On the hill stability of motion in the three-body problem
We consider the special case of the three-body problem where the mass of one of the bodies is considerably smaller than the masses of the other two bodies and investigate the relationship between the Lagrange stability of a pair of massive bodies and the Hill stability of the system of three bodies....
Gespeichert in:
| Datum: | 2008 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2008
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3258 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509312773783552 |
|---|---|
| author | Sosnitskii, S. P. Сосницький, С. П. |
| author_facet | Sosnitskii, S. P. Сосницький, С. П. |
| author_sort | Sosnitskii, S. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:15Z |
| description | We consider the special case of the three-body problem where the mass of one of the bodies is considerably smaller than the masses of the other two bodies and investigate the relationship between the Lagrange stability of a pair of massive bodies and the Hill stability of the system of three bodies. We prove a theorem on the existence of Hill stable motions in the case considered. We draw an analogy with the restricted three-body problem. The theorem obtained allows one to conclude that there exist Hill stable motions for the elliptic restricted three-body problem. |
| first_indexed | 2026-03-24T02:39:06Z |
| format | Article |
| fulltext |
UDK 531.36; 531.011
S. P. Sosnyc\kyj (In-t matematyky NAN Ukra]ny, Ky]v)
PRO STIJKIST| RUXU ZA XILLOM U ZADAÇI TR|OX TIL
We consider a special case of the three-body problem, where the mass of one of these bodies is
considerably less than masses of other ones, and explore relations between the Lagrange stability of the
pair of massive bodies and the Hill stability of the whole system. We prove a theorem, which states the
existence of Hill stable motions in the three-body problem under consideration. Additionally, we
suggest an analogy with the restricted three-body problem. The obtained theorem implies that Hill stable
motions exist also in the case of the elliptic restricted three-body problem.
Rassmotren çastn¥j sluçaj zadaçy trex tel, kohda massa odnoho yz nyx znaçytel\no men\ße
mass¥ kaΩdoho yz dvux druhyx tel. Yssledovana svqz\ meΩdu ustojçyvoj po LahranΩu paroj
massyvn¥x tel y ustojçyvost\g po Xyllu system¥ vsex trex tel. Dokazana teorema, ustanavly-
vagwaq v rassmatryvaemom sluçae suwestvovanye ustojçyv¥x po Xyllu dvyΩenyj. Provedena
analohyq s ohranyçennoj zadaçej trex tel. Poluçennaq teorema pozvolqet sdelat\ v¥vod o su-
westvovanyy ustojçyv¥x po Xyllu dvyΩenyj v sluçae πllyptyçeskoj ohranyçennoj zadaçy
trex tel.
1. Vstup. Pidxid, zaproponovanyj Xillom u vypadku obmeΩeno] zadaçi tr\ox
til, znajßov svo[ konstruktyvne zastosuvannq i v zahal\nij zadaçi tr\ox til
(dyv., napryklad, knyhy [1, 2] i navedenu v nyx bibliohrafig). U statti, wo pro-
ponu[t\sq, rozhlqda[t\sq vuΩça zadaça: doslidyty moΩlyvosti pidxodu Xilla u
vypadku, dosyt\ nablyΩenomu do obmeΩeno] zadaçi tr\ox til, koly masa m3
tret\oho tila [ nadto malog, wob istotno vplyvaty na rux pary dvox masyvnyx
til (m1, m2). Razom z tym u podal\ßomu, na vidminu vid obmeΩeno] zadaçi, ne bu-
demo nextuvaty masog m3 i vvaΩatymemo, wo xoç masa m3 i mala, prote m3 ≠
≠ 0. V ramkax takoho pidxodu vda[t\sq oderΩaty ßyrßi umovy stijkosti za Xil-
lom rozhlqduvano] systemy v porivnqnni z modellg obmeΩeno] zadaçi tr\ox til.
