First variation

In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional $\delta J(y)$ mapping the function h to

$\delta J(y,h)=\lim _{\varepsilon \to 0}{\frac {J(y+\varepsilon h)-J(y)}{\varepsilon }}=\left.{\frac {d}{d\varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0},$

where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.

Example

Compute the first variation of

$J(y)=\int _{a}^{b}yy'dx.$

From the definition above,

${\begin{aligned}\delta J(y,h)&=\left.{\frac {d}{d\varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&=\left.{\frac {d}{d\varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.{\frac {d}{d\varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}{\frac {d}{d\varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ dx\end{aligned}}$

See also

• Calculus of variations
• Functional derivative