# Definition:Euler's Equation for Vanishing Variation

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## 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.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.4$: The Simplest Variational Problem. Euler's Equation