View source for Pfaffian equation

An equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725103.png" /> is a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725104.png" />-form (cf. [[Differential form|Differential form]]), and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725106.png" />, are real-valued. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725107.png" /> and suppose that the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725108.png" /> does not have critical points in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p0725109.png" />.

A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251010.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251011.png" /> and of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251012.png" /> is called an integral manifold of the Pfaffian equation (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251014.png" />. The Pfaffian equation is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251015.png" /> through each point of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251016.png" />.

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251018.png" /> is the differential form of degree 2 obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251019.png" /> by exterior differentiation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251020.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251024.png" /> are functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251027.png" /> and condition (2) for complete integrability is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251029.png" /></td> </tr></table>

In this case there exist smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251031.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251032.png" />) such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251033.png" /></td> </tr></table>

and the integral surfaces of the Pfaffian equation (3) are given by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251035.png" /> is a certain force field, then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251036.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251039.png" /> are given, then (3) will be an ordinary differential equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251040.png" /> and the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251043.png" /> will be an integral curve.

It was J. Pfaff [[#References|[1]]] who posed the problem of studying equation (1) for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251044.png" /> and of reducing the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251045.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251046.png" /> to a canonical form. Condition (4) was first obtained by L. Euler in 1755 (see [[#References|[2]]]).

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251048.png" /> are the new independent variables (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251050.png" />). The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251051.png" /> is called the class of the Pfaffian equation; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251052.png" /> is the largest number such that the differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251053.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251054.png" /> is not identically zero. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251055.png" /> the Pfaffian equation is completely integrable. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251056.png" /> are called the first integrals of the Pfaffian equation (5) and the integral manifolds of maximum possible dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251057.png" /> are given by the equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251058.png" /></td> </tr></table>

A Pfaffian system is a system of equations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251061.png" /> are differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251062.png" />-forms:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251063.png" /></td> </tr></table>

The rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251064.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251065.png" /> is the rank of the Pfaffian system at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251066.png" />. A Pfaffian system is said to be completely integrable if there is one and only one integral manifold of maximum possible dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251067.png" /> through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251068.png" />.

Frobenius' theorem: A necessary and sufficient condition for a Pfaffian system (6) of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251069.png" /> to be completely integrable is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251070.png" /></td> </tr></table>

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 [[#References|[6]]]).

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251071.png" /></td> </tr></table>

of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251072.png" /> equations has been considered, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251074.png" /> are positive integers and the vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251076.png" /> are holomorphic at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251079.png" />; sufficient conditions have been given for the existence of a holomorphic solution at the origin (see [[#References|[7]]]); generalizations to a larger number of independent variables have also been given.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.F. Pfaff,   ''Berl. Abh.''  (1814–1815)  pp. 76–135</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Euler,   "Institutiones calculi differentialis"  G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; opera mat.'' , '''10''' , Teubner  (1980)  pp. Chapt. IX  ((in Latin))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,   "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.S. Bogdanov,   "Lectures on differential equations" , Minsk  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Cartan,   "Sur la théorie des systèmes en involution et ses applications à la relativité"  ''Bull. Soc. Math. France'' , '''59'''  (1931)  pp. 88–118</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.K. Rashevskii,   "Geometric theory of partial differential equations" , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  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)</TD></TR></table>



====Comments====
The article above describes the local situation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251080.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251081.png" />-dimensional manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251082.png" /> (part of) a coordinate chart. A differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251083.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251084.png" /> that is nowhere zero defines on the one hand a Pfaffian equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251085.png" /> and on the other hand a one-dimensional subbundle of the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251086.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251087.png" />. This leads to the modern global definition of a Pfaffian equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251088.png" /> as a vector subbundle of rank 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251089.png" />, cf. also [[Pfaffian structure|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|Pfaffian form]] defining a Pfaffian equation of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251090.png" /> may be either of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251091.png" /> or class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251092.png" />. Thus, Darboux's theorem (in its modern form) comes in two steps: i) let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251093.png" /> be a Pfaffian equation of constant class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251094.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251095.png" />; then everywhere locally there exists a Pfaffian form of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251096.png" /> defining that equation; and ii) a canonical form statement for Pfaffian forms of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251097.png" />, cf. [[Pfaffian form|Pfaffian form]].

Here the class of a Pfaffian equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251098.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251099.png" /> is defined by: let any differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510100.png" /> define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510101.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510102.png" />; then the class of the equation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510103.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p072510105.png" />. Cf. [[#References|[a1]]] for more details on all this.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Libermann,   C.-M. Marle,   "Symplectic geometry and analytical mechanics" , Reidel  (1987)  pp. Chapt. V  (Translated from French)</TD></TR></table>