# Joseph-Louis Lagrange

LagrangeJoseph Louis LagrangeJoseph LagrangeGiuseppe Luigi LagrangiaJ. L. LagrangeLagrangianJ.L. LagrangeJosef Louis LagrangeJoseph L. LagrangeLagrange, Joseph
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian Enlightenment Era mathematician and astronomer.wikipedia
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### Mécanique analytique

Lagrange's treatise on analytical mechanics (Mécanique analytique, 4.
Mécanique analytique (1788–89) is a two volume French treatise on analytical mechanics, written by Joseph-Louis Lagrange, and published 101 years following Isaac Newton's Philosophiæ Naturalis Principia Mathematica.

### Lagrange's four-square theorem

every positive integer is the sum of four squaresfour-square theoremanswered
He proved that every natural number is a sum of four squares.
This theorem was proven by Joseph Louis Lagrange in 1770.

### Euler–Lagrange equation

Euler–Lagrange equationsEuler-Lagrange equationLagrange's equation
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals.
It was developed by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange in the 1750s.

### Lagrangian mechanics

LagrangianLagrange's equationsLagrangians
But above all, he is best known for his work on mechanics, where he transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and presented the so-called mechanical "principles" as simple results of the variational calculus.
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

### Celestial mechanics

celestialcelestial dynamicscelestial mechanician
He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
After Newton, Lagrange (25 January 1736–10 April 1813) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points.

### Calculus of variations

variationalvariational calculusvariational methods
Lagrange was one of the creators of the calculus of variations, deriving the Euler–Lagrange equations for extrema of functionals.
Lagrange was influenced by Euler's work to contribute significantly to the theory.

### Lagrange polynomial

Lagrange interpolationLagrange formLagrange polynomials
In calculus, Lagrange developed a novel approach to interpolation and Taylor series.
Although named after Joseph Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring It is also an easy consequence of a formula published in 1783 by Leonhard Euler.

### Lagrangian point

Lagrange pointLagrange pointsLagrangian points
He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points.
The three collinear Lagrange points (L 1, L 2, L 3 ) were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two.

### Differential equation

differential equationsdifferentialsecond-order differential equation
Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations.
Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

### History of the metric system

decimalisationdevelopment of metricationevolved since the adoption of the original metric system
He was significantly involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.
Its members were Jean-Charles de Borda, Joseph-Louis Lagrange, Pierre-Simon Laplace, Gaspard Monge and Nicolas de Condorcet.

### Number theory

number theoristcombinatorial number theorytheory of numbers
He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to )—defining their equivalence relation, showing how to put them in reduced form, etc.

### Évariste Galois

GaloisEvariste GaloisGalois, Évariste
His treatise Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois.
At 15, he was reading the original papers of Joseph-Louis Lagrange, such as the Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.

In 1766, on the recommendation of Swiss Leonhard Euler and French d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences.
Frederick invited Joseph-Louis Lagrange to succeed Leonhard Euler as director; both were world-class mathematicians.

### Sénat conservateur

senatorSenateFrench senator
He was significantly involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.
The first Sénat conservateur included former members of the revolutionary assemblies (François de Neufchâteau, Garat, Lanjuinais), as well as scholars (Monge, Lagrange, Lacépède, Berthollet), philosophers (Cabanis), and even the explorer Bougainville and the painter Vien, member of the Institut.

### Three-body problem

restricted three-body problem3-body problemcircular restricted three-body problem
He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points.
In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant.

### François Daviet de Foncenex

de Foncenexde Foncenex, François Daviet
In this Academy one of his students was François Daviet de Foncenex.
Born in Savoy, he studied in the Scuola di Artilleria di Torino under the professorship of Lagrange, two years younger than him.

### Bureau des Longitudes

astronomical ephemeridesBureau of LongitudeBureau of Longitudes
He was significantly involved in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.

### Isaac Newton

NewtonSir Isaac NewtonNewtonian
Paris: Gauthier-Villars et fils, 1788–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.
The mathematician Joseph-Louis Lagrange said that Newton was the greatest genius who ever lived, and once added that Newton was also "the most fortunate, for we cannot find more than once a system of the world to establish."

### Virtual work

principle of virtual workD'Alembert's form of the principle of virtual workmechanical system
The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face was always turned to the earth, a problem which he treated by the aid of virtual work.
In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved.

### University of Turin

University of TorinoTurinTurin University
He studied at the University of Turin and his favourite subject was classical Latin.
A famous student of this age was Joseph-Louis Lagrange.

### Brook Taylor

Taylor, Brook
This volume also contains the complete solution of the problem of a string vibrating transversely; in this paper he points out a lack of generality in the solutions previously given by Brook Taylor, D'Alembert, and Euler, and arrives at the conclusion that the form of the curve at any time t is given by the equation.
The same work contained the well-known formula known as Taylor's formula, the importance of which remained unrecognized until 1772, when J. L. Lagrange realized its usefulness and termed it "the main foundation of differential calculus".

### Variation of parameters

method of variation of parametersvariation of constantsparameter variation
Lagrange invented the method of solving differential equations known as variation of parameters, applied differential calculus to the theory of probabilities and attained notable work on the solution of equations.
The method of variation of parameters was introduced by the Swiss-born mathematician Leonhard Euler (1707–1783) and completed by the Italian-French mathematician Joseph-Louis Lagrange (1736–1813).

### Tautochrone curve

tautochroneAbel integral equationtautochrone problem
Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions.
Later, the mathematicians Joseph Louis Lagrange and Leonhard Euler provided an analytical solution to the problem.

### Lagrange's theorem (group theory)

Lagrange's theoremLagrange's theorem for subgroupsLagrange's theorem in group theory
The theorem is named after Joseph-Louis Lagrange.

### Pell's equation

Pell equationPell equationsPellian equation
*Lagrange (1766–1769) was the first European to prove that Pell's equation
Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions.