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An equation of the form
 
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>
+
$$ \tag{1 }
 +
\omega  \equiv  a _ {1} ( x)  d x _ {1} + \dots + a _ {n} ( x) d x _ {n}  =  0 ,\  n \geq  3 ,
 +
$$
  
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" />.
+
where $  x \in D \subset  \mathbf R  ^ {n} $,  
 +
$  \omega $
 +
is a differential $  1 $-
 +
form (cf. [[Differential form|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 <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" />.
+
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
 
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>
+
$$ \tag{2 }
 +
d \omega \wedge \omega  \equiv  0 .
 +
$$
  
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.
+
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
 
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>
+
$$ \tag{3 }
 +
P  d x + Q  d y + R  d z  = 0 ,
 +
$$
 +
 
 +
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 (
  
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
+
\frac{\partial  R }{\partial  x }
 +
-
 +
\frac{\partial  P }{\partial  z }
  
<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>
+
\right ) +
 +
R \left (
 +
 
 +
\frac{\partial  P }{\partial  y }
 +
-  
 +
\frac{\partial  Q }{\partial  x }
 +
 
 +
\right )  = 0
 +
$$
  
 
or
 
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>
+
$$
 +
(  \mathop{\rm curl}  F , F  )  = 0 ,\  \textrm{ where } \
 +
= ( P , Q , R ) .
 +
$$
  
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
+
In this case there exist smooth functions $  \mu $,  
 +
$  U $(
 +
$  \mu \neq 0 $)  
 +
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>
+
$$
 +
P  d x + Q  d y + R  d z  \equiv  \mu  d U ,
 +
$$
  
and the integral surfaces of the Pfaffian equation (3) are given by the equations <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.
+
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 [[#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]]]).
+
It was J. Pfaff [[#References|[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 [[#References|[2]]]).
  
 
By a smooth change of variables any Pfaffian equation can locally be brought to the form
 
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>
+
$$ \tag{5 }
 +
d y _ {0} - \sum _ { j= } 1 ^ { p }  z _ {j}  d y _ {j}  = 0 ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
y _ {0} ( x)  = c _ {0} \dots y _ {p} ( x)  = c _ {p} .
 +
$$
  
 
A Pfaffian system is a system of equations of the form
 
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>
+
$$ \tag{6 }
 +
\omega _ {1}  = 0 \dots
 +
\omega _ {k}  = 0 ,\ \
 +
k < n ,
 +
$$
  
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:
+
where $  x \in D \subset  \mathbf R  ^ {n} $
 +
and $  \omega _ {i} $
 +
are differential $  1 $-
 +
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>
+
$$
 +
\omega _ {j}  = \
 +
\sum _ { q= } 1 ^ { n }
 +
\omega _ {jq} ( x)  d x _ {q} ,\ \
 +
j = 1 \dots k .
 +
$$
  
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" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072510/p07251069.png" /> to be completely integrable is
+
Frobenius' theorem: A necessary and sufficient condition for a Pfaffian system (6) of rank $  k $
 +
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>
+
$$
 +
d \omega _ {j} \wedge \omega _ {1} \wedge \dots \wedge \omega _ {k}  = \
 +
0 ,\  j = 1 \dots k .
 +
$$
  
 
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 [[#References|[6]]]).
 
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]]]).
Line 59: Line 176:
 
A number of results has been obtained on the analytic theory of Pfaffian systems. A completely-integrable Pfaffian system
 
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>
+
$$
 +
d y  = x  ^ {-} p f  d x + z  ^ {-} q g  d z
 +
$$
  
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.
+
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 [[#References|[7]]]); generalizations to a larger number of independent variables have also been given.
  
 
====References====
 
====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>
 
<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====
 
====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 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|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]].
+
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 $  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|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.
+
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. [[#References|[a1]]] for more details on all this.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:06, 6 June 2020


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)
How to Cite This Entry:
Pfaffian equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian_equation&oldid=18964
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article