# Poisson brackets

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The differential expression

$$\tag{1 } ( u , v ) = \sum_{i=1} ^ { n } \left ( \frac{\partial u }{\partial q _ {i} } \frac{\partial v }{\partial p _ {i} } - \frac{\partial u }{\partial p _ {i} } \frac{\partial v }{\partial q _ {i} } \right ) ,$$

depending on two functions $u ( q , p )$ and $v ( q , p )$ of $2n$ variables $q = ( q _ {1} \dots q _ {n} )$, $p = ( p _ {1} \dots p _ {n} )$. The Poisson brackets, introduced by S. Poisson [1], are a particular case of the Jacobi brackets. The Poisson brackets are a bilinear form in the functions $u$ and $v$, such that

$$( u , v ) = - ( v , u )$$

and the Jacobi identity

$$( u , ( v , w ) ) + ( v , ( w , u ) ) + ( w , ( u , v ) ) = 0$$

holds (see [2]).

The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see [3][5]). For example, if $q$ and $p$ are canonical variables and a transformation

$$\tag{2 } Q = Q ( q , p ) ,\ \ P = P ( q , p )$$

is given, where $Q = ( Q _ {1} \dots Q _ {n} )$, $P = ( P _ {1} \dots P _ {n} )$ and the $( n \times n )$- matrices

$$\tag{3 } ( P , P ) ,\ ( Q , Q ) ,\ ( Q , P )$$

are constructed with entries $( P _ {i} , P _ {j} )$, $( Q _ {i} , Q _ {j} )$, $( Q _ {i} , P _ {j} )$, respectively, then (2) is a canonical transformation if and only if the first two matrices in (3) are zero and the third is the unit matrix.

The Poisson brackets, computed for the case when $u$ and $v$ are replaced in (1) by some pair of coordinate functions in $q$ and $p$, are also called fundamental brackets.

#### References

 [1] S. Poisson, J. Ecole Polytechn. , 8 (1809) pp. 266–344 [2] C.G.J. Jacobi, "Nova methodus, aequationes differentiales partiales primi ordinis inter numurum variabilium quemcunque propositas integrandi" J. Reine Angew. Math. , 60 (1862) pp. 1–181 [3] E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944) [4] A.I. Lur'e, "Analytical mechanics" , Moscow (1961) (In Russian) [5] H. Goldstein, "Classical mechanics" , Addison-Wesley (1957)

Other basic properties of Poisson brackets are invariance under canonical transformations and the fact that $( F, H)$ expresses the derivative of $F( q, p)$ along trajectories, if $H$ is the Hamiltonian, so that the corresponding Hamiltonian equations are $\dot{q} _ {i} = ( q _ {i} , H)$, $\dot{p} _ {i} =( p _ {i} , H)$, which for a "standard" Hamiltonian of the form $H=( \sum p _ {i} ^ {2} )/2+ V( q)$ gives back $\dot{q} _ {i} = p _ {i}$, $\dot{p} _ {i} = - \partial H/ \partial q _ {i}$. Therefore $( F, H)$ expresses a conservation law, i.e. $F$ is a conserved quantity.

The Poisson brackets may be defined for functionals depending on a function $q( x)$, as

$$F[ q] = \int\limits _ {- \infty } ^ \infty \widetilde{F} ( q ,q ^ {(1)} , q ^ {(2)} ,\dots) dx,$$

with $q ^ {(n)} = d ^ {n} q/dx ^ {n}$.

One has

$$( F, G) = \int\limits _ {- \infty } ^ \infty \frac{\delta \widetilde{F} }{\delta q } \frac{d}{dx} \frac{\delta \widetilde{G} }{\delta q } dx,$$

with ${\delta \widetilde{F} } / {\delta q }$, ${\delta \widetilde{G} } / {\delta q }$ variational derivatives, i.e.

$$\frac{\delta \widetilde{F} }{\delta q } = \sum \left ( - \frac{d}{dx} \right ) ^ {n} \frac{\partial \widetilde{F} }{\partial q ^ {(n)} } .$$

#### References

 [a1] A.C. Newell, "Solitons in mathematical physics" , SIAM (1985) [a2] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) [a3] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978) [a4] F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)
How to Cite This Entry:
Poisson brackets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_brackets&oldid=55108
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article