Rozhlqnemo u tryvymirnomu evklidovomu prostori rux tr\ox material\nyx
toçok vidpovidno z masamy m1, m 2, m 3 pid di[g syl vza[mnoho hravitacijnoho
prytqhannq. V inercijnij systemi vidliku z poçatkom u centri mas mi , i = 1, 2,
3, vidpovidnyj lahranΩian ma[ vyhlqd
L = T + U = 1
2 1
3
2
i
i im
=
∑ ṙ + G
m m m m m m1 2
12
1 3
13
2 3
23r r r
+ +
. (1.1)
Tut ri — radiusy-vektory toçok u vybranij systemi vidliku, ri j = rj – ri , i, j =
= 1, 2, 3, G > 0 — hravitacijna stala. Rivnqnnq ruxu systemy na pidstavi vyrazu
dlq lahranΩiana L nabyragt\ vyhlqdu
˙̇r1 = G m m2
2 1
12
3 3
3 1
13
3
r r
r
r r
r
– –+
,
˙̇r2 = G m m–
– –
1
2 1
12
3 3
3 2
23
3
r r
r
r r
r
+
, (1.2)
˙̇r3 = G m m–
–
–
–
1
3 1
13
3 2
3 2
23
3
r r
r
r r
r
.
Perexodqçy u systemi (1.2) do bezrozmirnoho çasu
GM
r
t
0
3 2/ = τ, M =
i
im
=
∑
1
3
,
© S. P. SOSNYC|KYJ, 2008
1434 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
PRO STIJKIST| RUXU ZA XILLOM U ZADAÇI TR|OX TIL 1435
de r0 — parametr, wo ma[ rozmirnist\ odynyci dovΩyny, otrymu[mo
′′r1 = r0
3
2
2 1
12
3 3
3 1
13
3µ µr r
r
r r
r
– –+
,
′′r2 = r0
3
1
2 1
12
3 3
3 2
23
3–
– –µ µr r
r
r r
r
+
, (1.3)
′′r3 = r0
3
1
3 1
13
3 2
3 2
23
3–
–
–
–µ µr r
r
r r
r
.
Tut ßtryx oznaça[ dyferencigvannq za çasom τ, µi = m Mi / .
Qkwo v (1.3) perejty do vidnosnyx dovΩyn vektoriv
ρρi =
ri
r0
, (1.4)
to rivnqnnq ruxu (1.3) moΩna zapysaty u vyhlqdi
′′ρρ1 = µ2
12
12
3
ρρ
ρρ
+ µ3
13
13
3
ρρ
ρρ
,
′′ρρ2 = –µ1
12
12
3
ρρ
ρρ
+ µ3
23
23
3
ρρ
ρρ
, (1.5)
′′ρρ3 = –µ1
13
13
3
ρρ
ρρ
– µ2
23
23
3
ρρ
ρρ
.
Qk vidomo [3], dlq systemy (1.5) isnugt\ intehral enerhi]
T – U = 1
2 1
3
2
i
i i
=
∑ ′µ ρρ –
µ µ µ µ µ µ1 2
12
1 3
13
2 3
23ρρ ρρ ρρ
+ +
= h (1.6)
ta intehral momentu kil\kostej ruxu
M =
i
i i i
=
∑ ( × ′)
1
3
µ ρρ ρρ = C. (1.7)
Ostannij moΩna podaty u vyhlqdi
Mx = C1 = const, My = C2 = const, Mz = C3 = const. (1.8)
Velyçyny Mx , My , Mz v (1.8) oznaçagt\ proekci] momentu kil\kostej ruxu
M vidpovidno na osi Ox , Oy , Oz inercijno] systemy vidliku Oxyz .
Zrozumilo, wo vyraz
1
2 1
3
2
i
i i
=
∑ ′µ ρρ –
µ µ µ µ µ µ1 2
12
1 3
13
2 3
23ρρ ρρ ρρ
+ +
– Mz = h – C3 (1.9)
takoΩ [ intehralom systemy (1.5).
Oskil\ky dlq dano] systemy isnugt\ intehraly ruxu centra mas, to dali, vid-
povidno do vyboru systemy vidliku, bez obmeΩennq zahal\nosti rozhlqdu vvaΩa-
tymemo, wo
i
i i
=
∑ ′
1
3
µ ρρ = 0,
i
i i
=
∑
1
3
µ ρρ = 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1436 S. P. SOSNYC|KYJ
i, qk naslidok,
i
i i
=
∑
1
3
2µ ρρ =
i j
i j i j
<
∑µ µ ρρ 2
. (1.10)
2. Teorema pro stijkist\ za Xillom.
Oznaçennq 1. Fiksovanu paru til (material\nyx toçok) ( , )µ µi j , i < j,
systemy (1.5) nazvemo stijkog za LahranΩem, qkwo vidpovidna ]j vidstan\
ρρi j t( ) zadovol\nq[ nerivnist\
c
1
≤ ρρi j t( ) ≤ c
2
∀ ∈t R = – ,∞ ∞] [ , (2.1)
de c
1
i c
2
— dodatni stali.
