Editing Siméon Denis Poisson (section)

=== 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" />