## Sur une proprieté du système d'equations différentielles qui définit la rotation d'un corps solide autour d'un point fixe. Autograph manuscript signed, 11 leaves, written on recto only, with some corrections and additions. Undated, but first published in Acta Mathematica 14 (1890), pp. 81-93.

[1890].

Important autograph manuscript by “the greatest woman mathematician prior to the twentieth century” (DSB) and the first woman in Europe to obtain a doctorate in mathematics. This paper complements and completes her most famous work, ‘Sur le problème de la rotation d’un corps solide autour d’un point fixe’ (Acta Mathematica 12 (1889), pp. 177-232), for which she received the 1888 Bordin Prize from the French Academy of Sciences. “Prior to Sofya Kowalevsy’s work the only solutions to the motion of a rigid body about a fixed point had been developed for the two cases where the body is symnmetric. In the first case, developed by Euler, there are no external forces, and the center of mass is fixed within the body. This is the case that describes the motion of the earth. In the second case, derived by Lagrange, the fixed point and the center of gravity both lie of the axis of symmetry of the body. This case describes the motion of the top. Sofya Kowalevsy developed the first of the solvable special cases for an unsymmetrical top. In this case the center of mass is no longer on an axis in the body. She solved the problem by constructing coordinates explicitly as ultra-elliptic functions of time” (Rappaport, p. 570). Kovalevskaya continued her work in the present paper, which shows that the Euler, Lagrange and Kovalevskaya cases are, in fact, the only solvable cases of the motion of a rigid body. “Kovalevskaya's contribution was remarkable in many respects. First, she applied a recently developed and highly abstract mathematical theory (the theory of Abelian functions) to solve a physical problem [in the 1889 paper]. Second, she introduced one of the first proofs of the non-integrability of a physical system [in the present paper]. This result, together with Poincaré’s and Bruns’ work on the three-body problem, announced the failure of the program of classical mechanics to integrate exactly and for all times the equations of motion” (Goriely, p. 8).

“To appreciate Kovalevskaya’s contributions to the problem of a rotating rigid body it is necessary to understand the state of the problem before and after she worked on it. The physical problem – to express the position of the figure axes of a rigid body explicitly as functions of time – arises naturally from Newtonian mechanics. The first person to make progress on the problem was Euler, who over a period of twenty years gave various mathematical formulations of the problem and solved the resulting differential equations completely for the case of a body free of torque. Lagrange studied the problem, formulating it in terms of curvilinear coordinates that were better adapted to the problem than the rectangular coordinates used by Euler. Lagrange solved the case of greatest physical interest, when the body has the symmetry of a spinning top.

“In both the Euler and Lagrange cases the solutions are expressed by writing time as an elliptic integral of a spatial variable. By the time of Lagrange elliptic integrals had been extensively studied by Legendre … Further progress on the problem awaited advances in elliptic functions and sufficient liberation from physical intuition to regard time as a complex variable, at least for purposes of mathematical analysis …

“The necessary steps were taken by Jacobi, who had discovered the double-periodicity of the inverse functions and investigated them as functions of a complex variable. Jacobi’s researches on elliptic functions led him to the discovery that such functions are best represented as quotients of theta functions. Jacobi applied these theta functions to represent the elliptic functions which arise in the Euler case and produced some very elegant formulas to represent the motion. This work, published only two years before Jacobi’s death in 1851, clearly marked the beginning of a new epoch in the study of the rotation problem. For the first time since the equations of motion had been agreed upon a genuinely new tool – theta functions – was available for the study of the problem. Jacobi’s death prevented him from making the study himself. To encourage others to take up the work the Prussian Academy of Sciences posed the following problem for competition in July 1852: “To integrate the differential equations of a body rotating about a fixed point under the influence of gravity alone. All quantities necessary to express the motion to be represented explicitly as functions of time using uniformly progressing [converging] series” …

“Despite the interesting nature of the problem and the opportunity to use the new techniques discovered by Jacobi (not to mention the prize of 100 ducats), no one entered the competition. It is difficult to believe that no one worked on the problem, however. The problem is simply very difficult. It was not forgotten and was a prominent unsolved problem when Kovalevskaya became a student of Weierstrass fifteen years later.

“Like her interest in differential equations, Kovalevskaya’s interest in the Euler equations was constant from her days as a student, throughout her career. She studied it without much success while she was a student of Weierstrass, and then found her interest rekindled at a rather inopportune time, in 1881, when she was trying to solve the Lamé equations … The step she took at this time, however, was prophetically close to the solution she ultimately achieved. She was already considering the use of theta functions for certain values of the parameters … Kovalevskaya’s formulation of the problem led her to a physically weird special case that she knew was the only remaining case in which the solutions might be meromorphic functions of the time for all possible initial values …

“The next mention of Kovalevskaya’s work in a dated letter occurs in June 1886, when Hermite refers to “your beautiful discovery on rotation” … [but] there is an undated letter addressed to “Cher Monsieur”, in which the discovery is communicated. “Cher Monsieur” is probably Mittag-Leffler … in this letter Kovalevskaya states the Euler equations and says that she has succeeded in integrating them in a new case (now known as the Kovalevskaya case) and, “I can show that these three cases [Euler, Lagrange, Kovalevskaya] are the only ones in which the general integral is a single-valued analytic function having no singularities but poles for finite values of t” …

“Judging from the note just quoted, we see that Kovalevskaya’s work had progressed to such a point that she knew when the equations could have single-valued solutions and when they could not … but the harder part of her work still lay ahead: to find the explicit solution in the case she described (the Kovalevskaya case). The work on this part (essentially the last fifty pages of her memoirs on the subject took more than two years of intense concentration …

“Kovalevskaya’s description of her work is as follows. She finally reduced the problem to quadratures by Herculean labor, involving copious combinatorial tricks and heavy use of some transformations used by Weierstrass in his lectures on applications of elliptic functions. She was then faced with the problem of integrating some elliptic integrals of the first kind. As she remarked to Mittag-Leffler, “Now you will probably understand that these formulas led me to believe that I was really dealing with elliptic functions here. I searched and searched, but to no avail; [then] to my great joy and wonder, I discovered that every symmetric function of [the variables] is an ultraelliptic function of time, i.e. can be expressed by rational functions of quotients [of theta functions of two variables whose arguments are linear functions of time].”

“She was able to write the system of integrals as a complete system of integrals of the first kind in genus 2, i.e. involving the square root of a fifth-degree polynomial in the denominator of the integrand and a linear function in the numerator. At this glorious point she knew she had the result she wanted. Weierstrass’s advice, when it came, contained only some minor technical points on how to simplify the formulas. All that Kovalevskaya needed was time to prepare a clean manuscript, and the Bordin prize was assured.” (Cooke, pp. 37-45).

“Kovalevskaya was vividly criticized by the Russian academician Markov on a minor point related to the nonexistence of other cases where the equations could be integrated. This problem was settled by Lyapunov, who proved in 1894 that Kovalevskaya's claim was correct” (Goriely, p. 8).

Kovalevskaya (1850-91) “was the daughter of Vasily Korvin-Krukovsky, an artillery general, and Yelizaveta Shubert, both well-educated members of the Russian nobility … In Recollections of Childhood (and the fictionalized version, The Sisters Rajevsky), Sonya Kovalevsky vividly described her early life: her education by a governess of English extraction; the life at Palabino (the Krukovsky country estate); the subsequent move to St. Petersburg; the family social circle, which included Dostoevsky; and the general’s dissatisfaction with the “new” ideas of his daughters. The story ends with her fourteenth year. At that time the temporary wallpaper in one of the children’s rooms at Palabino consisted of the pages of a text from her father’s schooldays, namely, Ostrogradsky’s lithographed lecture notes on differential and integral calculus. Study of that novel wall-covering provided Sonya with her introduction to the calculus. In 1867 she took a more rigorous course under the tutelage of Aleksandr N. Strannolyubsky, mathematics professor at the naval academy in St. Petersburg, who immediately recognized her great potential as a mathematician.

