變分

變分是在應用數學與變分法中泛函應對與函數中的微分使用的概念。具體可以分為泛函的變分、函數的變分等。^[1]

函數的變分

設極值曲線為${\displaystyle {\hat {y}}={\hat {y}}(x)}$，可取曲線為${\displaystyle y=y(x)}$。定義${\displaystyle \delta {y}={\hat {y}}-y}$為y的一次變分，即函數y的增量。從而可得${\displaystyle \delta {y'}={\hat {y}}'-y'}$

對隱函數${\displaystyle \varphi (x,y)=0}$，其一次變分即為全微分：${\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}+\delta x{\frac {\partial {\varphi }}{\partial {x}}}}$。由於x無增量，即${\displaystyle \delta x=0}$，故有${\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}}$。

泛函的變分

對泛函${\displaystyle {\underset {y}{\min }}{J(y)}=\int _{x_{0}}^{x_{1}}F(x,y(x),y'(x))dx}$，

可得${\displaystyle J({\hat {y}})-J(y)=\int _{x_{0}}^{x_{1}}({\frac {\partial F}{\partial y}}\delta y+{\frac {\partial F}{\partial y'}}\delta y')dx+O(\delta y)}$，其一次變分是其Taylor級數的一次項，即${\displaystyle \delta J=\int _{x_{0}}^{x_{1}}({\frac {\partial F}{\partial y}}\delta y+{\frac {\partial F}{\partial y'}}\delta y')dx}$，或直接定義一次變分為 ${\displaystyle \delta J(y,h)={\frac {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{\varepsilon =0}\,}$ 。

故其二次變分為其Taylor級數的二次項，即${\displaystyle \delta ^{2}J={\frac {1}{2}}\int _{x_{0}}^{x_{1}}({\frac {\partial ^{2}F}{\partial y^{2}}}(\delta y)^{2}+{\frac {\partial ^{2}F}{\partial y\partial y'}}\delta y\delta y'+{\frac {\partial ^{2}F}{\partial y'^{2}}}(\delta y')^{2})dx}$。

需要注意，與二階微分${\displaystyle d^{2}y=d(dy)}$不同，泛函的二次變分不是對其一次變分再取變分。

實例

計算 ${\displaystyle J(y)=\int _{a}^{b}yy'dx\,}$的一次變分？

${\displaystyle \delta J(y,h)\,}$	${\displaystyle ={\frac {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{\varepsilon =0}}$
	${\displaystyle ={\frac {d}{d\varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ dx\left.\right|_{\varepsilon =0}}$
	${\displaystyle ={\frac {d}{d\varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\left.\right|_{\varepsilon =0}}$
	${\displaystyle =\int _{a}^{b}{\frac {d}{d\varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\left.\right|_{\varepsilon =0}}$
	${\displaystyle =\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ dx\left.\right|_{\varepsilon =0}}$
	${\displaystyle =\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ dx}$

參見

• 泛函
• 變分法
• 控制論

參考文獻