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:
- $\displaystyle \int_a^b \map F {x, y, y'} \rd x$
Then Euler's equation for vanishing variation is defined a differential equation, resulting from condition:
- $\displaystyle \delta \int_a^b \map F {x, y, y'} \rd x = 0$
In other words:
- $\displaystyle F_y - \dfrac \d {\d x} F_{y'} = 0$
Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms
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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