Pfaffian form
A differential form of degree 1.
Comments
A Pfaffian form defined on an open subset
,
a manifold, is of odd class
at
if it satisfies
![]() |
it is of even class at
if it satisfies
![]() |
Pfaffian forms of class and
both define a Pfaffian equation of class
.
Darboux's theorem on Pfaffian forms says the following.
1) If is a Pfaffian form of constant class
on an open subset
of a manifold
, then for every
there is a neighbourhood
with a family of independent functions
, such that on
,
![]() |
2) If is a Pfaffian form of constant class
on an open subset
of a manifold
, then for every
there is a neighbourhood
with a family of independent functions
such that on
,
![]() |
where the function is without zeros on
.
Thus, if , the functions
are canonical coordinates for the symplectic form
.
References
[a1] | P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V (Translated from French) |
Pfaffian form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_form&oldid=13779