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{{Short description|Russian mathematician (1857–1918)}}
{{refimprove|date=June 2009}}
{{Infobox Scientist
{{Infobox scientist
| image = Alexander Ljapunow jung.jpg
| name = Aleksandr Mikhailovich Lyapunov
| caption =
| image = Alexander Ljapunow jung.jpg
| birth_date = {{OldStyleDate|6 June|1857|25 May}}
| caption =
| birth_place = [[Yaroslavl]], Russian Empire
| birth_date = {{birth date|1857|6|6}}
| death_date = {{death date and age|1918|11|03|1857|06|06|df=y}}
| birth_place = [[Yaroslavl]], [[Russian Empire]]
| death_place = [[Odessa]], [[Ukrainian People's Republic]]
| death_date = {{death date and age|1918|11|3|1857|6|6}}
| fields = [[Applied mathematics]]
| death_place =
| workplaces = [[Saint Petersburg State University]]<br>[[Russian Academy of Sciences]]<br>[[Kharkov University]]
| residence = [[Russia]]
| alma_mater = [[Saint Petersburg State University]]
| citizenship =
| thesis_title = The general problem of the stability of motion
| nationality = [[Russia]]n
| thesis_year = 1892
| ethnicity =
| fields = [[Applied mathematics]]
| doctoral_advisor = [[Pafnuty Chebyshev]]
| doctoral_students = [[Vladimir Steklov (mathematician)|Vladimir Steklov]]
| workplaces = [[Saint Petersburg State University]]<br />[[Russian Academy of Sciences]]<br />[[Kharkov University]]
| known_for = [[Lyapunov function]]<br>[[Lyapunov stability]]<br>[[Lyapunov exponent]]<br>[[Central limit theorem#Lyapunov CLT|Lyapunov central limit theorem]]<br>[[Lyapunov vector]]<br>[[Qualitative theory of differential equations]]
| alma_mater = [[Saint Petersburg State University]]
| awards =
| doctoral_advisor =
| academic_advisors =
| doctoral_students =
| notable_students =
| known_for = [[Lyapunov function]]
| author_abbrev_bot =
| author_abbrev_zoo =
| influences =
| influenced =
| awards =
| religion =
| signature = <!-- (filename only) -->
| footnotes =
}}
}}
'''Aleksandr Mikhailovich Lyapunov'''{{efn|{{family name explanation|Mikhailovich|Lyapunov|lang=Eastern Slavic}} His surname is variously [[Romanization of Russian|romanized]] as '''Ljapunov''', '''Liapunov''', '''Liapounoff''' or '''Ljapunow'''.}}{{efn|{{lang-rus|Алекса́ндр Миха́йлович Ляпуно́в|p=ɐlʲɪkˈsandr mʲɪˈxajləvʲɪtɕ lʲɪpʊˈnof}}}} (Алекса́ндр Миха́йлович Ляпуно́в, {{OldStyleDate|6 June|1857|25 May}} – 3 November 1918) was a Russian [[mathematician]], [[mechanician]] and [[physicist]]. 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]] of a [[dynamical system]], as well as for his many contributions to [[mathematical physics]] and [[probability theory]].
'''Aleksandr Mikhailovich Lyapunov''' ({{lang-ru|Александр Михайлович Ляпунов}}; {{OldStyleDate|June 6|1857|May 25}} – November 3, 1918) was a [[Russians|Russian]] [[mathematician]], [[mechanician]] and [[physicist]]. His surname is sometimes [[Romanization of Russian|romanized]] as Ljapunov, Liapunov or Ljapunow.


== Biography ==
== Biography ==


=== Early life ===
=== Early life ===
Lyapunov was born in [[Yaroslavl]], [[Russian Empire]]. His father [[Mikhail Lyapunov|Mikhail Vasilyevich Lyapunov]] (1820–1868) was an [[astronomer]] 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]]). 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]], where he started the third class of the [[gymnasium (school)|gymnasium]]. He graduated from the gymnasium with distinction in 1876.{{sfn|Smirnov|1992}}

