Chaotic particle motion around a homogeneous circular ring

We consider test particle motion in a gravitational field generated by a homogeneous circular ring placed in $n$-dimensional Euclidean space. We observe that there exist no stable stationary orbits in $n=6,7,\dots ,10$ but exist in $n=3$, 4, 5 and clarify the regions in which they appear. In $n=3$, we show that the separation of variables of the Hamilton-Jacobi equation does not occur though we find no signs of chaos for stable bound orbits. Since the system is integrable in $n=4$, no chaos appears. In $n=5$, we find some chaotic stable bound orbits. Therefore, this system is nonintegrable at least in $n=5$ and suggests that the timelike geodesic system in the corresponding black ring spacetimes is nonintegrable.

Figure 1

Typical shapes of stable bound orbits in ${\mathrm{\Phi }}_3$ (upper panels) and Poincaré sections with the same energy and angular momenta but different initial positions and velocities (lower panels). Units in which $R=1$, $m=1$, and $GM=1$ are used. The local minimum point of $V_3$ is located at $(\zeta ,\rho )=({\zeta }_0,0)$ in each case. In the upper panels, the black and the red solid curves show contours of $V_3$; in particular, the red corresponds to $V_3=\mathcal{E}$. Each blue solid curve shows a stable bound orbits with energy $\mathcal{E}$. Each point in the lower panels show a value ($\rho$, $p_{\rho }$) of a particle that passes through a constant-$\zeta$ surface with $p_{\zeta }>0$. Thirty orbits with different initial conditions are superposed in each plot. (a) $({\zeta }_0,0)=1.8,\mathcal{E}=-0.23698$, (b) $({\zeta }_0,0)=2.0,\mathcal{E}=-0.21130$ and (c) $({\zeta }_0,0)=2.2,\mathcal{E}=-0.17415$.

Figure 2

Region $D_4$, the allowed region for stable stationary orbits in ${\mathrm{\Phi }}_4$. Units in which $R=1$ are used. The circular ring source is located at $(\zeta ,\rho )=(1,0)$. The shaded region denotes $D_4$. The blue solid and dashed curve are the boundaries of $D_4$ and are determined by $L_0=0$ and $h_0=0$, respectively. The region $D_4$ is colored in blue when ${\mathcal{E}}_0<0$ and orange when ${\mathcal{E}}_0>0$.

Figure 3

Region $D_5$, the allowed region for stable stationary orbits in ${\mathrm{\Phi }}_5$. Units in which $R=1$ are used. The circular ring source is located at $(\zeta ,\rho )=(1,0)$. The shaded region denotes $D_5$. The blue solid and dashed curves are the boundaries of $D_5$ and are determined by $L_0=0$ and $h_0=0$, respectively. The region $D_5$ is colored in blue when ${\mathcal{E}}_0<0$ and in orange when ${\mathcal{E}}_0>0$.

Figure 4

Typical shapes of stable bound orbits in ${\mathrm{\Phi }}_5$ (upper panels) and Poincaré sections with the same energy and angular momenta but different initial positions and velocities (lower panels). Units in which $R=1$, $m=1$, and $GM=1$ are used. The values $({\zeta }_0,{\rho }_0)$ denote the location of the local minimum point of $V_5$ in each case. The black and red solid curves of each upper panel are contours of the effective potential $V_5$, which take 1, ${10}^{-1}$, ${10}^{-2}$ (black), and 0 (red). The blue curves show stable bound orbits with $\mathcal{E}=0$. The plot in the $\rho \text{−}{p}_{\rho }$ plane in each lower panel show the Poincaré sections. Thirty orbits with different initial conditions are superposed in each plot.