Sofia Kovalevskaya

Leigh Ellison

French Mathematics

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Mittag-Leffler viewed mathematics very much as an international enterprise, and so it was because of both him and her work for Acta Mathematica that Sofia first came into contact with many of the great French mathematical minds of the day. She was to develop close working relationships with many of them, and the astronomer Hugo Gyldén was to refer to Charles Hermite, Mittag-Leffler, Charles émily Picard, Weierstrass and Sofia as, the mutual admiration society of mathematics in the late nineteenth century. In 1886 the fact that Sofia had made a significant breakthrough in her work was a fact known to many of the more prominent French mathematicians. At this time she managed to clarify the situation with regard to the use of theta functions in the solution of the fixed point problem, and was greatly assisted by the ideas of Weierstrass on ultra-elliptic integrals. It was becoming increasingly clear to everyone at this time that Sofia and her work would never receive the recognition which they deserved due largely to her sex. As a result the French mathematical community decided to honour her as best they could.

The Prix Bordin

The Prix Bordin was the second most prestigious in a very long line of awards given out by the French Academy of Sciences. The prominent French mathematicians of the time felt as though Sofia deserved some kind of reward for her work, and the Prix Bordin seemed to fit the bill perfectly. The topic on which entries could be submitted for the prize to be awarded in 1888 was thus chosen with Sofia in mind. The aim of anyone interested in obtaining the prize was to, perfect in one important point the theory of the movement of a solid body round an immovable point. It could be argued that as a result of this the competition was not as objective or anonymous as the rules requested, but this was true of many such competitions at the time and was in no way an unusual situation. As early as June, 1886, Sofia wrote to Mittag-Leffler that she had been discussing her project with members of the judging committee which included Hermite, Bertrand, and Darboux, and that they, seemed to find it so interesting that they believed it has a good chance of winning. As the deadline approached by which entries had to be submitted, it seemed to be clear to everyone except perhaps Sofia herself that she would in fact win this prestigious prize for her solution to a problem which had remained unsolved on each of the previous occasions it had been set.

By 1886 Sofia had really solved as much of the fixed point problem as she could hope to, yet as a faithful student of Weierstrass and his mathematical style she felt that she had to, do more than simply say that the integrals could be inverted using theta functions. 68 She wanted to go through this inversion in detail, and show how to express the parameters which describe the motions as functions of time. It was these steps which would occupy her time until the competition deadline in 1888.

During 1887 when she should really have been fully focussed on her manuscript for the competition, Sofia was distracted not only by her sister Anuita's death, her work for Acta Mathematica and her literary interests, but by a man called Maxim Kovalevsky with whom she became reacquainted on one of her Paris visits. He was a Russian jurist and sociologist whose friendship with Sofia progressed very rapidly, with her soon saying of him that, It is simply impossible for me, in his presence, to think of anybody and anything but him. Maxim was quite happy living his life as a bachelor, and it seems clear from the letters which they exchanged that Sofia was far more invested in the relationship than he was. Nevertheless he did propose to Sofia during this time on the condition that she give up her research. It is possible however that he only asked because he knew that this would not even be a consideration for Sofia whilst she was still working on her fixed point manuscript. Mittag-Leffler had become somewhat frustrated by the many distractions in Sofia's life at this time and invited Maxim to stay at his summer home in Uppsala so that she could complete her work in peace. She was not able to submit a complete solution in time for the deadline, but sent a first draft and requested some extra time to allow her to send a fuller version in time for judging. Her work was felt to be of such a high standard that had she needed even more time to complete it then the judging of the prize would have been delayed accordingly. In the late summer of 1888 the revised memoir which she submitted for judgement was still not complete, but she was nevertheless satisfied with its content and presentation.

It was in December of 1888 that Sofia learned she was to receive, the highest scientific recognition ever accorded a women to that day, and for long after. Her work was described by Hermite as, an ingenious application of mathematics to a system of equations of great mathematical interest, while the President of the Academy of Sciences said of it in his congratulatory speech that it, bears witness not only to profound and broad knowledge, but to a mind of great inventiveness. The originality and standard of her work also saw the prize money being raised from three to five thousand francs.

