# Poisson bracket

Let $M$ be a symplectic manifold^{} with symplectic form $\mathrm{\Omega}$. The *Poisson bracket* is a bilinear operation on the set of differentiable functions on $M$. In terms of local Darboux coordinates ${p}_{1},\mathrm{\dots},{p}_{n},{q}_{1},\mathrm{\dots},{q}_{n}$, the Poisson bracket of two functions is defined as follows:

$$[f,g]=\sum _{i=1}^{n}\frac{\partial f}{\partial {q}_{i}}\frac{\partial g}{\partial {p}_{i}}-\frac{\partial f}{\partial {p}_{i}}\frac{\partial g}{\partial {q}_{i}}$$ |

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:

$$[f,g]={\mathrm{\Omega}}^{-1}(df,dg)=\sum _{i=1}^{2n}{\mathrm{\Omega}}^{ij}\frac{\partial f}{\partial {x}_{i}}\frac{\partial g}{\partial {x}_{j}}$$ |

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

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

$$\frac{dX}{dt}=[X,H]$$ |

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

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

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 |