﻿ The Calculus of Variations in Functionals
San José State University

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The Calculus of Variations
in Functionals

This field of mathematics should probably be called The Calculus of Functionals. But the interest in this field arose from an interest in extremes and the conditions for extremes involve variations and thus arose the name.

## The Notion of Functionals

A functional is just a function from a set of functions on an interval of real numbers to the set of real nmbers. It could be as simple as the maximum of a particular set of functions defined on the interval [0, 1]. More typical are the functionals defined in terms of an integral; e.g.,

#### F(f) = ∫abf(x)dx

Functionals could be defined for more than one set of functions. For example let A and B be two sets of functions. The F: A×B → R would be a binary functional. Not much has bring done with multinary functionals.

Much of the Calculus of Variations involves functionals which are dependent upon the derivative of the function as well as the function itself. For example, Let g(x, y, z) be a function of three variables. Then

#### F[y] = ∫abg(x, y(x), y'(x))dx

is a functional. Functionals are customarily expressed using square brackets [.] instead of parentheses (. ).

## Function Spaces

Functional can be subjected to fuller analysis if the sets of functions they are defined over constitute a generalization of a vector space. This means that for any f, g and h in the set and α and β real numbers

• f + g = g + f
(Commutation)
• (f + g) + h = f + (g + h)
(Associativity)
• There exists an element 0
such that
f + 0 = f
for all f.
• For each f there exist an element −f
such that f + (−f) = 0
• 1·f = f
• α·(f + g) = α·f + g (Distributivity)
• (α + β)·f = α·f + β·f
• α·(β·f) = (α·β)·f

A function space may be considered a vector space with the dimensionality which is the cardinality of the continuum.

A function space is normed if there exists a real valued nonnegative function ||. || over the set of functions such that

• ||f|| = 0 if and only if f=0
• ||α·f|| = |α|·||f||
• ||f + g|| ≤ ||f|| + ||g||

The distance between f and g is ||f −g||= ||f +(−g)||

A functional F[f] is continuous at a point f=f0 if
for any ε>0 there exists a δ>0
such that if ||f−f0|| < δ
then |F[f] − F[f0]| < ε.

## Variations of Functionals

• Let δ be a function in a function space and F[f] a functional over that same space. The variation in F[f] with respect to δ is

## Necessary Conditions for Extrema

• A necessary condition for a function f0 to produce an extreme of a functional F[f] is that

#### Δ*δF[f0] = 0 for all admissible δ

• Euler's Equation:

Let F[f] be a functional of of the form

#### F[f] = ∫abg(x, f(x), f' (x))dx

defined over a set of functions {f} having continuous first derivatives over the interval [a, b] and having the boundary conditions f(a)=A and f(b)=B. A nessary condition for F[f] to have an extreme for f=f0 is that

#### (∂F/∂f(x)) − (d(∂F/∂f)/dx)) = 0 evaluated at f0 for all x.

(To be continued.)