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Recent advances of celestial mechanics in the soviet union
Recent Advances of Celestial Mechanics in the Soviet Union
Y. HAOr~A~A Department of Astronomy, University of Tokyo, Japan
INTRODUCTION Apart from the recent progress of Celestial Mechanics in the survey of periodic orbits in the restricted three-body problem and in the Trojan motion, not to speak of the modern development in the theory of motion of artificial celestial bodies with moderate eccentricities and inclinations, and apart from the development of modern computer techniques applied to the motion of in~vidual stars in cluster dynamics, and even adaptedfor analytical treatment, the main problems in Celestial Mechanics has been cultivated in the Soviet Union, such as: ( I ) - the capture of a distant star by a binary system (p. 15), I I - t h e existence of, quasi-periodic solutions (p. 20), ( H I ) - a n approximation theory with error estimation (p. 29) and ( I V ) - t h e stability theory (p. 36), all of which have opened up fresh aspects in Celestial Mechanics.*
I. PROBLEM OF CAPTURES Since Yu. Schmidt (1) proposed a new cosmogonical theory on the origin of the solar system in 1947, various discussions have been published on the capture of planets b y binary systems. Consider three bodies P0, P1, P~, denote the distances PoPx, P1P~, P~Po, respectively, b y r 2 --- r, to, rl, and the distance of Ps from the centre of mass G of P0, P1 b y ~. Suppose t h a t r 1 -~ ~ for t -~ - ~ while r 2 remains finite, and t h a t there exists an instant T* and a value R > 0 such t h a t 0 < r~ < R for all t > T*; then we say t h a t there is a capture of P2 b y Po. Numerical examples for captures have been computed b y several astronomers, Merman, (~) Proskurin, (s) Sizova, (4)Khrapovitskaya, (5)Kochina, (6) Merman and Kochina, (~) and for temporary captures b y Proskurin, (s) and Proskurln and Runjazewa. (°) Years ago Becket (1°) and recently Szebehely and Peters (n) computed numerical examples of captures with exchange of bodies, t h a t is, P~ is captured by Po and P1, but P1 recedes to infiuity later b y leaving Pz in a finite distance from Po * When my publications ~qtability in Cde~ial Mechanics (1957) and the chapter "Stability in the Solar System" in vol. IV of The ~dar System (edited by Kuiper and Middlehuret, 1963) were printed, the new literature of Soviet Russia had not been accessible to me. Now that it has become available, I am thrilled to see the brilliant successes that have been achieved; and I wish to dedicate this supple. mentary article to my Soviet colleagues in admiration of their outstanding recent advances in Celestial Mechanics. (Y. H.) 15
16
Celestial Mechanics m the Soviet Union
Poinear6, c12) Sehwarzschild, ¢18) a n d y o n Zeipel, ¢14~on t h e o t h e r hand, showed t h e impossibility of eternal c a p t u res,and t h e existence of ergochcity of their m o t i o n on the basis of conservative d y n a m i c s b y a n a l o g y to t h e m o t i o n of liquid molecules of an incompressible fluid enclosed in a vessel of finite volume. Chazy, C15) b y following up their ideas, showed t h a t a m o t i o n of hyperbolic-elliptic t y p e for t -~ + oo c a n n o t become b o u n d e d or oscillatory for t ~ - o o , a n d vice versa. According to Chazy Cle) a m o t i o n is of the " h y p e r b o h c elliptic t y p e " if one of three bodies recedes to infinity, while the o t h e r two are at a finite distance; of " b o u n d e d t y p e " if the three m u t u a l distances are all bounded, of " h y p e r b o l i c t y p e " if all t h r e e m u t u a l distances become mfimte; and of "oscillatory t y p e " if at least a n y one of t h e bodies oscillates indefinitely with mfimte amplitude. T h u s t h e possibility of captures has become a serious p r o b l e m in Celestial Mechanics. K h i l m i ¢17) defined a capture to be such t h a t r 0 > R1, r 1 > R~, r2 > R1 and t h e three velocities are hyperbolic for t = t~; and t h a t r 2 < R 2, 0 > R~ a n d t h e velocity of P1 relative to Po is elliptic, b u t t h e velocity of Pz relative to G as hyperbolic at t 2 > tl, w h e r e R 1 represents the largest dastance of t h e bodies from a n y star of a g a l a x y and R , t h e m u t u a l distance of a binary star P u t r m = m m { r o , r 1,%},
M* =mill{morn l + mlm2, M* = m a x
{
¢ = m i n { # o , rx,r2},
mlm2 + mom 2, mora2+morn1}, morn,
,
re,m2
,
morn,
},
M = mo + ml + ra~, M *-
M*
m 0 -~ m I m 1 + m~ m o + m~ M* Consider two cases" (A) if r(tl) > 0, ~(tl) < - ~/[8M*/rm(tl) ] at t h e initial epoch tl, t h e n to, rl, rz increase to infinity for t -* - oo, (B) if for t h e positive e n e r g y - c o n s t a n t h of t h e t h r e e - b o d y s y s t e m we can find R > 0 a n d ~ < R a f t 2 > tl s u c h t h a t
%(t~) < R, 0(t2) > 2R, ~(t2) > e(t2 )
16M ~(t2)
2_2_h ~1
2m2(mo + ml) /~l(R - ~)
0,
(too + m l ) ~ 2 M
t h e n P2 reeedes m o n o t o n o u s l y to infimty for all t > t 2, b u t r 2 remains smaller t h a n R, and the motaon as called hyperbolie-elhptic. I f for t h e m o t i o n of t h e three-bodies (A) holds for t 1 and (B) holds for t2, t h e n there is a capture. K h i l m i p r o v e d t h a t the measure in t h e phase-space of the point-set which represents t h e imtial state leading to a capture is n o t zero. K h i l m i ' s condition for a weak capture, so t h a t P~ recedes m o n o t o n o u s l y to infinity from G a n d the distance PoP1 does n o t exceed R in t h e case h > 0, is t h a t ~(0) > 2R, Q(0) > 0, ½ 0(0) < ro(0 ), rl(0 ) and [~(0)] 2 8M* 2 2morn1 - - > h + - e( 0 ) ]~1 /~1R I f the three m u t u a l distances t e n d to infinity as t ~ oo, t h e system is said to be eompletely dissolvable. W h e n a b o d y enters t h e sphere of radius ~ a r o u n d a n o t h e r body, it is called an c¢-approaeh. Khilml derived the conditions for each of these phenomena, and extended t h e m to the ease of n gravitating masses in order to a p p l y this to cosmogony. Sitmkov Cls~ and Alekseev (19) derived conditions for some particular cases. M e r m a n (~°~ criticized a n d supplemented t h e propositions of Chazy ¢2~) t h a t a hyperbolicelliptic m o t i o n for t --, + co can neither ehange to a b o u n d e d or an oscillatory motion,
Y. HXGIHX~A
17
nor to a hyperbolic motion for t -* - o o in t h e case h > 0 There are several numerical examples against these propositions According to Merman, t h e impossibility of a capture for h > 0 has n o t been estabhshed b y Chazy, because Chazy transferred the expansions of a m o t i o n from t h e d o m a i n of t -- + oo to a domain t = - oo, which is generally impossible. C h a z y ' s proof on t h e imposslbihty of a capture for h < 0 m a y be true, a l t h o u g h this m a y be Impossible to exist with positive measure for a bundle of trajectomes for h < 0, which are b o u n d e d for t -~ + oo b u t hyperbolic-elliptic for t ~ - oo, or which are hyperbolic-elliptic for t -* - o o b u t oscillatory for t -* + oo, or vice versa Merman (s°) discussed at first the characteristics of hyperbohc-elliptic motions for h < 0 and t h e error estimates. B y the energy integral we have min {ro, rl, rs} <
s,/Ihl, s~
= morn 1 -{- m l m ~ -b m s m o.
The quantity I -
(too + m l ) m s @ + M
morn1
rS
mo + m1
is either b o u n d e d or n o n - b o u n d e d for t ~ + ~ . If I -~ c~ for t ~ ~ , t h e n ~ becomes mfmite. T h e n Merman p r o v e d C h a z y ' s t h e o r e m s on a more rigorous basis, a m o n g others t h e t h e o r e m s t h a t t h e probability o f a capture m t h e region h < 0 is equal to zero, and t h a t if the area velocity constant / > 0, h < 0, and ff I is sufficiently small, t h e n the motion is hyperbolic-elhptie b o t h for t - , + oo and t -* After discussing various hyperbolic approaches (ss) and parabolic approaches(~) to approximate t h e three-body problem, Merman(u) gave four necessary conditions for a parabohcelhptic motion, two necessary conditions for a parabolic-hyperbolic, five sufficient conditions for a capture, one necessary condition for a capture, a necessary and sufficient condition for t h e existence of a parabolic-elliptic motion, and a necessary and sufficient condition for a t e m p o r a r y capture in the hyperbolic case. Alekseev (~5) studied the perturbations to a hyperbolic m o t i o n in a different m a n n e r and gave examples for an exchange of t h e distant bodies in t h e case of a capture. (se) M e r m a n (st) derived t h e criterion of a capture at first for a restricted problem (~) and t h e n for t h e general t h r e e - b o d y problem. (se) I n the former case there is m o -- 0; and in t h e hyperbolic m o t i o n we have 1),
ro=a(ecoshH-
t a n ~-- = 2
~(_i_oo)=2tan_ 1 /(e--l~
/(e-
~/\e
l~coth
+ lJ
H
2 '
nSaa
= m, + ms,
~(--~)=2~--tan-' /{e--l~.
Merman has also given eight necessary conditions for a hyperbolm-elliptic motion, one necessary condition for hyperbolic motion, and five necessary conditions for a capture. I f two n u m b e r s R > 0 and ~ > 1 are such t h a t to(0 ) >_ ~R, to(0 ) < air1(0), and m 1
r(0)
[1"(0)] z >-- m 1
2
R
2B
q-
na3(e s -
m a x {[e sin ~ - ~]~ [e sin ~ -- ~]0_ o0}, 1) 3/2
t h e n t h e motion is hyperholic-elhptic and r < R, where a1~
(~1--'-) -1 , 1 --
2B--msx/(SmlR
) . ( s a ~ q- a~ -{- ai)
18
Celestxal Mechamcs m the Soviet Unmn
From these conditions it is seen t h a t a capture can occur only for an extremely close approach. P u t . ml A =
[i(0)p
r(0)
+
2
~(0)
- ~o(0)
m1 + ms
+ ~(0)
~/(eS -
1) - ~o(0)
e
+
ml B ---- r ( 0 )
[r(0)]S -{-
- -
2
ms
{-- ~(O)[, na
m1 + ms
+ ~o(O)
]
+ ~(o)
e
I na
to(0 )
'
~ / ( e s _ x) - ~o(O)
]
+
'
e
ro(0 )
where ~, ~ represent the rectangular coordinates of P3 relative to G. At t -- 0, if A > 0 and B > 0, then the motaon is hyperbolic-elliptic; ff A < 0, B < 0 it is hyperbolic; and if A > 0, B < 0 or A < 0, B > 0, t h e n a capture can occur. A sufficient condition for a capture can be obtained b y combming these conditions. The exact conditions would be slightly different from these. For getting better insight into the problem, Merman (sS) provided new proofs of Birkhoff's theorems (2" on the behaviour of motion in the three-body problem and extended some of the theorems. Sibahara and Yoshida (") generalized some of the theorems. Merm a n (8°) proceeded to the qualitative s t u d y of the problem. H e considered in the general problem the possibility of resolving the motion into two almost independent motions of approximately Keplerian character, and found the distance and the velocity of the distant body from the other two at finite distances. More general and simpler criteria t h a n those of Merman are desirable. I f for h < 0 we have
K/s > =
27MsS~ ~3, m = m i n { m o , m l , m s } , 4 x/[m(M - m)]
K=
-h,
then we get for all t
Merman derived the m a x i m u m Qm and the m m i m u m ~m of ~, as well as the upper bounds of o, ]~[ and Sitnlkov (al) proved the existence of hyperbolic, hyperbolic-elliptic, and bounded motions for m n - - ml. Sibahara (Ss) has given a sufficient condition for Lagrangian stability b y sharpening Merman s theorems. H e also derived a criterion f the exchange of the distant body m an encounter process, and examined (u) the cosmogonical theory of Lyttleton (s,) Merman CSs)criticized on the basis of his inequality for K p the smallness of the convergence radius of Sundman's series (a~) derived b y Belorizky, (88) and estimated the error of Sundm a n ' s polynomial series in Sundman's regularizing variable. Let us consider m,,
'
(83)
/(co) = ~. a,eJ,
or
/,,(w) ----Z c~")a,w',
1=0
1=0
where c~") are the coefficients of the expansmn
m.
g.(o~) = ~ - ( "~,) ' c ~ , ~=o
1
g ( ~ ) = ~ ,
1 -co
Y. H ~ o ~
19
and m~ -- (~ + 1)~ - 1 (when we need ~ terms in the expansions); then we get
I1(:~) - 1.(~)I <
i
2~t~'(-
1)
1- (z/m)
~"
d~
max --
2:~
r'
where I" is the length of the integration contour ]~. Brumberg ~ ) applied this summation procedure to the solution of the three-body problem, by referring to the method of solution of differential equations by Painlev6 ~°) and Picard. ~ ) The result has been compared with the numerical solution of Zumk|ey ~ ) and with the solution by Sundman's series. ~ ) On the other hand, Cesco(~) started vath the summation method of Borel, ~t° and Zelmer (~) with t h a t of Euler-Knopp and found t h a t the convergence radius can be made smaller t h a n t h a t of Sundman's series by such an artifice. Such polynomial expansions are considered to be convenient for the numerical treatment of the problem. I t is mentioned in this connection t h a t Petrovskaya (~e) derived the convergence radius of Hill's series in his intermediary orbit to be Inn] < 0.21 .. on the basis of the infinite nonlinear analysis of Wmtner, (~7) who gave Inn] <0.0833 .... Further Petrovskaya (~) discussed the convergency of the Fourier series representing the disturbing function in the restricted three-body problem and found the domain of convergence in the complex planes in terms of undisturbed orbital elements. Years ago Liapounov~ ) derived the upper bound of convergence radius for Hill's series 1 to be nn < ~ b y a majorant method of his own. Merman ~ ) improved the inequalities and deduced m < 0.18. Rjabov (~) further estimated m < 1/~/15 = 0,258 ... by Liapounov's method. On the other hand, Petrovskaya (~) constructed a Poincar~ periodic solution of the first Sort in the form of a series m m for the satellite three-body problem, and derived the upper bound of convergence for m and for the mass-ratio.
REFEREI~CES*
1. YU. SCWM~T,Dold. Akad. Numb, 58 (1947) 213. 2. G. A. MEP~A~, Dold. A k ~ . Na~k, 86 (1952) 727; •Tu//. 1 ~ . Teor. A~r. 5 (1953) 373. 3. V. F. PROS~XN, B~uZl. Inst. Teor. Aatr. 5 (1953) 429. 4. O. A. SIzovA, Dokl. A k ~ . Nauk, 86 (1952) 485. 5. G. E. gm~POVITS~rA, BjUU. Inst. Teor. A~r. 5 (1953) 435. 6. N. G. Koam~x,/~Tu/l. In~. Teor A~r. 5 (1953) 445. 7. G. A. M ~ a ~ and N. G. K o c ~ A , B~u//. Inst. Teor Ast~. 6 (1956) 85, 349. 8. V. F. PBOSXURr~,B~uU. Inst. Teor. Astr. 5 (1953) 429; 8 (1962) 264. 9. V. 1~. PROSKU~I~and L. Yu. RuI~JAZ~WA,B~U. /n~. Teor. A~r. 7 (1959) 287. 10. L B~CKER, M. N. R. A. S. 80 (1920) 590. 11. V. Sz~.B~.~Y and F. P ~ R S , AsSt. J. 72 (1967) 876. 12. H. PoI~c~R~, Le~ M~thodes ~o~vd~e.s de h~ M&ani~ue C~,e.~e, Tome HI, 1899, Gauthter-Vdlars, Paris; C~lcul des ProbabilitY, 2nd edn., chap. XVI, 1912, Gauthier.Vfilars, Pans. 13 K. SCHW.~S~----.~, Astr. Nachr. 141 (1896) 1. 14. H. vo~ Z~I~r~, Bu//. A~r. 22 (1905) 449; V. M~t~a~, J. des Obse~t~urs, 20 (1937) 121. 15. J. C~azy, C. R. Acad. 8d. Par~, 179 (1924) 1307; 197 (1933) 1193. 16. J. C~*zv, A~n. Ec~e Norm. ~ p . , (3) 89 (1922) 29. 17. G.F. gm~.~L Doldady Akad. Nauk, 62 (1948) 39; 71 (1950) 847; 77 (1951) 589; 78 (1951) 653; 79 (1951) 419; Qualitative Methods in the Many-Body Problem, 1961, Gordon Breach, New York. 18. K. Sm~uxov, Ma~. ~bo~. 82 (1953) 693.
20
Celestml Mechanics in the Sowet Unmn
19. V. M. ALEKSEEV,DO~:~.Akad. Nauk, 108 (1956) 599. 20. G. A. M _ ~ , B~ull. Inst. Teor. Astr. 5 (1954) 594. 21. J. C'~xzY, J. de Math. Pure Appl. 8 (9) (1929) 353; Bull. Sc*. Math. (2) 56 (1932) 79; Bull. Astr. (2) 16 (1952) 175. 22. G. A. M_Em~x~,.B~ull. lnst Teor Astr 6 (1955) 73 23. G. A M E ~ , JB~ull.Inst Teor. Astr. 5 (1954) 606. 24. G. A. M~mwa~., Astr. Zhur 8() (1953) 332; B?ull. Inst. Teor. Astr. 5 (1953) 325. 25. V. M. AL~KS~V, Dokl. Akad. Nauk, 1(}8 (1956) 599; Astr. Zhur. 88 (1961) 325, 726, 1099; 89 (1962) 714, 1102. 26. G A. M~.~MAN, Dokl. Akad. Nauk, 99 (1954) 925. 27. G. I). B r m ~ o ~ , Dynamw,al ~ystems, Amer. Math. Soc., 1927. 28. G. A. ~ A N , JB~ull.Inst. Teor. Astr. 6 (1955) 69 29 R. S ~ m ~ x and J. YoSmDA, Pub. Astr. Soc. JaTan, 15 (1963) 87. 30. G. A. M E ~ , JB~ull.Inst. Teor. Aar. 6 (1958) 687. 31. K. SIT~OV, Dold. Akad. iVauIc, 188 (1960) 303. 32. R. S~A~A~, Pub. Astr. Soc. Japan, 18 (1961) 229. 33. R. S n ) ~ A , Pub. Astr. ~oc. Japan, 14 (1962) 90. 34. R. S ~ A ~ a ~ , Pub. Astr. ~oc. Japan, 18 (1961) 108. 35. R. A. LYTT~TO~, M. N. R..4. S., 96 (1936) 559. 36. G. A. ~ , B?ull. Inst. Teor Aar. 6 (1958) 713. 37. K. SU~D~X~, Acta Math. 86 (1912) 105. 38. D. B]:/~o~z~Y, J. Ob* 16 (1933) 109. 39. V. A. B~V~BV.~, ~B~ull.Inst. Teor. Astr. 9 (1963) 234. 40. P. P~I~L]~V~, Com. R. Acad. Sc~ Paros, 128 (1899) 1505 41. E. I~C~D, Tra~td d'Analyse, Tome II, chap XI, 1925, Gauthier-Vfllars. 42. J. ZUMK~Y, Astr. Naehr 272 (1941) 66. 43. R. P. C~sco, Ob,. Astr. Unw. Nae. l_,a Plata, set Astr. ~1 no. 14 {1965). 44. E. BOREr., Lefons sur les ~dmes Dwergentes, 1901, Gauthmr.Vfllars. 45. G. K. Z v . L ~ , Summatum Methods in the Two- and Three-Body Problem; Thesm, Umv. Br~tmh Columbia, 1967. 46. M. S. P~TROWS~YX, B?ull. Inst. Teor Astr. 7 (1959) 441,9 (1963) 257. 47. A. W ~ R , Math. Z. 24 (1925) 259, 80 (1929) 211; `4met J. Math. 59 0937) 795. 48. M. S. P~TROVSK~YX,JB~ull. Inst. Teor. `4,tr. 1(} (1965) 385; 11 (1968) 411. 49. A. M. L ~ o v ~ o v , Trud~ Otd Fys Nauk Absh. L?obit. Estestvoznan, 8 (1896} 1 = L~apounov ~obranie ~ochinsnii, I, 418. Akad. Nauk USSR, 1954. 5(}. G. A. M ~ x ~ , JB~ull.Inst. Teor. Astr. 5 (1952) 185. 51 J. A. I~,~BOV, B~ull. Inst. Teor. Astr. $ {1962) 772. 52. M. S P~TROVSKAY~,B~ull. Inst. Teor. Astr 8 (1962) 733 * l~ote: references appear at the end of each section.
I I . QI~ASI-PEI~IODIC SOLVTIO:NS K o l m o g o r o v (1) m n o n - d e g e n e r a t e cases a n d A r n o l d (s) i n d e g e n e r a t e cases of H a m i l t o m a n d y n a m i c a l systems, i n whmh t h e t h r e e - b o d y p r o b l e m is included, succeeded m p r o v i n g t h e existence of quasi-periodic solutions. L e t / z 1 . . . . /t m be non-zero l i n e a r l y i n d e p e n d e n t n u m b e r s a n d Uj be r a t i o n a l f u n c t i o n s of sin2xepjt, c o s 2 z p j t , j = 1 , 2 . . . . m. W e assume t h a t U 1 + U s + • + Uj + • is u n i f o r m l y c o n v e r g e n t i n t h e whole d o m a i n T of t, t h a t is, for a g i v e n e > 0 we h a v e IU,+I + "'" + Ut+s-ll < s for l > n , s > 0 for all t e T ; i n o t h e r words, we assume t h a t this serms represents a f u n c t i o n ~v(t) i n T a n d t h a t ~ > 0 can be f o u n d so t h a t I~v(t 4- v) ~v(t)l < 8 for all e > 0 a n d t, T E T, a n d that/z~v, i --- 1 . . . . m, differ a t m o s t b y ~ from t h e corresponding integers. T h e n t h e r e exists a u n i f o r m l y c o n t i n u o u s f u n c t i o n ~ ( x 1. . . . xm) -
Y. HAomARA
21
of period 1, such t h a t ~ l t ..... #=t) = ~(t) for all t. ~(% 8) is called the translation number. Suppose t h a t there are given e > 0 and a one-valued function ](xl, ..., xm) , which is continuous everywhere and periodic with period 1 for all the variables in the argument. Then we can find a uniformly and absolutely convergent trigonometric series ZA~'1 ~'.. cos (2rrn 1 xx - slur) cos (2rrn,x~ - 8,~r) ... cos (2gn,.x,. - r,.t) (81 . . . .
e m ----" 0 ,
½;
n I .....
nra --'-- 0 , l , 2 ,
. ),
such t h a t this series differs from [ at most b y e everywhere. Such a class of functions is called q~i-periodic and was s m d m d b y Bohl (8) and Esclangon. (4) Thin class of functions is a special case of Bohr's almost-periodic functions. The necessary and sufficient condition for
F(x) = f [(x) dx 0
to be almost-periodic is t h a t F(x) should be bounded. There exists the m e a n value of an almost-periodic function such t h a t T
0
The Fourier representation of the almost-periodic functions is unique and converges in the mean N
lim M {ll(~) where
~ A.e'/(-1)~"~P} = O,
/V
~. IA.P - M{If(x)p}
(Parseval).
n-----1
Consider a dynamical system
@ dt
OH
@
Op
dt
---,
-
~H
--,
Oq
l o , = (Pl . . . . p . ) , q = ( q ,
. . . . . q.),
(1)
with the a n a l y h c Hamiltonian
H(p, q) = Ho(p) + / d / l ( p , q) + ..., where Hi(p, q) is periodic with period 2~z in each q. For # = 0 we get
dp = 0 , dt
dq =o~(p), dt
o ~ - - - -OH (~1
..... w.),
Op
and every torus p -- eonst, is invanant. I f w is incommensurable, t h e n the motion is quasi-periodic with frequencies w 1 . . . . . ~o,. The trajectory [p(t), q(t)] Falls the torus everywhere densely. I n order t h a t the terms depending on q of order # be eliminated from the Hamiltonian
Ho(~) + #m~) + p~l(p, q) + . . . . Mo(p') + ~n,(p') + ~ F 0Ho ~a + h i ] + l op Oq _l
•
• 4 ~
22
Celestial Mechanics in the Soviet Union
where a bar denotes the average over q, after the Birkhoff transformation (6) ~S p = p' + /.t.-~-q
~S -= q + ,u Op '
S(p; q) = ~ Sk(p') e "/(-~)(~ ") k¢0
( k ' q ) = klq 1 + ... + k,q,,
it is necessary t h a t to, --~-/ ~S + /~t = 0 '
or ~k(~') --'~ N/(-- 1)hk(~D') , (to" k)
Ht = E htc'~/(-l'(k a),
(2)
k,o
The motion [p~(t), q'(t)] with the transformed Hamiltonian
it(p, q) = H'o(p') + ~H'~(p; q') + ... differs b y order/u from the motion represented by the H a m l l t o m a n Hg(p') during the internal t ~ / ~ - t . B y repeating this operation we arrive at the invarlant torus p(s)(p, q) = const, filled everywhere densely b y the trajectories of quasi-periodic motmns wath Hamiltortian H(p, q) = H(oS)p(s) up to the order/~s. By diophantme approximation almost every vector to = (tot . . . . ~o,) satisfies the inequalities
{(to" k)l -> Klk{ -("+2),
Ik{ = Ik,{ + .- + {k.{,
(3)
for all integers k :~ 0 for some K(to) > 0. The small denominator (to • k) can be estimated from below b y the degree of k. Let H , be analytic for 1~ ql < P and thus IH,(P, q){ < M. Then we have {hk{ _< M e - I ~l~. B y (3) the coefficmnts S k = ~ / ( - 1)h~/(to" k) decrease in geometric progression almost as fast as hk since for a n y 5 > 0 we have [Ski < ~ M L
8_lk{(0_~)'
L = const.
