# Lagrangian dynamics of the coupled field-medium state of light

## TL;DR

This paper develops a Lagrangian formulation of the mass-polariton theory of light, providing a symmetric, action-based description of light-medium interactions and deriving analytical solutions for Gaussian pulses.

## Contribution

It introduces a Lagrangian approach to the MP theory, deriving the optical Abraham force and coupled dynamics without additional assumptions.

## Key findings

- Derivation of the optical Abraham force from first principles
- Analytical solutions for Gaussian light pulses in the coupled system
- Consistent energy and momentum description of light-medium interactions

## Abstract

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 within the special theory of relativity. The Lagrangian densities and the resulting Euler-Lagrange equations lead directly and without any further postulates to the unique expression of the optical Abraham force that dynamically couples the electromagnetic field and the medium in the MP theory of light. The field-medium coupling is symmetric and bi-directional and it fulfills the law of action and counteraction. The coupled dynamical 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. It is astonishing how the simple analytic results for the dynamical equations, the optical force, and the stress-energy-momentum tensor of the MP theory follow ab initio from the Lagrangian densities that have been well known for almost a century.