  A simple mathematical procedure is introduced which allows redefining in an
exact way divergent integrals and limits that appear in the basic equations of
classical electrodynamics with point charges. In this way all divergences are
at once removed without affecting the locality and the relativistic covariance
of the theory, and with no need for mass renormalization. The procedure is
first used to obtain a finite expression for the electromagnetic
energy-momentum of the system. We show that the relativistic Lorentz-Dirac
equation can be deduced from the conservation of this electromagnetic
energy-momentum plus the usual mechanical term. Then we derive a finite
lagrangian, which depends on the particle variables and on the actual
electromagnetic potentials at a given time. From this lagrangian the equations
of motion of both particles and fields can be derived via Hamilton's
variational principle. The hamiltonian formulation of the theory can be
obtained in a straightforward way. This leads to an interesting comparison
between the resulting divergence-free expression of the hamiltonian functional
and the standard renormalization rules for perturbative quantum
electrodynamics.
