def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
x, y = symbols('x y', real=True)
m = Manifold('M', 2)
assert str(m) == "M"
p = Patch('P', m)
assert str(p) == "P"
rect = CoordSystem('rect', p, [x, y])
assert str(rect) == "rect"
b = BaseScalarField(rect, 0)
assert str(b) == "x"
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
m = Manifold('M', 2)
p = Patch('P', m)
rect = CoordSystem('rect', p)
assert srepr(
rect
) == "CoordSystem(Str('rect'), Patch(Str('P'), Manifold(Str('M'), Integer(2))), ('rect_0', 'rect_1'))"
b = BaseScalarField(rect, 0)
assert srepr(
b
) == "BaseScalarField(CoordSystem(Str('rect'), Patch(Str('P'), Manifold(Str('M'), Integer(2))), ('rect_0', 'rect_1')), Integer(0))"
def test_covD():
M = Manifold('Reisner-Nordstrom', 4)
p = Patch('origin', M)
cs = CoordSystem('spherical', p, ['t', 'r', 'theta', 'phi'])
t, r, theta, phi = cs.coord_functions()
dt, dr, dtheta, dphi = cs.base_oneforms()
f = Function('f')
metric = f(r)**2 * TP(dt, dt) - f(r)**(-2) * TP(dr, dr) - r**2 * TP(
dtheta, dtheta) - r**2 * sin(theta)**2 * TP(dphi, dphi)
ch_2nd = metric_to_Christoffel_2nd(metric)
G = ToArray(metric)
assert (G.covD(ch_2nd).to_tensor() == 0)
def define_coord_sys():
global rect
global polar
"""定义Manifold, Patch, CoordSystem"""
r, theta = symbols('r, theta')
x, y = symbols('x, y')
m = Manifold('M', 2)
# 定义一个patch
patch = Patch('P', m)
# 为这个patch建立坐标系，一个是笛卡尔，一个是极坐标
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
polar.connect_to(rect, [r, theta], [r * cos(theta), r * sin(theta)])
def test_deprecations():
m = Manifold('M', 2)
p = Patch('P', m)
with warns_deprecated_sympy():
CoordSystem('Car2d', p, names=['x', 'y'])
with warns_deprecated_sympy():
c = CoordSystem('Car2d', p, ['x', 'y'])
with warns_deprecated_sympy():
list(m.patches)
with warns_deprecated_sympy():
list(c.transforms)
def mobius_strip():
import find_metric as fm
import tensor as t
g, diff = fm.mobius_strip()
R = t.Riemann(g, dim=2, sys_title='mobius_strip', flat_diff=diff)
#metric=R.metric
from sympy.diffgeom import TensorProduct, Manifold, Patch, CoordSystem
manifold = Manifold("M", 2)
patch = Patch("P", manifold)
system = CoordSystem('mobius_strip', patch, ["u", "v"])
u, v = system.coord_functions()
du, dv = system.base_oneforms()
from sympy import cos
metric = (cos(u / 2)**2 * v**2 / 4 + cos(u / 2) * v + v**2 / 16 +
1) * TensorProduct(du, du) + 0.25 * TensorProduct(dv, dv)
C = Christoffel_2nd(metric=metric)
return C
def __new__(cls, *args):
obj = super(Manifold, cls).__new__(cls)
if len(args) >= 1:
dependent = args[0]
if len(args) >= 2:
name = args[1]
else:
name = 'manifold'
obj.name = name
obj.dimension = len(dependent)
obj._manifold = sympyManifold(obj.name, obj.dimension)
obj._patch = Patch('Atlas', obj._manifold)
obj._coordsystem = CoordSystem('Coordinates',
obj._patch,
names=[str(d) for d in dependent])
return obj
def toroidal_coords(manifold: Manifold = None, dim: int = 4):
"""Create a Toroidal Coordinate system (t ~ R and (x,y,z) ~ S^3) with maximal
dimension 4 (can create subspaces too)
Args:
manifold:
Manifold, default None. If specified use this Manifold to take a coordinate patch
dim:
int, the dimension of the resulting coordinate system. Must be less than or equal to 4,
and must match the dimension of the given Manifold if specified.
