343 lines
14 KiB
Python
343 lines
14 KiB
Python
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from sympy.core import Lambda, Symbol, symbols
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from sympy.diffgeom.rn import R2, R2_p, R2_r, R3_r, R3_c, R3_s, R2_origin
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from sympy.diffgeom import (Manifold, Patch, CoordSystem, Commutator, Differential, TensorProduct,
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WedgeProduct, BaseCovarDerivativeOp, CovarDerivativeOp, LieDerivative,
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covariant_order, contravariant_order, twoform_to_matrix, metric_to_Christoffel_1st,
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metric_to_Christoffel_2nd, metric_to_Riemann_components,
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metric_to_Ricci_components, intcurve_diffequ, intcurve_series)
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from sympy.simplify import trigsimp, simplify
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from sympy.functions import sqrt, atan2, sin
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from sympy.matrices import Matrix
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from sympy.testing.pytest import raises, nocache_fail
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from sympy.testing.pytest import warns_deprecated_sympy
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TP = TensorProduct
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def test_coordsys_transform():
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# test inverse transforms
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p, q, r, s = symbols('p q r s')
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rel = {('first', 'second'): [(p, q), (q, -p)]}
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R2_pq = CoordSystem('first', R2_origin, [p, q], rel)
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R2_rs = CoordSystem('second', R2_origin, [r, s], rel)
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r, s = R2_rs.symbols
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assert R2_rs.transform(R2_pq) == Matrix([[-s], [r]])
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# inverse transform impossible case
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a, b = symbols('a b', positive=True)
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rel = {('first', 'second'): [(a,), (-a,)]}
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R2_a = CoordSystem('first', R2_origin, [a], rel)
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R2_b = CoordSystem('second', R2_origin, [b], rel)
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# This transformation is uninvertible because there is no positive a, b satisfying a = -b
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with raises(NotImplementedError):
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R2_b.transform(R2_a)
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# inverse transform ambiguous case
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c, d = symbols('c d')
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rel = {('first', 'second'): [(c,), (c**2,)]}
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R2_c = CoordSystem('first', R2_origin, [c], rel)
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R2_d = CoordSystem('second', R2_origin, [d], rel)
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# The transform method should throw if it finds multiple inverses for a coordinate transformation.
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with raises(ValueError):
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R2_d.transform(R2_c)
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# test indirect transformation
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a, b, c, d, e, f = symbols('a, b, c, d, e, f')
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rel = {('C1', 'C2'): [(a, b), (2*a, 3*b)],
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('C2', 'C3'): [(c, d), (3*c, 2*d)]}
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C1 = CoordSystem('C1', R2_origin, (a, b), rel)
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C2 = CoordSystem('C2', R2_origin, (c, d), rel)
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C3 = CoordSystem('C3', R2_origin, (e, f), rel)
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a, b = C1.symbols
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c, d = C2.symbols
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e, f = C3.symbols
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assert C2.transform(C1) == Matrix([c/2, d/3])
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assert C1.transform(C3) == Matrix([6*a, 6*b])
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assert C3.transform(C1) == Matrix([e/6, f/6])
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assert C3.transform(C2) == Matrix([e/3, f/2])
