Dynamics of Coulombic and gravitational periodic systems

We study the dynamics and the phase-space structures of Coulombic and self-gravitating versions of the classical one-dimensional three-body system with periodic boundary conditions. We demonstrate that such a three-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 nonchaotic 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 Rapid Communication from the well-understood free-boundary versions of the three-body problem. Our treatment of the three-body systems 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.

Figure 1

Poincaré plots for the Coulombic system with (a) ${\mathcal{H}}_c=-1.098$ and (b) ${\mathcal{H}}_c=2.031$. The boxes denote the areas magnified in the corresponding insets. (c) and (d) depict the periodic trajectories on the rhombic plane for a P2 orbit and a P6 orbit, respectively, each with ${\mathcal{H}}_c=-1.098$. (e) Position vs time for the three-particle system corresponding to the rhombic trajectory shown in (c). $\alpha$ and $\beta$ have been expressed in the units of $\sqrt{2}L$, primitive-cell positions in the units of $\frac{2}{3}L$, and time in the units of $\sqrt{\frac{mL}{6\pi k{q}^2}}$.

Figure 2

Simulation results for the gravitating system: (a)–(e) are for ${\mathcal{H}}_g=0.226$ whereas (f)–(h) are for ${\mathcal{H}}_g=0.624$. (h) and (f) show Poincaré plots for the two values of ${\mathcal{H}}_g$. The boxes denote the areas magnified in the corresponding insets. The three-particle evolution and the corresponding trajectories on the rhombic plane for (b), (c) a P1 orbit; (d), (e) a P3 orbit; (g), (h) a P2 orbit; (i), (j) another P2 orbit. $\alpha$ and $\beta$ have been expressed in the units of $\sqrt{2}L$, primitive-cell positions in the units of $\frac{2}{3}L$, and time in the units of $\sqrt{\frac{L}{6\pi Gm}}$.