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)

Comments

The Poisson brackets are an essential tool in classical mechanics, cf. e.g. [a1].

References

[a1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)