sp
# the relevant cartesian and spherical symbols
x, y, z, r = sp.symbols('x y z r', real=True, positive=True)
xhat, yhat, zhat = sp.symbols('xhat yhat zhat', real=True, positive=True)
xpyhat = sp.Symbol('xpyhat', complex=True)
phi, theta = sp.symbols('phi theta')
defs = [(sp.sin(phi), y/sp.sqrt(x**2+y**2)),
(sp.cos(phi), x/sp.sqrt(x**2+y**2)),
(sp.cos(theta), z/sp.sqrt(x**2 + y**2 + z**2))
]
# the cos(theta) dependence encoded by the associated Legendre poly
expr = sp.assoc_legendre(l, m, sp.cos(theta))
# the exp(i*m*phi) dependence
expr *= sp.expand_trig(sp.cos(m*phi)) + sp.I*sp.expand_trig(sp.sin(m*phi))
# simplifying optimizations
expr = sp.together(expr.subs(defs)).subs(x**2 + y**2 + z**2, r**2)
expr = expr.expand().subs([(x/r, xhat), (y/r, yhat), (z/r, zhat)])
expr = expr.factor().factor(extension=[sp.I]).subs(xhat+sp.I*yhat, xpyhat)
expr = expr.subs(xhat**2 + yhat**2, 1-zhat**2).factor()
# and finally add the normalization
amp = sp.sqrt((2*l+1) / (4*numpy.pi) * sp.factorial(l-m) / sp.factorial(l+m))
expr *= amp
return expr
def challenge16_9():
a, b, c = symbols('a b c')
expr1 = (b * ((2 * a * cos(theta)) - b))
expr2 = ((a - c) * (c + a))
expr = (Eq(expr1, expr2))
print(solve(expr1))
print(expand_trig(expr1))
print(solve(expr2))
print(expand_trig(expr2))
print(solve(expr))
print(expand_trig(expr))
def form_to_fform(form, M=16):
"""Convert instance of `FormContainer` to instance of `fFormContainer`
Parameters
----------
form : FormContainer
Instance of :class:`~shgpy.core.data_handler.FormContainer`
M : int, optional
Number of Fourier frequencies, defaults to `16`.
Returns
-------
fform : fFormContainer
Instance of :class:`~shgpy.core.data_handler.fFormContainer`
Notes
-----
This function operates by taking an instance of the `FormContainer`
class and performing a Fourier transform.
"""
iterable = {}
for k,v in form.get_items():
iterable[k] = np.zeros((2*M+1,), dtype=object)
_logger.info(f'Currently computing {k}.')
for m in np.arange(-M, M+1):
_logger.debug(f'Currently computing m={m}.')
expr = sp.expand_trig(v*(sp.cos(-m*S.phi)+1j*sp.sin(-m*S.phi))).expand()
expr_re = _no_I_component(expr)
expr_im = _I_component(expr)
iterable[k][n2i(m, M)] = 1/2/sp.pi*sp.integrate(expr_re, (S.phi, 0, 2*sp.pi)) + 1/2/sp.pi*sp.I*sp.integrate(expr_im, (S.phi, 0, 2*sp.pi))
return fFormContainer(iterable, M=M)
def dddd(self, mu: int, nu: int, rho: int, sigma: int):
"""
Calculates the component of the Riemann tensor R_mu,nu,rho,sigma.
Parameters
----------
mu : int
The first lower index
nu : int
The second lower index
rho : int
The third lower index
sigma : int
The fourth lower index
Returns
-------
sympy expression
the expression representing the specified component of the Riemann tensor
"""
if self.dddd_evaluated[mu, nu, rho, sigma]:
return self.dddd_values[mu, nu, rho, sigma]
R = 0
for lam in range(self.g.dim()):
R += self.g.dd(mu, lam) * self.uddd(lam, nu, rho, sigma)
self.dddd_values[mu, nu, rho,
sigma] = sym.simplify(sym.expand_trig(R.simplify()))
self.dddd_evaluated[mu, nu, rho, sigma] = True
return self.dddd_values[mu, nu, rho, sigma]
def test_csch():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert csch(nan) is nan
assert csch(zoo) is nan
assert csch(oo) == 0
assert csch(-oo) == 0
assert csch(0) is zoo
assert csch(-1) == -csch(1)
assert csch(-x) == -csch(x)
assert csch(-pi) == -csch(pi)
assert csch(-2**1024 * E) == -csch(2**1024 * E)
assert csch(pi * I) is zoo
assert csch(-pi * I) is zoo
assert csch(2 * pi * I) is zoo
assert csch(-2 * pi * I) is zoo
assert csch(-3 * 10**73 * pi * I) is zoo
assert csch(7 * 10**103 * pi * I) is zoo
assert csch(pi * I / 2) == -I
assert csch(-pi * I / 2) == I
assert csch(pi * I * Rational(5, 2)) == -I
assert csch(pi * I * Rational(7, 2)) == I
assert csch(pi * I / 3) == -2 / sqrt(3) * I
assert csch(pi * I * Rational(-2, 3)) == 2 / sqrt(3) * I
assert csch(pi * I / 4) == -sqrt(2) * I
assert csch(-pi * I / 4) == sqrt(2) * I
assert csch(pi * I * Rational(7, 4)) == sqrt(2) * I
assert csch(pi * I * Rational(-3, 4)) == sqrt(2) * I
assert csch(pi * I / 6) == -2 * I
assert csch(-pi * I / 6) == 2 * I
assert csch(pi * I * Rational(7, 6)) == 2 * I
assert csch(pi * I * Rational(-7, 6)) == -2 * I
