def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.length == 1:
if f.is_constant:
return []
else:
return [S(0)] * f.degree
if f.length == 2:
if f.degree == 1:
return roots_linear(f)
else:
return roots_binomial(f)
x, result = f.symbols[0], []
for i in [S(-1), S(1)]:
if f(i).expand().is_zero:
f = poly_div(f, x - i)[0]
result.append(i)
break
n = f.degree
if n == 1:
result += roots_linear(f)
elif n == 2:
result += roots_quadratic(f)
elif n == 3 and flags.get('cubics', True):
result += roots_cubic(f)
elif n == 4 and flags.get('quartics', False):
result += roots_quartic(f)
return result
def angular_momentum(point, frame, *body):
"""Angular momentum of a system
This function returns the angular momentum of a system of Particle's and/or
RigidBody's. The angular momentum of such a system is equal to the vector
sum of the angular momentum of its constituents. Consider a system, S,
comprised of a rigid body, A, and a particle, P. The angular momentum of
the system, H, is equal to the vector sum of the linear momentum of the
particle, H1, and the linear momentum of the rigid body, H2, i.e-
H = H1 + H2
Parameters
==========
point : Point
The point about which angular momentum of the system is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose kinetic energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, angular_momentum
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> angular_momentum(O, N, Pa, A)
10*N.z
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
if not isinstance(point, Point):
raise TypeError('Please specify a valid Point')
else:
angular_momentum_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
angular_momentum_sys += e.angular_momentum(point, frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return angular_momentum_sys
def kinetic_energy(frame, *body):
"""Kinetic energy of a multibody system.
This function returns the kinetic energy of a system of Particle's and/or
RigidBody's. The kinetic energy of such a system is equal to the sum of
the kinetic energies of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The kinetic energy of the system, T,
is equal to the vector sum of the kinetic energy of the particle, T1, and
the kinetic energy of the rigid body, T2, i.e.
T = T1 + T2
Kinetic energy is a scalar.
Parameters
==========
frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose kinetic energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, kinetic_energy
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> kinetic_energy(N, Pa, A)
350
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
ke_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
ke_sys += e.kinetic_energy(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return ke_sys
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = S.Bernoulli(n + 1)
F = C.Factorial(n + 1)
return (-1)**((n + 1) // 2) * 2**(n + 1) * B / F * x**n
def potential_energy(*body):
"""Potential energy of a multibody system.
This function returns the potential energy of a system of Particle's and/or
RigidBody's. The potential energy of such a system is equal to the sum of
the potential energy of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The potential energy of the system, V,
is equal to the vector sum of the potential energy of the particle, V1, and
the potential energy of the rigid body, V2, i.e.
V = V1 + V2
Potential energy is a scalar.
Parameters
==========
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose potential energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, potential_energy
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> Pa = Particle('Pa', P, m)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> a = ReferenceFrame('a')
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))
>>> Pa.set_potential_energy(m * g * h)
>>> A.set_potential_energy(M * g * h)
>>> potential_energy(Pa, A)
M*g*h + g*h*m
"""
pe_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
pe_sys += e.potential_energy
else:
raise TypeError('*body must have only Particle or RigidBody')
return pe_sys
def linear_momentum(frame, *body):
"""Linear momentum of the system.
This function returns the linear momentum of a system of Particle's and/or
RigidBody's. The linear momentum of a system is equal to the vector sum of
the linear momentum of its constituents. Consider a system, S, comprised of
a rigid body, A, and a particle, P. The linear momentum of the system, L,
is equal to the vector sum of the linear momentum of the particle, L1, and
the linear momentum of the rigid body, L2, i.e-
L = L1 + L2
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose kinetic energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, linear_momentum
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = Point('Ac')
>>> Ac.set_vel(N, 25 * N.y)
>>> I = outer(N.x, N.x)
>>> A = RigidBody('A', Ac, N, 20, (I, Ac))
>>> linear_momentum(N, A, Pa)
10*N.x + 500*N.y
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please specify a valid ReferenceFrame')
else:
linear_momentum_sys = S(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
linear_momentum_sys += e.linear_momentum(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return linear_momentum_sys
def get_motion_params(frame, **kwargs):
"""
Returns the three motion parameters - (acceleration, velocity, and
position) as vectorial functions of time in the given frame.
