Joseph-Louis Lagrange
Joseph-Louis Lagrange | |
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Joseph-Louis Lagrange[a] (born Giuseppe Luigi Lagrangia[5][b] or Giuseppe Ludovico De la Grange Tournier;[6][c] 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange[7] or Lagrangia,[8] was an Italian mathematician, physicist and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
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. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. 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.
In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the
Scientific contribution
Lagrange was one of the creators of the
Biography
In appearance he was of medium height, and slightly formed, with pale blue eyes and a colourless complexion. In character he was nervous and timid, he detested controversy, and to avoid it willingly allowed others to take the credit for what he had himself done.
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.
W.W. Rouse Ball[9]
Early years
Firstborn of eleven children as Giuseppe Lodovico Lagrangia, Lagrange was of Italian and French descent.
His father, who had charge of the
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 Edmond Halley from 1693[12] 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. Charles Emmanuel III appointed Lagrange to serve as the "Sostituto del Maestro di Matematica" (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army's early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler. In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D'Antoni, the academy's military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications.[13] In this academy one of his students was François Daviet.[14]
Variational calculus
Lagrange is one of the founders of the
Euler was very impressed with Lagrange's results. It has been stated that "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"; however, this chivalric view has been disputed.[16] Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.
Miscellanea Taurinensia
In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the
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
The third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on the
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 generalised equations of motion, equations which he first formally proved in 1780.
Berlin
Already by 1756,
- It seems to me that Berlin would not be at all suitable for me while M.Euler is there.
In 1766, after Euler left Berlin for Saint Petersburg, Frederick himself wrote to Lagrange expressing the wish of "the greatest king in Europe" to have "the greatest mathematician in Europe" resident at his court. Lagrange was finally persuaded. He spent the next twenty years in Prussia, where he produced a long series of papers published in the Berlin and Turin transactions, and composed his monumental work, the Mécanique analytique. In 1767, he married his cousin Vittoria Conti.
Lagrange was a favourite of the king, who frequently lectured him on the advantages of perfect regularity of life. The lesson was accepted, and Lagrange studied his mind and body as though they were machines, and experimented to find the exact amount of work which he could do before exhaustion. 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 the subject-matter were capable of improvement. He carefully planned his papers before writing them, usually without a single erasure or correction.
Nonetheless, during his years in Berlin, Lagrange's health was rather poor, and that of his wife Vittoria was even worse. She died in 1783 after years of illness and Lagrange was very depressed. In 1786, Frederick II died, and the climate of Berlin became difficult for Lagrange.[10]
Paris
In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and
It was about the same time, 1792, that the unaccountable sadness of his life and his timidity moved the compassion of 24-year-old Renée-Françoise-Adélaïde Le Monnier, daughter of his friend, the astronomer Pierre Charles Le Monnier. She insisted on marrying him and proved a devoted wife to whom he became warmly attached.
In September 1793, the
- It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.[10]
Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time,
Units of measurement
Lagrange was involved in the development of the metric system of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures (la Commission des Poids et Mesures) when he was preparing to escape. After Lavoisier's death in 1794, it was largely Lagrange who influenced the choice of the metre and kilogram units with decimal subdivision, by the commission of 1799.[18] Lagrange was also one of the founding members of the Bureau des Longitudes in 1795.
École Normale
In 1795, Lagrange was appointed to a mathematical chair at the newly established
École Polytechnique
In 1794, Lagrange was appointed professor of the
However, Lagrange does not seem to have been a successful teacher. Fourier, who attended his lectures in 1795, wrote:
- his voice is very feeble, at least in that he does not become heated; he has a very marked Italian accent and pronounces the s like z [...] The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professeurs make amends for it.[21]
Late years
In 1810, Lagrange started a thorough revision of the Mécanique analytique, but he was able to complete only about two-thirds of it before his death in Paris in 1813, in 128 rue du Faubourg Saint-Honoré. Napoleon honoured him with the Grand Croix of the Ordre Impérial de la Réunion just two days before he died. He was buried that same year in the Panthéon in Paris. The inscription on his tomb reads in translation:
JOSEPH LOUIS LAGRANGE. Senator. Count of the Empire. Grand Officer of the Legion of Honour. Grand Cross of the Imperial Order of the Reunion. Member of the Institute and the Bureau of Longitude. Born in Turin on 25 January 1736. Died in Paris on 10 April 1813.
