Poisson bracket

Let $M$ be a symplectic manifold with symplectic form $\Omega$ . The Poisson bracket is a bilinear operation on the set of differentiable functions on $M$ . In terms of local Darboux coordinates $p_{1},\ldots,p_{n},q_{1},\ldots,q_{n}$ , the Poisson bracket of two functions is defined as follows:

[f,g]=\sum_{i=1}^{n}{\partial f\over\partial q_{i}}{\partial g\over\partial p_% {i}}-{\partial f\over\partial p_{i}}{\partial g\over\partial q_{i}}

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 system as follows:

[f,g]=\Omega^{-1}(df,dg)=\sum_{i=1}^{2n}\Omega^{ij}{\partial f\over\partial x_% {i}}{\partial g\over\partial x_{j}}

Here $\Omega^{-1}$ is the inverse of the symplectic form, and its components in an arbitrary coordinate system are denoted $\Omega^{ij}$ .

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

[f,g]=-[g,f]

It is a derivation:

[fg,h]=f[g,h]+g[f,h]

It satisfies Jacobi’s identitity:

[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0

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

{dX\over dt}=[X,H]

If $X$ is a smooth function on $\mathbb{R}\times M$ , we can describe the time-evolution of $X$ by the more general equation

{dX\over dt}={\partial X\over\partial t}-[X,H]

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