Hence the series S converges for {~ q[ < Q and we have
I~l <= ~
ML
~.+-----~-;
(4)
for a smaller region ]~ ql =< ~ -- 2 ~. Thus (2) converges almost for allto. B u t the functions thus defined depend on p everywhere discontinuously, and the convergence is doubtful I t is wor¢h mentioning t h a t Sibahara (6) showed t h a t the series is convergent if the denominator tot[tea for n -- 2 is not a U-number of Mahler (7) or not a Ihouville's number; and Wintner (s) showed the almost-everywhere convergence. A transcendental number ~ ~s called a LaouviUe n u m b e r if we can find a rational fraction such t h a t for a n y integer we have
t
~_
ga
<
1
h--~
ha>l'
where ga/ha is the 2-th convergent of She continued-fraction representation of ~. I n order to avoid the difficulty of small denominators, Kolmogorov aimed at only one invariant torus T~. of the perturbed motion, on which a quasi-periodic motion with frequency to* occurs. The set of frequencies to* satisfying (3) m fixed m advance, and the
Y. HAOrWARA
23
torus T~, is determined in the neighbourhood of the corresponding toms of the unperturbed system p = T* +/~P~ + "", ~Ho/~P* = co*. In (2) we replace Wo(p) ~H° by w*. The additional term which appears is of order/~ 2 for IP - P*[ ~ F, and the ~p new approximation is shown to be of order M s. The procedure is repeated b y Newton's approximation method of tangents. The rapid convergence overcomes the influence of small denominators and the series is convergent for the majority of the values of w. Thus an invariant torus T~. can be found. Accordingly, if we assume t h a t -
ii0o1
then there are,in the small neighbourhood of any point p, points at which the frequencies w are commensurable as well as points at which w(p) = w* can be estimated by (3). On certain t o n p = constant, canonical equations with the Hamiltonian Ho(p) determine everywhere dense quasi-periodic trajectories, but on the other tori t h e y do not. l~or small perturbations H = Ho(p)+ p h i ( p, q),/J < 1, the majority of invariant tori with incommensurable frequencies (o* satisfying (3) with a fixed K do not vanish, and are only somewhat deformed. Trajectories of the perturbed motion starting on the deformed tori T ¢ , fill everywhere densely and quasi-periodically. The tori T~.~form a closed nowhere-dense set and between them remain slits filled with the remnants of the destroyed tori with commensurable w, This invariant nowhere-dense set has a positive measure which tends to the measure of the whole phase space for K -* 0, F -* 0. For n = 2 the two-dimensional tori T~. divide the three.dimensional energy level H = const., and the trajectory beginning in a slit between two tori T~. cannot escape from this slit. Thus the existence of invariant tori allows us to draw conclusions on the stability. For n > 2 the n-dimensional tori T~. do not divide the (2n - 1)-dimensional manifold H = const., and slits can extend to infinity. The behaviour of trajectories on a two-dlmensional torus has been studied by Poincar6, Denjoy, (9) Cherry, and Ura. If the condition (5) is not satisfied in the case of natural degeneracy in which det ]1~HJapl] - 0, then the non-perturbed motion is described b y a smaller number of frequencies than n. Arnold (s) divided the Hamiltonian into the secular part P0 = (Pl ..... P,o), q0 = (ql . . . . q,o) with
1 f H1dqo,
R~(~, q) = (2~)--~
and the periodic part Pl = (P.o+l ..... P,), ql = (q.o+l ..... q.) with
I-I~ = Ha(p, q) - Fi(p, q). The canonical system -
dt
,u
+
8qx
...,
- -
dt
--
+ ...,
p
apx
determines the slowly varying variable Pl, qx definiug the invariant tori. The periodic part //1 leads only to additional oscdlations of the perturbed trajectory around the quasiperiodic motion with slowly varying variables described b y H o + P//1. The decomposition is possible b y the existence of adiabatic invariants proved b y Kasuga. (1°)
24
Celestml Mechanics in the Soviet Union
Let us assume t h a t the secular p a r t KtI does not depend on the phase ql of the slow variation R i = Hi(p) , and t h a t ~CO0 [I
agHO
*°'
[1 C3ql I]
~'1,/Zn--no
,0
Then, for sufficmntly small I/~[ and for a majority of initial conditions from the F of the phase space (p, q), a motion determined b y H(p, q) ,s quasi-periodic, and for these initial conditions the motion is near the quasi-periodic morton dpo _ 0 , dt
,~o = ~ O o ( P o ) - ~g0 dt ~Po
dPi - O, dt
dql -- ql(P) = ~ t dt Opi
(6)
for-- ~ 0. I f the 2~ are independent, then the motion is quasi-periodic. The trajectory fills the torus everywhere densely. By repeating the transformation s times we arrive at H(p, q) = R(s)(v (s)) +/~(~)(p(5), q(,)),
2 ~ s) = (p~))~ + (qi~))~, R (~) = (2. ~(s)) + • 2,jC)g~) + ... l.J
(2,j = 2j~). If (2" k) # 0,
Ikl = Ikll + "'" + Iknl _-< 2s - 1,
then A s -- 0(Iv(s)]s) begins with s-degree t e r m s and is convergent. The system with the H a m i l t o u i a n / 7 (~) is mtegrable. The invariant tom are given b y z (s) = const. The frequencies 2~) ----Jh + 2 E 2 , ~ ') ÷ "'" v a r y from a torus to another l,J
I f at least one of the inequalities
12~,A=
~
,0,
-~
0
ii_
~R
,0 0
is satisfied, the Hamiltonian (6) is called of the general elliptic type The trajectory is near t h a t of the integrable system with R (s) for t ~ s -~, the latter of which is in the e-neighbourhood of the origin for - oo < t < + oo. I f (2 • k) # 0, k # 0, then a n y trajectory starting at an ~-neighbourhood remains in the neighbourhood of the origin for t ~ C~-~. Arnold (n) proved the stability of motion of the general elhptic type for n = 2 near the equilibrium point. Moser (u) showed t h a t it is sufficient to have s = 5/2 and klX~ + k ~ 2 # 0
Y. HAGma~A
25
only for [kll + Ik~l >__4. Deprit (in) p r o v e d the stability in the case of t h e exceptional massratio for the T r o j a n p r o b l e m b y LeontoviS's (m criterion of stability based on this idea. I f for the one-to-one t r a n s f o r m a t i o n / ( A ) -- B b o t h / and its i n v e r s e / - 1 are continuous, t h e n / is said to be a topological trans/ormation or homeomorphism. I f such a t r a n s f o r m a t i o n exists for two sets A and B t h e n these sets are h o m e o m o r p h i c . I f ~ is not analytic b u t differentiable t h e n t h e t r a n s f o r m a t i o n is called di~eomorphism. We represent a conservative d y n a m i c a l s y s t e m with n degrees of freedom (1) in a 2ndimensional Euclidean space with coordinates x = x 1, x 2 . . . . . x, and with grad F, the vector gradient F ~ . . . . F~, of F(x), in the form
dX-IgradH, dt
1=(0
I.
O17 )
'
(7)
where I T is the unit m a t r i x of order n I n t r o d u c e t h e skew-scalar p r o d u c t of two vectors x and y b y [x, y] = (Ix, y) = -[y, x], which represents t h e s u m of t h e areas of t h e projections of the parallelogram with sides x, y onto the coordinate planes Linear transformations S which preserve the skew-scalar product such t h a t [Sx, Sy] = Ix, y], for all x, y, are called simplicial. The skew-scalar p r o d u c t of the gra4tent [grad F, grad G] is called t h e Poisson bracket (/7, G) of F and G. 2' is t h e first integral of (7), if and only if its Poisson b r a c k e t (F, H) with t h e t t a m i l t o n i a n H vanishes identically. If the Poisson b r a c k e t of two functions vanishes identically t h e functmns are said to be in involution. I f (7) admits n single-valued first integrals in involution F, = / t = const., i = 1 . . . . n, so t h a t H = F1, . , F , , (Ft, ~'~) = 0, and t h e equations Ft = It = const, define in t h e 2n. dimension space x and on the n-dimensional c o m p a c t manifold M = M~, at each point of which grad F , , (s = 1 . . . . . n), are linearly independent, t h e n M is an n.dimensional torus and the point x(t), representing a solution of (7), a d m i t s a quasi-periodic motion along it. I n particular Arnold (15) d e m o n s t r a t e d this t h e o r e m of K o l m o g o r o v in the case of a Liouville's d y n a m i c a l system.
Arnold's Fundamental Theorem Consider a function H, a domain G o and positive n u m b e r s ~, R,C, and suppose t h a t (I) t h e function H(p, q) where p = Po + Pl, q -- q0 + ql, P0 is a vector of dimension n o, and Pl is a vector of dimension nl, while n -- n o + n 1 , and qo, ql are angular variables. Yurthermore, H(po, Pl; qo -t- 2~t, ql) -- H(P0, Pl; q0, ql) is analytic in t h e d o m a i n F : p0et~0, I~ qo] =< Q, [Xll =< R, where x 1 = (Pl, ql), and depends on the p a r a m e t e r #, 0 < / x __
H,(p, q) = H,(po, px, q,) + [Ii(po, qo, Pl, ql),
Hi(po,
ql) = nl
Yl(P0'
x(po,
+ 71
f /-tl dqo = 0,
l(pO, nl
T) = ~'0 + Z ~|Tf| + Z ~IJ~"TJ + Z ~lJkT(t'J"~k' l=l l,J=l l,j.k
2 where ~o, ~l, ~ j = Jtjt, ~tjk are functions Po and 2~ l = p2,o+~ + q,o+~, i = 1, . , n, (iii) in P we h a v e for a certain C >__ 1
2 B-VIkVol 13
26
Celestml Mechamcs m the Soviet Unmn (iv) in GOwe have det
~H°
4 0,
det][~o(po)[[ =~ 0.
Then for any z > 0 it is possible to find ~(x; H o, ~ , Go; 9, R, C,/~o) > 0 such t h a t for 0 < ~ < 8 9 , 0 < / ~ < 84 (1) the domain ~ F ~ ' p e ~G 0, ]~ q0] = 0, 0 < v~ < 8, consists of two sets F, and ]8, of which F~ is invariant w~th respect to the canonical equation with the above Hamiltonian and/~ is small so t h a t rues ]~ < ~ . rues F~; (il) F~ consists of invarzant n-dimensional analytic t o n T~ given b y the parametric equations po = po~ +/o~(Q),
qo = Q9 + ao~(Q),
~ = ~ / { e [ ~ +/I~(Q)]} cos [Q1 + al~(Q)],
ql = ~ / { e [ ~ + t~(Q)]} sm [Q~ + ~,~(Q)],
where Q = Qo, QI are angular parameters and P0~ and ~ are constants depending on the number of the tori o~. (in) the mvariant tori Ts differ slightly from the tori I/,AQ)I < ~ ,
Iq,AQ)I < x~,
(iv) the motion determined by the above Hamiltonian on the torus T~ is quasi-periodm with n frequencies o)
dQo dt
--
( D 0~
dQz dt
~
(D 1,
~Ho , ~Po,o
( D O --~
(O 1
=
/.~
~Hz a~,
Belaga (le) studied the reductibility to a linear system in the neighbourhood of an invamant point. Melmkov (17) discussed the behaviour of trajectories near an autonomous Hamlltoman system, Pins (19) of a non-autonomous system; and Beljustma (is) discussed the focal, nodal, and saddle points of the trajectories Hagihara (~°) some time ago discussed a canomcal system with slow variatmns according to Birkhoff's transformations. (5) Moser (~a) discussed a rapidly convergent iteration method on the basis of Siegel's theorem. Cu) Moser, (~1) Arnold, (25) Berkovitz and Gardner, (28) and Ghmm (27) studied the ring transformations of Poincar6 (2s) and Blrkhoff (29) by demonstrating the existence of periodic solutions. We now consider the differential equations
dx = o~ + 8/(x, y, ~), dt
dy dt
-
~ y + 8g(x, y, 8),
where the matrix Y2 is diagonahzed and [, g are real analytic in all variables. Suppose that the equations admit x - cot + eonst, and y -- 0 as a solution with the characteristic numbers co1..... co,, Q1 ..... ~m, satisfying n
• =I
/~=I
with a positive constant ~. Then there exists a unique analytic power series for an n-vector ~(~), an m-vecter ~u(s), and an m x m-matrix M(8) satisfying ..Qp --- O,
,.QM = M,.Q,
Y. H ~ G ~ . ~
27
such t h a t the modified system dx dt
- o,) + ~/(x, y, ~) + ,~,
dy dt
= ~ y + ~g(x, y, ~) + p + Hy
admits a quasi-periodic solution with the same characteristic numbers as the non-perturbed motion. This is Moser's theorem (a°) on the existence of quasi-periodic solutions. Moser ~a°) applied the 1des of Lie groups (81) to the existence proof. Sternberg (8~) discussed the solutions of differential equations on a torus. Merman (aS~ showed the exmtence of a quasi-periodic solution m the plane restricted problem of three bodies. Such a solution is the limit of periodic solutions of the second genus of Poincar~ ca4~ whose periods are multiples of the period of the non-isoperiodie periodic soluhons of Schwarzschild. (aS~He referred to Birkhoff's formal series (~9~representation of the solution, and to Poincar~'s invariant point theorem (28) The existence theorem of Moser of quasi-periodic solutions has been applied to the threebody problem by Jefferys and Moser, (se~ and Arenstorf. (~7~ Deprit Henrard (aS) studied the genealogy of periodic solutions of the Trojan problem in the restricted three-body problem. Jefferys~ag~further apphed the area-preserving mapping of a torus into itself by Birkhoff, and recently cultivated by NIoser~1~) and Arnold, c2) with numerical application to periodic orbits in the restricted three-body problem. B y judging from the behavlour of the trajectories in the nelghbourhood of an mvariant point Jefferys studied the stability and the ergodlcity of the motion and the transition between stability and instability. The ergodic theorem of Poincar~ (~s~ and Birkhoff ~29~ has been discussed recently by several authors, such as Dowker, (~°~ Brunk, (4~) Blum and Hanson, C42~ Chacon and 0rnstem, c43~and Arnold and Avez.(4~) Furstenberg ~46) studied the strict ergodicity and transformations in almost-periodic processes. Let A be a diffeomorphism of a plane ring which conserves the area and 7 be a simple curve in the ring which is not homologous to zero. Then Poincar~'s mvariant point theorem tells t h a t the curves 7 and A 7 have at least two common points. Arnold (46~ calls the operation A" ~ -~ ~ g~oba~y canonical, ff it is homotopic to the identity and if +...
for each closed curve 7 c ~ . Let P(x) = p(~c), Q(x) = q ( ~ ) ; the operator A is globally canomcal if and only if
A(x)
- - . I (Q"
q) dp + ( p - P )
Xo
defines a monovalent function A(x). Let T be a torus p = 0 in ~ , and A T be the image of T by a globally canonical operation A. The tori T and A T have at least 2 ~ common points, provided t h a t A T satisfies p = p(q), ]dp/dq] < oo. Arnold proved t h a t i~ A is a globally canonical operation sufficiently near Ao: p, q -* p, q Jr (o(I0), such t h a t for II~li
detll~]l.
0 there exists a point Po for which o~,(po) are commensurable, i.e.,
0)~(~0)
ii - z - |1
=
2:~mdNw i t h
integers N and m~, i = 1, 2 ..... n , - t h e n
the operation A e has at least
2" fixed points in the neighbourhood of the torus p Po. Arnold (47) discussed stationary motions and their stability on the modern version of differential geometry of Lie groups of infinite dimensions, in contrast to the groups of diffeomorphism, and applied it to the hydrodynamics of a perfect fluid. 2* =
28
Celestial Mechanics m the Sower Uinon
Krassinsky (4s) reduced a canonical system in a simple form in the ne~ghbourhood of a quasi-periodic solutmn and p r o v e d the existence of quasi-periodic solutions in an a r b i t r a r y n e i g h b o u r h o o d of t h e generating quasi-periodic solution on the basis of K o l m o gorov-Arnold's theory. The existence of m e a n motions m the secular motions of t h e periheha and nodes of m a j o r planets has been d~seussed b y Skripmehenko (40) on t h e bas~s of W e y l ' s t h e o r y (a°) and of B o h l ' s work (a~)
REFERENCES 1. A. N. KOI~OGOROV,Dokl. Akad. Nauk, 98 (1953) 763; 99 (1954) 527. 2. V. I. A~Or.D, Dokl. Akad. Nauk, 187 (1961) 255; 188 (1961) 13; 142 (1962) 758; 146 (1962) 487; 156 (1964) 9, Uspekh~ Mat. Nauk, 18 (1963) 13, 91 3 P. BOHL, J. reme u. angew. Math 181 (1906) 268; Dissertation, Dorpat, 1893. 4. E. Escnx~oo~¢, C. R. Acad. Svi. Parts, 185 (1902) 891 ; 187 (1903) 30; Ann. Obs. Bordeaux, 16 (1916) 53. 5. G. D. BImr~OrF, Amer. J. Math. 49 (1927) 1 ; Dyuctmwal Systems, Amer. Math. Soc., 1927. See also C L. Smgel, Nachr. Akad. Wtss. Ctott~ngen, Math.-Phys. Kl., IIa, (1952) 21; Annals o/Math. 42 (1942) 607, 647; Math. Ann. 128 (1954) 144; Vorlesungen uber H~mmelsmechan~k, Springer, 1956. 6. R. SIBA~-RA, Pub. Astr. Soc. Japan, 8 (1952) 144. 7. K. MAImS.R,Math. Ze~tschr. 81 (1930) 729; J. re,he u. angew. Math. 166 (1932) 118, 137; Math Ann. 106 (1932) 131; J. ~'. Koksma, D~opha~t~sche Approx~matwnen, Spnnger, 1936. 8. A. Wr~TN~R, Math. Ze~tschr. 81 (1029) 434 9. H. Poi~exR~, J. Math. _Pure Appl. 1 (4) 161 (1885). A. DENJO:¢, J. Math. Pure Appl. 11 (9) 333 (1932); C. R Aead. Sci Parts, 194 (1932) 830, 2014; T. ~I. CHERRY, Amer. J. Math. 59 (1937) 957; Proc. £,ondon Math. got., 44 (1938) 175; T. URA, Ann Ecole Norm. Sup. 69 (3) 259 (1951); T. URA and ¥ HIR~SAWA,Pro~. Japan Acad. 80 (1954) 726. 10. T KAsuGA, Proc Japan Aead. 87 (1961), 366, 372, 377 11 V.I. AR~COnD,Dokl Akad. Nauk, 187 (1961) 255. 12. J. MOS~R, Nachr. Akad Wtss. Gottmgen, 1962, 1. 13. A. D~PRIT, Boeing Scl. Res. Lab. Document D 1-82-0580 (1966). 14. A M LEO~CTOVIS,Dokl. Akad. Nauk, 148 (1962) 525; 146 (1962) 523. 15. V. I. ARNOLD,S~b~rsk Mat Zhur. 4 (1963) 471. 16. F. G. B~LAGA,Dokl. Akad. Nauk, 148 (1962) 255. 17. V. K. M~L~I~OV, Dokl. Akad. Nauk, 189 (1961) 31; 142 (1962) 542; 165 (1965) 245. 18. L. N. BELJUSTr~CA,Soviet Mathematics, 4 (1962) 48 19. L. J. PIUS, B?uU. Inst. Teor. Astr 11 (1967) 84. 20. Y. HXGI~r_ARA,Proc. Inst. Acad. Japan 7 (1931) 44, 20 (1944) 617, 622. 21. J. MOSER, Nachr. A]~ad. Wiss. Gottingen, Math "-Phys K1. (1962) 1. 22. J NASH, Ann. Math. 63 (1956) 20. 23. J. MOS~R, Annall Scuola Norm. Sup. Ptsa, 20 (3) 265, 499 (1966). 24. C. L. S~EG~L, Ann Math. 48 (1942) 647. 25. V. I. ARNOLD,Dokl Akad. Naul¢, 156 (1964) 9. 26. J. BERKOVITZ,C. S. GXRD~R, Comm. Pure Appl. Math. 12 (1959) 501. 27. J. G L I ~ , Comm Pure A~apl. Math. 17 (1964) 509. 28 H P o ~ N c ~ , Rend~c.ont~ Cir. Mat. Palermo, 88 (1912) 375 29. G. D. BIRKHOYl~,Trans. Amer. Math. Soc. 14 (1913) 14; Rendiconti Cir. Mat. Palermo, 89 (1915) 271; Trans Amer. Math. Soc. 18 (1917) 199; Acta Math. 48 (1920) 113; Dynamical Systems, Ainer. Math Soc, 1927. 30. J MOSER,Math. A~n. 169 (1967) 136. 31. P M CoHEre, L~e (troups, Cainbndge Umv. Press, 1957; 1~. J ~ c o ~ s ~ , L~e Algebra, Interscmnce Publ., 1962; L. S. PO~TRJXOr~, Topologische Gruppen, Teubner, 1958. 32. S. STER~B~R~, Amer. J. Math 79 (1957) 397, 809; 80 (1958) 623; 81 (1959) 578; J. Math. Mech. 10 (1961) 451. 33. G. A. iW_ER~X~¢,Trudi Inst. Teor. Astr., no. 8 (1960). 34. H P o r s c ~ , Mdthodes Nouvelles de let Md.canulue Cdleste, Tome III, 1899, Gauthier-Vdlars. 35. K. SC~rW~zscHm~), Astr. Nachr. 147 (1898) 17, 289.
Y. H~om~a~
29
36. W. H. J~TE~YS and J. Mosv.a, AsSt. J. 71 (1966) 568. 37. R. F. A~STOBF, Amer. J. Math. 87 (1967) 27; J. reme u. anqew. Math. 221 (1966) 113. 38. A. D~PR~ and H. H~m~RD, Boeing ~c~. Lab. Document D 1-82-0623, 1967. 39. W. H. J~.mr~aYs, A~r. J. 71 (1966) 306. 40. Y. N. DOwz.~R,/)u/ce Math. J. 14 (1947) 105. 41. H. D. BRU~K, Trans. Amer. MaSh. 8oc. 78 (1955) 482. 42. J. R. B~.~ and D. L. ~ s o ~ , Prec. Amer. MaSh. ~oc.. 16 (1965) 413. 43. R. V. CH~eo~Tand D. S. 0R~s~r~, Ilhno~8 J. Math. 4 (1960) 153. 44. V. I. Av~o~.D and A. AVEZ, Probl~mes Er~od~es de la Mgcani~ue Class~ue. Gauthier-Vfllars 1967. 45. H. Fm~ST~B~RO, Amer, J. MaSh. 83 (1961) 573. 46. V. I. ARNOLD,C. R. Acad. 8el. Paris, 261 (1965) 3719. 47. V. I. AR~O~.D,C. R. Acad. 8ci. Paris, 260 (1965) 5668; 261 (1965) 17; Ann. In~. Fourier, Unw. Grenoble, 16 (1966) 319. 48. G. A. KR~SS~SKY, B]ull. Inst. Teor. AsSt. U (1968) 411 ; Trudi In~. Teor. AsSt. no. 13. 49. W. I. SKRn~IC~S~EO, B~ull. In~. Teor. AsSt. 11 (1968) 441. 50. H. W~YL, Math. Ann. 77 (1916) 313; Amer. J. Math. 60 (1938) 889; 61 (1939) 143. 51. P. Bo~., J. reine u. an~ew. Math. 185 (1909) 189.
