  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.
