Maxwell's equations as a special case of deformation of a solid lattice in Euler's coordinates

TL;DR

This paper demonstrates that Maxwell's equations can be derived as a special case of a tensor-based deformation theory of solid lattices, revealing a deep analogy between electromagnetism and lattice mechanics, and suggesting broader physical applications.

Contribution

It introduces an Eulerian tensor deformation framework for solid lattices that generalizes Maxwell's equations, establishing a novel analogy between electromagnetism and lattice deformation phenomena.

Findings

Maxwell's equations are derivable from lattice deformation theory.

The analogy extends to dielectric and magnetic phenomena.

The tensor approach suggests potential applications beyond electromagnetism.

Abstract

It is shown that the set of equations known as Maxwell's equations perfectly describe two very different systems: (1) the usual electromagnetic phenomena in vacuum or in the matter and (2) the deformation of isotropic solid lattices, containing topological defects as dislocations and disclinations, in the case of constant and homogenous expansion. The analogy between these two physical systems is complete, as it is not restricted to one of the two Maxwell's equation couples in the vacuum, but generalized to the two equation couples as well as to the diverse phenomena of dielectric polarization and magnetization of matter, just as to the electrical charges and the electrical currents. The eulerian approach of the solid lattice developed here includes Maxwell's equations as a special case, since it stems from a tensor theory, which is reduced to a vector one by contraction on the tensor indices. Considering the tensor aspect of the eulerian solid lattice deformation theory, the analogy can be extended to other physical phenomena than electromagnetism, a point which is shortly discussed at the end of the paper.