Sofia Kovalevskaya

How to use this course

Sofia Kovalevskaya is remembered for two results. Every student of partial differential equations meets the Cauchy–Kovalevskaya theorem, the basic existence theorem for analytic solutions. Every student of integrable systems meets the Kovalevskaya top, a third solvable case of the spinning top after those of Leonhard Euler and Joseph-Louis Lagrange. Around them lie papers on Abelian integrals, on Saturn’s rings and on light in crystals, and a life that crossed the Russian nihilist movement, the Paris Commune, Berlin’s mathematical school and the new university in Stockholm. She also wrote fiction and a memoir.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–07 follow her three dissertation papers of 1874 and the work that grew from them; Lessons 08–10 treat the rigid-body memoir that won the Prix Bordin in 1888.

Timeline

Palibino, 1850–1867

Sofia Vasilyevna Korvin-Krukovskaya was born in Moscow on 15 January 1850. Her father, Vasily Korvin-Krukovsky, was an artillery general from the minor nobility; her mother, Elizaveta Fyodorovna Shubert, came from a family of German descent. Sofia was the middle of three children, between her elder sister Anna (Anyuta) and her brother Fyodor. In 1858 her father retired and the family settled at Palibino, an estate in Vitebsk province near the Lithuanian border.

In her memoir she tells how, when she was about eleven, the walls of the nursery were papered with sheets left over from her father’s student days: lithographed lectures on differential and integral calculus by Mikhail Ostrogradsky. She spent hours puzzling out the formulas and symbols, and later felt that they had prepared her for calculus when she met it properly. Her education was in the hands of a tutor, Iosif Malevich, who taught her arithmetic and geometry with care.

A neighbour, a professor of physics, gave the family a copy of his elementary physics textbook. Sofia, who had not studied trigonometry, worked out for herself what the sine of an angle must mean in order to follow the optics chapter, essentially reconstructing the subject. The professor urged her father to let her study further. In St Petersburg she then took private lessons in analytic geometry and calculus from Aleksandr Strannolyubsky.

A marriage of convenience and Heidelberg, 1868–1870

Russian universities did not admit women, and an unmarried woman could not travel abroad without her father’s consent. Among the young radicals of the 1860s, the nihilists, a fictitious marriage was a recognized way out. Anyuta and Sofia looked for a willing partner. In September 1868 Sofia married Vladimir Kovalevsky, a young publisher with progressive views who later became a distinguished palaeontologist. The marriage gave her a husband’s permission to leave Russia.

In the spring of 1869 they went to Heidelberg. Women could not matriculate, but she obtained permission from individual professors to attend lectures for three semesters. She studied mathematics with Königsberger and Paul du Bois-Reymond and physics with Kirchhoff and Helmholtz. Königsberger had been a student of Karl Weierstrass in Berlin, and he told her that Weierstrass was the person to learn from.

Berlin, Weierstrass and the doctorate, 1870–1874

In 1870 she went to Berlin. The university senate refused to let her attend lectures, so she approached Weierstrass directly. He set her a group of problems to test her; her solutions impressed him, and he agreed to teach her privately. For four years he went over his lectures with her, on the theory of analytic functions, elliptic and Abelian functions, and the calculus of variations, and the two became close friends. Their correspondence lasted until her death.

In spring 1871 she and Vladimir travelled to Paris, then under siege during the Commune, where Anyuta was active with her partner Victor Jaclard; after the Commune fell, the family helped secure Jaclard’s escape.

By 1874 she had written three papers, each sufficient for a doctorate in Weierstrass’s judgement: one on partial differential equations (Lessons 04–05), one on the reduction of a class of Abelian integrals to elliptic integrals (Lesson 06), and one on Laplace’s theory of the shape of Saturn’s rings (Lesson 07). Berlin would not grant a degree to a woman. Weierstrass wrote to Göttingen, and in 1874 the University of Göttingen awarded her the doctorate summa cum laude, without the usual oral examination. It was the first doctorate in mathematics in the modern sense earned by a woman.

The Cauchy–Kovalevskaya theorem

For the general reader. A differential equation relates a function to its rates of change. If you know the function and enough of its derivatives along an initial surface, can you continue it uniquely away from that surface? Kovalevskaya proved that the answer is yes, for a broad class of equations, as long as everything is given by convergent power series. It is the first general existence theorem for partial differential equations.

