Editing Siméon Denis Poisson (section)

==Contributions==
=== Potential theory ===

==== Poisson's equation ====
[[File:Front cover of Griffiths' Electrodynamics.jpg|thumb|Poisson's equations for electricity (top) and magnetism (bottom) in SI units on the front cover of [[Introduction to Electrodynamics|an undergraduate textbook]].]]
In the theory of potentials, [[Poisson's equation]], 

: <math> \nabla^2 \phi = - 4 \pi \rho, \; </math>

is a well-known generalization of [[Laplace's equation]] of the second order [[partial differential equation]] <math> \nabla^2 \phi = 0</math> for [[potential]] <math>\phi</math>. 

If <math> \rho(x, y, z) </math> is a [[continuous function]] and if for <math> r \rightarrow \infty </math> (or if a point 'moves' to [[Extended real number line|infinity]]) a function <math> \phi </math> goes to 0 fast enough, the solution of Poisson's equation is the [[Newtonian potential]]

:<math> \phi = - {1\over 4 \pi} \iiint \frac{\rho (x, y, z)}{ r} \, dV, \; </math>

where <math> r </math> is a distance between a volume element <math> dV </math>and a point <math> P </math>. The integration runs over the whole space.

Poisson's equation was first published in the ''Bulletin de la société philomatique'' (1813).<ref name="EB1911" /> 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).<ref name="EB1911" />

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.<ref>{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|location=United States of America|pages=682–4|chapter=28.4: The Potential Equation and Green's Theorem}}</ref>

==== 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 [[Charge conservation|conservation of charge]]; [[Charles-Augustin de Coulomb]] had enunciated his [[Coulomb's law|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>.<ref name=":13">{{Cite book|last=Baigrie|first=Brian|title=Electricity and Magnetism: A Historical Perspective|publisher=Greenwood Press|year=2007|isbn=978-0-313-33358-3|location=United States of America|pages=47|chapter=Chapter 5: From Effluvia to Fluids}}</ref> Poisson's work on potential theory inspired [[George Green (mathematician)|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|Ampère's law]] and the [[Biot–Savart law|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.<ref name=":132">{{Cite book|last=Baigrie|first=Brian|title=Electricity and Magnetism: A Historical Perspective|publisher=Greenwood Press|year=2007|isbn=978-0-313-33358-3|location=United States of America|pages=72|chapter=Chapter 7: The Current and the Needle}}</ref> 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.<ref name=":02">{{Cite book|last=Baigrie|first=Brian|title=Electricity and Magnetism: A Historical Perspective|publisher=Greenwood Press|year=2007|isbn=978-0-313-33358-3|location=United States of America|pages=88|chapter=Chapter 8: Forces and Fields}}</ref>

=== Optics ===
[[File:A_photograph_of_the_Arago_spot.png|thumb|right|200px|Photo of the Arago spot in a shadow of a 5.8 mm circular obstacle.]]
Poisson was a member of the academic "old guard" at the [[French Academy of Sciences|Académie royale des sciences de l'Institut de France]], who were staunch believers in the [[Wave–particle duality|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]]. 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]].<ref name="fresnel1868">{{Citation|last=Fresnel|first=A.J.|title=OEuvres Completes 1|url=https://books.google.com/books?id=3QgAAAAAMAAJ|year=1868|publication-place=Paris|publisher=Imprimerie impériale}}</ref>

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, [[François Arago|Dominique-François-Jean Arago]], performed the experiment. He molded a 2&nbsp;mm metallic disk to a glass plate with wax.<ref name="fresnel1868_arago">{{Citation|last=Fresnel|first=A.J.|title=OEuvres Completes 1|url=https://books.google.com/books?id=3QgAAAAAMAAJ|page=369|year=1868|publication-place=Paris|publisher=Imprimerie impériale}}</ref> 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]]<ref name="fresnel1868_arago" /> and [[Giacomo F. Maraldi]]<ref name="maraldi1723">{{Citation|last=Maraldi|first=G.F.|title='Diverses expèriences d'optique' in Mémoires de l'Académie Royale des Sciences|url=http://gallica.bnf.fr/ark:/12148/bpt6k3592w/f300.image.langFR|page=111|year=1723|publisher=Imprimerie impériale}}</ref> a century earlier.

