On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities
We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities....
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| Datum: | 2010 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2010
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| author | Azhymbaev, D. T. Tleubergenov, M. I. Ажымбаев, Д. Т. Тлеубергенов, М. И. Ажымбаев, Д. Т. Тлеубергенов, М. И. |
| author_facet | Azhymbaev, D. T. Tleubergenov, M. I. Ажымбаев, Д. Т. Тлеубергенов, М. И. Ажымбаев, Д. Т. Тлеубергенов, М. И. |
| author_sort | Azhymbaev, D. T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:27Z |
| description | We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces. |
| first_indexed | 2026-03-24T02:33:03Z |
| format | Article |
| fulltext |
UDK 531.31; 519.21
M. Y. Tleuberhenov, D. T. AΩ¥mbaev
(Yn-t matematyky M-va obrazovanyq y nauky Respublyky Kazaxstan, Almat¥)
O POSTROENYY MNOÛESTVA
STOXASTYÇESKYX DYFFERENCYAL|NÁX URAVNENYJ
PO ZADANNOMU YNTEHRAL|NOMU MNOHOOBRAZYG,
NE ZAVYSQWEMU OT SKOROSTEJ
We construct the Lagrange equation, the Hamilton equation, and the Birkhoff equation on the basis of
given properties of motion under random perturbations. The random disturbing forces are assumed to
belong to the class of Wiener processes and the given properties of motion are assumed to be inde-
pendent of velocities. The obtained results are illustrated by the example of motion of Earth's satellite
under the action of gravitation and aerodynamic forces.
Pobudovano rivnqnnq LahranΩa, Hamil\tona ta Birkhofa za zadanymy vlastyvostqmy ruxu pry
naqvnosti vypadkovyx zburen\. Pry c\omu prypuskagt\, wo vypadkovi zburni syly naleΩat\
klasu vinerovyx procesiv, a zadani vlastyvosti ruxu ne zaleΩat\ vid ßvydkostej. Otrymani re-
zul\taty proilgstrovano na prykladi ruxu ßtuçnoho suputnyka Zemli pid di[g syl tqΩinnq ta
aerodynamiçnyx syl.
V rabote [1] postroeno mnoΩestvo ob¥knovenn¥x dyfferencyal\n¥x uravne-
nyj, ymegwyx zadannug yntehral\nug kryvug. ∏ta rabota vposledstvyy oka-
zalas\ osnovopolahagwej v stanovlenyy y razvytyy teoryy obratn¥x zadaç dy-
namyky system, opys¥vaem¥x ob¥knovenn¥my dyfferencyal\n¥my uravnenyq-
my (ODU) (sm., naprymer, [2, 3]). Sleduet otmetyt\, çto odyn yz obwyx metodov
reßenyq obratn¥x zadaç dynamyky v klasse ODU predloΩen v rabote [3].
Postanovka zadaçy. Po zadannomu mnoΩestvu
Λ( ) : ( , )t x tλ = 0 , λ ∈ Rm , x Rm∈ , λ ∈Cxt
22 , (1)
trebuetsq postroyt\ stoxastyçeskye uravnenyq lahranΩevoj, hamyl\tonovoj y
byrkhofovoj struktur
d
dt
L
x
L
x
x x tj
j∂
∂
−
∂
∂
= ′
�
� �
ν ν
νσ ξ( , , ) , ν = 1, n , j r= 1, , (2)
�q
H
p
k
k
=
∂
∂
,
(3)
�pk = −
∂
∂
+ ′
H
q
q p t
k
j
jσ ξν ( , , ) � , k n= 1, ,
∂
∂
−
∂
∂
R z t
z
R z t
z
zi
l
l
i
i
( , ) ( , ) � –
∂
∂
+
∂
∂
B z t
z
R z t
tl
l( , ) ( , )
= Tlµ µψ� ,
(4)
i, l n= 1 2, , µ = +1, n r ,
tak, çtob¥ mnoΩestvo Λ( )t b¥lo yntehral\n¥m mnohoobrazyem postroenn¥x
uravnenyj. Zdes\ ξ ω1( , )t{ , … , ξ ωk t( , )} y ψ ω1( , )t{ , … , ψ ωn r t+ }( , ) — syste-
m¥ nezavysym¥x vynerovskyx processov [4] , B = B z t( , ) — funkcyq Byrkhofa,
a W = ( )Wil — tenzor Byrkhofa s komponentamy
© M. Y. TLEUBERHENOV, D. T. AÛÁMBAEV, 2010
1002 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
O POSTROENYY MNOÛESTVA STOXASTYÇESKYX DYFFERENCYAL|NÁX … 1003
W
R z t
z
R z t
z
il
i
l
l
i
=
∂
∂
−
∂
∂
( , ) ( , )
.
