Classical Dynamics near the Triple Collision in a Three-Body Coulomb Problem

Abstract

We investigate the classical motion of three charged particles with both attractive and repulsive interactions. The triple collision is a main source of chaos in such three-body Coulomb problems. By employing the McGehee scaling technique, we analyze here for the first time in detail the three-body dynamics near the triple collision in 3 degrees of freedom. We reveal surprisingly simple dynamical patterns in large parts of the chaotic phase space. The underlying degree of order in the form of approximate Markov partitions may help in understanding the global structures observed in quantum spectra of two-electron atoms.

Photoionization of two-electron atoms via highly doubly excited states: Numerical and semiclassical results

“…[1,58]; more recent results on the dynamics near the triple collision can be found in Refs. [29,30], new insights into the global structure of the phase space were presented in Refs. [31,32,59].…”

“…Expressing L in scaled coordinates, we havẽ L → 0 as E → 0 when L remains constant. We can thus restrict the analysis to the three degrees of freedom subspacẽ L = 0 if we work at fixed L and in the limit E → 0 [30,60]. Following standard semiclassical arguments, the Green function G(R, ,R , ;E) can be approximated by contributions from classical trajectories at energy E with initial and final coordinates given by (R , ) and (R, ), respectively [61].…”

“…Asymptotically, these trajectories converge to a set of triple collision orbits starting and ending exactly in the singularity-we will refer to these orbits as closed triple collision orbits (CTCO's). Triple collision orbits can only occur in the so-called eZe space [30], a collinear subspace of the full three-body dynamics where the two electrons are on opposite sides of the nucleus [58]. As R 0 → 0, only orbits coming close to the eZe space can start and return to˜ and they will do so in the vicinity of a CTCO.…”

“…The chaotic part of the classical dynamics, on the other hand, is dominated by the complex folding patterns of the stable and unstable manifolds of the triple collision, which was analyzed in detail in Refs. [29][30][31][32].…”

Periodic orbits and binary collisions in the classical three-body Coulomb problem

“…Main difficulties are due to (1) the mixture of chaos and tori, concept of the triple collision manifold [7] are important examples [8][9][10][11][12][13][14]. Thanks to these tools, the dynamics near triple collision and the structure of stable and unstable manifolds were elucidated to some extent.…”