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In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras , where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this ...
A Poisson bracket (or Poisson structure) on is an -bilinear map ... Examples of regular Poisson structures include trivial and nondegenerate structures (see below).
The Poisson bracket acts as a derivation of the associative product ⋅, so that for any three elements x, y and z in the algebra, one has {x, y ⋅ z} = {x, y} ⋅ z + y ⋅ {x, z}. The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.
The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra. If F and G are smooth functions on M then the smooth function ω(J(dF), J(dG)) is properly defined; it is called a Poisson bracket of functions F and G and is denoted {F, G}. The Poisson bracket ...
Another useful result is Poisson's theorem, which states that if two quantities and are constants of motion, so is their Poisson bracket {,}. A system with n degrees of freedom, and n constants of motion, such that the Poisson bracket of any pair of constants of motion vanishes, is known as a completely integrable system .
In a constrained Hamiltonian system, a dynamical quantity is second-class if its Poisson bracket with at least one constraint is nonvanishing. A constraint that has a nonzero Poisson bracket with at least one other constraint, then, is a second-class constraint. See Dirac brackets for diverse illustrations.
It is an associative, non-commutative product, ★, on the functions on , equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra.
In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups.