Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants
Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation is constructed and its description is given in terms of covariant derivatives...
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| Date: | 2008 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3161 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509202869387264 |
|---|---|
| author | Prykarpatsky, Ya. A. Прикарпатський, Я. А. |
| author_facet | Prykarpatsky, Ya. A. Прикарпатський, Я. А. |
| author_sort | Prykarpatsky, Ya. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:47:10Z |
| description | Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation
is constructed and its description is given in terms of covariant derivatives and the curvature form of the corresponding connection. |
| first_indexed | 2026-03-24T02:37:22Z |
| format | Article |
| fulltext |
UDK 517.9
Q. A. Prykarpats\kyj
(In-t matematyky NAN Ukra]ny, Ky]v; Ped. un-t, Krakiv, Pol\wa)
HAMIL|TONOVA HEOMETRYÇNA ZV’QZNIST|,
ASOCIJOVANA Z ADIABATYÇNO ZBURENYMY
HAMIL|TONOVYMY SYSTEMAMY,
TA ISNUVANNQ ADIABATYÇNYX INVARIANTIV
Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the
adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection
on the main foliation is constructed and its description is given in terms of covariant derivatives and the
curvature form of the corresponding connection.
Yzuçagtsq dyfferencyal\no-heometryçeskye svojstva hamyl\tonov¥x svqznostej na symplek-
tyçeskyx mnohovydax dlq adyabatyçesky vozmuwennoj hamyl\tonovoj system¥. V çastnosty,
skonstruyrovana assocyyrovannaq hamyl\tonovaq svqznost\ na hlavnom rassloenyy y pryvedeno
ee opysanye v termynax kovaryantn¥x proyzvodn¥x y form¥ kryvyzn¥ sootvetstvugwej svqz-
nosty.
1. Hamil\tonova zv’qznist\. Rozhlqnemo symplektyçnyj mnohovyd M n2( ,
ω( )2 ) , na qkomu zadano hladku hamil\tonovu dig G M× → M hrupy Li G .
ProdovΩymo pryrodnym çynom cg dig na mnohovyd P : = M A× , de A [ tak
zvanym mnohovydom adiabatyçnyx parametriv, tobto:
i) koΩne volokno M a× { }, a A∈ , [ invariantnym;
ii) diq G × M a× { }( ) → M a× { } [ symplektyçnog;
iii) isnu[ hladka sim’q vidobraΩen\ momentu l : M A× → G∗
, de G∗
—
sprqΩenyj prostir do alhebry Li G.
Oskil\ky mnohovyd P :;= M A× [ tryvial\nym rozßaruvannqm P pA →
pA → A, na n\omu moΩna vyznaçyty klasyçnu zv’qznist\ H (tak zvanu zv’qz-
nist\ Kartana – Eresmana) [1, 2] za dopomohog horyzontal\nyx rozpodiliv
Hor ( )P � T P( ) .
Oznaçennq. Hamil\tonovog zv’qznistg budemo nazyvaty zv’qznist\ Kar-
tana – Eresmana H na rozßaruvanni P ApA → , asocijovanu z vidobraΩen-
nqm momentu l : P
pa → ∗G , qka zadovol\nq[ taki umovy:
1) paralel\ne perenesennq vzdovΩ bud\-qkoho kuskovo-hladkoho ßlqxu v A
zberiha[ mnoΩyny rivniv ( ; )u A P∈{ : l u A( ; ) = ξ ∈ }∗G vidobraΩennq momentu
l : M × A → G∗
;
2) paralel\ne perenesennq [ hamil\tonovym;
3) dlq koΩnoho vektornoho polq K T A∈ ( ) userednena velyçyna hK
horyzontal\noho lifta hK P∈Hor( ) [ postijnym vektornym polem na
M a× { }, a A∈ .
Tut dlq bud\-qkoho tenzornoho polq β na P vyznaçeno userednenu vely-
çynu
β β:= ∗∫ g dg
G
� ,
G
dg∫ = 1, (1)
de dg , g G∈ , [ standartnog invariantnog mirog Xaara na hrupi Li G , pryço-
mu cg hrupu Li vvaΩa[mo kompaktnog i zv’qznog.
