Definition:Euler's Equation for Vanishing Variation

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Definition

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:

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

This article is incomplete. In particular: Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Stub}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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