First variation explained

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} (J(y + \varepsilon h) - J(y))/(\varepsilon) = \left. (d)/(d\varepsilon) J(y + \varepsilon h)\right| _{\varepsilon} ,

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 ^{b} _{a} yy' dx.

From the definition above,

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

See also

• Calculus of variations
• Functional derivative