# Difference between revisions of "Completely-integrable differential equation"

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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |

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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |

## Revision as of 16:56, 15 April 2012

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].

#### 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 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.

#### 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