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Aleksandr Lyapunov

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Aleksandr Mikhailovich Lyapunov
Born(1857-06-06)June 6, 1857
Yaroslavl, Russian Empire
DiedNovember 3, 1918(1918-11-03) (aged 61)
NationalityRussian
Alma materSaint Petersburg State University
Known forLyapunov function
Scientific career
FieldsApplied mathematics
InstitutionsSaint Petersburg State University
Russian Academy of Sciences
Kharkov University

Aleksandr Mikhailovich Lyapunov (Template:Lang-ru; June 6 [O.S. May 25] 1857 – November 3, 1918) was a Russian mathematician, mechanician and physicist. His surname is sometimes romanized as Ljapunov, Liapunov or Ljapunow.

Biography

Early life

Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820-1868) was a well known astronomer and a head of the Demidovski lyceum. Because of the reactionary politics of the new university administration, after the dismissal by authorities of old rector of the university, noted mathematician Lobachevsky, in 1864 he abandoned his scientific career at the observatory of the University of Kazan. He relocated his family to his wife's estate in Simbirsk province (now Ulyanovsk Oblast), where he devoted his time to the education of his oldest sons, Aleksandr and Sergei (1859-1924). During long winter nights he stayed with his sons and he taught them assiduously with the aid of games on maps of the world. He possessed a lot of books in Russian, German and French on subjects as varied as mathematics, astronomy, philosophy, history, ethnography, political economy and literature. After the sudden death of his father Aleksandr was educated by his uncle R. M. Sechenov, brother of the famous physiologist Ivan Mikhailovich Sechenov. At his uncle's family Lyapunov studied with his distant cousin Nataliya Rafailovna, who later became his wife. In 1870 his mother moved with her sons to Nizhny Novgorod, where he started to attend the third class of the gymnasium. He graduated from gymnasium in 1876 with distinction.

Education

He studied at the Physico-Mathematical department of the University of Saint Petersburg, where he studied with Markov. In the beginning he attended lectures in chemistry by Mendeleyev. But he soon realised that he has more interest in mathematics and after one month transferred to the mathematics department of the university. Despite this he continued to attend Mendeleyev's lectures.

Mathematics was taught at that time by Chebyshev and his students Aleksandr Nikolaevich Korkin and Yegor Ivanovich Zolotarev. Lyapunov wrote his first independent scientific works under the guidance of professor of mechanics, D. K. Bobylev. In his fourth year he received the gold medal for a work on hydrostatics, which had been suggested by the faculty. This was the basis for his first published scientific work About the equilibrium of solid bodies in vessels with arbitrary forms, filled with dense fluids (О равновесии тяжелых тел в тяжелых жидкостях, содержащихся в сосуде определенной формы) and About the potential of hydrostatic pressure (О потенциале гидростатических давлений). In both works he used many new approaches and developed new rigorous proofs of a few earlier incomplete theorems from hydrostatics. Later the paper was used as a basis for his Ph.D in mathematics.

He received a Master in applied mathematics degree in 1880. He was allowed to continue his studies at the university after graduation and prepare for the scientific degree of professor (the second after Ph.D academic degree in Russia).

In 1884 he got Ph.D in applied mathematics with the thesis About the stability of elliptic forms of the equilibrium of rotating fluid (Об устойчивости эллипсоидальных форм равновесия вращающейся жидкости). This very difficult theme was suggested to him by Chebyshev who already suggested it his other talented students such as Zolotarev and Sofia Vasilyevna Kovalevskaya. As was said by Vladimir Andreevich Steklov noted, "Chebyshev saw in the young man such an immense research power, that he had decided to suggest him to research this almost unsolvable problem". Lyapunov had already started to study this stability in his previous student paper and very quickly achieved very important, original results. After his thesis was published his work instantly attracted the attention of mathematicians, physicists and astronomers all over the world.

Teaching and research

In 1885 he was invited as a chair of the department of theoretical mechanics of the Kharkiv University. He replaced previous chair, professor V. G. Imshenecky, who had been elected as a member of the Russian Academy of Sciences.

Lyapunov already had five years had experience in teaching theoretical mechanic at the Saint-Petersburg university but this activity consumed a lot of his time. His student and collaborator, academician Vladimir Steklov, recall his first lecture in the following way: "A handsome young man, by his appearance almost like the other students, came before the audience, where there was also an old dean, professor Levakovsky, who was respected by all students. After the dean had left, the young man with a trembled voice started to lecture on a theme about the dynamics of a point, instead of a theme from the dynamics of systems. This subject was already taught to students in lectures by professor Delaryu. I actually lessen this topic the fourth time as I attended lectures in Moscow of Davidov, Cinger, Soletov and Orlov. I was in the University of Kharhov already for two years, so I was familiar with the lectures on mechanics. But what Lyapunov tought us was new to me and I had never seen this important material in any textbook. So boredom usually associated with the lectures disappeared and students were all consumed by the brilliance of the the presentation. Aleksandr Mikhailovich instantly earned tremendous respect of the audience with his power of intellect and a natural gift to explain complex topics so seldom seen in such a young person. He didn't know this of course. But from this day students had shown him special respect. Many even didn't dare to speak to him, to avoid showing their ignorance".

