First variation

Remarks

It is also called the Gâteaux variation; it is a directional derivative of $J$. When the Gateaux derivative 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 $J(y) = \int_a^b L(x, y, y') \, dx$ 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 $\delta J(y; \eta) = 0$.