Verified interactive research releaseProduct v1.0.0 · frozen v0.1 study

Two nearly identical starts.
One diverges. Watch it happen.

The planar three-body problem has no general closed-form solution — but it does have a measurable chaos boundary. This laboratory integrates a reference trajectory and an almost-identical perturbed twin, watches them diverge, and estimates a maximal Lyapunov exponent from the divergence rate. A frozen, preregistered sweep places the figure-eight orbit, the Lagrange equilateral triangle, and the Euler collinear solution on that map — and the honest result is more interesting than a clean stable/chaotic split.

The figure-eight orbit tracing itself, with a barely-perturbed twin (dashed) already visibly peeling away — the phenomenon this lab measures, looping live.

Run the simulatorRead the chaos mapRead the findings
Preregistered protocolDOP853, 1e-9 conservationBenettin two-trajectory methodFrozen, byte-checked registryNo general 3-body solution claimed
01

Live simulator

Watch two nearly
identical starts diverge.

Base configuration preset
Reference (solid) vs. perturbed twin (dashed)Figure-eight trajectory
live simulation, browser-side leapfrog integrator
Body 1 Body 2 Body 3 Perturbed twin (dashed trail)
Elapsed simulated time0.0
Twin separation (state-space)0.00e+0
Energy drift since start0.00e+0
Twin trajectory viewSeparation distance vs. time
Separation between reference and perturbed twin over timeA rising curve means the twin trajectory is diverging from the reference; a flat or oscillating curve means it is staying close. The vertical scale rescales to the largest separation seen so far.

This browser view uses a fixed-step leapfrog integrator for real-time responsiveness, not the DOP853 adaptive integrator validated to 1e-9 conservation tolerance in the Python package. Treat it as illustrative, not as a source of the frozen registry numbers below — those come only from scripts/generate_registry.py.

04

Frozen registry

The chaos map,
exactly as measured.

Read this before the classification labels below

  • The stable / borderline / chaotic labels come from one disclosed, arbitrary threshold (λ̄ < 0.02 stable, ≥ 0.10 chaotic) — not a physical law.
  • Every window here is short (3–4 orbital periods). A positive exponent is evidence of measured divergence within that window, not proof of unbounded long-term chaos.
  • In this frozen v0.1 registry, every one of the 42 cells classified as “chaotic,” including two of the three classical special solutions — see the report for why that is a real, explainable finding and not a bug.

Full reasoning: docs/research-report.md

Cells in registry4236 perturbation-magnitude + 6 mass-ratio
Classified chaotic42λ̄ ≥ 0.10
Classified stable0λ̄ < 0.02
Borderline0not confidently classified
View 1 of 2Perturbation-magnitude sweep — 6 base configurations × 6 magnitudes
Special solution (figure-eight, Lagrange, Euler collinear) Generic scattered configuration
Perturbation-magnitude sweep scatter plotMean estimated Lyapunov exponent for each of the six base configurations across six perturbation magnitudes. Special solutions are marked with a filled diamond; generic configurations with a filled circle. The complete values follow in a table.1.860smallest δ₀largest δ₀chaotic cutoff (0.10)
Read the complete perturbation-magnitude sweep data
Perturbation-magnitude sweep — mean λ̂ across 3 direction seeds
ConfigurationPerturbation magnitude δ₀Mean λ̂Std across seedsClassificationEnergy drift
Figure-eight1.00e-80.21580.0275chaotic9.50e-11
Figure-eight1.58e-70.21580.0275chaotic9.50e-11
Figure-eight2.51e-60.21580.0275chaotic9.50e-11
Figure-eight3.98e-50.21580.0275chaotic9.50e-11
Figure-eight6.31e-40.21510.0277chaotic9.50e-11
Figure-eight1.00e-20.20990.0307chaotic9.50e-11
Lagrange equilateral1.00e-81.16830.0209chaotic2.00e-12
Lagrange equilateral1.58e-71.16830.0209chaotic2.00e-12
Lagrange equilateral2.51e-61.16830.0209chaotic2.00e-12
Lagrange equilateral3.98e-51.16830.0209chaotic2.00e-12
Lagrange equilateral6.31e-41.16830.0210chaotic2.00e-12
Lagrange equilateral1.00e-21.16830.0218chaotic2.00e-12
Euler collinear1.00e-81.85940.0965chaotic1.00e-12
Euler collinear1.58e-71.85940.0965chaotic1.00e-12
Euler collinear2.51e-61.85940.0965chaotic1.00e-12
Euler collinear3.98e-51.85940.0965chaotic1.00e-12
Euler collinear6.31e-41.85930.0966chaotic1.00e-12
Euler collinear1.00e-21.85870.0987chaotic1.00e-12
Generic A1.00e-80.27870.0210chaotic2.92e-10
Generic A1.58e-70.27870.0210chaotic2.92e-10
Generic A2.51e-60.27890.0210chaotic2.92e-10
Generic A3.98e-50.28150.0220chaotic2.92e-10
Generic A6.31e-40.31670.0335chaotic2.92e-10
Generic A1.00e-20.54940.0713chaotic2.92e-10
Generic B1.00e-80.80060.0057chaotic1.74e-8
Generic B1.58e-70.79570.0117chaotic1.74e-8
Generic B2.51e-60.79590.0114chaotic1.74e-8
Generic B3.98e-50.79880.0116chaotic1.74e-8
Generic B6.31e-40.88700.0199chaotic1.74e-8
Generic B1.00e-21.13140.0084chaotic1.74e-8
Generic C1.00e-80.56590.0398chaotic6.89e-9
Generic C1.58e-70.56240.0379chaotic6.89e-9
Generic C2.51e-60.56190.0390chaotic6.89e-9
Generic C3.98e-50.56220.0389chaotic6.89e-9
Generic C6.31e-40.56610.0361chaotic6.89e-9
Generic C1.00e-20.64080.0367chaotic6.89e-9
View 2 of 2Mass-ratio sweep — Lagrange equilateral family × 6 mass ratios
Read the complete mass-ratio sweep data
Mass-ratio sweep — Lagrange equilateral family, mean λ̂ across 3 direction seeds
Mass ratio m₂/m₁Mean λ̂Std across seedsClassificationEnergy drift
0.501.02590.0291chaotic1.00e-12
0.751.09460.0280chaotic1.00e-12
1.001.14950.0279chaotic1.00e-12
1.501.23800.0282chaotic1.00e-12
2.001.31930.0304chaotic2.00e-12
3.001.43280.0299chaotic2.00e-12

