Derive the ray-tracing equations

$$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$$

for wave propagation through a slowly-varying medium with local dispersion relation $\omega=\Omega(\mathbf{k}, \mathbf{x}, t)$. The meaning of the notation $d / d t$ should be carefully explained.

A non-dispersive slowly varying medium has a local wave speed $c$ that depends only on the $z$ coordinate. State and prove Snell's Law relating the angle $\psi$ between a ray and the $z$-axis to $c$.

Consider the case of a medium with wavespeed $c=A \cosh \beta z$, where $A$ and $\beta$ are positive constants. Find the equation of the ray that passes through the origin with wavevector $\left(k_{0}, 0, m_{0}\right)$, and show that it remains in the region $\beta|z| \leqslant \sinh ^{-1}\left(m_{0} / k_{0}\right)$. Sketch several rays passing through the origin.