Poisson's equation

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]

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 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]

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.^[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]

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. 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]

Wave propagation

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