Oznaçennq 2. Fiksovanyj rux ρρ( )t = ρρ1( )t( , ρρ2( )t , ρρ3( )t T) systemy (1.5)
nazvemo stijkym za Xillom, qkwo vykonu[t\sq umova
ρρi j t( ) ≤ c
3
∀ ∈t R ∀ <i j , (2.2)
de c
3
— dodatna stala.
ZauvaΩennq 1. Oskil\ky, zhidno z (1.4), velyçyny ρρi [ vidnosnymy dovΩy-
namy, to bez porußennq zahal\nosti rozhlqdu moΩemo vvaΩaty, wo v (2.1) c
2
=
= 1. Zokrema, qkwo rux pary ( , )µ µi j , i < j, [ ruxom po kolu, to ρρi j = 1, qkwo
eliptyçnym z malym ekscentrysytetom, to ρρi j kolyva[t\sq z malog amplitu-
dog v okoli userednenoho znaçennq ρρi j
0
, wo malo vidriznq[t\sq vid odynyci.
Teorema. Nexaj u systemi (1.5) vykonugt\sq umovy:
1) T – U = h < 0;
2) µ
1
≥ µ
2
j isnu[ take male çyslo ε > 0, wo pry µ µ3 2/ < ε tilo z masog
µ
3
istotno ne vplyva[ na qkisnyj xarakter ruxu pary til ( , )µ µ1 2 .
Todi qkwo para til ( , )µ µ1 2 [ stijkog za LahranΩem, to oblast\ stijkyx
za Xillom ruxiv systemy (1.5) ne [ poroΩn\og.
Dovedennq. ZauvaΩugçy, wo ρρi = (xi , yi , zi
T) i, otΩe,
Mz =
i
i i i i ix y y x
=
∑ ′ ′( )
1
3
µ – , (2.3)
zobraΩu[mo intehral (1.9) u formi
i
i i i i i i ix y y x
=
∑ ′ ′ ′( ) +[ ]
1
3
2 22µ ρρ ρρ– – –
i
i i
=
∑
1
3
2µ ρρ – 2
i j
i j
i j<
∑
µ µ
ρρ
= 2 3( – )h C . (2.4)
Perepyßemo teper (2.4) u vyhlqdi rivnosti
i
i i i i i i ix y y x
=
∑ ′ ′ ′( ) +[ ]
1
3
2 22µ ρρ ρρ– – =
i
i i
=
∑
1
3
2µ ρρ + 2
i j
i j
i j<
∑
µ µ
ρρ
+ 2 3( – )h C . (2.5)
Beruçy do uvahy rivnist\ (1.10), a takoΩ vraxovugçy nevid’[mnist\ livo] ças-
tyny rivnosti (2.5), otrymu[mo
i j
i j i j
i j<
∑ +
µ µ ρρ
ρρ
2 2
≥ 2 3( – )C h . (2.6)
Za umovy stijkosti za LahranΩem pary til ( , )µ µ1 2 nerivnist\ (2.6) zruçno zapy-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, #10
PRO STIJKIST| RUXU ZA XILLOM U ZADAÇI TR|OX TIL 1437
saty u vyhlqdi
µ1 13
2ρρ + µ2 23
2ρρ +
2 1
13
µ
ρρ
+
2 2
23
µ
ρρ
≥
1
2
2
3
3 1 2 12
2
12µ
µ µ( – ) –C h ρρ
ρρ
+
. (2.7)
ZauvaΩugçy, wo (– h) > 0, vybyra[mo stalu C3 takoΩ dodatnog i, zokrema, ta-
kog, wo stala ( – )C h3 [ dostatn\o velykog, wob zabezpeçyty dodatnist\ vyra-
zu v kvadratnyx duΩkax u pravij çastyni nerivnosti (2.7).
