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# Contact and symplectic geometry in electromagnetics

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Master's thesis, September 2002,

Matias Dahl,

## Abstract

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 *R*^{3} 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 *R*^{3}, 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.

## Notes

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.