# 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=12643

This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article