Generalized coordinates - Rotating pendulum

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SUMMARY

The discussion centers on the use of generalized coordinates in the context of a rotating pendulum. Specifically, the participant questions why the angle of rotation about the axis is not considered a generalized coordinate when formulating the Hamiltonian for the system. It is established that if the rotation around the axis is not influenced by gravity, the coordinate is cyclic, leading to the conservation of its time-derivative. This insight clarifies the relationship between angular velocity and the inclusion of the rotational angle in the Hamiltonian formulation.

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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)
 
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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.
 

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