She found the third and last integrable top by asking where the solution misbehaves in complex time — and demanding that it misbehave politely. Classify by the singularity, not by the regular part.
She contracted a marriage she did not want in order to leave Russia, because an unmarried woman could not. Berlin would not admit her, so Weierstrass taught her privately for four years and then pushed three papers through Göttingen for a doctorate in absentia, 1874. She could not get a post. She taught arithmetic in a girls' school, and for six years did no mathematics at all. Stockholm took her in 1884, made her full professor in 1889, and she died of influenza in 1891, forty-one years old. CITED
In 1888 the Paris Academy set its Bordin Prize on rigid-body motion. Entries were anonymous. Hers was judged so far beyond the others that the Academy raised the prize from 3 000 francs to 5 000 before opening the envelope.
A heavy rigid body turning about a fixed point has six state variables and, generically, three conserved quantities: energy, the vertical component of angular momentum, and $|\gamma|^2 = 1$. Three integrals for six variables is one short of what Liouville integrability needs. A fourth integral exists only for special bodies.
Two were known and had been for a century. Euler's case: no gravity term — the body pivots about its centre of mass. Lagrange's case: $A = B$ with the centre of mass on the symmetry axis — the top every child has spun.
Kovalevskaya found the third, and it is not a symmetry anyone would have guessed:
$A = B = 2C$, centre of mass in the equatorial plane
$K = \big[(p^2 - q^2) - c\gamma_1\big]^2 + \big[2pq - c\gamma_2\big]^2$
In 1906 Husson proved there are no others. Three cases, and that is the complete list for a heavy rigid body about a fixed point. CITED
Integrate the Euler–Poisson equations and watch the four quantities. Energy, vertical angular momentum and $|\gamma|^2$ hold to $10^{-14}$ whatever the body, because they are conserved for every body. $K$ is the one that discriminates:
A = B = 2, C = 1.0 (A = B = 2C) K drift 3.3 × 10−14
A = B = 2, C = 1.3 K drift 9.8 × 10−1
A = B = 2, C = 0.8 K drift 1.2
A = B = 2, C = 0.5 K drift 7.2 × 10−1 COMPUTED
Thirteen orders of magnitude between $C = 1$ and $C = 1.3$. Integrability is not a matter of degree. The quantity is either conserved or it is not, and the set of bodies for which it is conserved has measure zero in the space of bodies.
The method is the part this corpus should be reading, and it is stranger than the result.
She did not look for a conserved quantity. She asked where the solutions have their singularities in complex time, and demanded that they be poles — that the solution be meromorphic, single-valued, with no branch points. Then she solved for which bodies that is true. The answer was $A = B = 2C$ with the centre of mass off the axis, a condition nobody had reason to write down.
This is the Painlevé property, and she is using it in 1888, before Painlevé. The logic is: the singularity structure of the continued solution decides the qualitative behaviour of the real one. Go out into the complex plane, look at what kind of singularity you find, come back and know something about the motion you can actually see.
The dm³ chain reads a system by its fold — the place where the map degenerates — and infers the branch structure from it. Kovalevskaya's move is the same move in a different setting: classify by the singularity, not by the regular behaviour. What she adds is that the singularity worth classifying may not be anywhere on the real trajectory at all.
And the threshold is exact. $A = B = 2C$ is a codimension-one condition in the space of inertia tensors; a body one percent off it has no fourth integral, not a slightly worse one. SHOWN
The other half of her name sits in the corpus's foundations rather than its gallery. The Cauchy–Kovalevskaya theorem — analytic Cauchy data, analytic coefficients, a unique analytic solution locally — is the existence statement under every initial-value problem written in this series, and it is the one theorem here that is quoted more often than it is attributed.
| Operator | In this chapter | In dm³ |
|---|---|---|
| C | the inertia tensor — the body, fixed once | compression: the constraint |
| K | $C$ moved toward $A/2$ | approach to the condition |
| F | $A = B = 2C$: the fourth integral appears, or does not | the fold — threshold, not scale SHOWN |
| U | the motion, now foliated by $K$ into invariant tori | the branch the system settles on COMPUTED |
Every number on this page is produced by book7/ch-kovalevskaya-verify.py. It records in its own closing block what it establishes and what it does not.
S. Kowalevski, “Sur le problème de la rotation d’un corps solide autour d’un point fixe”, Acta Mathematica 12, 1889, 177–232 — the Bordin Prize memoir.
S. v. Kowalevsky, “Zur Theorie der partiellen Differentialgleichungen”, Crelle 80, 1875 — the Cauchy–Kovalevskaya theorem.
É. Husson, “Recherche des intégrales algébriques dans le mouvement d’un solide pesant autour d’un point fixe”, Ann. Fac. Sci. Toulouse 8, 1906 — the completeness of the three cases.
V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body about a Fixed Point, Moscow, 1953.
A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001 — chapter 2 for Kovalevskaya exponents and the Painlevé test.
R. Cooke, The Mathematics of Sonya Kovalevskaya, Springer, 1984.