def test_euler_sineg():
psi = Function('psi')
t = Symbol('t')
x = Symbol('x')
L = D(psi(t, x), t)**2/2 - D(psi(t, x), x)**2/2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [Eq(-sin(psi(t, x)) -
D(psi(t, x), t, t) +
D(psi(t, x), x, x), 0)]
def test_euler_sineg():
psi = Function("psi")
t = Symbol("t")
x = Symbol("x")
L = D(psi(t, x), t)**2 / 2 - D(psi(t, x), x)**2 / 2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [
Eq(-sin(psi(t, x)) - D(psi(t, x), t, t) + D(psi(t, x), x, x), 0)
]
def test_euler_sineg():
psi = Function('psi')
t = Symbol('t')
x = Symbol('x')
L = D(psi(t, x), t)**2 / 2 - D(psi(t, x), x)**2 / 2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [
Eq(-sin(psi(t, x)) - D(psi(t, x), t, t) + D(psi(t, x), x, x), 0)
]
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol('m')
k = Symbol('k')
x = Function('x')
y = Function('y')
t = Symbol('t')
L = (m * D(x(t), t)**2 / 2 + m * D(y(t), t)**2 / 2 -
k * D(x(t), t) * D(y(t), t, t) + k * D(y(t), t) * D(x(t), t, t))
assert euler(L, [x(t), y(t)]) == [
Eq(2 * k * D(y(t), t, t, t) - m * D(x(t), t, t), 0),
Eq(-2 * k * D(x(t), t, t, t) - m * D(y(t), t, t), 0)
]
w = Symbol('w')
L = D(x(t, w), t, w)**2 / 2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)]
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol('m')
k = Symbol('k')
x = Function('x')
y = Function('y')
t = Symbol('t')
L = (m*D(x(t), t)**2/2 + m*D(y(t), t)**2/2 -
k*D(x(t), t)*D(y(t), t, t) + k*D(y(t), t)*D(x(t), t, t))
assert euler(L, [x(t), y(t)]) == [Eq(2*k*D(y(t), t, t, t) -
m*D(x(t), t, t), 0),
Eq(-2*k*D(x(t), t, t, t) -
m*D(y(t), t, t), 0)]
w = Symbol('w')
L = D(x(t, w), t, w)**2/2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)]
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol("m")
k = Symbol("k")
x = Function("x")
y = Function("y")
t = Symbol("t")
L = (m * D(x(t), t)**2 / 2 + m * D(y(t), t)**2 / 2 -
k * D(x(t), t) * D(y(t), t, t) + k * D(y(t), t) * D(x(t), t, t))
assert euler(L, [x(t), y(t)]) == [
Eq(2 * k * D(y(t), t, t, t) - m * D(x(t), t, t), 0),
Eq(-2 * k * D(x(t), t, t, t) - m * D(y(t), t, t), 0),
]
w = Symbol("w")
L = D(x(t, w), t, w)**2 / 2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)]
def test_euler_henonheiles():
x = Function('x')
y = Function('y')
t = Symbol('t')
L = sum(D(z(t), t)**2 / 2 - z(t)**2 / 2 for z in [x, y])
L += -x(t)**2 * y(t) + y(t)**3 / 3
assert euler(L, [x(t), y(t)], t) == [
Eq(-2 * x(t) * y(t) - x(t) - D(x(t), t, t), 0),
Eq(-x(t)**2 + y(t)**2 - y(t) - D(y(t), t, t), 0)
]
def test_euler_henonheiles():
x = Function('x')
y = Function('y')
t = Symbol('t')
L = sum(D(z(t), t)**2/2 - z(t)**2/2 for z in [x, y])
L += -x(t)**2*y(t) + y(t)**3/3
assert euler(L, [x(t), y(t)], t) == [Eq(-2*x(t)*y(t) - x(t) -
D(x(t), t, t), 0),
Eq(-x(t)**2 + y(t)**2 -
y(t) - D(y(t), t, t), 0)]
