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

:$\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h)\_,$

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,

:$\begin \delta J(y,h)&=\left.\frac J(y + \varepsilon h)\_\\ &= \left.\frac \int_a^b (y + \varepsilon h)(y^\prime + \varepsilon h^\prime) \ dx\_\\ &= \left.\frac \int_a^b (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\_\\ &= \left.\int_a^b \frac (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\_\\ &= \left.\int_a^b (yh^\prime + y^\prime h + 2\varepsilon hh^\prime) \ dx\_\\ &= \int_a^b (yh^\prime + y^\prime h) \ dx \end$

See also

*Calculus of variations
*Functional derivative

Calculus of variations

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