# Lagrangian Mechanics - Bead sliding on a rotating rod

Say i have a bead of mass $$m$$ sliding on a friction-less rode (or wire) that is rotating with a permanent angular velocity $$ω$$. The whole system is under the influence of a uniform gravitational field $$F=mg$$.

How many degrees of freedom do the system have?

Assuming we use polar coordinates, it seems as if we have only one degree of freedom and that is the radial coordinate - r, that is because the polar angle theta is set to be ω*t. But on the other hand, r cant be smaller than 0, while it is obvious that in the physical system - if the angular velocity is slow and gravity is strong, the bead can slide to the other side of the rotation axis.

So will this system's Lagrangian depend on both r and theta and so the Euler–Lagrange equation should be applied once for each of them?

• The radial coordinate can be smaller than 0. I'd say there's only 1 DOF. You don't need to apply the Euler–Lagrange equation to theta. – Laff70 Apr 6 at 14:29