Simplest Variational Problem

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Problem

Let $ F\paren{x,y,z}$ be a function of a differentiability class $C^2$ WRT all its arguments.

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

$y\paren a=A$
$y\paren b=B$


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

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

has a weak extremum.



Sources