If a higher order differential function is provided, the lower order
functions are used as boundary conditions. For example, given the
acceleration, the velocity and position parameters are taken as
boundary conditions.
The values of time at which the boundary conditions are specified
are taken from timevalue1(for position boundary condition) and
timevalue2(for velocity boundary condition).
If any of the boundary conditions are not provided, they are taken
to be zero by default (zero vectors, in case of vectorial inputs). If
the boundary conditions are also functions of time, they are converted
to constants by substituting the time values in the dynamicsymbols._t
time Symbol.
This function can also be used for calculating rotational motion
parameters. Have a look at the Parameters and Examples for more clarity.
Parameters
==========
frame : ReferenceFrame
The frame to express the motion parameters in
acceleration : Vector
Acceleration of the object/frame as a function of time
velocity : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
position : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
timevalue1 : sympyfiable
Value of time for position boundary condition
timevalue2 : sympyfiable
Value of time for velocity boundary condition
Examples
========
>>> from sympy.physics.mechanics import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z
"""
##Helper functions
def _process_vector_differential(vectdiff, condition, \
variable, ordinate, frame):
"""
Helper function for get_motion methods. Finds derivative of vectdiff wrt
variable, and its integral using the specified boundary condition at
value of variable = ordinate.
Returns a tuple of - (derivative, function and integral) wrt vectdiff
"""
#Make sure boundary condition is independent of 'variable'
if condition != 0:
condition = frame.express(condition)
#Special case of vectdiff == 0
if vectdiff == Vector(0):
return (0, 0, condition)
#Express vectdiff completely in condition's frame to give vectdiff1
vectdiff1 = frame.express(vectdiff)
#Find derivative of vectdiff
vectdiff2 = frame.dt(vectdiff)
#Integrate and use boundary condition
vectdiff0 = Vector(0)
lims = (variable, ordinate, variable)
for dim in frame:
function1 = vectdiff1.dot(dim)
abscissa = dim.dot(condition).subs({variable: ordinate})
# Indefinite integral of 'function1' wrt 'variable', using
# the given initial condition (ordinate, abscissa).
vectdiff0 += (integrate(function1, lims) + abscissa) * dim
#Return tuple
return (vectdiff2, vectdiff, vectdiff0)
##Function body
_check_frame(frame)
#Decide mode of operation based on user's input
if 'acceleration' in kwargs:
mode = 2
elif 'velocity' in kwargs:
mode = 1
else:
mode = 0
#All the possible parameters in kwargs
#Not all are required for every case
#If not specified, set to default values(may or may not be used in
#calculations)
conditions = [
'acceleration', 'velocity', 'position', 'timevalue', 'timevalue1',
'timevalue2'
]
for i, x in enumerate(conditions):
if x not in kwargs:
if i < 3:
kwargs[x] = Vector(0)
else:
kwargs[x] = S(0)
elif i < 3:
_check_vector(kwargs[x])
else:
kwargs[x] = sympify(kwargs[x])
if mode == 2:
vel = _process_vector_differential(kwargs['acceleration'],
kwargs['velocity'],
dynamicsymbols._t,
kwargs['timevalue2'], frame)[2]
pos = _process_vector_differential(vel, kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)[2]
return (kwargs['acceleration'], vel, pos)
elif mode == 1:
return _process_vector_differential(kwargs['velocity'],
kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)
else:
vel = frame.dt(kwargs['position'])
acc = frame.dt(vel)
return (acc, vel, kwargs['position'])
def _eval_rewrite_as_cot(self, arg):
return 1 / S.Cot(arg)
def _eval_rewrite_as_cot(self, arg):
cot_half = S.Cot(S.Half * arg)**2
return (cot_half - 1) / (cot_half + 1)
def _eval_rewrite_as_cot(self, arg):
cot_half = S.Cot(S.Half * arg)
return 2 * cot_half / (1 + cot_half**2)
def express(expr, frame, frame2=None, variables=False):
"""
Global function for 'express' functionality.
Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame.
Refer to the local methods of Vector and Dyadic for details.
If 'variables' is True, then the coordinate variables (CoordinateSym
instances) of other frames present in the vector/scalar field or
dyadic expression are also substituted in terms of the base scalars of
this frame.
