Extremal of Functional/Examples/Brachistochrone

Examples of Extremals of Functionals

The brachistochrone is the extremal for the functional:

$\map \phi f = \ds \int_a^b \sqrt {\paren {\dfrac {1 + \paren {\map {f'} x}^2} {2 g \map f x} } } \rd x$

where $g$ denotes acceleration due to gravity.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} 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

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): extremal