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Consider a system of many point particles each having a certain mass and electric charge and certain initial velocity. This system is completely governed by the laws of gravitation and electromagnetic field. If this system is left on its own without any external influence, any physical quantity like velocity of a particle or electric field at a point, of this system will vary smoothly with time and space. (keeping aside any singularities that could occur eventually). My question is whether any such system which is governed by such laws of physics (except ofcourse QM) would remain smooth until the occurrence of a singularity, in the absence of any external influence.

PS : please feel free to edit/suggest changes to the question in case if it could make it more meaningful.

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Classical point charges always cause a singularity in the electric field, don't they? – leftaroundabout May 21 '11 at 11:45
@leftaroundabout : Ok....they are given an exception. – Rajesh D May 21 '11 at 11:47
If by singularity you mean "not smooth" then isn't it necessary that the solutions remain smooth "until the occurrence of a singularity"? – Greg P May 21 '11 at 16:29
@Greg P : I admit the mistake i made in the question. We know that there exist singularities in electric and gravitational fields at the points where there are point masses and charges. What I intended was that singularities apart from these...anyway the question doesn't seem reasonable now..I got this doubt as a part of a larger doubt bugging my mind for a long time....I'd like to post it ,ore openly in another question. – Rajesh D May 21 '11 at 16:37

2 Answers

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I think that singularities can occur in the electromagnetic fields (other than the ones occurring at the location of particles), if one allows the particles to move at speed $c$. That is, one can arrange for sonic booms or Cherenkov radiation.

This raises the question "is Cherenkov radiation smooth?" I suspect that the non smoothness only occurs at distances larger than the distance between the individual atoms; that is, that the lack of smoothness is in the math approximation rather than the physics.

In any case, if one does not allow the particles to approach the speed of light, I believe that the sum of all their electromagnetic fields will be a sum of smooth functions and so will be mathematically smooth.

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You can always create discontinuous waves (which are solutions that exists even in the total absence of sources (be them gravitational or electromagnetic)

You can always write a discontinuous function and use it to define, say, the electric field at some time $t_1$, and ask what do the fields look at time $t_0 < t_1$. Of course such solutions are a bit ad hoc in the sense that they are simply a wave solution with no actual sources

You certainly can generate fields that approach a discontinuous one by adding enough particles that you can approximate them by discontinuous functions $\rho$ for densities and currents, but of course the discontinuity is only an approximation

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