- Poisson bracket
In

mathematics andclassical mechanics , the**Poisson bracket**is an important operator inHamiltonian mechanics , playing a central role in the definition of the time-evolution of adynamical system in the Hamiltonian formulation. In a more general setting, the Poisson bracket is used to define aPoisson algebra , of which thePoisson manifold s are a special case. These are all named in honour ofSiméon-Denis Poisson .**Canonical coordinates**In

canonical coordinates $(q\_i,p\_j)$ on thephase space , given two functions $f(p\_i,q\_i,t),$ and $g(p\_i,q\_i,t),$, the Poisson bracket takes the form:$\{f,g\}\; =\; sum\_\{i=1\}^\{N\}\; left\; [\; frac\{partial\; f\}\{partial\; q\_\{i\; frac\{partial\; g\}\{partial\; p\_\{i\; -frac\{partial\; f\}\{partial\; p\_\{i\; frac\{partial\; g\}\{partial\; q\_\{i\; ight]\; .$

**Equations of motion**The

Hamilton-Jacobi equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that $f(p,q,t)$ is a function on the manifold. Then one has:$frac\; \{mathrm\{d\{mathrm\{d\}t\}\; f(p,q,t)\; =\; frac\{partial\; f\}\{partial\; t\}\; +frac\; \{partial\; f\}\{partial\; p\}\; frac\; \{mathrm\{d\}p\}\{mathrm\{d\}t\}\; +\; frac\; \{partial\; f\}\{partial\; q\}\; frac\; \{mathrm\{d\}q\}\{mathrm\{d\}t\}.$

Then, by taking $p=p(t)$ and $q=q(t)$ to be solutions to the Hamilton-Jacobi equations $dot\{q\}=\{partial\; H\}/\{partial\; p\}$ and $dot\{p\}=-\{partial\; H\}/\{partial\; q\}$, one may write

:$frac\; \{mathrm\{d\{mathrm\{d\}t\}\; f(p,q,t)\; =\; frac\{partial\; f\}\{partial\; t\}\; +frac\; \{partial\; f\}\{partial\; q\}\; frac\; \{partial\; H\}\{partial\; p\}\; -frac\; \{partial\; f\}\{partial\; p\}\; frac\; \{partial\; H\}\{partial\; q\}\; =\; frac\{partial\; f\}\{partial\; t\}\; +\{f,H\}.$

Thus, the time evolution of a function "f" on a symplectic manifold can be given as a one-parameter family of

symplectomorphism s, with the time "t" being the parameter. Dropping the coordinates, one has:$frac\{mathrm\{d\{mathrm\{d\}t\}\; f=left(frac\{partial\; \}\{partial\; t\}\; -\; \{,H,\; cdot,\}\; ight)f.$

The operator $-\; \{,H,\; cdot,\}$ is known as the

Liouvillian .**Constants of motion**An

integrable dynamical system will haveconstants of motion in addition to the energy. Such constants of motion will commute with theHamiltonian under the Poisson bracket. Suppose some function $f(p,q)$ is a constant of motion. This implies that if $p(t),q(t)$ is atrajectory or solution to theHamilton-Jacobi equations of motion , then one has that $0=frac\{mathrm\{d\}f\}\{mathrm\{d\}t\}$ along that trajectory. Then one has:$0\; =\; frac\; \{mathrm\{d\{mathrm\{d\}t\}\; f(p,q)\; =\; frac\; \{partial\; f\}\{partial\; p\}\; frac\; \{mathrm\{d\}p\}\{mathrm\{d\}t\}\; +\; frac\; \{partial\; f\}\{partial\; q\}\; frac\; \{mathrm\{d\}q\}\{mathrm\{d\}t\}\; =frac\; \{partial\; f\}\{partial\; q\}\; frac\; \{partial\; H\}\{partial\; p\}\; -frac\; \{partial\; f\}\{partial\; p\}\; frac\; \{partial\; H\}\{partial\; q\}\; =\; \{f,H\}$

where, as above, the intermediate step follows by applying the equations of motion. This equation is known as the

