Siméon Denis Poisson

Baron Siméon Denis Poisson (, ; ; 21 June 1781 – 25 April 1840) was a French mathematician and physicist 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 the Arago spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel.

While his interpretations of physical phenomena were often proven wrong by later researchers, his contributions to mathematics have stood the test of time. He demonstrated that physical problems could suggest new mathematical ideas. His father was Siméon Poisson, a retired soldier in the French Army who had served in the Seven Years' War.

In 1798, Poisson, then seventeen years of age, matriculated at the École Polytechnique in Paris after scoring in first place in the highly competitive entrance examination.

The latter of the two 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 an eighteen-year-old. This success at once gave Poisson admittance 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, 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. A list of Poisson's works, drawn up by himself, is given at the end of Arago's biography. His greatest services to science were performed in the application of mathematics to the study of physics. Some of the most influential were his memoirs on electricity and magnetism. in 1822 a Foreign Honorary Member of the American Academy of Arts and Sciences, and in 1823 a foreign member of the Royal Swedish Academy of Sciences.

Poisson died on 25 April 1840 in Sceaux. He was 58 years old.

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 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 <math>V</math> to the electric charge density <math>\rho</math>. 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 a deluge of published papers attempting to explain the phenomenon. Ampère's force 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. As a member of the examination committee, Poisson sought a way to disprove it. He calculated that Fresnel's predicted 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. For Poisson, this was absurd and had to be wrong. a century earlier.

Pure mathematics and statistics

thumb|Sample [[Poisson distributions. For large values of the mean <math>\lambda</math>, the Poisson distribution approaches the Gaussian or normal distribution.]]

Poisson's very first paper, written while he was still a student, was an "Addition" to a previous publication by Gaspard Monge and Jean Nicolas Pierre Hachette on the classification of quadrics. It was his last paper written in collaboration. Hachette co-signed this contribution.

Let<blockquote><math>S = \int\limits_{a}^{b} f (x, y(x), y'(x)) \, dx,</math></blockquote>where <math>y' = \frac{dy}{dx}</math>. Then <math>S</math> is extremized if <math>f(x,y(x),y'(x))</math> satisfies the Euler–Lagrange equations<blockquote><math>\frac{\partial f}{\partial y} - \frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0.</math></blockquote>But if <math>S</math> depends on higher-order derivatives of <math>y(x)</math>, that is, if <blockquote><math>S = \int\limits_{a}^{b} f \left(x, y(x), y'(x), ..., y^{(n)}(x) \right) \, dx,</math></blockquote>then <math>f</math> must satisfy the Euler–Poisson equation,<blockquote><math>\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.</math></blockquote>Poisson's Traité de mécanique (Treatise on Mechanics), in two volumes, is one of his most important scientific publications, Unlike Lagrange, however, Poisson did make use of diagrams. Let <math>u</math> and <math>v</math> be functions of the canonical variables of motion <math>q</math> and <math>p</math>. Then their Poisson bracket is given by<blockquote><math>[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}.</math></blockquote>Evidently, the operation anti-commutes. More precisely, <math>[u, v] = -[v, u]</math>. But it was Jacobi who first recognized the utility Poisson brackets in theoretical mechanics. In a series of lectures on dynamics delivered at the University of Königsberg during the 1842–43 academic year, Jacobi also presented his identity for Poisson brackets, which can be used to prove Poisson's theorem.

Continuum mechanics and fluid flow

In 1821, using an analogy with elastic bodies, Claude-Louis Navier arrived at the basic equations of motion for viscous fluids, now identified as the Navier–Stokes equations. In 1829 Poisson independently obtained the same result. George Gabriel Stokes re-derived them in 1845 using continuum mechanics.

Wave propagation

Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825). Poisson published his Théorie mathématique de la chaleur (Mathematical Theory of Heat) in 1835.

Other works

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 are:

Nouvelle théorie de l'action capillaire (4to, 1831);
Recherches sur la probabilité des jugements en matières criminelles et matière civile (4to, 1837), all published at Paris.
A catalog of all of Poisson's papers and works can be found in Oeuvres complétes de François Arago, Vol. 2
Mémoire sur l'équilibre et le mouvement des corps élastiques (v. 8 in Mémoires de l'Académie Royale des Sciences de l'Institut de France, 1829), digitized copy from the Bibliothèque nationale de France
Recherches sur le Mouvement des Projectiles dans l'Air, en ayant égard a leur figure et leur rotation, et a l'influence du mouvement diurne de la terre (1839)

File:Poisson-2.jpg|Title page to Recherches sur le Mouvement des Projectiles dans l'Air (1839)

File:Poisson - Mémoire sur le calcul numerique des integrales définies, 1826 - 744791.tif|Mémoire sur le calcul numerique des integrales définies (1826)

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Interaction with Évariste Galois

After political activist Évariste Galois had returned to mathematics after his expulsion from the École Normale, Poisson asked him to submit his work on the theory 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." 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, after which he was somehow persuaded to participate in what proved to be a fatal duel.