Returns:
CoordSystem, the coordinate system
"""
if manifold is None:
manifold = Manifold('M', dim)
origin_patch = Patch('o', manifold)
return CoordSystem('toroidal', origin_patch, SPHERICAL_SYMBOLS[:dim])
def flat_kerr(a=0, G=1, M=0.5):
import find_metric as fm
from sympy.diffgeom import CoordSystem, Manifold, Patch, TensorProduct
manifold = Manifold("M", 3)
patch = Patch("P", manifold)
kerr = CoordSystem("kerr", patch, ["u", "v", "w"])
u, v, w = kerr.coord_functions()
du, dv, dw = kerr.base_oneforms()
g11 = (a**2 * sym.cos(v) + u**2) / (-2 * G * M * u + a**2 + u**2)
g22 = a**2 * sym.cos(v) + u**2
g33 = -(1 - 2 * G * M * u / (u**2 + a**2 * sym.cos(v)))
# time independent : unphysical ?
#g33 = 2*G*M*a**2*sym.sin(v)**4*u/(a**2*sym.cos(v) + u**2)**2 + a**2*sym.sin(v)**2 + sym.sin(v)**2*u**2
metric = g11 * TensorProduct(du, du) + g22 * TensorProduct(
dv, dv) + g33 * TensorProduct(dw, dw)
C = Christoffel_2nd(metric=metric)
return C
def _set_coordinates(self, sys_title):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M", self.dim)
patch = Patch("P", manifold)
if self.dim == 4:
system = CoordSystem(sys_title, patch, ["u", "v", "w", "t"])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim == 3:
system = CoordSystem(sys_title, patch, ["u", "v", "w"])
u, v, w = system.coord_functions()
self.w = w
if self.dim == 2:
system = CoordSystem(sys_title, patch, ["u", "v"])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def _dummy_cs(self):
"""Create a dummy CoordSystem"""
manifold = Manifold('M', 3)
origin_patch = Patch('o', manifold)
return coords.CoordSystem('euclidean', origin_patch, ['x', 'y', 'z'])
from sympy.diffgeom import Manifold, Patch
m = Manifold('M', 3)
p = Patch('P', m)
p in m.patches
# https://krastanov.wordpress.com/2012/06/30/part-1-what-is-a-tensor-and-how-is-it-implemented-in-the-diffgeom-sympy-module/
import numpy
import scipy
from sympy.diffgeom import Manifold, Patch, CoordSystem
m = Manifold('my_manifold', 2) # A 2D manifold called 'my_manifold'
p = Patch('my_patch', m) # A patch called 'my_patch'
cs_r = CoordSystem('R', p) # A coordinate system called 'R' (for rectangular)
point = cs_r.point([1, 1]) # A point with coordinates (1, 1)
print(point)
# https://krastanov.wordpress.com/2012/06/30/part-1-what-is-a-tensor-and-how-is-it-implemented-in-the-diffgeom-sympy-module/
import numpy
import scipy
from sympy.diffgeom import CoordSystem, Manifold, Patch
m = Manifold("my_manifold", 2) # A 2D manifold called 'my_manifold'
p = Patch("my_patch", m) # A patch called 'my_patch'
cs_r = CoordSystem("R", p) # A coordinate system called 'R' (for rectangular)
point = cs_r.point([1, 1]) # A point with coordinates (1, 1)
print(point)
from sympy.physics.vector import gradient
from sympy.physics.vector import ReferenceFrame
from sympy import *
from sympy.tensor import IndexedBase, Idx
from sympy import symbols
from sympy import Matrix, I
from sympy import symbols, sin, cos, pi
from sympy.diffgeom import Manifold, Patch, CoordSystem
if __name__ == '__main__':
#f = Function('f')(x, y)
#g1 = Function('g')(x, y)
a1, a2, a3, ao1, ao2, theta1, theta2, theta3, theta4 = symbols(
'a1,a2,a3,ao1,ao2,theta1,theta2,theta3,theta4')