Horyzontal\nyj lift hK P∈Hor( ) bud\-qkoho vektornoho polq K T A∈ ( )
© Q. A. PRYKARPATS|KYJ, 2008
382 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HAMIL|TONOVA HEOMETRYÇNA ZV’QZNIST|, ASOCIJOVANA Z ADIABATYÇNO … 383
vyznaçymo za dopomohog userednennq (1) takym çynom:
h K :;= i KA,∗ =
dg i K ds u e hA
sh
g
G
g
, , ˆ∗ ∗( ) ( )
∫∫ �
0
1
+ i KA,∗ , (2)
de iA : A → P — vidobraΩennq vkladennq, hg ∈G — takyj element alhebry Li
G hrupy Li G, wo exphg = g G∈ , a ˆ; ˆu a( ): G → P — hladke vidobraΩennq,
indukovane di[g hrupy Li G na mnohovydi P zhidno z komutatyvnog diahramog:
T G T P T P
G P P
u a g
u a g
( ) ( ) ( )
.
( ˆ, ˆ)
( ˆ, ˆ)
∗ ∗ → ←
↓ ↓ ↓
→ ←
(3)
Lehko perekonatysq, wo vyraz (2) parametryçno ne zaleΩyt\ vid zobraΩennq
g = ehg
elementa hrupy g G∈ , wo vaΩlyvo dlq pobudovy vidpovidno] hamil\-
tonovo] zv’qznosti H na rozßaruvanni P. Dijsno, na osnovi (2) otrymu[mo
i KA,∗ = dg g i KA
G
∗ ∗∫ , = dg
u g
a
i K
G
A
∂
∂
�∫ ∗, + i KA,∗ =
= dg ds
a
d
ds
u e i K
G
sh
A
g∂
∂
0
1
∫∫
∗( ) ,� � + i KA,∗ =
=
dg ds
a
u e h i K
G
sh
A
g∂
∂
0
1
∫∫ ( ) ( )
∗
∗ˆ ,� + i KA,∗ =
= dg i K ds u g hA
s
g
G
, , ˆ∗ ∗( ) ( )
∫∫ �
0
1
+ i KA,∗ , (4)
de gs
:;= exp h sg( ) ∈ G dlq dovil\nyx s ∈[ ]0 1, , i zvidsy na pidstavi invari-
antnosti miry Xaara otrymu[mo (2), a otΩe, i nezaleΩnist\ vid vybrano] paramet-
ryzaci].
ZauvaΩennq 1. Z toçky zoru zastosuvan\ opysano] vywe konstrukci] ha-
mil\tonovo] zv’qznosti H do problemy opysu adiabatyçnyx invariantiv adiaba-
tyçno zburenyx hamil\tonovyx system vidpovidnyj mnohovyd parametriv A
moΩna podaty qk orbitu deqkoho hladkoho vidobraΩennq σ : R → A na svij
obraz imσ. Todi dlq neavtonomno] funkci] Hamil\tona Hσ : P → R moΩna
vyznaçyty funkcig H: M × imσ → R taku, wo H u tσ( ; ) = H u t; ( )σ( ) dlq vsix
( ; )u t ∈ M × R . Funkcig H: M × imσ → R moΩna vvaΩaty heneratorom
vektornoho polq i KM H,∗ + iA, ˙∗σ ∈ T P( ) na P, de σ̇ : = d dtσ , t ∈ R.
Oskil\ky horyzontal\ne vektorne hamil\tonove pole hK = i KA,∗ ∈ T P( )
[ rezul\tatom userednennq (1) za di[g hrupy G na P, to oçevydno, wo vidpovid-
na funkciq Hamil\tona H PK ∈D( ) bude stalog na M × a{ } dlq koΩnoho
a A∈ , qku dlq zruçnosti moΩna vvaΩaty rivnog nulevi. Prointerpretu[mo
umovu 1, qka zabezpeçu[ invariantnist\ mnohovydu rivniv vidobraΩennq momentu
l: P → G
∗ pry paralel\nomu perenesenni. Ostann[ zadamo za dopomohog kova-
riantno] poxidno]
∇
∗i KA
f
,
:;= df hK( ) = df i KA,∗( ) (5)
dlq dovil\no] funkci] f P∈D( ) . Todi umovu 1 moΩna zapysaty tak:
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
384 Q. A. PRYKARPATS|KYJ
∇
∗i KA
l
,
= 0 (6)
dlq vsix K T A∈ ( ) . Vraxovugçy umovu ekvivariantnosti vidobraΩennq momentu
l : P → G∗
, z (6) otrymu[mo
dl i KA,∗( ) = d i l KA A
∗ ( ) = 0 (7)
dlq vsix K T A∈ ( ) . Takym çynom, funkciq i lA
∗ : A → G
∗
[ stalog na mno-
hovydi parametriv A, v (7) çerez dA poznaçeno zovnißnij dyferencial vidnosno
zminnyx a A∈ .
