# User Contributed Dictionary

### Etymology

After Siméon Denis Poisson (1781-1840), French mathematician.### Noun

- The Poisson distribution.

### Proper noun

- Siméon Denis Poisson (1781-1840), French mathematician.

### Proper noun

Poisson# Extensive Definition

Siméon-Denis Poisson (June 21, 1781 –
April
25, 1840),
was a French
mathematician,
geometer, and physicist. The name is in
French.

## Biography

Poisson was born in Pithiviers, south of Paris.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 make his own choices as to what
he would study. In 1800, 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. The latter 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 honour 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 (the
Mathematics Genealogy Project lists Lagrange as his advisor,
but this may be an approximation); 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 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 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 was instituted in 1809 he was appointed 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 [several?] children. His father, whose early experiences had
led him to hate aristocrats, bred him in the stern creed of the
First Republic. Throughout the Revolution, the Empire, and the
following restoration, Poisson was not interested in politics,
concentrating on mathematics. He was appointed to the dignity of
baron in 1821; but he
neither took out the 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.

Like many scientists of his time, he was an
atheist.

As a teacher of mathematics Poisson is said to
have been extraordinarily successful, as might have been expected
from his early promise as a répétiteur at the École Polytechnique.
As a scientific worker, his productivity 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 mathematics, applied
mathematics, mathematical
physics, and rational mechanics.

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 of
electricity and
magnetism, which
virtually created a new branch of mathematical physics.

Next (or in 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 Connaissances 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 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 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.

## Contributions

Poisson's well-known correction of Laplace's second order partial differential equation for potential:- \nabla^2 \phi = - 4 \pi \rho \;

today named after him Poisson's
equation or the potential
theory equation, was first published in the Bulletin de la
société philomatique (1813). If a function of a given point ρ = 0,
we get Laplace's
equation:

- \nabla^2 \phi = 0 \; .

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. Poisson's equation for the divergence
of the gradient of a scalar
field, φ in 3-dimensional space is:

- \nabla^2 \phi = \rho (x, y, z) \; .

Consider for instance Poisson's equation for
surface electrical
potential, Ψ as a function of the density of electric
charge, ρe at a particular point:

- \nabla^2 \Psi = +

The distribution of a charge in a fluid is unknown and we have to
use the Poisson-Boltzmann
equation:

- \nabla^2 \Psi =

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

- \left( r^ \right) =

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

- \sqrt \phi_ = \rho (x, y, z, ct) \; .

If ρ(x, y, z) is a continuous
function 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):

- \phi_M = - \int \;

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

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

- \phi(\xi \eta) = \int _0^

where

- \xi = \rho \cos \psi, \;

- \quad \eta = \rho \sin \psi, \;

- φ is a boundary condition holding on the disk's boundary.

In the same manner, we define the Green function
for the Laplace equation with Dirichlet condition, ∇² φ = 0 over a
sphere of radius R. This time the Green function is:

- G(x,y,z;\xi,\eta,\zeta) = - \; ,

where

- \rho = \sqrt is the distance of a point (ξ, η, ζ) from the center of a sphere,

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

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

Poisson's integral now has a form:

- \phi(\xi, \eta, \zeta) = \int\!\!\!\int_S \phi\, ds \; .

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, 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. 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 the style of Laplace and Lagrange
and was long a standard work, he showed many novelties such as an
explicit usage of impulsive
coordinates:

- p_i = \;

which influenced 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

- Nouvelle théorie de l'action capillaire (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 matière civile (4to, 1837), all published at Paris. A translation of Poisson's Treatise on Mechanics was published in London in 1842.

In 1815 Poisson studied integrations along paths
in the complex plane. In 1831 he derived the Navier-Stokes
equations independently of Claude-Louis
Navier.

## See also

- Poisson process
- Poisson equation
- Screened Poisson equation
- Poisson kernel
- Poisson distribution
- Poisson regression
- Poisson summation formula
- Poisson's spot
- Poisson's ratio
- Poisson (crater) (named after Siméon Denis Poisson)
- Poisson bracket
- Euler–Poisson–Darboux equation
- Poisson Zeros
- Conway-Maxwell-Poisson distribution

## External links

## References

Poisson in Bulgarian: Симеон Дени Поасон

Poisson in Czech: Siméon Denis Poisson

Poisson in German: Siméon Denis Poisson

Poisson in Spanish: Siméon Denis Poisson

Poisson in French: Siméon Denis Poisson

Poisson in Italian: Siméon-Denis Poisson

Poisson in Japanese: シメオン・ドニ・ポアソン

Poisson in Marathi: सिमिओन-डेनिस पॉइसॉन

Poisson in Dutch: Siméon Poisson

Poisson in Norwegian: Siméon Denis Poisson

Poisson in Polish: Siméon Denis Poisson

Poisson in Portuguese: Siméon Denis
Poisson

Poisson in Romanian: Siméon Denis Poisson

Poisson in Russian: Пуассон, Симеон Дени

Poisson in Slovak: Siméon-Denis Poisson

Poisson in Slovenian: Siméon-Denis Poisson

Poisson in Swedish: Siméon Denis Poisson

Poisson in Ukrainian: Пуассон Сімеон-Дені

Poisson in Chinese: 泊松