Simplest Variational Problem

Problem

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

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

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


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

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

has a weak extremum.


This needs considerable tedious hard slog to complete it.
In particular: minor refinements
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Sources

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