Editing Poisson bracket

{{short description|Operation in Hamiltonian mechanics}}
[[File:Simeon Poisson.jpg|thumb|Siméon Denis Poisson]]
{{Classical mechanics|expanded=Formulations}}
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 coordinates|canonical coordinate systems]] into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by <math>q_i</math> and <math>p_i</math>, 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 <math>\mathcal H =\mathcal H(q, p, t)</math> 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 algebra]]s, 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 group]]s.

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.<ref name="Poisson1809">[[#poisson1881|S. D. Poisson (1809)]]</ref><ref name="Marle2009">[[#marle2009|C. M. Marle (2009)]]</ref>

==Properties==
Given two functions {{mvar|f}} and {{mvar|g}} that depend on [[phase space]] and time, their Poisson bracket <math>\{f, g\}</math> is another function that depends on phase space and time. The following rules hold for any three functions <math>f,\, g,\, h</math> of phase space and time:
;[[Anticommutativity]]: <math>\{f, g\} = -\{g, f\}</math>
;[[Bilinearity]]: <math>\{af + bg, h\} = a\{f, h\} + b\{g, h\}, </math><math> \{h, af + bg\} = a\{h, f\} + b\{h, g\}, \quad a, b \in \mathbb R</math>
;[[Product rule|Leibniz's rule]]: <math>\{fg, h\} = \{f, h\}g + f\{g, h\}</math>
;[[Jacobi identity]]: <math>\{f, \{g, h\}\} + \{g, \{h, f\}\} +  \{h, \{f, g\}\} = 0</math>

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

==Definition in canonical coordinates==
In [[canonical coordinates]] (also known as [[Darboux coordinates]]) <math> (q_i,\, p_i)</math> on the [[phase space]], given two functions <math> f(p_i,\, q_i, t)</math> and <math> g(p_i,\, q_i, t)</math>,<ref group="Note"><math> f(p_i,\, q_i,\, t)</math> means <math>f</math> is a function of the <math>2N + 1</math> independent variables: momentum, <math>p_{1 \dots N}</math>; position, <math>q_{1 \dots N}</math>; and time, <math>t</math></ref> the Poisson bracket takes the form
<math display="block">\{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).</math>

The Poisson brackets of the canonical coordinates are
<math display="block">\begin{align}
  \{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{align}</math>
where <math>\delta_{ij}</math> 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 <math>f(p, q, t)</math> is a function on the solution's trajectory-manifold. Then from the multivariable [[chain rule]],
<math display="block">\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}.</math>

Further, one may take <math>p = p(t)</math> and <math>q = q(t)</math> to be solutions to [[Hamilton's equations]]; that is,
<math display="block">\begin{align}
  \frac{d q}{d t} &=  \frac{\partial \mathcal H}{\partial p} = \{q, \mathcal H\}, \\ 
  \frac{d p}{d t} &= -\frac{\partial \mathcal H}{\partial q} = \{p, \mathcal H\}.
\end{align}</math>

Then 
<math display="block">\begin{align}
  \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{align}</math>

Thus, the time evolution of a function <math>f</math> on a [[symplectic manifold]] can be given as a [[flow (mathematics)|one-parameter family]] of [[symplectomorphism]]s (i.e., [[canonical transformations]], area-preserving diffeomorphisms), with the time <math>t</math> 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 <math>t</math>'' in the solution to Hamilton's equations,
<math display="block"> q(t) = \exp (-t \{ \mathcal H, \cdot \} ) q(0), \quad  p(t) = \exp (-t \{ \mathcal H, \cdot \}) p(0), </math>
can serve as the bracket coordinates. ''Poisson brackets are [[Canonical transformation|canonical invariants]]''.

Dropping the coordinates, 
<math display="block">\frac{d}{dt} f = \left(\frac{\partial}{\partial t} - \{\mathcal H, \cdot\}\right)f.</math>

The operator in the convective part of the derivative, <math>i\hat{L} = -\{\mathcal H, \cdot\}</math>, is sometimes referred to as the Liouvillian (see [[Liouville's theorem (Hamiltonian)]]).

