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Encyclopedia > Joseph Louis Lagrange
Joseph Louis, comte de Lagrange

Joseph Louis Lagrange
Born January 25, 1736(1736-01-25)
Turin, Italy
Died April 10, 1813 (aged 77)
Paris, France
Residence Flag of Italy Italy

Flag of France France
Image File history File links Langrange_portrait. ... is the 25th day of the year in the Gregorian calendar. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ... For other uses, see Turin (disambiguation). ... is the 100th day of the year (101st in leap years) in the Gregorian calendar. ... Year 1813 (MDCCCXIII) was a common year starting on Friday (link will display the full calendar). ... This article is about the capital of France. ... Image File history File links Flag_of_Italy. ... Image File history File links This is a lossless scalable vector image. ...

 Prussia
Nationality Flag of Italy Italian
Flag of France French
Field Mathematics
Mathematical physics
Institutions École Polytechnique
Academic advisor   Leonhard Euler
Notable students   Joseph Fourier
Giovanni Plana
Simeon Poisson
Known for Analytical mechanics
Celestial mechanics
Mathematical analysis
Number theory
Religion Roman Catholic
Note he did not have a doctoral advisor but academic genealogy authorities link his intellectual heritage to Leonhard Euler, who played the equivalent role.

Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. It is said that he was able to write out his papers complete without a single correction required. Before the age of 20 he was professor of geometry at the royal artillery school at Turin. By his mid-twenties he was recognized as one of the greatest living mathematicians because of his papers on wave propagation and the maxima and minima of curves. His greatest work, Mécanique Analytique (Analytical Mechanics) (4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888-89. First Edition: 1788), was a mathematical masterpiece and the basis for all later work in this field. On the recommendation of Euler and D'Alembert, Lagrange succeeded the former as the director of mathematics at the Prussian Academy of Sciences in Berlin. Under the First French Empire, Lagrange was made both a senator and a count; he is buried in the Panthéon. Image File history File links This is a lossless scalable vector image. ... For other uses, see Prussia (disambiguation). ... Image File history File links Flag_of_Italy. ... Image File history File links This is a lossless scalable vector image. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories. ... This article does not cite any references or sources. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Giovanni Antonio Amedeo Plana (November 6, 1781–January 20, 1864) was an Italian astronomer and mathematician. ... Simeon Poisson. ... Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The Roman Catholic Church, most often spoken of simply as the Catholic Church, is the largest Christian church, with over one billion members. ... An academic, or scientific, genealogy, is an attempt to organise a family tree of scientists and scholars according to dissertation supervision relationships. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... is the 25th day of the year in the Gregorian calendar. ... Events January 26 - Stanislaus I of Poland abdicates his throne. ... For other uses, see Turin (disambiguation). ... Kingdom of Sardinia, in 1839: Mainland Piedmont with Savoy, Nice, and Sardinia in the inset. ... is the 100th day of the year (101st in leap years) in the Gregorian calendar. ... Year 1813 (MDCCCXIII) was a common year starting on Friday (link will display the full calendar). ... This article is about the capital of France. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... An astronomer or astrophysicist is a person whose area of interest is astronomy or astrophysics. ... Analysis has its beginnings in the rigorous formulation of calculus. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ... Analytical mechanics is a term used for a refined, highly mathematical form of classical mechanics, constructed from the eighteenth century onwards as a formulation of the subject as founded by Isaac Newton. ... 1788 was a leap year starting on Tuesday (see link for calendar). ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... The Prussian Academy of Sciences (German: ) was an academy established in Berlin on July 11, 1700. ... This article is about the capital of Germany. ... Map of the First French Empire in 1811, with the Empire in dark blue and satellite states in light blue Capital Paris Language(s) French Government Monarchy Emperor  - 1804 - 1814/1815 Napoleon I  - 1814/1815 Napoleon II Legislature Parliament  - Upper house Senate  - Lower house Corps législatif Historical era Napoleonic... The Senate (in French : le Sénat) is the upper house of the Parliament of France. ... A count is a nobleman in most European countries, equivalent in rank to a British earl, whose wife is also still a countess (for lack of an Anglo-Saxon term). ... The Panthéon Interior Dome of the Panthéon Entrance of the Panthéon Voltaires statue and tomb in the crypt of the Panthéon The Panthéon (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ...


