Simeon Denis Poisson

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Siméon Denis Poisson (French: /simeɔ̃ dəni pwasɔ̃/; Pithiviers, France, June 21, 1781 - Sceaux (Hauts-de-Seine), France, April 25, 1840) was a French physicist and mathematician who is known for his different works in the field of electricity and for his publications about differential geometry and probability theory.

Biography

Poisson was born in Pithiviers, Loiret, the son of Siméon Poisson. His father served as a private in the Hanoverian Wars, but disgusted by the abusive treatment he received from noble officers, he deserted. When his son was born, he held various administrative positions, and apparently headed the local government during the revolutionary period.

Siméon Denis was first sent to his uncle, a Fontainebleau surgeon, and began to learn the trade, but made little progress. After showing the first signs of his talent as a mathematician, he was sent to the Central School of Fontainebleau, where he had the opportunity to have a class with a receptive teacher, M. Billy, who quickly realized that he was outclassed by his student, he encouraged him to learn the more difficult branches of mathematics, and predicted his future fame by recalling lines from the famous fabulist Jean de La Fontaine, playing on the meaning of his last name in French:

«Petit Poisson deviendra grand // Pourvu que Dieu lui prête vie».
(‘The small fish becomes big // As long as God gives him life’).

In 1798, he entered the École Polytechnique in Paris at the top of his class, and immediately began to attract the attention of the school's professors, who left him free to make his own decisions as to what he was going to do. study. In 1800, less than two years after his entry, he published two memoirs, one on Étienne Bézout's method of elimination, and the other on the number of integrals of a finite difference equation. The latter was reviewed by Sylvestre François Lacroix and by Adrien-Marie Legendre, who recommended that it be published in the Recueil des savants étrangers (Report of Foreign Scientists), an honor without precedents for an eighteen year old. This success introduced Poisson into the most distinguished 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 (the Mathematics Genealogy Project identifies Lagrange as his adviser, but this may be a mere simplification); while Pierre-Simon Laplace, who was attentive to Poisson's footsteps, regarded him almost as his son. The remainder of his career, until his death at Sceaux near Paris, was almost wholly occupied with the composition and publication of his numerous works and with the discharge of the duties of the numerous educational posts for which he was named successively.

Immediately after completing his studies at the École Polytechnique, he was appointed répétiteur (teaching assistant) at the school itself, a position he had held without pay while still a student at the school; His classmates would often visit him in his room after a particularly difficult class to hear him repeat it and explain it. He was appointed assistant professor (professeur suppléant) in 1802, and in 1806 full professor after the departure of Jean Baptiste Joseph Fourier, whom Napoleon had sent to Grenoble. In 1808 he became an astronomer at the Bureau des Longitudes; and when the Faculté des sciences de Paris was instituted in 1809, he was appointed professor of rational mechanics ( professeur de mécanique rationelle ). Later he became a member of the Institute in 1812, examiner at the military school (École Militaire) of Saint-Cyr in 1815, graduate examiner at the Polytechnic School in 1816, advisor to the university in 1820, and geometer of the Bureau des Longitudes succeeding Pierre-Simon Laplace in 1827.

In 1817, he married Nancy de Bardi and with her had four children. His father, whose early experiences had led him to hate aristocrats, raised him in the principles of the First Republic. Throughout the Revolution, the Empire, and the following restoration, Poisson was not interested in politics, concentrating on mathematics. During the First Empire he remained loyal to the Republic, refusing to swear allegiance to Napoleon.

He was made a baron in 1821; but he did not obtain the diploma nor use the title. In March 1818 he was elected a Fellow of the Royal Society and in 1823 a foreign member of the Royal Swedish Academy of Sciences. The Revolution of 1830 could mean the loss of all his honors; but this disdain for the government of Louis-Philippe of Orleans was skilfully avoided by François Arago: while the "revocation" of Poisson's charges was being dealt with by the Council of Ministers, Arago provided him with an invitation to dine at the Royal Palace, where he was openly and effusively received by the citizen king, who "remembered" him. After this, of course, his demotion was impossible, and seven years later he was made a Peer of France, not for political reasons, but as a scientific representative.

Become a staunch supporter of the monarchy, his dislike of the physicist and mathematician Louis Poinsot is known, with whom he had bitter scientific discussions for years, which took on political overtones and ended with Poinsot's removal from his academic posts when Poisson he was appointed responsible for education of the new monarchical government in 1830.

As a mathematics teacher, Poisson is said to have been extraordinarily successful, unsurprisingly after his early tenure as répétiteur at the École Polytechnique. As a scientific worker, his productivity is practically second to none. Despite his many official duties, he found time to publish more than three hundred works, several of them lengthy treatises; and many other memoirs devoted to dealing with the most abstruse branches of pure mathematics, applied mathematics, mathematical physics, and rational mechanics.

Arago attributes the following quote to him:

"Life is good only for two things: to do mathematics; and to teach them."