Lyapunov was born in [[Yaroslavl]], [[Russian Empire]]. His father [[Mikhail Lyapunov|Mikhail Vasilyevich Lyapunov]] (1820-1868) was a well known [[astronomer]] and a head of the Demidovski [[lyceum]]. Because of the reactionary politics of the new university administration, after the dismissal by authorities of old rector of the university, noted mathematician [[Nikolai Ivanovich Lobachevsky|Lobachevsky]], in 1864 he abandoned his scientific career at the [[observatory]] of the [[Kazan State University|University of Kazan]]. He relocated his family to his wife's estate in Simbirsk province (now [[Ulyanovsk Oblast]]), where he devoted his time to the education of his oldest sons, Aleksandr and [[Sergei Lyapunov|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 language|Russian]], [[German language|German]] and [[French language|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 physiologist [[Ivan Mikhailovich Sechenov]]. At his uncle's family Lyapunov studied with his distant cousin Nataliya Rafailovna, who later became his wife. In 1870 his mother moved with her sons to [[Nizhny Novgorod]], where he started to attend the third class of the [[gymnasium (school)|gymnasium]]. He graduated from gymnasium in 1876 with distinction.


=== Education ===
=== Education ===


He studied at the Physico-Mathematical department of the [[Saint Petersburg State University|University of Saint Petersburg]], where he studied with [[Andrei Andreevich Markov|Markov]]. In the beginning he attended lectures in [[chemistry]] by [[Dmitriy Mendeleyev|Mendeleyev]]. But he soon realised that he has more interest in mathematics and after one month transferred to the mathematics department of the university. Despite this he continued to attend Mendeleyev's lectures.
In 1876, Lyapunov entered the Physico-Mathematical department at the [[Saint Petersburg State University|University of Saint Petersburg]], but after one month he transferred to the Mathematics department of the university.


Mathematics was taught at that time by [[Pafnuty Chebyshev|Chebyshev]] and his students [[Aleksandr Korkin|Aleksandr Nikolaevich Korkin]] and [[Yegor Ivanovich Zolotarev]]. Lyapunov wrote his first independent scientific works under the guidance of professor of mechanics, D. K. Bobylev. 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 approaches and developed new rigorous proofs of a few earlier incomplete theorems from hydrostatics. Later the paper was used as a basis for his Ph.D in mathematics.
Among the Saint Petersburg mathematics professors were [[Pafnuty Chebyshev|Chebyshev]] and his students [[Aleksandr Korkin|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]]. 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 ===
He received a Master in [[applied mathematics]] degree in 1880. He was allowed to continue his studies at the university after graduation and prepare for the scientific degree of professor (the second after Ph.D academic degree in Russia).
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 [[Pafnuty Chebyshev|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)'.<ref>{{cite book |last1=A.M. Lyapunov |title=The general problem of the stability of motion |date=1892 |publisher=Kharkov Mathematical Society |location=Kharkov |pages=1–251}}</ref> This led on to his 1892 doctoral thesis ''The general problem of the stability of motion''.{{sfn|Lyapunov|1892}} The thesis was defended in Moscow University on 12 September 1892, with [[Nikolay Yegorovich Zhukovsky|Nikolai Zhukovsky]] and V. B. Mlodzeevski as opponents. In 1908, the Kharkov 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'.
In 1885, Lyapunov became [[privatdozent]] and was proposed to accept the chair of mechanics at [[Kharkov University]], where he went the same year. About the initial stay at [[Kharkov]], Smirnov writes in his biography of Lyapunov:


{{quote|text=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.{{sfn|Smirnov|1992}}}}
In 1884 he got Ph.D in applied mathematics with the thesis ''About the stability of elliptic forms of the equilibrium of rotating fluid (Об устойчивости эллипсоидальных форм равновесия вращающейся жидкости''). This very difficult theme was suggested to him by Chebyshev who already suggested it his other talented students such as Zolotarev and [[Sofia Vasilyevna Kovalevskaya]]. As was said by [[Vladimir Andreevich Steklov]] noted, "Chebyshev saw in the young man such an immense research power, that he had decided to suggest him to research this almost unsolvable problem". Lyapunov had already started to study this [[stability theory|stability]] in his previous student paper and very quickly achieved very important, original results. After his thesis was published his work instantly attracted the attention of mathematicians, physicists and astronomers all over the world.