Po povtorqgwymsq yndeksam predpolahaetsq summyrovanye.
Budem hovoryt\, çto mnoΩestvo Λ( )t qvlqetsq yntehral\n¥m mnohoobrazy-
em stoxastyçeskoho dyfferencyal\noho uravnenyq
��x = f x x t( , , )� + σ ξ( , , )x x t� � , (5)
esly yz x0 , �x t0 0∈Λ( ) sleduet P x t t x x( , , , )0 0 0�{ ∈ Λ( )t } = 1 pry vsex t t≥ 0 .
Postavlennaq zadaça v klasse ob¥knovenn¥x dyfferencyal\n¥x uravnenyj
rassmatryvalas\ v [5]. V rabote [6] rassmotren¥ zadaçy postroenyq po zadanno-
mu stoxastyçeskomu uravnenyg Yto vtoroho porqdka πkvyvalentnoho stoxasty-
çeskoho uravnenyq lahranΩevoj, hamyl\tonovoj y byrkhofovoj struktur. V
stat\e [7] reßagtsq stoxastyçeskye zadaçy postroenyq funkcyy LahranΩa, Ha-
myl\tona y Byrkhofa po zadann¥m svojstvam dvyΩenyq (1), zavysqwym kak ot
obobwenn¥x koordynat, tak y obobwenn¥x ot skorostej.
V otlyçye ot [7] v dannoj rabote rassmatryvagtsq zadaçy postroenyq funk-
cyy LahranΩa, Hamyl\tona y Byrkhofa po zadann¥m svojstvam dvyΩenyq (1), ne
zavysqwym ot skorostej.
Dlq reßenyq postavlenn¥x zadaç na pervom πtape po zadannomu mnoΩestvu
metodom kvazyobrawenyq [3] v soçetanyy s metodom Eruhyna [1] y v sylu stoxas-
tyçeskoho dyfferencyrovanyq sloΩnoj funkcyy [4] stroytsq uravnenye Yto
(5) tak, çtob¥ mnoΩestvo Λ( )t b¥lo yntehral\n¥m mnohoobrazyem postroen-
noho uravnenyq. Y, dalee, po postroennomu uravnenyg Yto vtoroho porqdka
stroqtsq πkvyvalentn¥e emu stoxastyçeskye uravnenyq lahranΩevoj, hamyl\-
tonovoj y byrkhofovoj struktur.
1. Postroenye stoxastyçeskoho uravnenyq lahranΩevoj struktur¥ (2)
po zadann¥m svojstvam dvyΩenyq (1). Predvarytel\no po pravylu stoxasty-
çeskoho dyfferencyrovanyq Yto sostavlqem uravnenyq vozmuwennoho dvy-
Ωenyq
��λ =
∂
∂
+
λ
σξ
x
f( )� + � �x
x x
xT ∂
∂ ∂
2λ
+ 2
2∂
∂ ∂
λ
x t
+
∂
∂
2
2
λ
t
. (6)
Dalee, sleduq metodu Eruhyna [1], vvodym vektor-funkcyg A y matrycu-funk-
cyg B, kotor¥e obladagt svojstvom A x x t( , , , , )0 0 � ≡ 0, B x x t( , , , , )0 0 � ≡ 0,
takye, çto ymeet mesto ravenstvo
��λ = A x x t( , , , , )λ λ� � + B x x t( , , , , )λ λ ξ� � � . (7)
Sravnyvaq uravnenyq (6) y (7), pryxodym k sootnoßenyqm
∂
∂
λ
x
f = A – � �x
x x
xT ∂
∂ ∂
2λ
– 2
2∂
∂ ∂
λ
x t
–
∂
∂
2
2
λ
t
,
(8)
∂
∂
=
λ
σ
x
B ,
yz kotor¥x metodom kvazyobrawenyq [3, s. 12] opredelqem vektor-funkcyg f y
matrycu σ :
f = k
x
C
∂
∂
λ
+
∂
∂
−
∂
∂ ∂
−
∂
∂ ∂
−
∂
∂
+λ λ λ λ
x
A x
x x
x
x t t
T� �
2 2 2
22
, (9)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 7
1004 M. Y. TLEUBERHENOV, D. T. AÛÁMBAEV
σ j = s
x
Cj
∂
∂
λ
+
∂
∂
+λ
x
Bj , j = 1, r , (10)
hde σ j = (σ1 j , σ2 j , … , σnj
T) — j-j stolbec matryc¥ σ = ( )σνj , ν = 1, n , j =
= 1, r ; Bj = (B j1 , B j2 , … , B j
T
µ ) — j-j stolbec matryc¥ B = ( )B jµ , µ = 1, m ,
j = 1, r ; s j , k — proyzvol\n¥e skalqrn¥e velyçyn¥, a pod v¥raΩenyqmy
∂
∂
λ
x
C y
∂
∂
+λ
x
, sleduq rabote [3, s. 12], ponymagtsq sootvetstvenno