m = Manifold('M', 4)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
rect in patch.coord_systems
polar.connect_to(rect, [theta1, theta2, theta3, theta4], [
a1 * theta1 + a2 * theta2 + a3 * theta3 +
(theta1 * ao1 + theta2 * ao2) * theta4
])
print(polar.jacobian(rect, [theta1, theta2, theta3, theta4]))
g = polar.jacobian(rect, [theta1, theta2, theta3, theta4])
patch2 = Patch('P', m)
rect2 = CoordSystem('rect2', patch2)
polar2 = CoordSystem('polar2', patch2)
rect2 in patch2.coord_systems
polar2.connect_to(rect2, [theta1, theta2, theta3, theta4], g)
# http://docs.sympy.org/latest/modules/diffgeom.html
from sympy import cos, pi, sin, symbols
from sympy.diffgeom import CoordSystem, Manifold, Patch
from sympy.simplify import simplify
r, theta = symbols("r, theta")
m = Manifold("M", 2)
patch = Patch("P", m)
rect = CoordSystem("rect", patch)
polar = CoordSystem("polar", patch)
print(rect in patch.coord_systems)
polar.connect_to(rect, [r, theta], [r * cos(theta), r * sin(theta)])
print(polar.coord_tuple_transform_to(rect, [0, 2]))
print(polar.coord_tuple_transform_to(rect, [2, pi / 2]))
print(rect.coord_tuple_transform_to(polar, [1, 1]).applyfunc(simplify))
print(polar.jacobian(rect, [r, theta]))
p = polar.point([1, 3 * pi / 4])
print(rect.point_to_coords(p))
print(rect.coord_function(0)(p))
print(rect.coord_function(1)(p))
v_x = rect.base_vector(0)
x = rect.coord_function(0)
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
from sympy import symbols, Function
m = Manifold("m", 2)
p = Patch("p", m)
cs = CoordSystem("cs", p, ["a", "b"])
cs_noname = CoordSystem("cs", p)
x, y = symbols("x y")
f = Function("f")
s1, s2 = cs.coord_functions()
v1, v2 = cs.base_vectors()
f1, f2 = cs.base_oneforms()
def test_point():
point = Point(cs, [x, y])
assert point == point.func(*point.args)
assert point != Point(cs, [2, y])
# TODO assert point.subs(x, 2) == Point(cs, [2, y])
# TODO assert point.free_symbols == set([x, y])
def test_rebuild():
assert m == m.func(*m.args)
assert p == p.func(*p.args)
assert cs == cs.func(*cs.args)
assert cs_noname == cs_noname.func(*cs_noname.args)
assert s1 == s1.func(*s1.args)
assert v1 == v1.func(*v1.args)
assert f1 == f1.func(*f1.args)
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
from sympy import symbols, Function
m = Manifold('m', 2)
p = Patch('p', m)
cs = CoordSystem('cs', p, ['a', 'b'])
cs_noname = CoordSystem('cs', p)
x, y = symbols('x y')
f = Function('f')
s1, s2 = cs.coord_functions()
v1, v2 = cs.base_vectors()
f1, f2 = cs.base_oneforms()
def test_point():
point = Point(cs, [x, y])
assert point == point.func(*point.args)
assert point != Point(cs, [2, y])
#TODO assert point.subs(x, 2) == Point(cs, [2, y])
#TODO assert point.free_symbols == set([x, y])
def test_rebuild():
assert m == m.func(*m.args)
assert p == p.func(*p.args)
assert cs == cs.func(*cs.args)
assert cs_noname == cs_noname.func(*cs_noname.args)
assert s1 == s1.func(*s1.args)
assert v1 == v1.func(*v1.args)
assert f1 == f1.func(*f1.args)
# http://docs.sympy.org/latest/modules/diffgeom.html
from sympy import symbols, sin, cos, pi
from sympy.diffgeom import Manifold, Patch, CoordSystem
from sympy.simplify import simplify
r, theta = symbols('r, theta')
