Pfaffian form

A differential form of degree 1.

Comments

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)