assert csch(pi * I * Rational(-5, 6)) == 2 * I
assert csch(pi * I / 105) == -1 / sin(pi / 105) * I
assert csch(-pi * I / 105) == 1 / sin(pi / 105) * I
assert csch(x * I) == -1 / sin(x) * I
assert csch(k * pi * I) is zoo
assert csch(17 * k * pi * I) is zoo
assert csch(k * pi * I / 2) == -1 / sin(k * pi / 2) * I
assert csch(n).is_real is True
assert expand_trig(csch(x +
y)) == 1 / (sinh(x) * cosh(y) + cosh(x) * sinh(y))
def is_equal(self, eq1, eq2):
"""Compare answers"""
#answer=eq1, solution=eq2
equation_types = [
Equality, Unequality, StrictLessThan, LessThan, StrictGreaterThan,
GreaterThan
]
#Symbolic equality/Perfect match
if self._comparison_type == "perfect_match":
return eq1 == eq2
eq1 = factor(
simplify(eq1)
) #simplify is mandatory to counter expand_trig and expand_log weaknesses
eq2 = factor(simplify(eq2))
#Trigonometric simplifications
if self._use_trigo:
eq1 = expand_trig(eq1)
eq2 = expand_trig(eq2)
#Logarithmic simplifications
if self._use_log:
if self._use_complex:
eq1 = expand_log(eq1)
eq2 = expand_log(eq2)
else:
eq1 = expand_log(eq1, force=True)
eq2 = expand_log(eq2, force=True)
if self._tolerance:
eq1 = eq1.subs([(E, math.e), (pi, math.pi)])
eq2 = eq2.subs([(E, math.e), (pi, math.pi)])
#Numbers
if (isinstance(eq1, Number)
and isinstance(eq2, Number)) or self._tolerance:
return round(float(abs(N(eq1 - eq2))), 10) <= round(
float(self._tolerance), 10) if self._tolerance else abs(
N(eq1 - eq2)) == 0
#Numerical Evaluation
if not type(eq1) == type(eq2):
return N(eq1) == N(eq2)
#Equality and inequalities
if type(eq1) in equation_types:
return eq1 == eq2 or simplify(eq1) == simplify(eq2)
#Direct match
if eq1 == eq2 or simplify(eq1) == simplify(eq2):
return True
#Uncaught
return abs(N(eq1 - eq2)) == 0
def expands_trig(expression):
'''
Expand a trig function
Syntax:
sympy.expand_trig(expression)
'''
return expand_trig(expression)
def sin(x):
if isinstance(x, sp.core.basic.Basic):
s = sp.sin(x)
s = sp.expand_trig(s)
s = sp.simplify(s)
s = sp.trigsimp(s)
return s
else:
return np.sin(x)
def cos(x):
if isinstance(x, sp.core.basic.Basic):
c = sp.cos(x)
c = sp.expand_trig(c)
c = sp.simplify(c)
c = sp.trigsimp(c)
return c
else:
return np.cos(x)
def test_sech():
x, y = symbols('x, y')
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) is nan
assert sech(zoo) is nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi * I) == sec(pi)
assert sech(-pi * I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi * I / 2) is zoo
assert sech(-pi * I / 2) is zoo
assert sech((-3 * 10**73 + 1) * pi * I / 2) is zoo
assert sech((7 * 10**103 + 1) * pi * I / 2) is zoo
assert sech(pi * I) == -1
assert sech(-pi * I) == -1
assert sech(5 * pi * I) == -1
assert sech(8 * pi * I) == 1
assert sech(pi * I / 3) == 2
assert sech(pi * I * Rational(-2, 3)) == -2
assert sech(pi * I / 4) == sqrt(2)
assert sech(-pi * I / 4) == sqrt(2)
assert sech(pi * I * Rational(5, 4)) == -sqrt(2)
assert sech(pi * I * Rational(-5, 4)) == -sqrt(2)
assert sech(pi * I / 6) == 2 / sqrt(3)
assert sech(-pi * I / 6) == 2 / sqrt(3)
assert sech(pi * I * Rational(7, 6)) == -2 / sqrt(3)
assert sech(pi * I * Rational(-5, 6)) == -2 / sqrt(3)
assert sech(pi * I / 105) == 1 / cos(pi / 105)
assert sech(-pi * I / 105) == 1 / cos(pi / 105)
assert sech(x * I) == 1 / cos(x)
assert sech(k * pi * I) == 1 / cos(k * pi)
assert sech(17 * k * pi * I) == 1 / cos(17 * k * pi)
assert sech(n).is_real is True
assert expand_trig(sech(x +
y)) == 1 / (cosh(x) * cosh(y) + sinh(x) * sinh(y))
def __init__(self,
shape,
wcs,
theory=None,
theory_norm=None,
lensed_cls=None):
ModeCoupling.__init__(self, shape, wcs)
self.Lx, self.Ly, self.L = get_Ls()
self.Ldl1 = (self.Lx * self.l1x + self.Ly * self.l1y)
self.Ldl2 = (self.Lx * self.l2x + self.Ly * self.l2y)
self.l1dl2 = (self.l1x * self.l2x + self.l2x * self.l2y)
phi1 = Symbol('phi1')
phi2 = Symbol('phi2')
cos2t12 = sympy.cos(2 * (phi1 - phi2))
sin2t12 = sympy.sin(2 * (phi1 - phi2))
simpcos = sympy.expand_trig(cos2t12)
simpsin = sympy.expand_trig(sin2t12)
self.cos2t12 = sympy.expand(
sympy.simplify(
simpcos.subs([(sympy.cos(phi1), self.l1x / self.l1),
(sympy.cos(phi2), self.l2x / self.l2),
(sympy.sin(phi1), self.l1y / self.l1),
