ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 7 / UIF:8.2 / MODERNSCIENCE.UZ
484
UDC 514763
COMPLETE INTEGRABLE VECTOR FIELD GEOMETRY
Ibodov Nabijon Muzaffarovich
Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, National
Research University, Bukhara Institute of Nature Management, Assistant of the Department of
Mathematics and Natural Sciences.
Email:
Toyirov Muhriddin Zoir o’g’li
Tashkent Institute of irrigation and agricultural mechanization engineers National
research university student of Bukhara Institute of Natural Resources Management.
Email:
https://doi.org/10.5281/zenodo.8203169
Abstract.
This article is the result of research on operations on integrable vector fields on
the plane, namely: finding their integral lines, finding the Lie bracket and finding the flow.
Keywords:
Integral line, vector field, vector field flow, Lie bracket (Lie commutator),
integrable vector field, manifold, field, orbit.
ПОЛНАЯ ИНТЕГРИРУЕМАЯ ГЕОМЕТРИЯ ВЕКТОРНОГО ПОЛЯ
Аннотация.
Эта статья является результатом исследования операций над
интегрируемыми векторными полями на плоскости, а именно: нахождения их целых линий,
нахождения скобки Ли и нахождения потока.
Ключевые слова:
Интегральная прямая, векторное поле, поток векторного поля,
скобка Ли (коммутатор Ли), интегрируемое векторное поле, многообразие, поле, орбита.
INTRODUCTION
Definition-1.
Let
𝑀
be a smooth manifold of dimension
𝑛
,
𝑇
𝑥
𝑀
be a tangent space at the
point
𝑥 ∈ 𝑀
. The map
P
, which assigns to each point
𝑥 ∈ 𝑀
some subspace
𝑃(𝑥) ⊂ 𝑇
𝑥
𝑀
, is called
a distribution. If
𝑑𝑖𝑚𝑃(𝑥) = 𝑘
for all
𝑥 ∈ 𝑀
, then
𝑃
is called a
𝑘
−dimensional distribution.
Definition-2.
A distribution
𝑃
is called smooth if for each point
𝑥 ∈ 𝑀
there exists a
neighborhood of this point
𝑈(𝑥)
, and smooth vector fields x
1
, x
2
, … , x
m
given on
𝑈(𝑥)
, such that
vectors x
1
(y), x
2
(y), … , x
m
(y) form a basis for subspaces
𝑃(𝑦)
for each
𝑦 ∈ 𝑈(𝑥).
Definition-3.
A distribution
𝑃
is called completely integrable if for every point
𝑥 ∈ 𝑀
there
exists a connected submanifold
N
x
of a manifold
M
containing a point
𝑥
such that
𝑇
𝑥
𝑁
𝑦
= 𝑃(𝑦)
for all
𝑦 ∈ 𝑁
𝑥
.
MATERIALS AND METHODS
The submanifold
𝑁
𝑥
is called the integral submanifold of the distribution
P
.
If a family
D
of smooth vector fields is given, then a smooth distribution naturally arises.
Indeed, if
D
consists of smooth vector fields, then for each point x
∈
M the set of vectors
𝐷(𝑥) =
{𝑋(𝑥): 𝑋 ∈ 𝐷}
generates some subspace P
D
(x) of the tangent space
𝑇
𝑥
𝑀
. Of course, the
dimensions of the subspace P
D
(x) can vary from point to point. This distribution is denoted by P
D
.
It is said that the vector field X belongs to the distribution P if
𝑋(𝑥) ∈ 𝑃(𝑥)
for all
𝑥 ∈ 𝑀
.
Recall that the distribution of P on a manifold M is called involutive if
𝑋, 𝑌 ∈ 𝑃
, then
[𝑋, 𝑌] ∈ 𝑃
holds.
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 7 / UIF:8.2 / MODERNSCIENCE.UZ
485
RESULTS
A necessary and sufficient condition for the complete integrability of a distribution of
constant dimension is given in Frobenius ' theorem .
Frobenius' theorem.
In order for the distribution
P
to be completely integrable, it is
necessary and sufficient that it be involutive.
Definition-4
. A family of vector fields D is called completely integrable if the
corresponding P
D
distribution generated by D is completely integrable.