Oskil\ky zhidno z vyborom h i C3 vykonu[t\sq nerivnist\
2
2
3 1 2 12
2
12
( – ) –C h µ µ ρρ
ρρ
+
≥ h∗
, 0 < h∗ = const, (2.8)
to (2.7) zruçno perepysaty u vyhlqdi
µ1 13
2ρρ + µ2 23
2ρρ +
2 1
13
µ
ρρ
+
2 2
23
µ
ρρ
≥ 1
3µ
h∗
. (2.9)
Vraxovugçy teper malyznu parametra µ3, zavΩdy moΩna dosqhty toho, wob
prava çastyna nerivnosti (2.9) perevywuvala bud\-qke napered zadane dodatne
çyslo γ:
1
3µ
h∗ ≥ γ > 0, γ = const. (2.10)
Pry dostatn\o velykomu γ, v zaleΩnosti vid vidstani material\no] toçky z
masog µ3 vid material\nyx toçok z masamy µ1 i µ2 vidpovidno, na pidstavi
(2.9) vyznaça[mo oblasti moΩlyvyx ruxiv toçky z masog µ3. Zokrema, qkwo vid-
stani ρ13, ρ23 dostatn\o velyki, to oblast\ moΩlyvyx ruxiv malo] çastky
moΩna zobrazyty vyhlqdi
µ1 13
2ρρ + µ2 23
2ρρ ≥ h1
∗ > 0, h1
∗ = const. (2.11)
Oskil\ky velykij stalij γ moΩut\ vidpovidaty i mali vidstani ρ13, ρ23 , to v
zaleΩnosti vid toho, v okoli qko] z toçok pary ( , )µ µ1 2 znaxodyt\sq mala çast-
ka, oblast\ ]] moΩlyvyx ruxiv vyznaça[mo z dopomohog nerivnostej
2 1
13
µ
ρ
≥ h2
∗
, (2.12)
2 2
23
µ
ρ
≥ h3
∗
, (2.13)
de h2
∗
i h3
∗
— dostatn\o velyki dodatni stali.
Vidpovidno do nerivnostej (2.12), (2.13) oblast\ obmeΩenyx po koordynatax
ruxiv toçky z masog µ3 ne [ poroΩn\og. Dlq bil\ß toçnoho vyznaçennq ob-
lasti obmeΩenyx ruxiv, osoblyvo koly vidstani ρ13 i ρ23 [ malymy, ale sumir-
nymy, cilkom pryrodno skorystatysq bil\ß strohog nerivnistg (2.9).
Beruçy do uvahy oznaçennq 2, robymo vysnovok pro spravedlyvist\ teoremy.
ZauvaΩennq 2. Qk baçymo, otrymana teorema zalyßa[ v syli klgçovi polo-
Ωennq klasyçno] modeli obmeΩeno] zadaçi tr\ox til, za vynqtkom toho momentu,
wo masa µ3 3( )m xoç i mala, ale ne dorivng[ nulg. V svog çerhu, vidminnist\
vid nulq masy µ3 3( )m dozvolq[ skorystatysq vsima intehralamy systemy (1.5).
Nextugçy Ω v obmeΩenij zadaçi masog µ3 3( )m , my istotno zming[mo vyxidnu
matematyçnu model\ i, mabut\, ne zavΩdy adekvatno vidobraΩa[mo sutnist\ vy-
xidno] zadaçi.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1438 S. P. SOSNYC|KYJ
ZauvaΩennq 3. Rivnqnnq
µ1 13
2ρρ + µ2 23
2ρρ +
2 1
13
µ
ρρ
+
2 2
23
µ
ρρ
= 1
3µ
h∗
u rozhlqduvanomu vypadku vyznaça[ meΩu oblasti moΩlyvyx ruxiv (rivnqnnq po-
verxon\ Xilla) lyße nablyΩeno. Ce poqsng[t\sq tym, wo u pravu çastynu vy-
xidno] nerivnosti (2.7) vxodyt\ zminna vidstan\ ρρ12 .
