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Simeon Poisson.
Siméon-Denis Poisson (June 21, 1781 – April 25, 1840), was
a French mathematician,
geometer and physicist.
Poisson was born at Pithiviers in
the département of Loiret, France.
His father, Siméon Poisson, served as a common soldier in the Hanoverian wars; but, disgusted by the ill-treatment he received from his patrician
officers, he deserted. About the time of the birth of his son, Siméon-Denis, he occupied a small administrative post at
Pithiviers, and seems to have been at the head of the local government of the place during the revolutionary period. Poisson was
first sent to an uncle, a surgeon at Fontainebleau, and began to take
lessons in bleeding and blistering, but made little progress. Having given promise of mathematical talent, he was sent to the
École Centrale of Fontainebleau, and was fortunate in having a kind and sympathetic teacher, M. Billy, who, when he speedily
found that his pupil was becoming his master, devoted himself to the study of higher mathematics in order to follow and appreciate him, and predicted his future fame by the punning quotation from
Jean de La Fontaine:
- "Petit Poisson deviendra grand
- Pourvu que Dieu lui prête vie."
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 the school, who left him free to follow the studies of his
predilection. In 1800, less than two years after his entry, he published two memoirs, one
on Etienne Bézout's method of elimination, the other on the number of
integrals of an equation of finite differences. The latter of these 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 unparalleled honour for a youth of eighteen. This success at once
procured for Poisson an entry into scientific circles. Joseph-Louis de Lagrange, whose lectures on the theory of functions he attended at the École
Polytechnique, early recognized his talent, and became his friend; while Pierre-Simon Laplace, in whose footsteps Poisson followed, regarded him almost as his son. The rest of
his career, till his death in Sceaux near
Paris, was almost entirely occupied in the composition and publication of his many works, and in discharging the duties of the
numerous educational offices to which he was successively appointed.
Immediately after finishing his course at the École Polytechnique he was appointed repetiteur there, an office which he had
discharged as an amateur while still a pupil in the school; for it had been the custom of his comrades often to resort to 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
in succession to Jean-Baptiste-Joseph
Fourier, who went to Grenoble. In 1808 he
became astronomer to the Bureau des Longitudes; and when the Faculté des Sciences was
instituted in 1809 he was appointed professor of rational mechanics (professeur de la mécanique rationelle). He further became member of the
Institute in 1812, examiner at the military school (École Militaire) at Saint-Cyr in 1815, leaving examiner at the École Polytechnique in 1816,
councillor of the university in 1820, and geometer to the Bureau des Longitudes in
succession to P. S. Laplace in 1827.
In 1817 he married Nancy de Bardi. His father, whose early experiences led him to hate
aristocrats, bred him in the stern creed of the first republic. Throughout the empire Poisson faithfully adhered to the family
principles, and refused to worship Napoleon I. When the Bourbons were restored, his hatred against Napoleon led him to become a
Legitimist — a conclusion which says more for the simplicity of his character than for the strength or logic of his
political creed. He was faithful to the Bourbons during the Hundred Days; in fact, was with difficulty dissuaded from
volunteering to fight in their cause. After the second restoration his fidelity was recognized by his elevation to the dignity of
baron in 1821; but he never either took out his
diploma or used the title. 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 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 more than ordinarily successful, as might have been expected from his
early promise as a repetiteur at the École Polytechnique. As a scientific worker his activity has rarely if ever been equalled.
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, applied mathematics, mathematical
physics and rational mechanics. There are two remarks of his, or perhaps two versions of the same remark, that explain how he
accomplished so much: one, "La vie n'est bonne que deux choses — à faire des mathématiques et à les professeurs;" the
other, "La vie c'est le travail." (Life is Work)
A list of Poisson's works, drawn up by himself, is given at the end of Arago's biography, a lengthened analysis of them would
be out of place here, and all that is possible is a brief mention of the more important. There are few branches of mathematics to
which he did not contribute something, but it was in the application of mathematics to physical subjects 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 (perhaps in the opinion of some first) in importance stand the memoirs on celestial mechanics, in which he proved himself a worthy successor to P.-S. 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 Connaiss. des temps (1821), etc.; and Sur la mouvement de la terre
autour de son centre de gravité, in Mém. d. l'acad. (1827), etc. In 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 the 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, viz, that 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.
