  In most optomechanical systems a movable mirror is a part of an optical
cavity, and its oscillation modulates either the resonance frequency of the
cavity, or its coupling to the environment. There exists the third option --
which we call a "coherent coupling" -- when the mechanical oscillation couples
several non-degenerate optical modes supported by the cavity. Identifying the
nature of the coupling can be an important step in designing the setup for a
specific application. In order to unambiguously distinguish between different
optomechanical couplings, we develop a general framework based on the
Hamiltonian of the system. Using this framework we give examples of different
couplings, and discuss in details one particular case of a purely coherent
coupling in a ring cavity with a movable mirror inside. We demonstrate that in
certain cases coherent coupling can be beneficial for cooling the motion of the
mechanical oscillator. Our general framework allows to approach the design of
optomechanical experiments in a methodological way, for precise exploitation of
the strengths of particular optomechanical couplings.
