# Pfaffian form

A differential form of degree 1.

A Pfaffian form $\omega = a _ {1} ( x) dx ^ {1} + \dots + a _ {n} ( x) dx ^ {n}$ defined on an open subset $U \subset M$, $M$ a manifold, is of odd class $2s+ 1$ at $x$ if it satisfies

$$\omega \wedge ( d \omega ) ^ {s} ( x) \neq 0 ,\ \ ( d \omega ) ^ {s+1} ( x) = 0 ;$$

it is of even class $2s+ 2$ at $x$ if it satisfies

$$\omega \wedge ( d \omega ) ^ {s} ( x) \neq 0 ,\ \ \omega \wedge ( d \omega ) ^ {s+1} ( x) = 0 ,\ \ ( d \omega ) ^ {s+1} ( x) \neq 0.$$

Pfaffian forms of class $2s+ 1$ and $2s+ 2$ both define a Pfaffian equation of class $2s+ 1$.

Darboux's theorem on Pfaffian forms says the following.

1) If $\omega$ is a Pfaffian form of constant class $2s+ 1$ on an open subset $U$ of a manifold $M$, then for every $x \in U$ there is a neighbourhood $V$ with a family of independent functions $x ^ {0} \dots x ^ {2s}$, such that on $V$,

$$\omega = dx ^ {0} - \sum_{k=1} ^ { s } x ^ {s+k} dx ^ {k} .$$

2) If $\omega$ is a Pfaffian form of constant class $2s+ 2$ on an open subset $U$ of a manifold $M$, then for every $x \in U$ there is a neighbourhood $V$ with a family of independent functions $x ^ {0} \dots x ^ {s} , z ^ {0} \dots z ^ {s}$ such that on $V$,

$$\omega = z ^ {0} dx ^ {0} - \sum_{k=1} ^ { s } z ^ {k} dx ^ {k} ,$$

where the function $z ^ {0}$ is without zeros on $V$.

Thus, if $\mathop{\rm dim} ( M) = 2s+ 2$, the functions $(- x ^ {0} , x ^ {1} \dots x ^ {s} , z ^ {0} \dots z ^ {s} )$ are canonical coordinates for the symplectic form $d \omega$.

#### References

 [a1] P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V (Translated from French)
How to Cite This Entry:
Pfaffian form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_form&oldid=54942