Parameters
==========
expr : Vector/Dyadic/scalar(sympyfiable)
The expression to re-express in ReferenceFrame 'frame'
frame: ReferenceFrame
The reference frame to express expr in
frame2 : ReferenceFrame
The other frame required for re-expression(only for Dyadic expr)
variables : boolean
Specifies whether to substitute the coordinate variables present
in expr, in terms of those of frame
Examples
========
>>> from sympy.physics.mechanics import ReferenceFrame, outer, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> from sympy.physics.mechanics import express
>>> express(d, B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
>>> express(B.x, N)
cos(q)*N.x + sin(q)*N.y
>>> express(N[0], B, variables=True)
B_x*cos(q(t)) - B_y*sin(q(t))
"""
_check_frame(frame)
if expr == 0:
return S(0)
if isinstance(expr, Vector):
#Given expr is a Vector
if variables:
#If variables attribute is True, substitute
#the coordinate variables in the Vector
frame_list = [x[-1] for x in expr.args]
subs_dict = {}
for f in frame_list:
subs_dict.update(f.variable_map(frame))
expr = expr.subs(subs_dict)
#Re-express in this frame
outvec = Vector([])
for i, v in enumerate(expr.args):
if v[1] != frame:
temp = frame.dcm(v[1]) * v[0]
if Vector.simp:
temp = temp.applyfunc(lambda x: \
trigsimp(x, method='fu'))
outvec += Vector([(temp, frame)])
else:
outvec += Vector([v])
return outvec
if isinstance(expr, Dyadic):
if frame2 is None:
frame2 = frame
_check_frame(frame2)
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += express(v[0], frame, variables=variables) * \
(express(v[1], frame, variables=variables) | \
express(v[2], frame2, variables=variables))
return ol
else:
if variables:
#Given expr is a scalar field
frame_set = set([])
expr = sympify(expr)
#Subsitute all the coordinate variables
for x in expr.atoms():
if isinstance(x, CoordinateSym) and x.frame != frame:
frame_set.add(x.frame)
subs_dict = {}
for f in frame_set:
subs_dict.update(f.variable_map(frame))
return expr.subs(subs_dict)
return expr
def poly_subresultants(f, g, *symbols, **flags):
"""Computes subresultant PRS of two univariate polynomials.
Polynomial remainder sequence (PRS) is a fundamental tool in
computer algebra as it gives as a sub-product the polynomial
greatest common divisor (GCD), provided that the coefficient
domain is an unique factorization domain.
There are several methods for computing PRS, eg.: Euclidean
PRS, where the most famous algorithm is used, primitive PRS
and, finally, subresultants which are implemented here.
The Euclidean approach is reasonably efficient but suffers
severely from coefficient growth. The primitive algorithm
avoids this but requires a lot of coefficient computations.
Subresultants solve both problems and so it is efficient and
have moderate coefficient growth. The current implementation
uses pseudo-divisions which is well suited for coefficients
in integral domains or number fields.
Formally, given univariate polynomials f and g over an UFD,
then a sequence (R_0, R_1, ..., R_k, 0, ...) is a polynomial
remainder sequence where R_0 = f, R_1 = g, R_k != 0 and R_k
is similar to gcd(f, g).
The result is returned as tuple (res, R) where R is the PRS
sequence and res is the resultant of the input polynomials.
If only polynomial remainder sequence is important, then by
setting res=False in keyword arguments expensive computation
of the resultant can be avoided (only PRS is returned).
