On the Lagrange Stability of Motion in the Three-Body Problem
For the three-body problem, we study the relationship between the Hill stability of a fixed pair of mass points and the Lagrange stability of a system of three mass points. We prove the corresponding theorem establishing sufficient conditions for the Lagrange stability and consider a corollary of th...
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| Datum: | 2005 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509798873694208 |
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| 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 | 2020-03-18T20:01:36Z |
| description | For the three-body problem, we study the relationship between the Hill stability of a fixed pair of mass points and the Lagrange stability of a system of three mass points. We prove the corresponding theorem establishing sufficient conditions for the Lagrange stability and consider a corollary of the theorem obtained concerning a restricted three-body problem. Relations that connect separately the squared mutual distances between mass points and the squared distances between mass points and the barycenter of the system are established. These relations can be applied to both unrestricted and restricted three-body problems. |
| first_indexed | 2026-03-24T02:46:50Z |
| format | Article |
| fulltext |
UDK 531.36; 531.011
S.�P.�Sosnyc\kyj (In-t matematyky NAN Ukra]ny, Ky]v)
PRO STIJKIST| RUXU ZA LAHRANÛEM
U ZADAÇI TR|OX TIL
For the three-body problem, we study the relation between the Hill stability of a fixed pair of mass points
and the Lagrange stability of the system of all three mass points. We prove a theorem establishing
sufficient conditions of the Lagrange stability. We consider a corollary of the theorem obtained
concerning the restricted three-body problem. The relations are established which connect separately
squared mutual distances between mass points and squared distances of mass points to the barycenter of
the system. These relations may work both for the unrestricted and restricted three-body problems.
U zadaçi tr\ox til rozhlqda[t\sq zv’qzok miΩ stijkistg za Xillom fiksovano] pary material\-
nyx toçok i stijkistg za LahranΩem systemy vsix tr\ox material\nyx toçok. Dovodyt\sq vidpo-
vidna teorema, wo vstanovlg[ dostatni umovy stijkosti za LahranΩem. Rozhlqda[t\sq naslidok
otrymano] teoremy stosovno obmeΩeno] zadaçi tr\ox til. Vstanovlggt\sq spivvidnoßennq, qki
zv’qzugt\ narizno kvadraty vza[mnyx vidstanej miΩ material\nymy toçkamy i kvadraty vidstanej
material\nyx toçok do barycentra systemy. Ci spivvidnoßennq moΩut\ vyqvytysq korysnymy qk
v neobmeΩenij, tak i v obmeΩenij zadaçax tr\ox til.
1. Vstup. Qk vidomo [1 – 3], zadaça tr\ox til (material\nyx toçok) polqha[ v
tomu, wo try material\ni toçky vidpovidno z masamy m1 , m2 , m3 zdijsnggt\ rux
u tryvymirnomu evklidovomu prostori pid di[g syl vza[mnoho hravitacijnoho
prytqhannq. Potribno vyznaçyty ]xni koordynaty i ßvydkosti v bud\-qkyj
moment çasu t , vyxodqçy z poçatkovyx danyx. U takij formi zadaça zalyßa-
[t\sq nerozv’qzanog i donyni, v zv’qzku z çym aktual\nym [ qkisne doslidΩennq
ruxu systemy, qke dozvolq[ prynajmni otrymaty v rqdi vypadkiv uqvu pro xarak-
ter tra[ktorij material\nyx toçok. Zokrema, stanovyt\ interes rozhlqd zv’qzku
miΩ stijkistg za Xillom fiksovano] pary toçok [4] i stijkistg za LahranΩem
vsi[] systemy.
