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Contact and symplectic geometry in electromagnetics

Master's thesis, September 2002,
Matias Dahl,


The present work is divided into two parts. In the first part, we show that, by working in Fourier space, the Bohren decomposition and the Helmholtz's decomposition can be combined into one decomposition. This yields a completely mathematical decomposition, which decomposes an arbitrary vector field on R3 into three components. The most important properties of this decomposition is that it commutes with both the curl operator and with the time derivative. We can therefore apply this decomposition to Maxwell's equations without assuming anything about the media. When we apply this decomposition to the traditional Maxwell's equations (for E, D, B, H) on R3, we will see that Maxwell's equations split into three completely uncoupled sets of equations. Also, when a medium is introduced, the decomposed Maxwell's equations either remain uncoupled, or become coupled depending on the complexity of the medium. As a special case, the decomposed Maxwell's equations contain the Bohren decomposition.

In the second part of this work, we begin with an introduction to contact and symplectic geometry and then study their relation to electromagnetism. By studying examples, we show that the decomposed fields in the decomposed Maxwell's equations always seem to induce contact structures. For instance, for a planewave, the decomposed fields are the right and left hand circulary polarized components and each of these induce their own contact structure. Moreover, we show that each contact structure induces its own Carnot-Carathéodory metric, and the path traversed by the circularly polarized waves seem to coincide with the geodesics of these metrics.


The paper Contact geometry in electromagnetism in PIER ( here ) contains an abridged version of this work. This might be easier to read.

Last modified 15.3.2006.

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