One-dimensional three-body problem via symbolic dynamics

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.