Siméon Denis Poisson

Poisson's equations for electricity (top) and magnetism (bottom) in SI units on the front cover of an undergraduate textbook.

Poisson's well-known generalization of Laplace's second order partial differential equation for potential ${\displaystyle \phi }$

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

is known as Poisson's equation after him, was first published in the Bulletin de la société philomatique (1813).[3] If ${\displaystyle \rho =0}$, we retrieve Laplace's equation

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

If ${\displaystyle \rho (x,y,z)}$ is a continuous function and if for ${\displaystyle r\rightarrow \infty }$ (or if a point 'moves' to infinity) a function ${\displaystyle \phi }$ goes to 0 fast enough, a solution of Poisson's equation is the Newtonian potential of a function ${\displaystyle \rho (x,y,z)}$

${\displaystyle \phi =-{1 \over 4\pi }\iiint {\frac {\rho (x,y,z)}{r}}\,dV\;}$

where ${\displaystyle r}$ is a distance between a volume element ${\displaystyle dV}$and a point ${\displaystyle P}$. The integration runs over the whole space.

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).[3]

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.[7]

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 conservation of charge; Charles-Augustin de Coulomb had enunciated his 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 ${\displaystyle V}$ to the electric charge density ${\displaystyle \rho }$.[8] 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.[9] 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.[10]

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 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. 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.[11]

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.[12] 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[12] and Giacomo F. Maraldi[13] a century earlier.

In pure mathematics, 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 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.[3]

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.[3]

In 1820 Poisson studied integrations along paths in the complex plane, becoming the first person to do so.[14]

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.[15]

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.[16]

Let

${\displaystyle S=\int \limits _{a}^{b}f(x,y(x),y'(x))\,dx,}$

where ${\displaystyle y'={\frac {dy}{dx}}}$. Then ${\displaystyle S}$ is extremized if it satisfies the Euler–Lagrange equations

${\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)=0.}$

But if ${\displaystyle S}$ depends on higher-order derivatives of ${\displaystyle y(x)}$, that is, if

${\displaystyle S=\int \limits _{a}^{b}f\left(x,y(x),y'(x),...,y^{(n)}(x)\right)\,dx,}$

then ${\displaystyle y}$ must satisfy the Euler–Poisson equation,

${\displaystyle {\frac {\partial f}{\partial y}}-{\frac {d}{dx}}\left({\frac {\partial f}{\partial y'}}\right)+...+(-1)^{n}{\frac {d^{n}}{dx^{n}}}\left[{\frac {\partial f}{\partial y^{(n)}}}\right]=0.}$[17]

Poisson's Traité de mécanique (2 vols. 8vo, 1811 and 1833) was written in the style of Laplace and Lagrange and was long a standard work.[3] Let ${\displaystyle q}$ be the position, ${\displaystyle T}$ be the kinetic energy, ${\displaystyle V}$ the potential energy, both independent of time ${\displaystyle t}$. Lagrange's equation of motion reads[16]

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial T}{\partial {\dot {q}}_{i}}}\right)-{\frac {\partial T}{\partial q_{i}}}+{\frac {\partial V}{\partial q_{i}}}=0,~~~~i=1,2,...,n.}$

Here, the dot notation for the time derivative is used, ${\displaystyle {\frac {dq}{dt}}={\dot {q}}}$. Poisson set ${\displaystyle L=T-V}$.[16] He argued that if ${\displaystyle V}$ is independent of ${\displaystyle {\dot {q}}_{i}}$, he could write

${\displaystyle {\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial T}{\partial {\dot {q}}_{i}}},}$

giving[16]

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)-{\frac {\partial L}{\partial q_{i}}}=0.}$

He introduced an explicit formula for momenta,[16]

${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}={\frac {\partial T}{\partial {\dot {q}}_{i}}}.}$

Thus, from the equation of motion, he got[16]

${\displaystyle {\dot {p}}_{i}={\frac {\partial L}{\partial q_{i}}}.}$

Poisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842. Let ${\displaystyle u}$ and ${\displaystyle v}$ be functions of the canonical variables of motion ${\displaystyle q}$ and ${\displaystyle p}$. Then their Poisson bracket is given by

${\displaystyle [u,v]={\frac {\partial u}{\partial q_{i}}}{\frac {\partial v}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial v}{\partial q_{i}}}.}$[18]

Evidently, the operation anti-commutes. More precisely, ${\displaystyle [u,v]=-[v,u]}$.[18] By Hamilton's equations of motion, the total time derivative of ${\displaystyle u=u(q,p,t)}$ is

${\displaystyle {\begin{aligned}{\frac {du}{dt}}&={\frac {\partial u}{\partial q_{i}}}{\dot {q}}_{i}+{\frac {\partial u}{\partial p_{i}}}{\dot {p}}_{i}+{\frac {\partial u}{\partial t}}\\[6pt]&={\frac {\partial u}{\partial q_{i}}}{\frac {\partial H}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial H}{\partial q_{i}}}+{\frac {\partial u}{\partial t}}\\[6pt]&=[u,H]+{\frac {\partial u}{\partial t}},\end{aligned}}}$

where ${\displaystyle H}$ is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as ${\displaystyle {\dot {q}}_{i}=[q_{i},H]}$ and ${\displaystyle {\dot {p}}_{i}=[p_{i},H]}$.[18] Suppose ${\displaystyle u}$ is a constant of motion, then it must satisfy

${\displaystyle [H,u]={\frac {\partial u}{\partial t}}.}$

Moreover, Poisson's theorem states the Poisson bracket of any two constants of motion is also a constant of motion.[18]

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.[19][20]

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. George Gabriel Stokes re-derived them in 1845 using continuum mechanics.[21] 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.[16]

Poisson also published a memoir on the theory of waves (Mém. ft. l'acad., 1825).[3]

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, 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 Théorie mathématique de la chaleur in 1835.[22]

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 ${\displaystyle c_{V}}$and at constant pressure ${\displaystyle c_{P}}$. 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 ${\displaystyle PV^{\gamma }={\text{constant}}}$, where ${\displaystyle P}$ is the pressure of the gas, ${\displaystyle V}$ its volume, and ${\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}$.[23]

Mémoire sur le calcul numerique des integrales définies, 1826

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:[3]

• 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