His student and collaborator, [[Vladimir Steklov (mathematician)|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]]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."{{sfn|Smirnov|1992}}
=== Teaching and research ===


=== Later years ===
In 1885 he was invited as a chair of the department of theoretical mechanics of the [[Kharkiv University]]. He replaced previous chair, professor V. G. Imshenecky, who had been elected as a member of the [[Russian Academy of Science]]s.
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, [[Pafnuty Chebyshev|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 Congress of Mathematicians|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.{{sfn|Smirnov|1992}}
Lyapunov already had five years had experience in teaching theoretical mechanic at the Saint-Petersburg university but this activity consumed a lot of his time. His student and collaborator, [[academician]] [[Vladimir Steklov]], recall his first lecture in the following way: "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 taught to students in lectures by professor Delaryu. I actually lessen this topic the fourth time as I attended 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 what Lyapunov tought us was new to me and I had never seen this important material in any textbook. So boredom usually associated with the lectures disappeared and students were all consumed by the brilliance of the the presentation. Aleksandr Mikhailovich instantly earned tremendous respect of the audience with his power of intellect and a natural gift to explain complex topics so seldom seen in such a young person. He didn't know this of course. But from this day students had shown him special respect. Many even didn't dare to speak to him, to avoid showing their ignorance".


=== Death ===
Lyapunov gave courses of lectures in several subjects such as theoretical mechanics, [[integral]]s of [[differential equation]]s and the [[probability|theory of probability]]. These lectures were never published and they remained only in the notes of students. He covered six areas of theoretical mechanics: [[kinematics]], the dynamics of a particle, the dynamics of systems of particles, the theory of attracting forces, the theory of the deformation of solid bodies, and hydrostatics. He also taught [[analytical mechanics]] between 1887 and 1893 at the [[National Technical University "Kharkiv Polytechnic Institute"|Technological Institute at Kharkiv]].


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.{{sfn|Shcherbakov|1992}} By that time, he was going blind from [[cataracts]].{{sfn|Smirnov|1992}}
In 1892 he was awarded the degree of Professor of Science (the second after PH.D academic degree of Russia) after defending 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 this, Lyapunov became a full professor at [[Kharkiv University]].

After the death of Chebyshev in 1894, Lyapunov moved back the University at Saint Petersburg and become the chair of the department of applied mathematics. He completly devoted himself to research work.


== Work ==
== Work ==
[[File:Aleksandr Lyapunov.jpg|thumb|Aleksandr Lyapunov in 1908]]
Lyapunov contributed to several fields, including [[differential equation]]s, [[potential theory]], [[dynamical system]]s and [[probability 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 of [[mathematical physics]] regarded the boundary value problem of the [[Laplace's equation|equation of Laplace]]. In the theory of potential, his work from 1897 ''On some questions connected with [[Peter Gustav Lejeune Dirichlet|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]] under more general conditions than his predecessors. The method of [[characteristic function]]s he used for the proof later found widespread use in probability theory.{{sfn|Smirnov|1992}}


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 [[French Academy of Sciences|Academy of Sciences]] in Paris.{{sfn|Smirnov|1992}}
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 equilibria and the motion of mechanical systems, the model theory for the stability of uniform turbulent liquid, and particles under the influence of [[gravity]]. His work in the field of [[mathematical physics]] was very important for subsequent advances of this field. His work from 1898 ''About some questions, connected with [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet's]] tasks'' (''О некоторых вопросах, связанных с задачей Дирихле'') contains a study of the properties of [[potential]] around [[electric charge|charges]] and [[dipole]]s, continuously distributed along any surface. His work in this field is in close connection with the work of Steklov. Lyapunov developed many important approximation 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 created the modern theory of the stability of dynamic system. In theory of probability, he generalised the works of Chebyshev and Markov, and proved the [[Central limit theorem]] using more general conditions than his forerunners. The method he used for the proof became 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 December 2, 1900 he was elected as a corresponding member of the [[Russian Academy of Sciences]], and on October 6, 1901 as a full member of the Academy in the field of applied mathematics (the highest academic degree in Russia, awarded only to top scientists).


Lyapunov's impact was significant, and the following mathematical concepts are named after him:
Like many mathematicians Lyapunov preferred to work alone and communicated mainly with few fellow mathematians 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 [[Académie des Sciences|Academy of Sciences]] in [[Paris, France|Paris]].
* [[Lyapunov equation]]
* [[Lyapunov exponent]]
* [[Lyapunov function]]
* [[Lyapunov fractal]]
* [[Lyapunov stability]]
* [[Lyapunov's central limit theorem]]
* [[Lyapunov vector]]


