Extensions to complex materials of the fitzgerald model for the solution of electromagnetic problems

Electromagnetic phenomena can be simulated by the dynamics of a mechanical system as long as the Hamiltonian of the electromagnetic and the mechanical systems coincide. In this paper we present a generalization of G.F. FitzGerald's pulleys and rubber-bands mechanical model for the interaction of electromagnetic waves with complex media. We show a direct analogy between the FitzGerald model and the electric vector potential formulation, at each stage of the extension of the original model: each mechanical observable has a unique correspondence in the vector potential formulation. This strict analogy allows further inductive developments of the mechanical model and extends the pedagogical importance of the original FitzGerald model. As a consequence very complex materials from the electromagnetic point of view, such as frequency dependent magneto dielectric materials are easily understood and implemented with simple modifications in the mechanical system. The condense node representation of the field in the vector potential formulation results in lower grid dispersion compared to other numerical techniques such as the Finite Difference Time Domain (FDTD), We describe several applications, such as classical scattering problems from dielectric, magnetically permeable, dielectritcally lossy and Debye materials. The simulations are validated with comparison to canonical solutions, or with FDTD calculations.