(sympy.sin(phi2), self.l2y / self.l2)])))
self.sin2t12 = sympy.expand(
sympy.simplify(
simpsin.subs([(sympy.cos(phi1), self.l1x / self.l1),
(sympy.cos(phi2), self.l2x / self.l2),
(sympy.sin(phi1), self.l1y / self.l1),
(sympy.sin(phi2), self.l2y / self.l2)])))
self.theory2d = None
if theory is not None:
self.theory2d = self._load_theory(theory, lensed_cls=lensed_cls)
if theory_norm is None:
self.theory2d_norm = self.theory2d
else:
self.theory2d_norm = self._load_theory(theory_norm,
lensed_cls=lensed_cls)
self.Als = {}
def secular_DF(e, e1, w, w1, order):
"""
Return the secular component of the disturbing function for coplanet planets up to
the specified order in the planets' eccentricities for an inner plaent with eccentricity
and longitude of periapse (e,w) and outer planet eccentricity/longitude of periape (e1,w1)
"""
s = 0
dw = w1 - w
# Introduce `eps' as order parameter multiplying eccentricities.
# 'eps' is set to 1 at teh end of calculation.
eps = S('epsilon')
maxM = int(np.floor(order / 2.))
for i in range(maxM + 1):
term = secular_DF_harmonic_term(eps * e, eps * e1, dw, i, order)
term = term.series(eps, 0, order + 1).removeO()
s = s + expand_trig(term)
return s.subs(eps, 1)
def fromTransformation(symbols, to_base_point):
"""
Creates a new metric from the given to_base_point transformation.
Parameters
----------
symbols : numpy array or array_like of sympy.core.symbol.Symbol
The symbols representing the different dimensions, e.g. `sympy.symbols('r theta z', real=True)`
to_base_point : numpy array or array_like
The transformation equation to get from the current coordinate system into the base coordinate system.
Should be expressed using the `symbols` and sympy functions for, e.g., the cosine.
Returns
-------
Metric
the (analytical) metric as derived from the transformation for a point into the base coordinate system
"""
if to_base_point is None:
raise Exception(
"You need to specify the point-to-base-transformation!")
print("Reconstructing metric from point-to-base-transformation...")
dim = len(to_base_point)
partial_derivatives = [[0 for i in range(dim)] for i in range(dim)]
for i in range(dim):
for j in range(dim):
partial_derivatives[i][j] = sym.diff(to_base_point[j],
symbols[i])
g = [[0 for i in range(dim)] for i in range(dim)]
for i in range(dim):
for j in range(dim):
for k in range(dim):
g[i][j] += partial_derivatives[i][k] * partial_derivatives[
j][k]
g[j][i] = g[i][j] = sym.simplify(
sym.expand_trig(g[i][j].simplify())) # catch some zeros
print("Metric successfully reconstructed!")
return Metric(symbols, sym.Matrix(g), to_base_point)
def quantization_form(expr, initial_expand=True):
"""Brings a sympy expression on a form that the quantizers can understand.
Parameters
----------
expr : sympy expression
The expression. Typically a symbolic potential or kinetic term
Returns
-------
sympy expr
The rewritten expression.
"""
if initial_expand:
expr = sp.expand(expr)
if expr.func == sp.Add:
expr = sp.Add(*[
quantization_form(term, initial_expand=False) for term in expr.args
])
if expr.func == sp.Mul:
expr = sp.Mul(*[
quantization_form(factor, initial_expand=False)
for factor in expr.args
])
if (expr.func == sp.cos) or (expr.func == sp.sin):
if expr.args[0].func == sp.Add:
arg_terms = expr.args[0].args
arg_terms_dummy = sp.symbols('arg_dummy0:' + str(len(arg_terms)))
# The reason for these subs shenanigans are to ensure that we don't mess with the coefficients of the arguments, so cos(2x)->cos(2x) and not cos(2x)->2*cos(x)**2-1
expr = expr.subs([
(arg, arg_dummy)
for arg, arg_dummy in zip(arg_terms, arg_terms_dummy)
])
expr = sp.expand_trig(expr)
expr = expr.subs([
(arg_dummy, arg)
for arg_dummy, arg in zip(arg_terms_dummy, arg_terms)
])
if initial_expand: # Used to cancel out terms
expr = sp.expand(expr)
return expr
def function_from_partials(df_dq, q, zero_constants=False):
"""Returns a function given a list of partial derivatives of the
function and a list of variables of which the partial derivative is
given. For a function f(q1, ..., qn):
'df_dq' is the list [∂ℱ/∂q1, ..., ∂ℱ/∂qn]
'q' is the list [q1, ..., qn]
'zero_constants' is True if zero should be used for integration constants.
