First variation

Let $𝑉$ be a function space over $𝕂$ ¹ and $𝐹 : 𝑉 \to 𝕂$ be a functional, e.g. an Action functional. Loosely speaking, the first variation $𝛿 𝐹$ tells you how the functional $𝐹$ varies in response to an infinitesimal variation $𝑓 \to 𝑓'$ in its input, #m/def/anal/fun/var i.e.

𝛿 𝐹 = 𝐹 [𝑓'] - 𝐹 [𝑓] .

This is made rigorous by considering $𝛿 𝐹$ to be a functional in its own right.

A variation of $𝑓$ is a map $𝛼 ? : (- 𝜖 0, 𝜖 0) \to 𝐹$ such that $𝛼 0 = 𝑓$ . Let $𝒱 𝑓$ denote the function space of all variations of $𝑓$ .² We define the functional $𝛿 𝐹 [𝑓] : 𝒱 𝑓 \to ℝ$ so that

𝛿 𝐹 [𝑓; 𝛼] = 𝛿 𝐹 [𝑓] [𝛼] = l i m 𝜖 \to 0 𝐹 [𝛼 𝜖] - 𝐹 [𝑓] 𝜖 = 𝐹 [𝛼 ?]' (0)

This generalizes easily to the $𝑛$ th variation

𝛿 𝑛 𝐹 [𝑓; 𝛼] = 𝐹 [𝛼 ?] (𝑛) (0) .

Extrema of functionals

The main utility of $𝑛$ th variation is for identifying extrema of functionals as a necessary (but in general insufficient) condition under certain hypotheses. Suppose $𝑉$ is topological and $𝐹 : 𝑉 \to 𝕂$ is continuous. Further let $𝑓 0 \in 𝑉$ be a local extremum of $𝐹$ .

If $\delta F[f_{0}]$ exists, then it is zero.
- If $𝑓 0$ is a local minimum and $𝛿 2 𝐹 [𝑓 0]$ exists, then it is strictly positive.
- If $𝑓 0$ is a local maximum and $𝛿 2 𝐹 [𝑓 0]$ exists, then it is strictly negative.

Proof

#missing/proof

First variation

Extrema of functionals

See also

Footnotes