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Difference between revisions of "Pfaffian form"

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$$  
 
$$  
 
\omega \wedge ( d \omega )  ^ {s} ( x)  \neq  0 ,\ \  
 
\omega \wedge ( d \omega )  ^ {s} ( x)  \neq  0 ,\ \  
( d \omega )  ^ {s+} 1 ( x)  =  0 ;
+
( d \omega )  ^ {s+1} ( x)  =  0 ;
 
$$
 
$$
  
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$$  
 
$$  
 
\omega \wedge ( d \omega )  ^ {s} ( x)  \neq  0 ,\ \  
 
\omega \wedge ( d \omega )  ^ {s} ( x)  \neq  0 ,\ \  
\omega \wedge ( d \omega )  ^ {s+} 1 ( x)  =  0 ,\ \  
+
\omega \wedge ( d \omega )  ^ {s+1} ( x)  =  0 ,\ \  
( d \omega )  ^ {s+} 1 ( x)  \neq  0.
+
( d \omega )  ^ {s+1} ( x)  \neq  0.
 
$$
 
$$
  
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$$  
 
$$  
\omega  =  dx  ^ {0} - \sum _ { k= } 1 ^ { s }  x  ^ {s+} k dx  ^ {k} .
+
\omega  =  dx  ^ {0} - \sum_{k=1} ^ { s }  x  ^ {s+k}  dx  ^ {k} .
 
$$
 
$$
  
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$$  
 
$$  
\omega  =  z  ^ {0}  dx  ^ {0} - \sum _ { k= } 1 ^ { s }  z  ^ {k}  dx  ^ {k} ,
+
\omega  =  z  ^ {0}  dx  ^ {0} - \sum_{k=1} ^ { s }  z  ^ {k}  dx  ^ {k} ,
 
$$
 
$$
  

Latest revision as of 07:53, 9 January 2024


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)
How to Cite This Entry:
Pfaffian form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_form&oldid=54942