 Jacobi integral

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 becomesFor $z=r^2/(2a)$ in uniform gravity, division by mass yieldsThis 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.