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Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
He was born in
Life and work
He was born in Saint-Omer
Saint-Omer (; vls, Sint-Omaars) is a commune and sub-prefecture of the Pas-de-Calais department in France.
It is west-northwest of Lille on the railway to Calais, and is located in the Artois province. The town is named after Saint Audomar, ...
in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse Liouville (née Balland).
Liouville gained admission into the École Polytechnique
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
in 1825 and graduated in 1827. Just like Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
before him, Liouville studied engineering at École des Ponts et Chaussées
École may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* École, Savoi ...
after graduating from the Polytechnique, but opted instead for a career in mathematics. After some years as an assistant at various institutions including the École Centrale Paris
École Centrale Paris (ECP; also known as École Centrale or Centrale) was a French grande école in engineering and science. It was also known by its official name ''École Centrale des Arts et Manufactures''. In 2015, École Centrale Paris mer ...
, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France
The Collège de France (), formerly known as the ''Collège Royal'' or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment (''grand établissement'') in France. It is located in Paris ne ...
in 1850 and a chair in mechanics at the Faculté des Sciences in 1857.
Besides his academic achievements, he was very talented in organisational matters. Liouville founded the ''Journal de Mathématiques Pures et Appliquées
The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. A ...
'' which retains its high reputation up to today, in order to promote other mathematicians' work. He was the first to read, and to recognize the importance of, the unpublished work of Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became a member of the Constituting Assembly in 1848. However, after his defeat in the legislative elections in 1849, he turned away from politics.
Liouville worked in a number of different fields in mathematics, including number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, differential geometry and topology
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, but also mathematical physics
Mathematical physics refers to 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 t ...
and even astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. He is remembered particularly for Liouville's theorem. In number theory, he was the first to prove the existence of transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
s by a construction using continued fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...
s (Liouville number
In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that
:0 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...
s). In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form:
for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
, which was joint work with Charles François Sturm
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was " ...
, and is now a standard procedure to solve certain types of integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
s by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure preserving for a Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
system. In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
as a description of completely integrable systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
. The modern formulation of this is sometimes called the Liouville–Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability.
In 1851, he was elected a foreign member of the Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
. In 1853, he was elected as a member of the American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
.
The crater Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
on the Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
is named after him. So is the Liouville function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function.
Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...
, an important function in number theory.
See also
*List of things named after Joseph Liouville
{{refimprove, date=February 2016
Several concepts from mathematics and physics are named after the French mathematician Joseph Liouville.
*Euler–Liouville equation
* Liouville–Arnold theorem
*Liouville–Bratu–Gelfand equation
* Liouville– ...
* Liouville's theorem (disambiguation) Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville:
* In complex analysis, see Liouville's theorem (complex analysis)
** There is also a related theorem on harmonic functions
* In conformal mappings, s ...
Notes
References
* * * Lutzen J., "Liouville's differential calculus of arbitrary order and its electrodynamical origin", in ''Proc. 19th Nordic Congress Mathematicians''. 1985. Icelandic Mathematical Society, Reykjavik, pp. 149–160.Further reading
*External links
* * {{DEFAULTSORT:Liouville, Joseph École Polytechnique alumni École des Ponts ParisTech alumni Corps des ponts 1809 births 1882 deaths 19th-century French mathematicians Mathematical analysts Members of the French Academy of Sciences Members of the Royal Swedish Academy of Sciences Foreign Members of the Royal Society Members of the Göttingen Academy of Sciences and Humanities