  In this work, the theory of second gradient electrodynamics, which is an
important example of generalized electrodynamics, is proposed and investigated.
Second gradient electrodynamics is a gradient field theory with up to
second-order derivatives of the electromagnetic field strengths in the
Lagrangian density. Second gradient electrodynamics possesses a weak
nonlocality in space and time. In the framework of second gradient
electrodynamics, the retarded Green functions, first-order derivatives of the
retarded Green functions, retarded potentials, retarded electromagnetic field
strengths, generalized Lienard-Wiechert potentials and the corresponding
electromagnetic field strengths are derived for three, two and one spatial
dimensions. The behaviour of the electromagnetic fields is investigated on the
light cone. In particular, the retarded Green functions and their first-order
derivatives show oscillations around the classical solutions inside the forward
light cone and it is shown that they are singularity-free and regular on the
light cone in three, two and one spatial dimensions. In second gradient
electrodynamics, the self-force and the energy release rate are calculated and
the equation of motion of a charged point particle, which is an
integro-differential equation where the infamous third-order time-derivative of
the position does not appear, is determined.
