Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is a bilinear map turning two differentiable functions over a symplectic space into a function over that symplectic space. In particular, if we have two functions, A and B, then

$[A,B]=\tilde{\omega}(dA,dB)$

where ω is the symplectic form, $\tilde{\omega}$ is the two-vector such that if ω is viewed as a map from vectors to 1-forms, $\tilde{\omega}$ is the linear map from 1-forms to vectors satisfying $\omega(\tilde{\omega}(\alpha))= \alpha$ for all 1-forms α and d is the exterior derivative.

The Poisson bracket is used extensively in classical mechanics. See Hamiltonian mechanics for more details.

In phase space coordinates p,q it can be written

$[A,B] = \sum_{i=1}^{N}\left[ \frac{\partial A }{\partial q_{i}} \frac{\partial B}{\partial p_{i}}-\frac{\partial A }{\partial p_{i}}\frac{\partial B}{\partial {q}_{i}}\right]$

It then follows that, for some function f, the total time derivative can be written

$\frac{d}{dt} f=\frac{\partial }{\partial t} f + [\,f,H\,].$

One thing to note is that Poisson brackets are anticommutative and satisfy the Jacobi identity. This makes the space of smooth functions over a symplectic manifold an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket. The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations).

More generally, we can have Poisson brackets over Poisson algebras.

See also:

Categories: Symplectic topology