ZauvaΩennq 4. Beruçu do uvahy rivnist\ [4]
ρ3
2 = –µ µ ρ1 2 12
2 + µ µ µ ρ1 1 2 13
2( )+ + µ µ µ ρ2 1 2 23
2( )+ , (2.14)
moΩna zobrazyty (2.7) u vyhlqdi
ρρ3
2 +
2
2 2
1
1 2
1
13
2
23
µ
µ µ µ µ
( )+ +
ρρ ρρ
≥
≥
µ µ
µ
µ µ1 2
3
3 1 2 12
2
12
2
2+ +
( – ) –C h ρρ
ρρ
– µ µ1 2 12
2ρρ . (2.15)
Z uraxuvannqm malyzny parametra µ3 nerivnist\ (2.15) moΩna zapysaty u vyh-
lqdi
ρρ3
2 + ( )µ µ µ µ
1 2
1
13
2
23
2 2+ +
ρρ ρρ
≥ γ∗ > 0, γ∗ = const,
wo duΩe nahadu[ nerivnist\ dlq oblasti moΩlyvyx ruxiv u klasyçnij modeli ob-
meΩeno] zadaçi tr\ox til.
3. Pro obmeΩenu zadaçu tr\ox til. U vypadku obmeΩeno] zadaçi tr\ox til,
qk vidomo [3, 5], zamist\ (1.5) ma[mo rivnqnnq
′′ρρ1 = µ
ρρ
ρρ
12
12
3 ,
′′ρρ2 = –( – )1 12
12
3µ
ρρ
ρρ
, (3.1)
′′ρρ3 = –( – ) –1 13
13
3
23
23
3µ µ
ρρ
ρρ
ρρ
ρρ
,
de
µ =
m
m m
2
1 2+
, 0 < µ ≤ 1 / 2. (3.2)
Qk baçymo, perßi dva vektorni rivnqnnq systemy (3.1), utvorggçy zamknenu
systemu, stanovlqt\ sobog zadaçu dvox til, i tomu, pidbyragçy naleΩnym çynom
poçatkovi umovy, lehko sformuvaty stijku za LahranΩem paru toçok ( – )1 µ( ,
µ) . Takym çynom, malyzna masy m3 i, qk naslidok, neznaçnyj vplyv malo]
çastky z danog masog na rux dvox masyvnyx til vidtvorggt\sq v rozhlqduvanij
modeli z dopomohog rivnosti m3 = 0, xoça faktyçno masa m3 zavΩdy vidminna
vid nulq.
V podal\ßomu, vraxovugçy zauvaΩennq 1, vvaΩatymemo, wo ρρ12 ≤ 1.
Odni[g z perevah modeli obmeΩeno] zadaçi tr\ox til [ zvedennq ]] do systemy
menßo] rozmirnosti, qka do toho Ω ma[ perßyj intehral (intehral Qkobi) u vy-
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, #10
PRO STIJKIST| RUXU ZA XILLOM U ZADAÇI TR|OX TIL 1439
padku kruhovoho ruxu pary ( – )1 µ( , µ) :
1
2 3
2′ρρ – x y yx′ ′( )– –
1
13 23
– µ
ρ
µ
ρ
+
= h̃ = const. (3.3)
Na pidstavi (3.3) moΩemo otrymaty vΩe vidomyj rezul\tat pro isnuvannq
oblastej Xilla, wo mistqt\ obmeΩeni po koordynatax ruxy systemy (3.1). Dlq
c\oho dostatn\o zobrazyty rivnist\ (3.3) u vyhlqdi
′ρρ3
2 – 2 xy yx′ ′( )– + ρρ3
2 = 2 h̃ + ρρ3
2 + 2
1
13 23
– µ
ρ
µ
ρ
+
(3.4)
i pry umovi, wo h̃ < 0, dotrymuvatys\ standartno] sxemy Xilla [5]. Perexid do
rivnqn\ u systemi koordynat, wo oberta[t\sq, qk ce zazvyçaj robyt\sq pry vy-
znaçenni oblastej Xilla, ne [ obov’qzkovym.
