Theorem 3.1
The closed-form rotating solution at the heart of the document — and exactly what the verification ladder certifies about it.
Statement
For the vortex model with the Chaplygin constant C_Ch, the document
constructs (Section 5) the following solution, a Lagrange-type relative equilibrium
with radial modulation:
θ_k(t) = ω·t + 2π·k/3
Two structural facts follow immediately from the form itself and are the subject of ladder check V1:
- Equilateral choreography. The angular positions differ by exactly 2π/3 at every instant — the three vortices form an equilateral triangle that rotates rigidly, the vortex analogue of the classical Lagrange solution of the three-body problem.
- Periodicity. The radial modulation repeats with period T_r = 2π/ω; the closed form is a periodic orbit of the model.
Reference values (C_Ch = 1, T = 2π)
| Quantity | Value | Verified by |
|---|---|---|
| frequency ω = (2π/T)·e^(1/π) | 1.3748022274393588 | pytest pin, exact to float64 |
| amplitude ε = 1/(e^(1/π)−1) | 0.7276379117656014 | pytest pin, exact to float64 |
| angular separation (all times) | 2π/3 ± 6.9e-14 | V1, residual table |
| periodicity residual over 100·T | ≤ 9.4e-14 | V1, residual table |
What the ladder certifies — and what it only records
An honest boundary separates two kinds of statements about the closed form:
Certified (pass/fail)
The 2π/3 choreography and the periodicity are properties of the formulas themselves: substituting the closed form, the angular separation is 2π/3 up to float64 argument reduction, and r_k(t + T_r) = r_k(t) up to the same round-off. These hold for every C_Ch > 0.01, on every probed time window (V1), and the same quantities are pinned by independent pytest reference values.
Recorded (diagnostic)
The combination C_Ch(t) = r²(θ̇ − q·A_θ) with A_θ = 1/r oscillates together with the radial modulation — by construction of the model, not by a failure of conservation. Section 6 of the document records its endpoint drift over [0, 100·T] as a diagnostic; the window-independent conserved quantities of the dynamics are the vortex integrals H, P, Q, I, certified numerically in V3–V4.
The numerical twin: the Lagrange triangle
Theorem 3.1 has an exact numerical counterpart that needs no closed form at all. Take three equal point vortices on an equilateral triangle of side a and integrate the Kirchhoff equations with RK4. Classical theory says the triangle rotates rigidly with angular velocity
— for Γ = 1, a = 1 that is ω_L = 3/(2π) ≈ 0.4775. Ladder check V2 confirms both halves of this statement numerically: the side lengths stay equilateral to ≤ 1e-10 over multiple full rotations, and the measured rotation frequency matches 3Γ/(2πa²) to a relative error ≤ 1e-6. Checks V3–V4 then verify the conservation of the four classical integrals along the same trajectory — first for the symmetric configuration, then for unequal circulations Γ = (1, 2, 3) from a generic triangle, where no symmetry can carry the answer.
Why «Lagrange»
In the gravitational three-body problem, the equilateral configuration rotating rigidly is one of Euler–Lagrange's central configurations (1772, the essai sur le problème des trois corps); in the vortex world the same triangle is an exact relative equilibrium known since Kirchhoff. Theorem 3.1 is the TRIVORTEX document's version of this classical object, dressed in the Chaplygin gauge: the radial modulation ε·cos(...) is the model-specific addition that the topological integral controls. The resonance between the gravitational and vortex stories is the quiet engine of the whole document — the reason a "three-body problem" paper is allowed to compute Berry phases and Chern numbers on the side.