First variation

In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional [math]\displaystyle{ \delta J(y) }[/math] mapping the function h to

[math]\displaystyle{ \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}, }[/math]

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

[math]\displaystyle{ J(y)=\int_a^b yy' dx. }[/math]

From the definition above,

[math]\displaystyle{ \begin{align} \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{align} }[/math]

See also

• Calculus of variations
• Functional derivative