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a, b, c, d, e, f = symbols('a, b, c, d, e, f')
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rel = {('C1', 'C2'): [(a, b), (2*a, 3*b + 1)],
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('C3', 'C2'): [(e, f), (-e - 2, 2*f)]}
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C1 = CoordSystem('C1', R2_origin, (a, b), rel)
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C2 = CoordSystem('C2', R2_origin, (c, d), rel)
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C3 = CoordSystem('C3', R2_origin, (e, f), rel)
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a, b = C1.symbols
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c, d = C2.symbols
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e, f = C3.symbols
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assert C2.transform(C1) == Matrix([c/2, (d - 1)/3])
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assert C1.transform(C3) == Matrix([-2*a - 2, (3*b + 1)/2])
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assert C3.transform(C1) == Matrix([-e/2 - 1, (2*f - 1)/3])
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assert C3.transform(C2) == Matrix([-e - 2, 2*f])
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# old signature uses Lambda
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a, b, c, d, e, f = symbols('a, b, c, d, e, f')
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rel = {('C1', 'C2'): Lambda((a, b), (2*a, 3*b + 1)),
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('C3', 'C2'): Lambda((e, f), (-e - 2, 2*f))}
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C1 = CoordSystem('C1', R2_origin, (a, b), rel)
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C2 = CoordSystem('C2', R2_origin, (c, d), rel)
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C3 = CoordSystem('C3', R2_origin, (e, f), rel)
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a, b = C1.symbols
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c, d = C2.symbols
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e, f = C3.symbols
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assert C2.transform(C1) == Matrix([c/2, (d - 1)/3])
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assert C1.transform(C3) == Matrix([-2*a - 2, (3*b + 1)/2])
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assert C3.transform(C1) == Matrix([-e/2 - 1, (2*f - 1)/3])
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assert C3.transform(C2) == Matrix([-e - 2, 2*f])
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def test_R2():
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x0, y0, r0, theta0 = symbols('x0, y0, r0, theta0', real=True)
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point_r = R2_r.point([x0, y0])
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point_p = R2_p.point([r0, theta0])
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# r**2 = x**2 + y**2
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assert (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_r) == 0
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assert trigsimp( (R2.r**2 - R2.x**2 - R2.y**2).rcall(point_p) ) == 0
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assert trigsimp(R2.e_r(R2.x**2 + R2.y**2).rcall(point_p).doit()) == 2*r0
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# polar->rect->polar == Id
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a, b = symbols('a b', positive=True)
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m = Matrix([[a], [b]])
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#TODO assert m == R2_r.transform(R2_p, R2_p.transform(R2_r, [a, b])).applyfunc(simplify)
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assert m == R2_p.transform(R2_r, R2_r.transform(R2_p, m)).applyfunc(simplify)
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# deprecated method
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with warns_deprecated_sympy():
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assert m == R2_p.coord_tuple_transform_to(
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R2_r, R2_r.coord_tuple_transform_to(R2_p, m)).applyfunc(simplify)
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def test_R3():
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a, b, c = symbols('a b c', positive=True)
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m = Matrix([[a], [b], [c]])
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assert m == R3_c.transform(R3_r, R3_r.transform(R3_c, m)).applyfunc(simplify)
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#TODO assert m == R3_r.transform(R3_c, R3_c.transform(R3_r, m)).applyfunc(simplify)
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assert m == R3_s.transform(
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R3_r, R3_r.transform(R3_s, m)).applyfunc(simplify)