=== Pure mathematics and statistics ===
In [[pure mathematics]], Poisson's most important works were his series of memoirs on [[definite integral]]s 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 integral]]s.<ref name="EB1911" />

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]] is named after him.<ref name="EB1911" />

In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so.<ref>{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|pages=633|chapter=27.4: The Foundation of Complex Function Theory}}</ref>

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]].<ref name=":32">{{Cite journal|last=Katz|first=Victor|date=May 1979|title=A History of Stokes' Theorem|url=https://www.jstor.org/stable/2690275|journal=Mathematics Magazine|volume=52|issue=3|pages=146–156|doi=10.1080/0025570X.1979.11976770|jstor=2690275}}</ref>

=== Mechanics ===
{{Classical mechanics|cTopic=Scientists}}

==== 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.<ref name=":0">{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|chapter=Chapter 30: The Calculus of Variations in the Nineteenth Century}}</ref>

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 it 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>y</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><ref>{{Cite book|last=Kot|first=Mark|title=A First Course in the Calculus of Variations|publisher=American Mathematical Society|year=2014|isbn=978-1-4704-1495-5|chapter=Chapter 4: Basic Generalizations}}</ref></blockquote>Poisson's [http://catalog.hathitrust.org/Record/000387664 ''Traité de mécanique''] (2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work.<ref name="EB1911" />

Let <math> q</math> be the position, <math>T</math> be the kinetic energy, <math>V</math> the potential energy, both independent of time <math>t</math>. Lagrange's equation of motion reads<ref name=":0" /><blockquote><math>\frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}_i} \right) - \frac{\partial T}{\partial q_i} + \frac{\partial V}{\partial q_i} = 0, ~~~~ i = 1, 2, ... , n.</math></blockquote>Here, the dot notation for the time derivative is used, <math>\frac{dq}{dt} = \dot{q}</math>. Poisson set <math>L = T - V</math>.<ref name=":0" /> He argued that if <math>V</math> is independent of <math>\dot{q}_i</math>, he could write<blockquote><math>\frac{\partial L}{\partial \dot{q}_i} = \frac{\partial T}{\partial \dot{q}_i},</math></blockquote>giving<ref name=":0" /> <blockquote><math>\frac{d}{dt} \left (\frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0.</math></blockquote>He introduced an explicit formula for [[momentum|momenta]],<ref name=":0" /><blockquote><math> p_i = \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial T}{\partial \dot{q}_i}.</math></blockquote>Thus, from the equation of motion, he got<ref name=":0" /><blockquote><math> \dot{p}_i = \frac{\partial L}{\partial q_i}.</math></blockquote>Poisson's text influenced the work of [[William Rowan Hamilton]] and [[Carl Gustav Jacob Jacobi]]. A translation of Poisson's [https://books.google.com/books?id=lksn7QwUZsQC&q=Poisson+mechanics Treatise on Mechanics] was published in London in 1842. 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><ref name=":1">{{Cite book|last=Goldstein|first=Herbert|title=Classical Mechanics|title-link=Classical Mechanics (Goldstein book)|publisher=Addison-Wesley Publishing Company|year=1980|isbn=0-201-02918-9|pages=397, 399, 406–7|chapter=Chapter 9: Canonical Transformations|author-link=Herbert Goldstein}}</ref></blockquote>Evidently, the operation anti-commutes. More precisely, <math>[u, v] = -[v, u]</math>.<ref name=":1" /> By [[Hamiltonian mechanics|Hamilton's equations of motion]], the total time derivative of <math>u = u (q, p, t)</math> is<blockquote><math>\begin{align}
\frac{du}{dt} &= \frac{\partial u}{\partial q_i} \dot{q}_i + \frac{\partial u}{\partial p_i} \dot{p}_i + \frac{\partial u}{\partial t} \\[6pt]
&= \frac{\partial u}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial u}{\partial p_i} \frac{\partial H}{\partial q_i} + \frac{\partial u}{\partial t} \\[6pt]
&= [u, H] + \frac{\partial u}{\partial t},
\end{align}</math></blockquote>where <math>H</math> is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as <math>\dot{q}_i = [q_i, H]</math> and <math>\dot{p}_i = [p_i, H]</math>.<ref name=":1" />

