Definition:Euler's Equation for Vanishing Variation

Equation
Let $y \left({x}\right)$ be a real function.

Let $F \left({x, y, z}\right)$ be a real function belonging to $C^2$ w.r.t. all its variables.

Let $J \left[{y}\right]$ be a functional of the form:


 * $\displaystyle \int_a^b F \left({x, y, y'}\right) \mathrm d x$

Then Euler's equation for vanishing variation is defined a differential equation, resulting from condition:


 * $ \displaystyle \delta \int_a^b F \left ( { x, y, y' } \right ) \mathrm d x = 0$

In other words:


 * $\displaystyle F_y - \frac {\mathrm d} {\mathrm d x} F_{y'} = 0$