Siméon Denis Poisson

Poisson's well-known generalization of Laplace's second order partial differential equation for potential

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

is known as Poisson's equation after him, was first published in the Bulletin de la société philomatique (1813).[1] 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 ${\displaystyle \rho (x,y,z)}$

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

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

${\displaystyle \phi (\xi \eta )={1 \over 4\pi }\int _{0}^{2\pi }{R^{2}-\rho ^{2} \over R^{2}+\rho ^{2}-2R\rho \cos(\psi -\chi )}\phi (\chi )\,d\chi ,\;}$

where ${\displaystyle \xi =\rho \cos \psi }$, ${\displaystyle \eta =\rho \sin \phi }$, and ${\displaystyle \phi }$ 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:

${\displaystyle G(x,y,z;\xi ,\eta ,\zeta )={1 \over r}-{R \over r_{1}\rho }\;,}$

where

${\displaystyle \rho ={\sqrt {\xi ^{2}+\eta ^{2}+\zeta ^{2}}}}$ 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:

${\displaystyle \phi (\xi ,\eta ,\zeta )={1 \over 4\pi }\iint _{S}{R^{2}-\rho ^{2} \over Rr^{3}}\phi \,ds\;.}$

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).[1]

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, ${\displaystyle \phi }$ in 3-dimensional space is:

${\displaystyle \nabla ^{2}\phi =\rho (x,y,z)\;.}$

Consider for instance Poisson's equation for surface electrical potential, ${\displaystyle \Psi }$ as a function of the density of electric charge, ${\displaystyle \rho _{e}}$ at a particular point:

${\displaystyle \nabla ^{2}\Psi ={\partial ^{2}\Psi \over \partial x^{2}}+{\partial ^{2}\Psi \over \partial y^{2}}+{\partial ^{2}\Psi \over \partial z^{2}}=-{\rho _{e} \over \varepsilon \varepsilon _{0}}.\;}$

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

${\displaystyle \nabla ^{2}\Psi ={n_{0}e \over \varepsilon \varepsilon _{0}}\left(e^{e\Psi (x,y,z)/k_{B}T}-e^{-e\Psi (x,y,z)/k_{B}T}\right),\;}$

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

${\displaystyle {1 \over r^{2}}{d \over dr}\left(r^{2}{d\Psi \over dr}\right)={n_{0}e \over \varepsilon \varepsilon _{0}}\left(e^{e\Psi (r)/k_{B}T}-e^{-e\Psi (r)/k_{B}T}\right),\;}$

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

${\displaystyle {\sqrt {\phi _{ik}}}=\rho (x,y,z,ct).\;}$

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.[5]

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.[6] 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[6] and Giacomo F. Maraldi[7] a century earlier.

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).[1] The Poisson distribution in probability theory is named after him.

In his Traité de mécanique (2 vols. 8vo, 1811 and 1833), which was written in the style of Laplace and Lagrange and was long a standard work,[1] he showed many novelties such as an explicit usage of momenta:

${\displaystyle p_{i}={\partial T \over {\partial (\partial q_{i}/\partial t})},}$

which influenced the work of Hamilton and Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842.

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:[1]

• 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.

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