# 变分法

## 欧拉-拉格朗日方程

$A[f] = \int_{x_1}^{x_2} \sqrt{1 + [ f'(x) ]^2} \, dx,$

$f'(x) = \frac{df}{dx}, \,$ $f(x_1)=y_1$, $f(x_2)=y_2$

$A[f_0] \le A[f_0 + \epsilon f_1]$

$\frac{d}{d\epsilon} \int_{x_1}^{x_2} \left. \sqrt{1 + [ f_0'(x) + \epsilon f_1'(x)]^2} dx \right|_{\epsilon =0} = \int_{x_1}^{x_2} \left. \frac{(f_0'(x) + \epsilon f_1'(x)) f_1'(x)}{\sqrt{1 + [ f_0'(x) + \epsilon f_1'(x)]^2}}\right|_{\epsilon =0} dx = \int_{x_1}^{x_2} \frac{ f_0'(x)f_1'(x) } {\sqrt{1 + [ f_0'(x) ]^2}}\,dx = 0$

$\int_{x_1}^{x_2} f_1 (x) \frac{d}{dx}\left[ \frac{ f_0'(x) } {\sqrt{1 + [ f_0'(x) ]^2}} \right] \, dx =0,$

$I =\int_{x_1}^{x_2} f_1 (x) H (x) dx =0,$

$\frac{d}{dx}\left[ \frac{ f_0'(x) } {\sqrt{1 + [ f_0'(x) ]^2}} \right] =0$

$\frac{d^2 f_0}{dx^2}=0,$

$A[f] = \int_{x_1}^{x_2} L(x,f,f') dx$

$-\frac{d}{dx} \frac{\part L}{\part f'} + \frac{\part L}{\part f}=0$

## 費馬原理

$A[f] = \int_{x=x_0}^{x_1} n(x,f(x)) \sqrt{1 + f'(x)^2} dx,$

$\delta A[f_0,f_1] = \int_{x=x_0}^{x_1} \left[ \frac{ n(x,f_0)f_0'(x)f_1'(x)}{\sqrt{1 + f_0'(x)^2}} + n_y(x,f_0) f_1\sqrt{1 + f_0'(x)^2} \right] dx$

$-\frac{d}{dx} \left[\frac{ n(x,f_0)f_0'}{\sqrt{1 + f_0'^2}} \right] + n_y(x,f_0) \sqrt{1 + f_0'(x)^2} =0$

### 斯涅爾定律

$n(x,y) = n_- \quad \hbox{if} \quad x<0,$
$n(x,y) = n_+ \quad \hbox{if} \quad x>0,$

$\delta A[f_0,f_1] = f_1 (0)\left[ n_-\frac{f_0'(0_-)}{\sqrt{1 + f_0'(0_-)^2}} -n_+\frac{f_0'(0_+)}{\sqrt{1 + f_0'(0_+)^2}} \right]$

$n_-$相乘的係數是入射角的正弦值，和$n_+$相乘的係數則是折射角的正弦值。若依照斯涅爾定律，上述二項的乘積相等，因此上述的變分量為0。因此斯涅爾定律所得的路徑也就是要求光程一階變分量為0的路徑。

### 費馬原理在三維下的形式

$A[C] = \int_{t=t_0}^{t_1} n (X) \sqrt{ \dot X \cdot \dot X} dt$

$\frac{d}{dt} P = \sqrt{ \dot X \cdot \dot X} \nabla n,$

$P = \frac{n (X) \dot X}{\sqrt{\dot X \cdot \dot X} }$

$P \cdot P = n (X)^2$

$A[C] = \int_{t=t_0}^{t_1} P \cdot \dot X \, dt$

## 参考

1. ^ Gelfand, I. M.; Fomin, S. V.. Silverman, Richard A., 编. Calculus of variations Unabridged repr. Mineola, N.Y.: Dover Publications. 2000: 3. ISBN 978-0486414485.
• Fomin, S.V. and Gelfand, I.M.: Calculus of Variations, Dover Publ., 2000
• Lebedev, L.P. and Cloud, M.J.: The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics, World Scientific, 2003, pages 1-98
• Charles Fox: An Introduction to the Calculus of Variations, Dover Publ., 1987
• Forsyth, A.R.: Calculus of Variations, Dover, 1960
• Sagan, Hans: Introduction to the Calculus of Variations, Dover, 1992
• Weinstock, Robert: Calculus of Variations with Applications to Physics and Engineering, Dover, 1974
• Clegg, J.C.: Calculus of Variations, Interscience Publishers Inc., 1968
• Elsgolc, L.E.: Calculus of Variations, Pergamon Press Ltd., 1962