Main works

The Poisson correction of Laplace's second order differential equation for the potential is known:

{displaystyle nabla ^{2}phi =-4pi rho ;}

Today it bears his name (Poisson's Equation) or the equation of the theory of potential, first published in the Bulletin de la Société Philomatique de Paris (1813). If the function at a given point is ρ = 0, then Laplace's equation is obtained:

{displaystyle nabla ^{2}phi =0;.}

In 1812 Poisson discovered that Laplace's equation is valid only outside a solid. A rigorous proof of masses with variable density was first given by Carl Friedrich Gauss in 1839. Both equations have their equivalents in vector calculus. Poisson's equation for the Laplacian operator of a scalar field; φ in three-dimensional space is:

{displaystyle nabla ^{2}phi =rho (x,y,z);.}

Considering for example the Poisson equation for the electrical potential on a surface; Ψ as a function of electric charge density; ρ_e known at a particular point:

{displaystyle nabla ^{2}Psi ={partial ^{2}Psi over partial x^{2}}+{partial ^{2}Psi over partial y^{2}}+{partial ^{2}Psi over partial z^{2}}=-{rho _{e} over varepsilon varepsilon _{0}};.}

The distribution of charge in a fluid is unknown and the Poisson-Boltzmann Equation must be used:

{displaystyle nabla ^{2}Psi ={n_{0}e over varepsilon varepsilon _{0}}left(e^{ePsi (x,y,z)/k_{B}T}-e^{-ePsi (x,y,z)/k_{B}T}right),;}

which in most cases cannot be solved analytically. In polar coordinates the Poisson-Boltzmann equation has the form:

{displaystyle {1 over r^{2}}{d over dr}left(r^{2}{dPsi over dr}right)={n_{0}e over varepsilon varepsilon _{0}}left(e^{ePsi (r)/k_{B}T}-e^{-ePsi (r)/k_{B}T}right);}

which cannot be solved analytically either. If a field φ is not scalar, Poisson's equation is valid, as it can be for example in 4-dimensional Minkowski space:

{displaystyle {sqrt {phi }}_{ik}=rho (x,y,z,ct);.}

If ρ ( x , y , z ) is a continuous function and if when r → ∞ (or if a point 'moves' to infinity) the function φ tends to 0 fast enough, one solution of Poisson's equation is that of the Newtonian potential of a function ρ (x,y,z):

{displaystyle phi _{M}=-{1 over 4pi }int {rho (x,y,z),dv over r};}

where r is the distance between a volume element dv and a point M. The integration runs across the entire space.

Another "Poisson integral" is the solution for the Green function for Laplace's equation with the Dirichlet condition on a circular disk:

{displaystyle phi (xi eta)={1 over 4pi }int _{0}^{2pi }{R^{2}-rho ^{2} over R^{2}+rho ^{2}-2Rrho cos(psi -chi)}phi (chi),dchi ;}

where

{displaystyle xi =rho cos psi;}

{displaystyle quad eta =rho sin psi;}

φ is a condition of contour imposed on the contour of the disk.

In the same way, the Green function for Laplace's equation is defined with the Dirichlet condition, ∇² φ = 0 on a sphere of radius R. In this case, the Green function is:

{displaystyle G(x,y,z;xietazeta)={1 over r}-{R over r_{1}rho };,}

where

{displaystyle rho ={sqrt {xi ^{2}+eta ^{2}+zeta ^{2}}}}

is the distance of a point (POL, γ, γ) from the center of the sphere,

r is the distance between the points (x,y,z) and (ξ, η, ζ), and

r₁ is the distance between the point (x,y,z) and the point (Rξ/ρRη/ρ,Rζ/ρ), symmetrical to the point (ξ, η, ζ).

The Poisson integral now has the form:

{displaystyle phi (xietazeta)={1 over 4pi }iint _{S}{R^{2}-rho ^{2} over Rr^{3}}phi ,ds;.}

In 1815 Poisson studied integrals in the complex plane, and in 1831 he obtained the Navier-Stokes Equations independently of Claude-Louis Navier.

Wrong view of the wave theory of light

Experiment of the point of Arago. A light source punctual, casts the shadow of a circle over a screen. In the middle of the shadow appears a bright point due to defaction, contradicting the predictions of geometric optics.

Poisson showed surprising arrogance in rejecting the wave theory of light. He was a member of the academic & # 34;old guard & # 34; at the Royal Academy of Sciences of the Institute of France, staunch supporters of the corpuscular theory of light alarmed by the growing acceptance of the wave theory of light. In 1818, the Academy laid the foundation for its prize by devoting it to diffraction, certain that a particle theorist would be the winner. Poisson, relying on intuition rather than mathematics or scientific experiments, ridiculed civil engineer Augustin-Jean Fresnel when he submitted a thesis to the competition explaining diffraction by analyzing both the Fresnel-Huygens Principle and the double-slit experiment. by Thomas Young.

Poisson studied Fresnel's theory in detail and, of course, looked for a way to prove it wrong, as a dogmatic supporter of the particle theory of light. He thought he had found a mistake when he argued that a consequence of Fresnel's theory is that there should be a bright spot in the center of the shadow of a circular obstacle blocking a light source, where in turn there should be complete darkness according with the corpuscular theory of light. Therefore, Fresnel's theory could not be true, Poisson declared, according to this admittedly absurd result. (It should be noted that the Poisson's spot is not easily observed in everyday situations, because most common light sources are not adequate point sources of light.)

