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 interactions and self-interactions (EM-interacting
N-body systems). The correct solution to the question represents an overriding prerequisite for the consistency
between classical and quantum mechanics. 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 interactions 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 “no-interaction” theorem for the non-local Hamiltonian system considered here.
