On the Metamorphoses of Maxwell’s EquationsDuring the Last 150 Years — spotlights on the

history of classical electrodynamics —Alberto Favaro∗, Friedrich W. Hehl†, and Jonathan Lux‡

∗Imperial College, London, UK; e-mail: a.favaro@imperial.ac.uk†Univ. of Cologne, Germany, and Univ. of Missouri-Colombia,USA; e-mail: hehl@thp.uni-koeln.de

‡Univ. of Cologne, Germany; e-mail: lux@thp.uni-koeln.de

Abstract—We outline Maxwell’s five decisive papers on hisequations governing electrodynamics (1862-1868). We study themetamorphoses of these equations and find essentially twelvedifferent versions of them. We express our preference for theso-called premetric version of the Maxwell equations, which isparticularly useful in understanding the structure of electrody-namics. Some selected applications are discussed.

The twelve different versions of the Maxwell equations,which we will discuss chronologically, are the following:

1. In components: Maxwell 1862-65.2. In quaternions (Hamilton 1843): Maxwell 1865, 1873.3. In symbolic vector calculus: Heaviside 1885-88, Foppl

1894, Gibbs 1901.4. In components (compact, without vector potential A):

Hertz 1890, including ansatz for moving bodies.5. In components a la Maxwell-Hertzplus Lorentz trans-

formations: Einstein 1905.6. In 4d calculus (components and symbolic) for special

relativity: Minkowski 1907-08.gravity=⇒ 7. In 4d generally covariant tensor calculus: Einstein 1916

(shortly after establishing general relativity).8. In premetric/integral formulation up to about 1960:

(Maxwell 1868, Murnaghan 1920) Kottler 1922, Car-tan 1923-24 (in differential forms), van Dantzig,Schrodinger, Schouten, Truesdell-Toupin 1961, Post1962.

9. In spinor calculus (after Pauli 1927 and Dirac 1928and commencing in 1929): Fock & Ivanenko, London,Weyl, Infeld & van der Waerden, Corson⇒ reviewedby Penrose & Rindler 1984 in the context of generalrelativity.

10. In 4d Clifford algebra formalism, including quaternionsand octonions: Riesz 1958⇒ Lounesto 1997, Baylis1999, ... , Hitzer 2012.

11. In algebraic (discrete) formulation in terms of (co)chains⇒ Bossavit, Tonti (“Algebraic formulation of physicalfields”) ⇒ “Maxwell in chains,” direct implementationby means of algebraic computer code.

12. In 3d and 4d exterior calculus (premetric topologicalform of Maxwell’s equations): Kiehn, Post; Baldomir& Hammond, Kovetz, Russer, Lindell, H. & Obukhov;

Fig. 1. Maxwell Monument in Edinburgh, Scotland.

signatureof the 4d Lorentz metric linked to the Lenzrule and to the sign of the energy density⇒ Obukhov,Itin, Friedman, H..

We invite comments and critical remarks to our findings,particularly to the 12-fold way, which we presented, see ouremail addresses above.

Fig. 2. The Maxwell-Heaviside equations on the monument.

FURTHER READING

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