الأربعاء، 8 ديسمبر، 2010

Achievements of Joseph-Louis Lagrange

date of birth : 25 January 1736
date of death :10 April 1813
Nationality :   Italian , French
Fields : Mathematics,Mathematical physics
Known for :Analytical mechanics ,Celestial mechanics,Mathematical analysis and Number theory

Joseph-Louis Lagrange (born Giuseppe Lodovico Lagrangia) was a brilliant man who advanced to become a teen-age Professor shortly after first studying mathematics. He excelled in all fields of analysis and number theory; he made key contributions to the theories of determinants, continued fractions, and many other fields. He developed partial differential equations far beyond those of D. Bernoulli and d'Alembert, developed the calculus of variations far beyond that of the Bernoullis, and developed terminology and notation (e.g. the use of f'(x) and f''(x) for a function's 1st and 2nd derivatives). He proved a fundamental Theorem of Group Theory.

Number Theory



Several of his early papers also deal with questions of number theory.
Lagrange (1766–1769) was the first to prove that Pell's equation x2 − ny2 = 1 has a nontrivial solution in the integers for any non-square natural number n.
He proved the theorem, stated by Bachet without justification, that every positive integer is the sum of four squares, 1770.
He proved Wilson's theorem that if n is a prime, then (n − 1)! + 1 is always a multiple of n, 1771.
His papers of 1773, 1775, and 1777 gave demonstrations of several results enunciated by Fermat, and not previously proved.
His Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms to handle the general problem of when an integer is representable by the form ax2 + by2 + cxy.

Mécanique analytique





Over and above these various papers he composed his great treatise, the Mécanique analytique. In this he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids.


The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalized co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D'Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation. For example, in dynamics of a rigid system he replaces the consideration of the particular problem by the general equation, which is now usually written in the form


where T represents the kinetic energy and V represents the potential energy of the system. He then presented what we now know as the method of Lagrange multipliers—though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action. All the analysis is so elegant that Sir William Rowan Hamilton said the work could only be described as a scientific poem. It may be interesting to note that Lagrange remarked that mechanics was really a branch of pure mathematics analogous to a geometry of four dimensions, namely, the time and the three coordinates of the point in space; and it is said that he prided himself that from the beginning to the end of the work there was not a single diagram. At first no printer could be found who would publish the book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under his supervision in 1788.

Differential calculus and calculus of variations


Lagrange's lectures on the differential calculus at École Polytechnique form the basis of his treatise Théorie des fonctions analytiques, which was published in 1797. This work is the extension of an idea contained in a paper he had sent to the Berlin papers in 1772, and its object is to substitute for the differential calculus a group of theorems based on the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in the Residual Analysis, published in London in 1758. Lagrange believed that he could thus get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. The book is divided into three parts: of these, the first treats of the general theory of functions, and gives an algebraic proof of Taylor's theorem, the validity of which is, however, open to question; the second deals with applications to geometry; and the third with applications to mechanics. Another treatise on the same lines was his Leçons sur le calcul des fonctions, issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers, in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as the starting point for the researches of Cauchy, Jacobi, and Weierstrass.

Prizes and distinctions

  •  He was elected a Fellow of the Royal Society of Edinburgh in 1790, a Fellow of the Royal Society and a foreign member of the Royal Swedish Academy of Sciences in 1806.
  • In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Comte of the Empire.
  • He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris.
  • Lagrange was awarded the 1764 prize of the French Academy of Sciences for his memoir on the libration of the Moon
  • Lagrange is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower when it first opened.
  • . Rue Lagrange in the 5th Arrondissement in Paris is named after him. In Turin, the street where the house of his birth still stands is named via Lagrange. The lunar crater Lagrange also bears his name.

(1736-01-25

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