Aleksandr Lyapunov

Encyclopedia

**Aleksandr Mikhailovich Lyapunov**was a Russian

Russians

The Russian people are an East Slavic ethnic group native to Russia, speaking the Russian language and primarily living in Russia and neighboring countries....

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, mechanician

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...

and physicist

Physicist

A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

. His surname is sometimes romanized

Romanization of Russian

Romanization of the Russian alphabet is the process of transliterating the Russian language from the Cyrillic alphabet into the Latin alphabet...

as Ljapunov, Liapunov or Ljapunow.

Lyapunov is known for his development of the stability theory

Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions...

of a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

, as well as for his many contributions to mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

and probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

.

### Early life

Lyapunov was born in YaroslavlYaroslavl

Yaroslavl is a city and the administrative center of Yaroslavl Oblast, Russia, located northeast of Moscow. The historical part of the city, a World Heritage Site, is located at the confluence of the Volga and the Kotorosl Rivers. It is one of the Golden Ring cities, a group of historic cities...

, Russian Empire

Russian Empire

The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...

. His father Mikhail Vasilyevich Lyapunov

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.-References:...

(1820–1868) was an astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

and a head of the Demidovski lyceum

Lyceum

The lyceum is a category of educational institution defined within the education system of many countries, mainly in Europe. The definition varies between countries; usually it is a type of secondary school.-History:...

. His brother, Sergei Lyapunov

Sergei Lyapunov

Sergei Mikhailovich Lyapunov was a Russian composer and pianist.-Life:Lyapunov was born in Yaroslavl in 1859. After the death of his father, Mikhail Lyapunov, when he was about eight, Sergei, his mother, and his two brothers went to live in the larger town of Nizhny Novgorod...

, 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

Ulyanovsk Oblast is a federal subject of Russia . It is located in the Volga Federal District. Its administrative center is the city of Ulyanovsk...

). 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. 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

Nizhny Novgorod , colloquially shortened to Nizhny, is, with the population of 1,250,615, the fifth largest city in Russia, ranking after Moscow, St. Petersburg, Novosibirsk, and Yekaterinburg...

, where he started the third class of the gymnasium

Gymnasium (school)

A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

. He graduated from the gymnasium with distinction in 1876.

### Education

In 1876 Lyapunov entered the Physico-Mathematical faculty of the University of Saint PetersburgSaint Petersburg State University

Saint Petersburg State University is a Russian federal state-owned higher education institution based in Saint Petersburg and one of the oldest and largest universities in Russia....

, but after one month he transferred to the Mathematics department of the university.

Among the Saint Petersburg professors of mathematics were Chebyshev

Pafnuty Chebyshev

Pafnuty Lvovich Chebyshev was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff .-Early years:One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk,...

and his students Aleksandr Nikolaevich Korkin

Aleksandr Korkin

Aleksandr Nikolayevich Korkin was a Russian mathematician. He made contribution to the development of partial differential equations. After Chebyshev, Korkin was the most important initiator of the formation of the Saint Petersburg Mathematical School....

and Yegor Ivanovich Zolotarev

Yegor Ivanovich Zolotarev

Yegor Ivanovich Zolotarev was a Russian mathematician.- Biography :...

. 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. 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 completed his university course in 1880, two years after Andrey Markov

Andrey Markov

Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...

who had also graduated at Saint Petersburg University. Lyapunov would maintain a scientific contact with Markov during all his life.

In 1884 Lyapunov defended his Master's thesis

*On the stability of ellipsoidal forms of equilibrium of rotating fluids*. This 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 publicized in 1885 in the

*Bulletin Astronomique*. It was fully translated in French

French language

French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

in 1904 and attracted the attention of mathematicians, physicists and astronomer

Astronomer

An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

s in Europe.

### Teaching and research

In 1895 Lyapunov became privatdozentPrivatdozent

Privatdozent or Private lecturer is a title conferred in some European university systems, especially in German-speaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...

and was proposed to accept the chair of mechanics at Kharkiv University

Kharkiv University

The University of Kharkiv or officially the Vasyl Karazin Kharkiv National University is one of the major universities in Ukraine, and earlier in the Russian Empire and Soviet Union...

