Aleksandr Mikhailovich Lyapunov^[a][b] (Алекса́ндр Миха́йлович Ляпуно́в, 6 June [O.S. 25 May] 1857 – 3 November 1918) was a Russianmathematician,mechanician andphysicist. He was the son of the astronomerMikhail Lyapunov and the brother of the pianist and composerSergei Lyapunov.

Aleksandr Lyapunov
Born	6 June [O.S. 25 May] 1857 Yaroslavl, Russian Empire
Died	3 November 1918(1918-11-03) (aged 61) Odessa,Ukrainian People's Republic
Alma mater	Saint Petersburg State University
Known for	Lyapunov function Lyapunov stability Lyapunov exponent Lyapunov central limit theorem Lyapunov vector Qualitative theory of differential equations
Scientific career
Fields	Applied mathematics
Institutions	Saint Petersburg State University Russian Academy of Sciences Kharkov University
Thesis	The general problem of the stability of motion (1892)
Doctoral advisor	Pafnuty Chebyshev
Doctoral students	Vladimir Steklov

Lyapunov is known for his development of thestability theory of adynamical system, as well as for his many contributions tomathematical physics andprobability theory.

Lyapunov was born inYaroslavl,Russian Empire. His fatherMikhail Vasilyevich Lyapunov (1820–1868) was anastronomer employed by theDemidov Lyceum. His brother,Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (nowUlyanovsk Oblast). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of thephysiologistIvan Mikhailovich Sechenov. At his uncle's family, Lyapunov studied with his distant cousin Natalia Rafailovna, who became his wife in 1886. In 1870, his mother moved with her sons toNizhny Novgorod, where he started the third class of thegymnasium. He graduated from the gymnasium with distinction in 1876.^[1]

In 1876, Lyapunov entered the Physico-Mathematical department at theUniversity of Saint Petersburg, but after one month he transferred to the Mathematics department of the university.

Among the Saint Petersburg mathematics professors wereChebyshev and his studentsAleksandr Nikolaevich Korkin andYegor Ivanovich Zolotarev. Lyapunov wrote his first independent scientific works under the guidance of the professor of mechanics, D. K. Bobylev. In 1880 Lyapunov received a gold medal for a work onhydrostatics. This was the basis for his first published scientific worksOn the equilibrium of a heavy body in a heavy fluid contained in a vessel of a fixed form andOn the potential of hydrostatic pressure. Lyapunov also completed his university course in 1880, two years afterAndrey Markov who had also graduated at Saint Petersburg University. Lyapunov maintained scientific contact with Markov throughout his life.

A major theme in Lyapunov's research was the stability of a rotating fluid mass with possible astronomical application. This subject was proposed to Lyapunov byChebyshev as a topic for his masters thesis which he submitted in 1884 with the titleOn the stability of ellipsoidal forms of rotating fluids. The main contribution was published in the celebrated monograph 'A.M. Lyapunov, The general problem of the stability of motion. 1892. Kharkov Mathematical Society, Kharkov, 251p. (in Russian)'.^[2] This led on to his 1892 doctoral thesisThe general problem of the stability of motion.^[3] The thesis was defended in Moscow University on 12 September 1892, withNikolai Zhukovsky and V. B. Mlodzeevski as opponents. In 1908, the Kharkov edition was translated to French and republished by theUniversity of Toulouse: 'Probleme General de la Stabilite du Mouvement, Par M.A. Liapounoff. Traduit du russe par M.Edouard Davaux'.

In 1885, Lyapunov becameprivatdozent and was proposed to accept the chair of mechanics atKharkov University, where he went the same year. About the initial stay atKharkov, Smirnov writes in his biography of Lyapunov:

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.^[1]

His student and collaborator,Vladimir Steklov, recalled his first lecture in the following way: "A handsome young man, almost of the age of the other students, came before the audience, where there was also the 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 ondynamical systems. This subject was already known to the students from the lectures of professor Delarue. But what Lyapunov taught us was new to me and I had never seen this material in any textbook. All antipathy to the course was immediately blown to dust. From that day students would show Lyapunov a special respect."^[1]

Lyapunov returned to Saint Petersburg in 1902, after being elected acting member of the Academy of Science as well as ordinary professor in the Faculty of Applied Mathematics of the university. The position had been left vacant by the death of his former teacher,Chebyshev. Not having any teaching obligations, this allowed Lyapunov 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 took part to the FourthInternational Mathematical Congress in Rome. He also participated in the publication of Euler's selected works: he was an editor of the volumes 18 and 19.^[1]

By the end of June 1917, Lyapunov traveled with his wife to his brother's palace inOdessa. Lyapunov's wife was suffering from tuberculosis so they moved in accordance with her doctor's orders. She died on 31 October 1918. The same day, Lyapunov shot himself in the head, and three days later he died.^[4] By that time, he was going blind fromcataracts.^[1]

Lyapunov contributed to several fields, includingdifferential equations,potential theory,dynamical systems andprobability theory. His main preoccupations were the stability of equilibria and the motion of mechanical systems, especially rotating fluid masses, and the study of particles under the influence of gravity. His work in the field ofmathematical physics regarded the boundary value problem of theequation of Laplace. In the theory of potential, his work from 1897On some questions connected withDirichlet's problem 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, 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 dynamical system. In the theory of probability, he generalized the works of Chebyshev and Markov, and proved theCentral Limit Theorem under more general conditions than his predecessors. The method ofcharacteristic functions he used for the proof later found widespread use in probability theory.^[1]

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 theAcademy of Sciences in Paris.^[1]

Lyapunov's impact was significant, and the following mathematical concepts are named after him:

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

• 1884,On the stability of ellipsoidal figures of equilibrium of a rotating fluid (in Russian) Published inBulletin Astronomique 1885
• 1892,A.M. Lyapunov, The general problem of the stability of motion. 1892. Kharkov Mathematical Society, Kharkov, 251p. (in Russian)
• 1897,Sur certaines questions qui se rattachent au problème de Dirichlet
• 1901,Nouvelle forme du théorème sur la limite de probabilité
• 1901,Sur un théorème du calcul des probabilités
• 1902,Sur une série dans la théorie des équations différentielles linéaires du second ordre à coefficients périodiques
• 1903,Recherches dans la théorie de la figure des corps célestes
• 1904,Sur l'équation de Clairaut et les équations plus générales de la théorie de la figure des planètes

• O'Connor, John J.;Robertson, Edmund F.,"Aleksandr Lyapunov",MacTutor History of Mathematics Archive,University of St Andrews
• Aleksandr Lyapunov at theMathematics Genealogy Project
• Ляпунов Александр Михайлович at www. mathsoc.spb. ru (in Russian)
• Ляпунов Александр Михайлович (1857-1918)Archived 2007-02-23 at theWayback Machine at www.spbu. ru (in Russian)
• Ляпунов Александр Михайлович at www-mechmath. univer. kharkov. ua (in Russian)
• [1] Aleksandr M. Lyapunov = Ляпунов Александр Михайлович alive at scholar.google.com (live citations)