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