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Inmathematics andclassical mechanics, thePoisson bracket is an importantbinary operation inHamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltoniandynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, calledcanonical transformations, which mapcanonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by${\displaystyle q_i}$ and${\displaystyle p_i}$, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself${\displaystyle H=H(q,p,t)}$ as one of the new canonical momentum coordinates.

In a more general sense, the Poisson bracket is used to define aPoisson algebra, of which the algebra of functions on aPoisson manifold is a special case. There are other general examples, as well: it occurs in the theory ofLie algebras, where thetensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in theuniversal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion ofquantum groups.

All of these objects are named in honor ofSiméon Denis Poisson. He introduced the Poisson bracket in his 1809 treatise on mechanics.^[1][2]

Given two functionsf andg that depend onphase space and time, their Poisson bracket${\displaystyle \{f,g\}}$ is another function that depends on phase space and time. The following rules hold for any three functions${\displaystyle f,g,h}$ of phase space and time:

Anticommutativity

${\displaystyle \{f,g\}=-\{g,f\}}$

Bilinearity

${\displaystyle \{af+bg,h\}=a\{f,h\}+b\{g,h\},}$${\displaystyle \{h,af+bg\}=a\{h,f\}+b\{h,g\},a,b\in \mathbb{R}}$

Leibniz's rule

${\displaystyle \{fg,h\}=\{f,h\}g+f\{g,h\}}$

Jacobi identity

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

Also, if a function${\displaystyle k}$ is constant over phase space (but may depend on time), then${\displaystyle \{f,k\}=0}$ for any${\displaystyle f}$.

Incanonical coordinates (also known asDarboux coordinates)${\displaystyle (q_i,p_i)}$ on thephase space, given two functions${\displaystyle f(p_i,q_i,t)}$ and${\displaystyle g(p_i,q_i,t)}$,^{[Note 1]} the Poisson bracket takes the form$${\displaystyle \{f,g\}=\sum _{i=1}^N\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }{q}_i}\frac{\mathrm{\partial }g}{\mathrm{\partial }{p}_i}-\frac{\mathrm{\partial }f}{\mathrm{\partial }{p}_i}\frac{\mathrm{\partial }g}{\mathrm{\partial }{q}_i}\right).}$$

The Poisson brackets of the canonical coordinates arewhere${\displaystyle {\delta }_{ij}}$ is theKronecker delta.

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${\displaystyle f(p,q,t)}$ is a function on the solution's trajectory-manifold. Then from the multivariablechain rule,$${\displaystyle \frac{d}{dt}f(p,q,t)=\frac{\mathrm{\partial }f}{\mathrm{\partial }q}\frac{dq}{dt}+\frac{\mathrm{\partial }f}{\mathrm{\partial }p}\frac{dp}{dt}+\frac{\mathrm{\partial }f}{\mathrm{\partial }t}.}$$

Further, one may take${\displaystyle p=p(t)}$ and${\displaystyle q=q(t)}$ to be solutions toHamilton's equations; that is,

Then

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

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

The operator in the convective part of the derivative,${\displaystyle i\hat{L}=-\{H,\cdot \}}$, is sometimes referred to as the Liouvillian (seeLiouville's theorem (Hamiltonian)).

The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.

Consider the following canonical transformation:$${\displaystyle \eta =\left[\begin{array}{c}{q}_1\\ ⋮\\ q_N\\ p_1\\ ⋮\\ p_N\end{array}\right]\to \epsilon =\left[\begin{array}{c}{Q}_1\\ ⋮\\ Q_N\\ P_1\\ ⋮\\ P_N\end{array}\right]}$$Defining$M:=\frac{\mathrm{\partial }(\mathbf{Q},\mathbf{P})}{\mathrm{\partial }(\mathbf{q},\mathbf{p})}$, the Poisson matrix is defined as$\mathcal{P}(\epsilon )=MJ{M}^T$, where${\displaystyle J}$ is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:$${\displaystyle {\mathcal{P}}_{ij}(\epsilon )=[MJ{M}^T{]}_{ij}=\sum _{k=1}^N\left(\frac{\mathrm{\partial }{\epsilon }_i}{\mathrm{\partial }{\eta }_k}\frac{\mathrm{\partial }{\epsilon }_j}{\mathrm{\partial }{\eta }_{N+k}}-\frac{\mathrm{\partial }{\epsilon }_i}{\mathrm{\partial }{\eta }_{N+k}}\frac{\mathrm{\partial }{\epsilon }_j}{\mathrm{\partial }{\eta }_k}\right)=\sum _{k=1}^N\left(\frac{\mathrm{\partial }{\epsilon }_i}{\mathrm{\partial }{q}_k}\frac{\mathrm{\partial }{\epsilon }_j}{\mathrm{\partial }{p}_k}-\frac{\mathrm{\partial }{\epsilon }_i}{\mathrm{\partial }{p}_k}\frac{\mathrm{\partial }{\epsilon }_j}{\mathrm{\partial }{q}_k}\right)=\{{\epsilon }_i,{\epsilon }_j{\}}_{\eta }.}$$

The Poisson matrix satisfies the following known properties:

where the$\mathcal{L}(\epsilon )$ is known as a Lagrange matrix and whose elements correspond toLagrange brackets. The last identity can also be stated as the following:$${\displaystyle \sum _{k=1}^{2N}\{{\eta }_i,{\eta }_k\}[{\eta }_k,{\eta }_j]=-{\delta }_{ij}}$$Note that the summation here involves generalized coordinates as well as generalized momentum.

