text: 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)) == []
  