I I I . AVERAGINGMETHOD Beginning with the v a n der Pol equation (1) y - / ¢ ( 1 - y~) ~ + y --- b2k cos (~tt + ¢¢), or a generalized form b y Cartwright and Littlewood ~2) ij + qD(y, [/, t) ~ + G(y) = p(t),
where ~, G, p are continuous, p(t) periodic with period 2/2~, ~(y, ~, t) periodic with period ~/2~ in t and bounded inferiorly for all values of y, ~, t, the averaging method has been cultivated b y K r y l o v and Bogoliubov, (s) Bogoliubov and Mitropolski, (4) and Mitropolski. c5) F r o m the vector equation dx
- 8 X(t,
x),
x = (x~ . . . .
(1)
x.),
dt x ( t , x) = 5". e "/~-~)" X J r ) , v
t h e y derived
-~, at
-F-
M{F},
dt
,
where N denotes an integrating operator and M an averaging operator b y keeping x constant such t h a t
M{F(t, x)} = Fo(x), t
~(t, x) = Z 1)----? ~_e'/(-'" (. (_z } , ~ )~0
~(x,0 = E
e'/~-~" F,(0
,÷0 IX/(-- I)~]s
The equation T
(2) 0
30
Celestial Mechanics in the Sovmt Unmn
is taken as the first approximation If ~ is defined by
then
(_~_)
x = ~ + 8J~(t, 2) + 8 ~ j~2
X(t, ~) - ~20~'(t, ~) X.(~)
O~ satisfies (1) to the accuracy of e 2. The method can be apphed to the system in standard form dx - s X ( t , x) + 8 2 Y(t, x ) .
dt Suppose t h a t there exist in a domain D positive numbers M and ~ such t h a t
IX(t, x)[ <=M,
[X(t, x') - X(t, x")l <=2lx' - x"[
for all real values of t __> 0, and for any points limit
x, x', x"
of D, and t h a t there exists the
T
lim X ( t ' x- )- d~ tf = X ° ( t ) r ~ 0
in D, then it can be shown that, corresponding to any positive numbers ~ and ~ however small, and for any intervalL of t however large, a positive number 80 can be found such t h a t if ~ = ~(t) is a solution of the averaged equation
d~
dt
-sX0(~)
(0
and is in D together with its Q-neighbourhood; then fx(t) -
~(t)l <
holds for 0 < s < 8o in the interval 0 < t < L/8, where x(t) is the solution of (1) with the same lmtial condition as ~(t). Let X(t, x) in (1) satisfy (a) the first approximate equation (2) admits a quasi.static solution ~ -- ~0 such t h a t
X0(~0) = 0; (b) the real parts of all n roots of the characteristic equation det IpI - X~(~0)[ = 0, derived for the variational equation d 65
dt
-
,x~x(~o)
•~
corresponding to the quasi-static solution ~ = ~0, are chfferent from zero; (c) a ~-nelghbourhood D e of the point ~0 can be found such t h a t X(t, x) are almostperiodic in De, even with respect to x ~ De; (d) X(t, x) and its first-order derivatives with respect to x arc bounded and uniformly continuous m x in - oo < t < + oo, x e DQ. Then positive constants 8', ao, a~, ao < a~ < ~, c a n be found such that for positive values e < e'
Y.H.~ors.~A
31
(1) the equation (1) admits a unique solution x = x*(t) defined in - oo < t < + oo for which
lx*(O-~0l
(-~
+~);
(ii) this solution is almost-periodic with the basis of the functions X(t, x); (in) a function 5(x) can be found which tends to zero with s such t h a t
Ix*if) - ~oI ---< D(e), (*v) if x(t) m a n y solution of (1) different from x*(t) and satisfying Ix(t) - ~ol < a0 for t = t 0, and if the real parts of all roots of the characteristic equation are negative, t h e n
Ix(t) - x*(t)l < Ce -~'('-'°), with the constants C and 7, and hence Ix(t) - x*(t)l tends to zero for t ~ oo. If the real parts of all roots of the characteristic equation are pos,tive, t h e n we can find t, > t o for which Ix(t1) - ~0[ > al. If the real parts of s roots are negative and those of the r e m a i n m g n - s roots are positive, then in t h e a0-neighbourhood of ~0 there exists an s-dlmensional manifold M,o such t h a t Ix(t) - ~0] tends to zero exponentmlly for t -* oo for the initial condition X(to)e M, o and t h a t Ix(t1) - ~o] > al for t I > t O for the initial condition x(to) M, o The theorem has been generalized to perturbed motions near a periodic solution m place of a quasi-static solution. Consider
dO - 8F(t, O) dt
(3)
with the initial conditmn Oo = O(to) for t = to, so t h a t O(t) = O(to) + ~ [ t - to, to, 0(to)].
If O, = O(to + nT), t h e n
(4)
0.+ 1 = 0,, + q~(T, t o + nT, 0.) We have [ ~ ' ( T , t o + n T , 0)[ < s~(s),
where O(e) ~ 0 for e -* 0. Hence F(O) = 0 + ~5(0) increases m o n o t o n o u s l y so t h a t F(O + 2~) = F(O) + 2z~. The transformation 0 -~ F(O) is unique and continuous. The iteration equation (4) 0.+ 1 = F(O.) admits the limit = lim 0. n-~o¢ 2~n independently of 0 0. If ~ is irrational, t h e n the general solution of (4) is
O, = 2n~n + ~p + E(2n~n + ~), where ~p is an a r b i t r a r y constant, E(~) is continuous and periodic with t h e period 2~. We get
O(t) = 2z~--~(t-- t°) + V2
'[ 2~
t - t° -
,2z~n+v2],
32
Celestml Mechamcs m the Soviet Umon
where
t -
to -
nT
T and accordingly
l(n
-
2u¢,
~ + 2:~) =/(n, ~).
The solutmn of (3) is quasi-periodic w~th the external frequency ~ and the natural frequency ~ , such t h a t 2~
x(t) = qS(~et, a~t ÷ ~p),
~ = __T--, lim O(t) _ ,-~ T
ap -
2~
~---,
~p -- const.,
2~ T
If ~ is rational, ~ = r/s, then (4) admits a periodic solution 0,+s - O, = 2gr and all solutions with mereasing n approach one of these periodic solutions. I f Om is any solution of (4) contained in the interval (0, 2~), then constants ~m > 0, tim > 0, ~m + fl~ < 2~, can be found such t h a t - ~ , , _< 0ms - 2 ~ m r <= ft,,. This is the rotation number of Pomcar6 and of Birkhoff. (I'~9) The method has been slightly modified b y Cesari, (e) Hale, (v) and ]Allo. (s) Levinson (9) simplified the analysis of K r y l o v and Bogohubov. Itufford, (1°) Marcus, (m K y n e r (12) discussed the existence of perioche solutions on the basis of such an idea. Dlliberto Hufford (18) stuched the asymptotically stable cases in the eyastence of an invariant surface The averaging method has been applied to the motion of E a r t h satellites, w~th the error estimates of the approximations by Struble (m and Kyner. (~5) They discussed the existence and the nature of the periodic solutions. Burshtein and Solovev (le) apphed the averaging method to a Hamfltonian system of differentml equations with a constant p a r a m e t e r e, and formulated the averaged form of a canomeal system
W
'[C
'
')]
2
'
where 2/~ means the integral ~(H - 1i) dt and the bracket denotes the Poisson bracket. Bogohubov and Zubarev (17) extended the method to the system with rapidly rotating phase, such as dx _ ~ X (x, W, 8),
d~
dt
dt
_ co(x) + 8 ~ ( x , ~, ~),
where x varies slowly and ~0 rapidly. A n u m b e r of generalizations were worked out by Kyner, (12) Mltropolsl~, (is) Volosov, (19) Zadiraka, (2°) Lykova, (~) P o n t r j a g m and Rodygin. (~2) Mol~anov ¢28) considered a time-independent system Los, (24) Gikhman, (25) Domidovieh, (26) Fedorchenko (~) discussed systems with continuous dependence of equations on parameters. Volosov rediscussed the averaging method m the case in which the coefficients of the nonlinear differential equations are purely periodic, and then Lykova, (21) Mitropolskl, (~s) Fedorchenko, (2e) Makaeva, (2s) Volosov (ag) m the case in which the differential equations contain slowly varying parameters. Arnold (8°) discussed the motion for the applicabihty
Y. HxGn~a:tx
33
and the error estimation of the averaging method for systems which pass through states of resonance in the course of their evolution. Consider vector equations for ~0 = (~1 . . . . ~k), I = (I 1 . . . . Ii)
dI _ 8F(I, ~0, e), dt
d~ _ (o(I, e) + el(I, % ~),
dt
where 8 4 1, q0(mod 2~r) and ~o, f, F are analytic for I e G, a complex compact domain, 1~ (P] <~, I'%[ < 8 The corresponding averaged equation is
dI dt _ eF(J),
F(J) -
1 k f F(J, of, O) dcp, (2~)
J = lr
Assume t h a t in the time t -~ 1/~ the difference between the exact and the averaged solutions with the same imtial conditions remains small so t h a t
I/(0-
J(t~ I < C,e
(0 < t < l/e),
where Cl, i -- 1, 2 . . . . denote sufficiently large constants independent of 8, k, K, N, x. For k = 2, Arnold derived a sufficient condition for the smallness of [I - JI with the estimate
C ; l ~ / e < II(t) - J(t)[ < C s x / , log~
.
( 7 For ~o~(I) ~ 0 put ;t(I) = ml/e)2 and assume t h a t C4-]~ < I~,1 < C4e Grebemkov (ax) proved two theorems justifying the averaging method for systems of differential equations with commensurable initial frequencies. The theorems are apphed to the Hfll-Delaunay ease of the restricted three-body problem. Musen (a~) extended the method of Krylov-Bogohubov to higher order terms, espeeially the long-permd and secular terms, and compared this with the Poincarg and yon Zeipel methods The eanonicity of differential equatmns is not required in Krylov-Bogoliubov's method, but the number of partial differentml equatmns to be solved m the process of elimination of the short-period terms increases as compared with Poinear6's. I n PoincarCs method the ehmination of the shorg-period terms from the coordinates and momenta is reduced to the elimination of such terms from the I-Iamfltonian b y a proper choice of the canomeal transformation. Assuming t h a t the generatmg function S of this transformatmn is expanded into a power series of a small parameter, the determination of S is reduced to the solution of a set of partial differentml equations. Musen has written those partial dtfferentml equations m t e r m s of F a a de Bruno operators. The greatest advantage of KrylovBogohubov's method is t h a t the error estimate can be made at each step of the successive approximatmn, while m Delaunay-von Zeipel's method there is no means of checking the convergence Morrtson (38~compared the method of averaging with the method of yon Zelpel m the nonresonant cases and showed t h a t yon Zelpel's method is a particular case of the generalized method of averaging, corresponding to an appropriate choice of the arbitrary function arising of the averaged equations. Morrison (38) and Kevorkian (a4) compared yon Zeipel's method with the two-variable expansion procedure, which is a generalized asymptotxc expansion method applied by Kevorkian and his collaborators to the motion of E a r t h satellites and lunar orbiters I t is worth mentioning in thin connection t h a t Diliberto (sS) developed his permdxc surface theory, which is based on the theory of conjectured periodic surfaces whose existence has 2a
~-V//t Vol t3
34
Celestial Mechanics in the Somet Unmn
not yet been rigorously proved If a family of such periodic surfaces is to exist, then it would establish the stability of the orbits and the vahdity of Diliberto's algorithm for finding approximate solutions. Diliberto proved the existence and uniqueness of permehc surfaces for speemhzed condltmns. K y n e r (aa) studied the relationship among the classical Delaunay theory, the Dfllberto periodic-surface theory, and the Krylov-Bogohubov averaging method, and showed that yon Zeipel's simplification of Delaunay's method and the canonical method of averaging produce the same second-order approximation; and showed from the existence of a formal expansion of yon Zeipel's generating function t h a t there exists a formal expansion of a family of permdic two-surfaces, ff a monotonicity condition is satisfied. The Drliberto method has been applied to the motion of E a r t h satelhtes by Dfliberto, K y n e r and Freund, (87) Haseltine, (as) K y n e r (~) for periodic solutions around an oblate Earth on the basis of the mapping of the invariant manifolds according to Poincarfi and Blrkhoff, and of the twist mapping of Moser, (4°) Orlov, (41) and Conley. (42) Diliberto is attempting to settle the three-body problem in an affirmative manner by showing that Pomcar~'s non-existence theorem really does not exclude even the simplest stability possibilities, t h a t the system is formally stable, and t h a t the series used to define the formal stability is convergent. The asymptotic method by Ritz and Galerkin (4a) hkewise enables us to estimate the errors committed. The variational method of Ritz, as modified by Galerkin, can be applied for determining the eigenvalues and the eigenfunctions for the boundary value problem. Cesari (~) discussed the problem whether a nonlinear periodic vector equation dx dt
-
X(x,
O,
where X ( x , t) ~s periodic m t with the permd 2g, admits an exact solution x(t) in the neighbourhood of z~(~) = a0 + x/2" ~ (a. cos n~ + b. sin nO, i.e. a trigonometrical polynomial of order m with indeterminate coefficients, which is called the truncated trigonometric polynomial P~(t), obtained by the Galerkin approximation method, and determined by how much the exact solution x(t) differs from x~(t). Cesari ¢~) and Urabe (~) discussed the problem on the basis of the topological theory and estimated the errors.
REFERENCES 1. B. v~w 1)ERPOL, Phil. Mag. 48 (6) 700 (1922). 2. M. L. C~a.rw~o~r and J. E. L~TTI~WOOD,J. Zondon Math. Eoc. 20 (1940) 180; Ann. Math. 48 (1947) 472. 3. N. KRYLOVand N. N. BooorzuBov, C. tt Acad. Sci. Paris. 194 (1932)957, 1064, 1119; 199 (1934) 1592; 200 (1935) 113; 201 (1935) 1002, 1454; 202 (1936) 200; L'Applicatio~ de Mdthode de la Mdcamque No~ding.aire ~ la Thgor~e des Perturbation~ des ~yst~raes Canon~qnes, Acad. Sci. Ukraine, no. 4 (1934); Mdthodes Approchds de la Mdcanique iVo~dinga~re clans leurs Applwations ~ l']~tude de la Perturbat~o~ des Mouvement8 P~riodglues et des d~vers Phdnvm~nes des Itesonanees s'y rapl~ortants, Acad. Sc~. Ukraine, no. 14 (1935); Itetueil Math. 1 (1936) 707; Ann. of Math. 88 (1937) 65; Introduction to Nonlinear Mechan~, Princeton Univ. Press 1949. 4. N. N. BoooT.rmaovand Y. A. Mm'~OrOLS~, Asyraptotiv Methods ~n the Theory of lqordlnear Oacilla-
t/on~, Hindustan Publ. Corp., 1921; Gordon Breach, 1962; Gauthier-Villars, 1962. 5. Y. A. M2rRoPoT.s~,Probl~me de la Thdorse Asymptotiqu, des Oscillations l~on-~tationaires, GauthlerVfllars, 1966.
y. H~om,~
35
6. L. CES~I, Asymptoti~ Behavior and ~tability Problem in Ordinary Di~erential Equatwns, Springer, 1959. 7. J. K. I-IA~x~ Ann. Math. 78 (1961) 496; Oscillattona tn 1%nh~ar ~y~ema, McGraw-ttfll, 1963. 8. J. l,u.T.o, Acta Math. 108 (1960) 123. 9. N. LEv~sox, Ann. Math. 52 (1950) 727; Ac~a Math. 82 (1950) 71; J. Math. Phys. 28 (1950) 215. 10. G. HWTORV, Contributwns to the Theory of JVonlinear Oseillatwns, III, 173, Princeton Umv. Press, 1956. 11. M. D. ~ c u s , ~b~/. HI, pp. 243 and 261, Princeton Umv. Press, 1956. 12. W. T. K ~ R , ~bu/. III, p. 197, Princeton Univ. Press, 1956. 13. S. P. I)~v.RTO and G. H~TOV~, ~b~d. III, p. 207, Princeton Umv. Press, 1956. 14. R. A. S ~ u ~ , J. Math. Mech. 10 (1961) 691; Archive Itat. Mech. Anal. 7 (1961), 87. 15. W. T. KY~r~R, Tectm. Rep., ~ R 041-152, 1963; Comm. Pure Alrpl. Math., 17 (1964) 227; J. Eo¢. Ind. Appl. Math. 18 (1965) 136. 16. E. L. B v ~ x N and L. S. SOLOV~V,Dold. Akad. 1Vauk, 189 (1961) 855. 17. N. N. BO(~OLIUBOVand D. N. ZuB~a~Ev, Ukrain. Mat. Zhur. 7 (1955) 5. 18. Yu. A. MI~OPOLS~, Uk'ram. Mat. Zhur. 9 (1957) 296; 10 (1958) 270; 11 (1959) 366; Problhme de la Thdome Asy~tot~que des Osedlattons, Gauthier-Vfllars. 19. V. M. VoLosov, Dokl. Alaad. zVaul¢, 106 (1956) 7; 114 (1957) 1149; 115 (1957) 20; 117 (1957) 927; 121 (1958) 22, 959; 128 (1958) 587. 20. K. B. ZAvraAua, Ukrain..Mat. Zhr 16 (1968) 121. 21. O. B. L~KOVA, U]crain. Mat. Zhur. 9 (1957) 155, 281, 419, 10 (1958) 239. 22. L. S. Po~TnJ~OI~ and L. V. l~ol)YGr~, 1)old. Ahad. iVauk, 182 (1960) 537. 23. A. M. MOL~A~OV,Dold. Al¢~l. l~auk, 186 (1961) 1030. 24. F. S. Los, Ukrain. Mat. Zhur. 2 (1950) 87. 25. I. I. G r m m ~ , Ukra,n. Mat. Zhur. 4 (1952) 215. 26. S. F. F~DOR(m~NKO, Ulcrain. Mat. Zhur. 9 (1957) 248. 27. B. P. Do~mowcH, Dold. Alcad. ~auk, 96 (1954) 693. 28. G. S. M ~ v ~ , Dokl. Akad. l~au]¢, 121 (1958) 973. 29. V. M. VOLOSOV,Dold. Akad. l~aul¢, 188 (1960) 261 ; 187 (1961) 21 ; U~pekh, Mat. lgauk, 17 (1962) 3. 30. V. I. Av~oLv, Dold. Akad. ~au~, 166 (1964) 9; 161 (1965) 1. 31. E. A. G m ~ m x o v , A~r. Zhur. 42 (1965) 190; BTuU. Inst. Teor. Astr. 11 (1968) 293. 32. P. M u s ~ , J. AeronauL ~ci. 12 (1965) 129. 33. J. A. Mow~so~, 8oc. Industr. A ~ l Math. l~ev. 8 (1966) 66; Method~ in A~rodynamic,s and Cde~ia~ Mechanicz, Academic Press, 1966. 34. J. K ~ v o ~ , Aatr. J. 66 (1961) 878; 67 (1962) 264; R. A. L~Gv,RS~ao~ and K~vom~z~, Astr. J. 68 (1963) 84; Cdest/al Mechanics andAstrodynam,~, Academic Press, 1964; M. C. EcKs~n~, Y . Y . S ~ and J. K ~ v o m r r ~ , Astr. J. 71 (1966) 248, 301 ; Theory of Orbits in the Solar 8y~tem and in ~ellar 8ystem~, Academic Press, 1966; Y. Y. SHz and M. C. Ecxs~n~, A~r. J. 72 (1967) 685. 35. S. P. ~ ¢ R ~ O , Rendicont, Cir. Mat. Palermo, 9 (2) 265 (1960); 10 (1961) 111. 36. W. T. K Y ~ R , Theory o~ Orbits in the Eolar ~ystem and ,n Stellar 8yatems, Academic Press, 1966. 37. S. P. D ~ a ~ T O , W. T. KY~ER, R. B. l~R~vm), Astr. J. 66 (1961) 118; NASA Space Technology Lab., Calif. 1962; O.N R. Techn. Rep. Umv. Califorma, NR-641-157 (1957); Navord 6445, China Lake, 1959; O.I~.R. Techn. Rep. no. 2, 1962; ibid. NR041-255 (1966). 38. W. R. H~S~LT~% Quart. Appl. Math. 20 (1963) 183. 39. W. T. KYNE~, Comm. Pure ATrpl. Math. 17 (1964) 227. 40. J. Mosv,R, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Kl. 1962, no. 1. 41. A. A. ORLOV, B]ull. Inst. Teor. Astr. 7 (1960) 805. 42. C. C. C o ~ Y , Comm. Pure Appl. Math. 17 (1964) 237. 43. W. Rrrz, J. f. re,he angew. Math. l ~ i (1909) 1 ; V. L. K a ~ o ~ o v I c H and G. P. AKrMOV,$'un~ional Analy~s ®n Normed ~paces, Pergamon, 1964; N. M. K~YLOV, Lea M~thode~ de 8d~ion~ A~rochges de Probl$me de la Physique Mathdmatique, Gauthier-Vflh~rs 1931. 44. L CESARI,Aeymptoti¢ Behaviour and 8tabdity Problems in Ordinary Di~eren$ial E~ation~, Springer, 1959; Contribution to the Theory of Nonlinear Osdllation~, V (1960) 110; Contributions to Di~erc~ial Equat/ons, I (1963) 149; Michigan Math. J. 11 (1964) 384. L. C~sa_~ and J. K. HALE, Proc. Amer. Math. 8oe. 8 (1957) 7b'~7. J. K. H . ~ , Contrib~ion to the Theory of ~onlmear O~cillation~, ~ (1960) 55. E. A. G ~ and J. K. HALE, J. Itat. Mech. Anal. ~ (1956) 353. 45. ~I. Uw~.~, Intern. Syrup. Nonlinear Di~erential E~ation~ and JVonllnear M~hanic,s, Aea~lemic Press, 1963; Archive l~at Mech. Anal. 20 (1965) 120. 2a*
36
Celestial Mechamcs m the Soviet Umon IV
STABILITY T H E O R Y
I t is k n o w n t h a t L i a p o u n o v ' s second or d i r e c t m e t h o d for t h e c r i t e r i o n of s t a b i l i t y consists in f o r m m g t h e Liapounov's/unctwnal (1)
= d F _ ~V X i + dt ~x I
~V X~ + ~x~
+ ~V X . + ~x.
~V ~t
where t h e given d~fferential e q u a t i o n s are
dxt -- X t
(~ =
1,2,
, n)
dt L l a p o u n o v ' s c r i t e r i a for s t a b i l i t y a n d i n s t a b i l i t y a r e . (I) I f t h e differential e q u a t i o n s for a d i s t u r b e d m o t i o n a r e such t h a t i t is possible to find a sign-definite f u n c t i o n V, of w h i c h t h e d e r i v a t i v e is a f u n c t i o n of fixed sign w i t h t h e sign o p p o s i t e to V, or r e d u c e s i d e n t i c a l l y t o zero, t h e n t h e n o n - d i s t u r b e d m o t i o n is stable (II) L e t V be a function of xs a n d t such t h a t V a d m i t s a n i n f i n i t e l y s m a l l u p p e r b o u n d , t h a t V is a s l g m d e f i m t e function, a n d t h a t V can t a k e t h e sign of l~ for t h e v a l u e s of t b e y o n d a c e r t a i n b o u n d , h o w e v e r s m a l l xs m a y be in a b s o l u t e values. I f such a f u n c t i o n V can be f o r m e d b y m e a n s of t h e d i f f e r e n t i a l e q u a t i o n s of t h e d i s t u r b e d m o t i o n , t h e n t h e non. d i s t u r b e d m o t i o n rs unstable ( I I I ) I f for differential e q u a t i o n s of a d i s t u r b e d m o t i o n i t is possible to find a f u n c t i o n V of a fixed sign or w i t h an i n f i n i t e s i m a l u p p e r b o u n d , whose t o t a l d e r i v a t i v e l~ is also of fixed sign o p p o s i t e to t h a t of V, t h e n t h e n o n - d m t u r b e d m o t i o n is asymptotically stable. (IV) I f for t h e differential e q u a t i o n s of a d i s t u r b e d m o t i o n i t is possible to find a funct i o n V such t h a t l? is a f u n c t i o n of fixed sign, a n d V Itself is n o t of fixed sign o p p o s i t e t o t h a t of 17, t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable. (V) L e t V be a f u n c t i o n of xs a n d t s u c h t h a t V is a l i m i t e d f u n c t i o n ; t h a t I? is of t h e f o r m [? = ~V + W, w h e r e ~ is a p o s i t i v e c o n s t a n t a n d W IS a f u n c t i o n of fixed sign, a n d t h a t V m a y t a k e t h e sign of W for a l l v a l u e s of t b e y o n d a c e r t a m b o u n d , h o w s m a l l xs m a y be in a b s o l u t e v a l u e s I f such a f u n c t i o n c a n be f o r m e d b y m e a n s of t h e d i f f e r e n t i a l e q u a t i o n of a d i s t u r b e d m o t i o n , t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable C h e t a e v ' s (=) Criterion if t h e r e exists a f u n c t i o n V(t, x), such t h a t for an a r b i t r a r y large v a l u e of t in a n a r b i t r a r y s m a l l n e l g h b o u r h o o d of t h e origin, t h e r e exists a region V > 0, if in t h e region V > 0 t h e f u n c t i o n is l i m i t e d ; if in t h e region V > 0 t h e d e r i v a t i v e l? t a k e s positive v a l u e s for a l l v a l u e s of t, x~ so t h a t V > ~, ~ > O, a n d l~ > l, l = l(~) > O, t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable.
