[Paper Review] Existence of prograde double-double orbits in the equal-mass four-body problem

Wentian Kuang, Duokui Yan|arXiv (Cornell University)|Oct 27, 2017

Spacecraft Dynamics and Control13 references3 citations

TL;DR

This paper establishes the existence of prograde double-double orbits in the equal-mass four-body problem for rotation angles θ ∈ (0, π/7] using topological constraints and a binary decomposition method. By introducing a novel geometric argument, it proves that the action of the prograde minimizer strictly exceeds that of the retrograde minimizer for all θ ∈ (0, π/2), resolving a long-standing open problem in celestial mechanics through variational and topological methods.

ABSTRACT

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle $θ\in (0, π/7]$ in the equal-mass four-body problem. A new geometric argument is introduced to show that for any $θ\in (0, π/2)$, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

Motivation & Objective

• To resolve the open problem of proving the existence of prograde double-double orbits in the equal-mass four-body problem.
• To exclude collision singularities in action minimizers under specific topological constraints.
• To establish a geometric argument showing that the action of the prograde minimizer exceeds that of the retrograde minimizer for all θ ∈ (0, π/2).
• To extend the applicability of variational methods to symmetric periodic orbits in the N-body problem, including the three-body problem.
• To provide a constructive existence proof via level estimate arguments and binary decomposition, avoiding local deformation techniques.

Proposed method

• Introduce topological constraints on initial (collinear) and final (rotated rectangular) configurations to define a function space Σ(θ) of admissible paths.
• Apply a binary decomposition method to analyze the structure of potential collision configurations in the minimizer.
• Use a level estimate argument to exclude collision singularities in the action minimizer over Σ(θ) for θ ∈ (0, π/7].
• Define the symmetric subspace V ⊂ ℝ^{4×2} corresponding to the parallelogram four-body configuration with equal masses.
• Construct test paths P_test,θ over discrete time intervals to numerically estimate action values and compare them with lower bounds.
• Develop a new geometric argument comparing the action of prograde and retrograde orbits by analyzing the relative positions and symmetries of the bodies.

Experimental results

Research questions

• RQ1 Does a prograde double-double orbit exist in the equal-mass four-body problem for any θ ∈ (0, π/7]?
• RQ2 Can topological constraints and variational methods be used to exclude collisions in the minimizer of the action functional for such orbits?
• RQ3 Is the action of the prograde double-double orbit strictly greater than that of the retrograde double-double orbit for all θ ∈ (0, π/2)?
• RQ4 Can the geometric argument used to compare prograde and retrograde actions be generalized to other periodic orbits in the three-body problem?
• RQ5 What is the minimal interval of θ for which the existence of prograde double-double orbits can be rigorously proven using this method?

Key findings

• The paper proves the existence of prograde double-double orbits for all θ ∈ (0, π/7] in the equal-mass four-body problem.
• A new geometric argument demonstrates that the action of the prograde minimizer is strictly greater than that of the retrograde minimizer for all θ ∈ (0, π/2).
• The action of the test path P_test,θ is numerically computed and shown to lie above the lower bound g₁(θ) for θ ∈ [0.115π, 0.143π], supporting the existence result.
• For θ₀ = 0.125π, the test path P_test,θ is defined over θ ∈ [0.115π, 0.131π], and for θ₀ = π/7, it is extended to θ ∈ [0.131π, 0.143π], with consistent action behavior.
• The level estimate argument successfully excludes collision singularities in the minimizer under the given topological constraints, confirming regularity of the orbit.
• The method is generalizable and can be applied to study other orbits such as the Schubart orbit and Hénon orbit in the planar three-body problem.