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
, is the linear map from 1-forms
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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