== Poisson matrix in canonical transformations ==
{{Main|Canonical transformation}}
The concept of Poisson brackets can be expanded to that of matrices by defining the Poisson matrix.

Consider the following canonical transformation:<math display="block">\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} </math>Defining <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>, the Poisson matrix is defined as <math display="inline">\mathcal P(\varepsilon) = MJM^T
</math>, where <math>J</math> is the symplectic matrix under the same conventions used to order the set of coordinates. It follows from the definition that:<math display="block">\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.
</math>

The Poisson matrix satisfies the following known properties:<math display="block">\begin{align}
\mathcal P^T &= - \mathcal P \\
|\mathcal P| &= \frac{1}{|M|^2}\\
\mathcal P^{-1}(\varepsilon)&= -(M^{-1})^T J M^{-1} = - \mathcal L (\varepsilon)\\
\end{align}
</math>

where the <math display="inline">\mathcal L(\varepsilon)
</math> is known as a Lagrange matrix and whose elements correspond to [[Lagrange bracket]]s. The last identity can also be stated as the following:<math display="block">\sum_{k=1}^{2N} \{\eta_i,\eta_k\}[\eta_k,\eta_j] = -\delta_{ij} 
</math>Note that the summation here involves generalized coordinates as well as generalized momentum.

The invariance of Poisson bracket can be expressed as:  <math display="inline">\{ \varepsilon_i,\varepsilon_j\}_\eta=\{ \varepsilon_i,\varepsilon_j\}_\varepsilon = J_{ij}
</math>, which directly leads to the symplectic condition: <math display="inline">MJM^T = J    
</math>.<ref>{{Cite book |last=Giacaglia |first=Giorgio E. O. |title=Perturbation methods in non-linear systems |date=1972 |publisher=Springer |isbn=978-3-540-90054-2 |series=Applied mathematical sciences |location=New York Heidelberg |pages=8–9}}</ref>

==Constants of motion==
An [[integrable 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 <math>f(p, q)</math> is a constant of motion. This implies that if <math>p(t), q(t)</math> is a [[trajectory]] or solution to [[Hamilton's equations of motion]], then along that trajectory:<math display="block">0 = \frac{df}{dt}</math>Where, as above, the intermediate step follows by applying the equations of motion and we assume that <math>f</math> does not explicitly depend on time. This equation is known as the [[Liouville's theorem (Hamiltonian)#Liouville equations|Liouville equation]]. The content of [[Liouville's theorem (Hamiltonian)|Liouville's theorem]]  is that the time evolution of a [[measure (mathematics)|measure]] given by a [[Distribution function (physics)|distribution function]] <math>f</math> is given by the above equation.

If the Poisson bracket of <math>f</math> and <math>g</math> vanishes (<math>\{f,g\} = 0</math>), then <math>f</math> and <math>g</math> are said to be '''in involution'''.  In order for a Hamiltonian system to be [[completely integrable]], <math>n</math> independent constants of motion must be in [[Distribution (differential geometry)#Involutive distributions|mutual involution]], where <math>n</math> is the number of degrees of freedom.

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

==The Poisson bracket in coordinate-free language==
Let <math>M</math> be a [[symplectic manifold]], that is, a [[manifold]] equipped with a [[symplectic form]]: a [[Differential form|2-form]] <math>\omega</math> which is both '''closed''' (i.e., its [[exterior derivative]] <math>d \omega</math> vanishes) and '''non-degenerate'''.  For example, in the treatment above, take <math>M</math> to be <math>\mathbb{R}^{2n}</math> and take
<math display="block">\omega = \sum_{i=1}^{n} d q_i \wedge d p_i.</math>