== Selected publications ==
In 1908 he participated at the 4th Mathematical congress in [[Rome]]. He also participated in publication of [[Leonhard Euler|Euler's]] selected works: he was an editor of the volumes 18 and 19.
* 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)
By the end of June 1917, he traveled to Odessa with his brother Boris and his wife, who was suffering from tuberculosis, hoping the southern climate will improve her health. In spite of a very difficult economic situation in the country caused by the Bolshevik revolution, he managed to delivere seven lectures about the form of celestial bodies at the invitation of the Department of Physics and Mathematics of the University of Odessa. The last lecture took place on October 28, 1917. His wife died on October 31, 1917. The same day he shot himself. He was hospitalised and stayed unconscious for a few days before his death.<ref>{{cite web|url=http://www.janthor.com/Lyapunov/explained.html|title=Lyapunov exponents|last=Thor|first=Jan|date=30 April 2004|accessdate=13 June 2009}}</ref>
* 1897, ''Sur certaines questions qui se rattachent au problème de Dirichlet''

* 1901, ''Nouvelle forme du théorème sur la limite de probabilité''
His paper "Problème général de la stabilité du mouvement" (1892) (in French) marks the beginning of stability theory.
* 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''
In 1992, the International Journal of Control (Taylor & Francis) published a centennial issue (volume 55, number 3) in memory of Lyapunov. Subsequently, the company Taylor & Francis published an English translation of Lyapunov's 1892 work on stability of motion, together with a biography and bibliography.
* 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''
== Publications ==

Among others he wrote such works as:

* 1890, ''Concerning the constant rotational motion of rigid bodies in fluids''

Articles, which were published by the Russian Academy of Sciences:

* 1902, "About a series in the theory of linear differential equations"
* 1902, "Researches in the theory of celestial bodies"'
* 1904, "About Clairaut's equation, etc. "
* 1906, "A new form of the theorem on the limit of probability"
* 1906, "About a proposition in the probability theory"


== See also ==
== See also ==
{{div col|colwidth=20em}}

* [[Lyapunov's central limit theorem]]
* [[Central limit theorem#Lyapunov condition|Lyapunov's condition]]
* [[Lyapunov equation]]
* [[Lyapunov equation]]
* [[Lyapunov exponent]]
* [[Lyapunov exponent]]
Line 98: Line 83:
* [[Lyapunov stability]]
* [[Lyapunov stability]]
* [[Lyapunov time]]
* [[Lyapunov time]]
* [[Lyapunov's central limit theorem]]

* [[Central limit theorem#Lyapunov condition|Lyapunov's condition]]
== Bibliography ==
* [[Lyapunov–Malkin theorem]]

* [[Lyapunov–Schmidt reduction]]
A. M. Lyapunov, "The general problem of the stability of motion", Taylor & Francis, London, 1992. (The English translation was done by A. T. Fuller. The biography and bibliography were done by V. I. Smirnov and J. F. Barrett, respectively.) ISBN 0748400621.
{{div col end}}
The theory of stability has been studied extensively by many researchers in the Arabic world Prof. Jammal Hamandoosh and his co-workers Yehya Daboos, and Ayman Maktabi from the central lab of mathematics in Aleppo university.


== Notes ==
== Notes ==
{{reflist|2}}
{{Notelist}}


== References ==
== References ==
{{reflist}}

== Further reading ==
{{refbegin}}
{{refbegin}}
*{{Citation
|last=Lyapunov
|first=A. M.
|year=1892
|title=The general problem of the stability of motion
|location=London
|publisher=Taylor & Francis
|isbn=978-0-7484-0062-1
|translator=A. T. Fuller
}} Reviewed in detail by M. C. Smith: Automatica 1995 vol.3(2), pp.&nbsp;353–356
*{{Citation
|last=Parks
|first=Patrick C.
|year=1992
|title=A. M. Lyapunov's stability theory – 100 years on
|journal=IMA Journal of Mathematical Control & Information
|volume=9
|pages=275–303
|doi=10.1093/imamci/9.4.275
|issue=4
}}
*{{Citation
*{{Citation
|last=Shcherbakov
|last=Shcherbakov
|first=Pavel S.
|first=Pavel S.
|year=2003
|year=1992
|title=Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion
|title=Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion
|journal=Automatica
|journal=Automatica
|volume=28
|volume=28
|issue=5
|issue=5
|pages=865-871
|pages=865–871
|doi=10.1016/0005-1098(92)90140-B
|doi=10.1016/0005-1098(92)90140-B
}}
|ref=harv
*{{Citation
|last=Smirnov
|first=Vladimir Ivanovich
|year=1992
|title=Biography of A. M. Lyapunov
|journal=International Journal of Control
|volume=55
|issue=3
|pages=775–784
|doi=10.1080/00207179208934258
|url=http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap05Extras/LyapBio/lyapbio.pdf
|access-date=2009-10-26
|archive-date=2011-05-31
|archive-url=https://web.archive.org/web/20110531174049/http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap05Extras/LyapBio/lyapbio.pdf
|url-status=dead
}}
*{{Citation
|last=Barrett
|first=J. F.
|year=1992
|title=Bibliography of A. M. Lyapunov's work
|journal=International Journal of Control
|volume=55
|issue=3
|pages=785–790
|doi=10.1080/00207179208934259
}}
}}
*{{Citation
*{{Citation
Line 127: Line 162:
|title=Russian Mathematicians in the 20th Century
|title=Russian Mathematicians in the 20th Century
|publisher=World Scientific
|publisher=World Scientific
|isbn=978-9812383853
|isbn=978-981-238-385-3
|ref=harv
}}
}}
{{refend}}
{{refend}}


