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:

(17.1) |

(17.2) |

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 2015-08-23