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One-dimensional three-body problem via symbolic dynamics.

Kiyotaka Tanikawa¹, Seppo Mikkola

¹National Astronomical Observatory, Mitaka, Tokyo 181, Japan.

Chaos (Woodbury, N.Y.)

|June 5, 2003

Summary

Symbolic dynamics reveals that not all collision sequences are possible in the three-body problem. This study identifies impossible sequences and proves that possible ones form a Cantor set.

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Area of Science:

Celestial mechanics
Dynamical systems theory
Computational physics

Background:

The three-body problem is a fundamental challenge in classical mechanics.
Understanding the long-term behavior of systems with three interacting bodies is complex.
Symbolic dynamics offers a method to analyze complex orbital behaviors.

Purpose of the Study:

To apply symbolic dynamics to the one-dimensional three-body problem with equal masses.
To systematically identify inadmissible (unrealizable) sequences of binary collisions.
To investigate the structure of admissible collision sequences and their relation to periodic orbits.

Main Methods:

Symbolic dynamics was used to represent collision sequences.
Time reversibility and numerical data were employed to find inadmissible sequences.
A transition graph of Poincare sections was constructed.
The properties of admissible sequences were analyzed under specific assumptions.

Main Results:

Collision sequences were encoded using a two-symbol system.
An infinite number of periodic orbits, beyond the known Schubart orbit, were discovered.
Inadmissible collision sequences were systematically identified.
The set of admissible symbol sequences was proven to form a Cantor set.

Conclusions:

Symbolic dynamics provides a powerful framework for analyzing the complexity of the three-body problem.
The identification of inadmissible sequences simplifies the understanding of possible orbital evolutions.
The discovery of an infinite number of periodic orbits highlights the rich dynamics of this system.
The Cantor set structure of admissible sequences suggests a fractal nature in the collision dynamics.

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