# Source code for sympy.physics.mechanics.functions

from __future__ import print_function, division

from sympy.utilities import dict_merge
from sympy.utilities.iterables import iterable
from sympy.physics.vector import (Dyadic, Vector, ReferenceFrame,
Point, dynamicsymbols)
from sympy.physics.vector.printing import (vprint, vsprint, vpprint, vlatex,
init_vprinting)
from sympy.physics.mechanics.particle import Particle
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy import simplify
from sympy.core.backend import (Matrix, sympify, Mul, Derivative, sin, cos,
tan, AppliedUndef, S)

__all__ = ['inertia',
'inertia_of_point_mass',
'linear_momentum',
'angular_momentum',
'kinetic_energy',
'potential_energy',
'Lagrangian',
'mechanics_printing',
'mprint',
'msprint',
'mpprint',
'mlatex',
'msubs',
'find_dynamicsymbols']

# These are functions that we've moved and renamed during extracting the
# basic vector calculus code from the mechanics packages.

mprint = vprint
msprint = vsprint
mpprint = vpprint
mlatex = vlatex

def mechanics_printing(**kwargs):
"""
Initializes time derivative printing for all SymPy objects in
mechanics module.
"""

init_vprinting(**kwargs)

mechanics_printing.__doc__ = init_vprinting.__doc__

[docs]def inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0):
"""Simple way to create inertia Dyadic object.

If you don't know what a Dyadic is, just treat this like the inertia
tensor. Then, do the easy thing and define it in a body-fixed frame.

Parameters
==========

frame : ReferenceFrame
The frame the inertia is defined in
ixx : Sympifyable
the xx element in the inertia dyadic
iyy : Sympifyable
the yy element in the inertia dyadic
izz : Sympifyable
the zz element in the inertia dyadic
ixy : Sympifyable
the xy element in the inertia dyadic
iyz : Sympifyable
the yz element in the inertia dyadic
izx : Sympifyable
the zx element in the inertia dyadic

Examples
========

>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)

"""

if not isinstance(frame, ReferenceFrame):
raise TypeError('Need to define the inertia in a frame')
ol = sympify(ixx) * (frame.x | frame.x)
ol += sympify(ixy) * (frame.x | frame.y)
ol += sympify(izx) * (frame.x | frame.z)
ol += sympify(ixy) * (frame.y | frame.x)
ol += sympify(iyy) * (frame.y | frame.y)
ol += sympify(iyz) * (frame.y | frame.z)
ol += sympify(izx) * (frame.z | frame.x)
ol += sympify(iyz) * (frame.z | frame.y)
ol += sympify(izz) * (frame.z | frame.z)
return ol

[docs]def inertia_of_point_mass(mass, pos_vec, frame):
"""Inertia dyadic of a point mass relative to point O.

Parameters
==========

mass : Sympifyable
Mass of the point mass
pos_vec : Vector
Position from point O to point mass
frame : ReferenceFrame
Reference frame to express the dyadic in

Examples
========

>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass
>>> N = ReferenceFrame('N')
>>> r, m = symbols('r m')
>>> px = r * N.x
>>> inertia_of_point_mass(m, px, N)
m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)

"""

return mass * (((frame.x | frame.x) + (frame.y | frame.y) +
(frame.z | frame.z)) * (pos_vec & pos_vec) -
(pos_vec | pos_vec))

[docs]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 linear momentum 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 = Vector(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

[docs]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 angular momentum of the
particle, H1, and the angular 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 angular momentum 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 = Vector(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

[docs]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

[docs]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.potential_energy = m * g * h
>>> A.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 gravity(acceleration, *bodies):
"""
Returns a list of gravity forces given the acceleration
due to gravity and any number of particles or rigidbodies.

Example
=======

>>> from sympy.physics.mechanics import ReferenceFrame, Point, Particle, outer, RigidBody
>>> from sympy.physics.mechanics.functions import gravity
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> m, M, g = symbols('m M g')
>>> F1, F2 = symbols('F1 F2')
>>> po = Point('po')
>>> pa = Particle('pa', po, m)
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer(A.x, A.x)
>>> B = RigidBody('B', P, A, M, (I, P))
>>> forceList = [(po, F1), (P, F2)]
>>> forceList.extend(gravity(g*N.y, pa, B))
>>> forceList
[(po, F1), (P, F2), (po, g*m*N.y), (P, M*g*N.y)]
"""

gravity_force = []
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")

for e in bodies:
point = getattr(e, 'masscenter', None)
if point is None:
point = e.point

gravity_force.append((point, e.mass*acceleration))

return gravity_force

def center_of_mass(point, *bodies):
"""
Returns the position vector from the given point to the center of mass
of the given bodies(particles or rigidbodies).

