Jacobi brackets

Mayer brackets

The differential expression

(1)

in the functions and of independent variables and .

The main properties are:

1) ;

2) ;

3) if , and , then ;

4) .

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

The expression (1) is sometimes written in the form

where the symbolic notation

(2)

is used. If and are regarded as functions of , and , , then (2) gets the meaning of the total derivative with respect to .

If and are independent of , 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)