范函变分

范函变分的定义

设$\mathrm{F}\left(\mathrm{x}, \mathrm{y}(\mathrm{x}), \mathrm{y}^{\prime}(\mathrm{x})\right)$F(x,y(x),y′(x))是三个独立变量$x, y(x), y^{\prime}(x)$x,y(x),y′(x)在区间$\left[x_{0}, x_{1}\right]$[x0​,x1​]上的已知函数,且二阶连续可微,其中$y(x)$y(x)和$y^{\prime}(x)$y′(x)是x的未知函数,则范函$J[y(x)]=\int_{x_{0}}^{x_{1}} F\left(x, y(x), y^{\prime}(x)\right)dx$J[y(x)]=∫x0​x1​​F(x,y(x),y′(x))dx被称为最简范函,被积函数F称为范函的核.
在$y=y(x)$y=y(x)的一阶邻域内.任取一曲线$y=y_{1}(x)$y=y1​(x),则:$\delta y=y_{1}(x)-y(x), \delta y^{\prime}=y_{1}^{\prime}(x)-y^{\prime}(x)$δy=y1​(x)−y(x),δy′=y1′​(x)−y′(x)由泰勒展开式,最简范函$J[y(x)]$J[y(x)]的增量为:$\begin{aligned} \Delta J &=J\left[y_{1}(x)\right]-J[y(x)]=J[y(x)+\delta y]-J[y(x)] \\ &=\int_{x_{0}}^{x_{1}} F\left(x, y+\delta y, y^{\prime}+\delta y^{\prime}\right) d x-\int_{x_{0}}^{x_{1}} F\left(x, y, y^{\prime}\right) d x \\ &=\int_{x_{0}}^{x_{1}}\left[F\left(x, y+\delta y, y^{\prime}+\delta y^{\prime}\right)-F\left(x, y, y^{\prime}\right)\right] d x \\ &=\int_{x_{0}}^{x_{1}}\left(F_{y} \delta y+F_{y^{\prime}} \delta y^{\prime}\right) d x+\cdots+\int_{x_{0}}^{x_{1}}\left[\left(\delta y \frac{\partial}{\partial y}+\delta y^{\prime} \frac{\partial}{\partial y^{\prime}}\right)^{n} F\right] d x+R_{n} \end{aligned}$ΔJ​=J[y1​(x)]−J[y(x)]=J[y(x)+δy]−J[y(x)]=∫x0​x1​​F(x,y+δy,y′+δy′)dx−∫x0​x1​​F(x,y,y′)dx=∫x0​x1​​[F(x,y+δy,y′+δy′)−F(x,y,y′)]dx=∫x0​x1​​(Fy​δy+Fy′​δy′)dx+⋯+∫x0​x1​​[(δy∂y∂​+δy′∂y′∂​)nF]dx+Rn​​把$\int_{x_{0}}^{x_{1}}\left(F_{y} \delta y+F_{y^{\prime}} \delta y^{\prime}\right) d x$∫x0​x1​​(Fy​δy+Fy′​δy′)dx称为范函的变分,记作$\delta J$δJ.

