Definition:Extremal of Functional

Definition

An extremal is a function which minimizes or maximizes a functional (usually a definite integral).

Examples

Brachistochrone

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.

Also see

• Results about extremals of functionals can be found here.

Sources

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