Le/schetz' s Extension
a r e closed a n d if t h e space R is c o m p a c t t h e n
P
t h e y are c o n n e c t e d , t h a t t h e closure ~ of a n i n v a r i a n t set M is a n i n v a r i a n t s e t ; t h a t t h e b o u n d a r y of a n i n v a r l a n t set IS a n i n v a r i a n t s e t ; a n d t h a t if M , N a r e l n v a r l a n t sets so a r e t h e i r u n i o n M w N, t h e i r i n t e r s e c t i o n M n N, a n d t h e i r difference M - N ; a c c o r d i n g l y /2(?) a n d A ( ? ) are i n v a r i a n t sets.
Y. H~GmARA
37
L e t M be a closed m v a r i a n t set of the d y n a m i c a l s y s t e m ](p, t), and S(e) be a sphere a t centre M and with radius 8, and H(s) be t h e boundary. M is defined so t h a t (a) it is stable, whenever for a given e > 0, t h e r e m an ~ > 0 such t h a t if p e S(~/) t h e n {/(p, t)/t >= 0} c S(e), (b) it is a s y m p t o t m a l l y stable w h e n e v e r it is stable, and m addition under t h e preceding c o n d i t m n / ( p , t) - , M as t ~ oo, (c) it is unstable w h e n e v e r for a n y e > 0 and w h a t e v e r ~ > 0 and ~ < e t h e r e is a point T e S(~) such t h a t / ( p , t) reaches H(e) for some t > 0 T h e d o m a i n of a s y m p t o t i c stability A of M is an open m v a r i a n t set containing a sphere S(~) and the closure A, and accordingly its boundary, are m v a r i a n t sets. T h e necessary and sufficient condition for stability of M rs t h a t , if we are given a n y ~ ;> 0 and p outside S(8), t h e r e is a ~(s) > 0 such t h a t ?~ remains outside S(~). The necessary and sufficient condition for a s y m p t o t i c s t a b d l t y of M is, m a d d i t m n to t h e condltmns for stabihty, t h a t there exists a neighbourhood of M free from complete p a t h s o t h e r t h a n those in M Itself. T h e generalized ]_aapounov functionals of Lefschetz are functions defined in a certain set S(~), such t h a t (a) g~ven a n y 0 < e < ~, t h e r e exists a 1 > 0, so as to satmfy V > 2 for p outside S(a); (b) given ~ > 0 t h e r e exists an 7(2) > 0, so as to satisfy V < ~ for p ~ S(~); {c) V[/(p, t)] is in S(~) a non-increasing function of t. Such a f u n c t m n V is a generalizatmn of L i a p o u n o v ' s concept of t h e positive-definite functmnal V with the neg a t i v e I~. A necessary and sufficient conditmn for t h e s t a b i h t y of M is t h e existence of a generahzed L i a p o u n o v functional V. A necessary and sufficient condition for a s y m p t o t i c stability of M ~s t h a t t h e r e exists a V{p) defined for ~ m a certain S(~), such t h a t V[](p, t)] -~ 0 as t ~ oo. A necessary and sufficient condition for instability of M is t h e existence of a functional V(p) defined in a certain S(¢¢), such t h a t (a) IV(p)] is b o u n d e d in S(~); (b) e v e r y S(e) contains at least one point p where V(p) > 0, (c) a t e v e r y p o i n t p e S(c¢) along ~ts p a t h , the derivative l? is defined and [7 = i V + W(p), where 2 > 0 and W(p) is non.negative in
s(~) Converse o/Lsapounov's Theorems The equilibrium point of t h e differential equation
dx - l(x, t), dt
I(O, t) = o,
l eE,
is called uni]ormly stable if to each 8 > 0 we can find a n u m b e r ~ --- ~(e) > 0, depending on e, such t h a t f r o m I~1 < ($ follows Ip(t, ~, to) I < ~, t > t o for all t o _> 0 (Krasovskii). (1) (I) - An equilibrium point is u n i f o r m l y stable if and only if t h e r e exists a function ~(r) such t h a t (a) Q(r) is definite, continuous and m o n o t o n o u s l y increasing m 0 _< r < r,; (b) ~(r) v a m s h e s for r = 0, (c) for I~l < r we h a v e IT(t, ~, to)[ -< ~(~) (Persidskn) (1) I f IT(t, ~, to) ] <<_H holds for all points ~ at which a solution curve starts, t h e n the a s y m p totic stability is said to be globally. If the assumptions of this t h e o r e m are satisfied for a n y arbitrarily large r, t h e n the equihbrium point is said to be uniformly stable globally. Similarly uniform a s y m p t o t i c global stability is defined. ( I I ) - I f an equilibrium point of an a u t o n o m o u s or periodic differential equation is stable, t h e n it is also u ~ f o r m l y stable, and t h e globally uniform stability follows from the global s t a b i h t y (Hahn(*)). Let a(r) be a defined function, continuous m o n o t o n o u s l y decreasing for all r _>__0, so t h a t llm a(r) = 0 and [xs(t, ~, to) [ ___
38
Celestial Mechanics in the Soviet Union
a function satisfying the conditions (a) and (b) of theorem I. ( I I I ) - The necessary and sufficient condition for uniformly as~nptotic stability of the equilibrimn point is t h a t there exist two functions a(r) and ~(r), such t h a t [p(t, ~, to)[ __< <= ;~([~1)a(t - to) for ~ of a fixed sphere K , (Hahn¢l)). The converse of the stability theorems depends on the nature of the differential equa. tlons, on the assumptions of the equilibrium point, and on the nature of the Liapounov functional (especially the existence and the continuity of its partial derivatives). A function V(t) is called positively semi-definite or negatively semi-definite if V(0) = 0 and if in the neighbourhood of the sphere IxI _< h of the origin we have V[x[ ~_ 0 or V(x) <=O. If V(0) = 0 and V(x) > 0 or >__0 for x =~ 0, then the function V is positive-definite or negative-definite. ( I V ) - L e t / ( x , t) and ~/l/axj be continuous in the spherical domain Ix[ __ t o, and let the equilibrium point be stable. Then there exists a negative-definite function V(x, t) which possesses continuous first derivatives with respect to all variables and also a negative semi-definite derivative I~(x, t). A function V(x, t) is called descresvent if there exists a function ~(r), such t h a t with a function ~v(r)-whieh is real and continuous in 0 _< r _< h, vanishes for r -- 0, and increases strictly monotonously with r - w e have IV(x, t)[ <_ ~v([xl) in Ix[ _-< h, t _> t o ( V ) - L e t the equilibrium point be uniformly asymptotically stable and / e C 0 in [xl -< h, t _>_t o. Then there exists a positive-defiuite descrescent of V, whose derivative ~ is negative-definite and which possesses partial derivatives of arbitrary orders (Massera ¢5)) ( V I ) - I f the equilibrium point is uniformly asymptotically stable globally, and ] e C0, then there exists a positive.definite, deserescent, and radially bounded function V with a negative-definite derivative which possesses partial derivatives of any arbitrary order (Massera(5)). (VII) For the existence of IAapounov's functional V, whose derivative l? is definite, it is necessary and sufficient t h a t there exists a neighbourhood of the origin in the phasespace in which a complete phase trajectory is contained (Krasovskh¢l)). (VIII) Let the equilibrium point be unstable. For the existence of the functional V, which can take negative values in any neighbourhood of the origin and whose derivative is negative-definite, it is necessary and sufficient t h a t there exists a neighbourhood of the origm, in which no complete phase-trajectory - o o < t < + oo is contained (Kra. sovskii(1)). Thus, if for Liapounov's functional V, we have VI? ~ 0 and the motion is stable. Otherwise it is unstable. Dearman and Le May (*) have given Liapounov's funetionals for simple cases, by discussing several proposed methods, such as those by Amerman, Zubov, Ingwerson, Schui~, and Gibson. Purl and Ku, ~) Purl and Weygandt, (s) and BrayCon and Moser, (*) Brayton and Mirander (l°) applied the idea of Liapounov to electric circuit networks and others. J. K. Hale (n) and La Salle, (12) and Yoshizawa, (is) by following them, showed the existence of a bounded solution by constructing Liapounov functionals, and also discussed the asymptotic behaviour of the solution for a system of differential equations. Halkin (~4), Markin, (is) Phss (~e) extended Llapounov's theorems. 'Moisseiev(1:) introduced the idea of stability probability, as based on point-set theory. A motion is said to show intensive behaviour, if any solution can be extended so t h a t
[p(t,~,to) I > f l ( ~ ) e x p [ ~ ( t - t o )
],
t___to,
~ >0,
fl>O.
For a motion to show intensive behaviour it is necessary and sufficient t h a t there exist a Liapounov functional satisfying
V(x, t) < al[xl y,
I[?l > a~[x]r,
Y. l ~ o r ~ * ~
39
where a 1, a s, ? are positive constants (Krasovskii). For every motion, at least one branch, to admit an estimate of the form
fi~($) e x p (~lt - t01) < [p(t, ~, t0)l < fi~(~) e x p (~lt - t01), it xs necessary and sufficient t h a t there exists a function lz which satisfies
~ l x l ~ < V(x, t) < allxl ~,
a~lx] ~ < I~'(x, t)l < a'olxl"
for sufficiently small Ixl. Bohl, ~s~ Liapounov, ~l~ and Poincar6 ~9~ discussed the stability of the system dx~
- as~ ,x~ + a~2x ~ + ... + a~.x.
(s = 1, 2 , .
, n)
dt
with constant coefficients a~j. If the fundamental determinant formed by the coefficients admits only roots of real parts negative, then the non-perturbed motion is stable and the motion for which the perturbation is sufficiently small tends asymptotically to the nonperturbed motion. I f the fundamental determinant admits only roots of real parts positive, then the non-perturbed motion is unstable. Liapounov ~ treated the case of one zero-root and the ease of a pair of purely imaginary roots, while the remaining roots are all of real parts negative, as well as the case of double zero-roots in a posthumous memoir.C~0~ Moser ~2~ discussed Siegel's theorem. Bohl ¢~s~ and Harasahal cz~> treated the case of quasi-periodic coefficients. Cotton c~a~discussed the asymptotic solutions on the basis of integral equations. Moustakhiehev <~s~ derived sufficient conditions for the instability of an equihbrium of a t t a m f l ~ n i a n system, of which the characteristic equation admits two zero-roots Perron' s T h e o r y
Lefschetz ¢~) and Bellman ¢26)discussed stability under less stringent conditions. Perron ¢~6) explored the situation on stability further. Consider for real t __> 0 : dxs -
dt
-
aslX 1 + a~x~ + ... + a~.x. + qgs(x1 . . . . . x., t),
%(0,0 ..... 0, t) = 0
(s = 1,2 . . . . n),
where xs, as~, q~s m a y be complex, which admits the trivial solution xl = x2 . . . . . x. = 0. If we can assign a positive number a, suoh t h a t to any s between 0 and a a number 81, 0 < 81 < 8, can be found so t h a t any solution for which 0 < ~ Ix,(0)[ < 81 exists for all S
values of t ~ 0 and satisfies ~ [xs(t)] < e, then the trivial solution x 1 . . . .
= x, = 0 is
$
said to be stable and otherwise unstable. If there is a positive number 81 such t h a t any solution for which 0 < ~ Ix,(0)l < 8 ceases to exist once, necessarily for a finite value of t, or at least satisfies ~ Ix~(t)] = 8, then the trivial solution is said to be completely unstable. $
Between stability and complete instability there occurs conditional stability. If there exists a positive number a, such t h a t to any number 8, 0 < e < a, a number 81, 0 < 81 < 8 can be found, such t h a t a certain number o f - b u t not all - solutions with 0 < ~ Ix,(0)] < 81, $
continue to exist fort ~ 0 and satisfy ~ ]xs(t)l < 8, then the trivial solution is said to be S
conditionally stable.
40
Celestml Meehamcs m the Soviet Unmn
Ifq0is continuous for ]xll < a . . . . [x,I =< a, t __> 0 (a > O),and ff[~] ~ K(lxx[ + .. + Ix.I), K > O, t h e n q~ is said to satisfy the condition (A). I f ~/(Y.lx~l) -* o for Z Ix~l -~ o, t -. o, s
s
t h a t is, if to a n y positive n u m b e r we can assign (~, and T, such t h a t I~l _-< e(~lx~l) for s
Ix~l <= ~ , s = l,
., n,
and ~ [x~l > 0, t __> T~, t h e n ~v is said to satisfy t h e c o n d i t i o n (B) $
The conditmn (B) is satisfied if we have uniformly hm
for
g~
~,-.o . . . . .
- O,
or
lim
~
o Y.Ix~l $
Ix~l < a,
- 0,
s
s = 1,
,n,
Ix~l > 0. s
I f for a given positive n u m b e r ~ we can assign two positive constants 5~ and T~ such t h a t I~(x i,
. , x.,' t) - ~(x~',
, x .", t)l =< ~ (E Ix" - x~' J) $
for Ix~[ =< (~, t __> T~, t h e n ~ is said to satisfy t h e condition (C). I f a function satisfies (A) and (C), t h e n it satisfies (B). I f 9~ satisfies t h e condition (B), t h e n as t are one-valued and t h e characteristic roots ~, of Ilasj - 81jOll = 0 are uniquely determined I f x(t) defined m t o <: t __< t 1 satisfies
d~s
--~(x,t)
(8 = 1,
.,n),
dt
where ~ satisfies (A), t h e n for a n y two values of t in [t0, tl] we have
S=I
x=l
(I) Suppose t h a t b y a h n e a r t r a n s f o r m a t m n t h e system of differential e q u a t m n s is reduced to dxs
s- 1
- ~ x ~ + ~ c~lxl+~f,(x 1,
dt
.,x.,t),
(s = 1 . . . .
n),
~=1
and t h a t 9~ are of negative real parts. P u t --?=
max~(Qs),
C=max[c~l,
~ =
mm
1, 2(n
1)ff
1
if(n-
1) C = 0 ,
I f ~s satisfy (A) and ? I~)sl ~
~n
( I x l l -~ ~%1X21 ~- * " -~
~%n-1]Xnt])'
(8 ~--- l , . ,
n),
t h e n the trivial solution xs = 0 is stable, and a n y solution with t h e initial condition xS(0) sufficmntly small in absolute values tends to zero. limxs = 0
(s =- 1 . . . . n).
r---~o0
(II) I f ~s satisfy (A) and (B), and if Qs are all of negative real parts, t h e n x~ = 0 is stable and a n y solution x~(0) sufficiently small in absolute values satisfies hmx~ = 0 t ---~oO
(s = 1,
,n)
Y. H s o m , ~ x
41
( H I ) L e t 0~ be such t h a t ~(0s) - ? > 0 for s >_ k a n d ~R(0~) =< 2 for s < k, k ~ n. I f ~0~ satisfy (A) and
I~A _-< ~
4n
C(Ixd + ~lz~l + • • + o,"-~lz.l),
t h e n xs -- 0 is unstable. I f k = n, t h e n it is unstable. (IV) Suppose t h a t ~ satisfy (A) and (B). I f at least one of t h e characteristic roots ]s of positive real part, t h e n x~ = 0 is unstable If all roots are of positive real parts, t h e n t h e instabflxty ]s complete. (V) L e t ~R(Qs) > ~ f o r s > k < n a n d ~R(Os) < - ~ f o r s > k, w h e r e ~ > 0. I f ~ s s a t l s f y (A) and
t h e n x~ = 0 is conditionally stable. T h e a s y m p t o t i c b e h a v i o u r of t h e solution is described b y the t h e o r e m . If ~s satisfy (A) and (B) and if t h e real p a r t s of Q1. . . . . On are n o t all positive, t h e n t h e r e is for each solution x~, for which lira xs = 0, s = 1, 2, ..., n, an index 1 such t h a t lira log([xl] + . . -
+ Ix,I) = ~(0~)=<0.
~-~00
If ~s satisfy (C) further, t h e n conversely t h e r e exists to an index 1 for which ~(Ol) < 0 a solution of this nature, and such solutions for which lim log (Ix1[ +
-. + Ix.I) __< - r
< 0
form a family of just h a r b i t r a r y constants, where h is t h e n u m b e r of characteristic roots for which ~R(q~) <__ - r . Consider t h e equations -
0~,, + ~(x~,
. , x,, t),
(s =
1 .....
n).
dt I f ~0~ satisfy (A) and (B) and if ~R(QI) > ... > ~(Q,), t h e n t h e r e is for each solution x~ for which lma x~ = O, s = 1, ..., n, and index l, such t h a t t-*oo
for s # l, and /1
dx~.\
t_.®kX~ dt ] I f ~ satisfy (C) further, t h e n conversely t h e r e exists to an index l, for which ~ ( ~ ) < O, a solution of this nature, and such solutmns for l ~ j f o r m a family with just n - j + 1 a r b i t r a r y constants
Order Numbers A n u m b e r ~a, such t h a t
l ~ ~ l o g (Z IxA) = ~, t-.o0
~
S
42
Celestial Mechamcs in the Soviet Unmn
is called t h e order number (Ordnungszahl) b y Perron. The n u m b e r of the order n u m b e r s of linear differential equations
dx~
- ~ f~,(t) xi ~=1
dt
( s = 1 ..... n)
is just equal to n, if multiplicity is p r o p e r l y counted. L e t 71 -> 7~ -~ • • --> 7 , be t h e order n u m b e r s of t h e differential equations, t h e n t
~R[fz~(v)] dr.
7s => lim s=l
t-.co
~ J2=l o
Consider t h e a d j o i n t s y s t e m
dy~ dt
-
-T,/,,(t~
t
yt
(s = 1 . . . . . n),
t h e n t h e order n u m b e r s &l --< &2 =< ... =< ~, arc such t h a t ?,+&,>0
( s = 1 . . . . . n).
N e x t consider t h e generalized s y s t e m
~x. _ T, f.~(t) x, + ~,(x~ . . . . . x., t) dt s
(s = 1 . . . . . n),
(l)
and assume t h a t
dxs - ~/,,(t) dt s
x, + ~,(t),
(s = 1 . . . . . n)
(2)
admits a t least one b o u n d e d solution for a r b i t r a r y continuous b o u n d e d functions ~s(t). W h e t h e r (2) a d m i t s always a b o u n d e d solution is k n o w n f r o m the behaviour of the homogeneous s y s t e m (1) with ~s =- 0. If/st(t), 1 _< i _< s _< n, are continuous a n d b o u n d e d for t ~ 0, the necessary ¢ n d sufficient condition for (2) to a d m i t a t least one b o u n d e d solution for a n y choice of t h e continuous and b o u n d e d functions ~s (t) is t h a t / s t ( t ) should satisfy t h e conditions with t
;/,,(v) dv = H~(t), 0
t h a t for a n y indices s = 1 . . . . . n we should have either t
(A) eu'(t) is b o u n d e d and e ~'(t) f [e-H'(')[d~ is bounded, or (B) e u'(t) is unbounded, but t
0
e g~(t) ~ [e-H'(°[dv exists and is bounded. 0
I f (2) admits at least one b o u n d e d solution for a n y choice of t h e continuous b o u n d e d functions ~ps(t), t h e n for each b o u n d e d solution of t h e homogeneous equations (1) with ~ ---- 0 there should hold t h e relation
lira ~ Ixs(t)[ = 0 t._,c~ s = l
Y. H ~ o m ~
48
If (A) is satisfied for all s, then all solutions are bounded. If (B) is satisfied for all s, t h e n there is only a single bounded solution. I f k indices s satisfy (A) but n - k satisfy (B), t h e n the linear homogeneous equations (1) with ~ - 0 admit just k linearly independent bounded solutions. For k = 0 there is no bounded solution other t h a n the trivial solution $~ = 0.
Kinematic Similarity Let Mn be the set of all n x n matrices, whose elements are complex-valued functmns of a real variable t, continuous and bounded for 0 __< t < oo. For A, B e Mn we say A ,~ B, i.e. A is kinematically similar to B if there exists a matrix P with P, p - I and P e M , , such t h a t _ p - x [ p _ AP] -= B (3) for all 0 _< $ < ~ . The idea has been introduced b y Perron (~7) and extended b y Jacobsen, (~) and b y Cameron, (2°) Lille, (~) and Markus ¢8x) to the case of almost-periodic coefficients. Consider ~(=AX, AeM,. For each non-zero solution X~($) we define a characteristic number
lira t-~ log Ix,(t)] = ~,. t,..~ oO
Perron showed t h a t 2~ are bounded and t h a t there arc at most n distinct characteristic numbers of.4, and t h a t every matrix in M n is kinematically equivalent to a triangular matrix. For such a solution vector X(t) with characteristic n u m b e r ~ we define a type of A as
1,()t, X(t)) = l,(;t) --- lira [log IX(t) e-at]/log t]. !'--* oO
The set of all solution vectors of X = A X with characteristic n u m b e r ~ =< ~ , i = 1,..., k, forms a linear space E(~). E ( ~ H ) is properly included in E ( ~ ) . We make the convention E(~0) = 0. We define the type of a coset in the factor space E(~)/E(~_x) , taking the idcntity-coset to be of zero type. The collection of cosets of E(2~)/E(2~_l), whose t y p e =< rtj, is a linear subspace of E(,~t)[E(2~_I) called T ( ~ , r~j) where we set T(2t, rt0) -- 0. Let hi < ~ < ... < 4, be the distinct characteristic numbers of .4 e M , , with corresponding types ~x < ~ < .... ~ < ~ < .... ~ < ~ < ... The multiplicity of 2~, i = 1, ..., k, is m(2~) = dim ~(2~)/(E(2H). The multiphcity of ~ is m ( ~ ) = dim T(2~, r~)/T(2~, ~.~-1). The characteristic numbers 2~, the types r~, and the multiplicities m(2~), m ( ~ ) are the invariants of kinematic similarity. B y a non-singular coordinate transformation Y = P - i X , P e M , , we get ~" = B Y , B = - P - ~ [ P - .4P] This is kinematic similarity. The solution space for X = A X is isomorphic to the solution for 1~ = B Y and the corresponding solutions X(t), Y(t) are of the same type with the same characteristic numbers.
Strong Stability Consider a canonical system dY dt
-
IH(t)
Y,
Y =
l(yl,.
,y.),
(4)
44 where
Celestial Mechanics m the Soviet Umon
,:(
--I n
)
2n '
l,j=l
and hu(t ) = h jr(t) are piecewise continuous real functions with the common period eo A canomcal system is called strongly stable if it is stable and if there exists an e > 0, such t h a t for any real symmetric matrix H(t) wath pieeewise continuous elements satisfying
R(t + co) =//(t),
[IB(t) - H(t)lJ <
all solutions are bounded for t -~ ~ , where [[ I[ denotes the norm. Strongly stable canonical systems form an open set in the set of all canonical systems. We find the conditions under which there exists a sequence of symmetric matrices H(t, ~) subject to H ( t + w, v) = H(t, v), and plecewise continuous in t and ~ such t h a t Y= IH(t,v)Yis strongly stable for all 0 < ~ < 1, and such t h a t H ( t , O ) = H l ( t ), H(t, 1) = H~(t). Such systems are said to belong to one stability domain The necessary and sufficient condition for the matrices of solutions of canomcal systems to belong to one stabihty domain, accordingly the topological structure of the stability domain, has been studied by Krein, (a2) ¥abuboviS, (aa) Gelfand and Lidskd. (34) We normahze the solution such t h a t Y(0) = E2,. We have Y(t + neo) = Y(t) Y"(w), n = 1, 2, ..., ff the elements of Y(t) are bounded in 0 _< t _< w. For stability all elements of Yn(w) are necessarily bounded for n -* oo ; or all elementary divisors of Y(eo) should be linear with unit moduli, t h a t is, Y(eo) should be reducible to a diagonal matrix. Y(co) is a monodromy matrix for the monodromy group. The roots of the characteristic equation are called multipliers. Krasnoselski (aS) has given a method for computing the multipliers. Each stability domain is determined by one of the 2 n possible types of relative distributions of multipliers on the unit circle, and b y the serial number of the stability domain AT, -- ~ < N < ~ , which is called the index of rotation of the system. A matrix Y(t) satisfying Y*(t) I Y ( t ) = I for all t ]s called symplectw. E v e r y curve Y(t) in the group of real symplectic matrices m 0 < t _< co possessing a piecewase continuous derivative satisfies a matrix equation of the form (4) in [0, o~]. The symplectic matrix Y is a matrix of stable type ff and only if it can be represented in the form Y = G R G -1, where G and R are real symplectic matrices and R=(
cos0 s i n 0 ~
-sin0
cos0]'
where 0 is a real diagonal matrix of order n, whose diagonal elements 01, ..., 0n satisfy [02[ <:~, s = 1, ..-, n, and 0~, ~:0s,,, 1 < s ' , s " < n . The group of real symplectic matrices is homeomorphic to the topological product of the circumference of a circle and a simply-connected topological space. Two continuous curves Yl(z) and Y2(z), 0 _< z < 1, with common end-points, lying entirely in a connected component of the matrices of stable type, can always be continuously deformed one to the other The topology has been studied by Segel (3e) m a dtfferent manner (B) Control I n order to maintain the stability of the origin x = 0 we use a compensating mechanism caned a control. (aT-ag) Let ~ be a scalar parameter and ~ --- 0 be a state of no control. The new equations of motion, corresponding to the lineanzed system [t = A x and obtained from y = Y(y, t), are x = ~) = A y + ~b, ~ = /(a), a = c r y -- r~,
Y. HAOmAaA
45
where b, c are n-vectors and r a scalar, while a is introduced as an intermediary and ](a) is the characteristic, assumed to be continuous, of the servo-mechanism. Assume t h a t ±ao
,~l(a) > 0 for ~ 0¢ O, l(O) = O, f l(~) da = + oo. OV IAapounov's funetional 12" -- ~ + Y • grad V is t a k e n of the form
V ( x , c/) = X T" B x + f ](~)d(r,
B > 0,
B r -- B,
0
where V is positive-definite over the whole space (x, a), such t h a t V(0, 0) = 0 and V and its first-order derivatives are continuous in the whole space. Geiss and Abbate (4°) computed the optimal quadratic estimate of the domain of attraction of an equilibrium solution of a quasi-linear differential equation
= A x + [(x),
1(0) --'- O,
where A is a stable matrix. I f ~ = X, X(0) = 0, whereby the X satisfy the relation lim &(Xo, to, t) = 0, for x(xo, to, to) e/1, t __> t o, and for all x o in G~, when F e G is given in t-*oO
a closed bounded subregion G~ ~ G; and if the non-perturbed motion is asymptotically stable, t h e n G~ is called an attractive domain around x -- 0 (Krasovskh).
Stability in A~stract Space Fomin (aI) considered an operator differential equation with distributed p a r a m e t e r in a separable Hilbert space. 1)
dt
= [z +
x,
= 0t,
where F is a self-adjoint operator, ~ is a symmetric operator with period 2~, F-loEf(T) is supposed to be uniformly continuous in [0, 2~] with uniformly continuous derivatives, and 0 characterizes the perturbation frequency. The vector solution X(t), specified in terms of the solution x(t) b y x(t) = X(t) x(0), can be represented b y X(t) = U(t) + V(t), where the operator U(t) is unitary for any t and V(t) is completely continuous for any t. Almkvist ¢~) discussed the stability of the solution of du
dt
- A(t) u(t)
(t > O)
m a separable Hflbert space, where A(t) is symmetric and periodic with period unity. K a t o (4a) studied a linear differential equation in Banach space. Ura and K i m u r a (44) discussed the flow of an incompressible fluid in Gottsehalk and Hedlund's topological dynamics.(45) Arnold (46) showed t h a t if two vector fields a and b in M of divergence zero commute, i.e. (a, b) -- 0, and are not eollinear anywhere, i e. a • b ~= 0, then almost all trajectories of the field a are closed or everywhere dense on the analytic tori T 2 immersed in M. U r a (47) further discussed the stability of trajectories in abstract spaces.
Y. HAGmA~
47
37. R. B ~ , Introduction to the Mathematical Theory of Contrd Proc~aes, Academxc Press, 1967. 38. D. SwoP.v~B, Optimal Adaptive Control Systems, Academic Press, 1966. 39. G. L E ~ (Ed.), Twpic~ ,n Optimiz~ion, Academic Press, 1967. 40. G. It. G~Iss and J. W. AB~x~, Grummer Res. Dep. Rep RE-282, Aero-Astrodynamics Lab., Marshall Space Flight Center, Hunts, 1967. 41. U. N. I~OM]~, Do/d. Akad. Na~d¢, 168 (1965) 830. 42. G. hv.uwvisT, Pacific J. Math. 16 (1966) 383. 43. T. KATo, Comm. Pure Ap~pl. Math 9 (1956) 479. 44. T. U~A and I. Ku~m~, Proc. Ja~an Acad. 40 (1964) 703. 45. W. Go~sc~Ar.w and G. H~DLV~D, Topologlc~d Dynamics, Amer. Math. Soc., 1955. 46. V. I. AJt~OLD, Doklady A ~ . Nauk, 162 (1965) No. 5; C. R. Acad Sei. Paris, 260 (1965) 5668; 261 (1965) 3719, 17. 47. T.U~,A, ~unkciala~ Ekvacio?, 2 (1959) 143; 9 (1966) 171, Contrtbutton to Di~erential Equations, 8 (1965) 249. 48. K. M. M o u s ~ ' A ~ C ~ v , B~uU..Inst. Teor. Astr. 11 (1968) 453.
No~ In addition to the above-mentioned advances, the ergodic theory has further been amplified in the Soviet Union through the introduction of relevant concepts in the fields of entropy, spectra and KSystems, by Kolmogorov, Sinai, and Arnold; see also the new volume V (MIT Press) of Y. Hag~hara's
Celesttal Mech~nic~.
Y. HAOr~A~A Department of Astronomy, University of Tokyo, Japan
INTRODUCTION Apart from the recent progress of Celestial Mechanics in the survey of periodic orbits in the restricted three-body problem and in the Trojan motion, not to speak of the modern development in the theory of motion of artificial celestial bodies with moderate eccentricities and inclinations, and apart from the development of modern computer techniques applied to the motion of in~vidual stars in cluster dynamics, and even adaptedfor analytical treatment, the main problems in Celestial Mechanics has been cultivated in the Soviet Union, such as: ( I ) - the capture of a distant star by a binary system (p. 15), I I - t h e existence of, quasi-periodic solutions (p. 20), ( H I ) - a n approximation theory with error estimation (p. 29) and ( I V ) - t h e stability theory (p. 36), all of which have opened up fresh aspects in Celestial Mechanics.*
I. PROBLEM OF CAPTURES Since Yu. Schmidt (1) proposed a new cosmogonical theory on the origin of the solar system in 1947, various discussions have been published on the capture of planets b y binary systems. Consider three bodies P0, P1, P~, denote the distances PoPx, P1P~, P~Po, respectively, b y r 2 --- r, to, rl, and the distance of Ps from the centre of mass G of P0, P1 b y ~. Suppose t h a t r 1 -~ ~ for t -~ - ~ while r 2 remains finite, and t h a t there exists an instant T* and a value R > 0 such t h a t 0 < r~ < R for all t > T*; then we say t h a t there is a capture of P2 b y Po. Numerical examples for captures have been computed b y several astronomers, Merman, (~) Proskurin, (s) Sizova, (4)Khrapovitskaya, (5)Kochina, (6) Merman and Kochina, (~) and for temporary captures b y Proskurin, (s) and Proskurln and Runjazewa. (°) Years ago Becket (1°) and recently Szebehely and Peters (n) computed numerical examples of captures with exchange of bodies, t h a t is, P~ is captured by Po and P1, but P1 recedes to infiuity later b y leaving Pz in a finite distance from Po * When my publications ~qtability in Cde~ial Mechanics (1957) and the chapter "Stability in the Solar System" in vol. IV of The ~dar System (edited by Kuiper and Middlehuret, 1963) were printed, the new literature of Soviet Russia had not been accessible to me. Now that it has become available, I am thrilled to see the brilliant successes that have been achieved; and I wish to dedicate this supple. mentary article to my Soviet colleagues in admiration of their outstanding recent advances in Celestial Mechanics. (Y. H.) 15
16
Celestial Mechanics m the Soviet Union
Poinear6, c12) Sehwarzschild, ¢18) a n d y o n Zeipel, ¢14~on t h e o t h e r hand, showed t h e impossibility of eternal c a p t u res,and t h e existence of ergochcity of their m o t i o n on the basis of conservative d y n a m i c s b y a n a l o g y to t h e m o t i o n of liquid molecules of an incompressible fluid enclosed in a vessel of finite volume. Chazy, C15) b y following up their ideas, showed t h a t a m o t i o n of hyperbolic-elliptic t y p e for t -~ + oo c a n n o t become b o u n d e d or oscillatory for t ~ - o o , a n d vice versa. According to Chazy Cle) a m o t i o n is of the " h y p e r b o h c elliptic t y p e " if one of three bodies recedes to infinity, while the o t h e r two are at a finite distance; of " b o u n d e d t y p e " if the three m u t u a l distances are all bounded, of " h y p e r b o l i c t y p e " if all t h r e e m u t u a l distances become mfimte; and of "oscillatory t y p e " if at least a n y one of t h e bodies oscillates indefinitely with mfimte amplitude. T h u s t h e possibility of captures has become a serious p r o b l e m in Celestial Mechanics. K h i l m i ¢17) defined a capture to be such t h a t r 0 > R1, r 1 > R~, r2 > R1 and t h e three velocities are hyperbolic for t = t~; and t h a t r 2 < R 2, 0 > R~ a n d t h e velocity of P1 relative to Po is elliptic, b u t t h e velocity of Pz relative to G as hyperbolic at t 2 > tl, w h e r e R 1 represents the largest dastance of t h e bodies from a n y star of a g a l a x y and R , t h e m u t u a l distance of a binary star P u t r m = m m { r o , r 1,%},
M* =mill{morn l + mlm2, M* = m a x
{
¢ = m i n { # o , rx,r2},
mlm2 + mom 2, mora2+morn1}, morn,
,
re,m2
,
morn,
},
M = mo + ml + ra~, M *-
M*
m 0 -~ m I m 1 + m~ m o + m~ M* Consider two cases" (A) if r(tl) > 0, ~(tl) < - ~/[8M*/rm(tl) ] at t h e initial epoch tl, t h e n to, rl, rz increase to infinity for t -* - oo, (B) if for t h e positive e n e r g y - c o n s t a n t h of t h e t h r e e - b o d y s y s t e m we can find R > 0 a n d ~ < R a f t 2 > tl s u c h t h a t
%(t~) < R, 0(t2) > 2R, ~(t2) > e(t2 )
16M ~(t2)
2_2_h ~1
2m2(mo + ml) /~l(R - ~)
0,
(too + m l ) ~ 2 M
t h e n P2 reeedes m o n o t o n o u s l y to infimty for all t > t 2, b u t r 2 remains smaller t h a n R, and the motaon as called hyperbolie-elhptic. I f for t h e m o t i o n of t h e three-bodies (A) holds for t 1 and (B) holds for t2, t h e n there is a capture. K h i l m i p r o v e d t h a t the measure in t h e phase-space of the point-set which represents t h e imtial state leading to a capture is n o t zero. K h i l m i ' s condition for a weak capture, so t h a t P~ recedes m o n o t o n o u s l y to infinity from G a n d the distance PoP1 does n o t exceed R in t h e case h > 0, is t h a t ~(0) > 2R, Q(0) > 0, ½ 0(0) < ro(0 ), rl(0 ) and [~(0)] 2 8M* 2 2morn1 - - > h + - e( 0 ) ]~1 /~1R I f the three m u t u a l distances t e n d to infinity as t ~ oo, t h e system is said to be eompletely dissolvable. W h e n a b o d y enters t h e sphere of radius ~ a r o u n d a n o t h e r body, it is called an c¢-approaeh. Khilml derived the conditions for each of these phenomena, and extended t h e m to the ease of n gravitating masses in order to a p p l y this to cosmogony. Sitmkov Cls~ and Alekseev (19) derived conditions for some particular cases. M e r m a n (~°~ criticized a n d supplemented t h e propositions of Chazy ¢2~) t h a t a hyperbolicelliptic m o t i o n for t --, + co can neither ehange to a b o u n d e d or an oscillatory motion,
Y. HXGIHX~A
17
nor to a hyperbolic motion for t -* - o o in t h e case h > 0 There are several numerical examples against these propositions According to Merman, t h e impossibility of a capture for h > 0 has n o t been estabhshed b y Chazy, because Chazy transferred the expansions of a m o t i o n from t h e d o m a i n of t -- + oo to a domain t = - oo, which is generally impossible. C h a z y ' s proof on t h e imposslbihty of a capture for h < 0 m a y be true, a l t h o u g h this m a y be Impossible to exist with positive measure for a bundle of trajectomes for h < 0, which are b o u n d e d for t -~ + oo b u t hyperbolic-elliptic for t ~ - oo, or which are hyperbolic-elliptic for t -* - o o b u t oscillatory for t -* + oo, or vice versa Merman (s°) discussed at first the characteristics of hyperbohc-elliptic motions for h < 0 and t h e error estimates. B y the energy integral we have min {ro, rl, rs} <
s,/Ihl, s~
= morn 1 -{- m l m ~ -b m s m o.
The quantity I -
(too + m l ) m s @ + M
morn1
rS
mo + m1
is either b o u n d e d or n o n - b o u n d e d for t ~ + ~ . If I -~ c~ for t ~ ~ , t h e n ~ becomes mfmite. T h e n Merman p r o v e d C h a z y ' s t h e o r e m s on a more rigorous basis, a m o n g others t h e t h e o r e m s t h a t t h e probability o f a capture m t h e region h < 0 is equal to zero, and t h a t if the area velocity constant / > 0, h < 0, and ff I is sufficiently small, t h e n the motion is hyperbolic-elhptie b o t h for t - , + oo and t -* After discussing various hyperbolic approaches (ss) and parabolic approaches(~) to approximate t h e three-body problem, Merman(u) gave four necessary conditions for a parabohcelhptic motion, two necessary conditions for a parabolic-hyperbolic, five sufficient conditions for a capture, one necessary condition for a capture, a necessary and sufficient condition for t h e existence of a parabolic-elliptic motion, and a necessary and sufficient condition for a t e m p o r a r y capture in the hyperbolic case. Alekseev (~5) studied the perturbations to a hyperbolic m o t i o n in a different m a n n e r and gave examples for an exchange of t h e distant bodies in t h e case of a capture. (se) M e r m a n (st) derived t h e criterion of a capture at first for a restricted problem (~) and t h e n for t h e general t h r e e - b o d y problem. (se) I n the former case there is m o -- 0; and in t h e hyperbolic m o t i o n we have 1),
ro=a(ecoshH-
t a n ~-- = 2
~(_i_oo)=2tan_ 1 /(e--l~
/(e-
~/\e
l~coth
+ lJ
H
2 '
nSaa
= m, + ms,
~(--~)=2~--tan-' /{e--l~.
Merman has also given eight necessary conditions for a hyperbolm-elliptic motion, one necessary condition for hyperbolic motion, and five necessary conditions for a capture. I f two n u m b e r s R > 0 and ~ > 1 are such t h a t to(0 ) >_ ~R, to(0 ) < air1(0), and m 1
r(0)
[1"(0)] z >-- m 1
2
R
2B
q-
na3(e s -
m a x {[e sin ~ - ~]~ [e sin ~ -- ~]0_ o0}, 1) 3/2
t h e n t h e motion is hyperholic-elhptic and r < R, where a1~
(~1--'-) -1 , 1 --
2B--msx/(SmlR
) . ( s a ~ q- a~ -{- ai)
18
Celestxal Mechamcs m the Soviet Unmn
From these conditions it is seen t h a t a capture can occur only for an extremely close approach. P u t . ml A =
[i(0)p
r(0)
+
2
~(0)
- ~o(0)
m1 + ms
+ ~(0)
~/(eS -
1) - ~o(0)
e
+
ml B ---- r ( 0 )
[r(0)]S -{-
- -
2
ms
{-- ~(O)[, na
m1 + ms
+ ~o(O)
]
+ ~(o)
e
I na
to(0 )
'
~ / ( e s _ x) - ~o(O)
]
+
'
e
ro(0 )
where ~, ~ represent the rectangular coordinates of P3 relative to G. At t -- 0, if A > 0 and B > 0, then the motaon is hyperbolic-elliptic; ff A < 0, B < 0 it is hyperbolic; and if A > 0, B < 0 or A < 0, B > 0, t h e n a capture can occur. A sufficient condition for a capture can be obtained b y combming these conditions. The exact conditions would be slightly different from these. For getting better insight into the problem, Merman (sS) provided new proofs of Birkhoff's theorems (2" on the behaviour of motion in the three-body problem and extended some of the theorems. Sibahara and Yoshida (") generalized some of the theorems. Merm a n (8°) proceeded to the qualitative s t u d y of the problem. H e considered in the general problem the possibility of resolving the motion into two almost independent motions of approximately Keplerian character, and found the distance and the velocity of the distant body from the other two at finite distances. More general and simpler criteria t h a n those of Merman are desirable. I f for h < 0 we have
K/s > =
27MsS~ ~3, m = m i n { m o , m l , m s } , 4 x/[m(M - m)]
K=
-h,
then we get for all t
Merman derived the m a x i m u m Qm and the m m i m u m ~m of ~, as well as the upper bounds of o, ]~[ and Sitnlkov (al) proved the existence of hyperbolic, hyperbolic-elliptic, and bounded motions for m n - - ml. Sibahara (Ss) has given a sufficient condition for Lagrangian stability b y sharpening Merman s theorems. H e also derived a criterion f the exchange of the distant body m an encounter process, and examined (u) the cosmogonical theory of Lyttleton (s,) Merman CSs)criticized on the basis of his inequality for K p the smallness of the convergence radius of Sundman's series (a~) derived b y Belorizky, (88) and estimated the error of Sundm a n ' s polynomial series in Sundman's regularizing variable. Let us consider m,,
'
(83)
/(co) = ~. a,eJ,
or
/,,(w) ----Z c~")a,w',
1=0
1=0
where c~") are the coefficients of the expansmn
m.
g.(o~) = ~ - ( "~,) ' c ~ , ~=o
1
g ( ~ ) = ~ ,
1 -co
Y. H ~ o ~
19
and m~ -- (~ + 1)~ - 1 (when we need ~ terms in the expansions); then we get
I1(:~) - 1.(~)I <
i
2~t~'(-
1)
1- (z/m)
~"
d~
max --
2:~
r'
where I" is the length of the integration contour ]~. Brumberg ~ ) applied this summation procedure to the solution of the three-body problem, by referring to the method of solution of differential equations by Painlev6 ~°) and Picard. ~ ) The result has been compared with the numerical solution of Zumk|ey ~ ) and with the solution by Sundman's series. ~ ) On the other hand, Cesco(~) started vath the summation method of Borel, ~t° and Zelmer (~) with t h a t of Euler-Knopp and found t h a t the convergence radius can be made smaller t h a n t h a t of Sundman's series by such an artifice. Such polynomial expansions are considered to be convenient for the numerical treatment of the problem. I t is mentioned in this connection t h a t Petrovskaya (~e) derived the convergence radius of Hill's series in his intermediary orbit to be Inn] < 0.21 .. on the basis of the infinite nonlinear analysis of Wmtner, (~7) who gave Inn] <0.0833 .... Further Petrovskaya (~) discussed the convergency of the Fourier series representing the disturbing function in the restricted three-body problem and found the domain of convergence in the complex planes in terms of undisturbed orbital elements. Years ago Liapounov~ ) derived the upper bound of convergence radius for Hill's series 1 to be nn < ~ b y a majorant method of his own. Merman ~ ) improved the inequalities and deduced m < 0.18. Rjabov (~) further estimated m < 1/~/15 = 0,258 ... by Liapounov's method. On the other hand, Petrovskaya (~) constructed a Poincar~ periodic solution of the first Sort in the form of a series m m for the satellite three-body problem, and derived the upper bound of convergence for m and for the mass-ratio.
REFEREI~CES*
1. YU. SCWM~T,Dold. Akad. Numb, 58 (1947) 213. 2. G. A. MEP~A~, Dold. A k ~ . Na~k, 86 (1952) 727; •Tu//. 1 ~ . Teor. A~r. 5 (1953) 373. 3. V. F. PROS~XN, B~uZl. Inst. Teor. Aatr. 5 (1953) 429. 4. O. A. SIzovA, Dokl. A k ~ . Nauk, 86 (1952) 485. 5. G. E. gm~POVITS~rA, BjUU. Inst. Teor. A~r. 5 (1953) 435. 6. N. G. Koam~x,/~Tu/l. In~. Teor A~r. 5 (1953) 445. 7. G. A. M ~ a ~ and N. G. K o c ~ A , B~u//. Inst. Teor Ast~. 6 (1956) 85, 349. 8. V. F. PBOSXURr~,B~uU. Inst. Teor. Astr. 5 (1953) 429; 8 (1962) 264. 9. V. 1~. PROSKU~I~and L. Yu. RuI~JAZ~WA,B~U. /n~. Teor. A~r. 7 (1959) 287. 10. L B~CKER, M. N. R. A. S. 80 (1920) 590. 11. V. Sz~.B~.~Y and F. P ~ R S , AsSt. J. 72 (1967) 876. 12. H. PoI~c~R~, Le~ M~thodes ~o~vd~e.s de h~ M&ani~ue C~,e.~e, Tome HI, 1899, Gauthter-Vdlars, Paris; C~lcul des ProbabilitY, 2nd edn., chap. XVI, 1912, Gauthier.Vfilars, Pans. 13 K. SCHW.~S~----.~, Astr. Nachr. 141 (1896) 1. 14. H. vo~ Z~I~r~, Bu//. A~r. 22 (1905) 449; V. M~t~a~, J. des Obse~t~urs, 20 (1937) 121. 15. J. C~azy, C. R. Acad. 8d. Par~, 179 (1924) 1307; 197 (1933) 1193. 16. J. C~*zv, A~n. Ec~e Norm. ~ p . , (3) 89 (1922) 29. 17. G.F. gm~.~L Doldady Akad. Nauk, 62 (1948) 39; 71 (1950) 847; 77 (1951) 589; 78 (1951) 653; 79 (1951) 419; Qualitative Methods in the Many-Body Problem, 1961, Gordon Breach, New York. 18. K. Sm~uxov, Ma~. ~bo~. 82 (1953) 693.
