Joseph Liouville
Encyclopedia
Joseph Liouville was a French
mathematician
.
in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris
, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France
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
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 He was the first to read, and to recognize, the importance of the unpublished work of Évariste Galois
which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became member of the Constituting Assembly
in 1848. However, after the defeat in the Assembly elections in 1849, he turned away from politics.
Liouville worked in a number of different fields in mathematics, including number theory
, complex analysis
, differential geometry and topology
, but also mathematical physics
and even astronomy
. He is remembered particularly for Liouville's theorem
, a nowadays rather basic result in complex analysis. In number theory, he was the first to prove the existence of transcendental number
s by a construction using continued fraction
s (Liouville numbers). In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equation
s by developing into eigenfunctions, and the fact (also known as Liouville's theorem
) that time evolution is measure preserving for a Hamiltonian
system. In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems. 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 crater Liouville
on the Moon
is named after him. So is the Liouville function, an important function in number theory.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
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....
.
Life and work
Liouville graduated from the École PolytechniqueÉcole Polytechnique
The École Polytechnique is a state-run institution of higher education and research in Palaiseau, Essonne, France, near Paris. Polytechnique is renowned for its four year undergraduate/graduate Master's program...
in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris
École Centrale Paris
École Centrale Paris is a French university-level institution in the field of engineering. It is also known by its original name École centrale des arts et manufactures, or ECP. Founded in 1829, it is one of the oldest and most prestigious engineering schools in France and has the special status...
, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France
Collège de France
The Collège de France is a higher education and research establishment located in Paris, France, in the 5th arrondissement, or Latin Quarter, across the street from the historical campus of La Sorbonne at the intersection of Rue Saint-Jacques and Rue des Écoles...
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
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. It is published by Elsevier. According to the 2008 Journal Citation Reports, its impact factor is 1.204. Articles are written in English or French.- External...
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
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became member of the Constituting Assembly
National Assembly
National Assembly is either a legislature, or the lower house of a bicameral legislature in some countries. The best known National Assembly, and the first legislature to be known by this title, was that established during the French Revolution in 1789, known as the Assemblée nationale...
in 1848. However, after the defeat in the Assembly elections in 1849, he turned away from politics.
Liouville worked in a number of different fields in mathematics, including number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, differential geometry and topology
Differential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis...
, but also 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 even astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
. He is remembered particularly for Liouville's theorem
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
, a nowadays rather basic result in complex analysis. In number theory, he was the first to prove the existence of transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
s by a construction using continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s (Liouville numbers). In mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s by developing into eigenfunctions, and the fact (also known as Liouville's theorem
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...
) that time evolution is measure preserving for a Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
system. In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems. 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
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...
.
The crater Liouville
Liouville (crater)
Liouville is a small lunar impact crater that is located near the eastern limb of the Moon. It lies to the southeast of the larger crater Dubyago, and was previously designated Dubyago S before being given a name by the IAU....
on the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
is named after him. So is the Liouville function, an important function in number theory.
See also
- Liouville dynamical systemLiouville dynamical systemIn classical mechanics, a Liouville dynamical system is an exactly soluble dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:...
- Liouville's equation
- Liouville's formulaLiouville's formulaIn mathematics, Liouville's formula is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system...
- Liouville's theorem (disambiguation)Liouville's theoremLiouville's theorem has various meanings, all mathematical results named after Joseph Liouville:*In complex analysis, see Liouville's theorem ; there is also a related theorem on harmonic functions....