Symmetries and Maxwells equations

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Discussion Overview

The discussion revolves around the separability of solutions to Maxwell's equations in the context of symmetries, particularly focusing on time-invariant and radially invariant systems. Participants explore whether the presence of symmetry in Maxwell's equations implies that solutions can be expressed in a separable form, and whether this principle extends beyond Maxwell's equations.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant, Niles, proposes that solutions to Maxwell's equations can be separable when the system exhibits symmetry, questioning if this is a general rule applicable beyond Maxwell's equations.
  • Another participant counters that separability is only true for specific potentials, namely 1/r and r² potentials.
  • Niles inquires about the role of spatial dependence in Maxwell's equations, suggesting a connection to ε(r)=n(r)².
  • A later reply references Landau's Mechanics, noting a hidden SO(4) symmetry in the equations for the 1/r potential, which contributes to the separation of the angular component in solutions.

Areas of Agreement / Disagreement

Participants express differing views on the generality of separability in solutions to Maxwell's equations under symmetry, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

The discussion highlights potential limitations regarding the specific conditions under which separability holds, as well as the dependence on particular definitions of symmetry and potential forms.

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

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