Completely-integrable differential equation

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