# Simplest Variational Problem

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## Problem

Let $\map F {x, y, z}$ be a function of a differentiability class $C^2$ with respect to all its arguments.

Let $y: \R \to \R$ be a continuously differentiable function for $x \in \sqbrk {a, b}$ such that

- $\map y a = A$

- $\map y b = B$

Then among all functions $y$ find the one for which the functional

- $\displaystyle J \sqbrk y = \int_a^b \map F {x, y, y'} \rd x$

has a weak extremum.

## Sources

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