We are interested in deducing the dynamics of point charged particles in ``given'' (i. e. -- fixed) electromagnetic fields. We already ``know'' the answer, it is given by the covariant form of Newton's law, that is: d p^&alpha#alpha;d &tau#tau; = m dU^&alpha#alpha;d &tau#tau; = qc F^&alpha#alpha;&beta#beta; U_&beta#beta;. From this we can find the 4-acceleration, d U^&alpha#alpha;d &tau#tau; = qmc F^&alpha#alpha;&beta#beta; U_&beta#beta; which we can integrate (in principle) to find the 4-trajectory of the particle in question.
However, this is not useful to us. Real physicists don't use Newton's law anymore. This is nothing against Newton, it is just that we need Hamilton's or Lagrange's formulation of dynamics in order to construct a quantum theory (or even an elegant classical theory). Our first chore, therefore, will be to generalize the arguments that lead to the Euler-Lagrange or Hamilton equations of motion to four dimensions.