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I am a computational mathematician currently working on space-time finite element approximations to PDEs. I am reducing our model equations from (3+1)D to (1+1)D to test our algorithms.

Can we use the Lagrangian of using the electromagnetic potential in (3+1)D, and assume the spacial-component only depends on $x$, to derive the Euler-Lagrange equation of this Lagrangian?

Will we get wave equation $$ \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0 $$ in (1+1)D?

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  • $\begingroup$ I am also interested in what experts would say. I doubt it is possible, because the electric and magnetic fields are perpendicular to each other and to the direction of the wave. It is 4 dimensions already right there. $\endgroup$ – safesphere Sep 1 '17 at 19:59
  • $\begingroup$ Related: physics.stackexchange.com/q/32685/2451 and links therein. $\endgroup$ – Qmechanic Sep 1 '17 at 20:00
  • $\begingroup$ Good answers in the related link, except it is not clear how the electric and magnetic fields can be perpendicular to each other and to the direction of the wave in less than 3 spatial dimensions. Can anyone explain? $\endgroup$ – safesphere Sep 1 '17 at 20:24
  • $\begingroup$ The very short answer is that in 1 spatial dimension there is only an electric field with one component and no magnetic field. $\endgroup$ – Javier Sep 1 '17 at 20:48
  • $\begingroup$ @Javier This was clear from the answers in the related link, but how about 2 spatial dimensions? See the concern in my comment above. $\endgroup$ – safesphere Sep 1 '17 at 20:53
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Technically the potentials and fields do obey the 1D wave equation. The problem is that the field also obeys a stronger equation which makes it kind of trivial.

From $\mathcal{L} = -F_{\mu\nu}F^{\mu\nu}/4\pi$ and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ it can be deduced that both $F_{\mu\nu}$ and $A_\mu$ satisfy the wave equation in any number of dimensions. But the field also has to obey Maxwell's equation $\partial_\mu F^{\mu\nu} = 0$, and in 1+1D this gets tricky. Since $F$ is an antisymmetric tensor it has only one independent component; let's call it $E$. Then the above equation implies

$$\partial_t E = 0$$ $$\partial_x E = 0$$

That is, in empty space the field must be a constant. It does satisfy the wave equation, but in a kinda trivial way.

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