Liouville equation . The content ofLiouville's theorem is that the time evolution of a measure (or "distribution function " on the phase space) is given by the above.In order for a Hamiltonian system to be

completely integrable , all of the constants of motion must be in mutual involution.**Definition**Let "M" be

symplectic manifold , that is, amanifold on which there exists asymplectic form : a 2-form $omega$ which is both**closed**($domega\; =\; 0$) and**non-degenerate**, in the following sense: when viewed as a map $omega:\; xi\; in\; mathrm\{vect\}\; [M]\; ightarrow\; i\_xi\; omega\; in\; Lambda^1\; [M]$, $omega$ is invertible to obtain $ilde\{omega\}:\; Lambda^1\; [M]\; ightarrow\; mathrm\{vect\}\; [M]$. Here $d$ is theexterior derivative operation intrinsic to the manifold structure of "M", and $i\_xi\; heta$ is theinterior product or contraction operation, which is equivalent to $heta(xi)$ on 1-forms $heta$.Using the axioms of the

exterior calculus , one can derive::$i\_\{\; [v,\; w]\; \}\; omega\; =\; d(i\_v\; i\_w\; omega)\; +\; i\_v\; d(i\_w\; omega)\; -\; i\_w\; d(i\_v\; omega)\; -\; i\_w\; i\_v\; domega$

Here $[v,\; w]$ denotes the Lie bracket on smooth vector fields, whose properties essentially define the manifold structure of "M".

If "v" is such that $d(i\_v\; omega)\; =\; 0$, we may call it $omega$-coclosed (or just

**coclosed**). Similarly, if $i\_v\; omega\; =\; df$ for some function "f", we may call "v" $omega$-coexact (or just**coexact**). Given that $domega\; =\; 0$, the expression above implies that the Lie bracket of two coclosed vector fields is always a coexact vector field, because when "v" and "w" are both coclosed, the only nonzero term in the expression is $d(i\_v\; i\_w\; omega)$. And because the exterior derivative obeys $d\; circ\; d\; =\; 0$, all coexact vector fields are coclosed; so the Lie bracket is closed both on the space of coclosed vector fields and on the subspace within it consisting of the coexact vector fields. In the language ofabstract algebra , the coclosed vector fields form asubalgebra of theLie algebra of smooth vector fields on "M", and the coexact vector fields form analgebraic ideal of this subalgebra.Given the existence of the inverse map $ilde\{omega\}$, every smooth real-valued function "f" on "M" may be associated with a coexact vector field $ilde\{omega\}(df)$. (Two functions are associated with the same vector field if and only if their difference is in the kernel of "d", i. e., constant on each connected component of "M".) We therefore define the

**Poisson bracket**on $(M,\; omega)$, abilinear operation ondifferentiable functions, under which the $C^infty$ (smooth) functions form analgebra . It is given by::$\{f,g\}\; =\; i\_\{\; ilde\{omega\}(df)\}\; dg\; =\; -\; i\_\{\; ilde\{omega\}(dg)\}\; df\; =\; -\{g,f\}$

The skew-symmetry of the Poisson bracket is ensured by the axioms of the

exterior calculus and the condition $domega\; =\; 0$. Because the map $ilde\{omega\}$ is pointwise linear and skew-symmetric in this sense, some authors associate it with a bivector, which is not an object often encountered in the exterior calculus. In this form it is called thePoisson bivector or thePoisson structure on the symplectic manifold, and the Poisson bracket written simply $\{f,g\}\; =\; ilde\{omega\}(df,\; dg)$.The Poisson bracket on smooth functions corresponds to the Lie bracket on coexact vector fields and inherits its properties. It therefore satisfies the

Jacobi identity ::$\{f,\{g,h\}\}\; +\; \{g,\{h,f\}\}\; +\; \{h,\{f,g\}\}\; =\; 0$

The Poisson bracket $\{f,\_\}$ with respect to a particular scalar field "f" corresponds to the

Lie derivative with respect to $ilde\{omega\}(df)$. Consequently, it is a derivation; that is, it satisfiesLeibniz' law ::$\{f,gh\}\; =\; \{f,g\}h\; +\; g\{f,h\}$

