Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

Oscar Perdomo

arxiv: 2604.05653 · v1 · submitted 2026-04-07 · 🧮 math.DS · math-ph· math.MP

Families of periodic solutions of the 4- and 6-body problem using a gradient-free continuation method

Oscar Perdomo This is my paper

Pith reviewed 2026-05-10 19:30 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MP

keywords periodic solutionsn-body problemgradient-free optimizationcontinuation methodsymmetric configurationscelestial mechanicsstochastic search

0 comments

The pith

A gradient-free stochastic method computes families of periodic solutions for symmetric 4- and 6-body problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a gradient-free stochastic black-box procedure that solves systems of equations using only function evaluations. It applies the technique to the 4-body problem where two pairs of unit-mass bodies move in opposition and to the 6-body problem where two equilateral triangles carry masses 1 and m2. The search enforces return conditions that allow for overall rotation and relabeling of the bodies, producing continuous families of pseudo-periodic planar solutions indexed by the mass ratio m2. These families matter because they give explicit bounded motions in gravitational systems that standard gradient-based solvers may miss when symmetries reduce the problem but leave the equations underdetermined.

Core claim

Using a stochastic optimization approach that relies solely on function evaluations, the paper constructs two families of pseudo-periodic planar solutions for the 4-body problem with opposing equal-mass pairs and for the 6-body problem with two equilateral triangles. The solutions satisfy the Newtonian equations under the imposed symmetries and return to the initial configuration up to rotation and particle relabeling.

What carries the argument

The gradient-free stochastic black-box optimization procedure that locates points satisfying the periodic return conditions under the chosen rotational and relabeling symmetries.

If this is right

A continuous family of solutions exists for the 4-body problem as the second mass varies.
A similar family exists for the 6-body problem under the equilateral-triangle symmetry.
All solutions remain planar and preserve the specified opposition or triangular arrangements.
The imposed symmetries reduce the search space so that the stochastic method can locate entire curves of solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same stochastic continuation approach could be tested on other highly symmetric n-body configurations to locate additional periodic orbits.
Direct forward integration of the reported solutions would independently confirm whether they satisfy the gravitational equations to high accuracy.
The technique may extend to underdetermined boundary-value problems in celestial mechanics where gradient information is costly or unavailable.

Load-bearing premise

The stochastic search reliably identifies configurations that satisfy the exact differential equations rather than approximate or spurious points generated by the optimization process.

What would settle it

Numerically integrate one reported initial condition and period; the bodies should return to their starting positions and velocities within numerical tolerance after accounting for the allowed rotation and relabeling.

Figures

Figures reproduced from arXiv: 2604.05653 by Oscar Perdomo.

**Figure 1.** Figure 1: Solution with θ(T) = 60◦ . The top left image shows the trajectories of the 4 bodies from t = 0 to t = T. The top right image shows the trajectories of bodies 1 and 3 from t = 0 to t = 3T. Bodies 1 and 2 share the same trajectory, but bodies 3 and 4 do not. The bottom left image shows all the trajectories, and the bottom right image shows the trajectories of bodies 3 and 4. Therefore, once again, up to a … view at source ↗

**Figure 2.** Figure 2: Solution with θ(T) = 90◦ . The left image shows the trajectories of the 6 bodies from t = 0 to t = T. The right image shows the trajectories of all six bodies from t = 0 to t = 4T. Bodies 1, 2 and 3 have their own orbit while bodies 4, 5 and 6 share the same orbit In the present paper, we do not impose a condition of the form β(t) = θ(t)+c. As mentioned before, in order to compute each periodic solution, w… view at source ↗

**Figure 3.** Figure 3: Motion of bodies 1 and 3 from t = 0 to t = T for the periodic solution given in [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Orbits of the four bodies for the periodic solution given in [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Motion of bodies 1 and 4 from t = 0 to t = T for the periodic solution given in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Orbits of the six bodies for the periodic solution given in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

read the original abstract

In this paper, we describe a gradient-free method to solve a system of equations, and we use it to construct two families of pseudo-periodic planar solutions of the 4- and 6-body problem. The method is a stochastic black-box procedure that uses only function evaluations. For the 4-body problem, bodies 1 and 2 have mass 1 and move opposite to each other, and bodies 3 and 4 have mass $m_2$ and also move opposite to each other. For the 6-body problem, bodies 1, 2, and 3 have mass 1 and move on the vertices of an equilateral triangle centered at the origin, and bodies 4, 5, and 6 have mass $m_2$ and also move on the vertices of an equilateral triangle. In both cases, we compute families of periodic solutions by imposing return conditions up to rotation and relabeling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies a stochastic gradient-free optimizer to hunt for periodic orbits in symmetric 4- and 6-body setups, but offers no checks that the outputs actually solve the equations of motion.