Symbols C, α1, ..., αn are used for integration constants.
"""
alpha = symbols('α1:{0}'.format(len(q) + 1))
f, zeta = symbols('C ζ')
q_alpha = zip(q, alpha)
for i, df_dqr in enumerate(df_dq):
if hasattr(df_dqr, 'subs'):
integrand = df_dqr.subs(dict(q_alpha[i + 1:])).subs(q[i], zeta)
else:
integrand = df_dqr
f += integrate(expand_trig(integrand), (zeta, alpha[i], q[i]))
if zero_constants:
f = f.subs(dict(zip([symbols('C')] + list(alpha), [0] * (len(q) + 1))))
return f
def angles_to_complex(equations, angles, control_angles):
x = {theta: sp.symbols(f'z_{i}') for i, theta in enumerate(angles)}
u = {theta: sp.symbols(f'w_{i}') for i, theta in enumerate(control_angles)}
substitutions = []
X = list(x.values()) + list(u.values())
for i, x_i in enumerate(x):
z = x[x_i]
substitutions += [(sp.sin(x_i), (z - 1 / z) / 2j),
(sp.cos(x_i), (z + 1 / z) / 2)]
for i, u_i in enumerate(u):
z = u[u_i]
substitutions += [(sp.sin(u_i), (z - 1 / z) / 2j),
(sp.cos(u_i), (z + 1 / z) / 2)]
out_equations = [
sp.expand_trig(equation).subs(substitutions) for equation in equations
]
return X, out_equations, substitutions, (x, u)
def function_from_partials(df_dq, q, zero_constants=False):
"""Returns a function given a list of partial derivatives of the
function and a list of variables of which the partial derivative is
given. For a function f(q1, ..., qn):
'df_dq' is the list [∂ℱ/∂q1, ..., ∂ℱ/∂qn]
'q' is the list [q1, ..., qn]
'zero_constants' is True if zero should be used for integration constants.
Symbols C, α1, ..., αn are used for integration constants.
"""
alpha = symbols('α1:{0}'.format(len(q) + 1))
f, zeta = symbols('C ζ')
q_alpha = zip(q, alpha)
for i, df_dqr in enumerate(df_dq):
if hasattr(df_dqr, 'subs'):
integrand = df_dqr.subs(dict(q_alpha[i + 1:])).subs(q[i], zeta)
else:
integrand = df_dqr
f += integrate(expand_trig(integrand), (zeta, alpha[i], q[i]))
if zero_constants:
f = f.subs(dict(zip([symbols('C')] + list(alpha),
[0] * (len(q) + 1))))
return f
v = p.vel(N).express(N).subs(kde_map).subs(eq_gen_speed_map)
partials[p] = dict(zip(u, map(lambda x: v.diff(x, N), u)))
u1d, u2d, u3d = ud = [x.diff(t) for x in u]
for k, v in vc_map.items():
vc_map[k.diff(t)] = v.diff(t).subs(kde_map).subs(vc_map)
# generalized active/inertia forces
Fr, _ = generalized_active_forces(partials, forces + torques)
Fr_star, _ = generalized_inertia_forces(partials, bodies, kde_map)
print('Fr')
for i, fr in enumerate(Fr, 1):
print('{0}: {1}'.format(i, msprint(fr)))
print('Fr_star')
for i, fr in enumerate(Fr_star, 1):
fr = trigsimp(expand(expand_trig(fr)), deep=True, recursive=True)
print('{0}: {1}'.format(i, msprint(fr)))
# The dynamical equations would be of the form Fr = Fr* (r = 1, 2, 3).
Fr_expected = [
alpha2 + a * (R1 * cos(q1) - R3 * sin(q1)),
b * (R1 * cos(q2) + R3 * sin(q2)), Q3 - R3
]
Fr_star_expected = [
-mA * (a_star**2 + kA**2) * u1d,
-mB * ((b_star**2 + kB**2) * u2d - b_star * sin(q2) * u3d),
-1 * ((mB + mC) * u3d - mB * b_star * sin(q2) * u2d +
mB * b_star * cos(q2) * u2**2)
]
for x, y in zip(Fr, Fr_expected):
assert expand(x - y) == 0
def get_real_Ylm(l, m):
"""
Return a function that computes the real spherical
harmonic of order (l,m)
Parameters
----------
l : int
the degree of the harmonic
m : int
the order of the harmonic; abs(m) <= l
Returns
-------
Ylm : callable
a function that takes 4 arguments: (xhat, yhat, zhat)
unit-normalized Cartesian coordinates and returns the
specified Ylm
References
----------
https://en.wikipedia.org/wiki/Spherical_harmonics#Real_form
"""
import sympy as sp
# make sure l,m are integers
l = int(l)
m = int(m)
# the relevant cartesian and spherical symbols
x, y, z, r = sp.symbols('x y z r', real=True, positive=True)
xhat, yhat, zhat = sp.symbols('xhat yhat zhat', real=True, positive=True)
phi, theta = sp.symbols('phi theta')
defs = [(sp.sin(phi), y / sp.sqrt(x**2 + y**2)),
(sp.cos(phi), x / sp.sqrt(x**2 + y**2)),
(sp.cos(theta), z / sp.sqrt(x**2 + y**2 + z**2))]
# the normalization factors
if m == 0:
amp = sp.sqrt((2 * l + 1) / (4 * numpy.pi))
else:
amp = sp.sqrt(2 * (2 * l + 1) / (4 * numpy.pi) *
sp.factorial(l - abs(m)) / sp.factorial(l + abs(m)))
# the cos(theta) dependence encoded by the associated Legendre poly
expr = (-1)**m * sp.assoc_legendre(l, abs(m), sp.cos(theta))
# the phi dependence
if m < 0:
expr *= sp.expand_trig(sp.sin(abs(m) * phi))
elif m > 0:
expr *= sp.expand_trig(sp.cos(m * phi))
# simplify
expr = sp.together(expr.subs(defs)).subs(x**2 + y**2 + z**2, r**2)
expr = amp * expr.expand().subs([(x / r, xhat), (y / r, yhat),
(z / r, zhat)])
Ylm = sp.lambdify((xhat, yhat, zhat), expr, 'numexpr')
# attach some meta-data
Ylm.expr = expr
Ylm.l = l
Ylm.m = m
return Ylm
def simp(x): return simplify(expand_trig(expand_complex(expand(x)))) def sinc(x): return sin(pi*x)/(pi*x)
def prep_geodesic_eqns(self, parameters: Dict = None):
r"""
Define geodesic equations.
