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Physics

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  • $\begingroup$ Notes for later: The momentum is ${\bf p} = \frac{\partial (-R)}{\partial \dot{\bf r}}=\frac{d{\bf r}}{ds}n\frac{E}{c}$, so that $|{\bf p}|=n\frac{E}{c}$. The ray equation $\nabla n =\frac{d}{ds}\left(n\frac{d{\bf r}}{ds} \right) \propto \frac{d{\bf p}}{ds}$ implies Snell's law, which states that the momentum tangential to the interface is conserved. $\endgroup$
    – Qmechanic ♦
    Commented Dec 24, 2021 at 12:40
  • $\begingroup$ Notes of later: Scattering: $\quad S_H[r,\theta,\varphi,t,p_r,p_{\theta},p_{\varphi},E,e]= \int_{\lambda_i}^{\lambda_f}\! \mathrm{d}\lambda~(-E\dot{t}+p_r\dot{r}+p_{\theta}\dot{\theta}+p_{\varphi}\dot{\varphi}-H)$, $\quad H=\frac{e}{2}\left(-\frac{E^2}{c^2}+p_r^2+\frac{p_{\theta}^2}{r^2}+\frac{p_{\varphi}^2}{r^2\sin^2\theta}+(mc)^2+2mV(r)\right)$. $\endgroup$
    – Qmechanic ♦
    Commented Jul 8 at 12:07
  • $\begingroup$ Notes of later: Choose coordinates: $(\varphi=0,p_{\varphi}=0)$. Int. over $t\in[t_i,t_f]$ and $\theta\in\mathbb{R}$ (winding possible) in the Ham. path int. imposes $\dot{E}\approx 0$ and $\dot{p}_{\theta}\approx 0$. Int. over $e$ leads to radial action $\int_{\lambda_i}^{\lambda_f}\! \mathrm{d}\lambda~\left. p_r\right|_{\ldots} \dot{r}$. Hm, maybe better to use stereographic projection. $\endgroup$
    – Qmechanic ♦
    Commented Jul 8 at 12:30