Eqs of motion for 2-body problem in EM?

[pellman]

What are the equations of motion for two charges where the two charges are the only sources for the EM field? (No background field)

What I'm looking for is given two particles of mass m_1 and m_2 with respective position vectors x_1 and x_2, what are f_1 and f_2 such that

$$m_1\frac{d^2x_1}{dt^2}=f_1(x_1,\dot{x}_1,x_2,\dot{x}_2)$$

$$m_2\frac{d^2x_2}{dt^2}=f_2(x_1,\dot{x}_1,x_2,\dot{x}_2)$$

?

Anyone know of a text which covers this? A Lagrangian or Hamiltonian for the same situation would do as well.

I'm looking for the relativistic case in which the propagation time is taken into effect. That is, $$x_2,\dot{x}_2$$ in the expression for $$f_1$$ should be taken at the "retarded time", and similarly for $$x_1,\dot{x}_1$$ in $$f_2$$.

 

[Meir Achuz]

You must mean the second time derivative.
The equations for f are quite complicated, and just about useless in classical physics because they involve the retarded time and the acceleration itself.
The equations for constant velocity of each particle are simpler, but they are useless if the particles start to accelerate (or had been accelerating in the past).

 

[pellman]

You must mean the second time derivative.

Thanks! Of course. I corrected it.

The equations for f are quite complicated, and just about useless in classical physics because they involve the retarded time and the acceleration itself.

Oh, I know the solutions are useless. My purpose is that I want to understand certain aspects of the Lagrangian. And to make sure I have the correct Lagrangian, it needs to be able to yield the equations of motion. Or, as I say in OP, I'd be happy with the correct Lagrangian itself.