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95-522: [REDACTED] Look up poisson in Wiktionary, the free dictionary. Poisson may refer to: People [ edit ] Siméon Denis Poisson , French mathematician Eric Poisson , Canadian physicist Places [ edit ] Poissons , a commune of Haute-Marne, France Poisson, Saône-et-Loire , a commune of Saône-et-Loire, France Other uses [ edit ] Poisson (surname) ,

190-1126: A b f ( x , y ( x ) , y ′ ( x ) , . . . , y ( n ) ( x ) ) d x , {\displaystyle S=\int \limits _{a}^{b}f\left(x,y(x),y'(x),...,y^{(n)}(x)\right)\,dx,} then y {\displaystyle y} must satisfy the Euler–Poisson equation, ∂ f ∂ y − d d x ( ∂ f ∂ y ′ ) + . . . + ( − 1 ) n d n d x n [ ∂ f ∂ y ( n ) ] = 0. {\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)+...+(-1)^{n}{\frac {d^{n}}{dx^{n}}}\left[{\frac {\partial f}{\partial y^{(n)}}}\right]=0.} Poisson's Traité de mécanique (2 vols. 8vo, 1811 and 1833)

285-734: A Conti from the noble Roman family. Lagrange's father, Giuseppe Francesco Lodovico, was a doctor in Law at the University of Torino , while his mother was the only child of a rich doctor of Cambiano , in the countryside of Turin . He was raised as a Roman Catholic (but later on became an agnostic ). His father, who had charge of the King 's military chest and was Treasurer of the Office of Public Works and Fortifications in Turin, should have maintained

380-498: A répétiteur at the École Polytechnique. Notwithstanding his many official duties, he found time to publish more than three hundred works, several of them extensive treatises, and many of them memoirs dealing with the most abstruse branches of pure mathematics, applied mathematics , mathematical physics , and rational mechanics. ( Arago attributed to him the quote, "Life is good for only two things: doing mathematics and teaching it." ) A list of Poisson's works, drawn up by himself,

475-672: A French surname Poisson (crater) , a lunar crater named after Siméon Denis Poisson The French word for fish See also [ edit ] Adolphe-Poisson Bay , a body of water located to the southwest of Gouin Reservoir, in La Tuque, Mauricie, Quebec Poisson distribution , a discrete probability distribution named after Siméon Denis Poisson Poisson's equation , a partial differential equation named after Siméon Denis Poisson List of things named after Siméon Denis Poisson Poison (disambiguation) Topics referred to by

570-561: A brief historical insight into his reason for proposing undecimal or Base 11 as the base number for the reformed system of weights and measures. The lectures 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" ["Les professeurs aux Écoles Normales ont pris, avec les Représentants du Peuple, et entr'eux l'engagement de ne point lire ou débiter de mémoire des discours écrits" ]. The discourses were ordered and taken down in shorthand to enable

665-481: 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. 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,

760-519: A custom of visiting him in his room after an unusually difficult lecture to hear him repeat and explain it. He was made deputy professor ( professeur suppléant ) in 1802, and, in 1806 full professor succeeding Jean Baptiste Joseph Fourier , whom Napoleon had sent to Grenoble . In 1808 he became astronomer to the Bureau des Longitudes ; and when the Faculté des sciences de Paris was instituted in 1809 he

855-474: A deluge of published papers attempting to explain the phenomenon. Ampère's law and the Biot-Savart law were quickly deduced. The science of electromagnetism was born. Poisson was also investigating the phenomenon of magnetism at this time, though he insisted on treating electricity and magnetism as separate phenomena. He published two memoirs on magnetism in 1826. By the 1830s, a major research question in

950-421: A good social position and wealth, but before his son grew up he had lost most of his property in speculations. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the University of Turin and his favourite subject was classical Latin. At first, he had no great enthusiasm for mathematics, finding Greek geometry rather dull. It

1045-510: 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. Lastly, there are numerous papers on problems in astronomy . Of these the most important are the following: Over and above these various papers he composed his fundamental treatise, the Mécanique analytique . In this book, he lays down

