# 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

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.