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.

 

[sardar]

Hello Niles,

Thank you for your question. The relationship between symmetries and separability in Maxwells equations is a fascinating topic in physics. It is indeed a general rule that when a system described by Maxwells equations has a symmetry, the solutions become separable. This is because symmetries in a system correspond to conserved quantities, and separability is a manifestation of these conserved quantities.

This rule does go beyond Maxwells equations and applies to other physical systems as well. In fact, separability is a fundamental concept in mathematical physics and has been extensively studied in various fields, including quantum mechanics and general relativity.

In essence, symmetries and separability are closely related concepts. Symmetries provide a powerful tool for analyzing and solving physical systems, while separability allows for simplification and deeper understanding of these systems.

I hope this helps clarify the relationship between symmetries and separability in Maxwells equations. If you have any further questions, please feel free to ask.

Best,

Scientist