Rozhlqnemo inercijnu systemu vidliku, poçatok qko] vyberemo v centri mas
m1 , m2 , m3 . Nexaj ri , i = 1, 2, 3, — radiusy-vektory material\nyx toçok vidpo-
vidno z masamy m1 , m2 , m3 v danij systemi vidliku. Todi lahranΩian rozhlqdu-
vano] systemy toçok, wo prytqhugt\sq, nabyra[ vyhlqdu
L = T + U = 1
2
2
3
1 2 1 3 2 3m G
m m m m m m
i
i
ṙ
r r ri
12 13 23
∑ + + +
| | | | | |
,
(1)
rij = rj – ri , i , j = 1, 2, 3,
de G > 0 — hravitacijna stala.
Rivnqnnq ruxu systemy,>qki vidpovidagt\ lahranΩianu>(1),>zapyßemo u vyhlqdi
˙̇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
, (2)
˙̇r3 = G m m− − − −
| | | |1
3 1
13
3 2
3 2
23
3
r r
r
r r
r
.
Na pidstavi struktury lahranΩiana (1) rozhlqduvano] systemy ma[mo
E = T – U = h = const . (3)
Oskil\ky dlq dano] systemy isnugt\ intehraly ruxu centra mas
© S.>P.>SOSNYC|KYJ, 2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8 1137
1138 S.>P.>SOSNYC|KYJ
mi i
i
ṙ
=
∑
1
3
= c , mi i
i
r
=
∑
1
3
= c t + b , (4)
de c i b — stali vektory, to u vidpovidnosti z vyborom systemy vidliku, bez ob-
meΩennq zahal\nosti rozhlqdu, dali vvaΩatymemo, wo
mi i
i
ṙ
=
∑
1
3
= 0, mi i
i
r
=
∑
1
3
= 0, (5)
i, qk naslidok [3, 5],
mi i
i
r2
1
3
=
∑ = M m mi j ij
i j
−
<
| |∑1 2r , M = mi
i=
∑
1
3
. (6)
Oznaçennq 1. Rux r ( t ) = ( r1 , r2 , r3 )T systemy (2) nazvemo stijkym za La-
hranΩem, qkwo vykonu[t\sq umova
c1 ≤ | ri j ( t ) | ≤ c2 ∀t ∈ R = ] – ∞ , ∞ [ , ∀i < j ,
de c1 , c2 — dodatni stali.
Oznaçennq 2. Rux r ( t ) = ( r1 , r2 , r3 )T systemy (2) nazvemo dystal\nym,
qkwo vykonu[t\sq nerivnist\
| ri j ( t ) | > c3 ∀t ∈ R , ∀i < j , 0 < c3 = const .
Oznaçennq 3. Fiksovanu paru toçok ( mi , mj ) , i < j , systemy (2) zhidno z
[4] nazvemo stijkog za Xillom, qkwo vykonu[t\sq nerivnist\
| ri j ( t ) | < c4 ∀t ∈ R , 0 < c4 = const ,
pry fiksovanyx i ta j .
2. Pro zv’qzok dovΩyn radiusiv-vektoriv toçok r i z vza[mnymy
vidstanqmy miΩ toçkamy | ri j | . Zobrazymo rivnist\
mi i
i
r
=
∑
1
3
= 0
u vyhlqdi tr\ox ekvivalentnyx rivnostej
M r1 = – m2 r12 – m3 r13 ,
M r2 = m1 r12 – m3 r23 , (7)
M r3 = m1 r13 + m2 r23 .
Na pidstavi (7) ma[mo
M2
1
2r = m m2
2
12
2
3
2
13
2r r+ + 2m2 m3 r12 r13 ,
M2
2
2r = m m1
2
12
2
3
2
23
2r r+ – 2m1 m3 r12 r23 , (8)
M2
3
2r = m m1
2
13
2
2
2
23
2r r+ + 2m1 m2 r13 r23 ,
a oskil\ky ma[ misce totoΩnist\
r12 + r23 – r13 = 0, (9)
to spravdΩugt\sq rivnosti
r12 r13 = 1
2
r r r12 13 23
2 2 2+ −( ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
PRO STIJKIST| RUXU ZA LAHRANÛEM U ZADAÇI TR|OX TIL 1139
r12 r23 = 1
2
− + −( )r r r12 13 23
2 2 2 , (10)
r13 r23 = 1
2
− + +( )r r r12 13 23
2 2 2 .
