Generalized coordinates - Rotating pendulum

[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 that's 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 oscillation rotating)

 

[mfb]

What is the orientation of that axis? If rotation around it is not influenced by gravity, the coordinate is cyclic (I think this is what you are looking for?) and its time-derivative is conserved.