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?
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.
