Symmetries and Maxwells equations

[Niles]

Hi

Maxwells Equations for a time-invariant system are separable, hence we can write a solution as E(r, t) = E(r)E(t). They also mention that if the system is radially invariant, then that implies that the solution splits into a product of radial and angular functions (with 2π periodic angular functions).

Is it a general rule that when the system described by Maxwells equations has a symmetry, then the solutions become separable? If yes, does this go beyond Maxwells Equations?


Niles.

 

[Vanadium 50]

No. It's only true for 1/r and r² potentials.

 

[Niles]

Thanks. Where does that spatial dependence come into play when looking at Maxwells Equations? Is in through ε(r)=n(r)²?

Do you have a suggestion for a reference that explains this in more details?


Niles.

 

[Vanadium 50]

It's described in Volume 1 of Landau's Mechanics. Mathematically, it's because there is a hidden SO(4) symmetry in the equations describing the 1/r potential, and this symmetry (the same one that gives the n-l degeneracy in the hydrogen atom) ensures that the angular piece is separated out.