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 map canonical coordinate systems into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by q i {\displaystyle q_{i}} and p i {\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 H = H ( q , p , t ) {\displaystyle {\mathcal {H}}={\mathcal {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. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups. All of these objects are named in honor of French mathematician Siméon Denis Poisson. He introduced the Poisson bracket in his 1809 treatise on mechanics.
Properties Given two functions f and g that depend on phase space and time, their Poisson bracket { f , g } {\displaystyle \{f,g\}} is another function that depends on phase space and time. The following rules hold for any three functions f , g , h {\displaystyle f,\,g,\,h} of phase space and time:
Anticommutativity
{ f , g } = − { g , f } {\displaystyle \{f,g\}=-\{g,f\}}
Bilinearity
{ a f + b g , h } = a { f , h } + b { g , h } , {\displaystyle \{af+bg,h\}=a\{f,h\}+b\{g,h\},}
{ h , a f + b g } = a { h , f } + b { h , g } , a , b ∈ R {\displaystyle \{h,af+bg\}=a\{h,f\}+b\{h,g\},\quad a,b\in \mathbb {R} }
Leibniz's rule
{ f g , h } = { f , h } g + f { g , h } {\displaystyle \{fg,h\}=\{f,h\}g+f\{g,h\}}
Jacobi identity
{ f , { g , h } } + { g , { h , f } } + { h , { f , g } } = 0 {\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}
Also, if a function k {\displaystyle k} is constant over phase space (but may depend on time), then { f , k } = 0 {\displaystyle \{f,\,k\}=0} for any f {\displaystyle f} .
Definition in canonical coordinates In canonical coordinates (also known as Darboux coordinates) ( q i , p i ) {\displaystyle (q_{i},\,p_{i})} on the phase space, given two functions f ( p i , q i , t ) {\displaystyle f(p_{i},\,q_{i},t)} and g ( p i , q i , t ) {\displaystyle g(p_{i},\,q_{i},t)} , the Poisson bracket takes the form
{ f , g } = ∑ i = 1 N ( ∂ f ∂ q i ∂ g ∂ p i − ∂ f ∂ p i ∂ g ∂ q i ) . {\displaystyle \{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).}
The Poisson brackets of the canonical coordinates are
{ q k , q l } = ∑ i = 1 N ( ∂ q k ∂ q i ∂ q l ∂ p i − ∂ q k ∂ p i ∂ q l ∂ q i ) = ∑ i = 1 N ( δ k i ⋅ 0 − 0 ⋅ δ l i ) = 0 , { p k , p l } = ∑ i = 1 N ( ∂ p k ∂ q i ∂ p l ∂ p i − ∂ p k ∂ p i ∂ p l ∂ q i ) = ∑ i = 1 N ( 0 ⋅ δ l i − δ k i ⋅ 0 ) = 0 , { q k , p l } = ∑ i = 1 N ( ∂ q k ∂ q i ∂ p l ∂ p i − ∂ q k ∂ p i ∂ p l ∂ q i ) = ∑ i = 1 N ( δ k i ⋅ δ l i − 0 ⋅ 0 ) = δ k l , {\displaystyle {\begin{aligned}\{q_{k},q_{l}\}&=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial q_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial q_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot 0-0\cdot \delta _{li}\right)=0,\\\{p_{k},p_{l}\}&=\sum _{i=1}^{N}\left({\frac {\partial p_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial p_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(0\cdot \delta _{li}-\delta _{ki}\cdot 0\right)=0,\\\{q_{k},p_{l}\}&=\sum _{i=1}^{N}\left({\frac {\partial q_{k}}{\partial q_{i}}}{\frac {\partial p_{l}}{\partial p_{i}}}-{\frac {\partial q_{k}}{\partial p_{i}}}{\frac {\partial p_{l}}{\partial q_{i}}}\right)=\sum _{i=1}^{N}\left(\delta _{ki}\cdot \delta _{li}-0\cdot 0\right)=\delta _{kl},\end{aligned}}}
where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta.
