The Vortex Model
How the three-body problem becomes a problem about vortices, flux and a topological integral.
From bodies to vortices
The classical three-body problem asks for the motion of three masses under mutual
Newtonian attraction; its vortex twin replaces masses by vortices with circulations
and topological charges q_k = (1, −1, 1). The trade is deliberate:
point-vortex dynamics are Hamiltonian and first-order in time, their configuration
space is the plane, and their conserved quantities are algebraic rather than
transcendental. What is lost in literal gravitational realism is gained in exact
statements — the Lagrange equilateral rotation becomes a theorem one can pin to
machine precision, and the chaotic N-vortex regime remains available for study as
N grows.
TRIVORTEX goes one step further than plain point vortices: each vortex sits in an effective magnetic flux, so the model lives at the intersection of hydrodynamics and the Aharonov–Bohm physics of charged particles on a lattice. The single-vortex conserved combination is the Chaplygin topological integral C_Ch = r² · (θ̇ − q·A_θ), A_θ = 1/r which binds the angular momentum to the gauge flux — the quantity Section 6 of the core document records along trajectories.
The Hamiltonian
Section 4 of the document builds an L×L Hofstadter Hamiltonian with
N_v embedded vortices: a square lattice with periodic boundary conditions,
Peierls phases φ = 2πα(ix−1)dy from the uniform flux α plus
vortex-induced AB phases, and a diagonal potential that falls off as the inverse
square of the distance to each vortex core, weighted by its charge. The result is a
Hermitian matrix whose spectrum is analysed by the spectral-statistics machinery of
Section 7 (GUE/GOE/Poisson classification, ⟨r⟩ statistic, KS tests against the
classical random-matrix ensembles).
| Ingredient | Symbol | Default | Role |
|---|---|---|---|
| lattice size | L | 20 (20×20) | discretization of the plane |
| number of vortices | N_v | 3 | the three bodies |
| effective flux | α | 0.5 | uniform Peierls phase |
| disorder strength | W | 2.0 | vortex-core potential scale |
| topological charges | q_k | (1, −1, 1) | vortex orientations |
| Chaplygin constant | C_Ch | 1.0 | integral of motion |
Observables the document tracks
Spectral statistics
Unfolded spacings, ⟨r⟩ statistic against GUE (0.5996), GOE (0.5307) and Poisson (0.3863) references, KS tests, number variance and spectral form factor (Sections 7, 13, 20–21).
Quantum-topological
Berry phase, Chern numbers via Fukui–Hatsugai–Suzuki, TKNN Hall conductance, Dirac-cone analysis (Sections 16–19) — the quantum layer of the same model.
Chaos diagnostics
Lyapunov exponents for vortex vs Newtonian dynamics, a permutation test with Z-scores, KAM-tori fraction and Poincaré sections (Section 22).
Real systems
Sun–Earth–Moon, α Centauri and Pluto–Charon–Nix presets map astronomical masses, distances and periods into the model (Section 10, dedicated page).
Where the model stands in the literature
The three-body problem needs no introduction — from Newton through Poincaré's memoir and the discovery of chaos to Chenciner–Montgomery's figure-eight choreography (2000) and the modern zoo of choreographic solutions. The point-vortex formulation goes back to Kirchhoff and to the exact results of the 19th century (Thomson's vortex atoms, the Lagrange triangle of vortices); Chaplygin's name is attached to the invariants of vortex systems and to the gas-dynamical reduction bearing his name. TRIVORTEX borrows the choreography viewpoint (exact relative equilibria) and the vortex formalism (Hamiltonian, invariants), then adds its own layer: the AB-flux gauge potential that turns the angular momentum into a topological integral. The document is honest about this lineage — it is a model study with exact checks, not a claim about planetary motion.