Args:
parameters:
dictionary of model parameter values to be used for
equation substitutions
Attributes:
gstar_ij_tanbeta_mat (`Matrix`_):
:math:`\dots`
g_ij_tanbeta_mat (`Matrix`_):
:math:`\dots`
tanbeta_poly_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\dots` where :math:`a := \tan\alpha`
tanbeta_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\tan{\left(\beta \right)} = \dots` where
:math:`a := \tan\alpha`
gstar_ij_tanalpha_mat (`Matrix`_):
a symmetric tensor with components (using shorthand
:math:`a := \tan\alpha`)
:math:`g^*[1,1] = \dots`
:math:`g^*[1,2] = g^*[2,1] = \dots`
:math:`g^*[2,2] = \dots`
gstar_ij_mat (`Matrix`_):
a symmetric tensor with components
(using shorthand :math:`a := \tan\alpha`, and with a
particular choice of model parameters)
:math:`g^*[1,1] = \dots`
:math:`g^*[1,2] = g^*[2,1] = \dots`
:math:`g^*[2,2] = \dots`
g_ij_tanalpha_mat (`Matrix`_):
a symmetric tensor with components (using shorthand
:math:`a := \tan\alpha`)
:math:`g[1,1] = \dots`
:math:`g[1,2] = g[2,1] = \dots`
:math:`g[2,2] = \dots`
g_ij_mat (`Matrix`_):
a symmetric tensor with components
(using shorthand :math:`a := \tan\alpha`, and with a
particular choice of model parameters)
:math:`g[1,1] =\dots`
:math:`g[1,2] = g[2,1] = \dots`
:math:`g[2,2] = \dots`
g_ij_mat_lambdified (function) :
lambdified version of `g_ij_mat`
gstar_ij_mat_lambdified (function) :
lambdified version of `gstar_ij_mat`
"""
logging.info("gme.core.geodesic.prep_geodesic_eqns")
self.gstar_ij_tanbeta_mat = None
self.g_ij_tanbeta_mat = None
self.tanbeta_poly_eqn = None
self.tanbeta_eqn = None
self.gstar_ij_tanalpha_mat = None
self.gstar_ij_mat = None
self.g_ij_tanalpha_mat = None
self.g_ij_mat = None
self.g_ij_mat_lambdified = None
self.gstar_ij_mat_lambdified = None
mu_eta_sub = {mu: self.mu_, eta: self.eta_}
# if parameters is None: return
H_ = self.H_eqn.rhs.subs(mu_eta_sub)
# Assume indexing here ranges in [1,2]
def p_i_lambda(i):
return [px, pz][i - 1]
# r_i_lambda = lambda i: [rx, rz][i-1]
# rdot_i_lambda = lambda i: [rdotx, rdotz][i-1]
def gstar_ij_lambda(i, j):
return simplify(
Rational(2, 2) * diff(diff(H_, p_i_lambda(i)), p_i_lambda(j)))
gstar_ij_mat = Matrix([
[gstar_ij_lambda(1, 1),
gstar_ij_lambda(2, 1)],
[gstar_ij_lambda(1, 2),
gstar_ij_lambda(2, 2)],
])
gstar_ij_pxpz_mat = gstar_ij_mat.subs({varphi_r(rvec): varphi})
g_ij_pxpz_mat = gstar_ij_mat.inv().subs({varphi_r(rvec): varphi})
cosbeta_eqn = Eq(cos(beta), 1 / sqrt(1 + tan(beta)**2))
sinbeta_eqn = Eq(sin(beta), sqrt(1 - 1 / (1 + tan(beta)**2)))
sintwobeta_eqn = Eq(sin(2 * beta), cos(beta)**2 - sin(beta)**2)
self.gstar_ij_tanbeta_mat = expand_trig(
simplify(gstar_ij_pxpz_mat.subs(e2d(
self.px_pz_tanbeta_eqn)))).subs(e2d(cosbeta_eqn))
self.g_ij_tanbeta_mat = expand_trig(
simplify(g_ij_pxpz_mat.subs(e2d(self.px_pz_tanbeta_eqn)))).subs(
e2d(cosbeta_eqn))
tanalpha_beta_eqn = self.tanalpha_beta_eqn.subs(mu_eta_sub)
tanbeta_poly_eqn = Eq(
numer(tanalpha_beta_eqn.rhs) -
tanalpha_beta_eqn.lhs * denom(tanalpha_beta_eqn.rhs),
0,
).subs({tan(alpha): ta})
tanbeta_eqn = Eq(tan(beta), solve(tanbeta_poly_eqn, tan(beta))[0])
self.tanbeta_poly_eqn = tanbeta_poly_eqn
self.tanbeta_eqn = tanbeta_eqn
# Replace all refs to beta with refs to alpha
self.gstar_ij_tanalpha_mat = (self.gstar_ij_tanbeta_mat.subs(
e2d(sintwobeta_eqn)).subs(e2d(sinbeta_eqn)).subs(
e2d(cosbeta_eqn)).subs(e2d(tanbeta_eqn))).subs(mu_eta_sub)
self.gstar_ij_mat = (self.gstar_ij_tanalpha_mat.subs({
