Pfaffian equation

An equation of the form

$$ \tag{1 } \omega \equiv a _ {1} ( x) d x _ {1} + \dots + a _ {n} ( x) d x _ {n} = 0 ,\ n \geq 3 , $$

where $ x \in D \subset \mathbf R ^ {n} $, $ \omega $ is a differential $1$-form (cf. Differential form), and the functions $ a _ {j} ( x) $, $ j = 1 \dots n $, are real-valued. Let $ a _ {j} ( x) \in C ^ {1} ( D) $ and suppose that the vector field $ a ( x) = ( a _ {1} ( x) \dots a _ {n} ( x) ) $ does not have critical points in the domain $ D $.

A manifold $ M ^ {k} \subset \mathbf R ^ {n} $ of dimension $ k \geq 1 $ and of class $ C ^ {1} $ is called an integral manifold of the Pfaffian equation (1) if $ \omega \equiv 0 $ on $ M ^ {k} $. The Pfaffian equation is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension $ n - 1 $ through each point of the domain $ D $.

Frobenius' theorem: A necessary and sufficient condition for the Pfaffian equation (1) to be completely integrable is

$$ \tag{2 } d \omega \wedge \omega \equiv 0 . $$

Here $ d \omega $ is the differential form of degree 2 obtained from $ \omega $ by exterior differentiation, and $ \wedge $ is the exterior product. In this case the integration of the Pfaffian equation reduces to the integration of a system of ordinary differential equations.

In three-dimensional Euclidean space a Pfaffian equation has the form

$$ \tag{3 } P d x + Q d y + R d z = 0 , $$

where $ P $, $ Q $ and $ R $ are functions of $ x $, $ y $ and $ z $ and condition (2) for complete integrability is

$$ \tag{4 } P \left ( \frac{\partial Q }{\partial z } - \frac{\partial R }{\partial y } \right ) + Q \left ( \frac{\partial R }{\partial x } - \frac{\partial P }{\partial z } \right ) + R \left ( \frac{\partial P }{\partial y } - \frac{\partial Q }{\partial x } \right ) = 0 $$

or

$$ ( \mathop{\rm curl} F , F ) = 0 ,\ \textrm{ where } \ F = ( P , Q , R ) . $$

In this case there exist smooth functions $ \mu $, $ U $( $ \mu \neq 0 $) such that

$$ P d x + Q d y + R d z \equiv \mu d U , $$

and the integral surfaces of the Pfaffian equation (3) are given by the equations $ U ( x , y , z ) = \textrm{ const } $. If $ F $ is a certain force field, then the field $ \mu ^ {-1} F $ has $ U $ as a potential function. If the Pfaffian equation (3) is not completely integrable, then it does not have integral surfaces but can have integral curves. If arbitrary functions $ x = x ( t) $, $ y = y ( t) $ are given, then (3) will be an ordinary differential equation in $ z $ and the curve $ x = x ( t) $, $ y = y ( t) $, $ z = z ( t) $ will be an integral curve.

It was J. Pfaff [1] who posed the problem of studying equation (1) for arbitrary $ n \geq 3 $ and of reducing the differential $ 1 $- form $ \omega $ to a canonical form. Condition (4) was first obtained by L. Euler in 1755 (see [2]).

By a smooth change of variables any Pfaffian equation can locally be brought to the form

$$ \tag{5 } d y _ {0} - \sum _ {j=1} ^ { p } z _ {j} d y _ {j} = 0 , $$

where $ y _ {0} \dots y _ {p} , z _ {1} \dots z _ {p} $ are the new independent variables ( $ 2 p + 1 \leq n $, $ p \geq 0 $). The number $ 2 p + 1 $ is called the class of the Pfaffian equation; here $ p $ is the largest number such that the differential form $ \omega \wedge d \omega \wedge \dots \wedge d \omega $ of degree $ 2 p + 1 $ is not identically zero. When $ p = 0 $ the Pfaffian equation is completely integrable. The functions $ y _ {0} ( x) \dots y _ {p} ( x) $ are called the first integrals of the Pfaffian equation (5) and the integral manifolds of maximum possible dimension $ n - p - 1 $ are given by the equations

$$ y _ {0} ( x) = c _ {0} \dots y _ {p} ( x) = c _ {p} . $$

A Pfaffian system is a system of equations of the form

$$ \tag{6 } \omega _ {1} = 0 \dots \omega _ {k} = 0 ,\ \ k < n , $$

where $ x \in D \subset \mathbf R ^ {n} $ and $ \omega _ {i} $ are differential $ 1 $- forms:

$$ \omega _ {j} = \ \sum _ { q=1} ^ { n } \omega _ {jq} ( x) d x _ {q} ,\ \ j = 1 \dots k . $$