Example
=======

>>> from sympy import symbols, S
>>> from sympy.physics.vector import Point
>>> from sympy.physics.mechanics import Particle, ReferenceFrame, RigidBody, outer
>>> from sympy.physics.mechanics.functions import center_of_mass
>>> a = ReferenceFrame('a')
>>> m = symbols('m', real=True)
>>> p1 = Particle('p1', Point('p1_pt'), S(1))
>>> p2 = Particle('p2', Point('p2_pt'), S(2))
>>> p3 = Particle('p3', Point('p3_pt'), S(3))
>>> p4 = Particle('p4', Point('p4_pt'), m)
>>> b_f = ReferenceFrame('b_f')
>>> b_cm = Point('b_cm')
>>> mb = symbols('mb')
>>> b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
>>> p2.point.set_pos(p1.point, a.x)
>>> p3.point.set_pos(p1.point, a.x + a.y)
>>> p4.point.set_pos(p1.point, a.y)
>>> b.masscenter.set_pos(p1.point, a.y + a.z)
>>> point_o=Point('o')
>>> point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
>>> expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
>>> point_o.pos_from(p1.point)
5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
"""
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")

total_mass = 0
vec = Vector(0)
for i in bodies:
total_mass += i.mass

masscenter = getattr(i, 'masscenter', None)
if masscenter is None:
masscenter = i.point
vec += i.mass*masscenter.pos_from(point)

return vec/total_mass

[docs]def Lagrangian(frame, *body):
"""Lagrangian of a multibody system.

This function returns the Lagrangian of a system of Particle's and/or
RigidBody's. The Lagrangian of such a system is equal to the difference
between the kinetic energies and potential energies of its constituents. If
T and V are the kinetic and potential energies of a system then it's
Lagrangian, L, is defined as

L = T - V

The Lagrangian is a scalar.

Parameters
==========

frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined to determine the kinetic energy.

body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose Lagrangian is required.

Examples
========

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, Lagrangian
>>> 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)
>>> 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))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> Lagrangian(N, Pa, A)
-M*g*h - g*h*m + 350

"""

if not isinstance(frame, ReferenceFrame):
raise TypeError('Please supply a valid ReferenceFrame')
for e in body:
if not isinstance(e, (RigidBody, Particle)):
raise TypeError('*body must have only Particle or RigidBody')
return kinetic_energy(frame, *body) - potential_energy(*body)

[docs]def find_dynamicsymbols(expression, exclude=None, reference_frame=None):
"""Find all dynamicsymbols in expression.

If the optional exclude kwarg is used, only dynamicsymbols
not in the iterable exclude are returned.
If we intend to apply this function on a vector, the optional
''reference_frame'' is also used to inform about the corresponding frame
with respect to which the dynamic symbols of the given vector is to be
determined.

Parameters
==========

expression : sympy expression

exclude : iterable of dynamicsymbols, optional

reference_frame : ReferenceFrame, optional
The frame with respect to which the dynamic symbols of the
given vector is to be determined.

Examples
========

>>> from sympy.physics.mechanics import dynamicsymbols, find_dynamicsymbols
>>> from sympy.physics.mechanics import ReferenceFrame
>>> x, y = dynamicsymbols('x, y')
>>> expr = x + x.diff()*y
>>> find_dynamicsymbols(expr)
{x(t), y(t), Derivative(x(t), t)}
>>> find_dynamicsymbols(expr, exclude=[x, y])
{Derivative(x(t), t)}
>>> a, b, c = dynamicsymbols('a, b, c')
>>> A = ReferenceFrame('A')
>>> v = a * A.x + b * A.y + c * A.z
>>> find_dynamicsymbols(v, reference_frame=A)
{a(t), b(t), c(t)}

"""
t_set = {dynamicsymbols._t}
if exclude:
if iterable(exclude):
exclude_set = set(exclude)
else:
raise TypeError("exclude kwarg must be iterable")
else:
exclude_set = set()
if isinstance(expression, Vector):
if reference_frame is None:
raise ValueError("You must provide reference_frame when passing a "
"vector expression, got %s." % reference_frame)
else:
expression = expression.to_matrix(reference_frame)
return set([i for i in expression.atoms(AppliedUndef, Derivative) if
i.free_symbols == t_set]) - exclude_set

[docs]def msubs(expr, *sub_dicts, **kwargs):
"""A custom subs for use on expressions derived in physics.mechanics.

