Dynamics of Coulombic and Gravitational Periodic Systems

TL;DR

This paper investigates the complex dynamics of Coulombic and gravitational three-body systems with periodic boundaries, revealing diverse behaviors including chaos and quasiperiodicity, and introduces a novel isomorphic reduction to a single-particle 2D potential model.

Contribution

It provides the first analysis of spatially periodic 3-body systems, deriving exact Hamiltonians and demonstrating their rich dynamical behaviors through simulations.

Findings

No evidence of global chaos in either system.

Transition from non-chaotic to mixed behavior with energy changes.

Distinct chaotic transition behavior compared to free-boundary systems.

Abstract

We study the dynamics and the phase-space structures of Coulombic and self-gravitating versions of the classical one-dimensional 3-body system with periodic boundary conditions. We demonstrate that such a 3-body system may be reduced isomorphically to a spatially periodic system of a single particle experiencing a two-dimensional potential on a rhombic plane. For the case of both Coulombic and gravitational versions, exact expressions of the Hamiltonian have been derived in rhombic coordinates. We simulate the phase-space evolution through an event-driven algorithm that utilizes analytic solutions to the equations of motion. The simulation results show that the motion exhibits chaotic, quasiperiodic, and periodic behaviors in segmented regions of the phase space. While there is no evidence of global chaos in either the Coulombic or the gravitational system, the former exhibits a transition from a completely non-chaotic phase space at low energies to a mixed behavior. Gradual yet striking transitions from mild to intense chaos are indicated with changing energy, a behavior that differentiates the spatially periodic systems studied in this paper from the well-understood free-boundary versions of the 3-body problem. Our treatment of the 3-body systems is the first one of its kind and opens avenues for analysis of the dynamical properties exhibited by spatially periodic versions of various classes of systems studied in plasma and gravitational physics as well as in cosmology.