 Solution

Write $\omega$ for the imposed constant angular velocity. In inertial polar coordinates, the bead has $v^2={\dot{r}}^2+{\omega }^2{r}^2$. Its constrained Lagrangian isThe reduced Lagrangian has no explicit time dependence, so its Lagrangian energy function is conserved:This is the Jacobi integral $J=E-\omega L_z$, since the bead's actual inertial energy and axial angular momentum areThe motor maintaining the stick's angular velocity supplies torque and does work. Indeed $\dot{E}=2m{\omega }^{2}r\dot{r}=\omega {\dot{L}}_z$, generally nonzero. A time-independent reduced Lagrangian therefore need not imply conservation of the bead's inertial kinetic energy when its imposed constraint is moving.