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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/c/c023/c023970/c0239701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\omega  \equiv \
 +
\sum _ {i = 1 } ^ { n }
 +
P _ {i} ( x)  dx  ^ {i}  = 0,\ \
 +
P _ {i} \in C  ^ {1} ,
 +
$$
 +
 
 +
for which an  $  ( n - 1 ) $-
 +
dimensional integral manifold passes through each point of a certain domain in the space  $  \mathbf R  ^ {n} $.
 +
A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition  $  \omega \wedge d \omega = 0 $,
 +
where  $  \wedge $
 +
is the symbol of the [[Exterior product|exterior product]] [[#References|[1]]]. If  $  n = 3 $,
 +
this condition has the form:
 +
 
 +
$$
 +
P _ {1} \left (
 +
 
 +
\frac{\partial  P _ {3} }{\partial  x  ^ {2} }
 +
-
 +
 
 +
\frac{\partial  P _ {2} }{\partial  x  ^ {3} }
 +
 
 +
\right ) + P _ {2} \left (
  
for which an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c0239702.png" />-dimensional integral manifold passes through each point of a certain domain in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c0239703.png" />. A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c0239704.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c0239705.png" /> is the symbol of the [[Exterior product|exterior product]] [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c0239706.png" />, this condition has the form:
+
\frac{\partial  P _ {1} }{\partial  x  ^ {3} }
 +
-
  
<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/c/c023/c023970/c0239707.png" /></td> </tr></table>
+
\frac{\partial  P _ {3} }{\partial  x  ^ {1} }
  
<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/c/c023/c023970/c0239708.png" /></td> </tr></table>
+
\right ) + P _ {3} \left (
 +
 
 +
\frac{\partial  P _ {2} }{\partial  x  ^ {1} }
 +
-
 +
 
 +
\frac{\partial  P _ {1} }{\partial  x  ^ {2} }
 +
 
 +
\right ) =
 +
$$
 +
 
 +
$$
 +
=  
 +
0.
 +
$$
  
 
Instead of equation (*) the following system of equations is sometimes considered [[#References|[2]]]:
 
Instead of equation (*) the following system of equations is sometimes considered [[#References|[2]]]:
  
<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/c/c023/c023970/c0239709.png" /></td> </tr></table>
+
$$
 +
dx  ^ {i}  = \
 +
\sum _ {j = 1 } ^ { {n }  - 1 } a _ {j}  ^ {i}
 +
( x, t)  dt  ^ {j} ,\ \
 +
i = 1 \dots n.
 +
$$
  
 
In this case the conditions of complete integrability assume the form:
 
In this case the conditions of complete integrability assume 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/c/c023/c023970/c02397010.png" /></td> </tr></table>
+
$$
 +
\sum _ {l = 1 } ^ { n } 
 +
\frac{\partial  a _ {j}  ^ {i} }{\partial  x  ^ {l} }
  
<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/c/c023/c023970/c02397011.png" /></td> </tr></table>
+
( x, t) a _ {k}  ^ {l} ( x, t) +
 +
\frac{\partial  a _ {j}  ^ {i} }{\partial  t  ^ {k} }
 +
( x, t) =
 +
$$
  
<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/c/c023/c023970/c02397012.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {l = 1 } ^ { n } 
 +
\frac{\partial  a _ {k}  ^ {i} }{\partial
 +
x  ^ {l} }
 +
( x, t) a _ {j}  ^ {l} ( x, t) +
 +
 
 +
\frac{\partial  a _ {k}  ^ {i} }{\partial  t  ^ {j} }
 +
( x, t),
 +
$$
 +
 
 +
$$
 +
= 1 \dots n ; \  j , k  = 1 \dots n .
 +
$$
  
 
The family of integral manifolds of a completely-integrable differential equation is a [[Foliation|foliation]] [[#References|[3]]].
 
