= Solution
The physical rapid phase is
$$
\Theta(\mathbf x,t)=\frac{\theta(\mathbf x,t)}{\varepsilon}.
$$
Define the local <wavevector>, <angular frequency>, and <group velocity> by
$$
k_i=\frac{\partial\Theta}{\partial x_i}
=\frac1\varepsilon\frac{\partial\theta}{\partial x_i},
\qquad
\omega=-\frac{\partial\Theta}{\partial t}
=-\frac1\varepsilon\frac{\partial\theta}{\partial t},
\qquad
c_{g,i}=\frac{\partial\Omega}{\partial k_i}.
$$
The equality of mixed partial derivatives gives the phase-compatibility equations
$$
\frac{\partial k_i}{\partial t}
+\frac{\partial\omega}{\partial x_i}=0,
\qquad
\frac{\partial k_i}{\partial x_j}
=\frac{\partial k_j}{\partial x_i}.
$$
A ray is defined to move with the local group velocity:
$$
\boxed{\frac{dx_i}{dt}=\frac{\partial\Omega}{\partial k_i}.}
$$
Differentiate the local <dispersion relation>
$$
\omega=\Omega(\mathbf k;\mathbf x,t)
$$
with respect to $x_i$. Along a ray,
$$
\begin{aligned}
\frac{dk_i}{dt}
&=\frac{\partial k_i}{\partial t}
+\frac{dx_j}{dt}\frac{\partial k_i}{\partial x_j}\\
&=-\frac{\partial\omega}{\partial x_i}
+\frac{\partial\Omega}{\partial k_j}
\frac{\partial k_i}{\partial x_j}\\
&=-\frac{\partial\Omega}{\partial x_i}
-\frac{\partial\Omega}{\partial k_j}
\frac{\partial k_j}{\partial x_i}
+\frac{\partial\Omega}{\partial k_j}
\frac{\partial k_i}{\partial x_j}\\
&=-\frac{\partial\Omega}{\partial x_i}.
\end{aligned}
$$
The last two terms cancel because $\mathbf k$ is a <gradient>. Similarly,
$$
\begin{aligned}
\frac{d\omega}{dt}
&=\frac{\partial\Omega}{\partial t}
+\frac{\partial\Omega}{\partial x_i}\frac{dx_i}{dt}
+\frac{\partial\Omega}{\partial k_i}\frac{dk_i}{dt}\\
&=\frac{\partial\Omega}{\partial t}.
\end{aligned}
$$
Finally,
$$
\frac1\varepsilon\frac{d\theta}{dt}
=\frac{\partial\Theta}{\partial t}
+\frac{dx_j}{dt}\frac{\partial\Theta}{\partial x_j}
=-\omega+k_j\frac{\partial\Omega}{\partial k_j}.
$$
Thus the <Hamiltonian ray-tracing equations> are
$$
\boxed{
\frac{dx_i}{dt}=\frac{\partial\Omega}{\partial k_i},
\qquad
\frac{d\omega}{dt}=\frac{\partial\Omega}{\partial t},
\qquad
\frac{dk_i}{dt}=-\frac{\partial\Omega}{\partial x_i},
\qquad
\frac1\varepsilon\frac{d\theta}{dt}
=-\omega+k_j\frac{\partial\Omega}{\partial k_j}.}
$$
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