BaronSiméon Denis Poisson (/pwɑːˈsɒ̃/,[1]US also/ˈpwɑːsɒn/;French:[si.me.ɔ̃də.nipwa.sɔ̃]; 21 June 1781 – 25 April 1840) was a Frenchmathematician andphysicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted theArago spot in his attempt to disprove the wave theory ofAugustin-Jean Fresnel.
Poisson was born inPithiviers, now inLoiret, France, the son of Siméon Poisson, an officer in the French Army.
In 1798, he entered theÉcole Polytechnique, inParis, as first in his year, and immediately began to attract the notice of the professors of 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 onÉtienne Bézout's method of elimination, the other on the number ofintegrals of afinite difference equation. This was so impressive that he was allowed to graduate in 1800 without taking the final examination[2],.[3] The latter of the memoirs was examined bySylvestre-François Lacroix andAdrien-Marie Legendre, who recommended that it should be published in theRecueil 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 inSceaux, near Paris, 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.[4]
Immediately after finishing his studies at the École Polytechnique, he was appointedrépétiteur (teaching assistant) there, a position which he had occupied as an amateur while still a pupil in the school; for his schoolmates had made 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 succeedingJean Baptiste Joseph Fourier, whomNapoleon had sent toGrenoble. In 1808 he becameastronomer to theBureau des Longitudes; and when the Faculté des sciences de Paris was instituted in 1809 he was appointed a professor ofrational mechanics (professeur de mécanique rationelle). He went on to become a member of the Institute in 1812, examiner at the military school (École Militaire) atSaint-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.[4]
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 the stern creed of theFirst Republic. Throughout theRevolution, theEmpire, and the following restoration, Poisson was not interested in politics, concentrating instead on mathematics. He was appointed to the dignity ofbaron in 1825,[4] but he neither took out the diploma nor used the title. In March 1818, he was elected aFellow of the Royal Society,[5] in 1822 a Foreign Honorary Member of theAmerican Academy of Arts and Sciences,[6] and in 1823 a foreign member of theRoyal Swedish Academy of Sciences. Therevolution of July 1830 threatened him with the loss of all his honours; but this disgrace to the government ofLouis-Philippe was adroitly averted byFrançois Jean Dominique Arago, who, while his "revocation" was being plotted by the council of ministers, procured him an invitation to dine at thePalais-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 apeer of France, not for political reasons, but as a representative of Frenchscience.[4]
Poisson in 1804 by E. Marcellot
As a teacher of mathematics Poisson is said to have been extraordinarily successful, as might have been expected from his early promise as arépétiteur 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,[4]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."[7])
A list of Poisson's works, drawn up by himself, 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 ofelectricity andmagnetism, which virtually created a new branch of mathematical physics.[4]
Next (or in the opinion of some, first) in importance stand the memoirs oncelestial mechanics, in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important of these are his memoirsSur les inégalités séculaires des moyens mouvements des planètes,Sur la variation des constantes arbitraires dans les questions de mécanique, both published in theJournal of the École Polytechnique (1809);Sur la libration de la lune, inConnaissance des temps (1821), etc.; andSur le mouvement de la terre autour de son centre de gravité, inMémoires de l'Académie (1827), etc. In the first of these memoirs, Poisson discusses the famous question of the stability of the planetaryorbits, 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 the greatest of his memoirs, entitledSur 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.[4]
As a tribute to Poisson's scientific work, which stretched to more than 300 publications, he was awarded a Frenchpeerage in 1837.
where is a distance between a volume elementand a point. The integration runs over the whole space.
Poisson's equation was first published in theBulletin de la société philomatique (1813).[4] Poisson's two most important memoirs on the subject areSur l'attraction des sphéroides (Connaiss. ft. temps, 1829), andSur l'attraction d'un ellipsoide homogène (Mim. ft. l'acad., 1835).[4]
Poisson discovered thatLaplace's equation is valid only outside of a solid. A rigorous proof for masses with variable density was first given byCarl Friedrich Gauss in 1839. Poisson's equation is applicable in not just gravitation, but also electricity and magnetism.[8]
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 a deluge of published papers attempting to explain the phenomenon.Ampère's law and theBiot-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.[10] By the 1830s, a major research question in 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.[11]
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 apoint source of light, where the particle-theory of light predicts complete darkness. Poisson argued this 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.[13] To everyone's surprise he observed the predicted bright spot, which vindicated 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 theArago spot and the Poisson spot) had already been observed byJoseph-Nicolas Delisle[13] andGiacomo F. Maraldi[14] a century earlier.
