Definition:Euler's Equation for Vanishing Variation

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

Let $ F \left ( { x, y, z } \right ) $ be a real function belonging to $ C^2 $ 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 $