A major problem with magnetic materials in application is they naturally have high losses in a wide frequency range of interest (e.g., Faraday rotation using ferromagnets in optical frequencies). Composites can inherit significantly altered properties from those of their components. Does this apply to losses and magnetic properties? How can broadband absorption suppression in magnetic-dielectric composites be achieved? In this talk, I will discuss our developments toward a theory of broadband absorption suppression in magnetic composites based on a Lagrangian and Hamiltonian approach from classical mechanics. In the first part of the talk, I will introduce the hierarchy of models we are using, based on two-component composite linear systems with a high-loss and a lossless component, for studying the interplay of dissipation (losses) and gyroscopy (magnetism) as well as the dominant mechanisms of energy loss. Next, I will discuss our new results towards answering these questions related to the modal dichotomy and selective overdamping phenomena in systems with finite degrees-of-freedom (e.g., electric circuits with gyrators and resistors). I will conclude the talk with our recent developments of this theory for Maxwell's equations pertaining to the dissipative properties of electromagnetic fields in stratified magnetic-dielectric media. This is joint work with Alex Figotin (UCI) and Robert Viator (IMA).