PlanetMath: Poisson bracket

(more info)

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Poisson bracket

(Definition)

Let be a symplectic manifold with symplectic form $\Omega$ . The Poisson bracket is a bilinear operation on the set of differentiable functions on . 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:

$\displaystyle [f,g] = \sum_{i=1}^n {\partial f \over \partial q_i} {\partial g ... ...\partial p_i} - {\partial f \over \partial p_i} {\partial g \over \partial q_i}$

It can be shown that the value of

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

is what would be referred to as

here.

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

$\displaystyle [f,g] = \Omega^{-1} (df, dg) = \sum_{i=1}^{2n} \Omega^{ij} {\partial f \over \partial x_i} {\partial g \over \partial x_j}$

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:

$\displaystyle [f,g] = -[g,f]$

It is a derivation:

$\displaystyle [fg,h] = f[g,h] + g[f,h]$

It satisfies Jacobi's identitity:

$\displaystyle [f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0$

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