“Sonya and her sister Anyuta were part of a young people’s movement to promote the emancipation of women in Russia. A favorite method of escaping from bondage was to arrange a marriage of convenience which would make it possible to study at a foreign university. Thus, at age eighteen, Sonya contracted such a nominal marriage with Vladimir Kovalevsky, a young paleontologist, whose brother Aleksandr was already a renowned zoologist at the University of Odessa. In 1869 the couple went to Heidelberg, where Vladimir studied geology and Sonya took courses with Kirchhoff, Helmholtz, Koenigsberger, and du Bois-Reymond. In 1871 she left for Berlin, where she studied with Weierstrass, and Vladimir went to Jena to obtain his doctorate. As a woman, she could not be admitted to university lectures; consequently Weierstrass tutored her privately during the next four years. By 1874 she had completed three research papers on partial differential equations, Abelian integrals, and Saturn’s rings. The first of these was a remarkable contribution, and all three qualified her for the doctorate in absentia from the University of Göttingen.

“In spite of Kovalevsky’s doctorate and strong letters of recommendation from Weierstrass, she was unable to obtain an academic position anywhere in Europe. Hence she returned to Russia where she was reunited with her husband. The couple’s only child, a daughter, “Foufie,” was born in 1878. When Vladimir’s lectureship at Moscow University failed to materialize, he and Sonya worked at odd jobs, then engaged in business and real estate ventures. An unscrupulous company involved Vladimir in shady speculations that led to his disgrace and suicide in 1883. His widow turned to Weierstrass for assistance and, through the efforts of the Swedish analyst Gösta Mittag-Leffler, one of Weierstrass’ most distinguished disciples, Sonya Kovalevsky vas appointed to a lectureship in mathematics at the University of Stockholm. In 1889 Mittag-Leffler secured a life professorship for her.

“During Kovalevsky’s years at Stockholm she carried on her most important research and taught courses (in the spirit of Weierstrass) on the newest and most advanced topics in analysis. She completed research already begun on the subject of the propagation of light in a crystalline medium. Her memoir, On the Rotation of a Solid Body About a Fixed Point (1888), won the Prix Bordin of the French Academy of Sciences. The judges considered the paper so exceptional that they raised the prize from 3,000 to 5,000 francs. Her subsequent research on the same subject won the prize from the Swedish Academy of Sciences in 1889. At the end of that year she was elected to membership in the Russian Academy of Sciences. Less than two years later, at the height of her career, she died of influenza complicated by pneumonia …

“An unusual aspect of Sonya Kovalevsky’s life was that, along with her scientific work, she attempted a simultaneous career in literature. The titles of some of her novels arc indicative of their subject matter: The University Lecturer, The Nihilist (unfinished), The Woman Nihilist, and, finally, A Story of the Riviera. In 1887 she collaborated with her good friend and biographer, Mittag-Leffier’s sister, Anne Charlotte Leffler-Edgren (later Duchess of Cajanello), in writing a drama, The Struggle for Happiness, which was favorably received when it was produced at the Korsh Theater in Moscow. She also wrote a critical commentary on George Eliot, whom she and her husband had visited on a holiday trip to England in 1869” (DSB).

Cooke, ‘Sonya Kovalevskaya’s place in nineteenth century mathematics,’ pp. 17-52 in The Legacy of Sonya Kovalevskaya (Linda Keen, ed.), American Mathematical Society, 1987. Goriely, ‘A brief history of Kovalevskaya exponents and modern developments,’ Regular and Chaotic Dynamics 5 (1999), pp. 3-15. Rappaport, ‘S. Kovalevsky: a mathematical lesson,’ American Mathematical Monthly 88 (1981), pp. 564-574.

11 loose leaves, written on recto only, 290 x 232 mm, first leaf with extremely mild toning, some leaves with fingerprints in ink probably by the author. Three light horizontal creases from having been foled for postage. Very fine and clean.

Item #4276

Price: \$29,500.00