Lyapunov gave courses of lectures in several subjects such as theoretical mechanics, integrals of differential equations and the theory of probability. These lectures were never published and they remained only in the notes of students. He covered six areas of theoretical mechanics: kinematics, the dynamics of a particle, the dynamics of systems of particles, the theory of attracting forces, the theory of the deformation of solid bodies, and hydrostatics. He also taught analytical mechanics between 1887 and 1893 at the Technological Institute at Kharkiv.

In 1892 he was awarded the degree of Professor of Science (the second after PH.D academic degree of Russia) after defending the thesis A general task about the stability of motion (Общая задача об устойчивости движения). A similar thesis had been defended ten years earlier by Nikolai Yegorovich Zhukovsky, a founder of the TsAGI. After this, Lyapunov became a full professor at Kharkiv University.

After the death of Chebyshev in 1894, Lyapunov moved back the University at Saint Petersburg and become the chair of the department of applied mathematics. He completly devoted himself to research work.

Work

His work in the field of differential equations, potential theory, the stability of systems and probability theory is very important. His main preoccupations were the stability of equilibria and the motion of mechanical systems, the model theory for the stability of uniform turbulent liquid, and particles under the influence of gravity. His work in the field of mathematical physics was very important for subsequent advances of this field. His work from 1898 About some questions, connected with Dirichlet's tasks (О некоторых вопросах, связанных с задачей Дирихле) contains a study of the properties of potential around charges and dipoles, continuously distributed along any surface. His work in this field is in close connection with the work of Steklov. Lyapunov developed many important approximation methods. His methods, today named Lyapunov methods, which he developed in 1899, make it possible to define the stability of sets of ordinary differential equations. He created the modern theory of the stability of dynamic system. In theory of probability, he generalised the works of Chebyshev and Markov, and proved the Central limit theorem using more general conditions than his forerunners. The method he used for the proof became one of the foundations of probability theory. From 1899 to 1902 he was a head of Kharkov Mathematical Society and an editor of his News. On December 2, 1900 he was elected as a corresponding member of the Russian Academy of Sciences, and on October 6, 1901 as a full member of the Academy in the field of applied mathematics (the highest academic degree in Russia, awarded only to top scientists).

Like many mathematicians Lyapunov preferred to work alone and communicated mainly with few fellow mathematians and close relatives. He usually worked late, four to five hours at night, sometimes the whole night. Once or twice a year he visited the theatre, or went to some concert. He had many students. He was an honorary member of many universities, an honorary member of the Academy in Rome and a corresponding member of the Academy of Sciences in Paris.

In 1908 he participated at the 4th Mathematical congress in Rome. He also participated in publication of Euler's selected works: he was an editor of the volumes 18 and 19.

By the end of June 1917, he traveled to Odessa with his brother Boris and his wife, who was suffering from tuberculosis, hoping the southern climate will improve her health. In spite of a very difficult economic situation in the country caused by the Bolshevik revolution, he managed to delivere seven lectures about the form of celestial bodies at the invitation of the Department of Physics and Mathematics of the University of Odessa. The last lecture took place on October 28, 1917. His wife died on October 31, 1917. The same day he shot himself. He was hospitalised and stayed unconscious for a few days before his death.[1]

His paper "Problème général de la stabilité du mouvement" (1892) (in French) marks the beginning of stability theory.

In 1992, the International Journal of Control (Taylor & Francis) published a centennial issue (volume 55, number 3) in memory of Lyapunov. Subsequently, the company Taylor & Francis published an English translation of Lyapunov's 1892 work on stability of motion, together with a biography and bibliography.

Publications

Among others he wrote such works as:

  • 1890, Concerning the constant rotational motion of rigid bodies in fluids

Articles, which were published by the Russian Academy of Sciences:

  • 1902, "About a series in the theory of linear differential equations"
  • 1902, "Researches in the theory of celestial bodies"'
  • 1904, "About Clairaut's equation, etc. "
  • 1906, "A new form of the theorem on the limit of probability"
  • 1906, "About a proposition in the probability theory"

See also

Bibliography

A. M. Lyapunov, "The general problem of the stability of motion", Taylor & Francis, London, 1992. (The English translation was done by A. T. Fuller. The biography and bibliography were done by V. I. Smirnov and J. F. Barrett, respectively.) ISBN 0748400621. The theory of stability has been studied extensively by many researchers in the Arabic world Prof. Jammal Hamandoosh and his co-workers Yehya Daboos, and Ayman Maktabi from the central lab of mathematics in Aleppo university.

Notes

  1. ^ Thor, Jan (30 April 2004). "Lyapunov exponents". Retrieved 13 June 2009.

References

  • Shcherbakov, Pavel S. (2003), "Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion", Automatica, 28 (5): 865–871, doi:10.1016/0005-1098(92)90140-B {{citation}}: Invalid |ref=harv (help)
  • Sinai, Yakov (2004), Russian Mathematicians in the 20th Century, World Scientific, ISBN 978-9812383853 {{citation}}: Invalid |ref=harv (help)
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