Obtaining the equations of motion for parametrically coupled oscillators or waves

When the equations of motion for any parametric system are written in terms of an appropriate set of normal modes, these equations have a general coupled-mode form which depends on the number of frequency components present and on the order of nonlinearity considered, but not on any other details of the specific system considered. This paper presents a simple procedure, based on a Hamiltonian approach, for obtaining the most general set of parametric coupled-mode equations for any number of frequency components and any order of nonlinearity. The procedure applies to parametrically coupled normal modes which grow in time, or to parametrically coupled waves which grow in distance and/or in time. The general equations obtained in this way are useful for pedagogic purposes and for general discussions of parametric interactions; for checking or even for bypassing detailed derivations on specific physical systems; and for proving the Manley-Rowe relations, which emerge very directly from the resulting general equations.