Poisson bracket

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. It is named after Siméon-Denis Poisson.

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Definition

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

[itex]\{f,g\}=\tilde{\omega}(df,dg)[itex]

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

Canonical coordinates

In canonical coordinates [itex](q^i,p_j)[itex] on the phase space, the Poisson bracket takes the form

[itex]\{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}} \right] [itex].

Lie algebra

The Poisson brackets are anticommutative. Note also that they satisfy the Jacobi identity. This makes the space of smooth functions over 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 [itex]P_X[itex] be its conjugate momentum. The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:

[itex]\{P_X,P_Y\}=-P_{[X,Y]}[itex].

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

[itex]X_q=\sum_i X^i(q) \frac{\partial}{\partial q^i}[itex]

where the [itex]\partial /\partial q^i[itex] is the local coordinate frame. The conjugate momentum to X has the expression

[itex]P_X(q,p)=\sum_i X^i(q) \;p_i[itex]

where the [itex]p_i[itex] are the momentum functions conjugate to the coordinates. One then has, for a point [itex](q,p)[itex] in the phase space,

[itex]\{P_X,P_Y\}(q,p)= \sum_i \sum_j \{X^i(q) \;p_i, Y^j(q)\;p_j \}[itex]
[itex]=\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} [itex]

[itex]= - \sum_i p_i \; [X,Y]^i(q) [itex]
[itex]= - P_{[X,Y]}(q,p) \,[itex]

The above holds for all [itex](q,p)[itex], giving the desired result.

Time evolution

The time evolution of a function f on the symplectic manifold can be given as a one-parameter family of symplectomorphisms, with the time t being the parameter. The total time derivative can be written as

[itex]\frac{d}{dt} f=

\frac{\partial }{\partial t} f + \{\,f,H\,\} = \frac{\partial }{\partial t} f - \{\,H,f\,\} = \left(\frac{\partial }{\partial t} - \{\,H,\,\}\right)f.[itex]

Poisson algebra

More generally, we can have Poisson brackets over Poisson algebras.

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