Baron Baron is a rank of nobility or title of honour, often Hereditary title, hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than ...

Siméon Denis Poisson (, ; ; 21 June 1781 – 25 April 1840) was a French

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

and

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

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.

Biography

Poisson was born in Pithiviers, now in Loiret, France, the son of Siméon Poisson, an officer in the French Army. In 1798, he entered the École Polytechnique, in

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

, 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 on Étienne Bézout's method of elimination, the other on the number of

integrals In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Int ...

of a

finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...

equation. This was so impressive that he was allowed to graduate in 1800 without taking the final examination^,. The latter of the 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 a youth of eighteen. This success at once procured entry for Poisson into scientific circles.

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPierre-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. Immediately after finishing his studies at the École Polytechnique, he was appointed ''répétiteur'' (

teaching assistant A teaching assistant (TA) or education assistant (EA) is an individual who assists a professor or teacher with instructional responsibilities. TAs include ''graduate teaching assistants'' (GTAs), who are graduate students; ''undergraduate teach ...

) 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 Napoleon Bonaparte (born Napoleone di Buonaparte; 15 August 1769 – 5 May 1821), later known by his regnal name Napoleon I, was a French general and statesman who rose to prominence during the French Revolution and led Military career ...

had sent to

Grenoble Grenoble ( ; ; or ; or ) is the Prefectures in France, prefecture and List of communes in France with over 20,000 inhabitants, largest city of the Isère Departments of France, department in the Auvergne-Rhône-Alpes Regions of France, region ...

. In 1808 he became

astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...

to the Bureau des Longitudes; and when the Faculté des sciences de Paris was instituted in 1809 he was appointed a 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 four 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 In political science, a revolution (, 'a turn around') is a rapid, fundamental transformation of a society's class, state, ethnic or religious structures. According to sociologist Jack Goldstone, all revolutions contain "a common set of elements ...

, the

Empire An empire is a political unit made up of several territories, military outpost (military), outposts, and peoples, "usually created by conquest, and divided between a hegemony, dominant center and subordinate peripheries". The center of the ...

, and the following restoration, Poisson was not interested in politics, concentrating instead on mathematics. He was appointed to the dignity of

baron Baron is a rank of nobility or title of honour, often Hereditary title, hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than ...

in 1825, but he neither took out the diploma nor used the title. In March 1818, he was elected a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

, in 1822 a Foreign Honorary Member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

, and in 1823 a foreign member of the Royal Swedish Academy of Sciences. 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 Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

. 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. 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 Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

, and rational mechanics. ( Arago attributed to him the quote, "Life is good for only two things: doing mathematics and teaching it.") 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 Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...

and

magnetism Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, ...

, 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 Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

, 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 '' Connaissance des temps'' (1821), etc.; an
''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

orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...

s, 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. As a tribute to Poisson's scientific work, which stretched to more than 300 publications, he was awarded a French

peerage A peerage is a legal system historically comprising various hereditary titles (and sometimes Life peer, non-hereditary titles) in a number of countries, and composed of assorted Imperial, royal and noble ranks, noble ranks. Peerages include: A ...

in 1837. His is one of the 72 names inscribed on the Eiffel Tower.

Contributions

Potential theory

Poisson's equation

In the theory of potentials, Poisson's equation, :

\nabla^2 \phi = - 4 \pi \rho, \;

is a well-known generalization of Laplace's equation of the second order

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

\nabla^2 \phi = 0

for potential

\phi

. If

\rho(x, y, z)

is a continuous function and if for

r \rightarrow \infty

(or if a point 'moves' to

infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...

) a function

\phi

goes to 0 fast enough, the solution of Poisson's equation is the Newtonian potential :

\phi = -  \iiint \frac \, dV, \;

where

r

is a distance between a volume element

dV

and a point

P

. The integration runs over the whole space. Poisson's equation was first published in the ''Bulletin de la société philomatique'' (1813). 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). 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. Poisson's equation is applicable in not just gravitation, but also electricity and magnetism.

Electricity and magnetism

As the eighteenth century came to a close, human understanding of electrostatics approached maturity.

Benjamin Franklin Benjamin Franklin (April 17, 1790) was an American polymath: a writer, scientist, inventor, statesman, diplomat, printer, publisher and Political philosophy, political philosopher.#britannica, Encyclopædia Britannica, Wood, 2021 Among the m ...

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 Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia 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 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. 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.

Optics

Poisson was a member of the academic "old guard" at the Académie royale des sciences de l'Institut de France, who were staunch believers in the particle theory of light and were skeptical of its alternative, the wave theory. In 1818, the Académie set the topic of their prize as

diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...

. One of the participants, civil engineer and opticist Augustin-Jean Fresnel submitted a thesis explaining diffraction derived from analysis of both the Huygens–Fresnel principle and Young's double slit experiment. 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 a point 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. 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 the Arago spot and the Poisson spot) had already been observed by Joseph-Nicolas Delisle and Giacomo F. Maraldi a century earlier.