ZauvaΩennq 2. Lehko perekonatysq, wo z umovy invariantnosti (6), qk nas-
lidok, ma[mo [dynist\ hamil\tonovo] zv’qznosti H na rozßaruvanni P ApA → .
2. Adiabatyçno zbureni hamil\tonovi systemy. Rozhlqnemo dovil\nyj
element ξ( )a ∈ ∗G , qkyj parametryçno zaleΩyt\ vid zminno] a A∈ , oskil\ky
zadano netryvial\nu hamil\tonovu dig G × M a× { }( ) → M × a{ } . Todi vektoru
Vξ ∈ T M( ) na M vidpovida[ na P vektorne pole i VM,∗ ξ ∈ T P( ) . Qkwo dlq
bud\-qkoho K T A∈ ( ) znajty dig komutatora i VM,∗[ ξ , i KA,∗ ] na dovil\nu hlad-
ku funkcig f P∈D( ) , to otryma[mo
i V i K fM A, ,,∗ ∗[ ]ξ = df i V i KM A, ,,∗ ∗[ ]( )ξ = –d d f V KA M ξ( )( )( ) , (8)
pryçomu vraxovano, wo vykonu[t\sq umova V d f KMξ
˜( )( ) = 0 dlq koΩno] funk-
ci]
˜ ( )f A∈D .
ZauvaΩennq 3. Spivvidnoßennq (7) u formi
d i lA A
∗ = 0 (9)
pokladeno v osnovu [3] analizu vaΩlyvo] problemy adiabatyçno] stabil\nosti
povil\no zminnyx u çasi povnistg intehrovnyx hamil\tonovyx system. U vypadku
odnoho stupenq vil\nosti umova (9) ta umova dim A = 1 zbihagt\sq z adiabatyç-
nym invariantom. Ce oznaça[, wo na çasovomu intervali 0 1, ε( ], de ε ↓ 0 , vely-
çyna (9) zming[t\sq na porqdok velyçyny ε > 0. Cej rezul\tat bulo vstanovle-
no klasyçnym metodom userednennq za di[g abelevo] hrupy G = S1
. Vidomo ta-
koΩ [3], wo çerez naqvnist\ bahatoçastotnyx rezonansiv na toro]dal\nyx mnoho-
vydax teorema pro userednennq, na Ωal\, ne ma[ miscq ta intehraly di] vΩe ne [
adiabatyçnymy invariantamy, nezvaΩagçy na vykonannq umovy (9).
U bud\-qkomu vypadku standartna hrupova diq h∗: T P( ) → T P( ) , h G∈ , po-
rodΩu[ bazysne horyzontal\ne vektorne pole hKa ∈ Hor ( )P , de v zminnyx
„diq-kut”
hKa ; : = i
aA,∗
∂
∂
=
∂ϕ
∂
∂
∂ϕ
i
i
n
ia=
∑
1
+ ∂
∂a
, (10)
pryçomu a — lokal\na koordynata na mnohovydi A. Qkwo σ : R → A [ deqkog
hladkog zamknenog kryvog na M, to v lokal\nyx koordynatax „diq-kut” lehko
znajty vyraz
hσ̇ = dA i
i
n
i
ϕ σ ∂
∂ϕ
( ˙ )( )
=
∑
1
+ σ̇ . (11)
Todi vidpovidni rivnqnnq paralel\noho perenesennq magt\ vyhlqd
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HAMIL|TONOVA HEOMETRYÇNA ZV’QZNIST|, ASOCIJOVANA Z ADIABATYÇNO … 385
d
dt
iϕ
= dA iϕ σ̇ , i n= 1, , (12)
rezul\tatom qkyx [ cykliçna „faza” ψσ : M → R Zn n
2π( ) , de
ψσ ;: =
σ
ϕ∫ dA . (13)
Vyraz (13) ma[ zrozumilu heometryçnu interpretacig qk elementa vidpovidno]
hrupy holonomi]
Hol H( ) � G vidpovidno pobudovano] hamil\tonovo] zv’qznosti
H na P. Pry c\omu, qk vydno z (13), hrupa holonomi]
Hol H( ) [ komutatyvnog,
izomorfnog do odnovymirno] hrupy homolohij H n
1 T Z;( ) nevyrodΩenoho tora
Tn
qk kompaktnoho intehral\noho mnohovydu cilkom intehrovno] hamil\tonovo]
systemy na symplektyçnomu mnohovydi M n2
.
