View source for Completely-integrable differential equation

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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|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 (

\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 [[#References|[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|foliation]] [[#References|[3]]].

====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 {{MR|0200938}} {{ZBL|0247.57006}} </TD></TR></table>

====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|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  $  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====
<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>