You say "the problem cannot be solved in classical electrodynamics".
Firstly, I'm not sure I agree.
I would be happy to be proven wrong, but that's what I remember from studying physics many years ago. You can calculate the motion of a charged pointlike particle in given fields, and you can calculate the fields produced by charged pointlike particles following any motions, but you can't calculate the effects of the interaction of a particle with its own field.
This said, I vaguely remember some tricks that involved arbitrary dropping of some infinite terms, a bit like renormalization in QED only not precisely defined in any way. Perhaps with those tricks you can arrive to some kind of solution.
Secondly, can it be solved at all? Can you give me an algorithm that will solve this problem?
The interaction of a particle with its own field is tractable in quantum electrodynamics or so I was told. However, as you are no doubt aware, the state of a quantum-mechanical system is not really defined in terms of positions and velocities of a set of particles, so your problem taken literally doesn't make sense in this framework. And even if you re-stated it in an appropriate language, quantum field theory won't give you the precise evolution in time of any system, typically the best you can hope for is a transition amplitude between non-interacting particles in time minus infinity and non-interacting particles in time plus infinity.
Assume we do this experiment. We can do it as many times as we wish, changing the positions, velocities, etc. of the bodies as we wish, until we have as many cases as we want.
Do you believe we will not succeed in detecting a pattern and predicting how any such system will behave?
You mean, assume we take a bunch of charged particles with relativistic speeds and see what happens when they interact? That's what physicists do using accelerators. (Except they don't really observe the positions of particles at all times, they just see what goes in and what comes out, which is OK because as I mentioned that is all we can compute anyway.) When the velocities, and therefore energies, involved are really high you can't talk of any classical approximation because the quantum-field-theoretic effects take over and you observe phenomena like particle creation and annihilation etc. When the velocities are low you will probably observe non-relativistic two-body Coulomb scattering, which is mathematically no different from the Newtonian interaction between a planetoid and the Sun.
Firstly, I don't mean particles; Just small enough bodies.
And near-lightspeed velocities are not necessary; Maybe 5% of lightspeed will do.
When the velocities are low you will probably observe non-relativistic two-body Coulomb scattering, which is mathematically no different from the Newtonian interaction between a planetoid and the Sun.
Dude, what does that mean? Can you solve the problem I raised or not?
When the velocities are low you will probably observe non-relativistic two-body Coulomb scattering, which is mathematically no different from the Newtonian interaction between a planetoid and the Sun.
You say "the problem cannot be solved in classical electrodynamics".
Firstly, I'm not sure I agree. Secondly, can it be solved at all? Can you give me an algorithm that will solve this problem?