范函变分的另一种定义

对于任意定值$x \in\left[x_{0}, x_{1}\right]$x∈[x0​,x1​],设函数的增量为$\delta y=y(x)-y_{0}(x)=\epsilon \eta(x)$δy=y(x)−y0​(x)=ϵη(x),则范函$J[y(x)+\epsilon \eta(x)]$J[y(x)+ϵη(x)]可以看成关于$\epsilon$ϵ的函数$\Phi(\epsilon)$Φ(ϵ),将其在$\ \epsilon=0 $处泰勒展开:
$\Phi(\epsilon)=\Phi(0)+\Phi^{\prime}(0) \epsilon+\cdots+\frac{\Phi^{n}(0)}{n !} \epsilon^{n}+o\left(\epsilon^{n+1}\right)$Φ(ϵ)=Φ(0)+Φ′(0)ϵ+⋯+n!Φn(0)​ϵn+o(ϵn+1)因此:
$J[y+\epsilon \eta]=J[y]+\left.\frac{d[J(y+\epsilon \eta]}{d \epsilon}\right|_{\epsilon=0} \cdot \epsilon+\left.\left(\frac{d^{2} J}{d \epsilon^{2}}\right)\right|_{\epsilon=0} \cdot \frac{\epsilon^{2}}{2 !}+\cdots+\left.\left(\frac{d^{n} J}{d \epsilon^{n}}\right)\right|_{\epsilon=0} \cdot \frac{\epsilon^{n}}{n !}+o\left(\epsilon^{n+1}\right)$J[y+ϵη]=J[y]+dϵd[J(y+ϵη]​∣∣∣∣​ϵ=0​⋅ϵ+(dϵ2d2J​)∣∣∣∣​ϵ=0​⋅2!ϵ2​+⋯+(dϵndnJ​)∣∣∣∣​ϵ=0​⋅n!ϵn​+o(ϵn+1)记$\delta J=\left.\frac{d[J(y+\epsilon \eta)]}{d \epsilon}\right|_{\epsilon=0} \cdot \epsilon$δJ=dϵd[J(y+ϵη)]​∣∣∣​ϵ=0​⋅ϵ,并称之为范函的一阶变分.
可以看出,当范函是最简范函$J[y(x)]=\int_{x_{0}}^{x_{1}} F\left(x, y(x), y^{\prime}(x)\right) d x$J[y(x)]=∫x0​x1​​F(x,y(x),y′(x))dx,变分其实就是$\delta J=\int_{x_{0}}^{x_{1}}\left(F_{y} \eta(x)+F_{y} \eta^{\prime}(x)\right) \cdot \epsilon d x=\int_{x_{0}}^{x_{1}}\left(F_{y} \delta y+F_{y^{\prime}} \delta y^{\prime}\right) d x$δJ=∫x0​x1​​(Fy​η(x)+Fy​η′(x))⋅ϵdx=∫x0​x1​​(Fy​δy+Fy′​δy′)dx其中$\delta y=\epsilon \eta(x)$δy=ϵη(x),与第一种定义一样.

变分导数定义

范函$J[y(x)]$J[y(x)]的范函导数(也叫变分导数),记作$\frac{\delta J}{\delta y(x)}$δy(x)δJ​,它由一阶变分的式子定义:$\delta J=\left.\epsilon \cdot \frac{d[J(y+\epsilon \eta)]}{d \epsilon}\right|_{\epsilon=0}=\int \frac{\delta J}{\delta y(x)} \epsilon \eta(x) d x \tag{1}$δJ=ϵ⋅dϵd[J(y+ϵη)]​∣∣∣∣​ϵ=0​=∫δy(x)δJ​ϵη(x)dx(1)当范函是最简范函$J[y(x)]=\int_{x_{0}}^{x_{1}} F\left(x, y(x), y^{\prime}(x)\right) d x$J[y(x)]=∫x0​x1​​F(x,y(x),y′(x))dx时,$\delta J=\int_{x_{0}}^{x_{1}}\left(F_{y} \delta y+F_{y} \delta y^{\prime}\right) d x=\int_{x_{0}}^{x_{1}}\left(F_{y} \eta(x)+F_{y} \eta^{\prime}(x)\right) \cdot \epsilon d x=\int_{x_{0}}^{x_{1}}\left(F_{y}-\frac{d}{d x} F_{y}^{\prime}\right) \epsilon \eta(x) d x \tag{2}$δJ=∫x0​x1​​(Fy​δy+Fy​δy′)dx=∫x0​x1​​(Fy​η(x)+Fy​η′(x))⋅ϵdx=∫x0​x1​​(Fy​−dxd​Fy′​)ϵη(x)dx(2)综合式(1)和式(2),固定边界的最简范函的范函导数是:$\frac{\delta J}{\delta y(x)}=F_{y}-\frac{d}{d x} F_{y}^{\prime}$δy(x)δJ​=Fy​−dxd​Fy′​

something else…

函数的变分$\delta y$δy与函数的增量$\Delta y$Δy之间的区别,如图所示:

函数的变分$\delta y$δy是两个不同的函数$y(x)$y(x)和$y_{0}(x)$y0​(x)在自变量x固定时的差,这是函数发生了改变.

函数的增量$\Delta y$Δy是自变量$x$x的增量引发的函数$y(x)$y(x)的增量,函数依然是原来的函数.