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1140-472: A novel approach to interpolation and Taylor's theorem . He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter's satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points . Lagrange is best known for transforming Newtonian mechanics into a branch of analysis, Lagrangian mechanics . He presented

1235-487: A point P {\displaystyle P} . The integration runs over the whole space. Poisson's equation was first published in the Bulletin de la société philomatique (1813). Poisson's two most important memoirs on the subject are Sur l'attraction des sphéroides (Connaiss. ft. temps, 1829), and Sur l'attraction d'un ellipsoide homogène (Mim. ft. l'acad., 1835). Poisson discovered that Laplace's equation

1330-410: A problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. In this academy one of his students was François Daviet . Lagrange is one of the founders of the calculus of variations . Starting in 1754, he worked on the problem of the tautochrone , discovering a method of maximizing and minimizing functionals in

1425-456: A single erasure or correction. W.W. Rouse Ball Firstborn of eleven children as Giuseppe Lodovico Lagrangia , Lagrange was of Italian and French descent. His paternal great-grandfather was a French captain of cavalry, whose family originated from the French region of Tours . After serving under Louis XIV , he had entered the service of Charles Emmanuel II , Duke of Savoy , and married

1520-601: A successful teacher. Fourier , who attended his lectures in 1795, wrote: 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

1615-480: A way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler's earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis . Euler

1710-410: Is a constant of motion , then it must satisfy Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier ; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia , was an Italian mathematician , physicist and astronomer , later naturalized French. He made significant contributions to

1805-526: Is a continuous function and if for r → ∞ {\displaystyle r\rightarrow \infty } (or if a point 'moves' to infinity ) a function ϕ {\displaystyle \phi } goes to 0 fast enough, the solution of Poisson's equation is the Newtonian potential where r {\displaystyle r} is a distance between a volume element d V {\displaystyle dV} and

1900-512: Is different from Wikidata All article disambiguation pages All disambiguation pages Sim%C3%A9on Denis Poisson Poisson was born in Pithiviers , Loiret district in France, the son of Siméon Poisson, an officer in the French army. In 1798, he entered the École Polytechnique in Paris as first in his year, and immediately began to attract the notice of the professors of

1995-642: Is extremized if it satisfies the Euler–Lagrange equations ∂ f ∂ y − d d x ( ∂ f ∂ y ′ ) = 0. {\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)=0.} But if S {\displaystyle S} depends on higher-order derivatives of y ( x ) {\displaystyle y(x)} , that is, if S = ∫

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2090-437: Is given at the end of Arago's biography. All that is possible is a brief mention of the more important ones. It was in the application of mathematics to physics that his greatest services to science were performed. Perhaps the most original, and certainly the most permanent in their influence, were his memoirs on the theory of electricity and magnetism , which virtually created a new branch of mathematical physics. Next (or in

2185-417: 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

2280-506: Is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as q ˙ i = [ q i , H ] {\displaystyle {\dot {q}}_{i}=[q_{i},H]} and p ˙ i = [ p i , H ] {\displaystyle {\dot {p}}_{i}=[p_{i},H]} . Suppose u {\displaystyle u}

2375-467: Is valid only outside of a solid. A rigorous proof for masses with variable density was first given by Carl Friedrich Gauss in 1839. Poisson's equation is applicable in not just gravitation, but also electricity and magnetism. As the eighteenth century came to a close, human understanding of electrostatics approached maturity. Benjamin Franklin had already established the notion of electric charge and

2470-561: The American Academy of Arts and Sciences , and in 1823 a foreign member of the Royal Swedish Academy of Sciences . The revolution of July 1830 threatened him with the loss of all his honours; but this disgrace to the government of Louis-Philippe was adroitly averted by François Jean Dominique Arago , who, while his "revocation" was being plotted by the council of ministers, procured him an invitation to dine at

2565-484: The Huygens–Fresnel principle and Young's double slit experiment . Poisson studied Fresnel's theory in detail and looked for a way to prove it wrong. Poisson thought that he had found a flaw when he demonstrated that Fresnel's theory predicts an on-axis bright spot in the shadow of a circular obstacle blocking a point source of light, where the particle-theory of light predicts complete darkness. Poisson argued this