== External links ==
== External links ==
{{commons category|Aleksandr Lyapunov}}

* {{MacTutor Biography|id=Lyapunov}}
* {{MacTutor Biography|id=Lyapunov}}
* {{MathGenealogy |id=31015}}
* {{MathGenealogy |id=31015}}
* [http://www.mathsoc.spb.ru/pantheon/lyapunov/b-e.html Ляпунов Александр Михайлович<!-- bot-generated title -->] at www. mathsoc.spb. ru (in Russian)
* [http://www.mathsoc.spb.ru/pantheon/lyapunov/b-e.html Ляпунов Александр Михайлович<!-- bot-generated title -->] at www. mathsoc.spb. ru (in Russian)
* [http://www.spbu.ru/History/275/Chronicle/spbu/Persons/L_yapunov.html Ляпунов Александр Михайлович (1857-1918)<!-- bot-generated title -->] at www.spbu. ru (in Russian)
* [http://www.spbu.ru/History/275/Chronicle/spbu/Persons/L_yapunov.html Ляпунов Александр Михайлович (1857-1918)<!-- bot-generated title -->] {{Webarchive|url=https://web.archive.org/web/20070223213144/http://www.spbu.ru/History/275/Chronicle/spbu/Persons/L_yapunov.html |date=2007-02-23 }} at www.spbu. ru (in Russian)
* [http://www-mechmath.univer.kharkov.ua/theormech/lapunov.html Ляпунов Александр Михайлович<!-- bot-generated title -->] at www-mechmath. univer. kharkov. ua (in Russian)
* [https://web.archive.org/web/20020909205413/http://www-mechmath.univer.kharkov.ua/theormech/lapunov.html Ляпунов Александр Михайлович<!-- bot-generated title -->] at www-mechmath. univer. kharkov. ua (in Russian)
* [https://scholar.google.com/citations?user=8Rj2u78AAAAJ&hl=en ] Aleksandr M. Lyapunov = Ляпунов Александр Михайлович alive at scholar.google.com (live citations)

{{chaos theory}}
{{Fractals}}
{{Authority control}}


{{DEFAULTSORT:Lyapunov, Aleksandr}}
{{DEFAULTSORT:Lyapunov, Aleksandr}}

[[Category:1857 births]]
[[Category:1857 births]]
[[Category:1918 deaths]]
[[Category:1918 deaths]]
[[Category:1918 suicides]]
[[Category:19th-century mathematicians from the Russian Empire]]
[[Category:Ukrainian mathematicians]]
[[Category:20th-century Russian mathematicians]]
[[Category:Chaos theorists]]
[[Category:Control theorists]]
[[Category:Full Members of the Russian Academy of Sciences (1917–1925)]]
[[Category:Full members of the Saint Petersburg Academy of Sciences]]
[[Category:People from Yaroslavl]]
[[Category:People from Yaroslavl]]
[[Category:19th-century mathematicians]]
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[[Category:Saint Petersburg State University alumni]]
[[Category:Chaos theorists]]
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[[Category:Lyapunov family|Aleksandr]]
[[ar:ليابونوف]]
[[de:Alexander Michailowitsch Ljapunow]]
[[es:Aleksandr Liapunov]]
[[fr:Alexandre Liapounov]]
[[it:Aleksandr Michajlovič Ljapunov]]
[[ht:Aleksandr Lyapunov]]
[[nl:Aleksandr Ljapoenov]]
[[ja:アレクサンドル・リャプノフ]]
[[pl:Aleksandr Lapunow]]
[[pt:Aleksandr Lyapunov]]
[[ru:Ляпунов, Александр Михайлович]]
[[sl:Aleksander Mihajlovič Ljapunov]]
[[tr:Aleksandr Lyapunov]]
[[zh:亞歷山大·李亞普諾夫]]