Budemo vvaΩaty, wo na P vykonu[t\sq umova (9). Ce oznaça[, wo na rozßa-
ruvanni P = M × A → A moΩna vvesty zv’qznist\ Kartana – Eresmana H taku,
wo vidpovidnyj prostir horyzontal\nyx vektornyx poliv Hor( )P � T P( ) zada-
[t\sq vyrazom (2). Todi ma[ misce take tverdΩennq.
TverdΩennq. Nexaj hrupova diq G × P → P [ takog, wo vidpovidne vidob-
raΩennq momentu l : P → G∗
zadovol\nq[ umovu d i lA A,∗ = 0, pryçomu
hrupa Li G [ kompaktnog i zv’qznog. Todi na rozßaruvanni P ApA → i s -
nu[ [dyna hamil\tonova zv’qznist\ H. Krim c\oho, na P isnu[ 1-forma θ ∈
∈ Λ1( )P taka, wo dlq bud\-qkoho vektornoho polq K T A∈ ( ) horyzontal\nyj
lift hK ∈ Hor ( )P zada[t\sq vyrazom
hK = V i KAθ( ),∗
+ i KA,∗ , (14)
de iV iA Kθ
ω
( , )
( )
∗
2 = – ,d i KM Aθ ∗( ), pryçomu, zhidno z (2),
– ,d i KM Aθ ∗( );: = – ,( )
,ω 2 i KA ∗ ⋅( ) = dg ds d d H u g K
G
M A g
s
0
1
∫∫ ( )( )� =
= –d d H KM A ( )( ) , H ; : =
dg ds H u g
G
g
s
0
1
∫∫ ( )� , (15)
funkci] Hg : P → R isnugt\ zavdqky hamil\tonovosti di] hrupy Li G n a
mnohovydi M i vyznaçagt\sq z rivnostej
– ˆ ( )
( )i
u hg∗
ω 2 = dH ug( ) . (16)
Vidpovidna kovariantna poxidna ∇
∗i KA,
: D ( )P → D ( )P vzdovΩ horyzontal\-
noho vektornoho polq (2) zada[t\sq qk
∇
∗i KA
f
,
= df i KA,∗( ) + θ i K fA, ,∗( ){ } , (17)
de ⋅ ⋅{ }, [ vidpovidnog do symplektyçno] struktury ω( )2 ∈ Λ2( )M duΩkog
Puassona, a funkciq f P∈D ( ) . Kryvyna Ω ( )2 ∈ Λ2( )M zv’qznosti H zada-
[t\sq qk
Ω ( )
, ,,2
1 2i K i KA A∗ ∗( ) ; : = θ θi K i KA A, ,,∗ ∗( ) ( ){ }1 2 (18)
dlq bud\-qkyx vektornyx poliv K1 i K2 ∈ T A( ) .
Dovedennq. Neobxidno vstanovyty lyße vyrazy (15) dlq 1-formy θ ∈
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
386 Q. A. PRYKARPATS|KYJ
∈ Λ1( )P ta (18) dlq 2-formy kryvyny. Na osnovi vyznaçennq horyzontal\noho
vektornoho polq hK ∈ Hor ( )P , K T A∈ ( ) , znaxodymo
ihK ω ( )2 : = ω ( )( , )2 hK ⋅ =
ω ω( )
, ,( ˆ ) ( )
( ),
,
2
0
1
2i K dsiA i K u g h
G
A
s
g
∗ [ ]⋅( ) ⇒
∗ ∗∫∫ � =
=
ds L i L
i K u g h u g i K
G
A
i s
g
s A, ,( ˆ ) ( ) ( ˆ )
( )–
∗ ∗ ∗ ∗( )∫∫ � �
0
1
2ω =
= –
dg ds L d H u g
G
i K M g
s
A
0
1
∫∫ ∗ ( ),
( )� =
− ∫∫ ∗
d dg ds L H u gM
G
i K g
s
A
0
1
,
( )� =
=
− ( )∫∫ dgd ds d H u g KM
G
A g
s
0
1
( )( )� = – ( )d d H KM A( ) , (19)
de vraxovano poznaçennq (16) i vykorystano invariantnist\ symplektyçno]
struktury ω( )2 ∈ Λ2( )M vidnosno hamil\tonovo] di] G × M → M , tobto
g∗ω( )2 = ω( )2
dlq bud\-qkoho g G∈ . Teper na osnovi (19) otrymu[mo
d d H KM A ( )( ) = d i KM Aθ( ),∗ , (20)
de vraxovano, wo vidobraΩennq dH : T A( ) → R,
d H KA ( ) = θ( ),i KA ∗ (21)
dlq K T A∈ ( ) , [ linijnym funkcionalom na T A( ) � T A∗( ). Ostann[ na osnovi
klasyçnoho tverdΩennq Rissa [4] oznaça[, wo isnu[ (oçevydno, ne [dyna) 1-for-
ma θ ∈Λ1( )P taka, wo dlq vsix K T A∈ ( ) vykonu[t\sq (21). Takym çynom, po-
[dnugçy (19) ta (21), otrymu[mo, wo na P isnu[ take hamil\tonove vektorne
pole V i KAθ( ),∗
∈ T P( ) z funkci[g Hamil\tona θ( ),i KA ∗ ∈ D( )P , wo vykonu[t\-
sq (14). Pry dovedenni formuly (18) dlq 2-formy kryvyny zv’qznosti H neob-
xidno vraxuvaty, wo kovariantnu poxidnu (17) moΩna zapysaty v takij formi:
∇
∗i KA
f
,
= θ( ),,i K fA ∗{ } + i K fA,∗ (22)
dlq bud\-qkoho vektornoho polq K T A∈ ( ) i funkcij f P∈D( ) . Todi, vyko-
rystovugçy vyznaçennq [5 – 7] tenzora kryvyny Ω̂ zv’qznosti H na P u
vyhlqdi
ˆ ,, ,Ω i K i K fA A∗ ∗( )1 2 ; : = ∇ ∇[ ]∗ ∗i K i KA A
f
, ,
,
1 2
– ∇
∗[ ]i K KA
f
, ,1 2
, (23)
dlq dovil\nyx K1 i K2 ∈ T A( ) i f P∈D( ) znaxodymo
ˆ ,, ,Ω i K i K fA A∗ ∗( )1 2 = θ θ( ), ( ),, ,i K i K fA A∗ ∗{ }{ }1 2 +
+ θ( ),, ,i K i K fA A∗ ∗{ }1 2 + ( )( ), ,i K i K fA A∗ ∗1 2 +
+ ( ) ( ),, ,i K i K fA A∗ ∗{ }1 2θ – θ θ( ), ( ),, ,i K i K fA A∗ ∗{ }{ }2 1 –
– θ θ( ), ( ), ,i K i K fA A∗ ∗{ }2 1 – ( )( ), ,i K i K fA A∗ ∗2 1 –
– i K i K fA A, ,( ),∗ ∗{ }2 1θ – i K i K fA A, ,,∗ ∗[ ]1 2 – θ i K K fA, , ,∗ [ ]( ){ }1 2 , (24)
de vraxovano funktorial\nu vlastyvist\ [7] i K KA, ,∗ [ ]1 2 = i KA,∗[ 1 , i KA,∗ ]2 do-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
HAMIL|TONOVA HEOMETRYÇNA ZV’QZNIST|, ASOCIJOVANA Z ADIABATYÇNO … 387
tyçnoho vidobraΩennq iA,∗: T A( ) → T P( ) . Vykorystovugçy teper totoΩnist\
Qkobi dlq duΩky Puassona ⋅ ⋅{ }, i rivnist\
i K i K fA A, ,( ),∗ ∗{ }1 1θ – i K i K fA A, ,( ),∗ ∗{ }2 2θ =
= θ( ), ( ), ,i K i K fA A∗ ∗{ }1 1 – θ( ), ( ), ,i K i K fA A∗ ∗{ }2 2 + θ i K K fA, , ,∗ [ ]( ){ }1 2 , (25)
z (24) i (25) otrymu[mo
ˆ ,, ,Ω i K i K fA A∗ ∗( )1 2 = θ θ( ), ( ) ,, ,i K i K fA A∗ ∗{ }{ }1 2 = V f
i K i KA Aθ θ( ), ( ), ,∗ ∗{ }1 2
. (26)
Wob nadaty sens vyrazu (26), skorysta[mosq vidomog konstrukci[g z [7, 8].