Dali poznaçymo | ri j | = ri j , | ri | = ri . Vraxovugçy (10), zhidno z (8)
oderΩu[mo rivnosti
r
M
m m m r m m m r m m r1
2
2 2 2 3 12
2
3 2 3 13
2
2 3 23
21= + + + −[ ]( ) ( ) ,
r
M
m m m r m m r m m m r2
2
2 1 1 3 12
2
1 3 13
2
3 1 3 23
21= + − + +[ ]( ) ( ) , (11)
r
M
m m r m m m r m m m r3
2
2 1 2 12
2
1 1 2 13
2
2 1 2 23
21= − + + + +[ ]( ) ( ) ,
na pidstavi qkyx ma[mo
r
m m m r m m m r m r
m m12
2 1 1 2 1
2
2 1 2 2
2
3
2
3
2
1 2
= + + + −( ) ( )
,
r
m m m r m r m m m r
m m13
2 1 1 3 1
2
2
2
2
2
3 1 3 3
2
1 3
= + − + +( ) ( )
, (12)
r
m r m m m r m m m r
m m23
2 1
2
1
2
2 2 3 2
2
3 2 3 3
2
2 3
= − + + + +( ) ( )
.
Poznaçagçy ṙij = vi j , ṙi = vi , oderΩu[mo spivvidnoßennq, wo zv’qzugt\
vij
2 i vi
2 , analohiçni spivvidnoßennqm (11) i (12):
v v v v1
2
2 2 2 3 12
2
3 2 3 13
2
2 3 23
21= + + + −[ ]
M
m m m m m m m m( ) ( ) ,
v v v v2
2
2 1 1 3 12
2
1 3 13
2
3 1 3 23
21= + − + +[ ]
M
m m m m m m m m( ) ( ) , (13)
v v v v3
2
2 1 2 12
2
1 1 2 13
2
2 1 2 23
21= − + + + +[ ]
M
m m m m m m m m( ) ( ) ,
v
v v v
12
2 1 1 2 1
2
2 1 2 2
2
3
2
3
2
1 2
= + + + −m m m m m m m
m m
( ) ( )
,
v
v v v
13
2 1 1 3 1
2
2
2
2
2
3 1 3 3
2
1 3
= + − + +m m m m m m m
m m
( ) ( )
, (14)
v
v v v
23
2 1
2
1
2
2 2 3 2
2
3 2 3 3
2
2 3
= − + + + +m m m m m m m
m m
( ) ( )
.
Rivnosti (11), (12), wo zv’qzugt\ kvadraty vidstanej material\nyx toçok do
barycentra systemy i kvadraty vza[mnyx vidstanej, a takoΩ ]x analohy (13), (14),
qki stosugt\sq ßvydkostej, ne>dyvlqçys\ na prostyj sposib ]x oderΩannq,
raniße buly nevidomymy i, mabut\, stanovlqt\ samostijnyj interes.
3. Pro rivnqnnq vidstanej u zadaçi tr\ox til. Rivnqnnq ruxu (2) zapyße-
mo u vyhlqdi [6]
˙̇r
r
r
F12
12
12
+ =
| |
GM Gm3 3 ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1140 S.>P.>SOSNYC|KYJ
˙̇r
r
r
F13
13
13
+ = −
| |
GM Gm3 2 , (15)
˙̇r
r
r
F23
23
23
+ =
| |
GM Gm3 1 ,
de
F =
r
r
r
r
r
r
12
12
13
13
23
23| | | | | |
− +3 3 3 .