Traverses the expression tree once, performing the subs found in sub_dicts.
Terms inside Derivative expressions are ignored:

>>> from sympy.physics.mechanics import dynamicsymbols, msubs
>>> x = dynamicsymbols('x')
>>> msubs(x.diff() + x, {x: 1})
Derivative(x(t), t) + 1

Note that sub_dicts can be a single dictionary, or several dictionaries:

>>> x, y, z = dynamicsymbols('x, y, z')
>>> sub1 = {x: 1, y: 2}
>>> sub2 = {z: 3, x.diff(): 4}
>>> msubs(x.diff() + x + y + z, sub1, sub2)
10

If smart=True (default False), also checks for conditions that may result
in nan, but if simplified would yield a valid expression. For example:

>>> from sympy import sin, tan
>>> (sin(x)/tan(x)).subs(x, 0)
nan
>>> msubs(sin(x)/tan(x), {x: 0}, smart=True)
1

It does this by first replacing all tan with sin/cos. Then each
node is traversed. If the node is a fraction, subs is first evaluated on
the denominator. If this results in 0, simplification of the entire
fraction is attempted. Using this selective simplification, only
subexpressions that result in 1/0 are targeted, resulting in faster
performance.

"""

sub_dict = dict_merge(*sub_dicts)
smart = kwargs.pop('smart', False)
if smart:
func = _smart_subs
elif hasattr(expr, 'msubs'):
return expr.msubs(sub_dict)
else:
func = lambda expr, sub_dict: _crawl(expr, _sub_func, sub_dict)
if isinstance(expr, (Matrix, Vector, Dyadic)):
return expr.applyfunc(lambda x: func(x, sub_dict))
else:
return func(expr, sub_dict)

def _crawl(expr, func, *args, **kwargs):
"""Crawl the expression tree, and apply func to every node."""
val = func(expr, *args, **kwargs)
if val is not None:
return val
new_args = (_crawl(arg, func, *args, **kwargs) for arg in expr.args)
return expr.func(*new_args)

def _sub_func(expr, sub_dict):
"""Perform direct matching substitution, ignoring derivatives."""
if expr in sub_dict:
return sub_dict[expr]
elif not expr.args or expr.is_Derivative:
return expr

def _tan_repl_func(expr):
"""Replace tan with sin/cos."""
if isinstance(expr, tan):
return sin(*expr.args) / cos(*expr.args)
elif not expr.args or expr.is_Derivative:
return expr

def _smart_subs(expr, sub_dict):
"""Performs subs, checking for conditions that may result in nan or
oo, and attempts to simplify them out.

The expression tree is traversed twice, and the following steps are
performed on each expression node:
- First traverse:
Replace all tan with sin/cos.
- Second traverse:
If node is a fraction, check if the denominator evaluates to 0.
If so, attempt to simplify it out. Then if node is in sub_dict,
sub in the corresponding value."""
expr = _crawl(expr, _tan_repl_func)

def _recurser(expr, sub_dict):
# Decompose the expression into num, den
num, den = _fraction_decomp(expr)
if den != 1:
# If there is a non trivial denominator, we need to handle it
denom_subbed = _recurser(den, sub_dict)
if denom_subbed.evalf() == 0:
# If denom is 0 after this, attempt to simplify the bad expr
expr = simplify(expr)
else:
# Expression won't result in nan, find numerator
num_subbed = _recurser(num, sub_dict)
return num_subbed / denom_subbed
# We have to crawl the tree manually, because expr may have been
# modified in the simplify step. First, perform subs as normal:
val = _sub_func(expr, sub_dict)
if val is not None:
return val
new_args = (_recurser(arg, sub_dict) for arg in expr.args)
return expr.func(*new_args)
return _recurser(expr, sub_dict)

def _fraction_decomp(expr):
"""Return num, den such that expr = num/den"""
if not isinstance(expr, Mul):
return expr, 1
num = []
den = []
for a in expr.args:
if a.is_Pow and a.args < 0:
den.append(1 / a)
else:
num.append(a)
if not den:
return expr, 1
num = Mul(*num)
den = Mul(*den)
return num, den

def _f_list_parser(fl, ref_frame):
"""Parses the provided forcelist composed of items
of the form (obj, force).
Returns a tuple containing:
vel_list: The velocity (ang_vel for Frames, vel for Points) in
the provided reference frame.
f_list: The forces.

Used internally in the KanesMethod and LagrangesMethod classes.
"""
def flist_iter():
for pair in fl:
obj, force = pair
if isinstance(obj, ReferenceFrame):
yield obj.ang_vel_in(ref_frame), force
elif isinstance(obj, Point):
yield obj.vel(ref_frame), force
else:
raise TypeError('First entry in each forcelist pair must '
'be a point or frame.')

if not fl:
vel_list, f_list = (), ()
else:
unzip = lambda l: list(zip(*l)) if l else [(), ()]
vel_list, f_list = unzip(list(flist_iter()))
return vel_list, f_list