Aleksandr Lyapunov

Aleksandr Mikhailovich Lyapunov
Born	(1857-06-06)June 6, 1857 Yaroslavl, Russian Empire
Died	November 3, 1918(1918-11-03) (aged 61)
Nationality	Russian
Alma mater	Saint Petersburg State University
Known for	Lyapunov function
Scientific career
Fields	Applied mathematics
Institutions	Saint 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 1863 he retired from 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. 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 death of his father in 1868, 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 became his wife in 1886. 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 faculty of the University of Saint Petersburg. In the beginning he even attended lectures in chemistry by Mendeleyev. But he soon realised that he had more interest in mathematics and after one month transferred to the mathematics department of the university.

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 the professor of mechanics, D. K. Bobylev. In his fourth year he received the gold medal for a work on hydrostatics. This was the basis for his first published scientific works On the equilibrium of a heavy body in a heavy fluid contained in a vessel of a fixed form (О равновесии тяжелых тел в тяжелых жидкостях, содержащихся в сосуде определенной формы) and On the potential of hydrostatic pressure (О потенциале гидростатических давлений). Lyapunov completed his university course, graduating in 1880, two years after Andrey Markov, with whom Lyapunov would maintain a scientific relationship during all his life.

In 1884 Lyapunov defended his Master's thesis On the stability of ellipsoidal forms of equilibrium of rotating fluids (Об устойчивости эллипсоидальных форм равновесия вращающейся жидкости). This difficult theme was suggested to him by Chebyshev, who had already suggested it to other students of his, such as Zolotarev and Sofia Vasilyevna Kovalevskaya. The thesis was translated in French in 1904 and attracted the attention of mathematicians, physicists and astronomers in Europe.

Teaching and research

In 1895 Lyapunov became privatdozent and was proposed to accept the chair of mechanics at Kharkiv University, where he went the same year.

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. About the initial stay at Kharkiv, Lyapunov writes in his autobiography:

Here at first, the research activity of Lyapunov was cut short. It was necessary to work out courses and put together notes for students, which took up much 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 a course on the dynamics of material points, instead of a course on dynamical systems. This subject was already taught to students in lectures by professor Delarue – the course on mechanics already known to me. 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."

Lyapunov gave lectures in several topics such as theoretical mechanics, integrals of differential equations and the theory of probability. 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 defended his famous doctoral thesis The general problem of the stability of motion (Общая задача об устойчивости движения). The thesis was defended in Moscow on October 12, 1892, with Nikolai Zhukovsky and V. B. Mlodzeevski as opponents. The next year Lyapunov became a full professor at Kharkiv University.

Later years

Lyapunov returned to Saint Petersburg in 1902, when he was elected acting member of the Academy of Science.

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 in the head, and he died three days later.

Work

He contributed to several fields, including differential equations, potential theory, dynamical systems and probability theory. 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 the study of particles under the influence of gravity. His work in the field of mathematical physics regarded the boundary value problem of the equation of Laplace. In the theory of potential, his work from 1897 About some questions, connected with Dirichlet's tasks (О некоторых вопросах, связанных с задачей Дирихле) clarified several important aspects of the theory. 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 a dynamic system. In theory of probability, he generalised the works of Chebyshev and Markov, and proved the Central Limit Theorem under more general conditions than his predecessors. The method he used for the proof found later widespread use in probability theory.

From 1899 to 1902 he was the chair of the Kharkov Mathematical Society and an editor of its Communications. On December 2, 1900 he was elected as a corresponding member of the Russian Academy of Sciences, and on November 6, 1901 he became ordinary professor in the faculty of applied mathematics. The position had been left vacant by the death of Chebyshev. In 1902 he thus returned to Saint Petersburg. Not having any teaching obligations, this allowed him to focus on his studies and in particular he was able to bring to a conclusion the work on the problem of Chebyshev with which he started his scientific career.

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.

Like many mathematicians, Lyapunov preferred to work alone and communicated mainly with few colleagues 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.

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

• Lyapunov's central limit theorem
• Lyapunov's condition
• Lyapunov equation
• Lyapunov exponent
• Lyapunov fractal
• Lyapunov function
• Lyapunov stability
• Lyapunov time

References

• Lyapunov, A. M. (1992), The general problem of the stability of motion, London: Taylor & Francis, ISBN 978-0748400621 {{citation}}: Invalid |ref=harv (help); Unknown parameter |other= ignored (|others= suggested) (help)
• Parks, Patrick C. (1992), "A. M. Lyapunov's stability theory – 100 years on" (PDF), IMA Journal of Mathematical Control & Information, 9: 275–303, doi:10.1093/imamci/9.4.275 {{citation}}: Invalid |ref=harv (help)
• Shcherbakov, Pavel S. (1992), "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)
• Smirnov, Vladimir Ivanovich (1992), "Biography of A. M. Lyapunov" (PDF), International Journal of Control, 55 (3): 775–784, doi:10.1080/00207179208934258 {{citation}}: Invalid |ref=harv (help)
• Barrett, J. F. (1992), "Bibliography of A. M. Lyapunov's work", International Journal of Control, 55 (3): 785–790, doi:10.1080/00207179208934259 {{citation}}: Invalid |ref=harv (help)
• Sinai, Yakov (2004), Russian Mathematicians in the 20th Century, World Scientific, ISBN 978-9812383853 {{citation}}: Invalid |ref=harv (help)

External links

• O'Connor, John J.; Robertson, Edmund F., "Aleksandr Lyapunov", MacTutor History of Mathematics Archive, University of St Andrews
• Aleksandr Lyapunov at the Mathematics Genealogy Project
• Ляпунов Александр Михайлович at www. mathsoc.spb. ru (in Russian)
• Ляпунов Александр Михайлович (1857-1918) at www.spbu. ru (in Russian)
• Ляпунов Александр Михайлович at www-mechmath. univer. kharkov. ua (in Russian)