Pfaffian equation

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An equation of the form


where , is a differential -form (cf. Differential form), and the functions , , are real-valued. Let and suppose that the vector field does not have critical points in the domain .

A manifold of dimension and of class is called an integral manifold of the Pfaffian equation (1) if on . The Pfaffian equation is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension through each point of the domain .

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


Here is the differential form of degree 2 obtained from by exterior differentiation, and 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


where , and are functions of , and and condition (2) for complete integrability is



In this case there exist smooth functions , () such that

and the integral surfaces of the Pfaffian equation (3) are given by the equations . If is a certain force field, then the field has 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 , are given, then (3) will be an ordinary differential equation in and the curve , , will be an integral curve.

It was J. Pfaff [1] who posed the problem of studying equation (1) for arbitrary and of reducing the differential -form 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


where are the new independent variables (, ). The number is called the class of the Pfaffian equation; here is the largest number such that the differential form of degree is not identically zero. When the Pfaffian equation is completely integrable. The functions are called the first integrals of the Pfaffian equation (5) and the integral manifolds of maximum possible dimension are given by the equations

A Pfaffian system is a system of equations of the form


where and are differential -forms:

The rank of the matrix is the rank of the Pfaffian system at the point . A Pfaffian system is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension through each point .

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

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

of equations has been considered, where and are positive integers and the vector functions , are holomorphic at the point , , ; 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.


[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 be an -dimensional manifold, (part of) a coordinate chart. A differential -form on that is nowhere zero defines on the one hand a Pfaffian equation on and on the other hand a one-dimensional subbundle of the cotangent bundle over . This leads to the modern global definition of a Pfaffian equation on as a vector subbundle of rank 1 of , 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 may be either of class or class . Thus, Darboux's theorem (in its modern form) comes in two steps: i) let be a Pfaffian equation of constant class on a manifold ; then everywhere locally there exists a Pfaffian form of class defining that equation; and ii) a canonical form statement for Pfaffian forms of class , cf. Pfaffian form.

Here the class of a Pfaffian equation at is defined by: let any differential form define near ; then the class of the equation is if and only if , . Cf. [a1] for more details on all this.


[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 equation. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098