The rank $ r $ of the matrix $ \| \omega _ {jk} ( x) \| $ is the rank of the Pfaffian system at the point $ x $. A Pfaffian system is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension $ n - r $ through each point $ x \in U $.

Frobenius' theorem: A necessary and sufficient condition for a Pfaffian system (6) of rank $ k $ to be completely integrable is

$$ d \omega _ {j} \wedge \omega _ {1} \wedge \dots \wedge \omega _ {k} = \ 0 ,\ j = 1 \dots k . $$

The problem of integrating any finite non-linear system of partial differential equations is equivalent to the problem of integrating a certain Pfaffian system (see [6]).

A number of results has been obtained on the analytic theory of Pfaffian systems. A completely-integrable Pfaffian system

$$ d y = x ^ {-p} f d x + z ^ {-q} g d z $$

of $ m $ equations has been considered, where $ p $ and $ q $ are positive integers and the vector functions $ f ( x , y , z ) $, $ g ( x , y , z ) $ are holomorphic at the point $ x = 0 $, $ y = 0 $, $ z = 0 $; sufficient conditions have been given for the existence of a holomorphic solution at the origin (see [7]); generalizations to a larger number of independent variables have also been given.

References

[1]	J.F. Pfaff, Berl. Abh. (1814–1815) pp. 76–135
[2]	L. Euler, "Institutiones calculi differentialis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 10 , Teubner (1980) pp. Chapt. IX ((in Latin))
[3]	I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)
[4]	Yu.S. Bogdanov, "Lectures on differential equations" , Minsk (1977) (In Russian)
[5]	E. Cartan, "Sur la théorie des systèmes en involution et ses applications à la relativité" Bull. Soc. Math. France , 59 (1931) pp. 88–118
[6]	P.K. Rashevskii, "Geometric theory of partial differential equations" , Moscow-Leningrad (1947) (In Russian)
[7]	R. Gérard (ed.) J.-R. Ramis (ed.) , Equations différentielles et systèmes de Pfaff dans le champ complexe 1–2 , Lect. notes in math. , 712; 1015 , Springer (1979)

Comments

The article above describes the local situation. Let $ M $ be an $ n $- dimensional manifold, $ U $( part of) a coordinate chart. A differential $ 1 $- form on $ U $ that is nowhere zero defines on the one hand a Pfaffian equation on $ U $ and on the other hand a one-dimensional subbundle of the cotangent bundle $ T ^ {*} U $ over $ U $. This leads to the modern global definition of a Pfaffian equation on $ M $ as a vector subbundle of rank 1 of $ T ^ {*} M $, cf. also Pfaffian structure.

The statement embodied in formula (5) of the article above is known as Darboux's theorem on Pfaffian equations. There is a subtlety involved here. The Pfaffian form defining a Pfaffian equation of class $ 2s+ 1 $ may be either of class $ 2s+ 1 $ or class $ 2s+ 2 $. Thus, Darboux's theorem (in its modern form) comes in two steps: i) let $ \xi $ be a Pfaffian equation of constant class $ 2s+ 1 $ on a manifold $ M $; then everywhere locally there exists a Pfaffian form of class $ 2s+ 1 $ defining that equation; and ii) a canonical form statement for Pfaffian forms of class $ 2s+ 1 $, cf. Pfaffian form.

Here the class of a Pfaffian equation $ \xi $ at $ x \in M $ is defined by: let any differential form $ \omega $ define $ \xi $ near $ x $; then the class of the equation is $ 2s+ 1 $ if and only if $ ( \omega \wedge ( d \omega ) ^ {s} )( x) \neq 0 $, $ ( \omega \wedge ( d \omega ) ^ {s+1} )( x) = 0 $. Cf. [a1] for more details on all this.

References

[a1]	P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) pp. Chapt. V (Translated from French)