The purpose of this paper is to present a geometric approach to the problem of dimensional reduction. To derive (3 + 1) D formulations of 4D field problems in the relativistic theory of electromagnetism, as well as 2D formulations of 3D field problems with continuous symmetries.
The framework of differential‐form calculus on manifolds is used. The formalism can thus be applied in arbitrary dimension, and with Minkowskian or Euclidean metrics alike.
The splitting of operators leads to dimensionally reduced versions of Maxwell's equations and constitutive laws. In the metric‐incompatible case, the decomposition of the Hodge operator yields additional terms that can be treated like a magnetization and polarization of empty space. With this concept, the authors are able to solve Schiff's paradox without use of coordinates.
The present formalism can be used to generate concise formulations of complex field problems. The differential‐form formulation can be readily translated into the language of discrete fields and operators, and is thus amenable to numerical field calculation.
The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.
Kurz, S., Auchmann, B. and Flemisch, B. (2009), "Dimensional reduction of field problems in a differential‐forms framework", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 28 No. 4, pp. 907-921. https://doi.org/10.1108/03321640910959008
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