The Poisson bracket is a bilinear map turning two differentiable functionss over a symplectic space into a function over that symplectic space. In particular, if we have two functionss, A and B, then where ω is the symplectic form, is the two-vector such that if ω is viewed as a map from vectors to 1-forms, is the linear map from 1-forms to vectors satisfying 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.
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 space an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket.
More generally, we can have Poisson brackets over Poisson algebras.
See also Peierls bracket, superPoisson algebra, superPoisson bracket.
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