20
Celestml Mechanics in the Sowet Unmn
19. V. M. ALEKSEEV,DO~:~.Akad. Nauk, 108 (1956) 599. 20. G. A. M _ ~ , B~ull. Inst. Teor. Astr. 5 (1954) 594. 21. J. C'~xzY, J. de Math. Pure Appl. 8 (9) (1929) 353; Bull. Sc*. Math. (2) 56 (1932) 79; Bull. Astr. (2) 16 (1952) 175. 22. G. A. M_Em~x~,.B~ull. lnst Teor Astr 6 (1955) 73 23. G. A M E ~ , JB~ull.Inst Teor. Astr. 5 (1954) 606. 24. G. A. M~mwa~., Astr. Zhur 8() (1953) 332; B?ull. Inst. Teor. Astr. 5 (1953) 325. 25. V. M. AL~KS~V, Dokl. Akad. Nauk, 1(}8 (1956) 599; Astr. Zhur. 88 (1961) 325, 726, 1099; 89 (1962) 714, 1102. 26. G A. M~.~MAN, Dokl. Akad. Nauk, 99 (1954) 925. 27. G. I). B r m ~ o ~ , Dynamw,al ~ystems, Amer. Math. Soc., 1927. 28. G. A. ~ A N , JB~ull.Inst. Teor. Astr. 6 (1955) 69 29 R. S ~ m ~ x and J. YoSmDA, Pub. Astr. Soc. JaTan, 15 (1963) 87. 30. G. A. M E ~ , JB~ull.Inst. Teor. Aar. 6 (1958) 687. 31. K. SIT~OV, Dold. Akad. iVauIc, 188 (1960) 303. 32. R. S~A~A~, Pub. Astr. Soc. Japan, 18 (1961) 229. 33. R. S n ) ~ A , Pub. Astr. ~oc. Japan, 14 (1962) 90. 34. R. S ~ A ~ a ~ , Pub. Astr. ~oc. Japan, 18 (1961) 108. 35. R. A. LYTT~TO~, M. N. R..4. S., 96 (1936) 559. 36. G. A. ~ , B?ull. Inst. Teor Aar. 6 (1958) 713. 37. K. SU~D~X~, Acta Math. 86 (1912) 105. 38. D. B]:/~o~z~Y, J. Ob* 16 (1933) 109. 39. V. A. B~V~BV.~, ~B~ull.Inst. Teor. Astr. 9 (1963) 234. 40. P. P~I~L]~V~, Com. R. Acad. Sc~ Paros, 128 (1899) 1505 41. E. I~C~D, Tra~td d'Analyse, Tome II, chap XI, 1925, Gauthier-Vfllars. 42. J. ZUMK~Y, Astr. Naehr 272 (1941) 66. 43. R. P. C~sco, Ob,. Astr. Unw. Nae. l_,a Plata, set Astr. ~1 no. 14 {1965). 44. E. BOREr., Lefons sur les ~dmes Dwergentes, 1901, Gauthmr.Vfllars. 45. G. K. Z v . L ~ , Summatum Methods in the Two- and Three-Body Problem; Thesm, Umv. Br~tmh Columbia, 1967. 46. M. S. P~TROWS~YX, B?ull. Inst. Teor Astr. 7 (1959) 441,9 (1963) 257. 47. A. W ~ R , Math. Z. 24 (1925) 259, 80 (1929) 211; `4met J. Math. 59 0937) 795. 48. M. S. P~TROVSK~YX,JB~ull. Inst. Teor. `4,tr. 1(} (1965) 385; 11 (1968) 411. 49. A. M. L ~ o v ~ o v , Trud~ Otd Fys Nauk Absh. L?obit. Estestvoznan, 8 (1896} 1 = L~apounov ~obranie ~ochinsnii, I, 418. Akad. Nauk USSR, 1954. 5(}. G. A. M ~ x ~ , JB~ull.Inst. Teor. Astr. 5 (1952) 185. 51 J. A. I~,~BOV, B~ull. Inst. Teor. Astr. $ {1962) 772. 52. M. S P~TROVSKAY~,B~ull. Inst. Teor. Astr 8 (1962) 733 * l~ote: references appear at the end of each section.
I I . QI~ASI-PEI~IODIC SOLVTIO:NS K o l m o g o r o v (1) m n o n - d e g e n e r a t e cases a n d A r n o l d (s) i n d e g e n e r a t e cases of H a m i l t o m a n d y n a m i c a l systems, i n whmh t h e t h r e e - b o d y p r o b l e m is included, succeeded m p r o v i n g t h e existence of quasi-periodic solutions. L e t / z 1 . . . . /t m be non-zero l i n e a r l y i n d e p e n d e n t n u m b e r s a n d Uj be r a t i o n a l f u n c t i o n s of sin2xepjt, c o s 2 z p j t , j = 1 , 2 . . . . m. W e assume t h a t U 1 + U s + • + Uj + • is u n i f o r m l y c o n v e r g e n t i n t h e whole d o m a i n T of t, t h a t is, for a g i v e n e > 0 we h a v e IU,+I + "'" + Ut+s-ll < s for l > n , s > 0 for all t e T ; i n o t h e r words, we assume t h a t this serms represents a f u n c t i o n ~v(t) i n T a n d t h a t ~ > 0 can be f o u n d so t h a t I~v(t 4- v) ~v(t)l < 8 for all e > 0 a n d t, T E T, a n d that/z~v, i --- 1 . . . . m, differ a t m o s t b y ~ from t h e corresponding integers. T h e n t h e r e exists a u n i f o r m l y c o n t i n u o u s f u n c t i o n ~ ( x 1. . . . xm) -
Y. HAomARA
21
of period 1, such t h a t ~ l t ..... #=t) = ~(t) for all t. ~(% 8) is called the translation number. Suppose t h a t there are given e > 0 and a one-valued function ](xl, ..., xm) , which is continuous everywhere and periodic with period 1 for all the variables in the argument. Then we can find a uniformly and absolutely convergent trigonometric series ZA~'1 ~'.. cos (2rrn 1 xx - slur) cos (2rrn,x~ - 8,~r) ... cos (2gn,.x,. - r,.t) (81 . . . .
e m ----" 0 ,
½;
n I .....
nra --'-- 0 , l , 2 ,
. ),
such t h a t this series differs from [ at most b y e everywhere. Such a class of functions is called q~i-periodic and was s m d m d b y Bohl (8) and Esclangon. (4) Thin class of functions is a special case of Bohr's almost-periodic functions. The necessary and sufficient condition for
F(x) = f [(x) dx 0
to be almost-periodic is t h a t F(x) should be bounded. There exists the m e a n value of an almost-periodic function such t h a t T
0
The Fourier representation of the almost-periodic functions is unique and converges in the mean N
lim M {ll(~) where
~ A.e'/(-1)~"~P} = O,
/V
~. IA.P - M{If(x)p}
(Parseval).
n-----1
Consider a dynamical system
@ dt
OH
@
Op
dt
---,
-
~H
--,
Oq
l o , = (Pl . . . . p . ) , q = ( q ,
. . . . . q.),
(1)
with the a n a l y h c Hamiltonian
H(p, q) = Ho(p) + / d / l ( p , q) + ..., where Hi(p, q) is periodic with period 2~z in each q. For # = 0 we get
dp = 0 , dt
dq =o~(p), dt
o ~ - - - -OH (~1
..... w.),
Op
and every torus p -- eonst, is invanant. I f w is incommensurable, t h e n the motion is quasi-periodic with frequencies w 1 . . . . . ~o,. The trajectory [p(t), q(t)] Falls the torus everywhere densely. I n order t h a t the terms depending on q of order # be eliminated from the Hamiltonian
Ho(~) + #m~) + p~l(p, q) + . . . . Mo(p') + ~n,(p') + ~ F 0Ho ~a + h i ] + l op Oq _l
•
• 4 ~
22
Celestial Mechanics in the Soviet Union
where a bar denotes the average over q, after the Birkhoff transformation (6) ~S p = p' + /.t.-~-q
~S -= q + ,u Op '
S(p; q) = ~ Sk(p') e "/(-~)(~ ") k¢0
( k ' q ) = klq 1 + ... + k,q,,
it is necessary t h a t to, --~-/ ~S + /~t = 0 '
or ~k(~') --'~ N/(-- 1)hk(~D') , (to" k)
Ht = E htc'~/(-l'(k a),
(2)
k,o
The motion [p~(t), q'(t)] with the transformed Hamiltonian
it(p, q) = H'o(p') + ~H'~(p; q') + ... differs b y order/u from the motion represented by the H a m l l t o m a n Hg(p') during the internal t ~ / ~ - t . B y repeating this operation we arrive at the invarlant torus p(s)(p, q) = const, filled everywhere densely b y the trajectories of quasi-periodic motmns wath Hamiltortian H(p, q) = H(oS)p(s) up to the order/~s. By diophantme approximation almost every vector to = (tot . . . . ~o,) satisfies the inequalities
{(to" k)l -> Klk{ -("+2),
Ik{ = Ik,{ + .- + {k.{,
(3)
for all integers k :~ 0 for some K(to) > 0. The small denominator (to • k) can be estimated from below b y the degree of k. Let H , be analytic for 1~ ql < P and thus IH,(P, q){ < M. Then we have {hk{ _< M e - I ~l~. B y (3) the coefficmnts S k = ~ / ( - 1)h~/(to" k) decrease in geometric progression almost as fast as hk since for a n y 5 > 0 we have [Ski < ~ M L
8_lk{(0_~)'
L = const.
Hence the series S converges for {~ q[ < Q and we have
I~l <= ~
ML
~.+-----~-;
(4)
for a smaller region ]~ ql =< ~ -- 2 ~. Thus (2) converges almost for allto. B u t the functions thus defined depend on p everywhere discontinuously, and the convergence is doubtful I t is wor¢h mentioning t h a t Sibahara (6) showed t h a t the series is convergent if the denominator tot[tea for n -- 2 is not a U-number of Mahler (7) or not a Ihouville's number; and Wintner (s) showed the almost-everywhere convergence. A transcendental number ~ ~s called a LaouviUe n u m b e r if we can find a rational fraction such t h a t for a n y integer we have
t
~_
ga
<
1
h--~
ha>l'
where ga/ha is the 2-th convergent of She continued-fraction representation of ~. I n order to avoid the difficulty of small denominators, Kolmogorov aimed at only one invariant torus T~. of the perturbed motion, on which a quasi-periodic motion with frequency to* occurs. The set of frequencies to* satisfying (3) m fixed m advance, and the
Y. HAOrWARA
23
torus T~, is determined in the neighbourhood of the corresponding toms of the unperturbed system p = T* +/~P~ + "", ~Ho/~P* = co*. In (2) we replace Wo(p) ~H° by w*. The additional term which appears is of order/~ 2 for IP - P*[ ~ F, and the ~p new approximation is shown to be of order M s. The procedure is repeated b y Newton's approximation method of tangents. The rapid convergence overcomes the influence of small denominators and the series is convergent for the majority of the values of w. Thus an invariant torus T~. can be found. Accordingly, if we assume t h a t -
ii0o1
then there are,in the small neighbourhood of any point p, points at which the frequencies w are commensurable as well as points at which w(p) = w* can be estimated by (3). On certain t o n p = constant, canonical equations with the Hamiltonian Ho(p) determine everywhere dense quasi-periodic trajectories, but on the other tori t h e y do not. l~or small perturbations H = Ho(p)+ p h i ( p, q),/J < 1, the majority of invariant tori with incommensurable frequencies (o* satisfying (3) with a fixed K do not vanish, and are only somewhat deformed. Trajectories of the perturbed motion starting on the deformed tori T ¢ , fill everywhere densely and quasi-periodically. The tori T~.~form a closed nowhere-dense set and between them remain slits filled with the remnants of the destroyed tori with commensurable w, This invariant nowhere-dense set has a positive measure which tends to the measure of the whole phase space for K -* 0, F -* 0. For n = 2 the two-dimensional tori T~. divide the three.dimensional energy level H = const., and the trajectory beginning in a slit between two tori T~. cannot escape from this slit. Thus the existence of invariant tori allows us to draw conclusions on the stability. For n > 2 the n-dimensional tori T~. do not divide the (2n - 1)-dimensional manifold H = const., and slits can extend to infinity. The behaviour of trajectories on a two-dlmensional torus has been studied by Poincar6, Denjoy, (9) Cherry, and Ura. If the condition (5) is not satisfied in the case of natural degeneracy in which det ]1~HJapl] - 0, then the non-perturbed motion is described b y a smaller number of frequencies than n. Arnold (s) divided the Hamiltonian into the secular part P0 = (Pl ..... P,o), q0 = (ql . . . . q,o) with
1 f H1dqo,
R~(~, q) = (2~)--~
and the periodic part Pl = (P.o+l ..... P,), ql = (q.o+l ..... q.) with
I-I~ = Ha(p, q) - Fi(p, q). The canonical system -
dt
,u
+
8qx
...,
- -
dt
--
+ ...,
p
apx
determines the slowly varying variable Pl, qx definiug the invariant tori. The periodic part //1 leads only to additional oscdlations of the perturbed trajectory around the quasiperiodic motion with slowly varying variables described b y H o + P//1. The decomposition is possible b y the existence of adiabatic invariants proved b y Kasuga. (1°)
24
Celestml Mechanics in the Soviet Union
Let us assume t h a t the secular p a r t KtI does not depend on the phase ql of the slow variation R i = Hi(p) , and t h a t ~CO0 [I
agHO
*°'
[1 C3ql I]
~'1,/Zn--no
,0
Then, for sufficmntly small I/~[ and for a majority of initial conditions from the F of the phase space (p, q), a motion determined b y H(p, q) ,s quasi-periodic, and for these initial conditions the motion is near the quasi-periodic morton dpo _ 0 , dt
,~o = ~ O o ( P o ) - ~g0 dt ~Po
dPi - O, dt
dql -- ql(P) = ~ t dt Opi
(6)
for-- ~
2 ~ s) = (p~))~ + (qi~))~, R (~) = (2. ~(s)) + • 2,jC)g~) + ... l.J
(2,j = 2j~). If (2" k) # 0,
Ikl = Ikll + "'" + Iknl _-< 2s - 1,
then A s -- 0(Iv(s)]s) begins with s-degree t e r m s and is convergent. The system with the H a m i l t o u i a n / 7 (~) is mtegrable. The invariant tom are given b y z (s) = const. The frequencies 2~) ----Jh + 2 E 2 , ~ ') ÷ "'" v a r y from a torus to another l,J
I f at least one of the inequalities
12~,A=
~
,0,
-~
0
ii_
~R
,0 0
is satisfied, the Hamiltonian (6) is called of the general elliptic type The trajectory is near t h a t of the integrable system with R (s) for t ~ s -~, the latter of which is in the e-neighbourhood of the origin for - oo < t < + oo. I f (2 • k) # 0, k # 0, then a n y trajectory starting at an ~-neighbourhood remains in the neighbourhood of the origin for t ~ C~-~. Arnold (n) proved the stability of motion of the general elhptic type for n = 2 near the equilibrium point. Moser (u) showed t h a t it is sufficient to have s = 5/2 and klX~ + k ~ 2 # 0
Y. HAGma~A
25
only for [kll + Ik~l >__4. Deprit (in) p r o v e d the stability in the case of t h e exceptional massratio for the T r o j a n p r o b l e m b y LeontoviS's (m criterion of stability based on this idea. I f for the one-to-one t r a n s f o r m a t i o n / ( A ) -- B b o t h / and its i n v e r s e / - 1 are continuous, t h e n / is said to be a topological trans/ormation or homeomorphism. I f such a t r a n s f o r m a t i o n exists for two sets A and B t h e n these sets are h o m e o m o r p h i c . I f ~ is not analytic b u t differentiable t h e n t h e t r a n s f o r m a t i o n is called di~eomorphism. We represent a conservative d y n a m i c a l s y s t e m with n degrees of freedom (1) in a 2ndimensional Euclidean space with coordinates x = x 1, x 2 . . . . . x, and with grad F, the vector gradient F ~ . . . . F~, of F(x), in the form
dX-IgradH, dt
1=(0
I.
O17 )
'
(7)
where I T is the unit m a t r i x of order n I n t r o d u c e t h e skew-scalar p r o d u c t of two vectors x and y b y [x, y] = (Ix, y) = -[y, x], which represents t h e s u m of t h e areas of t h e projections of the parallelogram with sides x, y onto the coordinate planes Linear transformations S which preserve the skew-scalar product such t h a t [Sx, Sy] = Ix, y], for all x, y, are called simplicial. The skew-scalar p r o d u c t of the gra4tent [grad F, grad G] is called t h e Poisson bracket (/7, G) of F and G. 2' is t h e first integral of (7), if and only if its Poisson b r a c k e t (F, H) with t h e t t a m i l t o n i a n H vanishes identically. If the Poisson b r a c k e t of two functions vanishes identically t h e functmns are said to be in involution. I f (7) admits n single-valued first integrals in involution F, = / t = const., i = 1 . . . . n, so t h a t H = F1, . , F , , (Ft, ~'~) = 0, and t h e equations Ft = It = const, define in t h e 2n. dimension space x and on the n-dimensional c o m p a c t manifold M = M~, at each point of which grad F , , (s = 1 . . . . . n), are linearly independent, t h e n M is an n.dimensional torus and the point x(t), representing a solution of (7), a d m i t s a quasi-periodic motion along it. I n particular Arnold (15) d e m o n s t r a t e d this t h e o r e m of K o l m o g o r o v in the case of a Liouville's d y n a m i c a l system.
Arnold's Fundamental Theorem Consider a function H, a domain G o and positive n u m b e r s ~, R,C, and suppose t h a t (I) t h e function H(p, q) where p = Po + Pl, q -- q0 + ql, P0 is a vector of dimension n o, and Pl is a vector of dimension nl, while n -- n o + n 1 , and qo, ql are angular variables. Yurthermore, H(po, Pl; qo -t- 2~t, ql) -- H(P0, Pl; q0, ql) is analytic in t h e d o m a i n F : p0et~0, I~ qo] =< Q, [Xll =< R, where x 1 = (Pl, ql), and depends on the p a r a m e t e r #, 0 < / x __
H,(p, q) = H,(po, px, q,) + [Ii(po, qo, Pl, ql),
Hi(po,
ql) = nl
Yl(P0'
x(po,
+ 71
f /-tl dqo = 0,
l(pO, nl
T) = ~'0 + Z ~|Tf| + Z ~IJ~"TJ + Z ~lJkT(t'J"~k' l=l l,J=l l,j.k
2 where ~o, ~l, ~ j = Jtjt, ~tjk are functions Po and 2~ l = p2,o+~ + q,o+~, i = 1, . , n, (iii) in P we h a v e for a certain C >__ 1
2 B-VIkVol 13
26
Celestml Mechamcs m the Soviet Unmn (iv) in GOwe have det
~H°
4 0,
det][~o(po)[[ =~ 0.
Then for any z > 0 it is possible to find ~(x; H o, ~ , Go; 9, R, C,/~o) > 0 such t h a t for 0 < ~ < 8 9 , 0 < / ~ < 84 (1) the domain ~ F ~ ' p e ~G 0, ]~ q0] = 0, 0 < v~ < 8, consists of two sets F, and ]8, of which F~ is invariant w~th respect to the canonical equation with the above Hamiltonian and/~ is small so t h a t rues ]~ < ~ . rues F~; (il) F~ consists of invarzant n-dimensional analytic t o n T~ given b y the parametric equations po = po~ +/o~(Q),
qo = Q9 + ao~(Q),
~ = ~ / { e [ ~ +/I~(Q)]} cos [Q1 + al~(Q)],
ql = ~ / { e [ ~ + t~(Q)]} sm [Q~ + ~,~(Q)],
where Q = Qo, QI are angular parameters and P0~ and ~ are constants depending on the number of the tori o~. (in) the mvariant tori Ts differ slightly from the tori I/,AQ)I < ~ ,
Iq,AQ)I < x~,
(iv) the motion determined by the above Hamiltonian on the torus T~ is quasi-periodm with n frequencies o)
dQo dt
--
( D 0~
dQz dt
~
(D 1,
~Ho , ~Po,o
( D O --~
(O 1
=
/.~
~Hz a~,
Belaga (le) studied the reductibility to a linear system in the neighbourhood of an invamant point. Melmkov (17) discussed the behaviour of trajectories near an autonomous Hamlltoman system, Pins (19) of a non-autonomous system; and Beljustma (is) discussed the focal, nodal, and saddle points of the trajectories Hagihara (~°) some time ago discussed a canomcal system with slow variatmns according to Birkhoff's transformations. (5) Moser (~a) discussed a rapidly convergent iteration method on the basis of Siegel's theorem. Cu) Moser, (~1) Arnold, (25) Berkovitz and Gardner, (28) and Ghmm (27) studied the ring transformations of Poincar6 (2s) and Blrkhoff (29) by demonstrating the existence of periodic solutions. We now consider the differential equations
dx = o~ + 8/(x, y, ~), dt
dy dt
-
~ y + 8g(x, y, 8),
where the matrix Y2 is diagonahzed and [, g are real analytic in all variables. Suppose that the equations admit x - cot + eonst, and y -- 0 as a solution with the characteristic numbers co1..... co,, Q1 ..... ~m, satisfying n
• =I
/~=I
with a positive constant ~. Then there exists a unique analytic power series for an n-vector ~(~), an m-vecter ~u(s), and an m x m-matrix M(8) satisfying ..Qp --- O,
,.QM = M,.Q,
Y. H ~ G ~ . ~
27
such t h a t the modified system dx dt
- o,) + ~/(x, y, ~) + ,~,
dy dt
= ~ y + ~g(x, y, ~) + p + Hy
admits a quasi-periodic solution with the same characteristic numbers as the non-perturbed motion. This is Moser's theorem (a°) on the existence of quasi-periodic solutions. Moser ~a°) applied the 1des of Lie groups (81) to the existence proof. Sternberg (8~) discussed the solutions of differential equations on a torus. Merman (aS~ showed the exmtence of a quasi-periodic solution m the plane restricted problem of three bodies. Such a solution is the limit of periodic solutions of the second genus of Poincar~ ca4~ whose periods are multiples of the period of the non-isoperiodie periodic soluhons of Schwarzschild. (aS~He referred to Birkhoff's formal series (~9~representation of the solution, and to Poincar~'s invariant point theorem (28) The existence theorem of Moser of quasi-periodic solutions has been applied to the threebody problem by Jefferys and Moser, (se~ and Arenstorf. (~7~ Deprit Henrard (aS) studied the genealogy of periodic solutions of the Trojan problem in the restricted three-body problem. Jefferys~ag~further apphed the area-preserving mapping of a torus into itself by Birkhoff, and recently cultivated by NIoser~1~) and Arnold, c2) with numerical application to periodic orbits in the restricted three-body problem. B y judging from the behavlour of the trajectories in the nelghbourhood of an mvariant point Jefferys studied the stability and the ergodlcity of the motion and the transition between stability and instability. The ergodic theorem of Poincar~ (~s~ and Birkhoff ~29~ has been discussed recently by several authors, such as Dowker, (~°~ Brunk, (4~) Blum and Hanson, C42~ Chacon and 0rnstem, c43~and Arnold and Avez.(4~) Furstenberg ~46) studied the strict ergodicity and transformations in almost-periodic processes. Let A be a diffeomorphism of a plane ring which conserves the area and 7 be a simple curve in the ring which is not homologous to zero. Then Poincar~'s mvariant point theorem tells t h a t the curves 7 and A 7 have at least two common points. Arnold (46~ calls the operation A" ~ -~ ~ g~oba~y canonical, ff it is homotopic to the identity and if +...
for each closed curve 7 c ~ . Let P(x) = p(~c), Q(x) = q ( ~ ) ; the operator A is globally canomcal if and only if
A(x)
- - . I (Q"
q) dp + ( p - P )
Xo
defines a monovalent function A(x). Let T be a torus p = 0 in ~ , and A T be the image of T by a globally canonical operation A. The tori T and A T have at least 2 ~ common points, provided t h a t A T satisfies p = p(q), ]dp/dq] < oo. Arnold proved t h a t i~ A is a globally canonical operation sufficiently near Ao: p, q -* p, q Jr (o(I0), such t h a t for II~li
detll~]l.