Dijsno, na pidstavi (3.4) dlq oblastej moΩlyvyx ruxiv oderΩu[mo nerivnist\
ρρ3
2 + 2
1
13 23
– µ
ρ
µ
ρ
+
≥ – ˜2h . (3.5)
Zhidno z (3.5) moΩemo zrobyty vysnovok, wo koly vidstani ρ13, ρ23 i stala
(– ˜)h [ dostatn\o velykymy, to oblast\ moΩlyvyx ruxiv moΩna zobrazyty u vyh-
lqdi
ρ3
2 ≥ h1
∗
. (3.6)
Qkwo Ω stala (– ˜)h dostatn\o velyka, a vidstani ρ13 i ρ23 [ malymy, to, zau-
vaΩugçy, wo
ρ3
2 = –( – )1 µ µ + ( – )1 13
2µ ρ + µρ23
2 , (3.7)
v zaleΩnosti vid toho, v okoli qko] z toçok pary ( , )µ µ1 2 znaxodyt\sq mala
çastka, oblast\ ]] moΩlyvyx ruxiv vyznaça[t\sq z dopomohog nerivnostej
ρ13 ≤
2 1
2
( – )µ
h∗ , (3.8)
ρ23 ≤
2
3
µ
h∗ . (3.9)
Tut dodatni stali h2
∗
i h3
∗
[ dostatn\o velykymy.
Nareßti, qkwo vidstani ρ13 i ρ23 [ malymy, ale sumirnymy, dlq vyznaçennq
oblasti moΩlyvyx ruxiv vykorystovu[mo toçnu nerivnist\
( – )1 13
2µ ρ + µρ23
2 + 2
1
13 23
– µ
ρ
µ
ρ
+
≥ µ µ( – )1 – 2h̃ , (3.10)
qka duΩe sxoΩa na nerivnist\ (2.9) u ramkax neobmeΩeno] zadaçi.
Teper baçymo, wo prynajmni nerivnosti (3.8), (3.9) dozvolqgt\ zrobyty vys-
novok pro isnuvannq obmeΩenyx po koordynatax ruxiv. Pry c\omu nam vdalosq
unyknuty perexodu do systemy koordynat, wo oberta[t\sq.
Odnak rozhlqduvana sxema peresta[ pracgvaty u vypadku eliptyçnoho ruxu
pary ( – )1 µ( , µ) , koly intehral Qkobi ne isnu[. V cij sytuaci] istotno korys-
nog moΩe vyqvytysq qkraz otrymana vywe teorema, qka, qk pro ce vΩe zhaduva-
losq vywe, zalyßa[ v syli klgçovi poloΩennq klasyçno] modeli obmeΩeno] za-
daçi tr\ox til. Dijsno, zamist\ nerivnosti (3.5) v c\omu vypadku vykorystovu[mo
nerivnist\ (2.7), pidstavlqgçy u ]] pravu çastynu zamist\ ρ12
2
i 2 12/ρ velyçyny,
qki vidpovidagt\ eliptyçnomu ruxovi v zadaçi dvox til, wo cilkom vypravdano.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 10
1440 S. P. SOSNYC|KYJ
AdΩe vidpovidno do vyxidno] postanovky zadaçi material\na toçka z masog m3
slabko vplyva[ na rux dvox masyvnyx til. OtΩe, ostanni, utvorggçy stijku za
LahranΩem paru, zdijsnggt\ rux, blyz\kyj do toho, qkyj my otrymu[mo na
pidstavi zadaçi dvox til. Zrozumilo, wo v ramkax takoho pidxodu naqvna deqka
poxybka, oskil\ky para ( , )µ µ1 2 u systemi (1.5) v umovax teoremy toçno ne vid-
tvorg[ eliptyçnyj rux, qkyj vidpovida[ zadaçi dvox til. Prote cq poxybka, vra-
xovugçy, wo my operu[mo nerivnostqmy, ne zavaΩa[ nam zrobyty vysnovky qkis-
noho xarakteru pro rux malo] çastky.
OtΩe, u vypadku obmeΩeno] eliptyçno] zadaçi tr\ox til, perexodqçy do pov-
nyx rivnqn\ (1.5), na pidstavi navedeno] vywe teoremy pryxodymo do vysnovku,
wo mnoΩyna stijkyx za Xillom ruxiv ne [ poroΩn\og.
