Dynamics starting from zero velocities in the classical Coulomb three-body problem

A qualitative geometrical analysis of the classical Coulomb three-body problem is developed. Three ions and atom are investigated: namely, ${\mathrm{H}}^{-}$, He, and ${\mathrm{Li}}^{+}$. The dynamics of these atom and ions is treated in the approximation that the nucleus has infinite mass. Geometrical structures of the dynamics starting from the initial conditions with zero velocities are elucidated. In particular, the distribution of the final states in the initial condition space, its fractal structure, and binary collision orbits are specified. As a result, it is found that binary collision orbits form a lobelike structure. It is shown that the lobelike structure is, in fact, fractal. The lobelike structure is continuously connected to the collinear $eZe$ configuration space, in which a well-known binary symbolic dynamics exists. With these observations, it is expected that as well as triple-collision orbits, binary-collision orbits also serve us some clue to find a symbolic dynamics for the full dynamics with zero angular momentum.

Figure 1

Initial condition space of the DZV in $D_{1∕4}$: ${\mathrm{H}}^{-}$. Dark gray represents the electron-1 escape. Bright gray represents the electron-2 escape. Black represents the forbidden region $(E>0)$. White represents the trapped trajectory.

Figure 2

Initial condition space of the DZV in $D_{1∕4}$: He. Dark gray represents the electron-1 escape. Bright gray represents the electron-2 escape. Black represents the forbidden region $(E>0)$. White represents the trapped trajectory.

Figure 3

Initial condition space of the DZV in $D_{1∕4}$: ${\mathrm{Li}}^{+}$. Dark gray represents the electron-1 escape. Bright gray represents the electron-2 escape. Black represents the forbidden region $(E>0)$. White represents the trapped trajectory.

Figure 4

The first-triple encounter orbits in the initial condition space: ${\mathrm{H}}^{-}$. Recall that the initial conditions where the triple encounters occur are the boundaries of the bright gray and the dark gray regions in Fig. 1.

Figure 5

Classification of the initial condition space $D_{1∕4}$ for He: (a) $n_{\mathit{min}}=1$, (b) $n_{\mathit{min}}=2$, (c) $n_{\mathit{min}}=3$, and (d) $n_{\mathit{min}}=4$. We set $R={10}^{-3}$. $S_0$ (white), $S_1$ (black), $S_2$ (gray), $S_3$ (dark gray), and $S_4$ (none). The black region in lower right corner is a forbidden region $(E⩾0)$.

Figure 6

Binary collision orbits for He: (a) $n_{\mathit{min}}=1$, (b) $n_{\mathit{min}}=2$, (c) $n_{\mathit{min}}=3$, (d) $n_{\mathit{min}}=4$, (e) $n_{\mathit{min}}=5$, and (f) $n_{\mathit{min}}=6$. We set $R={10}^{-3}$. The black (bright gray) curve corresponds to the electron-1 (-2) collisions, respectively.

Figure 7

Binary collision orbits for He: the binary collision orbits from $n_{\mathit{min}}=1$ to $n_{\mathit{min}}=8$ are superposed in one figure. $R={10}^{-3}$. The black (bright gray) curve corresponds to the electron-1 (-2) collisions, respectively.

Figure 8

A schematic picture of the lobelike structure: The $(n+1)\text{th}$ lobes are nested in the $n\text{th}$ lobes. The black (bright gray) curve corresponds to the electron-1 (-2) collisions, respectively.