2660-489: The Palais-Royal , where he was openly and effusively received by the citizen king, who "remembered" him. After this, of course, his degradation was impossible, and seven years later he was made a peer of France , not for political reasons, but as a representative of French science . As a teacher of mathematics Poisson is said to have been extraordinarily successful, as might have been expected from his early promise as

2755-791: 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. Lagrange was extremely active scientifically during

2850-498: The Prussian Academy of Sciences . Several of them deal with questions in algebra . Several of his early papers also deal with questions of number theory. 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 . During the years from 1772 to 1785, he contributed

2945-439: The calculus of variations . 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 third volume includes the solution of several dynamical problems by means of the calculus of variations; some papers on

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3040-449: The conservation of charge ; Charles-Augustin de Coulomb had enunciated his inverse-square law of electrostatics . In 1777, Joseph-Louis Lagrange introduced the concept of a potential function that can be used to compute the gravitational force of an extended body. In 1812, Poisson adopted this idea and obtained the appropriate expression for electricity, which relates the potential function V {\displaystyle V} to

3135-510: The decimalisation process in Revolutionary France , became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes , and became Senator in 1799. Lagrange was one of the creators of the calculus of variations , deriving the Euler–Lagrange equations for extrema of functionals . He extended the method to include possible constraints, arriving at

3230-615: The divergence theorem . Founded mainly by Leonhard Euler and Joseph-Louis Lagrange in the eighteenth century, the calculus of variations saw further development and applications in the nineteenth. Let S = ∫ a b f ( x , y ( x ) , y ′ ( x ) ) d x , {\displaystyle S=\int \limits _{a}^{b}f(x,y(x),y'(x))\,dx,} where y ′ = d y d x {\displaystyle y'={\frac {dy}{dx}}} . Then S {\displaystyle S}

3325-456: The integral calculus ; a solution of a Fermat 's problem: given an integer n which is not a perfect square , to find a number x such that nx + 1 is a perfect square; and the general differential equations of motion for three bodies moving under their mutual attractions. The next work he produced was in 1764 on the libration of the Moon, and an explanation as to why the same face

3420-526: 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. Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics . The greater number of his papers during this time were, however, contributed to

3515-621: 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

3610-717: 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

3705-538: 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. Most of the papers sent to Paris were on astronomical questions, and among these, including his paper on the Jovian system in 1766, his essay on the problem of three bodies in 1772, his work on

3800-608: The Turin Society in 1762 and 1773. In 1758, with the aid of his pupils (mainly with Daviet), Lagrange established a society, which was subsequently incorporated as the Turin Academy of Sciences , 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

3895-430: The afternoon after the trial. Lagrange said on the death of Lavoisier: 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, Napoleon ) gave him honours and distinctions. This luckiness or safety may to some extent be due to his life attitude he expressed many years before: " I believe that, in general, one of

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3990-597: 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 Reign of Terror began. Under the intervention of Antoine Lavoisier , who himself was by then already thrown out of the academy along with many other scholars, Lagrange was specifically exempted by name in the decree of October 1793 that ordered all foreigners to leave France. On 4 May 1794, Lavoisier and 27 other tax farmers were arrested and sentenced to death and guillotined on

4085-421: The book; but Legendre at last persuaded a Paris firm to undertake it, and it was issued under the supervision of Laplace, Cousin, Legendre (editor) and Condorcet in 1788. 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

4180-685: The canonical variables of motion q {\displaystyle q} and p {\displaystyle p} . Then their Poisson bracket is given by [ u , v ] = ∂ u ∂ q i ∂ v ∂ p i − ∂ u ∂ p i ∂ v ∂ q i . {\displaystyle [u,v]={\frac {\partial u}{\partial q_{i}}}{\frac {\partial v}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial v}{\partial q_{i}}}.} Evidently,

4275-453: The choice of the metre and kilogram units with decimal subdivision, by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795. In 1795, Lagrange was appointed to a mathematical chair at the newly established École Normale , which enjoyed only a short existence of four months. His lectures there were elementary; they contain nothing of any mathematical importance, though they do provide

4370-413: The conclusion that the form of the curve at any time t is given by the equation y = a sin ⁡ ( m x ) sin ⁡ ( n t ) {\displaystyle y=a\sin(mx)\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

