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# 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

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

## Mathematics Subject Classification

53D05*no label found*

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