Alexandre Liapounov

See also: Lyapunov

Alexandre Mikhailovitch Liapounov (June 6th 1857 - November 3rd 1918) is a Russian Mathématicien .

Wire of Sofia Aleksandrovna Shilipova and Mikhail Vasilievitch Liapounov. Mikhail Vassilievich was Astronome with the Université of Kazan up to two years before the birth of Aleksandr Mikhailovich. Then named directing of the Demidovski college, he moved with his small family for Yaroslavl, where Aleksandr was born on June 6th, 1857.

The other children of Sofia and Mikhail, although it is not the subject here, also showed great intellectual abilities. Their two others wire, Sergei and Boris, became respectively type-setter and member of the Soviet academy of Sciences as an expert in Slavic languages.

Aleksandr Mikhailovich began its education at the house, where one of his/her uncles, Rafail Mr. Setchenov prepared it at the entry in Gymnasium. At that time, Natalia Rafailovna Setchenova, the proper girl of " professeur" also took part in the lessons. In fact, Natalia and Aleksandr were marrièrent a few years later when it was 29 years old.

Little time after the death of her husband, Sofia, accompanied by his/her children, left Yaroslavl for Nizhny Novgorod (called Groky between 1932 and 1990) in 1870, and Lyapunov integrated Gymnasium into it. It met there Andrei Markov which became one of its close relations, and all two entered to the faculty of Physics and Mathematics of Saint-Petersbourg after having been graduate of Gymnasium in 1876.

To the University, it followed the teaching of Pafnouti Tchebychev, which, as we will see it thereafter, had a great influence on him. Lyapunov obtained its diploma in 1880 and decided to remain in Saint-Petersbourg to continue his research. In 1881, it published its first two articles on the hydrostatic one: On the balance of the heavy bodies in heavy liquids contained in unquestionable type of containers , and On the potential of the hydrostatic pressures . In the years which followed, Chebychev raised a question with Lyapunov which will remain the discussion thread of its research during many years:

“It is well-known that at a certain angular velocity, the ellipsoidal form ceases being the shape of balance of the liquids in rotation. Consequently, for small increases angular velocity let us not have not of transformations into new forms of balance which different légèrment from the ellipsoidal form? ”

Although the work of thesis of Lyapunov did not answer this question, it was the main theme. It presented a report, On the stability of the ellipsoidal forms of the balance of a liquid in rotation , which it supported at the University of St-Petersbourg in 1884. Following what, it was named with the Université of Kharkiv there to teach mechanics and to continue its research in thesis of doctorate. The subject was the general problem of the stability of the movement defended with the Université of Moscow on October 12th, 1892 of the current calendar. This thesis, of essential importance for all the problems of balance and stability to him been worth the title of doctor. The following year it was named professor of the University of Kharkov, where it remained until 1902. He played an important role in the Mathematical Company of Kharkov, from which he was vice-president of 1891 to 1898, then president of 1899 until his departure of Kharkov, in 1902. He also published the Communication of the Mathematical Company of Kharkov.

The problem arising from Chebychev on the existence of figures of balance other than the ellipsoidal ones for a fluid in rotation subjected to variations angular velocity sufficiently weak, the first time by Lyapunov was solved at first approximation. It interressa later with the stability of the ellipsoids formed by fluids leaving for its research the principle variations Thomson-TAIT. It showed that a sufficient condition of stability was that the second and larger variations of the potential energy was positive. Lyapunov admitted that to impose a certain number of constraints on the first variation the general character of its method reduced. He wrote however:

“But in this respect, no other method could be qualified the fully satisfactory one. ”

Lyapunov showed that with a variation angular velocity of revolution, the ellipsoids of Mac-Laurin became ellipsoids of Jacobi. The point of transition is an ellipse, of Jacobi in this case.

In 1901, Lyapunov was elected with the Russian Academy of Science with St-Petersbourg, and became academician the following year in mathematics applied. According to Grigorian: “to St-Petersbourg, Lyapunov devoted body and heart to scientific research. It enquit of the problem in front of which Chebychev had placed it, and with a quantity of articles which appeared until after its death, developed the theory of the figures of balance of a heavy fluid in rotation. ”

In 1917, Lyapunov left St-Petersbourg for a station with Odessa, on the coasts of the Black Sea. He taught at the University, but in the spring of 1918, the health of his wife started to be degraded quickly. Natalia Rafailovna suffered from a variety of Tuberculose, and Lyapunov was very reached by the health issues of his wife. She died on October 31st, 1918, and Lyapunov committed suicide by ball the same day. He died in the hospital three days later.

We described principal work of Lyapunov which relates to the fluids in rotation. There is however, other aspects of its work which deserves that one is delayed there. There is its contribution to the theory of probability, in which it was interested because of the courses that it ammené to make on the subject. In particular two articles published in 1900 and 1901, where it gives a proof of the Théorème of the central limit using a technique based on characteristic functions. It also allowed the edition of two gathering volumes collected work of Euler.

It was honoured, for its excellent contributions, and was elected in many academies such as the Accademia dei Lincei (1909) or the French Academy of Science (1916). It was also promoted honorary under member of the Universities of St-Petersbourg, Kharkov and Kazan. Many tributes were versed to him at the time of the centenary of its birth. June 6th, 1957, for example, Sobolev reading On work of A.M. Lyapunov on the potential theory made, in Moscow, for a united session, of the president of the Academy of Science, divisions of physical sciences and technology of the Academy of Science, University of Moscow, Mathematical Company of Moscow, Institute of Mechanics of the Academy of Science, and Institute of Automatic and Telemechanics of the Academy of Science.

Certain articles show how much the contribution Lyapunov on the stability of the movement was large and influenced the development of the subject during long years. The concepts approached are: stability, and in particular that of the critical points; construction, and the use of the functions of Lyapunov; the stability of the equations differential-functional calculuses; second method of Lyapunov; finally method of the vector function of Lyapunov in the theory of stability and the nonlinear analysis.

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