First Variation

tags:definition |variational

First Variation

Context

Statements

Definition. (Citation Needed) Let $\mathbf{V}$ be a function space, and let $J:\mathbf{V}\to \mathbb{R}$ be a Functional. The linear functional $\delta J{|}_y:\mathbf{V}\to \mathbb{R}$ is called the first variational of $J$ at $y$ if, for every $\eta \in \mathbf{V}$ and for every $\alpha \in \mathbb{R}$, the equality

$$J(y+\alpha \eta )=J(y)+\delta J{|}_y(\eta )\alpha +o(\alpha ),$$

holds, where $o$ represents higher-order terms (i.e. $o(\alpha )/\alpha \to 0$, as $\alpha \to 0$.).

References

Liberzon’s book