**Aleksandr Mikhailovich Lyapunov** (Александр Михайлович Ляпунов) (June 6 1857 - November 3 1918) was a Russian mathematician, mechanician and physicist.

Lyapunov was born in Yaroslavl, Imperial Russia. His father Mikhail Vasilyevich Lyapunov (1820-1868) was a well known astronomer and a head of the Demidovski lyceum. Because of the reaction of the university administration, after the departure of Lobachevsky, he gave up his work entirely in 1864 at the observatory of the University of Kazan. He moved with his family on his wife's estate in Simbirska province, 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 philosopher Ivan Mikhailovich Sechenov. At his uncle's Lyapunov learned with his cousin, his intended Nataliya Rafailovna. In 1870 his mother moved with her sons to Nizhny Novgorod, where he started to attend the third class of the gymnasium. He passed gymnasium in 1876 with distinction.

He studied at the Physical mathematical faculty of the University at Saint Petersburg, where he was a schoolfellow of Markov. In the beginning he listened to Mendeleev's lectures on chemistry. After a month he transferred to the mathematics department of the university, but he continued attending the chemistry lectures. Mathematics was taught at that time by Chebyshev and his students Aleksandr Nikholaevich Korkin and Egor Ivanovich Zolotarev. Lyapunov wrote his first independent scientific works under the guidance of professor of mechanics, D. K. Bobilev. 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 approachess and developed new rigorous proofs of a few earlier incomplete theorems from hydrostatics. With the first work he gained the title of candidate in mathematical sciences. Now he was able to leave the university to prepare for a professorial calling.

He graduated in 1880. He received a Master's degree in applied mathematics in 1884 with the thesis *About the stability of elliptic forms in the equilibrium of turbulent fluid* *(Об устойчивости эллипсоидальных форм равновесия вращающейся жидкости*) This work treated an important and difficult task about understanding the shapes of celestial bodies. This task was offered by Chebyshev to Zolotarev and to Sofia Vasilyevna Kovalevskaya and Chebyshev was aware of the difficulty. As was said by Vladimir Andreevich Steklov, "Chebyshev saw in the young man such an immense research power, that he had dared to lay on him such a toilsome task". Lyapunov had already begun to study this stability in his previous two-years attempts at solving the task. After the public announcement his work instantly attracted the attention of mathematicians, mechanicians, physicists and astronomers all over the world.

In 1885 he became private reader of the University at Kharkov in the chair of mechanics, where he replaced V. G. Imshenecky, who had been chosen as a member of the Russian academy of Sciences. Lyapunov had lectured already from 1880 at the faculty for mechanics and this had taken him a lot of time. His student and collaborator, academician Steklov, said about his fine lectures: "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 in lectures by a professor Delaryu. I was in the fourth class. I had listened to the 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 I hadn't known the subject from the beginning and I had never seen it in any textbook. So in this way boredom with the lecture had collapsed in ruins. Alekdandr Mikhailovich had earned the respect of the audience for an hour with the power of a natural gift so seldom seen in such a youth. He didn't know this of course. From this day on students considered him with different eyes, and they showed him special respect. Many times they even didn't dare to speak with him, to avoid showing their ignorance". Lyapunov lectured at the university on themes from 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 lectured about mechanics in six areas: kinematics, the dynamics of a pointed body, the dynamics of systems of pointed bodies, the theory of attracting forces, the theory of the deformation of solid bodies, and hydrostatics. At the same time he lectured on analytical mechanics between 1887 and 1893 at the Technological institute at Kharkov.

In 1892 he was awarded his Ph.D. with 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 the doctorate Lyapunov became a full professor of the University at Kharkov. After the death of Chebyshev in 1894 Lyapunov became in 1901 head of applied mathematics at the University at Saint Petersburg, where he had entirely devoted to tutorage and research 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 equilibrium and the motion of mechanical system, the model theory for the stability of uniform turbulent liquid, and the particles under the influence of gravity. His work in the field of mathematical physics is very important for the subsequent advance 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 approximative 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 elaborated the modern rigorous theory of the stability of a system, and the motion of a mechanical system on the basis of a finite number of parameters. In probability theory he generalised the works of Chebyshev and Markov and he finally proved the Central limit theorem using more common conditions than his forerunners. The method he used for the proof is today 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 the December 2 1900 he was elected as a corresponding member of the Russian Academy of Sciences, and on the October 6 1901 as a fully entitled member of the Academy in the field of applied mathematics.

Among others he wrote such works as: *About constant spiral motion of a rigid body in a fluid* (*О постоянных винтовых движениях твердого тела в жидкости*) in 1890, and many articles, which were published by the Russian Academy of Sciences:

*About a series in the theory of linear differential equations*(*Sur une série dans la théorie des équations differentielles linéaires etc.)*1902,*Researches in the theory of celestial bodies*(*Recherches dans la théorie des corps célestes*) 1903,*About Clairaut's equation, etc.*(*Sur l'équation de Clairaut etc.*) 1904,*A new form of the theorem on the limit of probability*(*Nouvelle forme du théorème sur la limite de probabilité*),*About a proposition in the probability theory*(*Sur une proposition de la théorie des probabilités*) 1906.

He usually worked four to five hours at night, and many times even the whole night. Once or twice he visited the theatre or he went to some concert. He had many students. But for the few who really knew him, Lyapunov was a rather raptured man. He had a lean figure, outwardly he acted pretty rude, otherwise he had a hot-blooded and sensitive temper. He was an honorary member of many universities, an external member of the Academy in Rome and a corresponding member of the Academy of Sciences in Paris.

- Lyapunov's central limit theorem,
- Lyapunov's condition,
- Lyapunov's characteristic number,
- Lyapunov's characteristic exponent,
- Lyapunov's equation,
- Lyapunov's exponent,
- Lyapunov's fractal,
- Lyapunov's function,
- Lyapunov's stability,
- Lyapunov's test,
- Lyapunov's time.

- http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Lyapunov.html (in English)
- http://www.mathsoc.spb.ru/pantheon/lyapunov/b-e.html (in Russian)
- http://www.spbu.ru/History/275/Chronicle/spbu/Persons/L_yapunov.html (in Russian)
- http://www-mechmath.univer.kharkov.ua/theormech/lapunov.html (in Russian)