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<p>The free-fall three-body problem exhibits a well-known coexistence of chaotic and regular dynamics. While its geometric structure has traditionally been explored through the AA-map and its statistical behavior described by classical escape theories, these approaches have remained largely disconnected. In this work, we introduce the Shape-Space-Sphere (SSS) as a global, non-hierarchical parametrization of all triangular configurations at zero velocity and zero angular momentum, enabling a direct geometric sampling of the complete configuration space. Using high-precision IAS15 integrations, we demonstrate that the topology of regular islands evolves continuously under variations of the mass ratios, revealing structural transitions rather than abrupt bifurcations. We further show that the statistical escape law of Valtonen &amp; Karttunen is consistently recovered when the free-fall configurations are sampled on the SSS, establishing an explicit correspondence between phase-space geometry and probabilistic outcomes. To quantify local predictability, we introduce a Local Stability Index (LSI) and identify a compact Most Stable Region (MSR) characterized by minimal sensitivity, low scramble number, and high numerical accuracy. These results unify geometric and statistical descriptions of the free-fall three-body problem within a single framework, demonstrating that classical escape statistics are intrinsically linked to the underlying configuration-space geometry.</p>

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A global geometric framework for chaos in the free-fall three-body problem

  • 期刊论文

  • 发表日期：

    2026年7月4日

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  • José Pinto

摘要

The free-fall three-body problem exhibits a well-known coexistence of chaotic and regular dynamics. While its geometric structure has traditionally been explored through the AA-map and its statistical behavior described by classical escape theories, these approaches have remained largely disconnected. In this work, we introduce the Shape-Space-Sphere (SSS) as a global, non-hierarchical parametrization of all triangular configurations at zero velocity and zero angular momentum, enabling a direct geometric sampling of the complete configuration space. Using high-precision IAS15 integrations, we demonstrate that the topology of regular islands evolves continuously under variations of the mass ratios, revealing structural transitions rather than abrupt bifurcations. We further show that the statistical escape law of Valtonen & Karttunen is consistently recovered when the free-fall configurations are sampled on the SSS, establishing an explicit correspondence between phase-space geometry and probabilistic outcomes. To quantify local predictability, we introduce a Local Stability Index (LSI) and identify a compact Most Stable Region (MSR) characterized by minimal sensitivity, low scramble number, and high numerical accuracy. These results unify geometric and statistical descriptions of the free-fall three-body problem within a single framework, demonstrating that classical escape statistics are intrinsically linked to the underlying configuration-space geometry.