PomnoΩyvßy rivnqnnq systemy (15) vidpovidno na r12 , r13 , r23 i vraxuvavßy
(10), otryma[mo
1
2 2
1
2
112
2
12
2 1 2
12
3
13
23
2
12
2
13
2
3
23
13
2
12
2
23
2r G
m m
r
m
r
r r
r
m
r
r r
r
⋅⋅
= + − + + − −
+ − −
v ,
1
2 2
1
2
113
2
13
2 1 3
13
2
12
23
2
13
2
12
2
2
23
12
2
13
2
23
2r G
m m
r
m
r
r r
r
m
r
r r
r
⋅⋅
= + − + + − −
+ − −
v , (16)
1
2 2
1
2
123
2
23
2 2 3
23
1
12
13
2
23
2
12
2
1
13
12
2
23
2
13
2r G
m m
r
m
r
r r
r
m
r
r r
r
⋅⋅
= + − + + − −
+ − −
v .
Rivnqnnq (16), wo zv’qzugt\ vidstani miΩ toçkamy, [ tymy opornymy spivvidno-
ßennqmy, qki budemo vykorystovuvaty v podal\ßomu.
4. Teorema pro stijkist\ za LahranΩem. OderΩani vywe rivnqnnq (16), wo
mistqt\ vza[mni vidstani miΩ material\nymy toçkamy, sprobu[mo zastosuvaty
dlq vstanovlennq zv’qzku miΩ stijkistg za Xillom pary toçok i stijkistg za
LahranΩem vsi[] systemy.
Dali obmeΩymosq rozhlqdom ruxiv systemy (2), qki naleΩat\ mnoΩynam
vid’[mnoho rivnq intehrala enerhi] (3), oskil\ky same pry h < 0 pytannq pro stij-
kist\ za Xillom fiksovano] pary material\nyx toçok [ konstruktyvnym (de-
tal\niße dyv. [4]).
Teorema. Nexaj r ( t ) = ( r1 , r2 , r3 )T — dystal\nyj rux systemy (2), qkyj
zadovol\nq[ umovy:
1) T – U = h < 0 na danomu rusi;
2) para til ( m1 , m2 ) [ stijkog za Xillom;
3) | r23 ( t ) – r13 ( t ) | ≥ r0 ∀t ∈ R , 0 < r0 = const .
Todi rozhlqduvanyj rux [ stijkym za LahranΩem.
Dovedennq. Oskil\ky doslidΩuvanyj rux r ( t ) [ dystal\nym, to ma[ misce
nerivnist\
T = h + U ≤ T0 = const . (17)
Prypustymo, wo pry vykonanni umov teoremy rux r ( t ) = ( r1 , r2 , r3 )T [ nestij-
kym za LahranΩem. Todi, z uraxuvannqm obmeΩenosti kinetyçno] enerhi] T sys-
temy, isnu[ taka poslidovnist\ { tk } , k = 1, 2, … , wo
lim
k kt→∞
= ∞ , lim ( )
k ij k
i j
r t
→∞ <
∑ 2
3
= ∞ , (18)
i, takym çynom, vraxovugçy umovu 3 teoremy, ma[mo
lim
( ) ( )
( )k
k k
k
r t r t
r t→∞
| |−23
2
13
2
12
2 =
r t r t r t r t
r t
k k k k
k
23 13 23 13
12
2
( ) ( ) ( ) ( )
( )
+( ) −| |
= ∞ . (19)
Perepysugçy perßu z rivnostej (10) u vyhlqdi
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
PRO STIJKIST| RUXU ZA LAHRANÛEM U ZADAÇI TR|OX TIL 1141
r23
2 = r r12 13
2 2+ – 2 r12 r13 ,
pryxodymo do vysnovku, wo
lim
( )
( )k
k
k
r t
r t→∞
23
2
13
2 =
lim
( )
( )
( )
( )
cos ,
k
k
k
k
k
r t
r t
r t
r t→∞
−
+
12
2
13
2
12
13
2 1r r12 13
%
= 1. (20)
Bez obmeΩennq zahal\nosti rozhlqdu, vraxovugçy strukturu systemy (16),
vvaΩatymemo dali, wo r r23
2
13
2− < 0. Todi, beruçy do uvahy (17) – (20), na pidstavi
druhoho rivnqnnq systemy (16) oderΩu[mo nerivnist\
1
2 13
2r
t tk
⋅⋅
∈{ }
≤ – γ ∀k ≥ s , 0 < γ = const , (21)
de s vidpovida[ dostatn\o velykomu nomeru v poslidovnosti { tk } .