On hearing of Sofia's victory in the Prix Bordin, Weierstrass was unsurprisingly delighted and proud. He wrote to her from Berlin that,
your success has gladdened the hearts of myself and my sisters, also of your friends here. I particularly experienced a true satisfaction; competent judges have now delivered their verdict that my 'faithful pupil', my 'weakness' is indeed not a 'frivolous humbug.'

More much needed prize money was awarded to Sofia when a further refined version of her solution would see her receive an award from the Swedish Academy of Sciences who were doing their very best not to lose the woman they viewed as the star of their relatively new university.

The work for which Sofia won the Prix Bordin was further secured a place in the history of mathematics when R. Louiville showed that there are no cases other than those looked at by Sofia, Euler and Lagrange where the equations which they investigated have four independent algebraic integrals. As such her work, represents the final chapter in the story of closed-form solutions.

In was in 1750 that Euler first derived the equations of motion of a rigid body about a fixed point. When the torque is produced by gravity these equations are of the form
Adp/dt = (B - C)qr + Mg(y0γ'' - z0γ')
Adq/dt = (C - A)rp + Mg(z0γ'' - x0γ')
Adr/dt = (A - B)pq + Mg(x0γ'' - y0γ')

where M is the mass of the body, g the acceleration of gravity, (γ, γ', γ'') is a unit vector which points downwards, (x0, y0, z0) are the coordinates of the centre of gravity of the body with respect to axes which are fixed in the body, A, B, C are the principle axes relative to the fixed point and p, q, r are the components of the angular velocity along the principal axes. We then look at another three equations which are obtained due to the fact that the unit vertical vector is fixed in space. These are
dγ/dt = rγ - qγ''
dγ'/dt = pγ'' - rγ
dγ''/dt = qγ - pγ'

The problem which many people worked on was to integrate these equations such that the position of the moving body could be calculated at any time. Prior to Sofia's work, this had been completed for two cases. The first case which was investigated by both Euler and Poisson was for the condition (x0= y0= z0= 0) and deals with the motion of a symmetric body which is force-free.

The second case which was studied by Lagrange was for the condition when A = B, (x0= y0= 0), and the motion here takes the form of a spinning top. These both deal with cases where the rigid body is symmetrical, and it was Sofia who first developed things for an unsymmetrical top.

Three integrals are relatively easy to find. These are the total energy
(Ap2+ Bq2+ Cr2)/2+Mg (x0γ + y0γ' + z0γ'') = a constant,

the angular momentum
Apγ + Bqγ' + Crγ'' = a constant,

and from the use of geometry
γ2+ γ'2+ γ''2= 1

A complete solution requires a fourth integral which Sofia was to find.

As well as deriving this fourth integral under the conditions A = B = 2C, z0= 0, she showed that in order for a series to be a solution of Euler's equations the only condition is that the constants A, B, C, x, y, z satisfy one of the following
A = B = C
x0= y0= z0= 0
A = B, x0= y0= 0
A = B = 2C, z0= 0

Sofia found her solution by choosing a suitable unit of length such that y0= 0 and C = 1, and by rotating the axes in the xy plane. Using c0 to denote Mgx0 the Euler equations become the following:
2 dp/dr = qr
2 dq/dt = -pr - c0γ''
2 dr/dt= c0γ
dγ/dt = rγ' - qγ''
dγ'/dt = pγ'' - rγ
dγ''/dt = qγ - pγ'

and the three integrals which we know are then written as
2(p2+ q2) + r2= 2c0γ + 6l1
2(pγ + qγ' ) = 2l
γ2+ γ'2+ γ''2= 1

The fourth integral which Sofia then found was
(p + qi)2+ c0(γ + iγ) (p - qi)2+ c0(γ - iγ) = k2

She then used a series of changes of variable which allowed her to transform the integrals and equations to a form more suitable for the application of theta functions. The work of R. Liouville was later to show that there are no other cases where the equations have four independent algebraic integrals. For all intents and purposes Sofia had done as much work on this subject as it was possible to do. Her paper, represents the final chapter in the story of closed-form solutions. This work, astounded her contemporaries by its beauty and elegance, and is still of interest to physicists even today. It also stimulated a great deal of interest in the problem of the rotation of a rigid body amongst other mathematicians of the day.

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