Self-Similar Structure in the Linear Three-Body Problem

Abstract

What characterizes the self-similarity of a phase structure in the N-body problem? Chaos or collision is considered to be an essential factor of self-similarity. This paper shows that the answer is collision. In the gravitational three-body problem, the self-similar structure of phase trajectories are correlated with triple and binary collisions (Umehara and Tanikawa, 2000). Each set of continuous phase trajectories converges to a triple couision trajectory (Tanikawa and Umehara, 1998). An analysis has shown that collision singularity induces such a distribution (Umehara and Tanikawa, 1999). However, there is also a correlation between self-similarity and triple collision even in the thre body problem with non-singular attractive potential (Nakato and Aizawa, 2000). Here, the system is further simplified. The three-body problem with harmonic potential is analyzed. This is the linear system. Even in this non-chaotic system, the phase structure shows the self-similarity with convergence to a collision orbit.