read the letter

The main thing here is a stochastic black-box search used to locate families of periodic solutions in two symmetric N-body problems. For the 4-body case they fix two pairs of equal masses moving in opposition, and for the 6-body case they fix two equilateral triangles of masses 1 and m2. They search for initial conditions whose trajectories return close to the start after one period, allowing for overall rotation and particle relabeling, then trace how the solutions change with the free mass parameter m2. The symmetry reduction is handled cleanly and cuts the search space in a useful way. Applying a gradient-free stochastic procedure to these return conditions is a straightforward new use of the tool in this setting, and it avoids having to differentiate the return map, which can be messy in N-body systems. That part is practical and clearly thought through. The soft spot is validation. The description gives the setup and the optimizer but no sign of forward integration checks, residual norms on the trajectory, or comparison against a standard high-order solver to confirm closure within controlled tolerance. Stochastic searches can settle on points that look good under the objective yet fail to satisfy the differential equations once integrated. Without those post-search tests the families remain unverified numerical outputs rather than confirmed periodic solutions. This is the kind of work that would interest people who study numerical methods for periodic orbits or symmetric few-body dynamics. A reader already working on continuation techniques or looking for concrete examples in constrained N-body problems could pull ideas from the symmetry reductions and the search formulation. I would bring it to a reading group to talk about the method and the symmetry choices, but only to discuss what extra checks would be needed. It deserves peer review once the authors add explicit verification steps and error analysis; the core idea has enough structure to be worth referee time.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a stochastic gradient-free (black-box) optimization method to locate families of pseudo-periodic planar solutions in the 4-body and 6-body problems. Symmetries are imposed (opposing motions for equal-mass pairs in the 4-body case; equilateral-triangle motions for the two mass groups in the 6-body case), and periodic orbits are sought by enforcing return conditions up to rotation and particle relabeling while varying the mass parameter m2.

Significance. If the reported points are shown to satisfy the Newtonian equations of motion to controlled accuracy, the families would enlarge the known set of symmetric periodic solutions in few-body gravitational dynamics and illustrate a gradient-free continuation technique applicable when derivatives are unavailable or expensive. The approach could serve as a template for exploring other symmetric N-body configurations.

major comments (2)

[Abstract / method description] The description of the stochastic black-box procedure (Abstract and method outline) contains no post-optimization validation: no forward integration of candidate initial conditions, no residual norms on the return map, and no comparison against a high-order integrator or known equal-mass solutions. Without such checks, it is impossible to confirm that optimizer outputs solve the underlying ODEs rather than merely minimizing the imposed return objective.
[Results / numerical experiments] No error analysis, tolerance thresholds, or convergence diagnostics are supplied for the families as functions of m2. This is load-bearing for the central claim that families of periodic solutions have been constructed.

minor comments (1)

[Abstract] The term 'pseudo-periodic' appears in the Abstract but is not defined relative to the imposed symmetries or the return conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The points raised about validation and numerical diagnostics are well-taken and will be addressed through revisions to strengthen the presentation of our results.

read point-by-point responses

Referee: [Abstract / method description] The description of the stochastic black-box procedure (Abstract and method outline) contains no post-optimization validation: no forward integration of candidate initial conditions, no residual norms on the return map, and no comparison against a high-order integrator or known equal-mass solutions. Without such checks, it is impossible to confirm that optimizer outputs solve the underlying ODEs rather than merely minimizing the imposed return objective.

Authors: We agree that the current description lacks explicit post-optimization validation steps. Although the stochastic procedure is constructed to drive the return-map objective (enforcing periodicity up to rotation and relabeling) toward zero, independent verification is necessary to confirm that the resulting initial conditions satisfy the Newtonian equations. In the revised manuscript we will add a validation subsection that includes: forward integration of the optimized initial conditions with a high-order integrator, computation and reporting of residual norms on the return map, and direct comparisons against known equal-mass periodic orbits (when m2 = 1) to quantify accuracy. revision: yes
Referee: [Results / numerical experiments] No error analysis, tolerance thresholds, or convergence diagnostics are supplied for the families as functions of m2. This is load-bearing for the central claim that families of periodic solutions have been constructed.