However, the chairman of the commission, François Arago (later Prime Minister of France), did not have Poisson's arrogance and decided that the experiment needed to be carried out in more detail. He molded a 2 mm metal disk onto a glass plate with wax. To everyone's surprise, he was able to observe the bright spot predicted by Poisson (later named "Arago's point" in his memory), which convinced most scientists of the wave nature of light. In the end Fresnel won the Academy contest, much to Poisson's chagrin.

After this, the corpuscular theory of light was shelved, not to appear again (this time in a very different form) until the century XX, when it revived as part of the newly developed wave-particle duality. Arago later pointed out that the diffraction bright spot (the Poisson spot, later known as the Arago spot) had already been observed by Joseph-Nicolas Delisle and by Giacomo F. Maraldi a century earlier.

Treaties

Mémoire sur le calcul numerique des integrales définies, 1826

A list of Poisson's works, compiled by himself, is given at the end of his biography written by Arago. Given its enormous size, it is usually summarized in a brief mention of the most important ones. It was in the application of mathematics to physics that his most relevant contributions to science were focused.

Perhaps the most original, and certainly the most enduring in its influence, were his memoirs on the theory of electricity and magnetism, which practically created a new branch of mathematical physics.

Next in importance (if not more important than the aforementioned work) are his memoirs on celestial mechanics, in which he proved himself a worthy successor to Pierre-Simon Laplace. The most important are his memoirs entitled Sur les inégalités seculaires des moyens mouvements des Planetes, Sur la variation des constantes arbitraires dans les questions de mécanique , both published in the Journal of the Polytechnic School (1809); Sur la libration de la lune, in Connaissances des temps (1821); and Sur le mouvement de la terre autour de son center de gravité, in Mémoires de l'Académie (1827). In the first of these memoirs, Poisson discusses the famous question of the stability of planetary orbits, which had already been solved by Lagrange to a first degree of approximation of the perturbing forces. Poisson showed that the result could be extended to a second approximation, and thus made an important advance in planetary theory. The memory is remarkable in that it "woke up" to Lagrange after an interval of inactivity, to compose in his old age one of his most important memoirs, entitled Sur la théorie des variations des éléments des planetes, et en particulier des variations des grands axes de leurs orbites . Lagrange held Poisson's memoir in such high esteem that he made a copy of it in his own hand, which was found among his papers after his death. Poisson made important contributions to the theory of gravitational attraction.

Two of Poisson's most important memoirs on the subject are Sur l'attraction des sphéroides (Connaiss. Ft. Temps, 1829), and Sur l'attraction d'un ellipsoide homogene (Mim. Ft. L '. acad, 1835). To conclude the selection of his physical memories, one can speak of his book on the theory of waves (MEM. Ft. L & # 39; Acad., 1825).

In pure mathematics, his most important works were his series of memoirs on definite integrals and his discussion of Fourier series, the latter paving the way for Peter Gustav Lejeune Dirichlet's and Bernhard Riemann's classical investigations on the same subject; they were published in the Journal of the École Polytechnique 1813/23, and in the Memoirs of the Academy of 1823. He also studied Fourier integrals. One can also talk about his essay on the calculus of variations (Mem. Acad de l'., 1833), and his memoirs on the probability of the average results of observations ( Connaiss d. Temps, 1827). The Poisson distribution in probability theory is named after him.

His Traité de mécanique (2 vols 8.º, 1811-1833), written in the style of Laplace and Lagrange, was for a long time a reference work, showing numerous innovations, such as the explicit use of moments:

{displaystyle p_{i}={partial T over {partial q_{i}/partial t}},}

that influenced the work of Hamilton and Jacobi.

In addition to his many memoirs, Poisson published a series of treatises, most of which were destined to form part of a great work on mathematical physics, which he did not live long enough to complete: Formules relatives aux effets du tir d'un canon sur les différentes parties de son affût (1838).

Among these works, we can mention:

Nouvelle théorie de l'action chapelire (4th, 1831);
Théorie mathématique de la chaleur (4th, 1835);
Supplement for the same (4th, 1837);
Recherches sur la probabilité des jugements en matières criminelles et matière civile (4th, 1837), all published in Paris.

A translation of Poisson's Treatise on Mechanics was published in London in 1842.

Acknowledgments

Elected Member of the Royal Society in 1818.
Foreign Member of the Royal Swedish Academy of Sciences in 1823.
It is one of 72 scientists whose name is registered at the Eiffel Tower.
A federation in mathematical and theoretical physics bears its name.
A school in the French town of Pithiviers also bears its name.
In 1935, the International Astronomical Union gave the name of Poisson to a lunar crater.
The asteroid (12874) Poisson also commemorates its name.
The exhibition "S.-D. Poisson, les mathématiques au service de la science" (S. D. Poisson, mathematics at the service of science) was organized at the university Pierre & Marie Curie (Paris) in 2014.

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