Definition:Extremal of Functional
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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