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#TODO assert m == R3_r.transform(R3_s, R3_s.transform(R3_r, m)).applyfunc(simplify)
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assert m == R3_s.transform(
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R3_c, R3_c.transform(R3_s, m)).applyfunc(simplify)
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#TODO assert m == R3_c.transform(R3_s, R3_s.transform(R3_c, m)).applyfunc(simplify)
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with warns_deprecated_sympy():
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assert m == R3_c.coord_tuple_transform_to(
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R3_r, R3_r.coord_tuple_transform_to(R3_c, m)).applyfunc(simplify)
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#TODO assert m == R3_r.coord_tuple_transform_to(R3_c, R3_c.coord_tuple_transform_to(R3_r, m)).applyfunc(simplify)
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assert m == R3_s.coord_tuple_transform_to(
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R3_r, R3_r.coord_tuple_transform_to(R3_s, m)).applyfunc(simplify)
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#TODO assert m == R3_r.coord_tuple_transform_to(R3_s, R3_s.coord_tuple_transform_to(R3_r, m)).applyfunc(simplify)
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assert m == R3_s.coord_tuple_transform_to(
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R3_c, R3_c.coord_tuple_transform_to(R3_s, m)).applyfunc(simplify)
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#TODO assert m == R3_c.coord_tuple_transform_to(R3_s, R3_s.coord_tuple_transform_to(R3_c, m)).applyfunc(simplify)
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def test_CoordinateSymbol():
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x, y = R2_r.symbols
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r, theta = R2_p.symbols
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assert y.rewrite(R2_p) == r*sin(theta)
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def test_point():
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x, y = symbols('x, y')
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p = R2_r.point([x, y])
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assert p.free_symbols == {x, y}
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assert p.coords(R2_r) == p.coords() == Matrix([x, y])
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assert p.coords(R2_p) == Matrix([sqrt(x**2 + y**2), atan2(y, x)])
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def test_commutator():
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assert Commutator(R2.e_x, R2.e_y) == 0
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assert Commutator(R2.x*R2.e_x, R2.x*R2.e_x) == 0
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assert Commutator(R2.x*R2.e_x, R2.x*R2.e_y) == R2.x*R2.e_y
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c = Commutator(R2.e_x, R2.e_r)
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assert c(R2.x) == R2.y*(R2.x**2 + R2.y**2)**(-1)*sin(R2.theta)
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def test_differential():
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xdy = R2.x*R2.dy
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dxdy = Differential(xdy)
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assert xdy.rcall(None) == xdy
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assert dxdy(R2.e_x, R2.e_y) == 1
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assert dxdy(R2.e_x, R2.x*R2.e_y) == R2.x
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assert Differential(dxdy) == 0
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def test_products():
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assert TensorProduct(
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R2.dx, R2.dy)(R2.e_x, R2.e_y) == R2.dx(R2.e_x)*R2.dy(R2.e_y) == 1
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assert TensorProduct(R2.dx, R2.dy)(None, R2.e_y) == R2.dx
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assert TensorProduct(R2.dx, R2.dy)(R2.e_x, None) == R2.dy
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assert TensorProduct(R2.dx, R2.dy)(R2.e_x) == R2.dy
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assert TensorProduct(R2.x, R2.dx) == R2.x*R2.dx
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assert TensorProduct(
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R2.e_x, R2.e_y)(R2.x, R2.y) == R2.e_x(R2.x) * R2.e_y(R2.y) == 1
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assert TensorProduct(R2.e_x, R2.e_y)(None, R2.y) == R2.e_x
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assert TensorProduct(R2.e_x, R2.e_y)(R2.x, None) == R2.e_y
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assert TensorProduct(R2.e_x, R2.e_y)(R2.x) == R2.e_y
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assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