All six mass ratios sit deep in the Gascheau–Routh analytically unstable region for the Lagrange equilateral relative equilibrium. In this exact (1, r, 1) family, linear stability requires r > 25 + 18√2 ≈ 50.456—far outside the 0.5–3.0 range tested here. The monotonic increase visible in the table describes degrees of instability within one unstable regime, not a stability transition.

Inspect the raw frozen registry ↗
05

Scientific method

See exactly what
the estimate knows.

1Generate

Fix a base configuration: a special solution or a seeded generic scatter.

→
2Integrate

DOP853, adaptive 8th-order, rtol 1e-11, atol 1e-12.

→
3Perturb

Displace a twin by a fixed magnitude in a seeded random direction.

→
4Renormalize

Every 1.0 time unit, measure and rescale the twin's separation.

→
5Classify

Average the log-growth rate; apply the disclosed threshold.

Equations of motion (G = 1, softened)d²rᵢ/dt² = Σⱼ≠ᵢ G·mⱼ·(rⱼ − rᵢ) / (|rⱼ − rᵢ|² + ε²)^1.5λ̂ = (1 / N·Δt) · Σ ln(dₖ / δ₀)

The softening length ε = 1×10⁻⁹ only prevents a literal division by zero; at every separation reached in this study it changes the force by a negligible amount relative to integrator tolerance. λ̂ is the Benettin et al. (1980) two-trajectory estimator: the mean log-growth rate of the twin's separation, renormalized every Δt = 1.0 time unit.

Observation

The twin separated by X after N renormalizations.

A numerical statement about one specific run, seed, and window.

Supported inference

This configuration shows measured sensitive dependence within this window.

A reproducible statement bounded by the declared integrator and parameters.

Unsupported leap

“This solves the three-body problem” or “this proves asymptotic chaos.”

Not established by a finite-window, single-threshold numerical sweep.

06

Research trail

Built on arguments
you can inspect.

Annals of Math · 2000A remarkable periodic solution of the three-body problem in the case of equal masses

Chenciner & Montgomery prove existence of the figure-eight orbit.

↗Proc. 3rd ECM (Birkhäuser) · 2001Periodic orbits of the planar N-body problem with equal masses and all bodies on the same path

Simo numerically refines the figure-eight initial conditions and finds it linearly stable.

↗Ergodic Theory Dynam. Systems · 2007Linear stability analysis of the figure-eight orbit in the three-body problem

Roberts gives a rigorous, computer-assisted proof of linear stability.

↗Proc. LMS · 1875On Laplace's three particles, with a supplement on the stability of steady motion

Routh derives the criterion now commonly called the Gascheau–Routh condition; Gascheau had obtained it earlier.

↗Meccanica · 1980Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Part 1: Theory

Benettin, Galgani, Giorgilli & Strelcyn establish the two-trajectory renormalization method used here.

↗Acta Mathematica, vol. 13 · 1890Sur le probleme des trois corps et les equations de la dynamique

Poincare's prize memoir, the origin of chaos theory and the non-integrability result this project never claims to overturn.

↗