The family of integral manifolds of a completely-integrable differential equation is a [[Foliation|foliation]] [[#References|[3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Frobenius,   "Ueber das Pfaffsche Problem" ''J. Reine Angew. Math.'' , '''82''' (1877) pp. 230–315</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.V. Nemytskii,   "On the orbit theory of general dynamic systems" ''Mat. Sb.'' , '''23 (65)''' : 2 (1948) pp. 161–186 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.P. Novikov,   "Topology of foliations" ''Trans. Moscow Math. Soc.'' , '''14''' (1965) pp. 268–304 ''Trudy Moskov. Mat. Obshch.'' , '''14''' (1965) pp. 248–278</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Frobenius, "Ueber das Pfaffsche Problem" ''J. Reine Angew. Math.'' , '''82''' (1877) pp. 230–315</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.V. Nemytskii, "On the orbit theory of general dynamic systems" ''Mat. Sb.'' , '''23 (65)''' : 2 (1948) pp. 161–186 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.P. Novikov, "Topology of foliations" ''Trans. Moscow Math. Soc.'' , '''14''' (1965) pp. 268–304 ''Trudy Moskov. Mat. Obshch.'' , '''14''' (1965) pp. 248–278 {{MR|0200938}} {{ZBL|0247.57006}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The exterior product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397013.png" /> is also called the outer product.
+
The exterior product $  \wedge $
 +
is also called the outer product.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397014.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397016.png" /> is an integral manifold of (*) if the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397017.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397018.png" /> is zero; cf. also [[Pfaffian equation|Pfaffian equation]]. Another (dual) way to formulate this is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397019.png" /> be an open subset where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397020.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397021.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397022.png" /> be the set of all (tangent) vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397023.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397025.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397026.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397027.png" />-dimensional subspace and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397028.png" /> define a distribution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397029.png" />. An integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397031.png" /> (or of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397032.png" />) is now an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397033.png" />-dimensional submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397036.png" />. A distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397038.png" /> is called involutive if for all vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397042.png" /> also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397043.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397044.png" />. The Frobenius integrability condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397045.png" /> is equivalent in these terms to the condition that the distribution defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397046.png" /> be involutive. All this generalizes to systems of equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397048.png" />; cf. [[Integrable system|Integrable system]].
+
An $  ( n - 1 ) $-
 +
dimensional submanifold $  M $
 +
of $  \mathbf R  ^ {n} $
 +
is an integral manifold of (*) if the restriction of $  \omega $
 +
to $  M $
 +
is zero; cf. also [[Pfaffian equation|Pfaffian equation]]. Another (dual) way to formulate this is as follows. Let $  U $
 +
be an open subset where $  \omega \neq 0 $.  
 +
For each $  x \in U $
 +
let $  D _ {x} $
 +
be the set of all (tangent) vectors $  \xi $
 +
at $  x \in \mathbf R  ^ {n} $
 +
such that $  \omega ( \xi ) = 0 $.  
 +
Then $  D _ {x} \subset  T _ {x} ( \mathbf R  ^ {n} ) $
 +
is an $  ( n - 1 ) $-
 +
dimensional subspace and the $  D _ {x} \in U $
 +
define a distribution on $  U $.  
 +
An integral manifold $  M $
 +
of $  D $(
 +
or of the equation $  \omega = 0 $)  
 +
is now an $  ( n - 1 ) $-
 +
dimensional submanifold of $  U $
 +
such that $  T _ {x} M = D _ {x} $
 +
for all $  x \in M $.  
 +
A distribution $  D $
 +
on $  U $
 +
is called involutive if for all vector fields $  \xi , \eta $
 +
on $  U $
 +
such that $  \xi ( x) , \eta ( x) \in D _ {x} $
 +
for all $  x $
 +
also $  [ \xi , \eta ] ( x) \in D _ {x} $
 +
for all $  x $.  
 +
The Frobenius integrability condition $  \omega \wedge d \omega = 0 $
 +
is equivalent in these terms to the condition that the distribution defined by $  D $
 +
be involutive. All this generalizes to systems of equations $  \omega  ^ {i} = 0 $,  
 +
$  i = 1 \dots r $;  
 +
cf. [[Integrable system|Integrable system]].
  
The phase completely-integrable system (completely-integrable Hamiltonian system), completely-integrable Hamiltonian equation on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397049.png" />-dimensional manifold refers to a rather different property, viz. that of having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023970/c02397050.png" /> (including the Hamiltonian (function) itself) integrals in involution; cf. [[Hamiltonian system|Hamiltonian system]].
+
The phase completely-integrable system (completely-integrable Hamiltonian system), completely-integrable Hamiltonian equation on an $  n $-
 +
dimensional manifold refers to a rather different property, viz. that of having $  n $(
 +
including the Hamiltonian (function) itself) integrals in involution; cf. [[Hamiltonian system|Hamiltonian system]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Cartan,   "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1–2''' , Interscience (1963–1969)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1–2''' , Interscience (1963–1969) {{MR|1393941}} {{MR|1393940}} {{MR|0238225}} {{MR|1533559}} {{MR|0152974}} {{ZBL|0526.53001}} {{ZBL|0508.53002}} {{ZBL|0175.48504}} {{ZBL|0119.37502}} </TD></TR></table>