Setting

Augustin-Louis Cauchy had treated special cases around 1842 with his “calcul des limites”. Kovalevskaya, at Weierstrass’s suggestion, gave a general treatment in her 1875 paper. Any analytic system can be reduced to a first-order quasilinear system, so it suffices to state it in that form.

For a single equation of order \(k\) the theorem applies in normal form: \(\partial_t^k u=F\big(t,x,(\partial_t^{\,j}\partial_x^\alpha u)_{j<k,\ j+|\alpha|\le k}\big)\) with the Cauchy data \(\partial_t^j u(0,x)\), \(j<k\). The derivative in \(t\) of highest order must be solvable for, and no other derivative may have total order above \(k\). In geometric language, the initial surface must be non-characteristic.

Proof sketch

Uniqueness and formal solution. Differentiating the equation, every Taylor coefficient \(\partial_t^m\partial_x^\alpha u(0,0)\) is determined recursively by those of lower \(t\)-order and by the data. The recursion is a polynomial with non-negative coefficients in the Taylor coefficients of \(A_j\), \(b\) and \(\varphi\).

Majorants. A series \(\sum c_\beta z^\beta\) is majorized by \(\sum C_\beta z^\beta\) if \(|c_\beta|\le C_\beta\) for all \(\beta\). If the \(A_j\), \(b\), \(\varphi\) are majorized by \(\tilde A_j,\tilde b,\tilde\varphi\), the positivity of the recursion shows that the formal solution \(u\) is majorized by the formal solution \(\tilde u\) of the majorizing problem. Analytic functions on a polydisc satisfy Cauchy estimates \(|c_\beta|\le M r^{-|\beta|}\), so all data are majorized by the single function

With this choice the majorizing problem reduces to a single first-order equation in one combined variable, which can be solved explicitly by characteristics and is analytic near 0. Its Taylor series converges, hence so does that of \(u\).

Where analyticity fails: the heat equation

For the general reader. The theorem needs the equation to give the highest time derivative directly. The heat equation does not: it gives one time derivative in terms of two space derivatives. Kovalevskaya showed with an example that for such equations the formal power series solution can diverge, even for perfectly analytic initial data. Her counterexample marks the boundary of the theorem.

Proof. From the equation, \(\partial_t^k u(0,x)=\partial_x^{2k}u(0,x)=\frac{(2k)!}{(1-x)^{2k+1}}\), so the only formal solution is

At \(x=0\) the coefficients \(\frac{(2k)!}{k!}\ge k!\) grow faster than any geometric sequence, so the radius of convergence in \(t\) is 0. The line \(t=0\) is characteristic for the heat operator, because the principal part \(\partial_x^2\) contains no \(\partial_t\).

After Kovalevskaya

• Holmgren (1901): for linear equations with analytic coefficients and a non-characteristic surface, the Cauchy–Kovalevskaya solution is unique even among merely smooth solutions.
• Hans Lewy (1957): a linear equation with polynomial coefficients, \(-u_x-iu_y+2i(x+iy)u_z=f\), has no solution at all for suitable smooth \(f\). Analyticity cannot be dropped.
• Masaki Kashiwara (1983): a version for linear systems in the language of \(\mathcal D\)-modules.

Abelian integrals and their reduction

For the general reader. Integrals of square roots of polynomials of degree three or four give elliptic functions, a well-understood theory. Higher-degree polynomials give Abelian integrals, which are much harder. Sometimes a clever substitution turns an Abelian integral into an elliptic one. Kovalevskaya’s second dissertation paper studied when this happens for a class Weierstrass had singled out.

Elliptic and hyperelliptic integrals

An integral \(\int R(x,y)\,dx\), with \(R\) rational and \(y^2=f(x)\), \(f\) a polynomial without repeated roots, lives on the curve \(y^2=f(x)\). If \(\deg f\in\{3,4\}\) the curve has genus 1 and the integral is elliptic. If \(\deg f\in\{2g+1,2g+2\}\) the curve is hyperelliptic of genus \(g\), with \(g\) independent differentials of the first kind \(x^k\,dx/y\), \(0\le k\le g-1\).

Weierstrass asked when integrals of this kind on curves of higher rank (his word for the genus) reduce to elliptic integrals. Kovalevskaya’s paper, published in Acta Mathematica in 1884, showed how a certain class of Abelian integrals of rank three reduces to elliptic integrals. Her treatment built on Weierstrass’s theory of Abelian functions and theta functions.