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 ) {\displaystyle f(p,q,t)} is a function on the solution's trajectory-manifold. Then from the multivariable chain rule,
d d t f ( p , q , t ) = ∂ f ∂ q d q d t + ∂ f ∂ p d p d t + ∂ f ∂ t . {\displaystyle {\frac {d}{dt}}f(p,q,t)={\frac {\partial f}{\partial q}}{\frac {dq}{dt}}+{\frac {\partial f}{\partial p}}{\frac {dp}{dt}}+{\frac {\partial f}{\partial t}}.}
Further, one may take p = p ( t ) {\displaystyle p=p(t)} and q = q ( t ) {\displaystyle q=q(t)} to be solutions to Hamilton's equations; that is,
d q d t = ∂ H ∂ p = { q , H } , d p d t = − ∂ H ∂ q = { p , H } . {\displaystyle {\begin{aligned}{\frac {dq}{dt}}&={\frac {\partial {\mathcal {H}}}{\partial p}}=\{q,{\mathcal {H}}\},\\{\frac {dp}{dt}}&=-{\frac {\partial {\mathcal {H}}}{\partial q}}=\{p,{\mathcal {H}}\}.\end{aligned}}}
Then
d d t f ( p , q , t ) = ∂ f ∂ q ∂ H ∂ p − ∂ f ∂ p ∂ H ∂ q + ∂ f ∂ t = { f , H } + ∂ f ∂ t . {\displaystyle {\begin{aligned}{\frac {d}{dt}}f(p,q,t)&={\frac {\partial f}{\partial q}}{\frac {\partial {\mathcal {H}}}{\partial p}}-{\frac {\partial f}{\partial p}}{\frac {\partial {\mathcal {H}}}{\partial q}}+{\frac {\partial f}{\partial t}}\\&=\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}~.\end{aligned}}}
Thus, the time evolution of a function f {\displaystyle 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 {\displaystyle 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 {\displaystyle t} in the solution to Hamilton's equations,
q ( t ) = exp ( − t { H , ⋅ } ) q ( 0 ) , p ( t ) = exp ( − t { H , ⋅ } ) p ( 0 ) , {\displaystyle q(t)=\exp(-t\{{\mathcal {H}},\cdot \})q(0),\quad p(t)=\exp(-t\{{\mathcal {H}},\cdot \})p(0),}
can serve as the bracket coordinates. Poisson brackets are canonical invariants. Dropping the coordinates,
d d t f = ( ∂ ∂ t − { H , ⋅ } ) f . {\displaystyle {\frac {d}{dt}}f=\left({\frac {\partial }{\partial t}}-\{{\mathcal {H}},\cdot \}\right)f.}
The operator in the convective part of the derivative, i L ^ = − { H , ⋅ } {\displaystyle i{\hat {L}}=-\{{\mathcal {H}},\cdot \}} , is sometimes referred to as the Liouvillian (see Liouville's theorem (Hamiltonian)).
Poisson matrix in canonical transformations
The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix. Consider the following canonical transformation: η = [ q 1 ⋮ q N p 1 ⋮ p N ] → ε = [ Q 1 ⋮ Q N P 1 ⋮ P N ] {\displaystyle \eta ={\begin{bmatrix}q_{1}\\\vdots \\q_{N}\\p_{1}\\\vdots \\p_{N}\\\end{bmatrix}}\quad \rightarrow \quad \varepsilon ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{N}\\P_{1}\\\vdots \\P_{N}\\\end{bmatrix}}} Defining M := ∂ ( Q , P ) ∂ ( q , p ) {\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}} , the Poisson matrix is defined as P ( ε ) = M J M T {\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}} , where J {\displaystyle J} is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that: P i j ( ε ) = [ M J M T ] i j = ∑ k = 1 N ( ∂ ε i ∂ η k ∂ ε j ∂ η N + k − ∂ ε i ∂ η N + k ∂ ε j ∂ η k ) = ∑ k = 1 N ( ∂ ε i ∂ q k ∂ ε j ∂ p k − ∂ ε i ∂ p k ∂ ε j ∂ q k ) = { ε i , ε j } η . {\displaystyle {\mathcal {P}}_{ij}(\varepsilon )=[MJM^{T}]_{ij}=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial \eta _{k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{N+k}}}-{\frac {\partial \varepsilon _{i}}{\partial \eta _{N+k}}}{\frac {\partial \varepsilon _{j}}{\partial \eta _{k}}}\right)=\sum _{k=1}^{N}\left({\frac {\partial \varepsilon _{i}}{\partial q_{k}}}{\frac {\partial \varepsilon _{j}}{\partial p_{k}}}-{\frac {\partial \varepsilon _{i}}{\partial p_{k}}}{\frac {\partial \varepsilon _{j}}{\partial q_{k}}}\right)=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }.}
The Poisson matrix satisfies the following known properties: P T = − P | P | = 1 | M | 2