Hamiltonian structure of classical N-body systems of finite-size particles subject to EM interactions

Abstract

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). 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 nonlocal. 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 Poincarè 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.the conditions of validity of DGF itself and, in particular, whether such an approach actually applies at all to EMinteracting N -body systems. Goals of the paperPut all the previous motivations in perspective, in this paper a systematic solution to these issues is presented, for the case of EM-interacting N -body systems of classical finite-size charged particles. The theory is developed in the framework of classical electrodynamics and special relativity (i.e., assuming a flat Minkowski space-time) and is shown to satisfy all the prerequisites indicated above (#1-#5).Starting point is the determination of both Lagrangian and Hamiltonian equations of motion for classical charged particles belonging to an EM-interacting N -body system. By construction (see Section 2) the latter can be considered as a system of smooth hard sphere, namely in which hard collisions occurring between the particles, when their boundaries ∂Ω (i) (see below) come into contact, conserve each particle angular momentum. In particular, for simplicity, in the following all extended particles will be considered as acted upon only by EM inte...

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