New derivation for the equations of motion for particles in electromagnetism

We present equations of motion for charged particles using balanced equations, and without introducing explicitly divergent quantities. This derivation contains as particular cases some well known equations of motion, as the Lorentz-Dirac equations. An study of our main equations in terms of order of the interaction with the external field conduces us to the Landau-Lifshitz equations. We find that the analysis in second order show a special behavior. We give an explicit presentation up to third order of our main equations, and expressions for the calculation of general orders.

Figure 1

Two spacelike hypersurfaces, ${\Sigma }^{\prime }$ and $\Sigma$, are the boundary of the internal four volume $\mathcal{V}$. In turn, each of these spacelike hypersurfaces has the same two dimensional boundary $S$.

Figure 2

Two hypersurfaces reaching future null infinity.