It is a fundamental property of manifolds that the

commutator of the Lie derivative operations with respect to two vector fields is equivalent to the Lie derivative with respect to some vector field, namely, their Lie bracket. The parallel role of the Poisson bracket is apparent from a rearrangement of the Jacobi identity::$\{f,\{g,h\}\}\; -\; \{g,\{f,h\}\}\; =\; \{\{f,g\},h\}$

If the Poisson bracket of "f" and "g" vanishes ($\{f,g\}=0$), then "f" and "g" are said to be in

**mutual involution**, and the operations of taking the Poisson bracket with respect to "f" and with respect to "g" commute.**Lie algebra**The Poisson brackets are

anticommutative . Note also that they satisfy theJacobi identity . This makes the space ofsmooth function s on asymplectic manifold an infinite-dimensionalLie algebra with the Poisson bracket acting as the Lie bracket. The correspondingLie group is the group ofsymplectomorphisms of the symplectic manifold (also known ascanonical transformation s).Given a differentiable

vector field "X" on thetangent bundle , let $P\_X$ be itsconjugate momentum . The conjugate momentum mapping is aLie algebra anti-homomorphism from the Poisson bracket to theLie bracket ::$\{P\_X,P\_Y\}=-P\_\{\; [X,Y]\; \}.,$

This important result is worth a short proof. Write a vector field "X" at point "q" in the

configuration space as:$X\_q=sum\_i\; X^i(q)\; frac\{partial\}\{partial\; q^i\}$

where the $partial\; /partial\; q^i$ is the local coordinate frame. The conjugate momentum to "X" has the expression

:$P\_X(q,p)=sum\_i\; X^i(q)\; ;p\_i$

where the $p\_i$ are the momentum functions conjugate to the coordinates. One then has, for a point $(q,p)$ in the

phase space ,:$\{P\_X,P\_Y\}(q,p)=\; sum\_i\; sum\_j\; \{X^i(q)\; ;p\_i,\; Y^j(q);p\_j\; \}$:::$=sum\_\{ij\}\; p\_i\; Y^j(q)\; frac\; \{partial\; X^i\}\{partial\; q^j\}\; -\; p\_j\; X^i(q)\; frac\; \{partial\; Y^j\}\{partial\; q^i\}$:::$=\; -\; sum\_i\; p\_i\; ;\; [X,Y]\; ^i(q)$ :::$=\; -\; P\_\{\; [X,Y]\; \}(q,p).\; ,$

The above holds for all $(q,p)$, giving the desired result.

**ee also***

Lagrange bracket

*Moyal bracket

*Peierls bracket

*Poisson superalgebra

*Poisson superbracket

*Dirac bracket **References***

*

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Bracket (disambiguation)**— Bracket may refer to: * Bracket, one of a class of punctuation marks used in pairs to set apart or interject text within other text ** Square bracket, one type of punctuation bracket (primarily American usage) ** Parenthesis, another type of… … Wikipedia**Poisson manifold**— In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra. Since their introduction by André… … Wikipedia**Poisson algebra**— In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central… … Wikipedia**Poisson ring**— In mathematics, a Poisson ring A is a commutative ring on which a binary operation [,] , known as the Poisson bracket, is defined. Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms… … Wikipedia**Bracket**— 〈 redirects here. It is not to be confused with く, a Japanese kana. This article is about bracketing punctuation marks. For other uses, see Bracket (disambiguation). Due to technical restrictions, titles like :) redirect here. For typographical… … Wikipedia**Bracket (mathematics)**— In mathematics, various typographical forms of brackets are frequently used in mathematical notation such as parentheses ( ), square brackets [ ] , curly brackets { }, and angle brackets < >. In the typical use, a mathematical expression is… … Wikipedia**Dirac bracket**— The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to correctly treat systems with second class constraints in Hamiltonian mechanics and canonical quantization. It is an important part of Dirac s development of… … Wikipedia**Moyal bracket**— In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase space star product. The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a… … Wikipedia**Schouten-Nijenhuis bracket**— In differential geometry, the Schouten Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different… … Wikipedia**Schouten–Nijenhuis bracket**— In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different… … Wikipedia