# 變分法基本引理

## 敘述

${\displaystyle C^{k}}$ 代表${\displaystyle k}$阶导数连续（${\displaystyle k}$阶光滑）的函数空间，${\displaystyle C^{\infty }}$代表无限光滑的函数空间。

${\displaystyle f(x)\in C^{\infty }[a,\ b]\,\!}$

${\displaystyle \int _{a}^{b}f(x)\,h(x)\,dx=0\,\!}$

${\displaystyle {\mbox{∀}}x\in (a,\ b):f(x)=0\,\!}$

## 證明

${\displaystyle f(x)\in C^{\infty }[a,\ b]\,\!}$${\displaystyle f(x)\neq 0\,\!}$

${\displaystyle r(x)\,\!}$ 滿足下列兩個條件：

${\displaystyle r(a)=r(b)=0\,\!}$

${\displaystyle {\mbox{∀}}x\in (a,\ b):r(x)>0\,\!}$

${\displaystyle h(x)=r(x)f(x)\,\!}$ 可得到

${\displaystyle 0=\int _{a}^{b}f(x)h(x)\;dx=\int _{a}^{b}r(x)f(x)^{2}\;dx\,\!}$

${\displaystyle {\mbox{∀}}x\in (a,\ b):f(x)=0\,\!}$

## 應用

${\displaystyle J[f(t,y,{\dot {y}})]=\int _{x_{0}}^{x_{1}}f(t,y,{\dot {y}})\,dt\,\!}$
${\displaystyle {d \over dt}\left({\partial f(t,y,{\dot {y}}) \over \partial {\dot {y}}}\right)-{\partial f(t,y,{\dot {y}}) \over \partial y}=0\,\!}$

## 參考文獻

• Leitmann, George. The Calculus of Variations and Optimal Control: An Introduction. Springer. 1981. ISBN 0306407078.