m = Manifold('M', 2)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
print(rect in patch.coord_systems)
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
print(polar.coord_tuple_transform_to(rect, [0, 2]))
print(polar.coord_tuple_transform_to(rect, [2, pi/2]))
print(rect.coord_tuple_transform_to(polar, [1, 1]).applyfunc(simplify))
print(polar.jacobian(rect, [r, theta]))
p = polar.point([1, 3*pi/4])
print(rect.point_to_coords(p))
print(rect.coord_function(0)(p))
print(rect.coord_function(1)(p))
v_x = rect.base_vector(0)
x = rect.coord_function(0)
from sympy.diffgeom import Manifold, Patch, CoordSystem
from sympy.diffgeom import TensorProduct as TP
from sympy.diffgeom import metric_to_Riemann_components as Riemann
from sympy.diffgeom import metric_to_Christoffel_2nd as Christoffel
from sympy import Symbol, Function, latex, sin
n = 2
M = Manifold('M', n)
P = Patch('P', M)
coord = CoordSystem('coord', P, ['x%s'%i for i in range(n)])
x = coord.coord_functions()
dx = coord.base_oneforms()
g = [[1, 0], [0, sin(x[0])**2]]
metric = sum([g[i][j]*TP(dx[i], dx[j]) for i in range(n) for j in range(n)])
C = Christoffel(metric)
R = Riemann(metric)
# Basic imports and functions
from sympy import latex, symbols, sin, cos, pi, simplify
from scipy.integrate import solve_ivp
from sympy.diffgeom import (Manifold, Patch, CoordSystem,
metric_to_Christoffel_2nd, TensorProduct as TP)
# def lprint(v):
# display(Math(latex(v)))
# Create a manifold.
M = Manifold('M', 4)
# Create a patch.
patch = Patch('P', M)
# Basic symbols
c, r_s = symbols('c r_s')
# Coordinate system
schwarzchild_coord = CoordSystem('schwarzchild', patch,
['t', 'r', 'theta', 'phi'])
# Get the coordinate functions
t, r, theta, phi = schwarzchild_coord.coord_functions()
# Get the base one forms.
dt, dr, dtheta, dphi = schwarzchild_coord.base_oneforms()
# Auxiliar terms for the metric.
from sympy import symbols, sin, cos, pi
from sympy.diffgeom import Manifold, Patch, CoordSystem
r, theta = symbols('r, theta')
m = Manifold('M', 2)
patch = Patch('P', m)
rect = CoordSystem('rect', patch)
polar = CoordSystem('polar', patch)
rect in patch.coord_systems
polar.connect_to(rect, [r, theta], [r*cos(theta), r*sin(theta)])
polar.coord_tuple_transform_to(rect, [0, 2])
polar.coord_tuple_transform_to(rect, [2, pi/2])
rect.coord_tuple_transform_to(polar, [1, 1])
polar.jacobian(rect, [r, theta])
p = polar.point([1, 3*pi/4])
rect.point_to_coords(p)
#Define a basis scalar field (i.e. a coordinate function), that takes a point and returns its coordinates. It is an instance of BaseScalarField.
rect.coord_function(0)(p)
rect.coord_function(1)(p)
#Define a basis vector field (i.e. a unit vector field along the coordinate line). Vectors are also differential operators on scalar fields. It is an instance of BaseVectorField.
v_x = rect.base_vector(0)
x = rect.coord_function(0)
v_x(x)
v_x(v_x(x))
#Define a basis oneform field:
dx = rect.base_oneform(0)
dx(v_x)
#If you provide a list of names the fields will print nicely: - without provided names:
x, v_x, dx
#with provided names
rect = CoordSystem('rect', patch, ['x', 'y'])
rect.coord_function(0), rect.base_vector(0), rect.base_oneform(0)