0 there exists a point Po for which o~,(po) are commensurable, i.e.,
0)~(~0)
ii - z - |1
=
2:~mdNw i t h
integers N and m~, i = 1, 2 ..... n , - t h e n
the operation A e has at least
2" fixed points in the neighbourhood of the torus p Po. Arnold (47) discussed stationary motions and their stability on the modern version of differential geometry of Lie groups of infinite dimensions, in contrast to the groups of diffeomorphism, and applied it to the hydrodynamics of a perfect fluid. 2* =
28
Celestial Mechanics m the Sower Uinon
Krassinsky (4s) reduced a canonical system in a simple form in the ne~ghbourhood of a quasi-periodic solutmn and p r o v e d the existence of quasi-periodic solutions in an a r b i t r a r y n e i g h b o u r h o o d of t h e generating quasi-periodic solution on the basis of K o l m o gorov-Arnold's theory. The existence of m e a n motions m the secular motions of t h e periheha and nodes of m a j o r planets has been d~seussed b y Skripmehenko (40) on t h e bas~s of W e y l ' s t h e o r y (a°) and of B o h l ' s work (a~)
REFERENCES 1. A. N. KOI~OGOROV,Dokl. Akad. Nauk, 98 (1953) 763; 99 (1954) 527. 2. V. I. A~Or.D, Dokl. Akad. Nauk, 187 (1961) 255; 188 (1961) 13; 142 (1962) 758; 146 (1962) 487; 156 (1964) 9, Uspekh~ Mat. Nauk, 18 (1963) 13, 91 3 P. BOHL, J. reme u. angew. Math 181 (1906) 268; Dissertation, Dorpat, 1893. 4. E. Escnx~oo~¢, C. R. Acad. Svi. Parts, 185 (1902) 891 ; 187 (1903) 30; Ann. Obs. Bordeaux, 16 (1916) 53. 5. G. D. BImr~OrF, Amer. J. Math. 49 (1927) 1 ; Dyuctmwal Systems, Amer. Math. Soc., 1927. See also C L. Smgel, Nachr. Akad. Wtss. Ctott~ngen, Math.-Phys. Kl., IIa, (1952) 21; Annals o/Math. 42 (1942) 607, 647; Math. Ann. 128 (1954) 144; Vorlesungen uber H~mmelsmechan~k, Springer, 1956. 6. R. SIBA~-RA, Pub. Astr. Soc. Japan, 8 (1952) 144. 7. K. MAImS.R,Math. Ze~tschr. 81 (1930) 729; J. re,he u. angew. Math. 166 (1932) 118, 137; Math Ann. 106 (1932) 131; J. ~'. Koksma, D~opha~t~sche Approx~matwnen, Spnnger, 1936. 8. A. Wr~TN~R, Math. Ze~tschr. 81 (1029) 434 9. H. Poi~exR~, J. Math. _Pure Appl. 1 (4) 161 (1885). A. DENJO:¢, J. Math. Pure Appl. 11 (9) 333 (1932); C. R Aead. Sci Parts, 194 (1932) 830, 2014; T. ~I. CHERRY, Amer. J. Math. 59 (1937) 957; Proc. £,ondon Math. got., 44 (1938) 175; T. URA, Ann Ecole Norm. Sup. 69 (3) 259 (1951); T. URA and ¥ HIR~SAWA,Pro~. Japan Acad. 80 (1954) 726. 10. T KAsuGA, Proc Japan Aead. 87 (1961), 366, 372, 377 11 V.I. AR~COnD,Dokl Akad. Nauk, 187 (1961) 255. 12. J. MOS~R, Nachr. Akad Wtss. Gottmgen, 1962, 1. 13. A. D~PRIT, Boeing Scl. Res. Lab. Document D 1-82-0580 (1966). 14. A M LEO~CTOVIS,Dokl. Akad. Nauk, 148 (1962) 525; 146 (1962) 523. 15. V. I. ARNOLD,S~b~rsk Mat Zhur. 4 (1963) 471. 16. F. G. B~LAGA,Dokl. Akad. Nauk, 148 (1962) 255. 17. V. K. M~L~I~OV, Dokl. Akad. Nauk, 189 (1961) 31; 142 (1962) 542; 165 (1965) 245. 18. L. N. BELJUSTr~CA,Soviet Mathematics, 4 (1962) 48 19. L. J. PIUS, B?uU. Inst. Teor. Astr 11 (1967) 84. 20. Y. HXGI~r_ARA,Proc. Inst. Acad. Japan 7 (1931) 44, 20 (1944) 617, 622. 21. J. MOSER, Nachr. A]~ad. Wiss. Gottingen, Math "-Phys K1. (1962) 1. 22. J NASH, Ann. Math. 63 (1956) 20. 23. J. MOS~R, Annall Scuola Norm. Sup. Ptsa, 20 (3) 265, 499 (1966). 24. C. L. S~EG~L, Ann Math. 48 (1942) 647. 25. V. I. ARNOLD,Dokl Akad. Naul¢, 156 (1964) 9. 26. J. BERKOVITZ,C. S. GXRD~R, Comm. Pure Appl. Math. 12 (1959) 501. 27. J. G L I ~ , Comm Pure A~apl. Math. 17 (1964) 509. 28 H P o ~ N c ~ , Rend~c.ont~ Cir. Mat. Palermo, 88 (1912) 375 29. G. D. BIRKHOYl~,Trans. Amer. Math. Soc. 14 (1913) 14; Rendiconti Cir. Mat. Palermo, 89 (1915) 271; Trans Amer. Math. Soc. 18 (1917) 199; Acta Math. 48 (1920) 113; Dynamical Systems, Ainer. Math Soc, 1927. 30. J MOSER,Math. A~n. 169 (1967) 136. 31. P M CoHEre, L~e (troups, Cainbndge Umv. Press, 1957; 1~. J ~ c o ~ s ~ , L~e Algebra, Interscmnce Publ., 1962; L. S. PO~TRJXOr~, Topologische Gruppen, Teubner, 1958. 32. S. STER~B~R~, Amer. J. Math 79 (1957) 397, 809; 80 (1958) 623; 81 (1959) 578; J. Math. Mech. 10 (1961) 451. 33. G. A. iW_ER~X~¢,Trudi Inst. Teor. Astr., no. 8 (1960). 34. H P o r s c ~ , Mdthodes Nouvelles de let Md.canulue Cdleste, Tome III, 1899, Gauthier-Vdlars. 35. K. SC~rW~zscHm~), Astr. Nachr. 147 (1898) 17, 289.
Y. H~om~a~
29
36. W. H. J~TE~YS and J. Mosv.a, AsSt. J. 71 (1966) 568. 37. R. F. A~STOBF, Amer. J. Math. 87 (1967) 27; J. reme u. anqew. Math. 221 (1966) 113. 38. A. D~PR~ and H. H~m~RD, Boeing ~c~. Lab. Document D 1-82-0623, 1967. 39. W. H. J~.mr~aYs, A~r. J. 71 (1966) 306. 40. Y. N. DOwz.~R,/)u/ce Math. J. 14 (1947) 105. 41. H. D. BRU~K, Trans. Amer. MaSh. 8oc. 78 (1955) 482. 42. J. R. B~.~ and D. L. ~ s o ~ , Prec. Amer. MaSh. ~oc.. 16 (1965) 413. 43. R. V. CH~eo~Tand D. S. 0R~s~r~, Ilhno~8 J. Math. 4 (1960) 153. 44. V. I. Av~o~.D and A. AVEZ, Probl~mes Er~od~es de la Mgcani~ue Class~ue. Gauthier-Vfllars 1967. 45. H. Fm~ST~B~RO, Amer, J. MaSh. 83 (1961) 573. 46. V. I. ARNOLD,C. R. Acad. 8el. Paris, 261 (1965) 3719. 47. V. I. AR~O~.D,C. R. Acad. 8ci. Paris, 260 (1965) 5668; 261 (1965) 17; Ann. In~. Fourier, Unw. Grenoble, 16 (1966) 319. 48. G. A. KR~SS~SKY, B]ull. Inst. Teor. AsSt. U (1968) 411 ; Trudi In~. Teor. AsSt. no. 13. 49. W. I. SKRn~IC~S~EO, B~ull. In~. Teor. AsSt. 11 (1968) 441. 50. H. W~YL, Math. Ann. 77 (1916) 313; Amer. J. Math. 60 (1938) 889; 61 (1939) 143. 51. P. Bo~., J. reine u. an~ew. Math. 185 (1909) 189.
I I I . AVERAGINGMETHOD Beginning with the v a n der Pol equation (1) y - / ¢ ( 1 - y~) ~ + y --- b2k cos (~tt + ¢¢), or a generalized form b y Cartwright and Littlewood ~2) ij + qD(y, [/, t) ~ + G(y) = p(t),
where ~, G, p are continuous, p(t) periodic with period 2/2~, ~(y, ~, t) periodic with period ~/2~ in t and bounded inferiorly for all values of y, ~, t, the averaging method has been cultivated b y K r y l o v and Bogoliubov, (s) Bogoliubov and Mitropolski, (4) and Mitropolski. c5) F r o m the vector equation dx
- 8 X(t,
x),
x = (x~ . . . .
(1)
x.),
dt x ( t , x) = 5". e "/~-~)" X J r ) , v
t h e y derived
-~, at
-F-
M{F},
dt
,
where N denotes an integrating operator and M an averaging operator b y keeping x constant such t h a t
M{F(t, x)} = Fo(x), t
~(t, x) = Z 1)----? ~_e'/(-'" (. (_z } , ~ )~0
~(x,0 = E
e'/~-~" F,(0
,÷0 IX/(-- I)~]s
The equation T
(2) 0
30
Celestial Mechanics in the Sovmt Unmn
is taken as the first approximation If ~ is defined by
then
(_~_)
x = ~ + 8J~(t, 2) + 8 ~ j~2
X(t, ~) - ~20~'(t, ~) X.(~)
O~ satisfies (1) to the accuracy of e 2. The method can be apphed to the system in standard form dx - s X ( t , x) + 8 2 Y(t, x ) .
dt Suppose t h a t there exist in a domain D positive numbers M and ~ such t h a t
IX(t, x)[ <=M,
[X(t, x') - X(t, x")l <=2lx' - x"[
for all real values of t __> 0, and for any points limit
x, x', x"
of D, and t h a t there exists the
T
lim X ( t ' x- )- d~ tf = X ° ( t ) r ~ 0
in D, then it can be shown that, corresponding to any positive numbers ~ and ~ however small, and for any intervalL of t however large, a positive number 80 can be found such t h a t if ~ = ~(t) is a solution of the averaged equation
d~
dt
-sX0(~)
(0
and is in D together with its Q-neighbourhood; then fx(t) -
~(t)l <
holds for 0 < s < 8o in the interval 0 < t < L/8, where x(t) is the solution of (1) with the same lmtial condition as ~(t). Let X(t, x) in (1) satisfy (a) the first approximate equation (2) admits a quasi.static solution ~ -- ~0 such t h a t
X0(~0) = 0; (b) the real parts of all n roots of the characteristic equation det IpI - X~(~0)[ = 0, derived for the variational equation d 65
dt
-
,x~x(~o)
•~
corresponding to the quasi-static solution ~ = ~0, are chfferent from zero; (c) a ~-nelghbourhood D e of the point ~0 can be found such t h a t X(t, x) are almostperiodic in De, even with respect to x ~ De; (d) X(t, x) and its first-order derivatives with respect to x arc bounded and uniformly continuous m x in - oo < t < + oo, x e DQ. Then positive constants 8', ao, a~, ao < a~ < ~, c a n be found such that for positive values e < e'
Y.H.~ors.~A
31
(1) the equation (1) admits a unique solution x = x*(t) defined in - oo < t < + oo for which
lx*(O-~0l
(-~
+~);
(ii) this solution is almost-periodic with the basis of the functions X(t, x); (in) a function 5(x) can be found which tends to zero with s such t h a t
Ix*if) - ~oI ---< D(e), (*v) if x(t) m a n y solution of (1) different from x*(t) and satisfying Ix(t) - ~ol < a0 for t = t 0, and if the real parts of all roots of the characteristic equation are negative, t h e n
Ix(t) - x*(t)l < Ce -~'('-'°), with the constants C and 7, and hence Ix(t) - x*(t)l tends to zero for t ~ oo. If the real parts of all roots of the characteristic equation are pos,tive, t h e n we can find t, > t o for which Ix(t1) - ~0[ > al. If the real parts of s roots are negative and those of the r e m a i n m g n - s roots are positive, then in t h e a0-neighbourhood of ~0 there exists an s-dlmensional manifold M,o such t h a t Ix(t) - ~0] tends to zero exponentmlly for t -* oo for the initial condition X(to)e M, o and t h a t Ix(t1) - ~o] > al for t I > t O for the initial condition x(to) M, o The theorem has been generalized to perturbed motions near a periodic solution m place of a quasi-static solution. Consider
dO - 8F(t, O) dt
(3)
with the initial conditmn Oo = O(to) for t = to, so t h a t O(t) = O(to) + ~ [ t - to, to, 0(to)].
If O, = O(to + nT), t h e n
(4)
0.+ 1 = 0,, + q~(T, t o + nT, 0.) We have [ ~ ' ( T , t o + n T , 0)[ < s~(s),
where O(e) ~ 0 for e -* 0. Hence F(O) = 0 + ~5(0) increases m o n o t o n o u s l y so t h a t F(O + 2~) = F(O) + 2z~. The transformation 0 -~ F(O) is unique and continuous. The iteration equation (4) 0.+ 1 = F(O.) admits the limit = lim 0. n-~o¢ 2~n independently of 0 0. If ~ is irrational, t h e n the general solution of (4) is
O, = 2n~n + ~p + E(2n~n + ~), where ~p is an a r b i t r a r y constant, E(~) is continuous and periodic with t h e period 2~. We get
O(t) = 2z~--~(t-- t°) + V2
'[ 2~
t - t° -
,2z~n+v2],
32
Celestml Mechamcs m the Soviet Umon
where
t -
to -
nT
T and accordingly
l(n
-
2u¢,
~ + 2:~) =/(n, ~).
The solutmn of (3) is quasi-periodic w~th the external frequency ~ and the natural frequency ~ , such t h a t 2~
x(t) = qS(~et, a~t ÷ ~p),
~ = __T--, lim O(t) _ ,-~ T
ap -
2~
~---,
~p -- const.,
2~ T
If ~ is rational, ~ = r/s, then (4) admits a periodic solution 0,+s - O, = 2gr and all solutions with mereasing n approach one of these periodic solutions. I f Om is any solution of (4) contained in the interval (0, 2~), then constants ~m > 0, tim > 0, ~m + fl~ < 2~, can be found such t h a t - ~ , , _< 0ms - 2 ~ m r <= ft,,. This is the rotation number of Pomcar6 and of Birkhoff. (I'~9) The method has been slightly modified b y Cesari, (e) Hale, (v) and ]Allo. (s) Levinson (9) simplified the analysis of K r y l o v and Bogohubov. Itufford, (1°) Marcus, (m K y n e r (12) discussed the existence of perioche solutions on the basis of such an idea. Dlliberto Hufford (18) stuched the asymptotically stable cases in the eyastence of an invariant surface The averaging method has been applied to the motion of E a r t h satellites, w~th the error estimates of the approximations by Struble (m and Kyner. (~5) They discussed the existence and the nature of the periodic solutions. Burshtein and Solovev (le) apphed the averaging method to a Hamfltonian system of differentml equations with a constant p a r a m e t e r e, and formulated the averaged form of a canomeal system
W
'[C
'
')]
2
'
where 2/~ means the integral ~(H - 1i) dt and the bracket denotes the Poisson bracket. Bogohubov and Zubarev (17) extended the method to the system with rapidly rotating phase, such as dx _ ~ X (x, W, 8),
d~
dt
dt
_ co(x) + 8 ~ ( x , ~, ~),
where x varies slowly and ~0 rapidly. A n u m b e r of generalizations were worked out by Kyner, (12) Mltropolsl~, (is) Volosov, (19) Zadiraka, (2°) Lykova, (~) P o n t r j a g m and Rodygin. (~2) Mol~anov ¢28) considered a time-independent system Los, (24) Gikhman, (25) Domidovieh, (26) Fedorchenko (~) discussed systems with continuous dependence of equations on parameters. Volosov rediscussed the averaging method m the case in which the coefficients of the nonlinear differential equations are purely periodic, and then Lykova, (21) Mitropolskl, (~s) Fedorchenko, (2e) Makaeva, (2s) Volosov (ag) m the case in which the differential equations contain slowly varying parameters. Arnold (8°) discussed the motion for the applicabihty
Y. HxGn~a:tx
33
and the error estimation of the averaging method for systems which pass through states of resonance in the course of their evolution. Consider vector equations for ~0 = (~1 . . . . ~k), I = (I 1 . . . . Ii)
dI _ 8F(I, ~0, e), dt
d~ _ (o(I, e) + el(I, % ~),
dt
where 8 4 1, q0(mod 2~r) and ~o, f, F are analytic for I e G, a complex compact domain, 1~ (P] <~, I'%[ < 8 The corresponding averaged equation is
dI dt _ eF(J),
F(J) -
1 k f F(J, of, O) dcp, (2~)
J = lr
Assume t h a t in the time t -~ 1/~ the difference between the exact and the averaged solutions with the same imtial conditions remains small so t h a t
I/(0-
J(t~ I < C,e
(0 < t < l/e),
where Cl, i -- 1, 2 . . . . denote sufficiently large constants independent of 8, k, K, N, x. For k = 2, Arnold derived a sufficient condition for the smallness of [I - JI with the estimate
C ; l ~ / e < II(t) - J(t)[ < C s x / , log~
.
( 7 For ~o~(I) ~ 0 put ;t(I) = ml/e)2 and assume t h a t C4-]~ < I~,1 < C4e Grebemkov (ax) proved two theorems justifying the averaging method for systems of differential equations with commensurable initial frequencies. The theorems are apphed to the Hfll-Delaunay ease of the restricted three-body problem. Musen (a~) extended the method of Krylov-Bogohubov to higher order terms, espeeially the long-permd and secular terms, and compared this with the Poincarg and yon Zeipel methods The eanonicity of differential equatmns is not required in Krylov-Bogoliubov's method, but the number of partial differentml equatmns to be solved m the process of elimination of the short-period terms increases as compared with Poinear6's. I n PoincarCs method the ehmination of the shorg-period terms from the coordinates and momenta is reduced to the elimination of such terms from the I-Iamfltonian b y a proper choice of the canomeal transformation. Assuming t h a t the generatmg function S of this transformatmn is expanded into a power series of a small parameter, the determination of S is reduced to the solution of a set of partial differentml equations. Musen has written those partial dtfferentml equations m t e r m s of F a a de Bruno operators. The greatest advantage of KrylovBogohubov's method is t h a t the error estimate can be made at each step of the successive approximatmn, while m Delaunay-von Zeipel's method there is no means of checking the convergence Morrtson (38~compared the method of averaging with the method of yon Zelpel m the nonresonant cases and showed t h a t yon Zelpel's method is a particular case of the generalized method of averaging, corresponding to an appropriate choice of the arbitrary function arising of the averaged equations. Morrison (38) and Kevorkian (a4) compared yon Zeipel's method with the two-variable expansion procedure, which is a generalized asymptotxc expansion method applied by Kevorkian and his collaborators to the motion of E a r t h satellites and lunar orbiters I t is worth mentioning in thin connection t h a t Diliberto (sS) developed his permdxc surface theory, which is based on the theory of conjectured periodic surfaces whose existence has 2a
~-V//t Vol t3
34
Celestial Mechanics in the Somet Unmn
not yet been rigorously proved If a family of such periodic surfaces is to exist, then it would establish the stability of the orbits and the vahdity of Diliberto's algorithm for finding approximate solutions. Diliberto proved the existence and uniqueness of permehc surfaces for speemhzed condltmns. K y n e r (aa) studied the relationship among the classical Delaunay theory, the Dfllberto periodic-surface theory, and the Krylov-Bogohubov averaging method, and showed that yon Zeipel's simplification of Delaunay's method and the canonical method of averaging produce the same second-order approximation; and showed from the existence of a formal expansion of yon Zeipel's generating function t h a t there exists a formal expansion of a family of permdic two-surfaces, ff a monotonicity condition is satisfied. The Drliberto method has been applied to the motion of E a r t h satelhtes by Dfliberto, K y n e r and Freund, (87) Haseltine, (as) K y n e r (~) for periodic solutions around an oblate Earth on the basis of the mapping of the invariant manifolds according to Poincarfi and Blrkhoff, and of the twist mapping of Moser, (4°) Orlov, (41) and Conley. (42) Diliberto is attempting to settle the three-body problem in an affirmative manner by showing that Pomcar~'s non-existence theorem really does not exclude even the simplest stability possibilities, t h a t the system is formally stable, and t h a t the series used to define the formal stability is convergent. The asymptotic method by Ritz and Galerkin (4a) hkewise enables us to estimate the errors committed. The variational method of Ritz, as modified by Galerkin, can be applied for determining the eigenvalues and the eigenfunctions for the boundary value problem. Cesari (~) discussed the problem whether a nonlinear periodic vector equation dx dt
-
X(x,
O,
where X ( x , t) ~s periodic m t with the permd 2g, admits an exact solution x(t) in the neighbourhood of z~(~) = a0 + x/2" ~ (a. cos n~ + b. sin nO, i.e. a trigonometrical polynomial of order m with indeterminate coefficients, which is called the truncated trigonometric polynomial P~(t), obtained by the Galerkin approximation method, and determined by how much the exact solution x(t) differs from x~(t). Cesari ¢~) and Urabe (~) discussed the problem on the basis of the topological theory and estimated the errors.
REFERENCES 1. B. v~w 1)ERPOL, Phil. Mag. 48 (6) 700 (1922). 2. M. L. C~a.rw~o~r and J. E. L~TTI~WOOD,J. Zondon Math. Eoc. 20 (1940) 180; Ann. Math. 48 (1947) 472. 3. N. KRYLOVand N. N. BooorzuBov, C. tt Acad. Sci. Paris. 194 (1932)957, 1064, 1119; 199 (1934) 1592; 200 (1935) 113; 201 (1935) 1002, 1454; 202 (1936) 200; L'Applicatio~ de Mdthode de la Mdcamque No~ding.aire ~ la Thgor~e des Perturbation~ des ~yst~raes Canon~qnes, Acad. Sci. Ukraine, no. 4 (1934); Mdthodes Approchds de la Mdcanique iVo~dinga~re clans leurs Applwations ~ l']~tude de la Perturbat~o~ des Mouvement8 P~riodglues et des d~vers Phdnvm~nes des Itesonanees s'y rapl~ortants, Acad. Sc~. Ukraine, no. 14 (1935); Itetueil Math. 1 (1936) 707; Ann. of Math. 88 (1937) 65; Introduction to Nonlinear Mechan~, Princeton Univ. Press 1949. 4. N. N. BoooT.rmaovand Y. A. Mm'~OrOLS~, Asyraptotiv Methods ~n the Theory of lqordlnear Oacilla-
t/on~, Hindustan Publ. Corp., 1921; Gordon Breach, 1962; Gauthier-Villars, 1962. 5. Y. A. M2rRoPoT.s~,Probl~me de la Thdorse Asymptotiqu, des Oscillations l~on-~tationaires, GauthlerVfllars, 1966.
