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 %2B \varepsilon h)-J(y)}{\varepsilon} = \left.\frac{d}{d\varepsilon} J(y %2B \varepsilon h)\right|_{\varepsilon = 0},

where y and h are functions, and ε is a scalar.

Example

Compute the first variation of

J(y)=\int_a^b yy' dx.

From the definition above,

\begin{align} \delta J(y)(h)&=\left.\frac{d}{d\varepsilon} J(y %2B \varepsilon h)\right|_{\varepsilon = 0}\\ &= \left.\frac{d}{d\varepsilon} \int_a^b (y %2B \varepsilon h)(y^\prime %2B \varepsilon h^\prime) \ dx\right|_{\varepsilon = 0}\\ &= \left.\frac{d}{d\varepsilon} \int_a^b (yy^\prime %2B y\varepsilon h^\prime %2B y^\prime\varepsilon h %2B \varepsilon^2 hh^\prime) \ dx\right|_{\varepsilon = 0}\\ &= \left.\int_a^b \frac{d}{d\varepsilon} (yy^\prime %2B y\varepsilon h^\prime %2B y^\prime\varepsilon h %2B \varepsilon^2 hh^\prime) \ dx\right|_{\varepsilon = 0}\\ &= \left.\int_a^b (yh^\prime %2B y^\prime h %2B 2\varepsilon hh^\prime) \ dx\right|_{\varepsilon = 0}\\ &= \int_a^b (yh^\prime %2B y^\prime h) \ dx \end{align}

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