Definition:Euler's Equation for Vanishing Variation

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

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

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

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

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

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

In other words:

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

Source of Name

This entry was named for Leonhard Paul Euler.