= Geodesic equation in a uniformly rotating Minkowski chart
{title2=$\ddot r=p_\phi^2/(m^2r^3),\quad p_\phi=-mr^2(\dot\phi+\omega\dot t)$}
Set the inertial cylindrical angle to $\Phi=\phi+\omega t$. The <Minkowski metric> becomes $ds^2=(c^2-\omega^2r^2)dt^2-2\omega r^2dt\,d\phi-dr^2-r^2d\phi^2-dz^2$. For a massive <geodesic> parametrized by <proper time>, its <cyclic coordinates> give conserved covariant <four-momentum> components $p_t=m[(c^2-\omega^2r^2)\dot t-\omega r^2\dot\phi]$, $p_\phi=-mr^2(\dot\phi+\omega\dot t)$ and $p_z=-m\dot z$. Their combination gives $\dot t=(p_t-\omega p_\phi)/(mc^2)$. The radial <Euler-Lagrange equation> gives the displayed coordinate <acceleration>, although the free worldline is straight in inertial coordinates. The conserved inertial <energy> is $p_t-\omega p_\phi$, and the inertial axial <angular momentum> is $-p_\phi$.
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