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, called canonical transformations, which maps canonical coordinate systems into canonical coordinate systems. (A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations.) 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.

Canonical coordinates

In canonical coordinates (also known as Darboux coordinates) $(q_i,p_i)$ 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

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 from the multivariable chain rule, one has

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

Further, one may take p = p(t) and q = q(t) to be solutions to Hamilton's equations; that is,

$\begin{cases} \dot{q} = \frac{\partial H}{\partial p}= \{q, H\} \\ \dot{p} = -\frac{\partial H}{\partial q} = \{p, H\} \end{cases}$

Then, one has

\begin{align} \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} + \frac{\partial f}{\partial t} \\ &= \{f,H\}+ \frac{\partial f}{\partial t} ~. \end{align}

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e. canonical transformations, area-preserving diffeomorphisms), with the time t being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that any time t in the solution to Hamilton's equations, q(t)=exp(−t{H,•}) q(0), p(t)=exp(−t{H,•}) p(0), can serve as the bracket coordinates. Poisson brackets are canonical invariants.

Dropping the coordinates, one has

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

The operator in the convective part of the derivative, i = −{H, •} , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).

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

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

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.

If the Poisson bracket of f and g vanishes ({f,g} = 0), then f and g are said to be in involution. In order for a Hamiltonian system to be completely integrable, all of the constants of motion must be in mutual involution.

The Poisson bracket in coordinate-free language

Let M be symplectic manifold, that is, a manifold equipped with a symplectic form: a 2-form ω which is both closed (i.e. its exterior derivative dω = 0) and non-degenerate. For example, in the treatment above, take M to be $\mathbb R^{2n}$ and take

$\omega = \sum_{i=1}^{n}dq_i\wedge dp_i.$

If $\iota_v \omega$ is the interior product or contraction operation defined by $(\iota_v \omega)(w)=\omega(v,w)$, then non-degeneracy is equivalent to saying that for every one-form α there is a unique vector field Ωα such that $\iota_{\Omega_\alpha} \omega = \alpha$. Then if H is a smooth function on M, the Hamiltonian vector field XH can be defined to be $\Omega_{dH}$. It is easy to see that

$X_{p_i}=\frac{\partial}{\partial q_i}$
$X_{q_i}=-\frac{\partial}{\partial p_i}$

The Poisson bracket $\{\cdot,\cdot\}$ on (M, ω) is a bilinear operation on differentiable functions, defined by $\{f,g\} = \omega(X_f,X_g)$; the Poisson bracket of two functions on M is itself a function on M. The Poisson bracket is antisymmetric because:

$\{f,g\} = \omega(X_f,X_g) = -\omega(X_g,X_f) = -\{g,f\}$.

Furthermore,

$\{f,g\} = \omega(X_f,X_g) = \omega(\Omega_{df},X_g) = (\iota_{\Omega_{df}}\omega)(X_g) = df(X_g) = X_gf = \mathcal{L}_{X_g} f$.

(1)

Here Xgf denotes the vector field Xg applied to the function f as a directional derivative, and $\mathcal{L}_{X_g} f$ denotes the (entirely equivalent) Lie derivative of the function f.

If α is an arbitrary one-form on M, the vector field Ωα generates (at least locally) a flow $\phi_x(t)$ satisfying the boundary condition $\phi_x(0)=x$ and the first-order differential equation

$\frac{d\phi_x}{dt} = \Omega_\alpha|_{\phi_x(t)}.$

The $\phi_x(t)$ will be symplectomorphisms (canonical transformations) for every t as a function of x if and only if $\mathcal L_{\Omega_\alpha}\omega = 0$; when this is true, Ωα is called a symplectic vector field. Recalling Cartan's identity $\mathcal L_X\omega = d (\iota_X \omega) + \iota_X d\omega$ and dω = 0, it follows that $\mathcal L_{\Omega_\alpha}\omega = d (\iota_{\Omega_\alpha} \omega) = d\alpha$. Therefore Ωα is a symplectic vector field if and only if α is a closed form. Since $d(df)=d^2f=0$, it follows that every Hamiltonian vector field Xf is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From (1) above, under the Hamiltonian flow XH,

