A spinning top has three integrable cases — Euler's, Lagrange's, and one more that took a century and a woman's genius to find. Sofya Kovalevskaya discovered it by demanding the motion be single-valued in complex time, and it hides a fourth conserved quantity K that no other top has. This panel is runnable: it integrates the rigid body and watches K stay frozen for her top — and drift the instant you break the mass ratio that makes it hers.
source S. V. Kovalevskaya, "Sur le problème de la rotation d'un corps solide autour d'un point fixe," Acta Mathematica 12 (1889) — the Prix Bordin memoir. Room: THE NUMBER. Rendered, not quoted.
★ Sofya Kovalevskaya — the first woman to hold a doctorate in mathematics (Göttingen, 1874) and the first appointed a full professor (Stockholm, 1889). She carries the keeper's own name.
A heavy top spinning about a fixed point is chaotic in general. Only three cases can be solved:
| case | moments A:B:C | centre of mass |
|---|---|---|
| Euler (1758) | any | at the fixed point |
| Lagrange (1788) | A = B | on the symmetry axis |
| Kovalevskaya (1888) | A = B = 2C | in the equatorial plane |
Hers is the hardest and the last. For a century people assumed there were only two.
Kovalevskaya asked a strange question: for which tops is the solution a single-valued (meromorphic) function of complex time? That analytic demand — the Kovalevskaya exponents — selects exactly the mass ratio A = B = 2C, and out of it falls a new conserved integral. Integrability, found by looking into the complex plane.
Three integrals are shared by every heavy top: energy H, the vertical angular momentum L, and |γ|² = 1. Three is not enough to solve it. Kovalevskaya's top has a fourth:
K = (ω₁² − ω₂² − c·γ₁)² + (2ω₁ω₂ − c·γ₂)² — a quartic that stays constant only when A = B = 2C.
Fix the inertia and spin it up. Kovalevskaya: A = B = 2, C = 1 (the 2:2:1 ratio). Generic: break the ratio and it is just another chaotic top. Same initial spin, same integrator (RK4).
| integral | value at t=0 | drift after t=6 | conserved? |
|---|
Nothing is looked up — the body is integrated step by step, and each integral is recomputed and compared.
"There are only two solvable tops." Cut. The assumption stood for a century until 1888; hers is the third — and the Prix Bordin doubled its prize for it.
"She was a novelist who dabbled in maths." Cut. She wrote fiction and proved the Cauchy–Kovalevskaya theorem and this top; a full professor at Stockholm.
"A conserved K can be built for any top." Cut. Break 2:2:1 by a hair and K drifts — the tamper shows it live.
Set C = 1.4 instead of 1, so A = B = 2C no longer holds. The energy and angular momentum still conserve — but K starts to drift. The integrability was hers alone.
Off the 2:2:1 ratio, the fourth integral is no longer conserved: K wanders, the top is chaotic again, and the witness turns red. Only Kovalevskaya's ratio keeps it frozen.