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. The Poisson bracket also distinguishes a certain class of coordinate-transformations, the so-called "canonical transformations", which maps canonical coordinate systems into canonical coordinate systems. (A "canonical coordinate system" consists of canonical position/momentum variables, that satisfy canonical Poisson-bracket relations.) Note that the set of possible canonical transformations is always very rich. For instance, often it is possible to choose the Hamiltonian itself $H = H(q,p;t)\,$ as one of the new canonical momentum coordinates.

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. These are all named in honour of Siméon-Denis Poisson.

1 Canonical coordinates
2 Hamilton's Equations of motion
3 Constants of motion
4 Definition
5 Lie algebra
6 See also
7 References

Canonical coordinates

In canonical coordinates (also known as Darboux coordinates) $(q_i,p_j)\,$ on the phase space, given two functions $f(p_i,q_i,t)\,$ and $g(p_i,q_i,t)\,$ , the Poisson bracket takes the form

$\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right].$

Hamilton's Equations of motion

The Hamilton's 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

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t} %2B \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} %2B \frac{\partial f}{\partial t} .$

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

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} %2B \frac{\partial f}{\partial t} = \{f,H\}%2B \frac{\partial f}{\partial t}.$

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. Dropping the coordinates, one has

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

The operator $i{\hat{\mathbf{L}}} = - \{\,H, \cdot\,\}$ is known as the Liouvillian.

Constants of motion

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's equations of motion, then one has that $0=\frac{\mathrm{d}f}{\mathrm{d}t}$ along that trajectory. Then one has

$0 = \frac {\mathrm{d}}{\mathrm{d}t} f(p,q) = \{f,H\}%2B \frac{\partial f}{\partial t}$

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.

Definition

Let M be symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form $\omega$ which is both closed ( $\ d\omega = 0$ ) and non-degenerate, in the following sense: when viewed as a map $\ \omega: \xi \in \mathrm{vect}[M] \rightarrow i_\xi \omega \in \Lambda^1[M]$ , $\ \omega$ is invertible to obtain $\tilde{\omega}: \Lambda^1[M] \rightarrow \mathrm{vect}[M]$ . Here $\ d$ is the exterior derivative operation intrinsic to the manifold structure of M, and $\ i_\xi \theta$ is the interior product or contraction operation, which is equivalent to $\ \theta(\xi)$ on 1-forms $\ \theta.$

Using the axioms of the exterior calculus, one can derive:

$\ i_{[v, w]} \omega = d(i_v i_w \omega) %2B i_v d(i_w \omega) - i_w d(i_v \omega) - i_w i_v d\omega.$

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 $\ d\omega = 0$ , 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 of abstract algebra, the coclosed vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the coexact vector fields form an algebraic ideal of this subalgebra.

Given the existence of the inverse map $\tilde{\omega}$ , every smooth real-valued function f on M may be associated with a coexact vector field $\tilde{\omega}(df)$ . (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 $\ (M, \omega)$ , a bilinear operation on differentiable functions, under which the $C^\infty$ (smooth) functions form an algebra. It is given by:

$\{f,g\} = i_{\tilde{\omega}(df)} dg = - i_{\tilde{\omega}(dg)} df = -\{g,f\}.$

The skew-symmetry of the Poisson bracket is ensured by the axioms of the exterior calculus and the condition $d\omega = 0$ . Because the map $\tilde{\omega}$ 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 the Poisson bivector or the Poisson structure on the symplectic manifold, and the Poisson bracket written simply $\{f,g\} = \tilde{\omega}(df, dg)$ .

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:

$\ \{f,\{g,h\}\} %2B \{g,\{h,f\}\} %2B \{h,\{f,g\}\} = 0.$

The Poisson bracket $\{f,\_\}$ with respect to a particular scalar field f corresponds to the Lie derivative with respect to $\tilde{\omega}(df)$ . Consequently, it is a derivation; that is, it satisfies Leibniz' law:

$\ \{f,gh\} = \{f,g\}h %2B g\{f,h\},$

also known as the "Poisson property". 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 bracket is skewsymmetric/antisymmetric. (Equivalently, viewed as a binary product operation, it is anticommutative.) It also satisfies 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 smooth 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:

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

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.

References

Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics (2nd ed. ed.). New York: Springer. ISBN 978-0387968902.

Landau, L. D.; Lifshitz, E. M. (1982). Mechanics. Course of Theoretical Physics. Vol. 1 (3rd ed. ed.). Butterworth-Heinemann. ISBN 978-0750628969.

Karasëv, M. V.; Maslov, V. P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky [A. B. Sosinskiĭ] and M. Shishkova. Translations of Mathematical Monographs, 119. American Mathematical Society, Providence, RI, 1993.