In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so.[15]
In 1829, Poisson published a paper on elastic bodies that contained a statement and proof of a special case of what became known as thedivergence theorem.[16]
Founded mainly by Leonhard Euler and Joseph-Louis Lagrange in the eighteenth century, thecalculus of variations saw further development and applications in the nineteenth.[17]
Let
where. Then is extremized if satisfies the Euler–Lagrange equations
But if depends on higher-order derivatives of, that is, if
Poisson'sTraité de mécanique (2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work.[4]Let be the position, be the kinetic energy, the potential energy, both independent of time. Lagrange's equation of motion reads[17]
Here, the dot notation for the time derivative is used,. Poisson set.[17] He argued that if is independent of, he could write
where is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as and.[20] Suppose is aconstant of motion, then it must satisfy
Moreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion.[20] Poisson had introduced his brackets while attempting to integrate the equations of motion resulting from the theory of perturbations for planetary orbits. But it was Jacobi who first recognized their utility in theoretical mechanics. In a series of lectures on dynamics delivered at theUniversity of Königsberg during the 1842-43 academic year, Jacobi also presentedhis identity for Poisson brackets, which can be used to prove Poisson's theorem.[19]
In 1821, using an analogy with elastic bodies,Claude-Louis Navier arrived at the basic equations of motion for viscous fluids, now identified as theNavier–Stokes equations. In 1829 Poisson independently obtained the same result.George Gabriel Stokes re-derived them in 1845 using continuum mechanics.[23] Poisson,Augustin-Louis Cauchy, andSophie Germain were the main contributors to the theory of elasticity in the nineteenth century. The calculus of variations was frequently used to solve problems.[17]
In his work on heat conduction, Joseph Fourier maintained that the arbitrary function may be represented as an infinite trigonometric series and made explicit the possibility of expanding functions in terms ofBessel functions andLegendre polynomials, depending on the context of the problem. It took some time for his ideas to be accepted as his use of mathematics was less than rigorous. Although initially skeptical, Poisson adopted Fourier's method. From around 1815 he studied various problems in heat conduction. He published hisThéorie mathématique de la chaleur in 1835.[24]
During the early 1800s, Pierre-Simon de Laplace developed a sophisticated, if speculative, description of gases based on the oldcaloric theory of heat, to which younger scientists such as Poisson were less committed. A success for Laplace was his correction of Newton's formula for the speed of sound in air that gives satisfactory answers when compared with experiments. TheNewton–Laplace formula makes use of the specific heats of gases at constant volumeand at constant pressure. In 1823 Poisson redid his teacher's work and reached the same results without resorting to complex hypotheses previously employed by Laplace. In addition, by using the gas laws ofRobert Boyle andJoseph Louis Gay-Lussac, Poisson obtained the equation for gases undergoingadiabatic changes, namely, where is the pressure of the gas, its volume, and.[25]
Besides his many memoirs, Poisson published a number of treatises, most of which were intended to form part of a great work on mathematical physics, which he did not live to complete. Among these may be mentioned:[4]
After political activistÉvariste Galois had returned to mathematics after his expulsion from the École Normale, Poisson asked him to submit his work on thetheory of equations, which he did January 1831. In early July, Poisson declared Galois' work "incomprehensible," but encouraged Galois to "publish the whole of his work in order to form a definitive opinion."[26] While Poisson's report was made before Galois' 14 July arrest, it took until October to reach Galois in prison. It is unsurprising, in the light of his character and situation at the time, that Galois vehemently decided against publishing his papers through the academy and instead publish them privately through his friend Auguste Chevalier. Yet Galois did not ignore Poisson's advice. He began collecting all his mathematical manuscripts while still in prison, and continued polishing his ideas until his release on 29 April 1832,[27] after which he was somehow persuaded to participate in what proved to be a fatal duel.[28]
^François Arago (1786–1853) attributed to Poisson the quote: "La vie n'est bonne qu'à deux choses: à faire des mathématiques et à les professer." (Life is good for only two things: to do mathematics and to teach it.) See: J.-A. Barral, ed.,Oeuvres complétes de François Arago ..., vol. II (Paris, France: Gide et J. Baudry, 1854),page 662.
^Kline, Morris (1972). "28.4: The Potential Equation and Green's Theorem".Mathematical Thought from Ancient to Modern Times. United States of America: Oxford University Press. pp. 682–4.ISBN0-19-506136-5.
^Baigrie, Brian (2007). "Chapter 5: From Effluvia to Fluids".Electricity and Magnetism: A Historical Perspective. United States of America: Greenwood Press. p. 47.ISBN978-0-313-33358-3.
^Baigrie, Brian (2007). "Chapter 7: The Current and the Needle".Electricity and Magnetism: A Historical Perspective. United States of America: Greenwood Press. p. 72.ISBN978-0-313-33358-3.
^Baigrie, Brian (2007). "Chapter 8: Forces and Fields".Electricity and Magnetism: A Historical Perspective. United States of America: Greenwood Press. p. 88.ISBN978-0-313-33358-3.
^Kline, Morris (1972). "27.4: The Foundation of Complex Function Theory".Mathematical Thought from Ancient to Modern Times. Oxford University Press. p. 633.ISBN0-19-506136-5.
^abcdefgKline, Morris (1972). "Chapter 30: The Calculus of Variations in the Nineteenth Century".Mathematical Thought from Ancient to Modern Times. Oxford University Press.ISBN0-19-506136-5.
^Kot, Mark (2014). "Chapter 4: Basic Generalizations".A First Course in the Calculus of Variations. American Mathematical Society.ISBN978-1-4704-1495-5.
^abJammer, Max (1966).The Conceptual Development of Quantum Mechanics. McGraw-Hill. p. 233.
^Farmelo, Graham (2009).The Strangest Man: the Hidden Life of Paul Dirac, Mystic of the Atom. Great Britain: Basic Books. pp. 83–88.ISBN978-0-465-02210-6.
^Kline, Morris (1972). "28.7: Systems of Partial Differential Equations".Mathematical Thought from Ancient to Modern Times. United States of America: Oxford University Press. pp. 696–7.ISBN0-19-506136-5.
^Kline, Morris (1972). "28.2: The Heat Equation and Fourier Series".Mathematical Thought from Ancient to Modern Times. United States of America: Oxford University Press. pp. 678–9.ISBN0-19-506136-5.
^Lewis, Christopher (2007). "Chapter 2: The Rise and Fall of the Caloric Theory".Heat and Thermodynamics: A Historical Perspective. United States of America: Greenwood Press.ISBN978-0-313-33332-3.