It was Lagrange who created the calculus of variations which was later expanded by Weierstrass, solved the isoperimetrical problem on which the variational calculus is based in part, and made some important discoveries on the tautochrone which would contribute substantially to the then newly formed subject. Lagrange established the theory of differential equations, and provided many new solutions and theorems in number theory, including Wilson's theorem. Lagrange's classic Theorie des fonctions analytiques laid some of the foundations of group theory, anticipating Galois. Lagrange developed the mean value theorem which led to a proof of the fundamental theorem of calculus, and a proof of Taylor's theorem. Lagrange also 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. 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 are now known as Lagrangian points. But above all he impressed on mechanics, having transformed Newtonian mechanics into a branch of analysis, Lagrangian mechanics as it is now called, and exhibited the so-called mechanical "principles" as simple results of the variational calculus. Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ... Isoperimetry literally means having an equal perimeter. In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. ... A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ... Group theory is that branch of mathematics concerned with the study of groups. ... Galois at the age of fifteen from the pencil of a classmate. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In mathematics, variation of parameters or variation of constants is a method used to solve inhomogeneous linear ordinary differential equations. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... 1764 was a leap year starting on Sunday (see link for calendar). ... 1766 was a common year starting on Wednesday (see link for calendar). ... Year 1772 was a leap year starting on Wednesday (see link for calendar). ... In celestial mechanics, the Lagrangian points, (also Lagrange point, L-point, or libration point) are the five stationary solutions of the circular restricted three-body problem. ... It has been suggested that this article or section be merged with Classical mechanics. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ...

Contents

Biography

Early years

He was born, of French and Italian descent, as Giuseppe Lodovico Lagrangia in Turin. His father, who had charge of the Kingdom of Sardinia's military chest, was of good social position and wealthy, but before his son grew up he had lost most of his property in speculations, and young Lagrange had to rely on his own abilities for his position. He was educated at the college of Turin, but it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmund Halley which he came across by accident. Alone and unaided he threw himself into mathematical studies; at the end of a year's incessant toil he was already an accomplished mathematician, and was made a lecturer in the artillery school. For other uses, see Turin (disambiguation). ... Kingdom of Sardinia, in 1839: Mainland Piedmont with Savoy, Nice, and Sardinia in the inset. ... Edmond Halley. ...


Letters

The first fruit of Lagrange's labours here was his letter, written when he was still only nineteen, to Leonhard Euler, in which he solved the isoperimetrical problem which for more than half a century had been a subject of discussion. To effect the solution (in which he sought to determine the form of a function so that a formula in which it entered should satisfy a certain condition) he enunciated the principles of the calculus of variations. Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Isoperimetry literally means having an equal perimeter. In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...


Euler recognized the generality of the method adopted, and its superiority to that used by himself; and with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus. The name of this branch of analysis was suggested by Euler. This paper at once placed Lagrange in the front rank of mathematicians then living.


Miscellanea Taurinensia

In 1758, with the aid of his pupils, Lagrange established a society, which was subsequently incorporated as the Turin Academy, and most of his early writings are to be found in the five volumes of its transactions, usually known as the Miscellanea Taurinensia. Many of these are elaborate papers. The first volume contains a paper on the theory of the propagation of sound; in this he indicates a mistake made by Newton, obtains the general differential equation for the motion, and integrates it for motion in a straight line. 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 y = a sin (mx)cdot sin (nt). The article concludes with a masterly discussion of echoes, beats, and compound sounds. Other articles in this volume are on recurring series, probabilities, and the calculus of variations. Year 1758 (MDCCLVIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the 11-day slower Julian calendar). ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... Brook Taylor (August 18, 1685 – November 30, 1731) was an English mathematician. ... Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, a series is often represented as the sum of a sequence of terms. ... Probability is the likelihood that something is the case or will happen. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...


The second volume contains a long paper embodying the results of several papers in the first volume on the theory and notation of the calculus of variations; and he illustrates its use by deducing the principle of least action, and by solutions of various problems in dynamics. The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ...


