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If I recall correct you can say that e.g. the electric vectorfield is only a function of the radius if the source terms (charge) is spherical and uniform so that a group action that rotates space makes the source term invariant. Is there some general rule for this? How about when it comes to boundary conditions?

The teachers never taught us this so I am trying to reconcile it with what I've read about G-spaces, Lie group et cetera on my freetime.

Emil
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  • What kind of general rule are you looking for? If the equation obeys some symmetry, then a symmetric source term will yield a symmetric solution. If you have (for example) an anisotropic medium this is no longer the case. On the matter of boundary conditions: In such problems one usually assumes sufficiently fast decay to infinity. If one imposes asymmetric boundary conditions the solution can obviously no longer be symmetric. Are you looking for a precise mathematical treatment? As it stands the question seems to be very broad and unspecific. – Sebastian Riese Oct 13 '15 at 20:40
  • Maybe I am wondering something about ignorable parameters. Or how a symmetry of say the rhs of a equation manifests itself on the lhs if the lhs is some PDE of a vector field. Do they apply equally to all the components of the vector? Do the symmetries apply to constraints like boundary conditions too? I think I need some hints or keywords I can google to learn the mathematical machinery behind this. – Emil Oct 13 '15 at 20:59
  • I read sometimes that physical theories can abide by SO3 symmetry or something like that. Does that mean that the functions automatically have ignorable parameters? Or only if the source terms are uniform and SO3? Etc I am feeling lost. – Emil Oct 13 '15 at 21:02
  • A symmetry of a theory means, that you get transformed results from the transformed source (the applied transformation being the same in both cases). In such a setting a symmetric source will have a symmetric solution (for which in an apt choice of coordinates allows to reduce the number of parameters). – Sebastian Riese Oct 13 '15 at 21:06

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