Suppose <math>u</math> is a [[constant of motion]], then it must satisfy<blockquote><math>[H, u] = \frac{\partial u}{\partial t}.</math></blockquote>Moreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion.<ref name=":1" />

In September 1925, [[Paul Dirac]] received proofs of a seminal paper by [[Werner Heisenberg]] on the new branch of physics known as [[quantum mechanics]]. 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]]''.<ref>{{Cite book|last=Farmelo|first=Graham|title=The Strangest Man: the Hidden Life of Paul Dirac, Mystic of the Atom|publisher=Basic Books|year=2009|isbn=978-0-465-02210-6|location=Great Britain|pages=83–88}}</ref><ref name="Coutinho12">{{Cite journal|last=Coutinho|first=S. C.|date=1 May 2014|title=Whittaker's analytical dynamics: a biography|url=https://doi.org/10.1007/s00407-013-0133-1|journal=Archive for History of Exact Sciences|language=en|volume=68|issue=3|pages=355–407|doi=10.1007/s00407-013-0133-1|s2cid=122266762|issn=1432-0657}}</ref>

==== Continuum mechanics and fluid flow ====
{{unsolved|physics|Under what conditions do [[Navier–Stokes existence and smoothness|solutions to the Navier–Stokes equations exist and are smooth]]? This is a [[Millennium Prize Problems|Millennium Prize Problem]] in mathematics.}}
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. [[Sir George Stokes, 1st Baronet|George Gabriel Stokes]] re-derived them in 1845 using continuum mechanics.<ref>{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|location=United States of America|pages=696–7|chapter=28.7: Systems of Partial Differential Equations}}</ref> 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.<ref name=":0" />

==== Wave propagation ====
Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825).<ref name="EB1911" />

=== 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 function]]s and [[Legendre polynomials]], 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 his [http://catalog.hathitrust.org/Record/001988678 ''Théorie mathématique de la chaleur''] in 1835.<ref>{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|location=United States of America|pages=678–9|chapter=28.2: The Heat Equation and Fourier Series}}</ref>

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 [[Speed of sound#Speed of sound in ideal gases and air|Newton–Laplace formula]] makes use of the specific heats of gases at constant volume <math>c_V</math>and at constant pressure <math>c_P</math>. 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 process|adiabatic changes]], namely <math>PV^{\gamma} = \text{constant}</math>, where <math>P</math> is the pressure of the gas, <math>V</math> its volume, and <math>\gamma = \frac{c_P}{c_V}</math>.<ref name=":8">{{Cite book|last=Lewis|first=Christopher|title=Heat and Thermodynamics: A Historical Perspective|publisher=Greenwood Press|year=2007|isbn=978-0-313-33332-3|location=United States of America|chapter=Chapter 2: The Rise and Fall of the Caloric Theory}}</ref>

=== 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:<ref name=EB1911>{{EB1911|wstitle=Poisson, Siméon Denis|volume=21|page=896|inline=1}}</ref>
* [http://catalog.hathitrust.org/Record/001479713 ''Nouvelle théorie de l'action capillaire''] (4to, 1831);
* [http://catalog.hathitrust.org/Record/000577641 ''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 ''[https://babel.hathitrust.org/cgi/pt?id=ucm.5322769579;view=1up;seq=684 Oeuvres complétes de François Arago, Vol. 2]''
*[http://gallica.bnf.fr/ark:/12148/bpt6k3223j/f613.image 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]]
*''[https://libserv.aip.org/ipac20/ipac.jsp?session=X670S28934P65.732673&menu=search&aspect=power&npp=10&ipp=20&spp=20&profile=rev-all&ri=2&source=%7E%21horizon&index=.GW&term=Recherches+sur+le+Mouvement+des+Projectiles+dans+l%27Air%2C+en+ayant+%C3%A9gard+a+leur+figure+et+leur+rotation%2C+et+a+l%27influence+du+mouvement+diurne+de+la+terre+&x=2&y=17&aspect=power 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)
<gallery>
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)
</gallery>