# 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 PoissonBracket 2013-03-22 14:46:04 2013-03-22 14:46:04 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Definition msc 53D05 Quantization CanonicalQuantization