Zhidno z (17), (18) isnugt\ qk zavhodno velyki vidrizky çasu t tk k2 1
∗ ∗−[ ], k2 >
> k1 , na qkyx vykonu[t\sq nerivnist\ (21). Intehrugçy (21), otrymu[mo
1
2 13
2
1
r
t
t
⋅
≤ – γ ( t – t1 ) , t > t1 ,
zvidky
1
2 13
2
1
r
t
t
≤ 1
2 213
2
1 1
2
1
r t t t t
t t
⋅
−( ) − −( )
=
γ
. (22)
Tut [ t1 , t ] ⊆ t tk k1 2
∗ ∗[ ], .
Poznaçymo çerez t∗ ( k ) toçku v zamknutomu intervali t tk k1 2
∗ ∗[ ], , t∗ ( k ) > tk1
∗ , v
qkij velyçyna r13
2 2/ nabuva[ maksymal\noho znaçennq. Poklademo v rivnosti
(22) t1 = tk1
∗ , t = t∗ ( k ) i perepyßemo ]] u vyhlqdi
1
2 13
2
1
r
t
t k
k
∗
∗( )
≤ t k t r t k tk
t t
k
k
∗ ∗
=
∗ ∗−[ ]
⋅
− −[ ]
∗
( ) ( )
1
1
1
1
2 213
2 γ
. (23)
Dodanok r
t tk
13
2 2
1
/( )⋅
= ∗
v (23) vidpovida[ takomu momentu çasu t = tk1
∗ , wo suma
r tij ki j
23
1
∗
< ( )∑ dosqha[ v n\omu krytyçnoho znaçennq, pry qkomu
1
2 13
2
1
r
t tk
⋅⋅
= ∗
≤ – γ .
OtΩe, vybir velyçyny
1
2 13
2
1
r
t tk
⋅
= ∗
v rivnosti (23) zavΩdy moΩna zdijsnyty takym çynom, wob vona bula skinçennog
i ne>zaleΩala vid dovΩyny promiΩku t t kk1
∗ ∗[ ], ( ) . Razom z tym cq dovΩyna, z
uraxuvannqm rivnosti (18), pry k → ∞ prqmu[ do neskinçennosti. Oskil\ky za
umovy prqmuvannq dovΩyny promiΩku t t kk1
∗ ∗[ ], ( ) do neskinçennosti prava
çastyna nerivnosti (23) sta[ vid’[mnog, a liva — dodatnog, pryxodymo do
supereçnosti.
Takym çynom, zroblene prypuwennq pro nestijkist\ za LahranΩem ruxu
r ( t ) = ( r1 , r2 , r3 )T [ pomylkovym, zvidky vyplyva[ vysnovok pro spravedlyvist\
teoremy.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
1142 S.>P.>SOSNYC|KYJ
Naslidok 1. Rozv’qzky LahranΩa, wo vidpovidagt\ roztaßuvanng toçok na
odnij prqmij, zadovol\nqgt\ otrymanu teoremu.
