Jacobi brackets
From Encyclopedia of Mathematics
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) |
How to Cite This Entry:
Jacobi brackets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_brackets&oldid=47454
Jacobi brackets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_brackets&oldid=47454
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article