Lagrangian dynamics of the coupled field-medium state of light

In the recently introduced mass-polariton (MP) theory of light [Phys. Rev. A 95, 063850 (2017)], the optical force of light drives in a medium forward an atomic mass density wave. In this work, we present the Lagrangian formulation of the MP theory starting directly from the principle of least action and the well-known Lagrangian densities of the electromagnetic field and the medium. We write the Euler-Lagrange equations for the coupled state of the field and the medium and obtain from the first principles the unique stress-energy-momentum (SEM) tensor of the MP theory. We also show that the Lagrangian densities and the resulting dynamical equations lead directly and without any further postulates to the unique expression of the optical Abraham force density that couples the dynamics of the electromagnetic field to the dynamics of the medium in the MP theory of light. The coupled Euler-Lagrange equations also enable the exact description of the very small kinetic energy of the medium as a part of the total energy of the coupled state of light. Thus, the Lagrangian formulation of the present work is a complementary approach to Lorentz covariance properties of the MP theory discussed in our recent work [Phys. Rev. A 99, 033852 (2019)]. We show how the coupled dynamical equations of the field and the medium can be solved analytically for a Gaussian light pulse. The mathematical model of the bi-directional coupling of the field and the medium may also find applications elsewhere in the theory of dynamical systems. It is astonishing how the simple analytic results for the dynamical equations, the optical force, and the SEM tensor of the MP theory follow ab initio from the Lagrangian densities that have been well known for almost a century.