What can one infer from the Hamiltonian in one coordinate system to another? In particular, how is the Hamiltonian of a rotating system concocted?
In chapter 8.2.2 of Chaikin&Lubensky [1], they argue why the addition of a term $ -\Omega_e L$ to the Hamiltonian can be used to model a rotating system.
A simplified version of the argument, as I understand it, goes like this. Consider a collection of rods in the rest frame $(x,y)$ that can rotate around their individual centers of mass at fixed positions ${l}$. We can assign the coordinate $\theta_l$ to each rotor, which is its angle wrt. the x-axis. The Lagrangian for this system will thus be
$$ \mathcal L = \sum_l \frac{1}{2} I \dot \theta_l^2, $$
where $I$ is its moment of inertia of one rod. The Hamiltonian is found to be
$$ \mathcal H = \sum_l \frac{p_l^2}{2I}, $$
where $p_l = \frac{\partial \mathcal L}{\partial \dot \theta_l} = I \dot \theta_l$. Now, if we introduce a rotating coordinate system $(x',y')$, given by $\theta_l = \theta_l' + \Omega_e t$, we can express the same Lagrangian as follows
$$\mathcal L = \sum_l \frac{1}{2} I (\dot \theta_l' + \Omega_e)^2$$
while the Hamiltonian for these coordinates is given by
$$\mathcal H_T = \sum_l \frac{{p_l'}^2}{2I} - \Omega_e L$$
where $L = \sum_l p_l'$ and $p_l' = \frac{\partial \mathcal L}{\partial \dot \theta_l'} = I(\dot \theta_l' + \Omega_e) = I \dot \theta_l= p_l$. 101
Thus, $L$ is the angular momentum in the rest frame. So we can write
$$ \mathcal H_T = \mathcal H - \Omega_e L. $$
The question is, how can we infer meaning from the Hamiltonian of the $(x',y')$ system expressed in coordinates of the $(x,y)$ system? It seems to me that the authors claim that this shows the energetic contribution from rotating a system, but this is not obvious to me, as the Hamiltonian of the $(x',y')$ system does not describe the evolution of the system in the $(x,y)$-system.
But I have seen this type of argument before, so there is clearly some deeper understanding that I have not properly understood.
[1] <PERSON>, P. M., & <PERSON>, T. C. (1995). Principles of condensed matter physics. Cambridge University Press. 101