ta: tan(alpha)
}).subs(e2d(self.tanalpha_rdot_eqn)).subs(
e2d(self.varphi_rx_eqn.subs(
{varphi_r(rvec): varphi}))).subs(parameters)).subs(mu_eta_sub)
self.g_ij_tanalpha_mat = (expand_trig(self.g_ij_tanbeta_mat).subs(
e2d(sintwobeta_eqn)).subs(e2d(sinbeta_eqn)).subs(
e2d(cosbeta_eqn)).subs(e2d(tanbeta_eqn))).subs(mu_eta_sub)
self.g_ij_mat = (self.g_ij_tanalpha_mat.subs({
ta: rdotz / rdotx
}).subs(e2d(self.varphi_rx_eqn.subs(
{varphi_r(rvec): varphi}))).subs(parameters)).subs(mu_eta_sub)
self.g_ij_mat_lambdified = lambdify((rx, rdotx, rdotz, varepsilon),
self.g_ij_mat, "numpy")
self.gstar_ij_mat_lambdified = lambdify((rx, rdotx, rdotz, varepsilon),
self.gstar_ij_mat, "numpy")
# Space variable
x = sympy.Symbol('x')
u = 0
exp = []
eig = []
for i in range(ns + nf):
exp.append(expa(i))
eig.append(eigenfunctions(i))
u += coefficients[i] * exp[i]
# Calculation of the nonlinearity
nonlinear_term = sympy.collect(
sympy.expand(sympy.expand_trig(nonlinearity(u))),
[sympy.sin(x), sympy.cos(x)])
print("Nonlinearity:")
print(nonlinear_term)
# Projection of the nonlinearity
# TODO: justify division by sqrt(pi)
projections = []
for i in range(ns + nf):
product = sympy.expand(nonlinear_term * sympy.expand_trig(eig[i]))
result_integration = sympy.integrate(product,
(x, L, R)) / sympy.sqrt(sympy.pi)
projections.append(sympy.simplify(result_integration))
print(projections[i])
# 合并同类项 collect
expr = x**2 + 2 * x + x + 1
print(sympy.collect(expr, x))
# 分式化简 cancel
expr = (x**2 + 2 * x + 1) / (x**2 + x)
print(sympy.cancel(expr))
# 分式裂项 apart
expr = (4 * x**3 + 21 * x**2 + 10 * x + 12) / (x**4 + 5 * x**3 + 5 * x**2 +
4 * x)
print(sympy.apart(expr))
# 三角化简 trigsimp
expr = sympy.sin(x) / sympy.cos(x)
print(sympy.trigsimp(expr))
# 三角展开 expand_trig
expr = sympy.sin(x + y)
print(sympy.expand_trig(expr))
# 指数化简 powsimp / 指数展开 expand_power_exp
a, b = sympy.symbols('a b')
expr = x**a * x**b
print(sympy.powsimp(expr))
# 化简指数的指数 powdenest
# 必须满足条件 底数 positive=True
x = sympy.symbols('x', positive=True)
expr = (x**a)**b
print(sympy.powdenest(expr))
# 对数展开 expand_log / 对数合并 logcombine
# 需要指出 log ln 在 sympy 中都是自然对数
# symbol 也需要满足条件
x, y = sympy.symbols('x y', positive=True)
n = sympy.symbols('n', real=True)
print(sympy.expand_log(sympy.log(x**n)))
from sympy.physics.mechanics import dot, dynamicsymbols, inertia, msprint
m, B11, B22, B33, B12, B23, B31 = symbols('m B11 B22 B33 B12 B23 B31')
q1, q2, q3, q4, q5, q6 = dynamicsymbols('q1:7')
# reference frames
A = ReferenceFrame('A')
B = A.orientnew('B', 'body', [q4, q5, q6], 'xyz')
omega = B.ang_vel_in(A)
# points B*, O
pB_star = Point('B*')
pO = pB_star.locatenew('O', q1*B.x + q2*B.y + q3*B.z)
pO.set_vel(A, 0)
pO.set_vel(B, 0)
pB_star.v2pt_theory(pO, A, B)
# rigidbody B
I_B_O = inertia(B, B11, B22, B33, B12, B23, B31)
rbB = RigidBody('rbB', pB_star, B, m, (I_B_O, pO))
# kinetic energy
K = rbB.kinetic_energy(A)
print('K = {0}'.format(msprint(trigsimp(K))))
K_expected = dot(dot(omega, I_B_O), omega)/2
print('K_expected = 1/2*omega*I*omega = {0}'.format(
msprint(trigsimp(K_expected))))
assert expand(expand_trig(K - K_expected)) == 0
init_printing()
from sympy import (symbols, sin, expand, expand_trig)
#%% 1.-
print('1.-')
x, y = symbols('x y') # Consola: 'expr1'
expr1 = (x**3 + 3 * y + 2)**2
sol1 = expr1.expand() # Consola: 'sol1'
#%% 2.-
print('2.-')
expr2 = (3 * x**2 - 2 * x + 1) / (x - 1)**2 # Consola: 'expr2'
sol2 = expr2.apart() # Consola: 'sol2'