For more information on the implemented algorithm refer to:
[1] M. Bronstein, Symbolic Integration I: Transcendental
Functions, Second Edition, Springer-Verlang, 2005
[2] M. Keber, Division-Free computation of subresultants
using Bezout matrices, Tech. Report MPI-I-2006-1-006,
Saarbrucken, 2006
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
if f.is_multivariate:
raise MultivariatePolyError(f)
else:
symbols = f.symbols
n, m = f.degree, g.degree
if n < m:
f, g = g, f
n, m = m, n
R = [f, g]
d = n - m
b = S(-1)**(d + 1)
c = S(-1)
B, D = [b], [d]
h = poly_prem(f, g)
h = h.mul_term(b)
while not h.is_zero:
k = h.degree
R.append(h)
lc = g.LC
C = (-lc)**d / c**(d - 1)
c = Poly.cancel(C)
b = -lc * c**(m - k)
f, g, m, d = g, h, k, m - k
B.append(b)
D.append(d)
h = poly_prem(f, g)
h = h.div_term(b)
if not flags.get('res', True):
return R
if R[-1].degree > 0:
return (Poly((), *symbols), R)
if R[-2].is_one:
return (R[-1], R)
s, c, i = 1, S(1), 1
for b, d in zip(B, D)[:-1]:
u = R[i - 1].degree
v = R[i].degree
w = R[i + 1].degree
if u % 2 and v % 2:
s = -s
lc = R[i].LC
C = c * (b / lc**(1 + d))**v * lc**(u - w)
c = Poly.cancel(C)
i += 1
j = R[-2].degree
return (R[-1]**j * s * c, R)
def _eval_rewrite_as_exp(self, arg):
neg_exp, pos_exp = S.Exp(-arg), S.Exp(arg)
return (pos_exp + neg_exp) / (pos_exp - neg_exp)
def _eval_rewrite_as_exp(self, arg):
return (S.Exp(arg) + S.Exp(-arg)) / 2
def roots(f, *symbols, **flags):
"""Computes symbolic roots of an univariate polynomial.
Given an univariate polynomial f with symbolic coefficients,
returns a dictionary with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set cubics=False or quartics=False respectively.
To get roots from a specific domain set the 'domain' flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting domain='C').
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a tuple containing all
those roots set the 'multiple' flag to True.
>>> from sympy import *
>>> x,y = symbols('xy')
>>> roots(x**2 - 1, x)
{1: 1, -1: 1}
>>> roots(x**2 - y, x)
{y**(1/2): 1, -y**(1/2): 1}
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
if f.is_multivariate:
raise MultivariatePolyError(f)
def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors = poly_decompose(f)
result, g = {}, factors[0]
for i, h in enumerate(poly_sqf(g)):
for r in _try_heuristics(h):
_update_dict(result, r, i + 1)
for factor in factors[1:]:
last, result = result.copy(), {}
for last_r, i in last.iteritems():
g = factor.sub_term(last_r, (0, ))
for j, h in enumerate(poly_sqf(g)):
for r in _try_heuristics(h):
_update_dict(result, r, i * (j + 1))
return result
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.length == 1:
if f.is_constant:
return []
else:
return [S(0)] * f.degree
if f.length == 2:
if f.degree == 1:
return roots_linear(f)
else:
return roots_binomial(f)
x, result = f.symbols[0], []
for i in [S(-1), S(1)]:
if f(i).expand().is_zero:
f = poly_div(f, x - i)[0]
result.append(i)
break
n = f.degree
if n == 1:
result += roots_linear(f)
elif n == 2:
result += roots_quadratic(f)
elif n == 3 and flags.get('cubics', True):
result += roots_cubic(f)
elif n == 4 and flags.get('quartics', False):
result += roots_quartic(f)
return result
multiple = flags.get('multiple', False)
if f.length == 1:
if f.is_constant:
if multiple:
return []
else:
return {}
else:
result = {S(0): f.degree}
else:
(k, ), f = f.as_reduced()
if k == 0:
zeros = {}
else:
zeros = {S(0): k}
result = {}
if f.length == 2:
if f.degree == 1:
result[roots_linear(f)[0]] = 1
else:
for r in roots_binomial(f):
_update_dict(result, r, 1)
elif f.degree == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
else:
try:
_, factors = poly_factors(f)
if len(factors) == 1 and factors[0][1] == 1:
raise CoefficientError
for factor, k in factors:
for r in _try_heuristics(factor):
_update_dict(result, r, k)
except CoefficientError:
result = _try_decompose(f)
result.update(zeros)
domain = flags.get('domain', None)
if domain not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[domain]
except KeyError:
raise ValueError("Invalid domain: %s" % domain)
for zero in dict(result).iterkeys():
if not query(zero):
del result[zero]
predicate = flags.get('predicate', None)
if predicate is not None:
for zero in dict(result).iterkeys():
if not predicate(zero):
del result[zero]
if not multiple:
return result
else:
zeros = []
for zero, k in result.iteritems():
zeros.extend([zero] * k)
return zeros