$\frac{d}{dt}f(\phi_x(t)) = X_Hf = \{f,H\}.$

This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when {f,H} = 0, f is a constant of motion of the system. In addition, in canonical coordinates (with $\{p_i,p_j\} = \{q_i,q_j\} = 0$ and $\{q_i,p_j\} = \delta_{ij}$), Hamilton's equations for the time evolution of the system follow immediately from this formula.

It also follows from (1) that the Poisson bracket is a derivation; that is, it satisfies a non-commutative version of Leibniz's product rule:

$\{fg,h\} = f\{g,h\} + g\{f,h\}$, and $\{f,gh\} = g\{f,h\} + h\{f,g\}$

(2)

The Poisson bracket is intimately connected to the Lie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation,

$\mathcal L_v\iota_w\omega = \iota_{\mathcal L_vw}\omega + \iota_w\mathcal L_v\omega = \iota_{[v,w]}\omega + \iota_w\mathcal L_v\omega$.

Thus if v and w are symplectic, using $\mathcal L_v\omega = 0$, Cartan's identity, and the fact that $\iota_w\omega$ is a closed form,

$\iota_{[v,w]}\omega = \mathcal L_v\iota_w\omega = d(\iota_v\iota_w\omega) + \iota_vd(\iota_w\omega) = d(\iota_v\iota_w\omega) = d(\omega(w,v)).$

It follows that $[v,w] = X_{\omega(w,v)}$, so that

$[X_f,X_g] = X_{\omega(X_g,X_f)} = -X_{\omega(X_f,X_g)} = -X_{\{f,g\}}$.

(3)

Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of abstract algebra, the symplectic vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the Hamiltonian vector fields form an ideal of this subalgebra. The sympletic vector fields are the Lie algebra of the (infinite-dimensional) Lie group of symplectomorphisms of M.

It is widely asserted that the Jacobi identity for the Poisson bracket,

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

follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is sufficient to show that:

$\operatorname{ad}_{\{f,g\}}=[\operatorname{ad}_f,\operatorname{ad}_g]$

where the operator $\operatorname{ad}_g$ on smooth functions on M is defined by $\operatorname{ad}_g(\cdot) = \{\cdot,g\}$ and the bracket on the right-hand side is the commutator of operators, $[\operatorname A,\operatorname B] = \operatorname A\operatorname B-\operatorname B\operatorname A$. By (1), the operator $\operatorname{ad}_g$ is equal to the operator Xg. The proof of the Jacobi identity follows from (3) because the Lie bracket of vector fields is just their commutator as differential operators.

The algebra of smooth functions on M, together with the Poisson bracket forms a Poisson algebra, because it is a Lie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule (2). We have shown that every symplectic manifold is a Poisson manifold, that is a manifold with a "curly-bracket" operator on smooth functions such that the smooth functions form a Poisson algebra. However, not every Poisson manifold arises in this way, because Poisson manifolds allow for degeneracy which cannot arise in the symplectic case.

A result on conjugate momenta

Given a smooth vector field X on the configuration space, let PX 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 pi are the momentum functions conjugate to the coordinates. One then has, for a point (q,p) in the phase space,

\begin{align} \{P_X,P_Y\}(q,p) &= \sum_i \sum_j \left \{X^i(q) \;p_i, Y^j(q)\;p_j \right \} \\ &=\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). \end{align}

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

Quantization

Poisson brackets deform to Moyal brackets upon quantization, that is, they generalize to a different Lie algebra, the Moyal algebra, or, equivalently in Hilbert space, quantum commutators. The Wigner-İnönü group contraction of these (the classical limit, ħ→0) yields the above Lie algebra.