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

TL;DR

This paper develops a covariant Hamiltonian framework for finite-size classical charged particle systems with electromagnetic interactions, incorporating delay effects and challenging existing no-interaction theorems.

Contribution

It introduces a variational Hamiltonian formulation for N-body systems with finite-size particles, including delay effects and non-local Hamiltonian structure, advancing classical electrodynamics theory.

Findings

Derived covariant delay-type equations of motion.

Established a Hamiltonian formalism with classical Poisson brackets.

Discussed implications for Poincaré generators and no-interaction theorems.

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). 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.