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 To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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