A same, nexaj P = M × A
pA → A [ symplektyçnym rozßaruvannqm [6], struk-
turna hrupa Sp( ; )P G qkoho [ hrupog vsix takyx symplektomorfizmiv symp-
lektyçno] 2-formy ω ( )2 ∈ Λ2 M a× { }( ) dlq vsix a A∈ , qki komutugt\ z di[g
hrupy Li G: M a× { } → M a× { }, a A∈ . Oçevydno, wo cq diq vyznaça[ pry-
rodnyj homomorfizm hrupy Li G v hrupu Sp( ; )P G . Alhebra Li sp( ; )P G
hrupy Sp( ; )P G vyznaça[t\sq pryrodnym çynom qk alhebra Li C M aG × { }(
a A∈ ) � D( )P vsix G-invariantnyx funkcij na M a× { }, a A∈ , vidnosno vid-
povidno] duΩky Puassona.
Takym çynom, vyraz (26) oznaça[, wo di] tenzora kryvyny
ˆ
,Ω i KA ∗( 1,
i K fA,∗ )2 ∈ D( )P na bazysnomu mnohovydi A zv’qznosti H na P = M × A vid-
povida[ dyferencial\na 2-forma kryvyny Ω Λ( ) ( )2 2∈ P ⊗ C M aG × { }( a A∈ ) zi
znaçennqm v alhebri Li sp( ; )P G � C M aG × { }( a A∈ ), pryçomu
Ω ( )
, ,( , )2
1 2i K i KA A∗ ∗ ; : = θ( ), ( ), ,i K i KA A∗ ∗{ }1 2
dlq vsix K1 i K2 ∈ T A( ) , wo j dovodyt\ formulu (18).
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OderΩano 19.09.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 3
|
| id | umjimathkievua-article-3161 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:22Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/6f/5e0ade97cd62e9f25e6d42fc566a836f.pdf |
| spelling | umjimathkievua-article-31612020-03-18T19:47:10Z Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants Гамільтонова геометрична зв'язність, асоційована з адіабатично збуреними гамільтоновими системами, та існування адіабатичних інваріантів Prykarpatsky, Ya. A. Прикарпатський, Я. А. Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation is constructed and its description is given in terms of covariant derivatives and the curvature form of the corresponding connection. Изучаются дифференциально-геометрические свойства гамильтоновых связностей на симплектических многовидах для адиабатически возмущенной гамильтоновой системы. В частности, сконструирована ассоциированная гамильтоновая связность на главном расслоении и приведено ее описание в терминах ковариантных производных и формы кривизны соответствующей связности. Institute of Mathematics, NAS of Ukraine 2008-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3161 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 3 (2008); 382–387 Український математичний журнал; Том 60 № 3 (2008); 382–387 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3161/3070 https://umj.imath.kiev.ua/index.php/umj/article/view/3161/3071 Copyright (c) 2008 Prykarpatsky Ya. A. |
| spellingShingle | Prykarpatsky, Ya. A. Прикарпатський, Я. А. Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title | Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title_alt | Гамільтонова геометрична зв'язність, асоційована з адіабатично збуреними гамільтоновими системами, та існування адіабатичних інваріантів |
| title_full | Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title_fullStr | Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title_full_unstemmed | Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title_short | Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants |
| title_sort | hamiltonian geometric connection associated with adiabatically perturbed hamiltonian systems and the existence of adiabatic invariants |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3161 |
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