Joseph Louis Lagrange

Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia.

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Lagrange worked for Frederick II, in Berlin, for twenty years.

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It was Lagrange who developed the mean value theorem, solved the isoperimetrical problem, and developed Lagrangian mechanics.

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Early years

He was born (as Giuseppe Luigi 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.

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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.

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Euler recognized the generality of the method adopted, and its superiority to that used by himself; and with rare 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.

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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.

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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.

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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 (x?n + 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.

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Health suffers

In 1761 Lagrange stood without a rival as the foremost mathematician living; but the unceasing labour of the preceding nine years had seriously affected his health, and the doctors refused to be responsible for his reason or life unless he would take rest and exercise. Although his health was temporarily restored his nervous system never quite recovered its tone, and henceforth he constantly suffered from attacks of severe melancholy.

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Middle years

Moon's motion

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.

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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 M?canique 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.

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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.

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Treatises

His mental activity during these twenty years was amazing. Not only did he produce his splendid M?canique analytique, but he contributed between one and two hundred papers to the Academies of Berlin, Turin, and Paris. 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.

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First, his contributions to the fourth and fifth volumes, 1766–1773, 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, 1784–1785, 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.

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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 French Academy, and in each case the prize was awarded to him.

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Lagrangian mechanics

Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics.

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Algebra

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

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• 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 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.

Theory of numbers

Several of his early papers also deal with questions connected with the neglected but singularly fascinating subject of the theory of numbers. Among these are the following:

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• 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.
• And, lastly, his method for determining the factors of numbers of the form $x^2\; +\; ay^2.$

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.

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During the years from 1772 to 1785 he contributed a long series of papers which created the science of differential equations, at any rate as far as partial differential equations are concerned. I do not think that any previous writer had done anything beyond considering equations of some particular form. A large part of these results were collected in the second edition of Euler's integral calculus which was published in 1794.

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Lagrange's papers on mechanics require no separate mention here as the results arrived at are embodied in the M?chanique analytique which is described below.

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Astronomy

Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following:

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• Attempting to solving the three-body problem results 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.

M?canique analytique

Over and above these various papers he composed his great treatise, the M?canique 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.

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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

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rac{d}{dt}

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rac{partial T}{partial dot{ heta}}

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