Na zaverßennq zauvaΩymo, wo pytannq pro docil\nist\ zastosuvannq povnyx
rivnqn\ (1.5), koly rozhlqda[t\sq obmeΩena zadaça, vΩe rozhlqdalos\ avtorom u
statti [6]. Odnak tut, na vidminu vid [6], navodyt\sq menß Ωorstke oznaçennq
stijko] za LahranΩem pary, mabut\, bil\ß nablyΩene do real\no] kartyny ruxu
v ramkax zahal\no] zadaçi tr\ox til.
1. Neustojçyvosty v dynamyçeskyx systemax. PryloΩenyq k nebesnoj mexanyke / Pod red.
V.ODΩ. Sebexeq. – M.: Myr, 1982. – 168 s.
2. Holubev V. H., Hrebennykov E. A. Problema trex tel v nebesnoj mexanyke. – M.: Yzd-vo
Mosk. un-ta, 1985. – 240 s.
3. Roj A. E. DvyΩenye po orbytam. – M.: Myr, 1981. – 544 s.
4. Sosnyc\kyj S. P. Pro stijkist\ ruxu za LahranΩem u zadaçi tr\ox til // Ukr. mat. Ωurn. –
2005. – 57, # 8. – S. 1137 – 1143.
5. Sebexej V. DΩ. Teoryq orbyt. Ohranyçennaq zadaça trex tel. – M.: Nauka, 1982. – 656 s.
6. Sosnitskii S. P. On the Lagrange and Hill stability of the motion of certain systems with Newtonian
potential // Astron. J. – 1999. – 117, # 6. – P. 3054 – 3058.
OderΩano 12.06.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, #10
|
| id | umjimathkievua-article-3258 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:39:06Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/be/17e8e2b6283ba13de98fb5e19048bebe.pdf |
| spelling | umjimathkievua-article-32582020-03-18T19:49:15Z On the hill stability of motion in the three-body problem Про стійкість руху за Хіллом у задачі трьох тіл Sosnitskii, S. P. Сосницький, С. П. We consider the special case of the three-body problem where the mass of one of the bodies is considerably smaller than the masses of the other two bodies and investigate the relationship between the Lagrange stability of a pair of massive bodies and the Hill stability of the system of three bodies. We prove a theorem on the existence of Hill stable motions in the case considered. We draw an analogy with the restricted three-body problem. The theorem obtained allows one to conclude that there exist Hill stable motions for the elliptic restricted three-body problem. Рассмотрен частный случай задачи трех тел, когда масса одного из них значительно меньше массы каждого из двух других тел. Исследована связь между устойчивой по Лагранжу парой массивных тел и устойчивостью по Хиллу системы всех трех тел. Доказана теорема, устанавливающая в рассматриваемом случае существование устойчивых по Хиллу движений. Проведена аналогия с ограниченной задачей трех тел. Полученная теорема позволяет сделать вывод о существовании устойчивых по Хиллу движений в случае эллиптической ограниченной задачи трех тел. Institute of Mathematics, NAS of Ukraine 2008-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3258 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 10 (2008); 1434–1440 Український математичний журнал; Том 60 № 10 (2008); 1434–1440 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3258/3262 https://umj.imath.kiev.ua/index.php/umj/article/view/3258/3263 Copyright (c) 2008 Sosnitskii S. P. |
| spellingShingle | Sosnitskii, S. P. Сосницький, С. П. On the hill stability of motion in the three-body problem |
| title | On the hill stability of motion in the three-body problem |
| title_alt | Про стійкість руху за Хіллом у задачі трьох тіл |
| title_full | On the hill stability of motion in the three-body problem |
| title_fullStr | On the hill stability of motion in the three-body problem |
| title_full_unstemmed | On the hill stability of motion in the three-body problem |
| title_short | On the hill stability of motion in the three-body problem |
| title_sort | on the hill stability of motion in the three-body problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3258 |
| work_keys_str_mv | AT sosnitskiisp onthehillstabilityofmotioninthethreebodyproblem AT sosnicʹkijsp onthehillstabilityofmotioninthethreebodyproblem AT sosnitskiisp prostíjkístʹruhuzahíllomuzadačítrʹohtíl AT sosnicʹkijsp prostíjkístʹruhuzahíllomuzadačítrʹohtíl |