For lossless media, Hamilton's equations of geometrical optics can be derived from the dispersion equation either by the method of characteristics or by its combination with Sommerfeld-Runge's refraction law, Whitham's conservation law, and the expression ∂ω / ∂ k for the group velocity. The formal generalization to media with absorption leads to characteristics with complex space-time coordinates due to the now complex coefficients of the dispersion equation. For media with moderate absorption a real-valued generalization of Hamilton's equations is proposed. It is based on a dispersion equation with complex coefficients, on Sommerfeld-Runge's and Whitham's laws for the real parts of k, ω, on Connor and Felsen's condition for the imaginary part of k, and on the expression Re (∂ ω/∂k) for the velocity of a wave packet. These real-valued equations have been shown to hold in homogeneous media with moderate absorption.