Difference between revisions of "Completely-integrable differential equation"
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | c0239701.png | ||
+ | $#A+1 = 50 n = 0 | ||
+ | $#C+1 = 50 : ~/encyclopedia/old_files/data/C023/C.0203970 Completely\AAhintegrable differential equation | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
An equation of the form | 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]]]: | 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: | 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]]]. | The family of integral manifolds of a completely-integrable differential equation is a [[Foliation|foliation]] [[#References|[3]]]. | ||
Line 25: | Line 96: | ||
====References==== | ====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> | <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==== | ====Comments==== | ||
− | The exterior product | + | The exterior product $ \wedge $ |
+ | is also called the outer product. | ||
− | An | + | 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 | + | 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==== | ====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> | <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> |
Latest revision as of 17:45, 4 June 2020
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 |
Completely-integrable differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-integrable_differential_equation&oldid=46424