Work in Berlin
Lagrange was extremely active scientifically during the twenty years he spent in Berlin. Not only did he produce his Mécanique analytique, but he contributed between one and two hundred papers to the Academy of Turin, the Berlin Academy, and the French Academy. 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.
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
Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the
Lagrangian mechanics
Part of a series on |
Classical mechanics |
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Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics.
Algebra
The greater number of his papers during this time were, however, contributed to the Prussian Academy of Sciences. Several of them deal with questions in algebra.
- His discussion of representations of integers by quadratic forms (1769) and by more general algebraic forms (1770).
- His tract on the Theory of Elimination, 1770.
- Lagrange's theorem that the order of a subgroup H of a group G must divide the order of G.
- His papers of 1770 and 1771 on the general process for solving an Lagrange resolvents. This method fails to give a general formula for solutions of an equation of degree five and higher because the auxiliary equation involved has a higher degree than the original one. The significance of this method is that it exhibits the already known formulas for solving equations of second, third, and fourth degrees as manifestations of a single principle, and was foundational in Galois theory. The complete solution of a binomial equation (namely an equation of the form ± ) is also treated in these papers.
- In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian. He also proved the expression for the volume of a tetrahedron with one of the vertices at the origin as the one-sixth of the absolute value of the determinant formed by the coordinates of the other three vertices.
Number theory
Several of his early papers also deal with questions of number theory.
- Lagrange (1766–1769) was the first European to prove that Pell's equation x2 − ny2 = 1 has a nontrivial solution in the integers for any non-square natural number n.[22]
- 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 (for any integer n > 1): n is a prime if and only if (n − 1)! + 1 is 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 quadratic formsto handle the general problem of when an integer is representable by the form ax2 + by2 + cxy.
- He made contributions to the theory of continued fractions.
Other mathematical work
There are also numerous articles on various points of
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 was collected in the second edition of Euler's integral calculus which was published in 1794.
Astronomy
Lastly, there are numerous papers on problems in astronomy. Of these the most important are the following:
- Attempting to solve the Lagrangian points.
- 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.
Fundamental treatise
Over and above these various papers he composed his fundamental treatise, the Mécanique analytique.
In this book, 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 generalised 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 x, called generalized coordinates, 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
Work in France
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, relying in particular on the principle of the generality of algebra.
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
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1813 copy of "Theorie des fonctions analytiques"
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Title page to "Theorie des fonctions analytiques"
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Introduction to "Theorie des fonctions analytiques"
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First page of "Theorie des fonctions analytiques"
Infinitesimals
At a later period Lagrange fully embraced 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 Analytique, 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.
Number theory
His Résolution des équations numériques, published in 1798, was also the fruit of his lectures at École Polytechnique. There he gives the method of approximating 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 is
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.
Celestial mechanics
A theory of the
Prizes and distinctions
Euler proposed Lagrange for election to the Berlin Academy and he was elected on 2 September 1756. He was elected a Fellow of the
Lagrange was awarded the 1764 prize of the
Lagrange is one of the
See also
- List of things named after Joseph-Louis Lagrange
- Four-dimensional space
- Gauss's law
- History of the metre
- Lagrange's role in measurement reform
- Seconds pendulum
Notes
- ^ UK: /læˈɡrɒ̃ʒ/,[1] US: /ləˈɡreɪndʒ, ləˈɡrɑːndʒ, ləˈɡrɒ̃ʒ/,[2][3][4] French: [ʒozɛf lwi laɡʁɑ̃ʒ].
- ^ Italian: [dʒuˈzɛppe luˈiːdʒi laˈɡrandʒa].
- ^ Italian: [dʒuˈzɛppe ludoˈviːko de la ˈɡranʒ turˈnje], French: [də la ɡʁɑ̃ʒ tuʁnje].
References
Citations
- ^ "Lagrange, Joseph Louis". Lexico UK English Dictionary. Oxford University Press. Archived from the original on 23 April 2021.
- ^ "Lagrange". Random House Webster's Unabridged Dictionary.
- ^ "Lagrange". The American Heritage Dictionary of the English Language (5th ed.). HarperCollins. Retrieved 6 August 2019.
- ^ "Lagrange". Merriam-Webster.com Dictionary. Retrieved 6 August 2019.
- ^ Joseph-Louis Lagrange, comte de l’Empire, Encyclopædia Britannica
- ^ Angelo Genocchi (1883). "Luigi Lagrange". Il primo secolo della R. Accademia delle Scienze di Torino (in Italian). Accademia delle Scienze di Torino. pp. 86–95. Retrieved 2 January 2014.
