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Suppose you are given a configuration of electric charges and currents. How do you go about solving it? First you find the fields using Maxwell's equations. Then you solve for the forces and find the accelerations.

This changes the configuration and you have to go through the above mentioned procedure from start once again and then again. Is there some way in which you could find the entire future of the configuration all in one go? Some super-maxwell's equations where if you input the time it will tell you the future like all scientific theories are supposed to do?

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  • $\begingroup$ I would say that should work with classical tools for classical mechanics... to the same extend you can solve an n-body problem. $\endgroup$
    – mikuszefski
    Commented Mar 16, 2017 at 7:22
  • $\begingroup$ I see problems arising, since the field one electron produces will not affect itself: When you are given the trajectory of an electron, you can calculate the EM Field it produces, but this EM Field won't affect the electron. $\endgroup$
    – Quantumwhisp
    Commented Mar 16, 2017 at 8:53

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It depends on what you mean by solving for the future state. Even in Newtonian gravity, the three body problem requires approximations and with many bodies there really is no shortcut other than to calculate numerically with a simulation.

The equations of classical electrodynamics can become messy even faster than Newtonian gravity, as there is radiation. But changes in the fields propagate at a finite velocity, so you can use the equations to locally update the fields and forces on particles, take small step in time and repeat to do a simulation.

So if you were hoping for an awesome method that can give an exact formula solving the field equations and giving charge density changes, then no, we have no such thing. The closest we have to a "super solver" is a computer simulation.

Now, I should mention that there are situations in classical electrodynamics where the "feed-back" between charges and fields can run awry. In Griffith's EM textbook, he argues that point particles in classical electrodynamics lead to difficulties where accelerating the particle can make it interact with its own radiation in a way that are not fully resolvable in classical E&M. So even a single particle and fields can run into trouble.

Its not always possible to avoid approximations.

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