Latest revision as of 17:45, 4 June 2020


An equation of the form

$$ \tag{* } \omega \equiv \ \sum _ {i = 1 } ^ { n } P _ {i} ( x) dx ^ {i} = 0,\ \ P _ {i} \in C ^ {1} , $$

for which an $ ( n - 1 ) $- dimensional integral manifold passes through each point of a certain domain in the space $ \mathbf R ^ {n} $. A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition $ \omega \wedge d \omega = 0 $, where $ \wedge $ is the symbol of the exterior product [1]. If $ n = 3 $, this condition has the form:

$$ P _ {1} \left ( \frac{\partial P _ {3} }{\partial x ^ {2} } - \frac{\partial P _ {2} }{\partial x ^ {3} } \right ) + P _ {2} \left ( \frac{\partial P _ {1} }{\partial x ^ {3} } - \frac{\partial P _ {3} }{\partial x ^ {1} } \right ) + P _ {3} \left ( \frac{\partial P _ {2} }{\partial x ^ {1} } - \frac{\partial P _ {1} }{\partial x ^ {2} } \right ) = $$

$$ = 0. $$

Instead of equation (*) the following system of equations is sometimes considered [2]:

$$ dx ^ {i} = \ \sum _ {j = 1 } ^ { {n } - 1 } a _ {j} ^ {i} ( x, t) dt ^ {j} ,\ \ i = 1 \dots n. $$

In this case the conditions of complete integrability assume the form:

$$ \sum _ {l = 1 } ^ { n } \frac{\partial a _ {j} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {k} ^ {l} ( x, t) + \frac{\partial a _ {j} ^ {i} }{\partial t ^ {k} } ( x, t) = $$

$$ = \ \sum _ {l = 1 } ^ { n } \frac{\partial a _ {k} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {j} ^ {l} ( x, t) + \frac{\partial a _ {k} ^ {i} }{\partial t ^ {j} } ( x, t), $$

$$ i = 1 \dots n ; \ j , k = 1 \dots n . $$

The family of integral manifolds of a completely-integrable differential equation is a foliation [3].

References

[1] G. Frobenius, "Ueber das Pfaffsche Problem" J. Reine Angew. Math. , 82 (1877) pp. 230–315
[2] V.V. Nemytskii, "On the orbit theory of general dynamic systems" Mat. Sb. , 23 (65) : 2 (1948) pp. 161–186 (In Russian)
[3] S.P. Novikov, "Topology of foliations" Trans. Moscow Math. Soc. , 14 (1965) pp. 268–304 Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 248–278 MR0200938 Zbl 0247.57006

Comments

The exterior product $ \wedge $ is also called the outer product.

An $ ( n - 1 ) $- dimensional submanifold $ M $ of $ \mathbf R ^ {n} $ is an integral manifold of (*) if the restriction of $ \omega $ to $ M $ is zero; cf. also Pfaffian equation. Another (dual) way to formulate this is as follows. Let $ U $ be an open subset where $ \omega \neq 0 $. For each $ x \in U $ let $ D _ {x} $ be the set of all (tangent) vectors $ \xi $ at $ x \in \mathbf R ^ {n} $ such that $ \omega ( \xi ) = 0 $. Then $ D _ {x} \subset T _ {x} ( \mathbf R ^ {n} ) $ is an $ ( n - 1 ) $- dimensional subspace and the $ D _ {x} \in U $ define a distribution on $ U $. An integral manifold $ M $ of $ D $( or of the equation $ \omega = 0 $) is now an $ ( n - 1 ) $- dimensional submanifold of $ U $ such that $ T _ {x} M = D _ {x} $ for all $ x \in M $. A distribution $ D $ on $ U $ is called involutive if for all vector fields $ \xi , \eta $ on $ U $ such that $ \xi ( x) , \eta ( x) \in D _ {x} $ for all $ x $ also $ [ \xi , \eta ] ( x) \in D _ {x} $ for all $ x $. The Frobenius integrability condition $ \omega \wedge d \omega = 0 $ is equivalent in these terms to the condition that the distribution defined by $ D $ be involutive. All this generalizes to systems of equations $ \omega ^ {i} = 0 $, $ i = 1 \dots r $; cf. Integrable system.

The phase completely-integrable system (completely-integrable Hamiltonian system), completely-integrable Hamiltonian equation on an $ n $- dimensional manifold refers to a rather different property, viz. that of having $ n $( including the Hamiltonian (function) itself) integrals in involution; cf. Hamiltonian system.

References

[a1] E. Cartan, "Les systèmes différentielles extérieurs et leur applications géométriques" , Hermann (1945)
[a2] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1963–1969) MR1393941 MR1393940 MR0238225 MR1533559 MR0152974 Zbl 0526.53001 Zbl 0508.53002 Zbl 0175.48504 Zbl 0119.37502
How to Cite This Entry:
Completely-integrable differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-integrable_differential_equation&oldid=13469
This article was adapted from an original article by L.E. Reizin' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article