- ^ a b c d e f g h Luigi Pepe. "Giuseppe Luigi Lagrange". Dizionario Biografico degli Italiani (in Italian). Enciclopedia Italiana. Retrieved 8 July 2012.
- ^ [1] Encyclopedia of Space and Astronomy.
- A Short Account of the History of Mathematics, 4th ed. pp. 401–412. Complete article online, p.338 and 333: [2]
- ^ a b c d e f Lagrange Archived 25 March 2007 at the Wayback Machine St. Andrew University
- ISBN 978-0-19-504230-6.
Lagrange and Laplace, though of Catholic parentage, were agnostics.
- S2CID 186212029.
- ISBN 0-262-19516-X.
- ISBN 978-1-4357-1633-9.
- .
- MR1264671.
- ISBN 978-0-8176-4075-0.
- ^ Delambre, Jean Baptiste Joseph (1816). "Notice sur la vie et les ouvrages de M. Malus, et de M. le Comte Lagrange". Mémoires de la classe des Sciences mathématiques et physiques de l'Institut de France, Année 1812, Seconde Partie. Paris: Firmin Didot. pp. xxvii–lxxx.
- OCLC 780161317.
- ^ OCLC 490193660.
- ^ Ivor Grattan-Guinness. Convolutions in French Mathematics, 1800–1840. Birkhäuser 1990. Vol. I, p.108. [3]
- ^ Œuvres, t.1, 671–732
- ^ Marco Panza, "The Origins of Analytic Mechanics in the 18th Century", in Hans Niels Jahnke (editor), A History of Analysis, 2003, p. 149
Sources
The initial version of this article was taken from the
- Maria Teresa Borgato; Luigi Pepe (1990), Lagrange, appunti per una biografia scientifica (in Italian), Torino: La Rosa
- Columbia Encyclopedia, 6th ed., 2005, "Lagrange, Joseph Louis."
- W. W. Rouse Ball, 1908, "Joseph Louis Lagrange (1736–1813)" A Short Account of the History of Mathematics, 4th ed. also on Gutenberg
- Chanson, Hubert, 2007, "Velocity Potential in Real Fluid Flows: Joseph-Louis Lagrange's Contribution," La Houille Blanche 5: 127–31.
- Fraser, Craig G., 2005, "Théorie des fonctions analytiques" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 258–76.
- Lagrange, Joseph-Louis. (1811). Mécanique Analytique. Courcier (reissued by ISBN 978-1-108-00174-8)
- Lagrange, J.L. (1781) "Mémoire sur la Théorie du Mouvement des Fluides"(Memoir on the Theory of Fluid Motion) in Serret, J.A., ed., 1867. Oeuvres de Lagrange, Vol. 4. Paris" Gauthier-Villars: 695–748.
- Pulte, Helmut, 2005, "Méchanique Analytique" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 208–24.
- A. Conte; C. Mancinelli; E. Borgi.; L. Pepe, eds. (2013), Lagrange. Un europeo a Torino (in Italian), Torino: Hapax Editore, ISBN 978-88-88000-57-2
External links
- O'Connor, John J.; Robertson, Edmund F., "Joseph-Louis Lagrange", MacTutor History of Mathematics Archive, University of St Andrews
- ScienceWorld.
- Lagrange, Joseph Louis de: The Encyclopedia of Astrobiology, Astronomy and Space Flight
- Clerke, Agnes Mary (1911). . Encyclopædia Britannica. Vol. 16 (11th ed.). pp. 75–78.
- Joseph-Louis Lagrange at the Mathematics Genealogy Project
- The Founders of Classical Mechanics: Joseph Louis Lagrange
- The Lagrange Points
- Derivation of Lagrange's result (not Lagrange's method)
- Lagrange's works (in French) Oeuvres de Lagrange, edited by Joseph Alfred Serret, Paris 1867, digitized by Göttinger Digitalisierungszentrum (Mécanique analytique is in volumes 11 and 12.)
- Joseph Louis de Lagrange – Œuvres complètes Gallica-Math
- Inventaire chronologique de l'œuvre de Lagrange Persee
- Works by Joseph-Louis Lagrange at Project Gutenberg
- Works by or about Joseph-Louis Lagrange at Internet Archive
- Mécanique analytique (Paris, 1811-15)