Electrodynamics(?): finding position of particles in function of time?

[WindScars]

Im still on high school so I don't know where this question belongs to, I don't know even what is the exact subject of this question, so I would like you more experienced members to help me understand what exactly I am asking and where I can find information about it:

"In an empty universe, two particles, A and B, are instantly created at time=0s with an arbitrary position, mass, charge and velocity. They interact by electromagnetism and nothing else. How to find their positions in function of time?"

I have attempted solving it this way: their positions are the an integral of their velocities, right? Their velocities are an integral of their accelerations. Their accelerations are a function of their distances, that is a function of their positions. So in the end I had something like:
$posA(t) = \int_0^t \int_0^t(\frac{k*qA*qB}{|posA(t)-posB(t)|²*mA}) dtdt$
Well its probably wrong and even if it were right Id have no idea of how to solve it. But you got the idea. Thoughts please.

 

[jtbell]

It looks like you're including only the electric force. In general, you have to include the effect of the magnetic force which depends on the speed and direction of motion of both particles.

 

[Born2bwire]

Yeah I think this is going to be beyond high school mathematics and physics. The fact that you will have accelerating charges means that there will be electromagnetic fields as opposed to purely electric as jtbell stated. Normally the calculation of the trajectory of a charged particle in an electromagnetic field would not be too daunting. You would just take the Lorentz force,

$$\mathbf{F} = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B} \right) = m \frac{d \mathbf{v}}{dt} = m \frac{d^2 \mathbf{x}}{dt^2}$$

You could perform the integrations above and use your initial conditions to find the unknown constants. But since you have two charges that will both be accelerating and interacting with each other then it becomes a much more difficult problem. The most general way of solving this would probably to use Lagrangian mechanics to find the trajectory. This would include the interaction of the particles with the fields and each other as their trajectories evolve.