Latest revision as of 12:17, 16 December 2024

Aleksandr Lyapunov
Born6 June [O.S. 25 May] 1857
Yaroslavl, Russian Empire
Died3 November 1918(1918-11-03) (aged 61)
Odessa, Ukrainian People's Republic
Alma materSaint Petersburg State University
Known forLyapunov function
Lyapunov stability
Lyapunov exponent
Lyapunov central limit theorem
Lyapunov vector
Qualitative theory of differential equations
Scientific career
FieldsApplied mathematics
InstitutionsSaint Petersburg State University
Russian Academy of Sciences
Kharkov University
Thesis The general problem of the stability of motion  (1892)
Doctoral advisorPafnuty Chebyshev
Doctoral studentsVladimir Steklov

Aleksandr Mikhailovich Lyapunov[a][b] (Алекса́ндр Миха́йлович Ляпуно́в, 6 June [O.S. 25 May] 1857 – 3 November 1918) was a Russian mathematician, mechanician and physicist. 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 of a dynamical system, as well as for his many contributions to mathematical physics and probability theory.

Biography

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Early life

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Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer 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). 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, where he started the third class of the gymnasium. He graduated from the gymnasium with distinction in 1876.[1]

Education

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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. 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

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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)'.[2] This led on to his 1892 doctoral thesis The general problem of the stability of motion.[3] 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: 'Probleme General de la Stabilite du Mouvement, Par M.A. Liapounoff. Traduit du russe par M.Edouard Davaux'.

In 1885, Lyapunov became privatdozent and was proposed to accept the chair of mechanics at Kharkov University, where he went the same year. About the initial stay at Kharkov, 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 on dynamical 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]

Later years

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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.[1]

Death

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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.[4] By that time, he was going blind from cataracts.[1]

Work

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Aleksandr Lyapunov in 1908

Lyapunov contributed to several fields, including differential equations, potential theory, dynamical systems and probability 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 of mathematical physics 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 under more general conditions than his predecessors. The method of characteristic 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 the Academy of Sciences in Paris.[1]

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

Selected publications

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  • 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

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Notes

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  1. ^ In this name that follows Eastern Slavic naming customs, the patronymic is Mikhailovich and the family name is Lyapunov. His surname is variously romanized as Ljapunov, Liapunov, Liapounoff or Ljapunow.
  2. ^ Russian: Алекса́ндр Миха́йлович Ляпуно́в, IPA: [ɐlʲɪkˈsandr mʲɪˈxajləvʲɪtɕ lʲɪpʊˈnof]

References

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  1. ^ a b c d e f g Smirnov 1992.
  2. ^ A.M. Lyapunov (1892). The general problem of the stability of motion. Kharkov: Kharkov Mathematical Society. pp. 1–251.
  3. ^ Lyapunov 1892.
  4. ^ Shcherbakov 1992.

Further reading

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  • Lyapunov, A. M. (1892), The general problem of the stability of motion, translated by A. T. Fuller, London: Taylor & Francis, ISBN 978-0-7484-0062-1 Reviewed in detail by M. C. Smith: Automatica 1995 vol.3(2), pp. 353–356
  • Parks, Patrick C. (1992), "A. M. Lyapunov's stability theory – 100 years on", IMA Journal of Mathematical Control & Information, 9 (4): 275–303, doi:10.1093/imamci/9.4.275
  • Shcherbakov, Pavel S. (1992), "Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion", Automatica, 28 (5): 865–871, doi:10.1016/0005-1098(92)90140-B
  • Smirnov, Vladimir Ivanovich (1992), "Biography of A. M. Lyapunov" (PDF), International Journal of Control, 55 (3): 775–784, doi:10.1080/00207179208934258, archived from the original (PDF) on 2011-05-31, retrieved 2009-10-26
  • Barrett, J. F. (1992), "Bibliography of A. M. Lyapunov's work", International Journal of Control, 55 (3): 785–790, doi:10.1080/00207179208934259
  • Sinai, Yakov (2004), Russian Mathematicians in the 20th Century, World Scientific, ISBN 978-981-238-385-3
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