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

$\delta J(y,h)=\underset{\epsilon \to 0}{\lim }\frac{J(y+\epsilon h)-J(y)}{\epsilon }={\frac{\mathrm{d}}{\mathrm{d}\epsilon }J(y+\epsilon h)|}_{\epsilon =0},$

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

$J(y)=\underset{a}{\overset{b}{\int }}y{y}^{\prime }\mathrm{d}x.$

From the definition above,

$\begin{array}{cc}\delta J(y,h)& ={\frac{\mathrm{d}}{\mathrm{d}\epsilon }J(y+\epsilon h)|}_{\epsilon =0}\\ & ={\frac{\mathrm{d}}{\mathrm{d}\epsilon }\underset{a}{\overset{b}{\int }}(y+\epsilon h)(y^{\prime }+\epsilon h^{\prime })\mathrm{d}x|}_{\epsilon =0}\\ & ={\frac{\mathrm{d}}{\mathrm{d}\epsilon }\underset{a}{\overset{b}{\int }}(y{y}^{\prime }+y\epsilon h^{\prime }+y^{\prime }\epsilon h+{\epsilon }^{2}h{h}^{\prime })\mathrm{d}x|}_{\epsilon =0}\\ & ={\underset{a}{\overset{b}{\int }}\frac{\mathrm{d}}{\mathrm{d}\epsilon }(y{y}^{\prime }+y\epsilon h^{\prime }+y^{\prime }\epsilon h+{\epsilon }^{2}h{h}^{\prime })\mathrm{d}x|}_{\epsilon =0}\\ & ={\underset{a}{\overset{b}{\int }}(y{h}^{\prime }+y^{\prime }h+2\epsilon h{h}^{\prime })\mathrm{d}x|}_{\epsilon =0}\\ & =\underset{a}{\overset{b}{\int }}(y{h}^{\prime }+y^{\prime }h)\mathrm{d}x\\ \end{array}$

See also

• Calculus of variations
• Functional derivative