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:
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.