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''Mayer brackets''

The differential expression

$$ \tag{1 }
[ F, G]  = \ 
\sum _ {k = 1 } ^ { n } 
\left [

\frac{\partial  F }{\partial  p _ {k} }

\left (

\frac{\partial  G }{\partial  x _ {k} }
 +
p _ {k} 
\frac{\partial  G }{\partial  u }

\right ) \right . -
$$

$$ 
- \left .

\frac{\partial  G }{\partial  p _ {k} }

\left ( 
\frac{\partial  F }{\partial  x _ {k} }
 + p _ {k} 
\frac{\partial  F }{\partial  u }
 \right )  \right ]
$$

in the functions  $  F ( x, u , p) $
and  $  G ( x, u , p) $
of  $  2n + 1 $
independent variables  $  x = ( x _ {1} \dots x _ {n} ) $
and  $  p = ( p _ {1} \dots p _ {n} ) $.

The main properties are:

1)  $  [ F, G] = - [ G, F] $;

2)  $  [ F, GH] = G [ F, H] + H [ F, G] $;

3) if  $  G = g ( y) $, 
$  y = ( y _ {1} \dots y _ {s} ) $
and  $  y _ {i} = f _ {i} ( x) $, 
then  $  [ F, G] = \sum _ {i = 1 }   ^ {s} ( {\partial  g } / {\partial  y _ {i} } ) [ F, f _ {i} ] $;

4)  $  [ F, [ G, H]] + [ G, [ H, F]] + [ H, [ F, G]] = $
$  ( {\partial  F } / {\partial  u } ) [ G, H] + ( {\partial  G } / {\partial  u } ) [ H, F] + ( {\partial  H } / {\partial  u } ) [ F, G] $.

The last property is called the Jacobi identity (see [[#References|[1]]], [[#References|[2]]]).

The expression (1) is sometimes written in the form

$$ 
\sum _ {k = 1 } ^ { n } 
\left (

\frac{\partial  F }{\partial  p _ {k} }

\frac{dG }{dx _ {k} }
 -

\frac{\partial  G }{\partial  p _ {k} }

\frac{dF }{dx _ {k} }

\right ) ,
$$

where the symbolic notation

$$ \tag{2 }

\frac{dH }{dx _ {k} }
  = \ 

\frac{\partial  H }{\partial  x _ {k} }
 +
p _ {k} 
\frac{\partial  H }{\partial  u }

$$

is used. If  $  u $
and  $  p _ {k} $
are regarded as functions of  $  x = ( x _ {1} \dots x _ {n} ) $, 
and  $  p _ {k} = \partial  u/ \partial  x _ {k} $, 
$  1 \leq  k \leq  n $, 
then (2) gets the meaning of the total derivative with respect to  $  x _ {k} $.

If  $  F $
and  $  G $
are independent of  $  u $, 
then their Jacobi brackets (1) are [[Poisson brackets|Poisson brackets]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,   "Nova methodus, aequationes differentiales partiales primi ordinis inter numerum variabilium quemcunque propositas integrandi"  ''J. Reine Angew. Math.'' , '''60'''  (1862)  pp. 1–181</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Mayer,   "Ueber die Weiler'sche Integrationsmethode der partiellen Differentialgleichungen erster Ordnung"  ''Math. Ann.'' , '''9'''  (1876)  pp. 347–370</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.M. Gyunter,   "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,   "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>

====Comments====
The Poisson brackets are an essential tool in classical mechanics, cf. e.g. [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR></table>