= Jacobi integral
{c}
{title2=$J=E-\omega L_z$}
For a mechanical system whose imposed motion is uniform rotation with <angular velocity> $\omega$ about a fixed axis, a conserved rotating-frame <energy> can exist even when the inertial <mechanical energy> $E$ changes. With <angular momentum> $L_z$ about that axis, it is $J=E-\omega L_z$. For a bead constrained to rotate with $\dot\theta=\omega$, this becomes
$$
J=\frac m2(\dot r^2+\dot z^2)+V-\frac m2\omega^2r^2.
$$
For $z=r^2/(2a)$ in uniform <gravity>, division by <mass> yields
$$
\frac Jm=\frac12\left(1+\frac{r^2}{a^2}\right)\dot r^2+\frac12\left(\frac ga-\omega^2\right)r^2.
$$
This is the <first integral> that controls the bead's radial motion. Different normalizations and signs are used for the related Jacobi constant in celestial mechanics.
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