Three bodies, mutual gravitation, no closed-form solution. The oldest open problem in classical mechanics meets fractal discretization.

The Oldest Unsolved Problem

Newton solved the two-body problem in Principia. The three-body problem has resisted exact solution ever since. Poincaré proved it chaotic in 1890 — not merely difficult, but structurally unpredictable. Three gravitating masses in the plane define a 12-dimensional phase space; after conserved quantities reduce it, four essential dimensions remain.

The system is deceptively simple:

m_i ẍ_i = Σ G m_i m_j (x_j - x_i) / |x_j - x_i|³

Three masses. Inverse-square attraction. No friction. Yet from these ingredients emerge ejections, near-collisions, and the sensitive dependence that makes long-term prediction impossible.

Jacobi Coordinates and Reduction

The standard trick is to factor out the centre of mass. Define Jacobi coordinates:

• ρ₁ — the vector from body 1 to body 2 (relative separation)
• ρ₂ — the vector from the centre of mass of the (1,2) pair to body 3

These four real degrees of freedom (two complex coordinates) parametrise the reduced configuration space. The transformation diagonalises the kinetic energy and reveals the collision set Δ = (ρ₁ = 0) ∪ (ρ₂ = 0) as the singular locus.

The Morales-Ramis theorem then establishes the key result: for generic mass ratios, the three-body problem is meromorphically non-integrable. No hidden symmetry will save us.

Julia Sets as Structured Noise

If the dynamics are non-integrable, we need stochastic methods. But classical Brownian noise — white, Gaussian, memoryless — discards geometric structure. The alternative: drive the system with deterministic chaos generated by iterating f_c(z) = z² + c on its Julia set.

Why Julia sets?

1. Ergodic with computable statistics — Birkhoff sums approximate Brownian motion with error O(n^(1/2 - ε))
2. Tunable complexity — the parameter c controls fractal dimension, spectral gap, and diffusion coefficient
3. Natural discretization — the iterated function system z ↦ ±√(z - c) generates a geometry-conforming mesh
4. Canonical basis — Bessis-Geronimo-Moussa orthogonal polynomials on the equilibrium measure provide a spectral expansion

Two independent Julia sets J₁ and J₂ provide the four real noise channels needed for the reduced three-body configuration space.

Interactive Demo

Explore all four layers of the framework: Julia set noise generation, three-body gravitational simulation, transfer operator spectral analysis, and convergence rate theory.

The Transfer Operator

The Perron-Frobenius (transfer) operator ℒ_f encodes how densities evolve under the Julia set dynamics. For the product system J₁ × J₂, it factorises as a tensor product:

ℒ(f_c₁) ⊗ ℒ(f_c₂)

The spectral gap Δ(c) = 1 - ρ(c) controls everything:

• Mixing rate — exponential decay of correlations at rate ρ(c)
• CLT variance — Green-Kubo formula gives σ²(c) from the resolvent
• Galerkin convergence — error bounds scale as N^(-s^eff) where s^eff = 1 - dim_H(μ_c)/2

The Zdunik gap Δ = d_c - δ_c > 0 (strict for all non-Chebyshev parameters) means the equilibrium measure is always more regular than the Julia set itself — free accuracy from fractal geometry.

Convergence and the Regularity Trade-off

The effective Sobolev index s^eff governs the algebraic convergence rate of the Galerkin scheme. There is a fundamental tension:

• Far from the Mandelbrot boundary — sparse Cantor Julia sets, fast convergence, but low-dimensional noise
• Near the boundary — rich fractal structure providing more realistic noise, but poor convergence rates

The sweet spot lies in the moderately hyperbolic regime, where spectral gaps are large enough for practical convergence but the Julia set retains enough geometric complexity to drive meaningful diffusion.

What the Simulation Shows

• Julia Sets tab — Two independent Julia sets with adjustable parameters. Toggle the IFS mesh overlay to see the geometry-conforming discretization that replaces uniform grids.
• Three-Body tab — Classical gravitational simulation with three presets: Lagrange equilateral triangle (periodic), figure-8 choreography (discovered by Moore 1993), and asymmetric masses (chaotic). Enable Jacobi vectors to see the reduced coordinates.
• Transfer Operator tab — The Galerkin matrix, its eigenvalue spectrum in the complex plane, and key spectral quantities. Watch how the spectral gap varies as you move c through the hyperbolic component.
• Convergence tab — Log-log error plots showing how different Julia set parameters yield different algebraic convergence rates, all predicted by the Zdunik-corrected regularity index.

The three-body problem remains unsolved. But its chaos need not be modelled by structureless noise. The Julia set framework replaces randomness with deterministic complexity — fractal geometry all the way down.