, where he went the same year.

About the initial stay at Kharkiv

Kharkiv

Kharkiv or Kharkov is the second-largest city in Ukraine.The city was founded in 1654 and was a major centre of Ukrainian culture in the Russian Empire. Kharkiv became the first city in Ukraine where the Ukrainian Soviet Socialist Republic was proclaimed in December 1917 and Soviet government was...

, Lyapunov writes in his autobiography

Autobiography

An autobiography is a book about the life of a person, written by that person.-Origin of the term:...

:

His student and collaborator, Vladimir Steklov

Vladimir Steklov

Vladimir Andreevich Steklov was a Soviet/Russian mathematician, mechanician and physicist.Steklov was born in Nizhny Novgorod, Russia. In 1887, he graduated from the Kharkov University, where he was a student of Aleksandr Lyapunov. In 1889–1906 he worked at the Department of Mechanics of this...

, 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 dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

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."

In 1892 Lyapunov defended his doctoral thesis

*The general problem of the stability of motion*. The thesis was defended in Moscow University on September 12, 1892, with Nikolai Zhukovsky and V. B. Mlodzeevski as opponents. This dissertation was, as with the master's, also translated into French. The next year Lyapunov became a full professor at Kharkiv University

Kharkiv University

The University of Kharkiv or officially the Vasyl Karazin Kharkiv National University is one of the major universities in Ukraine, and earlier in the Russian Empire and Soviet Union...

.

### 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, 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 Fourth International Mathematical Congress

International Congress of Mathematicians

The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....

in Rome. He also participated in the publication of Euler's selected works: he was an editor of the volumes 18 and 19.

By the end of June 1917, Lyapunov traveled with his wife to his brother's place in Odessa. Lyapunov's wife was suffering from tuberculosis so they moved following her doctor's orders. She died on October 31, 1918. The same day, Lyapunov shot himself in the head, and three days later he died.

## Work

Lyapunov contributed to several fields, including differential equationDifferential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, potential theory

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

, dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s and probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. 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

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

regarded the boundary value problem of the equation of Laplace

Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...

. In the theory of potential, his work from 1897

*On some questions connected with Dirichlet's*

problemclarified 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 dynamic system. In the theory of probability, he generalised the works of Chebyshev and Markov, and proved the Central Limit Theorem

Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...

problem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

under more general conditions than his predecessors. The method he used for the proof found later 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 in Paris.

## Selected publications

- 1884,
*On the stability of ellipsoidal figures of equilibrium of a rotating fluid*(in Russian) - 1892,
*General problem of the stability of motion*(in Russian) - 1897,
*Sur certaines questions qui se rattachent au problème de Dirichlet* - 1901,
*Novelle 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*

## See also

- Lyapunov stabilityLyapunov stabilityVarious types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...
- Lyapunov functionLyapunov functionIn the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...
- Lyapunov exponentLyapunov exponentIn mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
- Lyapunov's central limit theorem
- Lyapunov's condition
- Lyapunov equation
- Lyapunov fractalLyapunov fractalIn mathematics, Lyapunov fractals are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.A Lyapunov fractal is constructed by mapping the regions of stability and chaotic...
- Lyapunov timeLyapunov timeIn mathematics, the Lyapunov time is the length of time for a dynamical system to become chaotic. The Lyapunov time reflects the limits of the predictability of the system. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor...
- Lyapunov–Malkin theoremLyapunov–Malkin theoremThe Lyapunov–Malkin theorem is a mathematical theorem detailing nonlinear stability of systems.-Theorem:In the system of differential equations,\dot x = Ax + X,\quad\dot y = Y\...

## External links

- 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)
- High Anxieties - The Mathematics of Chaos BBC documentary directed by David MaloneDavid Malone (independent filmmaker)David Malone, author of The Debt Generation, is also director of acclaimed documentaries on philosophy, science and religion originally broadcast in the UK by the BBC and Channel 4.-Work:Malone's work includes...

of which a part looks at Lyapunov's work on turbulence.