I have this problem from an exam but I don't know how to start any section, could you help me? We have a laboratory system rotating around $Z$-axis relative to an inertial referencial frame with frequence $\omega$. A particle with mass $m$ whose canonical variables are $q_{i}$ and $p_{i}$ ($i=1,2,3$) is moving in the laboratory system under potential $V(q_{1},q_{2},q_{3})$.
$a)$ Write the canonical variables transformation between these two reference frames $(q_{i},p_{i})\to (q'_{i},p'_{i})$ and the operator $U(t)$ that implements this transformation: $q'_{i}=U(t)q_{i}U^{\dagger}(t)$, $p'_{i}=U(t)p_{i}U^{\dagger}(t)$.
$b)$ Write the Hamiltonian for the particle in the two reference frames.
$c)$ Suppose the particle is in an eigenstate from $L_{z}$ with eigenvalue $l'_{z}$ in the rotating frame system. Verify the particle is in an eigenstate of $L_{z}$ in the laboratory system too and calculate the corresponding eigenvalue.
