Aleksandr Lyapunov
   HOME

TheInfoList



OR:

Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, – 3 November 1918) was a Russian
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, mathematical structure, structure, space, Mathematica ...
, mechanician 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 ...
. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. Lyapunov is known for his development of the
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. The heat equation, for example, is a stable partial differ ...
of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, as well as for his many contributions to
mathematical physics Mathematical physics is the development of mathematics, 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 de ...
and
probability theory Probability theory or probability calculus 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 expre ...
.


Biography


Early life

Lyapunov was born in
Yaroslavl Yaroslavl (; , ) 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 confluence of the Volga and the Kotorosl rivers. ...
,
Russian Empire The Russian Empire was an empire that spanned most of northern Eurasia from its establishment in November 1721 until the proclamation of the Russian Republic in September 1917. At its height in the late 19th century, it covered about , roughl ...
. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
employed by the Demidov 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 (now
Ulyanovsk Oblast Ulyanovsk Oblast () is a federal subjects of Russia, federal subject of Russia (an oblast). It is located in the Volga Federal District. Its administrative center is the types of inhabited localities in Russia, city of Ulyanovsk. It has a populat ...
). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of the
physiologist Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a subdiscipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out chemical and ...
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 ( ; rus, links=no, Нижний Новгород, a=Ru-Nizhny Novgorod.ogg, p=ˈnʲiʐnʲɪj ˈnovɡərət, t=Lower Newtown; colloquially shortened to Nizhny) is a city and the administrative centre of Nizhny Novgorod Oblast an ...
, 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, but after one month he transferred to the Mathematics department of the university. Among the Saint Petersburg mathematics professors were 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 1880 Lyapunov received a gold medal for a work on
hydrostatics Hydrostatics is the branch of fluid mechanics that studies fluids at hydrostatic equilibrium and "the pressure in a fluid or exerted by a fluid on an immersed body". The word "hydrostatics" is sometimes used to refer specifically to water and ...
. 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 Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. He was also a strong, close to mas ...
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 Chebyshev as a topic for his masters thesis which he submitted in 1884 with the title ''On 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)'. 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 Kharkov edition was translated to French and republished by the
University of Toulouse The University of Toulouse (, ) is a community of universities and establishments ( ComUE) based in Toulouse, France. Originally it was established in 1229, making it one of the earliest universities to emerge in Europe. Suppressed during the ...
: 'Probleme General de la Stabilite du Mouvement, Par M.A. Liapounoff. Traduit du russe par M.Edouard Davaux'. In 1885, Lyapunov became
privatdozent ''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualifi ...
and was proposed to accept the chair of mechanics at Kharkov University, where he went the same year. About the initial stay at
Kharkov Kharkiv, also known as Kharkov, is the second-largest List of cities in Ukraine, city in Ukraine.
, Smirnov writes in his biography of Lyapunov: 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 on
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
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."


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 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 ODESSA is an American codename (from the German language, German: ''Organisation der ehemaligen SS-Angehörigen'', meaning: Organization of Former SS Members) coined in 1946 to cover Ratlines (World War II aftermath), Nazi underground escape-pl ...
. 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 of the eye. Cataracts often develop slowly and can affect one or both eyes. Symptoms may include faded colours, blurry or double vision, halos around ligh ...
.


Work

Lyapunov contributed to several fields, including differential equations,
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 the two fundamental forces of nature known at the time, namely g ...
,
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s and
probability theory Probability theory or probability calculus 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 expre ...
. 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
mathematical physics Mathematical physics is the development of mathematics, 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 de ...
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) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
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 in Paris. Lyapunov's impact was significant, and the following mathematical concepts are named after him: * Lyapunov equation * 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 ...
* 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 * 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''


See also

* Lyapunov equation * Lyapunov exponent * Lyapunov fractal *
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 ...
*
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 time * Lyapunov's central limit theorem * Lyapunov's condition * Lyapunov–Malkin theorem * Lyapunov–Schmidt reduction


Notes


References


Further reading

* 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 deaths 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