# Poisson brackets

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 , 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 ).

The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see ). 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.

How to Cite This Entry:
Poisson brackets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_brackets&oldid=48215
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article