The simplest example

On \(y^2=x^6+ax^4+bx^2+c\), genus 2, the substitution \(u=x^2\) gives

an elliptic integral. The curve has the involution \((x,y)\mapsto(-x,y)\), and the quotient by it is the elliptic curve \(v^2=u^3+au^2+bu+c\). Symmetries of curves are the main source of such reductions.

The shape of Saturn’s rings

For the general reader. Pierre-Simon Laplace modelled Saturn’s rings as rotating fluid and found that a thin ring’s cross-section should be an ellipse. Kovalevskaya carried the calculation to the next order and found that the cross-section is an egg-shaped oval, symmetric about only one axis. The problem itself was soon overtaken by evidence that the rings are made of separate particles, but her method is a model of careful perturbation theory.

Equilibrium of a rotating fluid

A fluid ring rotating rigidly with angular velocity \(\Omega\) about Saturn’s axis is in equilibrium when its free surface is a level surface of the total potential

where \(\varrho\) is the distance from the axis. Let the ring have mean radius \(R\) and cross-section of size \(a\ll R\). Laplace kept only the leading terms in \(a/R\): the ring then looks locally like an infinite straight cylinder, the tidal field of Saturn plus the centrifugal term is quadratic in the local coordinates, and the classical theory of homogeneous fluid figures gives an elliptical cross-section.

In outline, the asymmetry has a simple source. The curvature of the ring makes the self-potential of a ring different from that of a straight cylinder by terms of relative size \(a/R\), and these include terms odd in the radial coordinate. In a polar description of the cross-section, \(r(\theta)=a\big(1+\epsilon_1\cos\theta+\epsilon_2\cos2\theta+\cdots\big)\), the \(\cos2\theta\) term is Laplace’s ellipse and the odd terms produce the egg shape. Kovalevskaya also studied the stability of such rings. James Clerk Maxwell had argued in 1859 that neither solid nor fluid rings could be stable, and observations later confirmed that the rings consist of many small bodies.

The heavy rigid body

For the general reader. A top spinning under gravity about a fixed point obeys six differential equations. Euler solved them when the fixed point is the centre of gravity; Lagrange solved them for a symmetric top whose centre of gravity lies on its axis. For a century nobody found another case that could be solved completely. The Paris Academy made the problem the subject of its Prix Bordin for 1888.

The Euler–Poisson equations

Work in the body frame along the principal axes, with principal moments \(A,B,C\), angular velocity \(\omega=(p,q,r)\), the centre of gravity at \((x_0,y_0,z_0)\), and \(\gamma=(\gamma_1,\gamma_2,\gamma_3)\) the unit vector pointing vertically upward, seen from the body. With \(M\) the mass and \(g\) the gravitational acceleration,

Three integrals always exist:

The system has six variables. By a theorem of Euler and Jacobi on systems with an invariant measure, one more independent integral makes it solvable by quadratures. In modern language, the phase space reduces to a four-dimensional symplectic manifold, and a fourth integral in involution with \(H\) gives Liouville integrability.

The Kovalevskaya top

For the general reader. Kovalevskaya found a new top that can be solved exactly: two equal moments of inertia, a third half as large, and the centre of gravity placed in the plane of the equal moments. The conserved quantity she discovered is not suggested by any visible symmetry. She then integrated the motion completely, using hyperelliptic functions of genus two.

Normalization

Take \(A=B=2\), \(C=1\) and rotate the axes so that the centre of gravity lies on the first axis, \(\mathbf r_0=(x_0,0,0)\). Write \(c=Mgx_0\). The Euler–Poisson equations become

with energy \(H=p^2+q^2+\tfrac12r^2+c\gamma_1\) and \(L=2p\gamma_1+2q\gamma_2+r\gamma_3\).

Why \(K\) is conserved

Introduce the complex variable \(\xi=(p+iq)^2-c(\gamma_1+i\gamma_2)\), so that \(K=|\xi|^2\). A direct computation from the equations gives

Then \(\frac{d}{dt}|\xi|^2=\dot\xi\bar\xi+\xi\dot{\bar\xi}=-ir|\xi|^2+ir|\xi|^2=0\). (Exercise 9.1 carries out the computation.)