∂
∂
+λ
x
=
∂
∂
λ
x
T ∂
∂
∂
∂
−
λ λ
x x
T 1
y
∂
∂
=
∂
∂
∂
∂
∂
∂
∂λ
λ λ
λ λ
x
C
e e
x x
x
n
n
m m
1
1
1
1
1
�
�
� � �
�
∂∂
+ +
− −
x
c c
c c
n
m m n
n n n
1 1 1
1 1 1
, ,
, ,
.
�
� � �
�
Takym obrazom, yz (9), (10) sleduet, çto mnoΩestvo dyfferencyal\n¥x urav-
nenyj Yto vtoroho porqdka, ymegwee zadannug yntehral\nug kryvug (1),
ymeet vyd
��x = k
x
C
∂
∂
λ
+
∂
∂
−
∂
∂ ∂
−
∂
∂ ∂
−
∂
∂
+λ λ λ λ
x
A x
x x
x
x t t
T� �
2 2 2
22
+ σξ� .
Dalee, po pravylu stoxastyçeskoho dyfferencyrovanyq Yto raskroem v¥ra-
Ωenye
d
dt
L
x
∂
∂
�ν
=
∂
∂ ∂
2L
x t�ν
+
∂
∂ ∂
2L
x xk
|
| id | umjimathkievua-article-2932 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:33:03Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fb/4db3574af2a8d3500eaef1fbdb6c98fb.pdf |
| spelling | umjimathkievua-article-29322020-03-18T19:40:27Z On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities О построении множества стохастических дифференциальных уравнений по заданному иптеїральному многообразию, не зависящему от скоростей Azhymbaev, D. T. Tleubergenov, M. I. Ажымбаев, Д. Т. Тлеубергенов, М. И. Ажымбаев, Д. Т. Тлеубергенов, М. И. We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces. Побудовано рівняння Лагранжа, Гамільтона та Біркгофа за заданими властивостями руху при наявності випадкових збурень. При цьому припускають, що випадкові збурні сили належать класу віперових процесів, а задані властивості руху не залежать від швидкостей. Отримані результати проілюстровано па прикладі руху штучного супутника Землі під дією сил тяжіння та аеродинамічних сил. Institute of Mathematics, NAS of Ukraine 2010-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2932 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 7 (2010); 1002–1008 Український математичний журнал; Том 62 № 7 (2010); 1002–1008 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/2932/2614 https://umj.imath.kiev.ua/index.php/umj/article/view/2932/2615 Copyright (c) 2010 Azhymbaev D. T.; Tleubergenov M. I. |
| spellingShingle | Azhymbaev, D. T. Tleubergenov, M. I. Ажымбаев, Д. Т. Тлеубергенов, М. И. Ажымбаев, Д. Т. Тлеубергенов, М. И. On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title | On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title_alt | О построении множества стохастических дифференциальных уравнений по заданному иптеїральному многообразию, не зависящему от скоростей |
| title_full | On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title_fullStr | On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title_full_unstemmed | On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title_short | On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| title_sort | on the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2932 |
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