Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a

Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (), Russian language term for all citizens and peo ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

mechanician A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or computational mechanics, applied mechanics, geomechanics, biomechanics, and mechanics of materials. Names other than mecha ...

and

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

. His surname is variously romanized as Ljapunov, Liapunov, Liapounoff or Ljapunow. He was the son of the astronomer

Mikhail Lyapunov Mikhail Vasilyevich Lyapunov () was a Russian astronomer and a head of the Demidov Lyceum in Yaroslavl. He was the father of Aleksandr and Sergei Lyapunov Sergei Mikhailovich Lyapunov (or Liapunov; russian: Серге́й Миха́йло� ...

and the brother of the pianist and composer

Sergei Lyapunov Sergei Mikhailovich Lyapunov (or Liapunov; russian: Серге́й Миха́йлович Ляпуно́в, ; 8 November 1924) was a Russian composer, pianist and conductor. Life Lyapunov was born in Yaroslavl in 1859. After the death of his fath ...

. Lyapunov is known for his development of the stability theory of a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

, as well as for his many contributions to

mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...

and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

Biography

Early life

Lyapunov was born in

Yaroslavl Yaroslavl ( rus, Ярослáвль, p=jɪrɐˈsɫavlʲ) is a city and the administrative center of Yaroslavl Oblast, Russia, located northeast of Moscow. The historic part of the city is a World Heritage Site, and is located at the confluenc ...

Russian Empire The Russian Empire was an empire and the final period of the Russian monarchy from 1721 to 1917, ruling across large parts of Eurasia. It succeeded the Tsardom of Russia following the Treaty of Nystad, which ended the Great Northern War. ...

. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an

astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...

employed by the

Demidov Lyceum The Yaroslavl Demidov State University ( Russian: ''Ярославский государственный университет имени П. Г. Демидова'') is an institution of higher education in Yaroslavl, Russia. In 1918, Yarosla ...

. His brother,

, 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 (now

Ulyanovsk Oblast Ulyanovsk Oblast (russian: Ульяновская область, ''Ul’janovskaja oblast’'') is a federal subject of Russia (an oblast). It is located in the Volga Federal District. Its administrative center is the city of Ulyanovsk. Populati ...

). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of the physiologist

Ivan Mikhailovich Sechenov Doctor Ivan Mikhaylovich Sechenov (russian: Ива́н Миха́йлович Се́ченов; , Tyoply Stan (now Sechenovo) near Simbirsk, Russia – , Moscow), was a Russian psychologist, physiologist, and medical scientist. The very f ...

. 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 to

Nizhny Novgorod Nizhny Novgorod ( ; rus, links=no, Нижний Новгород, a=Ru-Nizhny Novgorod.ogg, p=ˈnʲiʐnʲɪj ˈnovɡərət ), colloquially shortened to Nizhny, from the 13th to the 17th century Novgorod of the Lower Land, formerly known as Gork ...

, where he started the third class of the gymnasium. He graduated from the gymnasium with distinction in 1876.

Education

In 1876, Lyapunov entered the Physico-Mathematical department at the

University of Saint Petersburg A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, th ...

, but after one month he transferred to the Mathematics department of the university. Among the Saint Petersburg mathematics professors were

Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...

and his students Aleksandr Nikolaevich Korkin and

Yegor Ivanovich Zolotarev Yegor (Egor) Ivanovich Zolotarev (russian: Его́р Ива́нович Золотарёв) (31 March 1847, Saint Petersburg – 19 July 1878, Saint Petersburg) was a Russian mathematician. Biography Yegor was born as a son of Agafya Izoto ...

. 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 on

hydrostatics Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...

. 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'' and ''On the potential of hydrostatic pressure''. Lyapunov also completed his university course in 1880, two years after Andrey Markov who had also graduated at Saint Petersburg University. Lyapunov maintained scientific contact with Markov throughout his life.