ZauvaΩennq. Dana teorema da[ uqvu pro konfihuracig tr\ox material\nyx
toçok, qka zabezpeçu[ stijkist\ ruxu za LahranΩem. Z>ci[g metog provedemo
plowynu π çerez seredynu vidrizka P1 P2 , wo z’[dnu[ vidpovidno toçky z masa-
my m1 , m2 . Budemo vvaΩaty, wo π ⊥ P1 P2 . Todi bud\-qkyj trykutnyk, utvore-
nyj toçkamy P1 , P2 i toçkog na plowyni π, bude rivnobedrenym, i, vidpovidno,
sutnist\ teoremy polqha[ v tomu, wo ]] umovy vyklgçagt\ zitknennq mas m 1 ,
m2 , m 3 i moΩlyvist\ peretynu toçkog P3 , v qkij zoseredΩeno masu m 3 ,
plowyny π . Cikavym u c\omu sensi [ toj fakt, wo v systemi Sonce – Zemlq –
Misqc\, qkwo stijkog za Xillom vvaΩaty paru Sonce – Zemlq, rux Misqcq za-
dovol\nq[ umovu 3 teoremy.
5. Pro obmeΩenu zadaçu tr\ox til. Rozhlqnemo takyj vypadok zadaçi tr\ox
til, koly masa m3 tret\oho tila [ znaçno menßog za masy perßoho i druhoho til
( m1 ≥ m2 >> m3 ) . Zokrema, prypustymo, wo masa m3 tret\oho tila nastil\ky
mala, wo ]] vplyv na rux til z masamy m1 i m2 [ duΩe slabkym. Todi, qk vidomo
[6], pryxodymo do klasyçno] modeli obmeΩeno] zadaçi tr\ox til
˙̇r1 =
G m2
2 1
12
3
r r
r
−
| |
,
˙̇r2 = − −
| |
G m1
2 1
12
3
r r
r
, (24)
˙̇r3 = G − − − −
| | | |
m m1
3 1
13
3 2
3 2
23
3
r r
r
r r
r
.
Wob oderΩaty rivnqnnq (24), neobxidno v rivnqnnqx (2) poklasty m3 = 0.
Takym Ωe sposobom otrymu[mo dlq obmeΩeno] zadaçi tr\ox til i analoh rivnqn\
(16). Zokrema, ma[mo
1
2 12
2
12
2 1 2
12
r G
m m
r
⋅⋅
= + − +
v ,
1
2 2
1
2
113
2
13
2 1
13
2
12
23
2
13
2
12
2
2
23
12
2
13
2
23
2r G
m
r
m
r
r r
r
m
r
r r
r
⋅⋅
= + − + − −
+ − −
v , (25)
1
2 2
1
2
123
2
23
2 2
23
1
12
13
2
23
2
12
2
1
13
12
2
23
2
13
2r G
m
r
m
r
r r
r
m
r
r r
r
⋅⋅
= + − + − −
+ − −
v .
Oskil\ky naqvnist\ pary til ( m1 , m2 ) , stijko] za Xillom, [ orhaniçnog ças-
tynog obmeΩeno] zadaçi tr\ox til, to cilkom pryrodno sformulgvaty dlq ne]
naslidok oderΩano] vywe teoremy.
Naslidok 2. Nexaj r ( t ) = ( r1 , r2 , r3 )T — dystal\nyj rux systemy (24),
qkyj naleΩyt\ mnoΩyni vid’[mnoho rivnq intehrala Qkobi. Todi pry vykonanni
umovy
| r23 ( t ) – r13 ( t ) | ≥ r0 ∀t ∈ R , 0 < r0 = const ,
rozhlqduvanyj rux [ stijkym za LahranΩem.