Note that if the family D consists of a single vector field, then it is always completely
integrable, since according to the theorem on the existence and uniqueness of the solution of a
system of differential equations, a single integral curve of the vector field passes through each
point. If a family consists of more than one vector field, then it is not always completely integrable.
The Frobenius theorem generalized by Herman for distributions of non-constant dimension
gives a necessary and sufficient condition for the complete integrability of a family of vector fields
consisting of a finite number of vector fields.
Theorems (Hermann).
Let
𝐷 = {𝑋
1
, 𝑋
2
, . . . , 𝑋
𝑘
}
be a family of finitely many vector fields
on a manifold
M
. A family
D
is completely integrable if and only if it is involutive.
The involution of a family of vector fields
𝐷 = {𝑋
1
, 𝑋
2
, . . . , 𝑋
𝑘
}
means the following: for
any vector fields
𝑋, 𝑌 ∈ 𝐷
, there exist smooth functions
𝑓
𝑙
(𝑥)
,
𝑥 ∈ 𝑀
,
𝑙 = 1, . . . , 𝑘
such that
[𝑋, 𝑌] = ∑ 𝑓
𝑙
(𝑥)X
𝑙
𝑘
𝑙=1
In the case when the family consists of an infinite number of vector fields, as the existing
examples show, this theorem is incorrect.
We have proved the following theorem.
The theorem.
Let
𝑀 = 𝑅
3
be a Euclidean space with Cartesian coordinates
𝑥, 𝑦, 𝑧
. A family
𝐷
consisting of vector fields
𝑋 = 𝑦
𝜕
𝜕𝑥
+ 𝑥
𝜕
𝜕𝑦
,
𝑌 =
𝜕
𝜕𝑧
,
generates a completely integrable distribution, integral manifolds
which are hyperbolic cylinders, half-planes and straight.
Example.
For a function
2
2
2
(x, y, z)
4
3
5
x
y
z
H
the corresponding Hamiltonian vector field has the following form
2
x
x
H
2
3
y
y
H
.
The corresponding Hamiltonian system has the following form
H
x
y
X
H
H
y
x
2
2
3
H
x
y
X
y
x
ISSN:
2181-3906
2023
International scientific journal
«MODERN SCIENCE АND RESEARCH»
VOLUME 2 / ISSUE 7 / UIF:8.2 / MODERNSCIENCE.UZ
486
2
, , 0
3 2
4 , 3 , 0
H
y x
X
Y
y
x
3
1
, Y
(
)
i
i
i
H
j
j
j
j
j
Y
X
X
X
Y
x
x
1
, Y
0
H
X
2
, Y
0
H
X
3
, Y
0
H
X
, Y
0
H
X
2
, , 0
3
2
H
y x
X
0
(0)
x
x
0
y(0)
y
0
z(0)
z
The trajectories of which are circles:
0
0
0
0
2 3
cos 2 3
sin 2 3
3
3
sin 2 3
cos 2 3
2
0
x
x
t
y
t
y
x
t
y
t
z
Vector field flow:
2 3
cos 2 3
sin 2 3
3
3
sin 2 3
cos 2 3
2
0
x
x
t
y
t
y
x
t
y
t
z
CONCLUSIONS
The theorem.
Let
1
2
,
,..., v
r
v v
be smooth vector fields on a manifold
𝑀
. Then the system
1
2
,
,..., v
r
v v
is integrable if and only if it is in involution.
REFERENCES
1.
Олвер, П.Ж. Апплиcатионсоф Лие Групс то Дифферентиал Эқуатионс.
Спрингер,1993.
2.
Гольдфан И.А Векторны анализ и теория поляю. –M.:Наука, 1968.
3.
Азамов А.А.,Нарманов А. Я. О предельных множествах орбит систем векторных
полей. Дифференциальные уравнения. 2004. Т. 40, №2, С.
4.
A.Y. Narmanov, A. S. Sharipov, J. O. Aslonov Differensial geometriya va topologiya
kursidan masalalar to‘plami, T:Universitet, 2014 yil.
5.
Катанаев
М.О.
Геометрические
методы
в
математической
физике,
Математический институт имени и. А. Стеклова РАН, май 2010, 553 с.
6.
Аслонов Ж.О. Геометрия орбит векторных полей Киллинга. УзМЖ –Ташкент, 2011.–
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