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from sympy.core.function import (Derivative as D, Function) |
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from sympy.core.relational import Eq |
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from sympy.core.symbol import (Symbol, symbols) |
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from sympy.functions.elementary.trigonometric import (cos, sin) |
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from sympy.testing.pytest import raises |
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from sympy.calculus.euler import euler_equations as euler |
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def test_euler_interface(): |
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x = Function('x') |
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y = Symbol('y') |
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t = Symbol('t') |
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raises(TypeError, lambda: euler()) |
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raises(TypeError, lambda: euler(D(x(t), t)*y(t), [x(t), y])) |
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raises(ValueError, lambda: euler(D(x(t), t)*x(y), [x(t), x(y)])) |
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raises(TypeError, lambda: euler(D(x(t), t)**2, x(0))) |
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raises(TypeError, lambda: euler(D(x(t), t)*y(t), [t])) |
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assert euler(D(x(t), t)**2/2, {x(t)}) == [Eq(-D(x(t), t, t), 0)] |
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assert euler(D(x(t), t)**2/2, x(t), {t}) == [Eq(-D(x(t), t, t), 0)] |
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def test_euler_pendulum(): |
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x = Function('x') |
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t = Symbol('t') |
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L = D(x(t), t)**2/2 + cos(x(t)) |
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assert euler(L, x(t), t) == [Eq(-sin(x(t)) - D(x(t), t, t), 0)] |
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def test_euler_henonheiles(): |
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x = Function('x') |
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y = Function('y') |
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t = Symbol('t') |
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L = sum(D(z(t), t)**2/2 - z(t)**2/2 for z in [x, y]) |
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L += -x(t)**2*y(t) + y(t)**3/3 |
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assert euler(L, [x(t), y(t)], t) == [Eq(-2*x(t)*y(t) - x(t) - |
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D(x(t), t, t), 0), |
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Eq(-x(t)**2 + y(t)**2 - |
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y(t) - D(y(t), t, t), 0)] |
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def test_euler_sineg(): |
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psi = Function('psi') |
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t = Symbol('t') |
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x = Symbol('x') |
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L = D(psi(t, x), t)**2/2 - D(psi(t, x), x)**2/2 + cos(psi(t, x)) |
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assert euler(L, psi(t, x), [t, x]) == [Eq(-sin(psi(t, x)) - |
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D(psi(t, x), t, t) + |
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D(psi(t, x), x, x), 0)] |
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def test_euler_high_order(): |
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m = Symbol('m') |
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k = Symbol('k') |
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x = Function('x') |
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y = Function('y') |
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t = Symbol('t') |
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L = (m*D(x(t), t)**2/2 + m*D(y(t), t)**2/2 - |
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k*D(x(t), t)*D(y(t), t, t) + k*D(y(t), t)*D(x(t), t, t)) |
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assert euler(L, [x(t), y(t)]) == [Eq(2*k*D(y(t), t, t, t) - |
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m*D(x(t), t, t), 0), |
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Eq(-2*k*D(x(t), t, t, t) - |
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m*D(y(t), t, t), 0)] |
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w = Symbol('w') |
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L = D(x(t, w), t, w)**2/2 |
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assert euler(L) == [Eq(D(x(t, w), t, t, w, w), 0)] |
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def test_issue_18653(): |
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x, y, z = symbols("x y z") |
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f, g, h = symbols("f g h", cls=Function, args=(x, y)) |
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f, g, h = f(), g(), h() |
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expr2 = f.diff(x)*h.diff(z) |
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assert euler(expr2, (f,), (x, y)) == [] |
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