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assert TensorProduct(
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R2.dx, R2.e_y)(R2.e_x, R2.y) == R2.dx(R2.e_x) * R2.e_y(R2.y) == 1
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assert TensorProduct(R2.dx, R2.e_y)(None, R2.y) == R2.dx
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assert TensorProduct(R2.dx, R2.e_y)(R2.e_x, None) == R2.e_y
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assert TensorProduct(R2.dx, R2.e_y)(R2.e_x) == R2.e_y
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assert TensorProduct(R2.x, R2.e_x) == R2.x * R2.e_x
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assert TensorProduct(
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R2.e_x, R2.dy)(R2.x, R2.e_y) == R2.e_x(R2.x) * R2.dy(R2.e_y) == 1
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assert TensorProduct(R2.e_x, R2.dy)(None, R2.e_y) == R2.e_x
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assert TensorProduct(R2.e_x, R2.dy)(R2.x, None) == R2.dy
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assert TensorProduct(R2.e_x, R2.dy)(R2.x) == R2.dy
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assert TensorProduct(R2.e_y,R2.e_x)(R2.x**2 + R2.y**2,R2.x**2 + R2.y**2) == 4*R2.x*R2.y
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assert WedgeProduct(R2.dx, R2.dy)(R2.e_x, R2.e_y) == 1
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assert WedgeProduct(R2.e_x, R2.e_y)(R2.x, R2.y) == 1
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def test_lie_derivative():
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assert LieDerivative(R2.e_x, R2.y) == R2.e_x(R2.y) == 0
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assert LieDerivative(R2.e_x, R2.x) == R2.e_x(R2.x) == 1
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assert LieDerivative(R2.e_x, R2.e_x) == Commutator(R2.e_x, R2.e_x) == 0
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assert LieDerivative(R2.e_x, R2.e_r) == Commutator(R2.e_x, R2.e_r)
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assert LieDerivative(R2.e_x + R2.e_y, R2.x) == 1
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assert LieDerivative(
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R2.e_x, TensorProduct(R2.dx, R2.dy))(R2.e_x, R2.e_y) == 0
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@nocache_fail
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def test_covar_deriv():
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ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
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cvd = BaseCovarDerivativeOp(R2_r, 0, ch)
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assert cvd(R2.x) == 1
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# This line fails if the cache is disabled:
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assert cvd(R2.x*R2.e_x) == R2.e_x
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cvd = CovarDerivativeOp(R2.x*R2.e_x, ch)
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assert cvd(R2.x) == R2.x
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assert cvd(R2.x*R2.e_x) == R2.x*R2.e_x
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def test_intcurve_diffequ():
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t = symbols('t')
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start_point = R2_r.point([1, 0])
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vector_field = -R2.y*R2.e_x + R2.x*R2.e_y
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equations, init_cond = intcurve_diffequ(vector_field, t, start_point)
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assert str(equations) == '[f_1(t) + Derivative(f_0(t), t), -f_0(t) + Derivative(f_1(t), t)]'
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assert str(init_cond) == '[f_0(0) - 1, f_1(0)]'
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equations, init_cond = intcurve_diffequ(vector_field, t, start_point, R2_p)
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assert str(
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equations) == '[Derivative(f_0(t), t), Derivative(f_1(t), t) - 1]'
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assert str(init_cond) == '[f_0(0) - 1, f_1(0)]'
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def test_helpers_and_coordinate_dependent():
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one_form = R2.dr + R2.dx
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two_form = Differential(R2.x*R2.dr + R2.r*R2.dx)
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three_form = Differential(
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R2.y*two_form) + Differential(R2.x*Differential(R2.r*R2.dr))
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metric = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dy, R2.dy)
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metric_ambig = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dr, R2.dr)
