Definition:Euler's Equation for Vanishing Variation

Definition
Let $\map y x$ be a real function.

Let $\map F {x, y, z}$ be a real function belonging to $C^2$ all its variables.

Let $J \sqbrk y$ be a functional of the form:


 * $\displaystyle \int_a^b \map F {x, y, y'} \rd x$

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


 * $\displaystyle \delta \int_a^b \map F {x, y, y'} \rd x = 0$

In other words:


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