# Definition:Euler's Equation for Vanishing Variation

## Definition

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

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

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

$\displaystyle\int_a^b F \paren{x,y,y'}\d x$

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

$\displaystyle\delta\int_a^b F \paren{x,y,y'}\d x=0$

In other words:

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

## Source of Name

This entry was named for Leonhard Paul Euler.