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.

Robert G. Brown 2017-07-11