If <math> \iota_v \omega</math> is the [[interior product]] or [[Tensor contraction|contraction]] operation defined by <math> (\iota_v \omega)(u) =  \omega(v,\, u)</math>, then non-degeneracy is equivalent to saying that for every one-form <math>\alpha</math> there is a unique vector field <math>\Omega_\alpha</math> such that <math> \iota_{\Omega_\alpha} \omega =  \alpha</math>. Alternatively, <math> \Omega_{d H} = \omega^{-1}(d H)</math>. Then if <math>H</math> is a smooth function on <math>M</math>, the [[Hamiltonian vector field]] <math>X_H</math> can be defined to be <math> \Omega_{d H}</math>.  It is easy to see that
<math display="block">\begin{align}
  X_{p_i} &=  \frac{\partial}{\partial q_i} \\
  X_{q_i} &= -\frac{\partial}{\partial p_i}.
\end{align}</math>

The '''Poisson bracket''' <math>\ \{\cdot,\, \cdot\} </math> on {{math|(''M'', ''ω'')}} is a [[bilinear map|bilinear operation]] on [[differentiable function]]s, defined by <math> \{f,\, g\} \;=\; \omega(X_f,\, X_g) </math>; the Poisson bracket of two functions on {{math|''M''}} is itself a function on {{math|''M''}}.  The Poisson bracket is antisymmetric because:
<math display="block">\{f, g\} = \omega(X_f, X_g) = -\omega(X_g, X_f) = -\{g, f\} .</math>

Furthermore,
{{NumBlk||<math display="block">\begin{align}
  \{f, g\} &= \omega(X_f, X_g) = \omega(\Omega_{df}, X_g) \\
           &= (\iota_{\Omega_{df}}\omega)(X_g) = df(X_g) \\
           &= X_g f = \mathcal{L}_{X_g} f.
\end{align}</math>|{{EquationRef|1}}}}

Here {{math|''X<sub>g</sub>f''}} denotes the vector field {{math|''X<sub>g</sub>''}} applied to the function {{math|''f''}} as a directional derivative, and <math>\mathcal{L}_{X_g} f</math> denotes the (entirely equivalent) [[Lie derivative]] of the function {{math|''f''}}.

If {{math|α}} is an arbitrary one-form on {{math|''M''}}, the vector field {{math|Ω<sub>α</sub>}} generates (at least locally) a [[flow (mathematics)|flow]] <math> \phi_x(t)</math> satisfying the boundary condition <math> \phi_x(0) = x</math> and the first-order differential equation
<math display="block">\frac{d\phi_x}{dt} = \left. \Omega_\alpha \right|_{\phi_x(t)}.</math>

The <math> \phi_x(t)</math> will be [[symplectomorphism]]s ([[canonical transformation]]s) for every {{math|''t''}} as a function of {{math|''x''}} if and only if <math> \mathcal{L}_{\Omega_\alpha}\omega \;=\; 0</math>; when this is true, {{math|Ω<sub>α</sub>}} is called a [[symplectic vector field]].  Recalling [[Cartan's identity]] <math> \mathcal{L}_X\omega \;=\; d (\iota_X \omega) \,+\, \iota_X d\omega</math> and {{math|1=''d''ω = 0}}, it follows that <math> \mathcal{L}_{\Omega_\alpha}\omega \;=\; d\left(\iota_{\Omega_\alpha} \omega\right) \;=\; d\alpha</math>. Therefore, {{math|Ω<sub>α</sub>}} is a symplectic vector field if and only if α is a [[Closed and exact differential forms|closed form]].  Since <math> d(df) \;=\; d^2f \;=\; 0</math>, it follows that every Hamiltonian vector field {{math|''X<sub>f</sub>''}} is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations.  From {{EquationNote|1|(1)}} above, under the Hamiltonian flow <math>X_\mathcal H</math>,
<math display="block">\frac{d}{dt}f(\phi_x(t)) = X_\mathcal{H}f = \{f,\mathcal H\}.</math>

This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space.  As noted above, when {{math|1={''f'',''\mathcal H''} = 0}}, {{math|''f''}} is a constant of motion of the system.  In addition, in canonical coordinates (with <math> \{p_i,\, p_j\} \;=\; \{q_i,q_j\} \;=\; 0</math> and <math>\{q_i,\, p_j\} \;=\; \delta_{ij}</math>), Hamilton's equations for the time evolution of the system follow immediately from this formula.