#%% 3.-
print('3.-')
expr3 = x**3 + 9 * x**2 + 27 * x + 27 # Consola: 'expr3'
sol3 = expr3.factor() # Consola: 'sol3'
#%% 4.-
print('4.-')
expr4 = sin(x + 2 * y) # Consola: 'expr4'
expr4_exp = expand(expr4)
expr4_exp2 = expand_trig(expr4)
sol4 = expand(expr4, trig=True) # Consola: 'sol4'
for k, v in kde_map.items():
kde_map[k.diff(t)] = v.diff(t)
# f1, f2 are forces the panes of glass exert on P1, P2 respectively
R1 = f1*B.z + C*E.x - m1*g*B.y
R2 = f2*B.z - C*E.x - m2*g*B.y
forces = [(pP1, R1), (pP2, R2)]
system = [Particle('P1', pP1, m1), Particle('P2', pP2, m2)]
partials = partial_velocities([pP1, pP2], u, A, kde_map)
Fr, _ = generalized_active_forces(partials, forces)
Fr_star, _ = generalized_inertia_forces(partials, system, kde_map)
# dynamical equations
dyn_eq = [x + y for x, y in zip(Fr, Fr_star)]
u1d, u2d, u3d = ud = [x.diff(t) for x in u]
dyn_eq_map = solve(dyn_eq, ud)
for x in ud:
print('{0} = {1}'.format(msprint(x),
msprint(cancel(trigsimp(dyn_eq_map[x])))))
u1d_expected = (-g*sin(q3) + omega**2*q1*cos(q3) + u2*u3 +
(omega**2*cos(q3)**2 + u3**2)*L*m2/(m1 + m2))
u2d_expected = -g*cos(q3) - (omega**2*q1*sin(q3) + u3*u1)
u3d_expected = -omega**2*sin(q3)*cos(q3)
assert expand(cancel(expand_trig(dyn_eq_map[u1d] - u1d_expected))) == 0
assert expand(cancel(expand_trig(dyn_eq_map[u2d] - u2d_expected))) == 0
assert expand(expand_trig(dyn_eq_map[u3d] - u3d_expected)) == 0
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def expand_trig(self):
return SymbolicVector(expand_trig(self.x),
expand_trig(self.y),
expand_trig(self.z))
v = p.vel(N).express(N).subs(kde_map).subs(eq_gen_speed_map)
partials[p] = dict(zip(u, map(lambda x: v.diff(x, N), u)))
u1d, u2d, u3d = ud = [x.diff(t) for x in u]
for k, v in vc_map.items():
vc_map[k.diff(t)] = v.diff(t).subs(kde_map).subs(vc_map)
# generalized active/inertia forces
Fr, _ = generalized_active_forces(partials, forces + torques)
Fr_star, _ = generalized_inertia_forces(partials, bodies, kde_map)
print('Fr')
for i, fr in enumerate(Fr, 1):
print('{0}: {1}'.format(i, msprint(fr)))
print('Fr_star')
for i, fr in enumerate(Fr_star, 1):
fr = trigsimp(expand(expand_trig(fr)), deep=True, recursive=True)
print('{0}: {1}'.format(i, msprint(fr)))
# The dynamical equations would be of the form Fr = Fr* (r = 1, 2, 3).
Fr_expected = [
alpha2 + a*(R1*cos(q1) - R3*sin(q1)),
b*(R1*cos(q2) + R3*sin(q2)),
Q3 - R3]
Fr_star_expected = [
-mA*(a_star**2 + kA**2)*u1d,
-mB*((b_star**2 + kB**2)*u2d - b_star*sin(q2)*u3d),
-1*((mB + mC)*u3d - mB*b_star*sin(q2)*u2d + mB*b_star*cos(q2)*u2**2)]
for x, y in zip(Fr, Fr_expected):
assert expand(x - y) == 0
for x, y in zip(Fr_star, Fr_star_expected):
assert trigsimp(expand(expand_trig((x - y).subs(vc_map)))) == 0
def _expand_trig():
# 展开三角函数
x = sympy.Symbol('x')
y = sympy.Symbol('y')
print(sympy.expand_trig(sympy.sin(x + y)))
pO = Point('O')
pS_star = pO.locatenew('S*', b*A.y)
pS_hat = pS_star.locatenew('S^', -r*B.y) # S^ touches the cone
pS1 = pS_star.locatenew('S1', -r*A.z) # S1 touches horizontal wall of the race
pS2 = pS_star.locatenew('S2', r*A.y) # S2 touches vertical wall of the race
pO.set_vel(R, 0)
pS_star.v2pt_theory(pO, R, A)
pS1.v2pt_theory(pS_star, R, S)
pS2.v2pt_theory(pS_star, R, S)
# Since S is rolling against R, v_S1_R = 0, v_S2_R = 0.
vc = [dot(p.vel(R), basis) for p in [pS1, pS2] for basis in R]
pO.set_vel(C, 0)
pS_star.v2pt_theory(pO, C, A)
pS_hat.v2pt_theory(pS_star, C, S)