Dovedennq. Oskil\ky vidpovidno do postanovky obmeΩeno] zadaçi tr\ox til
dva masyvnyx tila prytqhugt\ tret[ tilo z neskinçenno malog masog, ale sami
nym ne prytqhugt\sq, to velyçyny v13 i v23 v rivnqnnqx (25) na pidstavi dy-
stal\nosti rozhlqduvanoho ruxu [ obmeΩenymy. Tomu, skorystavßys\ dali riv-
nqnnqmy (25) i vykladenog vywe sxemog dovedennq teoremy, robymo vysnovok
pro spravedlyvist\ naslidku 2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
PRO STIJKIST| RUXU ZA LAHRANÛEM U ZADAÇI TR|OX TIL 1143
Pry | r1 2 | = | r1 2 | 0 = const, a takoΩ vid’[mnij i dostatn\o velykij za modulem
stalij intehrala Qkobi, z ohlqdu na strukturu poverxon\ Xilla [6], v obmeΩenij
zadaçi isnugt\ stijki za LahranΩem ruxy, wo zadovol\nqgt\ naslidok 2.
Rivnist\ m 3 = 0 v obmeΩenij zadaçi tr\ox til bil\ßog mirog [ matematyç-
nog abstrakci[g, niΩ vidobraΩennqm real\no] sytuaci], koly masa m 3 xoç i
mala, prote skinçenna. A ce oznaça[, wo za umovy skinçennosti masy m3 znextu-
vani v rivnqnnqx (2) dodanky ne [ malymy prynajmni v toçkax zitknennq masy m3
z masamy m1 i m2 , vklgçagçy j deqki ]x okoly. OtΩe, qkwo masa m3 [ skin-
çennog, to dlq zabezpeçennq korektnosti modeli obmeΩeno] zadaçi tr\ox til
postulgvannq dystal\nosti rozhlqduvanyx ruxiv vyhlqda[ cilkom pryrodnym.
ZauvaΩymo, wo pytannq skinçennosti malo] masy m3 v obmeΩenij zadaçi tr\ox
til i naslidky, qki zvidsy vyplyvagt\, bil\ß dokladno rozhlqnuto v roboti [7].
U zv’qzku z [7] interes stanovlqt\ spivvidnoßennq (11), wo dozvolqgt\ otryma-
ty velyçyny r13
2 i r23
2 qk funkci] parametriv, qki xarakteryzugt\ rux til z
masamy m1 , m2 :
r
M m m r m r
m
m m m r
m
13
2 1 3 1
2
2 2
2
3
2
2 1 3 12
2
3
2=
+ +[ ] − +( ) ( )
,
(26)
r
M m r m m r
m
m m m r
m
23
2 1 1
2
2 3 2
2
3
2
1 2 3 12
2
3
2=
+ +[ ] − +( ) ( )
.
Oskil\ky v obmeΩenij zadaçi tr\ox til vvaΩa[t\sq, wo rux mas m 1 i m 2 [
malym zburennqm zadaçi dvox til, to v konkretnij sytuaci], qka modelg[ obme-
Ωenu zadaçu, vyrazy (26) moΩna rozhlqdaty qk kryterij pryjnqtnosti modeli
obmeΩeno] zadaçi tr\ox til. Ce oznaça[, wo koly rux mas m1 i m 2 dijsno vid-
povida[ malomu zburenng zadaçi dvox til, to vyrazy (26), qki vyznaçagt\ kvad-
raty vidstanej r13
2 i r23
2 , povynni dostatn\o toçno uzhodΩuvatysq z real\nym
ruxom tila z malog masog m3 .
Porqd z formulamy (26) korysnym moΩe vyqvytysq takoΩ rozhlqd vyrazu
r
m
m m m r m r m m r3
2
3
2 1 2 1 1
2
2 2
2
1 2 12
21= + +( ) −[ ]( ) ,
qkyj otrymu[mo z perßo] rivnosti (12).
Dana robota [ dewo dopovnenym variantom preprynta [8].
1. Pars3L.3A. Analytyçeskaq dynamyka. – M.: Nauka, 1971. – 635>s.
2. Uytteker3E.3T. Analytyçeskaq dynamyka. – M.; L.: ONTY, 1937. – 500>s.
3. Duboßyn3H.3N. Nebesnaq mexanyka. Analytyçeskye y kaçestvenn¥e metod¥. – M.: Nauka,
1964. – 560>s.