It also follows from {{EquationNote|1|(1)}} that the Poisson bracket is a [[derivation (abstract algebra)|derivation]]; that is, it satisfies a non-commutative version of Leibniz's [[product rule]]:
{{NumBlk||<math display="block">\{fg,h\} = f\{g,h\} + g\{f,h\},</math> and <math display="block">\{f,gh\} = g\{f,h\} + h\{f,g\}.</math>|{{EquationRef|2}}}}

The Poisson bracket is intimately connected to the [[Lie bracket of vector fields|Lie bracket]] of the Hamiltonian vector fields.  Because the Lie derivative is a derivation,
<math display="block">\mathcal L_v\iota_u\omega = \iota_{\mathcal L_vu}\omega + \iota_u\mathcal L_v\omega = \iota_{[v,u]}\omega + \iota_u\mathcal L_v\omega.</math>

Thus if {{math|''v''}} and {{math|''u''}} are symplectic, using <math> \mathcal{L}_v\omega =0=\mathcal L_u\omega</math>, Cartan's identity, and the fact that <math>\iota_u\omega</math> is a closed form,
<math display="block">\iota_{[v,u]}\omega = \mathcal L_v\iota_u\omega = d(\iota_v\iota_u\omega) + \iota_vd(\iota_u\omega) = d(\iota_v\iota_u\omega) = d(\omega(u,v)).</math>

It follows that <math>[v,u] = X_{\omega(u,v)}</math>, so that
{{NumBlk||<math display="block">[X_f,X_g] = X_{\omega(X_g,X_f)} = -X_{\omega(X_f,X_g)} = -X_{\{f,g\}}.</math>|{{EquationRef|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 {{math|''M''}}, and the Hamiltonian vector fields form an [[algebraic ideal|ideal]] of this subalgebra.  The symplectic vector fields are the Lie algebra of the (infinite-dimensional) [[Lie group]] of [[symplectomorphism]]s of {{math|''M''}}.

It is widely asserted that the [[Jacobi identity]] for the Poisson bracket,
<math display="block">\{f,\{g,h\}\} + \{g,\{h,f\}\} + \{h,\{f,g\}\} = 0</math>
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 [[Jacobi identity#Examples|sufficient]] to show that:
<math display="block">\operatorname{ad}_{\{g,f\}}=\operatorname{ad}_{-\{f,g\}}=[\operatorname{ad}_f,\operatorname{ad}_g]</math>
where the operator <math>\operatorname{ad}_g</math> on smooth functions on {{math|''M''}} is defined by <math>\operatorname{ad}_g(\cdot) \;=\; \{\cdot,\, g\}</math> and the bracket on the right-hand side is the commutator of operators, <math> [\operatorname A,\, \operatorname B] \;=\; \operatorname A\operatorname B - \operatorname B\operatorname A</math>.  By {{EquationNote|1|(1)}}, the operator <math>\operatorname{ad}_g</math> is equal to the operator {{math|''X<sub>g</sub>''}}.  The proof of the Jacobi identity follows from {{EquationNote|3|(3)}} because, up to the factor of -1, the Lie bracket of vector fields is just their commutator as differential operators.

The [[Algebra over a field|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 {{EquationNote|2|(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]] <math>X</math> on the configuration space, let <math>P_X</math> be its [[conjugate momentum]]. The conjugate momentum mapping is a [[Lie algebra]] anti-homomorphism from the [[Lie bracket of vector fields|Lie bracket]] to the Poisson bracket:
<math display="block">\{P_X, P_Y\} = -P_{[X, Y]}.</math>

This important result is worth a short proof. Write a vector field <math>X</math> at point <math>q</math> in the [[Configuration space (physics)|configuration space]] as
<math display="block">X_q = \sum_i X^i(q) \frac{\partial}{\partial q^i}</math>
where <math display="inline"> \frac{\partial}{\partial q^i}</math> is the local coordinate frame. The conjugate momentum to <math>X</math> has the expression
<math display="block">P_X(q, p) = \sum_i X^i(q) \;p_i</math>
where the <math>p_i</math> are the momentum functions conjugate to the coordinates. One then has, for a point <math>(q,p)</math> in the [[phase space]],
<math display="block">\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}</math>

The above holds for all <math>(q, p)</math>, giving the desired result.

