= Hamiltonian ray equations for a local dispersion relation
{c}
{title2=$\dot{\mathbf X}=\Omega_{\mathbf k},\quad\dot{\mathbf k}=-\Omega_{\mathbf X}$}
A local <dispersion relation> $\omega=\Omega(\mathbf k;\mathbf X,T)$ yields
$$
\frac{d\mathbf X}{dT}=\nabla_{\mathbf k}\Omega,\quad\frac{d\mathbf k}{dT}=-\nabla_{\mathbf X}\Omega\big|_{\mathbf k},\quad\frac{d\omega}{dT}=\Omega_T\big|_{\mathbf k}.
$$
The spatial and temporal derivatives on the right hold the <wavevector> fixed. Translation symmetries conserve the corresponding wavevector components, while time independence conserves frequency.
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