Hamiltonian Relativistic Two-Body Problem: Center of Mass and Orbit Reconstruction

TL;DR

This paper explores the relativistic two-body problem by defining the center of mass and reconstructing particle orbits within a Hamiltonian framework, providing explicit solutions for Coulomb-like interactions.

Contribution

It offers explicit formulas for relativistic center of mass and orbit reconstruction, advancing the understanding of two-body dynamics in relativistic Hamiltonian mechanics.

Findings

Explicit expressions for relativistic center of mass.

Generators of Poincaré group with interactions.

Explicit integration of Coulomb-like relative motion.

Abstract

After a short review of the history and problems of relativistic Hamiltonian mechanics with action-at-a-distance inter-particle potentials, we study isolated two-body systems in the rest-frame instant form of dynamics. We give explicit expressions of the relevant relativistic notions of center of mass, we determine the generators of the Poincare' group in presence of interactions and we show how to do the reconstruction of particles' orbits from the relative motion and the canonical non-covariant center of mass. In the case of a simple Coulomb-like potential model, it is possible to integrate explicitly the relative motion and show the two dynamical trajectories.