Dimensional reduction of field problems in a differential‐forms framework

Dimensional reduction of field problems in a differential‐forms framework Purpose – 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. Design/methodology/approach – 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. Findings – 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. Practical implications – 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. Originality/value – The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Emerald Publishing

Dimensional reduction of field problems in a differential‐forms framework

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Publisher
Emerald Publishing
Copyright
Copyright © 2009 Emerald Group Publishing Limited. All rights reserved.
ISSN
0332-1649
DOI
10.1108/03321640910959008
Publisher site
See Article on Publisher Site

Abstract

Purpose – 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. Design/methodology/approach – 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. Findings – 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. Practical implications – 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. Originality/value – The approach is an evolution of recent work, striving for a generalization of different approaches, and deliberately avoiding a mix of paradigms.

Journal

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic EngineeringEmerald Publishing

Published: Jul 10, 2009

Keywords: Dimensional measurement; Electromagnetism; Differential geometry

References

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