Pure mathematics and statistics

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...

, Poisson's 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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

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. Poisson wrote an essay on the calculus of variations (''Mem. de l'acad.,'' 1833), and memoirs on the probability of the mean results of observations (''Connaiss. d. temps,'' 1827, &c). The Poisson distribution in

probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

is named after him. In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so. In 1829, Poisson published a paper on elastic bodies that contained a statement and proof of a special case of what became known as the divergence theorem.

Mechanics

Analytical mechanics and the calculus of variations

Founded mainly by Leonhard Euler and Joseph-Louis Lagrange in the eighteenth century, the calculus of variations saw further development and applications in the nineteenth. Let

$S = \int\limits_^ f (x, y(x), y'(x)) \, dx,$

where

y' = \frac

. Then

S

is extremized if it satisfies the Euler–Lagrange equations

$\frac - \frac \left( \frac \right) = 0.$

But if

S

depends on higher-order derivatives of

y(x)

, that is, if

$S = \int\limits_^ f \left(x, y(x), y'(x), ..., y^(x) \right) \, dx,$

then

y

must satisfy the Euler–Poisson equation,

$\frac - \frac \left( \frac \right) + ... + (-1)^ \frac \left \frac \right 0.$

Poisson'
''Traité de mécanique''
(2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work. Let

q

be the position,

T

be the kinetic energy,

V

the potential energy, both independent of time

t

. Lagrange's equation of motion reads

$\frac \left( \frac \right) - \frac + \frac = 0, ~~~~ i = 1, 2, ... , n.$

Here, the dot notation for the time derivative is used,

\frac = \dot

. Poisson set

L = T - V

. He argued that if

V

is independent of

\dot_i

, he could write

$\frac = \frac,$

giving

$\frac \left (\frac \right) - \frac = 0.$

He introduced an explicit formula for momenta,

$p_i = \frac = \frac.$

Thus, from the equation of motion, he got

$\dot_i = \frac.$

Poisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi. A translation of Poisson'
Treatise on Mechanics
was published in London in 1842. In a paper read at the Institut de France in 1809, Poisson introduced a bracket now named after him. Let

u

and

v

be functions of the canonical variables of motion

q

and

p

. Then their Poisson bracket is given by

$= \frac \frac - \frac \frac.$

Evidently, the operation anti-commutes. More precisely,

, v = -, u /math>. By Hamilton's equations of motion, the total time derivative of u = u (q, p, t) is

$+ \frac, \end$

where

H

is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as

\dot_i =_i, H /math> and \dot_i =_i, H /math>. Suppose u is a constant of motion, then it must satisfy

$= \frac.$

Moreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion. 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 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. In September 1925, Paul Dirac received proofs of a seminal paper by Werner Heisenberg on the new branch of physics known as

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

. Soon he realized that the key idea in Heisenberg's paper was the anti-commutativity of dynamical variables and remembered that the analogous mathematical construction in classical mechanics was Poisson brackets. He found the treatment he needed in E. T. Whittaker's '' Analytical Dynamics of Particles and Rigid Bodies''.

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 The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...

. In 1829 Poisson independently obtained the same result. George Gabriel Stokes re-derived them in 1845 using continuum mechanics. Poisson, Augustin-Louis Cauchy, and Sophie Germain were the main contributors to the theory of elasticity in the nineteenth century. The calculus of variations was frequently used to solve problems.

Wave propagation

Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825).

Thermodynamics

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 of Bessel functions and

Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...

, 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 hi
''Théorie mathématique de la chaleur''
in 1835. During the early 1800s, Pierre-Simon de Laplace developed a sophisticated, if speculative, description of gases based on the old caloric 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. The Newton–Laplace formula makes use of the specific heats of gases at constant volume

c_V

and at constant pressure

c_P

. 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 of Robert Boyle and Joseph Louis Gay-Lussac, Poisson obtained the equation for gases undergoing adiabatic changes, namely

PV^ = \text

, where

P

is the pressure of the gas,

V

its volume, and

\gamma = \frac

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 may be mentioned:
''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)

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.

References

External links

* * * * {{DEFAULTSORT:Poisson, Simeon 1781 births 1840 deaths People from Pithiviers 19th-century French mathematicians École Polytechnique alumni Fellows of the American Academy of Arts and Sciences Fellows of the Royal Society French agnostics French mathematical analysts French fluid dynamicists Members of the French Academy of Sciences Members of the Royal Swedish Academy of Sciences Members of the Chamber of Peers of the July Monarchy French probability theorists Recipients of the Copley Medal Burials at Père Lachaise Cemetery Recipients of the Lalande Prize People associated with electricity

Biography

Contributions

Potential theory

Poisson's equation

Electricity and magnetism

Optics

Pure mathematics and statistics

Mechanics

Analytical mechanics and the calculus of variations

Continuum mechanics and fluid flow

Wave propagation

Thermodynamics

Other works

Interaction with Évariste Galois

See also

References

External links