Three Classes of Newtonian Three-Body Planar Periodic Orbits (1303.0181v1)

Abstract: We present the results of a numerical search for periodic orbits of three equal masses moving in a plane under the influence of Newtonian gravity, with zero angular momentum. A topological method is used to classify periodic three-body orbits into families, which fall into four classes, with all three previously known families belonging to one class. The classes are defined by the orbits geometric and algebraic symmetries. In each class we present a few orbits initial conditions, 15 in all; 13 of these correspond to distinct orbits.

• The paper introduces three new classes of periodic orbits by applying a refined topological classification method.
• It employs gradient descent refinement on a two-parameter subset of initial conditions to achieve high-precision orbit detection.
• The findings deepen our understanding of classical three-body dynamics and suggest potential applications in gravitational wave analysis and dynamical research.

Three Classes of Newtonian Three-Body Planar Periodic Orbits

Milovan Šuvakov and V. Dmitrašinović present a comprehensive numerical investigation into periodic orbits in a classical three-body problem, focusing on equal masses in a plane influenced by Newtonian gravity. Their paper is particularly concerned with configurations having zero angular momentum, a condition providing insights into potential orbit families. By employing a topological classification method, the authors elucidate periodic orbits more thoroughly, organizing them into distinct categories based on geometric and algebraic symmetries not apparent in traditional classification models.

Background and Methods

The complexity of the three-body problem, arising from limited integrals of motion compared to the degrees-of-freedom, has intrigued researchers since the late 19th century. Periodic solutions, first extensively analyzed by Lagrange and Euler, form an important subset due to their implications in dynamically understanding the problem. These solutions guide researchers, offering a feasible entry point into a historically inaccessible area of classical mechanics, as famously noted by Poincaré.

Prior methods, like the braid group approach used by Moore and others, have provided frameworks for identifying individual trajectories but lacked efficiency in categorizing orbit families. Montgomery introduced a novel topological approach, increasingly shaping the systematic classification and assessment of these orbits.

Šuvakov and Dmitrašinović expanded this methodology, examining a two-parameter subset of the initial conditions in the four-dimensional space associated with zero angular momentum. Implementing gradient descent refinement, they achieved precision in orbit detection, presenting a set of 15 solutions encompassing 13 topologically distinct families. These are further grouped into three new classes and one pre-existing class based on symmetry considerations. The class categorization provides deeper insight into the structure of periodic orbits.

Results and Classification

The authors introduce classes with both geometric and algebraic characteristics:

1. Class I.A: Features two-axis reflection symmetry, with distinctive members like the Butterfly and Bumblebee orbits. For instance, the Butterfly's free group element topology suggests robustness akin to the Figure-8 orbit, albeit with variations in spatial and temporal dimensions.
2. Class I.B: Contains orbits with sub-symmetries, such as moths and butterfly variants, showcasing unique geometric trajectory patterns and free group elements that typify the complexity of these orbits.
3. Class II.B: Identified through central point reflection symmetry, orbits in this class, such as the Yarn orbit, offer an alternative symmetry structure while maintaining mathematical distinctness.
4. Class II.C: These solutions lack algebraic symmetrical simplicity and include the Yin-Yang I and II orbits. Noteworthy is the multi-initial condition association with single topological patterns, indicating underlying connections in their dynamical evolution.

Implications and Future Directions

These findings pave the way for expanding theoretical and practical understandings of the Newtonian three-body problem by refining the taxonomy of possible trajectories. The identified periodic solutions have potential analogs with nonzero angular momentum orbits, with implications for stability and further theoretical exploration.

While observational confirmation of such systems remains elusive due to the rarity of equal mass, low-angular-momentum celestial configurations, these theoretical constructs enrich the repository of known orbits, with possibilities for applications in gravitation wave analysis and general relativistic extensions.

This paper underscores the necessity of methodical numerical approaches in uncovering higher-order periodic solutions, suggesting a fertile ground for future analytic derivations and solutions in varied gravitational potentials. Evidently, the authors anticipate further exploration of other initial condition subspaces and potential discovery of additional orbit types, providing valuable insights into the classical and quantum dynamical realms.