The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the integral calculus; a solution of Fermat's problem mentioned above, to find a number x which will make (x²n + 1) a square where n is a given integer which is not a square; and the general differential equations of motion for three bodies moving under their mutual attractions. This article deals with the concept of an integral in calculus. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... This article is about the n-body problem in classical mechanics. ...


Health Problems

In 1761, Lagrange stood without a rival as the foremost living mathematician; but the unceasing labor of the preceding nine years had seriously affected his health. Furthermore, his doctors refused to be responsible for his life unless he would rest and exercise, temporarily abandoning the pursuit of further mathematical innovations. Although his health was temporarily restored, his nervous system never quite recovered, and thus, Lagrange constantly suffered from attacks of severe melancholy, which have been hypothesized to be the cause of his death. 1761 was a common year starting on Thursday (see link for calendar). ...


Middle years

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. His solution is especially interesting as containing the germ of the idea of generalized equations of motion, equations which he first formally proved in 1780. 1764 was a leap year starting on Sunday (see link for calendar). ... Not to be confused with Liberation. ... This article is about Earths moon. ... A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. ... 1780 was a leap year starting on Saturday (see link for calendar). ...


Royal court

He now set off on a visit to London, but on the way fell ill at Paris. There he was received with marked honour, and it was with regret that he left the brilliant society of that city to return to his provincial life at Turin. His further stay in the province of Piedmont was, however, short. In 1766 Euler left Berlin, and Frederick the Great wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange accepted the offer and spent the next twenty years in Prussia, where he produced not only the long series of papers published in the Berlin and Turin transactions, but his monumental work, the Mécanique analytique. His residence at Berlin commenced with an unfortunate mistake. Finding most of his colleagues married, and assured by their wives that it was the only way to be happy, he married; his wife soon died, but the union was not a happy one. This article is about the capital of England and the United Kingdom. ... This article is about the capital of France. ... 1766 was a common year starting on Wednesday (see link for calendar). ... This article is about the capital of Germany. ... Frederick the Great Frederick II of Prussia (Friedrich der Große, Frederick the Great, January 24, 1712 – August 17, 1786) was the Hohenzollern king of Prussia 1740–86. ... For other uses, see Prussia (disambiguation). ...


Lagrange was a favourite of the king, who used frequently to discourse to him on the advantages of perfect regularity of life. The lesson went home, and thenceforth Lagrange studied his mind and body as though they were machines, and found by experiment the exact amount of work which he was able to do without breaking down. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or in the subject-matter were capable of improvement. He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.


Treatises

His mental activity during these twenty years was amazing. Not only did he produce his splendid Mécanique analytique, but he contributed between one and two hundred papers to the Academy of Turin, the Royal Academy of Berlin, and the Académie française. Some of these are really treatises, and all without exception are of a high order of excellence. Except for a short time when he was ill he produced on average about one paper a month. Of these, note the following as amongst the most important. The Académie française In the French educational system an académie LAcadémie française, or the French Academy, is the pre-eminent French learned body on matters pertaining to the French language. ...


First, his contributions to the fourth and fifth volumes, 17661773, of the Miscellanea Taurinensia; of which the most important was the one in 1771, in which he discussed how numerous astronomical observations should be combined so as to give the most probable result. And later, his contributions to the first two volumes, 17841785, of the transactions of the Turin Academy; to the first of which he contributed a paper on the pressure exerted by fluids in motion, and to the second an article on integration by infinite series, and the kind of problems for which it is suitable. 1766 was a common year starting on Wednesday (see link for calendar). ... 1773 was a common year starting on Friday (see link for calendar). ... 1771 was a common year starting on Tuesday (see link for calendar). ... For other uses, see Astronomy (disambiguation). ... 1784 was a leap year starting on Thursday (see link for calendar). ... 1785 was a common year starting on Saturday (see link for calendar). ... In mathematics, a series is a sum of a sequence of terms. ...


