assed

My question is kinda simple but it has been causing me some trouble for a while. In the problem of the pendulum rotating about an axis, why isn't the angle of rotation about the axis a generalized coordinate? The doubt appears when i try to write the hamiltonian for the system and i don't know if i include the term relative to that angle.
How thats related to the fact that the angular velocity is or not constant.
Thanks in advance.
( The problem considered is of a simple pendulum, but with its axis of oscilation rotating)