4465-471: The deputies to see how the professors acquitted themselves. It was also thought the published lectures would interest a significant portion of the citizenry ["Quoique des feuilles sténographiques soient essentiellement destinées aux élèves de l'École Normale, on doit prévoir quיelles seront lues par une grande partie de la Nation" ]. In 1794, Lagrange was appointed professor of the École Polytechnique ; and his lectures there, described by mathematicians who had

4560-1597: The dot notation for the time derivative is used, d q d t = q ˙ {\displaystyle {\frac {dq}{dt}}={\dot {q}}} . Poisson set L = T − V {\displaystyle L=T-V} . He argued that if V {\displaystyle V} is independent of q ˙ i {\displaystyle {\dot {q}}_{i}} , he could write ∂ L ∂ q ˙ i = ∂ T ∂ q ˙ i , {\displaystyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial T}{\partial {\dot {q}}_{i}}},} giving d d t ( ∂ L ∂ q ˙ i ) − ∂ L ∂ q i = 0. {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-{\frac {\partial L}{\partial q_{i}}}=0.} He introduced an explicit formula for momenta , p i = ∂ L ∂ q ˙ i = ∂ T ∂ q ˙ i . {\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial T}{\partial {\dot {q}}_{i}}}.} Thus, from

4655-540: The electric charge density ρ {\displaystyle \rho } . Poisson's work on potential theory inspired George Green 's 1828 paper, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism . In 1820, Hans Christian Ørsted demonstrated that it was possible to deflect a magnetic needle by closing or opening an electric circuit nearby, resulting in

4750-538: The equation of motion, he got p ˙ i = ∂ L ∂ q i . {\displaystyle {\dot {p}}_{i}={\frac {\partial L}{\partial q_{i}}}.} Poisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi . A translation of Poisson's Treatise on Mechanics was published in London in 1842. Let u {\displaystyle u} and v {\displaystyle v} be functions of

4845-444: 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

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4940-612: The fields of analysis , number theory , and both classical and celestial mechanics . In 1766, on the recommendation of Leonhard Euler and 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 many 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), which

5035-471: The first of these memoirs, Poisson discusses the famous question of the stability of the planetary orbits , which had already been settled by Lagrange to the first degree of approximation for the disturbing forces. Poisson showed that the result could be extended to a second approximation, and thus made an important advance in planetary theory. The memoir is remarkable inasmuch as it roused Lagrange, after an interval of inactivity, to compose in his old age one of

5130-513: The first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable ". 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 the 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

5225-409: 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 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

5320-409: The good fortune to be able to attend them, were 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. However, Lagrange does not seem to have been

5415-463: The greatest of his memoirs, entitled Sur la théorie des variations des éléments des planètes, et en particulier des variations des grands axes de leurs orbites . So highly did he think of Poisson's memoir that he made a copy of it with his own hand, which was found among his papers after his death. Poisson made important contributions to the theory of attraction. As a tribute to Poisson's scientific work, which stretched to more than 300 publications, he

5510-507: The latter paving the way for the classic researches of Peter Gustav Lejeune Dirichlet and Bernhard Riemann on the same subject; these are to be found in the Journal of the École Polytechnique from 1813 to 1823, and in the Memoirs de l'Académie for 1823. He also studied Fourier integrals . Poisson wrote an essay on the calculus of variations ( Mem. de l'acad., 1833), and memoirs on

5605-418: 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

5700-466: The mechanical "principles" as simple results of the variational calculus. 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

5795-726: The memoirs was examined by Sylvestre-François Lacroix and Adrien-Marie Legendre , who recommended that it should be published in the Recueil des savants étrangers, an unprecedented honor for a youth of eighteen. This success at once procured entry for Poisson into scientific circles. Joseph Louis Lagrange , whose lectures on the theory of functions he attended at the École Polytechnique, recognized his talent early on, and became his friend. Meanwhile, Pierre-Simon Laplace , in whose footsteps Poisson followed, regarded him almost as his son. The rest of his career, until his death in Sceaux near Paris,