Most of the papers sent to Paris were on astronomical questions, and among these one ought to particularly mention his paper on the Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on the secular equation of the Moon in 1773, and his treatise on cometary perturbations in 1778. These were all written on subjects proposed by the Académie française, and in each case the prize was awarded to him. This siliqua of Jovian, ca 363, celebrates his fifth year of reign, as a good omen. ... Year 1772 was a leap year starting on Wednesday (see link for calendar). ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial or secular equation. ... 1773 was a common year starting on Friday (see link for calendar). ... The Académie française In the French educational system an académie LAcadémie française, or the French Academy, is the pre-eminent French learned body on matters pertaining to the French language. ...


Lagrangian mechanics

Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics. Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ...

Image File history File links Copyright-problem_paste. ... This page is about computer text editing. ...

Algebra

The greater number of his papers during this time were, however, contributed to the Royal Berlin Academy. Several of them deal with questions on algebra. In particular: This article is about the branch of mathematics. ...

  • His discussion of the solution in integers of indeterminate quadratics, 1769, and generally of indeterminate equations, 1770.
  • His tract on the theory of elimination, 1770.
  • His theorem that the order of a subgroup H of a group G must divide the order of G.
  • His papers on the general process for solving an algebraic equation of any degree, 1770 and 1771; this method fails for equations of an order above the fourth, because it then involves the solution of an equation of higher dimensions than the one proposed, but it gives all the solutions of his predecessors as modifications of a single principle.
  • The complete solution of a binomial equation of any degree; this is contained in the papers last mentioned.
  • Lastly, in 1773, his treatment of determinants of the second and third order, and of invariants.

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... 1769 was a common year starting on Sunday (see link for calendar). ... For the village in Queensland, see 1770, Queensland. ... In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. ... Lagranges theorem, in the mathematics of group theory, states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G. It is named after Joseph Lagrange. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... 1773 was a common year starting on Friday (see link for calendar). ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Number Theory

Several of his early papers also deal with questions of number theory. Among these are the following:

  • His proof of the theorem that every positive integer which is not a square can be expressed as the sum of two, three or four integral squares, 1770.
  • His proof of 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, which give the demonstrations of several results enunciated by Fermat, and not previously proved.
  • He was the first to prove that Pell's equation always has a solution.
  • And, lastly, his method for determining the factors of numbers of the form x2 + ay2.

Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ... Year 1775 (MDCCLXXV) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the 11-day slower Julian calendar). ... Year 1777 (MDCCLXXVII) was a common year starting on Wednesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Sunday of the 11-day slower Julian calendar). ... Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

Miscellaneous

There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical forms. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... 1792 was a leap year starting on Sunday (see link for calendar). ... 1793 was a common year starting on Tuesday (see link for calendar). ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... Generally, in mathematics, a canonical form is a function that is written in the most standard, conventional, and logical way. ...


During the years from 1772 to 1785, he contributed a long series of papers which created the science of partial differential equations. A large part of these results were collected in the second edition of Euler's integral calculus which was published in 1794. Year 1772 was a leap year starting on Wednesday (see link for calendar). ... 1785 was a common year starting on Saturday (see link for calendar). ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... 1794 was a common year starting on Wednesday (see link for calendar). ...


He made contributions to the theory of continued fractions. In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...


Astronomy

Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following: For other uses, see Astronomy (disambiguation). ...

  • Attempting to solve the three-body problem resulting in the discovery of Lagrangian points, 1772
  • On the attraction of ellipsoids, 1773: this is founded on Maclaurin's work.
  • On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777.
  • On the motion of the nodes of a planet's orbit, 1774.
  • On the stability of the planetary orbits, 1776.
  • Two papers in which the method of determining the orbit of a comet from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject.
  • His determination of the secular and periodic variations of the elements of the planets, 1781-1784: the upper limits assigned for these agree closely with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted.
  • Three papers on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it.

This article is about the n-body problem in classical mechanics. ... A contour plot of the effective potential (the Hills Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... Colin Maclaurin Colin Maclaurin (February, 1698 - June 14, 1746) was a Scottish mathematician. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... Comet Hale-Bopp Comet West For other uses, see Comet (disambiguation). ... The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two ideal masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction. ... Urbain Le Verrier. ...

