Completely-integrable differential equation

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An equation of the form


for which an -dimensional integral manifold passes through each point of a certain domain in the space . A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition , where is the symbol of the exterior product [1]. If , this condition has the form:

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

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

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


[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


The exterior product is also called the outer product.

An -dimensional submanifold of is an integral manifold of (*) if the restriction of to is zero; cf. also Pfaffian equation. Another (dual) way to formulate this is as follows. Let be an open subset where . For each let be the set of all (tangent) vectors at such that . Then is an -dimensional subspace and the define a distribution on . An integral manifold of (or of the equation ) is now an -dimensional submanifold of such that for all . A distribution on is called involutive if for all vector fields on such that for all also for all . The Frobenius integrability condition is equivalent in these terms to the condition that the distribution defined by be involutive. All this generalizes to systems of equations , ; cf. Integrable system.

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


[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)
How to Cite This Entry:
Completely-integrable differential equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.E. Reizin' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article