def test_euler_high_order():
# an example from hep-th/0309038
m = Symbol("m")
k = Symbol("k")
x = Function("x")
y = Function("y")
t = Symbol("t")
L = (
m * D(x(t), t) ** 2 / 2
+ m * D(y(t), t) ** 2 / 2
- k * D(x(t), t) * D(y(t), t, t)
+ k * D(y(t), t) * D(x(t), t, t)
)
assert euler(L, [x(t), y(t)]) == [
Eq(2 * k * D(y(t), t, t, t) - m * D(x(t), t, t)),
Eq(-2 * k * D(x(t), t, t, t) - m * D(y(t), t, t)),
]
w = Symbol("w")
L = D(x(t, w), t, w) ** 2 / 2
assert euler(L) == [Eq(D(x(t, w), t, t, w, w))]
def test_euler_interface():
x = Function('x')
y = Symbol('y')
t = Symbol('t')
raises(TypeError, lambda: euler())
raises(TypeError, lambda: euler(D(x(t), t) * y(t), [x(t), y]))
raises(ValueError, lambda: euler(D(x(t), t) * x(y), [x(t), x(y)]))
raises(TypeError, lambda: euler(D(x(t), t)**2, x(0)))
raises(TypeError, lambda: euler(D(x(t), t) * y(t), [t]))
assert euler(D(x(t), t)**2 / 2, {x(t)}) == [Eq(-D(x(t), t, t), 0)]
assert euler(D(x(t), t)**2 / 2, x(t), {t}) == [Eq(-D(x(t), t, t), 0)]
def test_euler_interface():
x = Function('x')
y = Symbol('y')
t = Symbol('t')
raises(TypeError, lambda: euler())
raises(TypeError, lambda: euler(D(x(t), t)*y(t), [x(t), y]))
raises(ValueError, lambda: euler(D(x(t), t)*x(y), [x(t), x(y)]))
raises(TypeError, lambda: euler(D(x(t), t)**2, x(0)))
assert euler(D(x(t), t)**2/2, {x(t)}) == [Eq(-D(x(t), t, t), 0)]
assert euler(D(x(t), t)**2/2, x(t), {t}) == [Eq(-D(x(t), t, t), 0)]
def test_euler_interface():
x = Function("x")
y = Symbol("y")
t = Symbol("t")
raises(TypeError, lambda: euler())
raises(TypeError, lambda: euler(D(x(t), t) * y(t), [x(t), y]))
raises(ValueError, lambda: euler(D(x(t), t) * x(y), [x(t), x(y)]))
raises(TypeError, lambda: euler(D(x(t), t) ** 2, x(0)))
assert euler(D(x(t), t) ** 2 / 2, set([x(t)])) == [Eq(-D(x(t), t, t))]
assert euler(D(x(t), t) ** 2 / 2, x(t), set([t])) == [Eq(-D(x(t), t, t))]
def test_issue_18653():
x, y, z = symbols("x y z")
f, g, h = symbols("f g h", cls=Function, args=(x, y))
f, g, h = f(), g(), h()
expr2 = f.diff(x) * h.diff(z)
assert euler(expr2, (f, ), (x, y)) == []
def test_euler_pendulum():
x = Function('x')
t = Symbol('t')
L = D(x(t), t)**2 / 2 + cos(x(t))
assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t), 0)]
def test_euler_pendulum():
x = Function('x')
t = Symbol('t')
L = D(x(t), t)**2/2 + cos(x(t))
assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t), 0)]
def test_euler_sineg():
psi = Function("psi")
t = Symbol("t")
x = Symbol("x")
L = D(psi(t, x), t) ** 2 / 2 - D(psi(t, x), x) ** 2 / 2 + cos(psi(t, x))
assert euler(L, psi(t, x), [t, x]) == [Eq(-sin(psi(t, x)) - D(psi(t, x), t, t) + D(psi(t, x), x, x))]
def test_euler_pendulum():
x = Function("x")
t = Symbol("t")
L = D(x(t), t) ** 2 / 2 + cos(x(t))
assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t))]
