Editing Poisson bracket (section)

{{short description|Operation in Hamiltonian mechanics}}
[[File:Simeon Poisson.jpg|thumb|Siméon Denis Poisson]]
{{Classical mechanics|expanded=Formulations}}
In [[mathematics]] and [[classical mechanics]], the '''Poisson bracket''' is an important [[binary operation]] in [[Hamiltonian mechanics]], playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian [[dynamical system]]. The Poisson bracket also distinguishes a certain class of coordinate transformations, called ''[[canonical transformations]]'', which map [[Canonical coordinates|canonical coordinate systems]] into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by <math>q_i</math> and <math>p_i</math>, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself <math>\mathcal H =\mathcal H(q, p, t)</math> as one of the new canonical momentum coordinates.

In a more general sense, the Poisson bracket is used to define a [[Poisson algebra]], of which the algebra of functions on a [[Poisson manifold]] is a special case. There are other general examples, as well: it occurs in the theory of [[Lie algebra]]s, where the [[tensor algebra]] of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the [[universal enveloping algebra]] article. Quantum deformations of the universal enveloping algebra lead to the notion of [[quantum group]]s.

All of these objects are named in honor of French mathematician [[Siméon Denis Poisson]]. He introduced the Poisson bracket in his 1809 treatise on mechanics.<ref name="Poisson1809">[[#poisson1881|S. D. Poisson (1809)]]</ref><ref name="Marle2009">[[#marle2009|C. M. Marle (2009)]]</ref>