  An open issue in classical relativistic mechanics is the consistent treatment
of the dynamics of classical $N$-body systems of mutually-interacting
particles. This refers, in particular, to charged particles subject to EM
interactions, including both binary and self interactions (EM-interacting
$N$-body systems). In this paper it is shown that such a description can be
consistently obtained in the context of classical electrodynamics, for the case
of a $N$-body system of classical finite-size charged particles. A variational
formulation of the problem is presented, based on the $N$-body hybrid
synchronous Hamilton variational principle. Covariant Lagrangian and
Hamiltonian equations of motion for the dynamics of the interacting $N$-body
system are derived, which are proved to be delay-type ODEs. Then, a
representation in both standard Lagrangian and Hamiltonian forms is proved to
hold, the latter expressed by means of classical Poisson Brackets. The theory
developed retains both the covariance with respect to the Lorentz group and the
exact Hamiltonian structure of the problem, which is shown to be intrinsically
non-local. Different applications of the theory are investigated. The first one
concerns the development of a suitable Hamiltonian approximation of the exact
equations that retains finite delay-time effects characteristic of the binary
and self EM interactions. Second, basic consequences concerning the validity of
Dirac generator formalism are pointed out, with particular reference to the
instant-form representation of Poincar\`{e} generators. Finally, a discussion
is presented both on the validity and possible extension of the Dirac generator
formalism as well as the failure of the so-called Currie \textquotedblleft
no-interaction\textquotedblright\ theorem for the non-local Hamiltonian system
considered here.
