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| from sympy.core.function import (Derivative as D, Function) | |
| from sympy.core.relational import Eq | |
| from sympy.core.symbol import (Symbol, symbols) | |
| from sympy.functions.elementary.trigonometric import (cos, sin) | |
| from sympy.testing.pytest import raises | |
| from sympy.calculus.euler import euler_equations as euler | |
| 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_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_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_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_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)) == [] | |