mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Poisson ring is a

commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...

on which an

anticommutative In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...

and distributive

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

cdot,\cdot /math> satisfying the

Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...

and the

product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...

is defined. Such an operation is then known as the

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

of the Poisson ring. Many important operations and results of

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...

and

Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...

may be formulated in terms of the Poisson bracket and, hence, apply to

Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central i ...

s as well. This observation is important in studying the

classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

—the

non-commutative algebra In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not a ...

operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

has the Poisson algebra of functions on a

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...

as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

Definition

The Poisson bracket must satisfy the identities *

,g = -,f /math> (skew symmetry)
* + g, h =,h +,h (distributivity)
* g,h = f,h +,h (

derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...

) *

,[g,h_+_[g,[h,f.html"_;"title=",h.html"_;"title=",[g,h">,[g,h_+_[g,[h,f">,h.html"_;"title=",[g,h">,[g,h_+_[g,[h,f_+_[h,[f,g.html" ;"title=",h">,[g,h_+_[g,[h,f.html" ;"title=",h.html" ;"title=",[g,h">,[g,h + [g,[h,f">,h.html" ;"title=",[g,h">,[g,h + [g,[h,f + [h,[f,g">,h">,[g,h_+_[g,[h,f.html" ;"title=",h.html" ;"title=",[g,h">,[g,h + [g,[h,f">,h.html" ;"title=",[g,h">,[g,h + [g,[h,f + [h,[f,g = 0

(

) for all

f,g,h

in the ring. A

is a Poisson ring that is also an algebra over a field. In this case, add the extra requirement :

[sf,g] = s[f,g]

for all scalars ''s''. For each ''g'' in a Poisson ring ''A'', the operation

ad_g

defined as

ad_g(f) = [f,g]

is a

. If the set

\

generates the set of derivations of ''A'', then ''A'' is said to be non-degenerate. If a non-degenerate Poisson ring is ring isomorphism, isomorphic as a commutative ring to the algebra of smooth functions on a manifold ''M'', then ''M'' must be a

and

cdot,\cdot /math> is the Poisson bracket defined by the

symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

References

* {{DEFAULTSORT:Poisson Ring Ring theory Symplectic geometry