When can the Lagrangian be used

dsdsuster

Hello. I have a question about when the Lagrangian can be used. In the textbook we are using it is shown that for constraint forces which do no work, the Lagrangian is of the form T-U. In this question though, it seems like rod would provide a normal force that is in the same direction as the bead's instantaneous velocity at all times.

A bead of mass m is threaded on a frictionless, straight rod which lies in a
horizontal (x-y so no change in gravitational potential energy of the bead) plane and is forced to spin with constant angular velocity ω about a vertical axis through the midpoint of the rod.

Writing out the Lagrangian equations of motion, I get the same result as if the bead were always subject to a radial centripetal force mrw^2. I would basically like to know why the Lagrangian can be used in this case, when the constraining force seems to do work.

Thanks for your help!

bp_psy

Centripetal forces do no work.

dsdsuster

The radius of the circular path the bead is taking is changing though. Doesn't this imply work is being done?