Integration

Kovalevskaya introduced new variables \(s_1,s_2\), built from \(p\pm iq\) and the integrals, in which the equations separate into

with \(\kappa\) a constant and \(R\) a polynomial of degree 5 whose coefficients depend on the values of \(H,L,K\). This is the Jacobi inversion problem on a genus-2 curve, solved by theta functions of two variables, exactly the theory she had learned from Weierstrass. The memoir was one of fifteen anonymous entries, and the Academy raised the prize from 3,000 to 5,000 francs. She received a further prize from the Swedish Academy of Sciences in 1889 for continued work on the problem.

Her method: poles, Laurent series, integrability

For the general reader. Kovalevskaya did not find her top by guessing an integral. She asked when the solutions, viewed as functions of complex time, have no singularities worse than poles. Requiring this for a general solution forced the special ratios of the moments of inertia. Her idea grew into the Painlevé test, now a standard tool for detecting integrable equations.

The idea

In the Euler and Lagrange cases, the solutions are elliptic functions of time: single-valued and meromorphic in the complex \(t\)-plane. Kovalevskaya asked for which \(A,B,C,\mathbf r_0\) the general solution has this property. She looked for solutions as Laurent series around a movable pole \(t_0\),

and demanded that the series contain as many free constants as a general solution of the system requires. The leading coefficients satisfy algebraic equations; the free constants can enter only at certain orders, the resonances, where the recursion fails to determine a coefficient. The count works only in the Euler and Lagrange cases and the new case \(A=B=2C\), \(z_0=0\).

Resonances in a model equation

The method is easiest to see on \(\ddot x=6x^2\). Try \(x\approx a(t-t_0)^{-2}\): then \(6a=6a^2\), so \(a=1\). Perturb: \(x=\tau^{-2}(1+\varepsilon\tau^{\rho})\) with \(\tau=t-t_0\). The linear terms in \(\varepsilon\) give \((\rho-2)(\rho-3)=12\), so \(\rho=-1\) or \(\rho=6\). The resonance \(\rho=-1\) corresponds to the free \(t_0\); the resonance \(\rho=6\) is where the second free constant enters. When the recursion at a resonance is consistent, no logarithms appear, and the equation passes the test. Paul Painlevé turned this into a classification of second-order equations, and in the twentieth century the test became a fast detector of integrable systems such as the Korteweg–de Vries equation.

Russia and Stockholm, 1874–1889

With a doctorate in hand, Kovalevskaya returned to Russia in 1874 and found no academic post open to a woman;. For several years she left mathematics aside. She and Vladimir lived as a married couple, moved in literary and scientific circles in St Petersburg, and she wrote journalism and theatre criticism. Her father died in 1879, and their daughter Sofia, known as Fufa, was born in October 1878. In 1880 Chebyshev invited her to speak at the Sixth Congress of Russian Natural Scientists and Physicians, and she returned to research. From 1882 she worked on the refraction of light in crystals, following Gabriel Lamé’s method; Vito Volterra later found that the method contained an error she had inherited.

Vladimir’s business ventures failed, and in the spring of 1883 he took his own life. Weierstrass’s student Gösta Mittag-Leffler, who admired her work, had been working to bring her to the new Stockholm University College, a private institution that could appoint a woman. She arrived late in 1883, began lecturing early in 1884, and in June 1884 was appointed professor for five years, on the understanding that she would soon lecture in Swedish. She did so within a year. She joined the editorial board of Mittag-Leffler’s journal Acta Mathematica and acted as its link with mathematicians in Paris and Berlin, including Charles Hermite and Henri Poincaré.

She also wrote. With Mittag-Leffler’s sister, the writer Anne Charlotte Leffler, she wrote the play The Struggle for Happiness, published in 1887. Her sister Anyuta died in Paris that autumn. The Prix Bordin in December 1888 brought her fame across Europe, and in June 1889 Stockholm made her professor for life. She was the first woman in modern Europe to hold a full chair of mathematics. At Chebyshev’s initiative, the Imperial Academy of Sciences in St Petersburg changed its rules and elected her a corresponding member at the end of 1889, though no Russian university post followed.

Last years, death, legacy

In her last years Kovalevskaya divided her time between Stockholm and visits to France and Italy, often with the Russian jurist and sociologist Maxim Kovalevsky, a distant relative of her husband. She wrote a memoir of her childhood (in English, A Russian Childhood), and a novel, Nihilist Girl, drawn from the lives of young Russian radicals. Returning to Stockholm from the Riviera in early 1891 she caught influenza, which turned to pneumonia. She died on 10 February 1891, aged 41, and is buried in the Norra begravningsplatsen in Solna, north of Stockholm.

