Variational treatment of the electromagnetic field in an optical cavity with a moving mirror

Optical cavities with moving mirrors provide a versatile platform for exploring radiation-matter interactions and optically mediated mechanical effects, whose control has wide technological implications. However, capturing the coupled dynamics of the electromagnetic field and of the mirror within a consistent theoretical framework remains challenging. We analyze the problem of the coupling between classical electromagnetic fields in a cavity and a movable mirror, considering both nonrelativistic and relativistic regimes of motion. Starting from the equations of motion for a mirror subject to a generic external potential, we provide a variational formulation of the mirror-radiation interaction. Within this framework, a single-mode variational approximation is introduced, which captures the essential dynamical features of the coupled system. In the special case of a mirror undergoing free motion, the variational method yields an exact solution. This unified treatment highlights the connection between different dynamical regimes and provides a basis for analyzing applications ranging from precision interferometry to relativistic radiation-pressure effects.