  The problem of the classical non-relativistic electromagnetically kicked
oscillator can be cast into the form of an iterative map on phase space. The
original work of Zaslovskii {\it et al} showed that the resulting evolution
contains a stochastic flow in phase space to unbounded energy. Subsequent
studies have formulated the problem in terms of a relativistically charged
particle in interaction with the electromagnetic field. We review the standard
derivation of the covariant Lorentz force, and review the structure of the
relativistic equations used to study this problem.
  We show that the Lorentz force equation can be derived as well from the
manifestly covariant mechanics of Stueckelberg in the presence of a standard
Maxwell field. We show how this agreement is achieved, and criticize some of
the fundamental assumptions underlying these derivations. We argue that a more
complete theory, involving ``off-shell'' electromagnetic fields should be
utilized. We then discuss the formulation of the off-shell electromagnetism
implied by the full gauge invariance of the Stueckelberg mechanics (based on
its quantized form), and show that a more general class of physical phenomena
can occur.
