Generalizing the Poynting Vector

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Abstract
A very general energy conservation law derived from a Lagrangian theory of dielectric crystals is presented. It includes energy propagation from electromagnetic, spin, and acoustic waves. Both linear and nonlinear waves are included as well as various polaritonic combinations. Waves involving nonlocal (wave-vector–dispersive) interactions are also included. An example of the latter for which the Poynting vector is invalid, but which is correctly handled by this theory, is presented.