Legacy

• Analysis: the Cauchy–Kovalevskaya theorem remains the basic local existence theorem for analytic PDE, and the notion of non-characteristic data it introduced runs through the modern theory.
• Mechanics and integrable systems: the Kovalevskaya top is the third classical integrable top. Her search for single-valued solutions led to the Painlevé property and the Kovalevskaya exponents, now standard tools; her top was revisited with Lax pairs and algebraic geometry in the 1980s.
• Women in mathematics: the Association for Women in Mathematics and SIAM hold an annual Sonia Kovalevsky Lecture; the Kovalevskaia Fund, founded in 1985 by Ann Hibner Koblitz and Neal Koblitz, supports women scientists in developing countries. The Russian Academy of Sciences awards a Kovalevskaya Prize, and the Humboldt Foundation ran a Sofja Kovalevskaja Award for young researchers.
• Commemoration: a lunar crater and the asteroid 1859 Kovalevskaya bear her name.

Connections

People linked to Sofia Kovalevskaya across this collection. A name without a link has no page here yet. See Sofia Kovalevskaya in the network of all 337 lives →

Teachers and advisors

• Karl Weierstrass 1870-1874

Influenced by

• Augustin-Louis Cauchy
• Gustav Kirchhoff 1869-1870
• Hermann von Helmholtz 1869-1870
• Joseph-Louis Lagrange
• Leonhard Euler
• Pierre-Simon Laplace
• Leo Königsberger 1869-1870

Influenced

• Aleksandr Lyapunov
• Ivar Fredholm
• Paul Painlevé

Colleagues and collaborators

• Charles Hermite
• Gösta Mittag-Leffler 1883-1891
• Henri Poincaré
• Pafnuty Chebyshev

Friends

• Anne Charlotte Leffler
• Maxim Kovalevsky 1888-1891

Family

• Vladimir Kovalevsky 1868-1883

Also mentioned in this life

• Niels Henrik Abel
• Jacques Hadamard
• Alexander von Humboldt
• Carl Gustav Jacob Jacobi
• Joseph Liouville
• James Clerk Maxwell
• Siméon Denis Poisson
• Bernhard Riemann
• Vito Volterra

Fields

• partial differential equations 8 lives
• mechanics 27 lives

On CoursesHub

• ODEs / PDEs / Green Functions · Physical Mathematics
• Classical Mechanics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Sofia Vasilyevna Kovalevskaya, MacTutor History of Mathematics.
2. Sofya Kovalevskaya, Wikipedia.
3. Stockholm University, Sonja Kovalevsky, Sveriges första kvinnliga professor.
4. P. Ya. Kochina, Kovalevskaya entry, Dictionary of Scientific Biography.
5. A. H. Koblitz, A Convergence of Lives. Sofia Kovalevskaia: Scientist, Writer, Revolutionary, Birkhäuser, 1983.
6. S. Kovalevskaya, A Russian Childhood, trans. B. Stillman, Springer, 1978.
7. M. Audin, Remembering Sofya Kovalevskaya, Springer, 2011. Author’s page.

Mathematics

1. S. von Kowalevsky, Zur Theorie der partiellen Differentialgleichungen, J. reine angew. Math. 80 (1875), 1–32.
2. S. Kowalevski, Über die Reduction einer bestimmten Klasse Abel’scher Integrale dritten Ranges auf elliptische Integrale, Acta Math. 4 (1884), 393–414.
3. S. Kowalevski, Zusätze und Bemerkungen zu Laplace’s Untersuchung über die Gestalt der Saturnsringe, Astronomische Nachrichten 111 (1885).
4. S. Kowalevski, Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Math. 12 (1889), 177–232.
5. Cauchy–Kovalevskaya theorem and Kovalevskaya top, Wikipedia.
6. G. B. Folland, Introduction to Partial Differential Equations, Princeton, 2nd ed. 1995, ch. 1 (Cauchy–Kovalevskaya and Holmgren).
7. M. Audin, Spinning Tops: A Course on Integrable Systems, Cambridge University Press, 1996.
8. A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, 2001 (Kovalevskaya exponents).