# Jacobi brackets

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 [1], [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.

#### References

 [1] 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 [2] A. Mayer, "Ueber die Weiler'sche Integrationsmethode der partiellen Differentialgleichungen erster Ordnung" Math. Ann. , 9 (1876) pp. 347–370 [3] N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian) [4] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)