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5890-472: The method of Lagrange multipliers . Lagrange invented the method of solving differential equations known as variation of parameters , applied differential calculus to the theory of probabilities and worked on solutions for algebraic equations . He proved that every natural number is a sum of four squares . His treatise Theorie des fonctions analytiques laid some of the foundations of group theory , anticipating Galois . In calculus , Lagrange developed

5985-435: The method of Lagrange multipliers —though this is not the first time that method was published—as a means to solve this equation. Amongst other minor theorems here given it may suffice to 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

6080-518: The offer. In 1765, d'Alembert interceded on Lagrange's behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin. He again turned down the offer, responding that 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

6175-1648: The operation anti-commutes. More precisely, [ u , v ] = − [ v , u ] {\displaystyle [u,v]=-[v,u]} . By Hamilton's equations of motion , the total time derivative of u = u ( q , p , t ) {\displaystyle u=u(q,p,t)} is d u d t = ∂ u ∂ q i q ˙ i + ∂ u ∂ p i p ˙ i + ∂ u ∂ t = ∂ u ∂ q i ∂ H ∂ p i − ∂ u ∂ p i ∂ H ∂ q i + ∂ u ∂ t = [ u , H ] + ∂ u ∂ t , {\displaystyle {\begin{aligned}{\frac {du}{dt}}&={\frac {\partial u}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial u}{\partial p_{i}}}{\dot {p}}_{i}+{\frac {\partial u}{\partial t}}\\[6pt]&={\frac {\partial u}{\partial q_{i}}}{\frac {\partial H}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}+{\frac {\partial u}{\partial t}}\\[6pt]&=[u,H]+{\frac {\partial u}{\partial t}},\end{aligned}}} where H {\displaystyle H}

6270-717: The opinion of some, first) in importance stand the memoirs on celestial mechanics , in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important of these are his memoirs Sur les inégalités séculaires des moyens mouvements des planètes , Sur la variation des constantes arbitraires dans les questions de mécanique , both published in the Journal of the École Polytechnique (1809); Sur la libration de la lune , in Connaissance des temps (1821), etc.; and Sur le mouvement de la terre autour de son centre de gravité , in Mémoires de l'Académie (1827), etc. In

6365-405: The probability of the mean results of observations ( Connaiss. d. temps, 1827, &c). The Poisson distribution in probability theory is named after him. In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so. In 1829, Poisson published a paper on elastic bodies that contained a statement and proof of a special case of what became known as

6460-411: The same term [REDACTED] This disambiguation page lists articles associated with the title Poisson . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Poisson&oldid=1133096330 " Category : Disambiguation pages Hidden categories: Short description

6555-417: The school, who left him free to make his own decisions as to what he would study. In his final year of study, less than two years after his entry, he published two memoirs, one on Étienne Bézout 's method of elimination, the other on the number of integrals of a finite difference equation and this was so impressive that he was allowed to graduate in 1800 without taking the final examination . The latter of

6650-545: The starting point for the researches of Cauchy , Jacobi , and Weierstrass . 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: His Résolution des équations numériques , published in 1798,

6745-638: The stern creed of the First Republic . Throughout the Revolution , the Empire , and the following restoration, Poisson was not interested in politics, concentrating instead on mathematics. He was appointed to the dignity of baron in 1825, but he neither took out the diploma nor used the title. In March 1818, he was elected a Fellow of the Royal Society , in 1822 a Foreign Honorary Member of

6840-417: The study of electricity was whether or not electricity was a fluid or fluids distinct from matter, or something that simply acts on matter like gravity. Coulomb, Ampère, and Poisson thought that electricity was a fluid distinct from matter. In his experimental research, starting with electrolysis, Michael Faraday sought to show this was not the case. Electricity, Faraday believed, was a part of matter. Poisson

6935-439: 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

7030-441: 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

7125-438: The third with applications to mechanics. Another treatise on the same lines was his Leçons sur le calcul des fonctions , issued in 1804, with the second edition in 1806. It is in this book that Lagrange formulated his celebrated method of Lagrange multipliers , in the context of problems of variational calculus with integral constraints. These works devoted to differential calculus and calculus of variations may be considered as