4. Holubev3V.3H., Hrebenykov3E.3A. Problema trex tel v nebesnoj mexanyke. – M.: Yzd-vo Mosk.
un-ta, 1985. – 240>s.
5. Arnol\d3V.3Y., Kozlov3V.3V., Nejßtadt3A.3Y. Matematyçeskye aspekt¥ klassyçeskoj y
nebesnoj mexanyky // Ytohy nauky y texnyky. Sovr. probl. matematyky. Fundam.
napravlenyq. – M.: VYNYTY, 1985. – T.>3. – 304>s.
6. Roj3A.3E. DvyΩenye po orbytam. – M.: Myr, 1981. – 544>s.
7. 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.
8. Sosnitskii S. P. On the Lagrange stability of the motion for the three-body problem. – Kyiv, 2003. –
13 p. – Preprint 2003.2.
OderΩano 27.07.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2005, t. 57, # 8
|
| id | umjimathkievua-article-3672 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:46:50Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0c/f672c5e65516c37b6b4947ed0bfdd10c.pdf |
| spelling | umjimathkievua-article-36722020-03-18T20:01:36Z On the Lagrange Stability of Motion in the Three-Body Problem Про стійкість руху за Лагранжем у задачі трьох тіл Sosnitskii, S. P. Сосницький, С. П. For the three-body problem, we study the relationship between the Hill stability of a fixed pair of mass points and the Lagrange stability of a system of three mass points. We prove the corresponding theorem establishing sufficient conditions for the Lagrange stability and consider a corollary of the theorem obtained concerning a restricted three-body problem. Relations that connect separately the squared mutual distances between mass points and the squared distances between mass points and the barycenter of the system are established. These relations can be applied to both unrestricted and restricted three-body problems. У задачі трьох тіл розглядається зв'язок між стійкістю за Хіллом фіксованої пари матеріальних точок і стійкістю за Лагранжем системи всіх трьох матеріальних точок. Доводиться відповідна теорема, що встановлює достатні умови стійкості за Лагранжем. Розглядається наслідок отриманої теореми стосовно обмеженої задачі трьох тіл. Встановлюються співвідношення, які зв'язують нарізно квадрати взаємних відстаней між матеріальними точками і квадрати відстаней матеріальних точок до барицентра системи. Ці співвідношення можуть виявитися корисними як в необмеженій, так і в обмеженій задачах трьох тіл. Institute of Mathematics, NAS of Ukraine 2005-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3672 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 8 (2005); 1137 – 1143 Український математичний журнал; Том 57 № 8 (2005); 1137 – 1143 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3672/4072 https://umj.imath.kiev.ua/index.php/umj/article/view/3672/4073 Copyright (c) 2005 Sosnitskii S. P. |
| spellingShingle | Sosnitskii, S. P. Сосницький, С. П. On the Lagrange Stability of Motion in the Three-Body Problem |
| title | On the Lagrange Stability of Motion in the Three-Body Problem |
| title_alt | Про стійкість руху за Лагранжем у задачі трьох тіл |
| title_full | On the Lagrange Stability of Motion in the Three-Body Problem |
| title_fullStr | On the Lagrange Stability of Motion in the Three-Body Problem |
| title_full_unstemmed | On the Lagrange Stability of Motion in the Three-Body Problem |
| title_short | On the Lagrange Stability of Motion in the Three-Body Problem |
| title_sort | on the lagrange stability of motion in the three-body problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3672 |
| work_keys_str_mv | AT sosnitskiisp onthelagrangestabilityofmotioninthethreebodyproblem AT sosnicʹkijsp onthelagrangestabilityofmotioninthethreebodyproblem AT sosnitskiisp prostíjkístʹruhuzalagranžemuzadačítrʹohtíl AT sosnicʹkijsp prostíjkístʹruhuzalagranžemuzadačítrʹohtíl |