Teaching and research

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 by

as a topic for his masters thesis which he submitted in 1884 with the title ''On the stability of ellipsoidal forms of rotating fluids''. In 1885, Lyapunov became privatdozent and was proposed to accept the chair of mechanics at

Kharkiv University The Kharkiv University or Karazin University ( uk, Каразінський університет), or officially V. N. Karazin Kharkiv National University ( uk, Харківський національний університет імені ...

, where he went the same year. About the initial stay at

Kharkiv Kharkiv ( uk, wikt:Харків, Ха́рків, ), also known as Kharkov (russian: Харькoв, ), is the second-largest List of cities in Ukraine, city and List of hromadas of Ukraine, municipality in Ukraine.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 on

s. 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." The main contribution was published in the celebrated monograph 'A.M. Lyapunov, The general problem of the stability of motion. 1892. Kharkiv Mathematical Society, Kharkiv, 251p. (in Russian)'. This led on to his 1892 doctoral thesis ''The general problem of the stability of motion''. The thesis was defended in Moscow University on 12 September 1892, with Nikolai Zhukovsky and V. B. Mlodzeevski as opponents. In 1908, the Kharkiv edition was translated to French and republished by the University of Toulouse: 'Probleme General de la Stabilite du Mouvement, Par M.A. Liapounoff. Traduit du russe par M.Edouard Davaux'.

Later years

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,

. 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 Fourth International 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.

Death

By the end of June 1917, Lyapunov traveled with his wife to his brother's palace in Odessa. 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. By that time, he was going blind from

cataracts A cataract is a cloudy area in the lens of the eye that leads to a decrease in vision. Cataracts often develop slowly and can affect one or both eyes. Symptoms may include faded colors, blurry or double vision, halos around light, trouble w ...

Work

Lyapunov contributed to several fields, including

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

s and

. 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 of

regarded the boundary value problem of the equation of Laplace. In the theory of potential, his work from 1897 ''On some questions connected with Dirichlet'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 the

Central Limit Theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...

under more general conditions than his predecessors. The method of

characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...

s he used for the proof later found widespread use in probability theory. 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 An academy of sciences is a type of learned society or academy (as special scientific institution) dedicated to sciences that may or may not be state funded. Some state funded academies are tuned into national or royal (in case of the Unit ...

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), ma ...

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

Lyapunov equation In control theory, the discrete Lyapunov equation is of the form :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A. The continuous Lyapunov equation is of the form :AX + XA^H + Q = 0. The Lyapunov equation o ...

* Lyapunov exponent *

Lyapunov function In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s s ...

* Lyapunov fractal *

Lyapunov stability Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. ...

Lyapunov's central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...

* Lyapunov vector

Selected publications

* 1884, ''On the stability of ellipsoidal figures of equilibrium of a rotating fluid'' (in Russian) Published in ''Bulletin 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''

Notes

References

* Reviewed in detail by M. C. Smith: Automatica 1995 vol.3(2), pp. 353–356 * * * * *

External links

* *
Ляпунов Александр Михайлович
at www. mathsoc.spb. ru (in Russian)

at www.spbu. ru (in Russian)

at www-mechmath. univer. kharkov. ua (in Russian)

Aleksandr M. Lyapunov = Ляпунов Александр Михайлович alive at scholar.google.com (live citations) {{DEFAULTSORT:Lyapunov, Aleksandr 1857 births 1918 suicides 19th-century mathematicians from the Russian Empire Ukrainian mathematicians 20th-century Russian mathematicians Chaos theorists Control theorists Full Members of the Russian Academy of Sciences (1917–1925) Full members of the Saint Petersburg Academy of Sciences People from Yaroslavl Saint Petersburg State University alumni Suicides by firearm in the Soviet Union Suicides by firearm in Ukraine

Aleksandr Alexander is a male given name. The most prominent bearer of the name is Alexander the Great, the king of the Ancient Greek kingdom of Macedonia who created one of the largest empires in ancient history. Variants listed here are Aleksandar, Al ...

1918 deaths