7220-594: 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

7315-402: 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

7410-664: The wave model. Fresnel won the competition. After that, the corpuscular theory of light was dead, but was revived in the twentieth century in a different form, wave-particle duality . Arago later noted that the diffraction bright spot (which later became known as both the Arago spot and the Poisson spot) had already been observed by Joseph-Nicolas Delisle and Giacomo F. Maraldi a century earlier. In pure mathematics , Poisson's most important works were his series of memoirs on definite integrals and his discussion of Fourier series ,

7505-409: The work could be described only as a scientific poem. 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

7600-525: Was a French mathematician, but the Italians continued to claim him as Italian. 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

7695-494: Was a member of the academic "old guard" at the Académie royale des sciences de l'Institut de France , who were staunch believers in the particle theory of light and were skeptical of its alternative, the wave theory. In 1818, the Académie set the topic of their prize as diffraction . One of the participants, civil engineer and opticist Augustin-Jean Fresnel submitted a thesis explaining diffraction derived from analysis of both

7790-400: Was absurd and Fresnel's model was wrong. (Such a spot is not easily observed in everyday situations, because most everyday sources of light are not good point sources.) The head of the committee, Dominique-François-Jean Arago , performed the experiment. He molded a 2 mm metallic disk to a glass plate with wax. To everyone's surprise he observed the predicted bright spot, which vindicated

7885-430: 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

7980-399: 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. Already by 1756, Euler and Maupertuis , seeing Lagrange's mathematical talent, tried to persuade Lagrange to come to Berlin, but he shyly refused

8075-553: Was appointed a professor of rational mechanics ( professeur de mécanique rationelle ). He went on to become a member of the Institute in 1812, examiner at the military school ( École Militaire ) at Saint-Cyr in 1815, graduation examiner at the École Polytechnique in 1816, councillor of the university in 1820, and geometer to the Bureau des Longitudes succeeding Pierre-Simon Laplace in 1827. In 1817, he married Nancy de Bardi and with her, he had four children. His father, whose early experiences had led him to hate aristocrats, bred him in

8170-605: Was awarded a French peerage in 1837. His is one of the 72 names inscribed on the Eiffel Tower . In the theory of potentials, Poisson's equation , is a well-known generalization of Laplace's equation of the second order partial differential equation ∇ 2 ϕ = 0 {\displaystyle \nabla ^{2}\phi =0} for potential ϕ {\displaystyle \phi } . If ρ ( x , y , z ) {\displaystyle \rho (x,y,z)}

8265-620: 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

8360-475: 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 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

8455-414: Was occupied by the composition and publication of his many works and in fulfilling the duties of the numerous educational positions to which he was successively appointed. Immediately after finishing his studies at the École Polytechnique, he was appointed répétiteur ( teaching assistant ) there, a position which he had occupied as an amateur while still a pupil in the school; for his schoolmates had made

8550-413: 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. In 1786, following Frederick's death, Lagrange received similar invitations from states including Spain and Naples , and he accepted the offer of Louis XVI to move to Paris. In France he

8645-514: 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 Institut de France (1795). At the beginning of his residence in Paris, he was seized with an attack of melancholy, and even the printed copy of his Mécanique on which he had worked for a quarter of

8740-468: 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. Appointed senator in 1799, he was the first signer of the Sénatus-consulte which in 1802 annexed his fatherland Piedmont to France. He acquired French citizenship in consequence. The French claimed he

8835-407: 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. Lagrange published his method in two memoirs of

8930-400: Was 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

9025-1000: Was written in the style of Laplace and Lagrange and was long a standard work. Let q {\displaystyle q} be the position, T {\displaystyle T} be the kinetic energy, V {\displaystyle V} the potential energy, both independent of time t {\displaystyle t} . Lagrange's equation of motion reads d d t ( ∂ T ∂ q ˙ i ) − ∂ T ∂ q i + ∂ V ∂ q i = 0 ,         i = 1 , 2 , . . . , n . {\displaystyle {\frac {d}{dt}}\left({\frac {\partial T}{\partial {\dot {q}}_{i}}}\right)-{\frac {\partial T}{\partial q_{i}}}+{\frac {\partial V}{\partial q_{i}}}=0,~~~~i=1,2,...,n.} Here,

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