变分

变分是在應用數學與變分法中泛函应对与函数中的微分使用的概念。具体可以分为泛函的变分、函数的变分等。^[1]

函数的变分

设极值曲线为${\displaystyle {\hat {y}}={\hat {y}}(x)}$，可取曲线为${\displaystyle y=y(x)}$。定义${\displaystyle \delta {y}={\hat {y}}-y}$为y的一次变分，即函数y的增量。从而可得${\displaystyle \delta {y'}={\hat {y}}'-y'}$

对隐函数${\displaystyle \varphi (x,y)=0}$，其一次变分即为全微分：${\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}+\delta x{\frac {\partial {\varphi }}{\partial {x}}}}$。由于x无增量，即${\displaystyle \delta x=0}$，故有${\displaystyle {\delta }{\varphi }=\delta y{\frac {\partial {\varphi }}{\partial {y}}}}$。

泛函的变分

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

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

故其二次变分为其Taylor级数的二次项，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，与二阶微分${\displaystyle d^{2}y=d(dy)}$不同，泛函的二次变分不是对其一次变分再取变分。

實例

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

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

参见

• 泛函
• 变分法
• 控制论

参考文献

脚注

1. 吴, 受章. 最优控制理论与应用.

外链

• Exampleproblems.com（页面存档备份，存于互联网档案馆）有更多計算泛函一次變分的實例。