How does the Hamiltonian change when going to a moving frame?

Question

The Hamiltonian of a free particle in a rotating frame is given by $$ H = H_0 - \omega \cdot J, $$ where $H_0$ is the Hamiltonian in the non-rotating frame, $\omega$ is the angular velocity of the frame and $J$ is the angular momentum of the particle.

This relation is too beautiful to be a coincidence. Does it hold for arbitrary systems with rotational invariance? Is a more general statement possible?

Answer 1

At least when the change of frame is given by some canonical flow the situation appears to be quite nice. If $\phi_t$ is the canonical flow generated by some $J$ and we make the change of frame $$ x' = \phi_t(x) \,, $$ the Hamiltonian in the new frame is given by $$ H' = H \circ \phi_t + J \,. $$ Given I didn't make any mistakes deriving this.