On the degrees of freedom of lattice electrodynamics

Abstract

Using Euler's formula for a network of polygons for 2D case (or polyhedra for 3D case), we show that the number of dynamic\textit{\}degrees of freedom of the electric field equals the number of dynamic degrees of freedom of the magnetic field for electrodynamics formulated on a lattice. Instrumental to this identity is the use (at least implicitly) of a dual lattice and of a (spatial) geometric discretization scheme based on discrete differential forms. As a by-product, this analysis also unveils a physical interpretation for Euler's formula and a geometric interpretation for the Hodge decomposition.Comment: 14 pages, 6 figure

Differential form approach to the analysis of electromagnetic cloaking and masking

“…The basic feature underlying this remarkable possibility is the invariance of Maxwell equations under diffeomorphisms of the metric (metric invariance) [1], [2], [3], [4], [5], i.e., the fact that a change on the metric of space can be mimicked by a proper change of the constitutive tensors. In this work, we discuss how this feature of Maxwell equations is obviated using differential forms and the exterior calculus framework [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. We then illustrate how this feature also allows for generic masking of objects (again, under idealized conditions) via appropriate metamaterial coatings.…”

The Second Order Finite Element Analysis of Eddy Currents Based on the T-Ω Method

“…where φ is a magnetic scalar potential 0-form, A a 2-form, χ the harmonic field component, and d the exterior derivative, and δ the codifferential operator, the adjoint of d [9]. The codifferential δ operator can be expressed as [10] …”

LATTICE MAXWELL'S EQUATIONS (Invited Paper)

“…The operator ∂ carries its ordinary meaning [20] as illustrated in Fig. 3, and on a lattice it maps any p-simplex to the set of (p − 1)-simplices comprising its geometrical boundary.…”