In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. In a more general setting, the Poisson bracket is used to define a Poisson algebra, of which the Poisson manifolds are a special case. These are all named in honour of Siméon-Denis Poisson.

Definition

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The Poisson bracket is a bilinear map turning two differentiable functions on a symplectic manifold into a function on that symplectic manifold. In particular, if we have two functions, f and g, then the Poisson bracket

$\backslash \{f,g\backslash \}=\backslash tilde\{\backslash omega\}(df,dg)$

where ω is the symplectic form, $\backslash tilde\{\backslash omega\}$ is the two-vector such that if ω is viewed as a map from vectors to 1-forms, $\backslash tilde\{\backslash omega\}$ is the linear map from 1-forms to vectors satisfying $\backslash omega(\backslash tilde\{\backslash omega\}(\backslash alpha))=\; \backslash alpha$ for all 1-forms α and d is the exterior derivative. The bivector $\backslash tilde\{\backslash omega\}$ is sometimes called the Poisson bivector or the Poisson structure on the symplectic manifold.

The Poisson brackets are anti-symmetric:

$\backslash \{f,g\backslash \}\; =\; -\backslash \{g,f\backslash \}$

and satisfy the Jacobi identity:

$\backslash \{f,\backslash \{g,h\backslash \}\backslash \}\; +\backslash \{g,\backslash \{h,f\backslash \}\backslash \}\; +\; \backslash \{h,\backslash \{f,g\backslash \}\backslash \}=0$

They are also a derivation; that is, they satisfy Leibniz' law:

$\backslash \{fg,h\backslash \}=f\backslash \{g,h\backslash \}\; +\; \backslash \{f,h\backslash \}g$

If the Poisson bracket of f and g vanishes: $\backslash \{f,g\backslash \}=0$, then f and g are said to be in mutual involution.

Canonical coordinates

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In canonical coordinates $(q^i,p\_j)$ on the phase space, the Poisson bracket takes the form

$\backslash \{f,g\backslash \}\; =\; \backslash sum\_\{i=1\}^\{N\}\; \backslash left[$

\frac{\partial f}{\partial q^{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q^{i}} \right].

Equations of motion

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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

$\backslash frac\; \{d\}\{dt\}\; f(p,q,t)\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +$

\frac {\partial f}{\partial p} \frac {dp}{dt} + \frac {\partial f}{\partial q} \frac {dq}{dt}

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

$\backslash frac\; \{d\}\{dt\}\; f(p,q,t)\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; t\}\; +$

\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 symplectomorphisms, with the time t being the parameter. Droping the coordinates, one has

$\backslash frac\{d\}\{dt\}\; f=$

\left(\frac{\partial }{\partial t} - \{\,H, \cdot\,\}\right)f.

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

Constants of motion

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An integrable dynamical system will have constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function $f(p,q)$ is a constant of motion. This implies that if $p(t),q(t)$ is a trajectory or solution to the Hamilton-Jacobi equations of motion, then one has that $0=df/dt$ along that trajectory. Then one has

$0\; =\; \backslash frac\; \{d\}\{dt\}\; f(p,q)\; =$

\frac {\partial f}{\partial p} \frac {dp}{dt} + \frac {\partial f}{\partial q} \frac {dq}{dt} = \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 of Liouville'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.

Lie algebra

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The Poisson brackets are anticommutative. Note also that they satisfy the Jacobi identity. This makes the space of smooth functions on a symplectic manifold an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket. The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations).

Given a differentiable vector field X on the tangent bundle, let $P\_X$ be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:

$\backslash \{P\_X,P\_Y\backslash \}=-P\_\{*\}.\backslash ,$

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

$X\_q=\backslash sum\_i\; X^i(q)\; \backslash frac\{\backslash partial\}\{\backslash partial\; q^i\}$

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

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

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

$\backslash \{P\_X,P\_Y\backslash \}(q,p)=\; \backslash sum\_i\; \backslash sum\_j\; \backslash \{X^i(q)\; \backslash ;p\_i,\; Y^j(q)\backslash ;p\_j\; \backslash \}$

$=\backslash sum\_\{ij\}$

p_i Y^j(q) \frac {\partial X^i}{\partial q^j} - p_j X^i(q) \frac {\partial Y^j}{\partial q^i}

$=\; -\; \backslash sum\_i\; p\_i\; \backslash ;*^i(q)$

$=\; -\; P\_\{*\}(q,p).\; \backslash ,$

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

See also

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• Lagrange bracket
• Moyal product
• Peierls bracket
• superPoisson algebra
• superPoisson bracket

Symplectic topology | Hamiltonian mechanics | Binary operations

Poisson-Klammer | Colchete de Poisson | Скобки Пуассона | 泊松括號