Authors: We concur that error analysis, tolerance thresholds, and convergence diagnostics are essential to support the claim of constructed families. In the revision we will supply these elements: plots or tables of the minimized return objective versus m2, the specific tolerance thresholds employed by the stochastic optimizer, and any observed convergence or stability diagnostics for the families. These additions will make the numerical reliability of the results transparent. revision: yes

Circularity Check

0 steps flagged

Numerical stochastic search for periodic orbits via imposed return conditions is self-contained with no definitional reduction

full rationale

The paper describes a direct numerical procedure: a gradient-free stochastic optimizer that evaluates a return-map objective (imposing closure up to rotation and relabeling) on the Newtonian equations of motion for symmetric 4- and 6-body configurations. No analytical derivation, fitted parameters, or self-citation chain is present; the outputs are simply the optimizer's reported points that minimize the objective. Because the method consists solely of function evaluations on the underlying ODEs without any intermediate quantity defined in terms of the final result, the computation does not reduce to its inputs by construction. This is the standard, non-circular workflow for numerical discovery of periodic orbits.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on standard Newtonian n-body dynamics and the assumption that the stochastic search converges to true periodic orbits under symmetry constraints. No new physical entities or ad-hoc axioms beyond domain standards.

free parameters (1)

m2
Mass ratio parameterizing the families of solutions for the second group of bodies.

axioms (1)

domain assumption Planar motion and Newtonian inverse-square gravity for point masses
Standard setup for the n-body problem invoked to define the equations being solved.

pith-pipeline@v0.9.0 · 5467 in / 1225 out tokens · 53948 ms · 2026-05-10T19:30:56.002846+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we compute families of periodic solutions by imposing return conditions up to rotation and relabeling... stochastic black-box procedure that uses only function evaluations
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

gradient-free continuation method... adaptive stochastic modification... Err(Z) = ||h(Z)||_∞
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

n=4 and n=6... planar solutions... conservation of angular momentum and total energy

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

and Zeffiro, D.Direct-search methods in the year 2025: Theo- retical guarantees and algorithmic paradigms, EURO J

Dzahini, K.J., Rinaldi, F., Royer, C.W. and Zeffiro, D.Direct-search methods in the year 2025: Theo- retical guarantees and algorithmic paradigms, EURO J. Comput. Optim.13Article 100110 (2025)

work page 2025
[2]

and Ize, J.Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for then-body problem, J

García-Azpeitia, C. and Ize, J.Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for then-body problem, J. Differential Equations254pp. 2033–2075 (2013)

work page 2033
[3]

and Perdomo, O.Robust implementation of generative modeling with parametrized quantum circuits, Quantum Machine Intelligence3Article 17 (2021)

Leyton-Ortega, V., Perdomo-Ortiz, A. and Perdomo, O.Robust implementation of generative modeling with parametrized quantum circuits, Quantum Machine Intelligence3Article 17 (2021). doi:10.1007/s42484-021-00040-2 GRADIENT-FREE CONTINUATION METHOD FOR THE 4- AND 6-BODY PROBLEM 15 Central Connecticut State University Email address:perdomoosm@ccsu.edu

work page doi:10.1007/s42484-021-00040-2 2021

[1] [1]

and Zeffiro, D.Direct-search methods in the year 2025: Theo- retical guarantees and algorithmic paradigms, EURO J

Dzahini, K.J., Rinaldi, F., Royer, C.W. and Zeffiro, D.Direct-search methods in the year 2025: Theo- retical guarantees and algorithmic paradigms, EURO J. Comput. Optim.13Article 100110 (2025)

work page 2025

[2] [2]

and Ize, J.Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for then-body problem, J

García-Azpeitia, C. and Ize, J.Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for then-body problem, J. Differential Equations254pp. 2033–2075 (2013)

work page 2033

[3] [3]

and Perdomo, O.Robust implementation of generative modeling with parametrized quantum circuits, Quantum Machine Intelligence3Article 17 (2021)

Leyton-Ortega, V., Perdomo-Ortiz, A. and Perdomo, O.Robust implementation of generative modeling with parametrized quantum circuits, Quantum Machine Intelligence3Article 17 (2021). doi:10.1007/s42484-021-00040-2 GRADIENT-FREE CONTINUATION METHOD FOR THE 4- AND 6-BODY PROBLEM 15 Central Connecticut State University Email address:perdomoosm@ccsu.edu

work page doi:10.1007/s42484-021-00040-2 2021