His well-known correction of Laplace's partial differential equation of the second degree for the potential:
-
today named after him the Poisson's equation or the
potential theory
equation, was first published in the Bulletin de in société philomatique (1813). If a function of a given point ρ = 0, we
get Laplace's equation:
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In 1812 Poisson discovered that Laplace's equation is valid only outside of a solid. A
rigorous proof for masses with variable density was first given by Carl Friedrich Gauss in 1839. Both equations have their
equivalents in vector algebra. The study of scalar field φ from a given divergence ρ(x, y, z) of its gradient leads to Poisson's equation in 3-dimensional space:
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For instance Poisson's equation for surface electrical
potential Ψ, which shows its dependence from the density of electrical charge ρe in particular place:
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The distribution of a charge in a fluid is
unknown and we have to use Poisson-Boltzmann equation:
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which in most cases can't be solved analytically but just for special cases. In polar coordinates the Poisson-Boltzmann equation is:
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which also can't be solved analytically. If a field φ is not a scalar, the Poisson equation is valid, as can be for example in 4-dimensional Minkowski space:
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If ρ(x, y, z) is continuous and if for r→∞ (or if a point 'moves' to infinity) a function φ goes to 0 fast enough, a solution of Poisson's equation is the Newtonian potential of a function ρ(x, y,
z):
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where r is a distance between the element with the volume dv and point M. Integration runs over the
whole space. The Poisson's integral in solving the Green's
function for the Dirichlet problem of the Laplace's equation, if circle is investigated domain:
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where:
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φ(χ) is prescribed function on a circular line, which defines bounding conditions of requested function φ of
Laplace's equation. In the same manner we define the Green's function for the Dirichlet problem of the Laplace's equation
2 φ = 0 in space, if we look to the investigated domain of a sphere with a radius
R. This time the Green's function is:
-
where: is a
distance of a point (ξ, η, ζ) from the center of a sphere, r a distance between points (x,
y, z), (ξ, η, ζ), r1 is a distance between the point (x,
y, z) and the point (Rξ/ρ, Rη/ρ, Rζ/ρ), symmetrical to
the point (ξ, η, ζ). The Poisson's integral now has a form:
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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). In concluding our selection from his physical memoirs we may mention his memoir on the
theory of waves (Mém. ft. l'acad., 1825).
In pure mathematics, his most important works were his series of
memoirs on definite integrals, and his discussion of Fourier series, which paved the way for the classical 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. In addition we may also mention his
essay on the calculus of variations (Mem. de
l'acad., 1833), and his memoirs on the probability of the mean results of observations (Connaiss. d. temps, 1827,
&c). The Poisson distribution in probability theory is named after him.
In his Traité de mécanique (2 vols. 8vo, 1811 arid 1833), which was written in Laplace and Lagrange style and was
long a standard work he showed many new grips such as an explicit usage of impulsive
coordinates:
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which has influenced on the work of William Rowan
Hamilton and Carl Gustav Jakob Jacobi.
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
- Théorie nouvelle de l'action cappillaire (4to, 1831);
- Théorie mathématique de la chaleur (4to, 1835);
- Supplement to the same (4to, 1837);
- Recherches sur la probabilité des jugements en matières criminelles et matiere civile (4to, 1837), all published at
Paris.
In 1815 Poisson carried out integrations along paths in the complex plane. In 1831 he
independently of Claude-Louis-Marie Henri Navier derived the Navier-Stokes equations.
This article incorporates text from the public domain 1911 Encyclopædia Britannica.
See also
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