---
tag: 7EY98D6MG0
kind: definition
name: First variation
msc: [49K05, 49J50, 46G05]
defines: 5B8NFYB2XQ
uses: [1EQM0DJ4NC, 000AYPA9AJ, 06KWGHDN2J, EAZF9NJW12]
sources:
  - url: https://en.wikipedia.org/wiki/First_variation
---

## Definition

Let $V$ be a real [[vector space|000AYPA9AJ]], let $A \subseteq V$, and let $J \colon A \to \R$ be a [[functional|1EQM0DJ4NC]].
Let $y \in A$ and $\eta \in V$, and suppose that $y + t\eta \in A$ for all real $t$ with $\abs{t}$ small enough.
The *first variation* of $J$ at $y$ in the direction $\eta$ is
$$\delta J(y; \eta) = \frac{d}{dt} J(y + t\eta) \Big|_{t = 0} = \lim_{t \to 0} \frac{J(y + t\eta) - J(y)}{t},$$
when this limit exists.

## Remarks

It is also called the Gâteaux variation; it is a [[directional derivative|06KWGHDN2J]] of $J$.
When the [[Gateaux derivative|EAZF9NJW12]] of $J$ at $y$ exists, $\delta J(y; \eta)$ is its value at $\eta$.
If $V$ is normed and $J$ is Fréchet differentiable at $y$, then $\delta J(y; \eta) = DJ(y)\eta$ for all $\eta$.

Example: on $V = C([0, 1])$ let $J(y) = \int_0^1 y^2 \, dx$.
Then $J(y + t\eta) = J(y) + 2t \int_0^1 y\eta \, dx + t^2 \int_0^1 \eta^2 \, dx$, so $\delta J(y; \eta) = 2 \int_0^1 y \eta \, dx$.

Non-example: for $J(y) = \int_0^1 \abs{y} \, dx$, $y = 0$ and $\eta = 1$, the quotient $(J(t\eta) - J(0))/t = \abs{t}/t$ has no limit as $t \to 0$, so the first variation does not exist.

For an [[integral functional|JRQZ803Q5Y]] $J(y) = \int_a^b L(x, y, y') \, dx$ with $L$ of class $C^1$, and $y, \eta \in C^1([a, b])$, differentiation under the integral sign gives
$$\delta J(y; \eta) = \int_a^b \big( L_z(x, y, y') \, \eta + L_p(x, y, y') \, \eta' \big) \, dx,$$
where $L_z$ and $L_p$ are the partial derivatives of $L(x, z, p)$.
If $y$ is a local minimum of $J$ along the line $t \mapsto y + t\eta$ and $\delta J(y; \eta)$ exists, then $\delta J(y; \eta) = 0$.