I am just wondering the 4 Maxwell equations (i.e Fadaray Law, Maxwell-Ampere) are Lagrangian or Eulerian description? Does it really matter?

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    $\begingroup$ I've only heard that distinction made with respect to fluid mechanics. What do you mean by an "Eulerian" description of E&M? $\endgroup$ – mike stone Nov 21 at 23:20
  • $\begingroup$ Magnetohydrodynamics combine Maxwell's equations with Navier-Stokes fluid equations en.wikipedia.org/wiki/Magnetohydrodynamics which is typically done in the Eulerian approach, I think, e.g. using bulk plasma $\endgroup$ – N. Steinle Nov 22 at 0:15

The distinction between the two descriptions you consider assumes that there is a field of velocities describing the motion of the particles of the continuous body.

Adopting one or the other description respectively means (a) to use the initial position of the particles to label the integral curves of the field (Lagrangian description) or (b) to refer to the istantaneous positions of particles in a given rest space (Eulerian description).

The electromagnetic field does not provide this field of velocities so that the choice does not make much sense. Or maybe, to some extent we can say that only the Eulerian description can be adopted.

The EM field is not a continuous body or a fluid, even if it shares some features with these physical systems.

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  • $\begingroup$ Thanks for your answer. Now for the electric field example, does it propagate accordingly to his vector direction or to the wave (perpendicular to the Magnetic and Electric Field)? $\endgroup$ – Abdoulaye ndiongue Nov 23 at 11:52
  • $\begingroup$ Yes, the Poyning vector field satisfies a conservation equation with respect to the density of electromagnetic energy that shares some similarities with the continuity equation of the mass. But it is not enough to describe the EM system as a continuum within the classical view. $\endgroup$ – Valter Moretti Nov 23 at 12:43

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