Poisson bracket

Let M be a symplectic manifoldMathworldPlanetmath with symplectic form Ω. The Poisson bracket is a bilinear operation on the set of differentiable functions on M. In terms of local Darboux coordinates p1,,pn,q1,,qn, the Poisson bracket of two functions is defined as follows:


It can be shown that the value of [f,g] does not depend on the choice of Darboux coordinates. Therefore, the Poisson bracket is a well-defined operation on the symplectic manifold. Also, some authors use a different sign convention — what they call [f,g] is what would be referred to as -[f,g] here.

The Poisson bracket can be defined without reference to a special coordinate systemMathworldPlanetmath as follows:


Here Ω-1 is the inverse of the symplectic form, and its componentsPlanetmathPlanetmathPlanetmath in an arbitrary coordinate system are denoted Ωij.

The Poisson bracket sastisfies several important algebraic identities. It is antisymmetric:


It is a derivation:


It satisfies Jacobi’s identitity:


The Hamilton equations can be expressed elegantly in terms of the Poisson bracket. If X is a smooth function on M, we can describe the time-evolution of X by the equation


If X is a smooth function on ×M, we can describe the time-evolution of X by the more general equation

Title Poisson bracket
Canonical name PoissonBracket
Date of creation 2013-03-22 14:46:04
Last modified on 2013-03-22 14:46:04
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 53D05
Related topic Quantization
Related topic CanonicalQuantization