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.
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References
Nakato, M. and Aizawa, Y.: 2000, Chaos, Solitons and Fractals 11, pp. 171–185.
Tanikawa, K. and Umehara, H.: 1998, Cel. Mech. Dyn. Astr. 70, pp. 167–180.
Umehara, H. and Tanikawa, K.: 1999, Cel. Mech. Dyn. Astr. 74, pp. 69–94.
Umehara, H. and Tanikawa, K.: 2000, Cel. Mech. Dyn. Astr.,in press.
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Umehara, H. (2001). Self-Similar Structure in the Linear Three-Body Problem. In: Pretka-Ziomek, H., Wnuk, E., Seidelmann, P.K., Richardson, D.L. (eds) Dynamics of Natural and Artificial Celestial Bodies. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1327-6_37
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DOI: https://doi.org/10.1007/978-94-017-1327-6_37
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