Calculus of Variations
Calculus of variations is the study of extrema of functionals. This page provides a brief introduction to some elements of the theory.
Variation and differentiability
Consider a linear functional , where is a normed linear space.
We define the differential of at as
- for all
We say that is differentiable at if there exist a linear functional and so that
and as .
Here, is called the variation or differential of at .
Uniqueness of variation
Lemma: Let denote a linear functional, where is defined as above.
If as , then is identically zero for all .
Proof: Suppose for some . Then, for
and hence does not approach zero as , a contradiction. QED
Theorem: For defined as above, is unique.
Proof: Suppose, if possible, that there exist two distinct variations and . Then,
Comparing the two equations, we get
- as .
Hence, by the lemma, or , a contradiction. QED
Function space norm
Consider a space consisting of all functions continuous on . We define the norm as
Generally, a space consists of all functions that are continuously differentiable times. In this case, we define the norm as
It can be shown that satisfies the conditions for a norm.
Strong and weak extrema
Consider some linear functional , where is a normed function space.
Then, we say that has a strong minimum at some if there exists some such that
for all , where .
Similarly, we define a weak minimum at some with in place of .
Notice that a strong minimum is also a weak minimum.
Strong and weak maximum are defined similarly with in place of .
We may also define extrema using norm for .
A necessary condition for an extremum
Theorem: If a differentiable functional has an extremum for some , then
- for all .
Proof: Suppose, if possible, that for some , where .
Let and consider .
As , we have and hence .
That is, if is sufficiently small, then and share the same sign.
However, . Hence, within any neighbourhood of , we have taking on either sign, a contradiction. QED
- Gelfand, I.M. and Fomin, S.V.: Calculus of Variations, Dover Publ., 2000.