From Wikipedia, the free encyclopedia
In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional
mapping the function h to
![{\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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e87cc49bbf44c62510635a5921ee9e4a119565b)
where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.
Example[edit]
Compute the first variation of
![{\displaystyle J(y)=\int _{a}^{b}yy'dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98c26454f0caf3b648c547a3a084c6647022112c)
From the definition above,
![{\displaystyle {\begin{aligned}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66f630c776bd16b0d89c60a3417eb7d469a192d5)
See also[edit]