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

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?