The invariance of Poisson bracket can be expressed as:$\{{\epsilon }_i,{\epsilon }_j{\}}_{\eta }=\{{\epsilon }_i,{\epsilon }_j{\}}_{\epsilon }=J_{ij}$, which directly leads to the symplectic condition:$MJ{M}^T=J$.^[3]

Anintegrable system will haveconstants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function${\displaystyle f(p,q)}$ is a constant of motion. This implies that if${\displaystyle p(t),q(t)}$ is atrajectory or solution toHamilton's equations of motion, then along that trajectory:$${\displaystyle 0=\frac{df}{dt}}$$Where, as above, the intermediate step follows by applying the equations of motion and we assume that${\displaystyle f}$ does not explicitly depend on time. This equation is known as theLiouville equation. The content ofLiouville's theorem is that the time evolution of ameasure given by adistribution function${\displaystyle f}$ is given by the above equation.

If the Poisson bracket of${\displaystyle f}$ and${\displaystyle g}$ vanishes (${\displaystyle \{f,g\}=0}$), then${\displaystyle f}$ and${\displaystyle g}$ are said to bein involution. In order for a Hamiltonian system to becompletely integrable,${\displaystyle n}$ independent constants of motion must be inmutual involution, where${\displaystyle n}$ is the number of degrees of freedom.

Furthermore, according toPoisson's Theorem, if two quantities${\displaystyle A}$ and${\displaystyle B}$ are explicitly time independent (${\displaystyle A(p,q),B(p,q)}$) constants of motion, so is their Poisson bracket${\displaystyle \{A,B\}}$. This does not always supply a useful result, however, since the number of possible constants of motion is limited (${\displaystyle 2n-1}$ for a system with${\displaystyle n}$ degrees of freedom), and so the result may be trivial (a constant, or a function of${\displaystyle A}$ and${\displaystyle B}$.)

Let${\displaystyle M}$ be asymplectic manifold, that is, amanifold equipped with asymplectic form: a2-form${\displaystyle \omega }$ which is bothclosed (i.e., itsexterior derivative${\displaystyle d\omega }$ vanishes) andnon-degenerate. For example, in the treatment above, take${\displaystyle M}$ to be${\displaystyle {\mathbb{R}}^{2n}}$ and take$${\displaystyle \omega =\sum _{i=1}^{n}d{q}_i\wedge d{p}_i.}$$

If${\displaystyle {\iota }_v\omega }$ is theinterior product orcontraction operation defined by${\displaystyle ({\iota }_v\omega )(u)=\omega (v,u)}$, then non-degeneracy is equivalent to saying that for every one-form${\displaystyle \alpha }$ there is a unique vector field${\displaystyle {\mathrm{\Omega }}_{\alpha }}$ such that${\displaystyle {\iota }_{{\mathrm{\Omega }}_{\alpha }}\omega =\alpha }$. Alternatively,${\displaystyle {\mathrm{\Omega }}_{dH}={\omega }^{-1}(dH)}$. Then if${\displaystyle H}$ is a smooth function on${\displaystyle M}$, theHamiltonian vector field${\displaystyle X_H}$ can be defined to be${\displaystyle {\mathrm{\Omega }}_{dH}}$. It is easy to see that

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

Furthermore,

1

HereX_gf denotes the vector fieldX_g applied to the functionf as a directional derivative, and${\displaystyle {\mathcal{L}}_{X_g}f}$ denotes the (entirely equivalent)Lie derivative of the functionf.

Ifα is an arbitrary one-form onM, the vector fieldΩ_α generates (at least locally) aflow${\displaystyle {\varphi }_x(t)}$ satisfying the boundary condition${\displaystyle {\varphi }_x(0)=x}$ and the first-order differential equation$${\displaystyle \frac{d{\varphi }_x}{dt}={{\mathrm{\Omega }}_{\alpha }|}_{{\varphi }_x(t)}.}$$

The${\displaystyle {\varphi }_x(t)}$ will besymplectomorphisms (canonical transformations) for everyt as a function ofx if and only if${\displaystyle {\mathcal{L}}_{{\mathrm{\Omega }}_{\alpha }}\omega =0}$; when this is true,Ω_α is called asymplectic vector field. RecallingCartan's identity${\displaystyle {\mathcal{L}}_X\omega =d({\iota }_X\omega )+{\iota }_{X}d\omega }$ anddω = 0, it follows that${\displaystyle {\mathcal{L}}_{{\mathrm{\Omega }}_{\alpha }}\omega =d\left({\iota }_{{\mathrm{\Omega }}_{\alpha }}\omega \right)=d\alpha }$. Therefore,Ω_α is a symplectic vector field if and only if α is aclosed form. Since${\displaystyle d(df)=d^{2}f=0}$, it follows that every Hamiltonian vector fieldX_f is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From(1) above, under the Hamiltonian flowX_H,$${\displaystyle \frac{d}{dt}f({\varphi }_x(t))=X_{H}f=\{f,H\}.}$$