# Since S is rolling against C, v_S^_C = 0.
# Cone has only angular velocity in R.z direction.
vc += [dot(pS_hat.vel(C), basis).subs(vc_map) for basis in A]
vc += [dot(C.ang_vel_in(R), basis) for basis in [R.x, R.y]]
vc_map = solve(vc, u)
# Pure rolling between S and C, dot(ω_C_S, B.y) = 0.
b_val = solve([dot(C.ang_vel_in(S), B.y).subs(vc_map).simplify()], b)[0][0]
print('b = {0}'.format(msprint(collect(cancel(expand_trig(b_val)), r))))
b_expected = r*(1 + sin(theta))/(cos(theta) - sin(theta))
assert trigsimp(b_val - b_expected) == 0
def test_coth():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert coth(nan) is nan
assert coth(zoo) is nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) is zoo
assert unchanged(coth, 1)
assert coth(-1) == -coth(1)
assert unchanged(coth, x)
assert coth(-x) == -coth(x)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == cot(pi) * I
assert unchanged(coth, 2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi * I) == -I * cot(pi)
assert coth(-pi * I) == I * cot(pi)
assert coth(2 * pi * I) == -I * cot(2 * pi)
assert coth(-2 * pi * I) == I * cot(2 * pi)
assert coth(-3 * 10**73 * pi * I) == I * cot(3 * 10**73 * pi)
assert coth(7 * 10**103 * pi * I) == -I * cot(7 * 10**103 * pi)
assert coth(pi * I / 2) == 0
assert coth(-pi * I / 2) == 0
assert coth(pi * I * Rational(5, 2)) == 0
assert coth(pi * I * Rational(7, 2)) == 0
assert coth(pi * I / 3) == -I / sqrt(3)
assert coth(pi * I * Rational(-2, 3)) == -I / sqrt(3)
assert coth(pi * I / 4) == -I
assert coth(-pi * I / 4) == I
assert coth(pi * I * Rational(17, 4)) == -I
assert coth(pi * I * Rational(-3, 4)) == -I
assert coth(pi * I / 6) == -sqrt(3) * I
assert coth(-pi * I / 6) == sqrt(3) * I
assert coth(pi * I * Rational(7, 6)) == -sqrt(3) * I
assert coth(pi * I * Rational(-5, 6)) == -sqrt(3) * I
assert coth(pi * I / 105) == -cot(pi / 105) * I
assert coth(-pi * I / 105) == cot(pi / 105) * I
assert unchanged(coth, 2 + 3 * I)
assert coth(x * I) == -cot(x) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(17 * k * pi * I) == -cot(17 * k * pi) * I
assert coth(k * pi * I) == -cot(k * pi) * I
assert coth(log(tan(2))) == coth(log(-tan(2)))
assert coth(1 + I * pi / 2) == tanh(1)
assert coth(x).as_real_imag(
deep=False) == (sinh(re(x)) * cosh(re(x)) /
(sin(im(x))**2 + sinh(re(x))**2), -sin(im(x)) *
cos(im(x)) / (sin(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert coth(x).as_real_imag(deep=False) == (coth(x), 0)
assert expand_trig(coth(2 * x)) == (coth(x)**2 + 1) / (2 * coth(x))
assert expand_trig(coth(
3 * x)) == (coth(x)**3 + 3 * coth(x)) / (1 + 3 * coth(x)**2)
assert expand_trig(
coth(x + y)) == (1 + coth(x) * coth(y)) / (coth(x) + coth(y))
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
# print(a)
x = sym.symbols("x")
y = sym.symbols("y")
z = sym.symbols("z")
# simplify function
a = sym.simplify(sym.sin(x)**2 + sym.cos(x)**2)
b = sym.simplify((x**3 + x**2 - x - 1) / (x**2 + 2 * x + 1))
print(f"simplification of sin(x)**2 + cos(x)**2= {a}")
print(f"simplification of (x**3 + x**2 - x - 1)/(x**2 + 2*x + 1) = {b}")
# expand function
c = sym.expand((x + 1)**2)
d = sym.expand((x + 2) * (x + 3))
e = sym.expand_trig(sym.sin(x + y))
print(f"Development of (x + 1)**2 = {c}")
print(f"Development of (x+2)*(x+3) = {d}")
print(f"Development of sin(x+y) = {e}")
# factor function
f = sym.factor(x**2 * z + 4 * x * y * z + a * y**2 * z)
g = sym.factor(sym.cos(x)**2 + 2 * sym.cos(x) * sym.sin(x) + sym.sin(x)**2)
print(f"Factor of x**2*z + 4*x*y*z + a*y**2*z = {f}")
print(f"Factor of cos(x)**x + 2*cos(x)*sin(x) + sin(x)**2 = {g}")
# derivative function
h = sym.diff(sym.cos(x), x)
i = sym.diff(x**4, x, 3)
print(f"The derivative of cos(x) = {h}")
print(f"The 3rd derivative of x**4 = {i}")