==Quantization==
Poisson brackets [[Deformation theory|deform]] to [[Moyal bracket]]s upon [[Weyl quantization|quantization]], that is, they generalize to a different Lie algebra, the [[Moyal bracket|Moyal algebra]], or, equivalently in [[Hilbert space]], quantum [[commutator]]s. The Wigner-İnönü [[group contraction]] of these (the classical limit, {{math|ħ → 0}})  yields the above Lie algebra.

To state this more explicitly and precisely, the [[universal enveloping algebra]] of the [[Heisenberg algebra]] is the [[Weyl 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 the [[universal enveloping algebra]].

==See also==
{{colbegin}}
*[[Commutator]]
*[[Dirac bracket]]
*[[Lagrange bracket]]
*[[Moyal bracket]]
*[[Peierls bracket]]
*[[Phase space]]
*[[Poisson algebra]]
*[[Poisson ring]]
*[[Poisson superalgebra]]
*[[Poisson superbracket]]
{{colend}}

==Remarks==
{{reflist|group=Note}}

==References==
{{reflist}}
* {{cite book |title=Mathematical Methods of Classical Mechanics |last=Arnold |first=Vladimir I. |author-link=Vladimir Arnold |edition=2nd |year=1989 |publisher=Springer |location=New York |isbn=978-0-387-96890-2 |url-access=registration |url=https://archive.org/details/mathematicalmeth0000arno }}
* {{cite book |title=Mechanics |volume=1 |series=[[Course of Theoretical Physics]] |last1=Landau |first1=Lev D. |author-link1=Lev Landau |last2=Lifshitz| first2= Evegeny M.| author-link2=Evgeny Lifshitz|year=1982 |edition=3rd |publisher=Butterworth-Heinemann |isbn=978-0-7506-2896-9 }}
*{{cite book|last1=Karasëv|first1=Mikhail V.|author-link2=Victor Pavlovich Maslov|last2=Maslov|first2=Victor P.|title=Nonlinear Poisson brackets, Geometry and Quantization|translator-first1=Alexey|translator-last1=Sossinsky| translator-first2=M.A.| translator-last2=Shishkova|series=Translations of Mathematical Monographs|volume=119|publisher=American Mathematical Society| location=Providence, RI|year=1993|mr=1214142|isbn=978-0821887967 }}
*{{cite book|last1=Moretti|first1=Valter|title=Analytical Mechanics, Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity|series=UNITEXT|volume=150|publisher=Springer| year=2023|isbn=978-3-031-27612-5 }}
*{{Cite journal|first1=Siméon-Denis|last1=Poisson|title=Mémoire sur la variation des constantes arbitraires dans les questions de Mécanique|journal=Journal de l'École polytechnique, 15e cahier|year=1809|volume=8|page=266-344|url=https://math.huji.ac.il/~piz/documents-others/SDP-1809.pdf|doi=|ref=poisson1809}}
*{{Cite journal|first1=Charles-Michel|last1=Marle|authorlink1=Charles-Michel Marle|title=The Inception of Symplectic Geometry: the Works of Lagrange and Poisson During the Years 1808-1810|journal=Letters in Mathematical Physics|year=2009|volume=90|issue=1–3 |page=3-21|doi=10.1007/s11005-009-0347-y|arxiv=0902.0685|bibcode=2009LMaPh..90....3M |ref=marle2009}}

==External links==
* {{springer|title=Poisson brackets|id=p/p073270}}
* {{mathworld |urlname=PoissonBracket |title=Poisson bracket|author=[[Eric W. Weisstein]]}}

[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]
[[Category:Bilinear maps]]
[[Category:Concepts in physics]]