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${\displaystyle \{p_i,p_j\}=\{q_i,q_j\}=0}$ and${\displaystyle \{q_i,p_j\}={\delta }_{ij}}$), 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 aderivation; that is, it satisfies a non-commutative version of Leibniz'sproduct rule:

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

2

The Poisson bracket is intimately connected to theLie bracket of the Hamiltonian vector fields. Because the Lie derivative is a derivation,$${\displaystyle {\mathcal{L}}_v{\iota }_u\omega ={\iota }_{{\mathcal{L}}_{v}u}\omega +{\iota }_u{\mathcal{L}}_v\omega ={\iota }_{[v,u]}\omega +{\iota }_u{\mathcal{L}}_v\omega .}$$

Thus ifv andu are symplectic, using${\displaystyle {\mathcal{L}}_v\omega =0={\mathcal{L}}_u\omega }$, Cartan's identity, and the fact that${\displaystyle {\iota }_u\omega }$ is a closed form,$${\displaystyle {\iota }_{[v,u]}\omega ={\mathcal{L}}_v{\iota }_u\omega =d({\iota }_v{\iota }_u\omega )+{\iota }_{v}d({\iota }_u\omega )=d({\iota }_v{\iota }_u\omega )=d(\omega (u,v)).}$$

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

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

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 ofabstract algebra, the symplectic vector fields form asubalgebra of theLie algebra of smooth vector fields onM, and the Hamiltonian vector fields form anideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional)Lie group ofsymplectomorphisms ofM.

It is widely asserted that theJacobi identity for the Poisson bracket,$${\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}$$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 issufficient to show that:$${\displaystyle {\mathrm{ad}}_{\{g,f\}}={\mathrm{ad}}_{-\{f,g\}}=[{\mathrm{ad}}_f,{\mathrm{ad}}_g]}$$where the operator${\displaystyle {\mathrm{ad}}_g}$ on smooth functions onM is defined by${\displaystyle {\mathrm{ad}}_g(\cdot )=\{\cdot ,g\}}$ and the bracket on the right-hand side is the commutator of operators,${\displaystyle [\mathrm{A},\mathrm{B}]=\mathrm{A}\mathrm{B}-\mathrm{B}\mathrm{A}}$. By(1), the operator${\displaystyle {\mathrm{ad}}_g}$ is equal to the operatorX_g. The proof of the Jacobi identity follows from(3) because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.

Thealgebra of smooth functions on M, together with the Poisson bracket forms aPoisson algebra, because it is aLie algebra under the Poisson bracket, which additionally satisfies Leibniz's rule(2). We have shown that everysymplectic manifold is aPoisson 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.

Given a smoothvector field${\displaystyle X}$ on the configuration space, let${\displaystyle P_X}$ be itsconjugate momentum. The conjugate momentum mapping is aLie algebra anti-homomorphism from theLie bracket to the Poisson bracket:$${\displaystyle \{P_X,P_Y\}=-P_{[X,Y]}.}$$

This important result is worth a short proof. Write a vector field${\displaystyle X}$ at point${\displaystyle q}$ in theconfiguration space as$${\displaystyle X_q=\sum _i{X}^i(q)\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}}$$where$\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}$ is the local coordinate frame. The conjugate momentum to${\displaystyle X}$ has the expression$${\displaystyle P_X(q,p)=\sum _i{X}^i(q)p_i}$$where the${\displaystyle p_i}$ are the momentum functions conjugate to the coordinates. One then has, for a point${\displaystyle (q,p)}$ in thephase space,

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

Poisson bracketsdeform toMoyal brackets uponquantization, that is, they generalize to a different Lie algebra, theMoyal algebra, or, equivalently inHilbert space, quantumcommutators. The Wigner-İnönügroup contraction of these (the classical limit,ħ → 0) yields the above Lie algebra.

To state this more explicitly and precisely, theuniversal enveloping algebra of theHeisenberg algebra is theWeyl algebra (modulo the relation that the center be the unit). The Moyal product is then a special case of the star product on the algebra of symbols. An explicit definition of the algebra of symbols, and the star product is given in the article on theuniversal enveloping algebra.

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• Marle, Charles-Michel (2009). "The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808-1810".Letters in Mathematical Physics.90 (1–3): 3-21.arXiv:0902.0685.Bibcode:2009LMaPh..90....3M.doi:10.1007/s11005-009-0347-y.

• "Poisson brackets",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Eric W. Weisstein."Poisson bracket".MathWorld.

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