The usefulness of Brown's polarization formalism as a paradigm for the dynamics of analogous linear systems is demonstrated. By connecting the rotations of the Lorentz group with birefringence and the Lorentz group boosts with dichroism, this formalism has offered closed-form solutions, expressed in the language of the Lorentz group, for the 4 X 4 Mueller matrices of polarization optics. First, we discuss the dynamics of a generic two-state system satisfying a Schrodinger-like differential equation. This equation is cast into a four dimensional form. Next, we show the solution of the latter to be a Lorentz transformation. Then, we analyze several specific examples. The examples include: (1) precession, (2) a magnetic dipole in a magnetic field, (3) a charged particle in an electric field, and (4) time- independent perturbation theory with phenomenological damping. We argue the approach of Brown's formalism provides a convenient conceptual framework to describe the physics and to interpret the measurement results of such systems. In addition, this approach broadens the role of the Lorentz group to areas other than special relativity and electromagnetic scattering.