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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 [[#References|[1]]], are a particular case of the [[Jacobi brackets|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 [[#References|[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 [[#References|[3]]]–[[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Poisson,   ''J. Ecole Polytechn.'' , '''8'''  (1809)  pp. 266–344</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.T. Whittaker,   "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Lur'e,   "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Goldstein,   "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>

====Comments====
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|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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.C. Newell,   "Solitons in mathematical physics" , SIAM  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Abraham,   J.E. Marsden,   "Foundations of mechanics" , Benjamin  (1978)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR>
</table>