# Pfaffian equation

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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)

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.