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misform_a = TensorProduct(R2.dr, R2.dr) + R2.dr
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misform_b = R2.dr**4
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misform_c = R2.dx*R2.dy
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twoform_not_sym = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dx, R2.dy)
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twoform_not_TP = WedgeProduct(R2.dx, R2.dy)
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one_vector = R2.e_x + R2.e_y
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two_vector = TensorProduct(R2.e_x, R2.e_y)
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three_vector = TensorProduct(R2.e_x, R2.e_y, R2.e_x)
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two_wp = WedgeProduct(R2.e_x,R2.e_y)
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assert covariant_order(one_form) == 1
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assert covariant_order(two_form) == 2
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assert covariant_order(three_form) == 3
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assert covariant_order(two_form + metric) == 2
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assert covariant_order(two_form + metric_ambig) == 2
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assert covariant_order(two_form + twoform_not_sym) == 2
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assert covariant_order(two_form + twoform_not_TP) == 2
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assert contravariant_order(one_vector) == 1
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assert contravariant_order(two_vector) == 2
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assert contravariant_order(three_vector) == 3
|
||
|
|
assert contravariant_order(two_vector + two_wp) == 2
|
||
|
|
|
||
|
|
raises(ValueError, lambda: covariant_order(misform_a))
|
||
|
|
raises(ValueError, lambda: covariant_order(misform_b))
|
||
|
|
raises(ValueError, lambda: covariant_order(misform_c))
|
||
|
|
|
||
|
|
assert twoform_to_matrix(metric) == Matrix([[1, 0], [0, 1]])
|
||
|
|
assert twoform_to_matrix(twoform_not_sym) == Matrix([[1, 0], [1, 0]])
|
||
|
|
assert twoform_to_matrix(twoform_not_TP) == Matrix([[0, -1], [1, 0]])
|
||
|
|
|
||
|
|
raises(ValueError, lambda: twoform_to_matrix(one_form))
|
||
|
|
raises(ValueError, lambda: twoform_to_matrix(three_form))
|
||
|
|
raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: metric_to_Christoffel_1st(twoform_not_sym))
|
||
|
|
raises(ValueError, lambda: metric_to_Christoffel_2nd(twoform_not_sym))
|
||
|
|
raises(ValueError, lambda: metric_to_Riemann_components(twoform_not_sym))
|
||
|
|
raises(ValueError, lambda: metric_to_Ricci_components(twoform_not_sym))
|
||
|
|
|
||
|
|
|
||
|
|
def test_correct_arguments():
|
||
|
|
raises(ValueError, lambda: R2.e_x(R2.e_x))
|
||
|
|
raises(ValueError, lambda: R2.e_x(R2.dx))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: Commutator(R2.e_x, R2.x))
|
||
|
|
raises(ValueError, lambda: Commutator(R2.dx, R2.e_x))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: Differential(Differential(R2.e_x)))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: R2.dx(R2.x))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: LieDerivative(R2.dx, R2.dx))
|
||
|
|
raises(ValueError, lambda: LieDerivative(R2.x, R2.dx))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: CovarDerivativeOp(R2.dx, []))
|
||
|
|
raises(ValueError, lambda: CovarDerivativeOp(R2.x, []))
|
||
|
|
|
||
|
|
a = Symbol('a')
|
||
|
|
raises(ValueError, lambda: intcurve_series(R2.dx, a, R2_r.point([1, 2])))
|
||
|
|
raises(ValueError, lambda: intcurve_series(R2.x, a, R2_r.point([1, 2])))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: intcurve_diffequ(R2.dx, a, R2_r.point([1, 2])))
|
||
|
|
raises(ValueError, lambda: intcurve_diffequ(R2.x, a, R2_r.point([1, 2])))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: contravariant_order(R2.e_x + R2.dx))
|
||
|
|
raises(ValueError, lambda: covariant_order(R2.e_x + R2.dx))
|
||
|
|
|
||
|
|
raises(ValueError, lambda: contravariant_order(R2.e_x*R2.e_y))
|
||
|
|
raises(ValueError, lambda: covariant_order(R2.dx*R2.dy))
|
||
|
|
|
||
|
|
def test_simplify():
|
||
|
|
x, y = R2_r.coord_functions()
|
||
|
|
dx, dy = R2_r.base_oneforms()
|
||
|
|
ex, ey = R2_r.base_vectors()
|
||
|
|
assert simplify(x) == x
|
||
|
|
assert simplify(x*y) == x*y
|
||
|
|
assert simplify(dx*dy) == dx*dy
|
||
|
|
assert simplify(ex*ey) == ex*ey
|
||
|
|
assert ((1-x)*dx)/(1-x)**2 == dx/(1-x)
|
||
|
|
|
||
|
|
|
||
|
|
def test_issue_17917():
|
||
|
|
X = R2.x*R2.e_x - R2.y*R2.e_y
|
||
|
|
Y = (R2.x**2 + R2.y**2)*R2.e_x - R2.x*R2.y*R2.e_y
|
||
|
|
assert LieDerivative(X, Y).expand() == (
|
||
|
|
R2.x**2*R2.e_x - 3*R2.y**2*R2.e_x - R2.x*R2.y*R2.e_y)
|
||
|
|
|
||
|
|
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)
|