Fermat’s Principle

Fermat’s Principle is

$$\delta s=\delta\int n\dd \ell=0$$

If we choose the path connecting two points to be

$$\bm r=\bm r(t),\quad t_0\leq t\leq t_1$$

We can write

$$s=\int n\frac{\dd \ell}{\dd t}\dd t, \quad \frac{\dd \ell}{\dd t}=|\dot{\bm r}|$$

The Lagrangian about parameter \(t\) is

$$L=n\frac{\dd \ell}{\dd t}=n(\bm r)|\bm{\dot r}|$$

then the Euler-Lagrange Equation is

$$\begin{aligned} &\left(\frac{\dd}{\dd t}\frac{\pp}{\pp\bm{\dot r}}-\frac{\pp}{\pp\bm{r}}\right)L=0\\ \bm p&=\frac{\pp L}{\pp \bm{\dot r}}=n \hat\tau,\quad\hat{\tau}=\frac{\pp |\dot{\bm r}|}{\pp \bm{\dot r}}=\bm{\dot r}/|\bm{\dot r}|=\frac{\dd\bm r}{\dd\ell}\\ \frac{\dd \bm p}{\dd t}&=\frac{\pp L}{\pp \bm{r}}=\frac{\dd\ell}{\dd t}\nabla n\\ \frac{\dd \bm p}{\dd \ell}&=\nabla n=\bm F\end{aligned}$$

Generalized Momentum \(\bm p\) is independent of the choice of \(t\), and


Rectangular Coordinates

$$n(x,y,z)=f(x,y),\quad\bm F=\frac{\pp f}{\pp x}\hat x+\frac{\pp f}{\pp y}\hat y$$

It is obvious that \(p_z\) are conserved.

Central Gradient

Define \(\bm L=\bm r\times \bm p\),

$$\begin{aligned} \frac{\dd \bm L}{\dd \ell}&=\frac{\dd\bm r}{\dd \ell}\times\bm p+\bm r\times\frac{\dd\bm p}{\dd \ell} \\&=\hat{\tau}\times \bm p+\bm r\times\bm F \\&=\bm r\times\bm F\end{aligned}$$

\(\bm L\) is conserved for a spherical symmetric distribution

$$n(\bm r)=f(r),\quad \bm r\times\bm F=\bm 0$$

\(L_z\) is conserved for

$$n(\bm r)=f(r,\theta),\quad (\bm r\times\bm F)_z=0$$


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