Poisson Bracket

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, the so-called "canonical transformations", which maps canonical coordinate systems into canonical coordinate systems. (A "canonical coordinate system" consists of canonical position/momentum variables, that satisfy canonical Poisson-bracket relations.) Note that the set of possible canonical transformations is always very rich. For instance, often it is possible to choose the Hamiltonian itself H = H(q,p;t) 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. These are all named in honour of Siméon-Denis Poisson.