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. For other uses, see Mechanic (disambiguation). ...


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

 frac{d}{dt} frac{partial T}{partial dot{theta}} + frac{partial V}{partial theta} = 0.

T for the Kinetic energy and V for the Potential energy. 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. The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ...


Later years

France

In 1786, Frederick died, and Lagrange, who had found the climate of Berlin trying, gladly accepted the offer of Louis XVI to migrate to Paris. He received similar invitations from Spain and Naples. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences, which later became part of the National Institute. At the beginning of his residence in Paris he was seized with an attack of the melancholy, and even the printed copy of his Mécanique on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the French revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed. 1786 was a common year starting on Sunday (see link for calendar). ... Louis XVI, born Louis-Auguste de France (23 August 1754 – 21 January 1793) ruled as King of France and Navarre from 1774 until 1791, and then as King of the French from 1791 to 1792. ... For other uses, see Naples (disambiguation). ... Louis XIV visiting the Académie in 1671 The French Academy of Sciences (Académie des sciences) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. ... The Institut de France (French Institute) is a French learned society, grouping five académies, the most famous of which is probably the Académie française. ... The French Revolution (1789–1815) was a period of political and social upheaval in the political history of France and Europe as a whole, during which the French governmental structure, previously an absolute monarchy with feudal privileges for the aristocracy and Catholic clergy, underwent radical change to forms based on...


It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of a young girl who insisted on marrying him, and proved a devoted wife to whom he became warmly attached. Although the decree of October 1793 that ordered all foreigners to leave France specifically exempted him by name, he was preparing to escape when he was offered the presidency of the commission for the reform of weights and measures. The choice of the units finally selected was largely due to him, and it was mainly owing to his influence that the decimal subdivision was accepted by the commission of 1799. In 1795, Lagrange was one of the founding members of the Bureau des Longitudes. 1792 was a leap year starting on Sunday (see link for calendar). ... For other uses, see October (disambiguation). ... 1793 was a common year starting on Tuesday (see link for calendar). ... 1799 was a common year starting on Tuesday (see link for calendar). ... The Bureau des Longitudes is a French scientific institution, founded by decree of June 25, 1795 and charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation. ...


Though Lagrange had determined to escape from France while there was yet time, he was never in any danger; and the different revolutionary governments (and at a later time, Napoleon) loaded him with honours and distinctions. A striking testimony to the respect in which he was held was shown in 1796 when the French commissary in Italy was ordered to attend in full state on Lagrange's father, and tender the congratulations of the republic on the achievements of his son, who "had done honour to all mankind by his genius, and whom it was the special glory of Piedmont to have produced." It may be added that Napoleon, when he attained power, warmly encouraged scientific studies in France, and was a liberal benefactor of them. Napoléon I, Emperor of the French (born Napoleone di Buonaparte, changed his name to Napoléon Bonaparte)[1] (15 August 1769; Ajaccio, Corsica – 5 May 1821; Saint Helena) was a general during the French Revolution, the ruler of France as First Consul (Premier Consul) of the French Republic from... Year 1796 (MDCCXCVI) was a leap year starting on Friday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Monday of the 11-day slower Julian calendar). ... “Italian Republic” redirects here. ...


École normale

In 1795, Lagrange was appointed to a mathematical chair at the newly-established École normale, which enjoyed only a brief existence of four months. His lectures here were quite elementary, and contain nothing of any special importance, but they were published because the professors had to "pledge themselves to the representatives of the people and to each other neither to read nor to repeat from memory," and the discourses were ordered to be taken down in shorthand in order to enable the deputies to see how the professors acquitted themselves. 1795 was a common year starting on Thursday (see link for calendar). ... See also École Normale de Musique de Paris. ...


École Polytechnique

On the establishment of the École Polytechnique in 1797, Lagrange was made a professor; and his lectures there are described by mathematicians who had the good fortune to be able to attend them, as almost perfect both in form and matter. Beginning with the merest elements, he led his hearers on until, almost unknown to themselves, they were themselves extending the bounds of the subject: above all he impressed on his pupils the advantage of always using general methods expressed in a symmetrical notation. This article does not cite any references or sources. ... 1797 (MDCCXCVII) was a common year starting on Sunday (see link for calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 11-day-slower Julian calendar). ...


