Centre-of-mass and internal symmetries in classical relativistic systems

TL;DR

This paper explores the internal symmetries of relativistic systems, revealing that Lorentz-Poincaré symmetry implies the existence of Laplace-Runge-Lenz vectors as internal moments, with applications to two-body and many-body systems.

Contribution

It demonstrates that Lorentz-Poincaré symmetry universally implies internal Laplace-Runge-Lenz vectors in relativistic systems.

Findings

LRL vectors are internal moments linked to Lorentz boosts

Internal symmetries are analogous to spatial rotation symmetries

Applications include two-body and many-body relativistic systems

Abstract

The internal symmetry of composite relativistic systems is discussed. It is demonstrated that Lorentz-Poincar\'e symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace-Runge-Lenz (LRL) vectors. The LRL symmetry is thus found to be the internal symmetry universally associated with the global Lorentz transformations, in much the same way as internal spatial rotations are associated with global spatial rotations. Two applications are included, for an interacting 2-body system and for an interaction-free many-body system of particles. The issue of localizability of the relativistic CM coordinate is also discussed.