His lectures on the differential calculus form the basis of his 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. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass. John Landen (23 January 1719 - 15 January 1790) was an English mathematician, He was born at Peakirk near Peterborough in Northamptonshire, and died at Milton in the same county. ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... 1804 was a leap year starting on Sunday (see link for calendar). ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Karl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (December 10, 1804 - February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...


Infinitesimals

At a later period Lagrange reverted to the use of infinitesimals in preference to founding the differential calculus on the study of algebraic forms; and in the preface to the second edition of the Mécanique, which was issued in 1811, he justifies the employment of infinitesimals, and concludes by saying that: :"when we have grasped the spirit of the infinitesimal method, and have verified the exactness of its results either by the geometrical method of prime and ultimate ratios, or by the analytical method of derived functions, we may employ infinitely small quantities as a sure and valuable means of shortening and simplifying our proofs." Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... 1811 was a common year starting on Tuesday (see link for calendar). ...


Continued fractions

His "Résolution des équations numériques", published in 1798, was also the fruit of his lectures at the Polytechnic. In this he gives the method of approximating to the real roots of an equation by means of continued fractions, and enunciates several other theorems. In a note at the end he shows how Fermat's little theorem that Year 1798 (MDCCXCVIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday of the 11-day slower Julian calendar). ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by...

ap−1 − 1 ≡ 0 (mod p)

where p is a prime and a is prime to p, may be applied to give the complete algebraic solution of any binomial equation. He also here explains how the equation whose roots are the squares of the differences of the roots of the original equation may be used so as to give considerable information as to the position and nature of those roots.


The theory of the planetary motions had formed the subject of some of the most remarkable of Lagrange's Berlin papers. In 1806 the subject was reopened by Poisson, who, in a paper read before the French Academy, showed that Lagrange's formulae led to certain limits for the stability of the orbits. Lagrange, who was present, now discussed the whole subject afresh, and in a letter communicated to the Academy in 1808 explained how, by the variation of arbitrary constants, the periodical and secular inequalities of any system of mutually interacting bodies could be determined. ... 1806 was a common year starting on Wednesday (see link for calendar). ... Simeon Poisson. ... Year 1808 (MDCCCVIII) was a leap year starting on Friday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Wednesday of the 12-day slower Julian calendar). ...


Death

Lagrange's tomb in the crypt of the Panthéon.
Lagrange's tomb in the crypt of the Panthéon.

In 1808, Napoleon made Lagrange a Grand Officer of the Legion of Honour and a Comte of the Empire. In 1810, Lagrange commenced a thorough revision of the Mécanique analytique, but he was able to complete only about two-thirds of it before his death. He was awarded the Grand Croix of the Ordre Impérial de la Réunion in 1813, a week before his death in Paris. He was buried that same year in the Panthéon in Paris. The French inscription on his tomb there reads: Image File history File links Size of this preview: 800 × 600 pixelsFull resolution (1600 × 1200 pixel, file size: 555 KB, MIME type: image/jpeg) Lagranges tomb at the Pantheon, Paris, France. ... Image File history File links Size of this preview: 800 × 600 pixelsFull resolution (1600 × 1200 pixel, file size: 555 KB, MIME type: image/jpeg) Lagranges tomb at the Pantheon, Paris, France. ... The Panthéon Interior Dome of the Panthéon Entrance of the Panthéon Voltaires statue and tomb in the crypt of the Panthéon The Panthéon (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ... Year 1808 (MDCCCVIII) was a leap year starting on Friday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Wednesday of the 12-day slower Julian calendar). ... For other uses, see Napoleon (disambiguation). ... French Legion of Honor The Légion dhonneur (in Legion of Honor (AmE) or Legion of Honour (ComE)) is an Order of Chivalry awarded by the President of France. ... For article about famous philosopher and sociologist, see Auguste Comte Comte is a title of French nobility. ... 1810 was a common year starting on Monday (see link for calendar). ... Year 1813 (MDCCCXIII) was a common year starting on Friday (link will display the full calendar). ... The Panthéon Interior Dome of the Panthéon Entrance of the Panthéon Voltaires statue and tomb in the crypt of the Panthéon The Panthéon (Latin Pantheon[1], from Greek Pantheon, meaning All the Gods) is a building in the Latin Quarter in Paris, France. ...

JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of Réunion. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813.

A street in Paris is named rue Lagrange in his honour. In Turin, the street where the house of his birth still stands is also named via Lagrange.


Appearance

He was of medium height and slightly formed, with pale blue eyes and a colorless complexion. He was nervous and timid, he detested controversy, and, to avoid it, willingly allowed others to take credit for what he had done himself.


Pure mathematics

Lagrange's interests were essentially those of a student of pure mathematics: he sought and obtained far-reaching abstract results, and was content to leave the applications to others. Indeed, no inconsiderable part of the discoveries of his great contemporary, Laplace, consists of the application of the Lagrangian formulae to the facts of nature; for example, Laplace's conclusions on the velocity of sound and the secular acceleration of the Moon are implicitly involved in Lagrange's results. The only difficulty in understanding Lagrange is that of the subject-matter and the extreme generality of his processes; but his analysis is "as lucid and luminous as it is symmetrical and ingenious." To meet Wikipedias quality standards, this article or section may require cleanup. ...


A recent writer speaking of Lagrange says truly that he took a prominent part in the advancement of almost every branch of pure mathematics. Like Diophantus and Fermat, he possessed a special genius for the theory of numbers, and in this subject he gave solutions of many of the problems which had been proposed by Fermat, and added some theorems of his own. He created the calculus of variations. To him, too, the theory of differential equations is indebted for its position as a science rather than a collection of ingenious artifices for the solution of particular problems. To the calculus of finite differences he contributed the formula of interpolation which bears his name (although the formula was known to Euler). But above all he impressed on mechanics (which it will be remembered he considered a branch of pure mathematics) that generality and completeness towards which his labours invariably tended. Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ...


Notes

References

See also: wikipedia:history One of Wikipedias public domain resources! There are a series of articles transcribed by Dr. David R. Wilkins (dwilkins@maths. ... Walter William Rouse Ball (1850 August 14–1925 April 4) was a Brtish mathematician, and a fellow at Trinity College, Cambridge from 1878 to 1905. ...

See also

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A contour plot of the effective potential (the Hills Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... Fig. ... In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. ... In mathematics, Lagranges theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange: Lagranges theorem in group theory Lagranges theorem in number theory Lagranges four-square theorem, which states that every positive integer can be expressed as the sum of four squares... Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. ... In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. ... Lagrange is a lunar crater that is attached to the northwestern rim of Piazzi crater. ...

External links

Persondata
NAME Lagrange, Joseph
ALTERNATIVE NAMES
SHORT DESCRIPTION Mathematician, Mathematical physicist
DATE OF BIRTH January 25, 1736
PLACE OF BIRTH Turin, Italy
DATE OF DEATH April 10, 1813
PLACE OF DEATH Paris, France

  Results from FactBites:
 
Joseph Louis Lagrange - LoveToKnow 1911 (3346 words)
JOSEPH LOUIS LAGRANGE (1736-1813), French mathematician, was born at Turin, on the 25th of January 1736.
His father, Joseph Louis Lagrange, married Maria Theresa Gros, only daughter of a rich physician at Cambiano, and had by her eleven children, of whom only the eldest (the subject of this notice) and the youngest survived infancy.
The prize was again awarded to Lagrange; and he earned the same distinction with essays on the problem of three bodies in 1772, on the secular equation of the moon in 1774, and in 1778 on the theory of cometary perturbations.
Lagrange biography (2944 words)
Lagrange was the eldest of their 11 children but one of only two to live to adulthood.
Lagrange was a major contributor to the first volumes of the Mélanges de Turin volume 1 of which appeared in 1759, volume 2 in 1762 and volume 3 in 1766.
Lagrange was greeted warmly by most members of the Academy and he soon became close friends with Lambert and Johann(III) Bernoulli.
  More results at FactBites »

 
 

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