y. H~om,~
35
6. L. CES~I, Asymptoti~ Behavior and ~tability Problem in Ordinary Di~erential Equatwns, Springer, 1959. 7. J. K. I-IA~x~ Ann. Math. 78 (1961) 496; Oscillattona tn 1%nh~ar ~y~ema, McGraw-ttfll, 1963. 8. J. l,u.T.o, Acta Math. 108 (1960) 123. 9. N. LEv~sox, Ann. Math. 52 (1950) 727; Ac~a Math. 82 (1950) 71; J. Math. Phys. 28 (1950) 215. 10. G. HWTORV, Contributwns to the Theory of JVonlinear Oseillatwns, III, 173, Princeton Umv. Press, 1956. 11. M. D. ~ c u s , ~b~/. HI, pp. 243 and 261, Princeton Umv. Press, 1956. 12. W. T. K ~ R , ~bu/. III, p. 197, Princeton Univ. Press, 1956. 13. S. P. I)~v.RTO and G. H~TOV~, ~b~d. III, p. 207, Princeton Umv. Press, 1956. 14. R. A. S ~ u ~ , J. Math. Mech. 10 (1961) 691; Archive Itat. Mech. Anal. 7 (1961), 87. 15. W. T. KY~r~R, Tectm. Rep., ~ R 041-152, 1963; Comm. Pure Alrpl. Math., 17 (1964) 227; J. Eo¢. Ind. Appl. Math. 18 (1965) 136. 16. E. L. B v ~ x N and L. S. SOLOV~V,Dold. Akad. 1Vauk, 189 (1961) 855. 17. N. N. BO(~OLIUBOVand D. N. ZuB~a~Ev, Ukrain. Mat. Zhur. 7 (1955) 5. 18. Yu. A. MI~OPOLS~, Uk'ram. Mat. Zhur. 9 (1957) 296; 10 (1958) 270; 11 (1959) 366; Problhme de la Thdome Asy~tot~que des Osedlattons, Gauthier-Vfllars. 19. V. M. VoLosov, Dokl. Alaad. zVaul¢, 106 (1956) 7; 114 (1957) 1149; 115 (1957) 20; 117 (1957) 927; 121 (1958) 22, 959; 128 (1958) 587. 20. K. B. ZAvraAua, Ukrain..Mat. Zhr 16 (1968) 121. 21. O. B. L~KOVA, U]crain. Mat. Zhur. 9 (1957) 155, 281, 419, 10 (1958) 239. 22. L. S. Po~TnJ~OI~ and L. V. l~ol)YGr~, 1)old. Ahad. iVauk, 182 (1960) 537. 23. A. M. MOL~A~OV,Dold. Al¢~l. l~auk, 186 (1961) 1030. 24. F. S. Los, Ukrain. Mat. Zhur. 2 (1950) 87. 25. I. I. G r m m ~ , Ukra,n. Mat. Zhur. 4 (1952) 215. 26. S. F. F~DOR(m~NKO, Ulcrain. Mat. Zhur. 9 (1957) 248. 27. B. P. Do~mowcH, Dold. Alcad. ~auk, 96 (1954) 693. 28. G. S. M ~ v ~ , Dokl. Akad. l~au]¢, 121 (1958) 973. 29. V. M. VOLOSOV,Dold. Akad. l~aul¢, 188 (1960) 261 ; 187 (1961) 21 ; U~pekh, Mat. lgauk, 17 (1962) 3. 30. V. I. Av~oLv, Dold. Akad. ~au~, 166 (1964) 9; 161 (1965) 1. 31. E. A. G m ~ m x o v , A~r. Zhur. 42 (1965) 190; BTuU. Inst. Teor. Astr. 11 (1968) 293. 32. P. M u s ~ , J. AeronauL ~ci. 12 (1965) 129. 33. J. A. Mow~so~, 8oc. Industr. A ~ l Math. l~ev. 8 (1966) 66; Method~ in A~rodynamic,s and Cde~ia~ Mechanicz, Academic Press, 1966. 34. J. K ~ v o ~ , Aatr. J. 66 (1961) 878; 67 (1962) 264; R. A. L~Gv,RS~ao~ and K~vom~z~, Astr. J. 68 (1963) 84; Cdest/al Mechanics andAstrodynam,~, Academic Press, 1964; M. C. EcKs~n~, Y . Y . S ~ and J. K ~ v o m r r ~ , Astr. J. 71 (1966) 248, 301 ; Theory of Orbits in the Solar 8y~tem and in ~ellar 8ystem~, Academic Press, 1966; Y. Y. SHz and M. C. Ecxs~n~, A~r. J. 72 (1967) 685. 35. S. P. ~ ¢ R ~ O , Rendicont, Cir. Mat. Palermo, 9 (2) 265 (1960); 10 (1961) 111. 36. W. T. K Y ~ R , Theory o~ Orbits in the Eolar ~ystem and ,n Stellar 8yatems, Academic Press, 1966. 37. S. P. D ~ a ~ T O , W. T. KY~ER, R. B. l~R~vm), Astr. J. 66 (1961) 118; NASA Space Technology Lab., Calif. 1962; O.N R. Techn. Rep. Umv. Califorma, NR-641-157 (1957); Navord 6445, China Lake, 1959; O.I~.R. Techn. Rep. no. 2, 1962; ibid. NR041-255 (1966). 38. W. R. H~S~LT~% Quart. Appl. Math. 20 (1963) 183. 39. W. T. KYNE~, Comm. Pure ATrpl. Math. 17 (1964) 227. 40. J. Mosv,R, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Kl. 1962, no. 1. 41. A. A. ORLOV, B]ull. Inst. Teor. Astr. 7 (1960) 805. 42. C. C. C o ~ Y , Comm. Pure Appl. Math. 17 (1964) 237. 43. W. Rrrz, J. f. re,he angew. Math. l ~ i (1909) 1 ; V. L. K a ~ o ~ o v I c H and G. P. AKrMOV,$'un~ional Analy~s ®n Normed ~paces, Pergamon, 1964; N. M. K~YLOV, Lea M~thode~ de 8d~ion~ A~rochges de Probl$me de la Physique Mathdmatique, Gauthier-Vflh~rs 1931. 44. L CESARI,Aeymptoti¢ Behaviour and 8tabdity Problems in Ordinary Di~eren$ial E~ation~, Springer, 1959; Contribution to the Theory of Nonlinear Osdllation~, V (1960) 110; Contributions to Di~erc~ial Equat/ons, I (1963) 149; Michigan Math. J. 11 (1964) 384. L. C~sa_~ and J. K. HALE, Proc. Amer. Math. 8oe. 8 (1957) 7b'~7. J. K. H . ~ , Contrib~ion to the Theory of ~onlmear O~cillation~, ~ (1960) 55. E. A. G ~ and J. K. HALE, J. Itat. Mech. Anal. ~ (1956) 353. 45. ~I. Uw~.~, Intern. Syrup. Nonlinear Di~erential E~ation~ and JVonllnear M~hanic,s, Aea~lemic Press, 1963; Archive l~at Mech. Anal. 20 (1965) 120. 2a*
36
Celestial Mechamcs m the Soviet Umon IV
STABILITY T H E O R Y
I t is k n o w n t h a t L i a p o u n o v ' s second or d i r e c t m e t h o d for t h e c r i t e r i o n of s t a b i l i t y consists in f o r m m g t h e Liapounov's/unctwnal (1)
= d F _ ~V X i + dt ~x I
~V X~ + ~x~
+ ~V X . + ~x.
~V ~t
where t h e given d~fferential e q u a t i o n s are
dxt -- X t
(~ =
1,2,
, n)
dt L l a p o u n o v ' s c r i t e r i a for s t a b i l i t y a n d i n s t a b i l i t y a r e . (I) I f t h e differential e q u a t i o n s for a d i s t u r b e d m o t i o n a r e such t h a t i t is possible to find a sign-definite f u n c t i o n V, of w h i c h t h e d e r i v a t i v e is a f u n c t i o n of fixed sign w i t h t h e sign o p p o s i t e to V, or r e d u c e s i d e n t i c a l l y t o zero, t h e n t h e n o n - d i s t u r b e d m o t i o n is stable (II) L e t V be a function of xs a n d t such t h a t V a d m i t s a n i n f i n i t e l y s m a l l u p p e r b o u n d , t h a t V is a s l g m d e f i m t e function, a n d t h a t V can t a k e t h e sign of l~ for t h e v a l u e s of t b e y o n d a c e r t a i n b o u n d , h o w e v e r s m a l l xs m a y be in a b s o l u t e values. I f such a f u n c t i o n V can be f o r m e d b y m e a n s of t h e d i f f e r e n t i a l e q u a t i o n s of t h e d i s t u r b e d m o t i o n , t h e n t h e non. d i s t u r b e d m o t i o n rs unstable ( I I I ) I f for differential e q u a t i o n s of a d i s t u r b e d m o t i o n i t is possible to find a f u n c t i o n V of a fixed sign or w i t h an i n f i n i t e s i m a l u p p e r b o u n d , whose t o t a l d e r i v a t i v e l~ is also of fixed sign o p p o s i t e to t h a t of V, t h e n t h e n o n - d m t u r b e d m o t i o n is asymptotically stable. (IV) I f for t h e differential e q u a t i o n s of a d i s t u r b e d m o t i o n i t is possible to find a funct i o n V such t h a t l? is a f u n c t i o n of fixed sign, a n d V Itself is n o t of fixed sign o p p o s i t e t o t h a t of 17, t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable. (V) L e t V be a f u n c t i o n of xs a n d t s u c h t h a t V is a l i m i t e d f u n c t i o n ; t h a t I? is of t h e f o r m [? = ~V + W, w h e r e ~ is a p o s i t i v e c o n s t a n t a n d W IS a f u n c t i o n of fixed sign, a n d t h a t V m a y t a k e t h e sign of W for a l l v a l u e s of t b e y o n d a c e r t a m b o u n d , h o w s m a l l xs m a y be in a b s o l u t e v a l u e s I f such a f u n c t i o n c a n be f o r m e d b y m e a n s of t h e d i f f e r e n t i a l e q u a t i o n of a d i s t u r b e d m o t i o n , t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable C h e t a e v ' s (=) Criterion if t h e r e exists a f u n c t i o n V(t, x), such t h a t for an a r b i t r a r y large v a l u e of t in a n a r b i t r a r y s m a l l n e l g h b o u r h o o d of t h e origin, t h e r e exists a region V > 0, if in t h e region V > 0 t h e f u n c t i o n is l i m i t e d ; if in t h e region V > 0 t h e d e r i v a t i v e l? t a k e s positive v a l u e s for a l l v a l u e s of t, x~ so t h a t V > ~, ~ > O, a n d l~ > l, l = l(~) > O, t h e n t h e n o n - d i s t u r b e d m o t i o n is unstable.
Le/schetz' s Extension
a r e closed a n d if t h e space R is c o m p a c t t h e n
P
t h e y are c o n n e c t e d , t h a t t h e closure ~ of a n i n v a r i a n t set M is a n i n v a r i a n t s e t ; t h a t t h e b o u n d a r y of a n i n v a r l a n t set IS a n i n v a r i a n t s e t ; a n d t h a t if M , N a r e l n v a r l a n t sets so a r e t h e i r u n i o n M w N, t h e i r i n t e r s e c t i o n M n N, a n d t h e i r difference M - N ; a c c o r d i n g l y /2(?) a n d A ( ? ) are i n v a r i a n t sets.
Y. H~GmARA
37
L e t M be a closed m v a r i a n t set of the d y n a m i c a l s y s t e m ](p, t), and S(e) be a sphere a t centre M and with radius 8, and H(s) be t h e boundary. M is defined so t h a t (a) it is stable, whenever for a given e > 0, t h e r e m an ~ > 0 such t h a t if p e S(~/) t h e n {/(p, t)/t >= 0} c S(e), (b) it is a s y m p t o t m a l l y stable w h e n e v e r it is stable, and m addition under t h e preceding c o n d i t m n / ( p , t) - , M as t ~ oo, (c) it is unstable w h e n e v e r for a n y e > 0 and w h a t e v e r ~ > 0 and ~ < e t h e r e is a point T e S(~) such t h a t / ( p , t) reaches H(e) for some t > 0 T h e d o m a i n of a s y m p t o t i c stability A of M is an open m v a r i a n t set containing a sphere S(~) and the closure A, and accordingly its boundary, are m v a r i a n t sets. T h e necessary and sufficient condition for stability of M rs t h a t , if we are given a n y ~ ;> 0 and p outside S(8), t h e r e is a ~(s) > 0 such t h a t ?~ remains outside S(~). The necessary and sufficient condition for a s y m p t o t i c s t a b d l t y of M is, m a d d i t m n to t h e condltmns for stabihty, t h a t there exists a neighbourhood of M free from complete p a t h s o t h e r t h a n those in M Itself. T h e generalized ]_aapounov functionals of Lefschetz are functions defined in a certain set S(~), such t h a t (a) g~ven a n y 0 < e < ~, t h e r e exists a 1 > 0, so as to satmfy V > 2 for p outside S(a); (b) given ~ > 0 t h e r e exists an 7(2) > 0, so as to satisfy V < ~ for p ~ S(~); {c) V[/(p, t)] is in S(~) a non-increasing function of t. Such a f u n c t m n V is a generalizatmn of L i a p o u n o v ' s concept of t h e positive-definite functmnal V with the neg a t i v e I~. A necessary and sufficient conditmn for t h e s t a b i h t y of M is t h e existence of a generahzed L i a p o u n o v functional V. A necessary and sufficient condition for a s y m p t o t i c stability of M ~s t h a t t h e r e exists a V{p) defined for ~ m a certain S(~), such t h a t V[](p, t)] -~ 0 as t ~ oo. A necessary and sufficient condition for instability of M is t h e existence of a functional V(p) defined in a certain S(¢¢), such t h a t (a) IV(p)] is b o u n d e d in S(~); (b) e v e r y S(e) contains at least one point p where V(p) > 0, (c) a t e v e r y p o i n t p e S(c¢) along ~ts p a t h , the derivative l? is defined and [7 = i V + W(p), where 2 > 0 and W(p) is non.negative in
s(~) Converse o/Lsapounov's Theorems The equilibrium point of t h e differential equation
dx - l(x, t), dt
I(O, t) = o,
l eE,
is called uni]ormly stable if to each 8 > 0 we can find a n u m b e r ~ --- ~(e) > 0, depending on e, such t h a t f r o m I~1 < ($ follows Ip(t, ~, to) I < ~, t > t o for all t o _> 0 (Krasovskii). (1) (I) - An equilibrium point is u n i f o r m l y stable if and only if t h e r e exists a function ~(r) such t h a t (a) Q(r) is definite, continuous and m o n o t o n o u s l y increasing m 0 _< r < r,; (b) ~(r) v a m s h e s for r = 0, (c) for I~l < r we h a v e IT(t, ~, to)[ -< ~(~) (Persidskn) (1) I f IT(t, ~, to) ] <<_H holds for all points ~ at which a solution curve starts, t h e n the a s y m p totic stability is said to be globally. If the assumptions of this t h e o r e m are satisfied for a n y arbitrarily large r, t h e n the equihbrium point is said to be uniformly stable globally. Similarly uniform a s y m p t o t i c global stability is defined. ( I I ) - I f an equilibrium point of an a u t o n o m o u s or periodic differential equation is stable, t h e n it is also u ~ f o r m l y stable, and t h e globally uniform stability follows from the global s t a b i h t y (Hahn(*)). Let a(r) be a defined function, continuous m o n o t o n o u s l y decreasing for all r _>__0, so t h a t llm a(r) = 0 and [xs(t, ~, to) [ ___
38
Celestial Mechanics in the Soviet Union
a function satisfying the conditions (a) and (b) of theorem I. ( I I I ) - The necessary and sufficient condition for uniformly as~nptotic stability of the equilibrimn point is t h a t there exist two functions a(r) and ~(r), such t h a t [p(t, ~, to)[ __< <= ;~([~1)a(t - to) for ~ of a fixed sphere K , (Hahn¢l)). The converse of the stability theorems depends on the nature of the differential equa. tlons, on the assumptions of the equilibrium point, and on the nature of the Liapounov functional (especially the existence and the continuity of its partial derivatives). A function V(t) is called positively semi-definite or negatively semi-definite if V(0) = 0 and if in the neighbourhood of the sphere IxI _< h of the origin we have V[x[ ~_ 0 or V(x) <=O. If V(0) = 0 and V(x) > 0 or >__0 for x =~ 0, then the function V is positive-definite or negative-definite. ( I V ) - L e t / ( x , t) and ~/l/axj be continuous in the spherical domain Ix[ __
[p(t,~,to) I > f l ( ~ ) e x p [ ~ ( t - t o )
],
t___to,
~ >0,
fl>O.
For a motion to show intensive behaviour it is necessary and sufficient t h a t there exist a Liapounov functional satisfying
V(x, t) < al[xl y,
I[?l > a~[x]r,
Y. l ~ o r ~ * ~
39
where a 1, a s, ? are positive constants (Krasovskii). For every motion, at least one branch, to admit an estimate of the form
fi~($) e x p (~lt - t01) < [p(t, ~, t0)l < fi~(~) e x p (~lt - t01), it xs necessary and sufficient t h a t there exists a function lz which satisfies
~ l x l ~ < V(x, t) < allxl ~,
a~lx] ~ < I~'(x, t)l < a'olxl"
for sufficiently small Ixl. Bohl, ~s~ Liapounov, ~l~ and Poincar6 ~9~ discussed the stability of the system dx~
- as~ ,x~ + a~2x ~ + ... + a~.x.
(s = 1, 2 , .
, n)
dt
with constant coefficients a~j. If the fundamental determinant formed by the coefficients admits only roots of real parts negative, then the non-perturbed motion is stable and the motion for which the perturbation is sufficiently small tends asymptotically to the nonperturbed motion. I f the fundamental determinant admits only roots of real parts positive, then the non-perturbed motion is unstable. Liapounov ~ treated the case of one zero-root and the ease of a pair of purely imaginary roots, while the remaining roots are all of real parts negative, as well as the case of double zero-roots in a posthumous memoir.C~0~ Moser ~2~ discussed Siegel's theorem. Bohl ¢~s~ and Harasahal cz~> treated the case of quasi-periodic coefficients. Cotton c~a~discussed the asymptotic solutions on the basis of integral equations. Moustakhiehev <~s~ derived sufficient conditions for the instability of an equihbrium of a t t a m f l ~ n i a n system, of which the characteristic equation admits two zero-roots Perron' s T h e o r y
Lefschetz ¢~) and Bellman ¢26)discussed stability under less stringent conditions. Perron ¢~6) explored the situation on stability further. Consider for real t __> 0 : dxs -
dt
-
aslX 1 + a~x~ + ... + a~.x. + qgs(x1 . . . . . x., t),
%(0,0 ..... 0, t) = 0
(s = 1,2 . . . . n),
where xs, as~, q~s m a y be complex, which admits the trivial solution xl = x2 . . . . . x. = 0. If we can assign a positive number a, suoh t h a t to any s between 0 and a a number 81, 0 < 81 < 8, can be found so t h a t any solution for which 0 < ~ Ix,(0)[ < 81 exists for all S
values of t ~ 0 and satisfies ~ [xs(t)] < e, then the trivial solution x 1 . . . .
= x, = 0 is
$
said to be stable and otherwise unstable. If there is a positive number 81 such t h a t any solution for which 0 < ~ Ix,(0)l < 8 ceases to exist once, necessarily for a finite value of t, or at least satisfies ~ Ix~(t)] = 8, then the trivial solution is said to be completely unstable. $
Between stability and complete instability there occurs conditional stability. If there exists a positive number a, such t h a t to any number 8, 0 < e < a, a number 81, 0 < 81 < 8 can be found, such t h a t a certain number o f - b u t not all - solutions with 0 < ~ Ix,(0)] < 81, $
continue to exist fort ~ 0 and satisfy ~ ]xs(t)l < 8, then the trivial solution is said to be S
conditionally stable.
40
Celestml Meehamcs m the Soviet Unmn
Ifq0is continuous for ]xll < a . . . . [x,I =< a, t __> 0 (a > O),and ff[~] ~ K(lxx[ + .. + Ix.I), K > O, t h e n q~ is said to satisfy the condition (A). I f ~/(Y.lx~l) -* o for Z Ix~l -~ o, t -. o, s
s
t h a t is, if to a n y positive n u m b e r we can assign (~, and T, such t h a t I~l _-< e(~lx~l) for s
Ix~l <= ~ , s = l,
., n,
and ~ [x~l > 0, t __> T~, t h e n ~v is said to satisfy t h e c o n d i t i o n (B) $
The conditmn (B) is satisfied if we have uniformly hm
for
g~
~,-.o . . . . .
- O,
or
lim
~
o Y.Ix~l $
Ix~l < a,
- 0,
s
s = 1,
,n,
Ix~l > 0. s
I f for a given positive n u m b e r ~ we can assign two positive constants 5~ and T~ such t h a t I~(x i,
. , x.,' t) - ~(x~',
, x .", t)l =< ~ (E Ix" - x~' J) $
for Ix~[ =< (~, t __> T~, t h e n ~ is said to satisfy t h e condition (C). I f a function satisfies (A) and (C), t h e n it satisfies (B). I f 9~ satisfies t h e condition (B), t h e n as t are one-valued and t h e characteristic roots ~, of Ilasj - 81jOll = 0 are uniquely determined I f x(t) defined m t o <: t __< t 1 satisfies
d~s
--~(x,t)
(8 = 1,
.,n),
dt
where ~ satisfies (A), t h e n for a n y two values of t in [t0, tl] we have
S=I
x=l
(I) Suppose t h a t b y a h n e a r t r a n s f o r m a t m n t h e system of differential e q u a t m n s is reduced to dxs
s- 1
- ~ x ~ + ~ c~lxl+~f,(x 1,
dt
.,x.,t),
(s = 1 . . . .
n),
~=1
and t h a t 9~ are of negative real parts. P u t --?=
max~(Qs),
C=max[c~l,
~ =
mm
1, 2(n
1)ff
1
if(n-
1) C = 0 ,
I f ~s satisfy (A) and ? I~)sl ~
~n
( I x l l -~ ~%1X21 ~- * " -~
~%n-1]Xnt])'
(8 ~--- l , . ,
n),
t h e n the trivial solution xs = 0 is stable, and a n y solution with t h e initial condition xS(0) sufficmntly small in absolute values tends to zero. limxs = 0
(s =- 1 . . . . n).
r---~o0
(II) I f ~s satisfy (A) and (B), and if Qs are all of negative real parts, t h e n x~ = 0 is stable and a n y solution x~(0) sufficiently small in absolute values satisfies hmx~ = 0 t ---~oO
(s = 1,
,n)
Y. H s o m , ~ x
41
( H I ) L e t 0~ be such t h a t ~(0s) - ? > 0 for s >_ k a n d ~R(0~) =< 2 for s < k, k ~ n. I f ~0~ satisfy (A) and
I~A _-< ~
4n
C(Ixd + ~lz~l + • • + o,"-~lz.l),
t h e n xs -- 0 is unstable. I f k = n, t h e n it is unstable. (IV) Suppose t h a t ~ satisfy (A) and (B). I f at least one of t h e characteristic roots ]s of positive real part, t h e n x~ = 0 is unstable If all roots are of positive real parts, t h e n t h e instabflxty ]s complete. (V) L e t ~R(Qs) > ~ f o r s > k < n a n d ~R(Os) < - ~ f o r s > k, w h e r e ~ > 0. I f ~ s s a t l s f y (A) and
t h e n x~ = 0 is conditionally stable. T h e a s y m p t o t i c b e h a v i o u r of t h e solution is described b y the t h e o r e m . If ~s satisfy (A) and (B) and if t h e real p a r t s of Q1. . . . . On are n o t all positive, t h e n t h e r e is for each solution x~, for which lira xs = 0, s = 1, 2, ..., n, an index 1 such t h a t lira log([xl] + . . -
+ Ix,I) = ~(0~)=<0.
~-~00
If ~s satisfy (C) further, t h e n conversely t h e r e exists to an index 1 for which ~(Ol) < 0 a solution of this nature, and such solutions for which lim log (Ix1[ +
-. + Ix.I) __< - r
< 0
form a family of just h a r b i t r a r y constants, where h is t h e n u m b e r of characteristic roots for which ~R(q~) <__ - r . Consider t h e equations -
0~,, + ~(x~,
. , x,, t),
(s =
1 .....
n).
dt I f ~0~ satisfy (A) and (B) and if ~R(QI) > ... > ~(Q,), t h e n t h e r e is for each solution x~ for which lma x~ = O, s = 1, ..., n, and index l, such t h a t t-*oo
for s # l, and /1
dx~.\
t_.®kX~ dt ] I f ~ satisfy (C) further, t h e n conversely t h e r e exists to an index l, for which ~ ( ~ ) < O, a solution of this nature, and such solutmns for l ~ j f o r m a family with just n - j + 1 a r b i t r a r y constants
Order Numbers A n u m b e r ~a, such t h a t
l ~ ~ l o g (Z IxA) = ~, t-.o0
~
S
42
Celestial Mechamcs in the Soviet Unmn
is called t h e order number (Ordnungszahl) b y Perron. The n u m b e r of the order n u m b e r s of linear differential equations
dx~
- ~ f~,(t) xi ~=1
dt
( s = 1 ..... n)
is just equal to n, if multiplicity is p r o p e r l y counted. L e t 71 -> 7~ -~ • • --> 7 , be t h e order n u m b e r s of t h e differential equations, t h e n t
~R[fz~(v)] dr.
7s => lim s=l
t-.co
~ J2=l o
Consider t h e a d j o i n t s y s t e m
dy~ dt
-
-T,/,,(t~
t
yt
(s = 1 . . . . . n),
t h e n t h e order n u m b e r s &l --< &2 =< ... =< ~, arc such t h a t ?,+&,>0
( s = 1 . . . . . n).
N e x t consider t h e generalized s y s t e m
~x. _ T, f.~(t) x, + ~,(x~ . . . . . x., t) dt s
(s = 1 . . . . . n),
(l)
and assume t h a t
dxs - ~/,,(t) dt s
x, + ~,(t),
(s = 1 . . . . . n)
(2)
admits a t least one b o u n d e d solution for a r b i t r a r y continuous b o u n d e d functions ~s(t). W h e t h e r (2) a d m i t s always a b o u n d e d solution is k n o w n f r o m the behaviour of the homogeneous s y s t e m (1) with ~s =- 0. If/st(t), 1 _< i _< s _< n, are continuous a n d b o u n d e d for t ~ 0, the necessary ¢ n d sufficient condition for (2) to a d m i t a t least one b o u n d e d solution for a n y choice of t h e continuous and b o u n d e d functions ~s (t) is t h a t / s t ( t ) should satisfy t h e conditions with t
;/,,(v) dv = H~(t), 0
t h a t for a n y indices s = 1 . . . . . n we should have either t
(A) eu'(t) is b o u n d e d and e ~'(t) f [e-H'(')[d~ is bounded, or (B) e u'(t) is unbounded, but t
0
e g~(t) ~ [e-H'(°[dv exists and is bounded. 0
I f (2) admits at least one b o u n d e d solution for a n y choice of t h e continuous b o u n d e d functions ~ps(t), t h e n for each b o u n d e d solution of t h e homogeneous equations (1) with ~ ---- 0 there should hold t h e relation
lira ~ Ixs(t)[ = 0 t._,c~ s = l
Y. H ~ o m ~
48
If (A) is satisfied for all s, then all solutions are bounded. If (B) is satisfied for all s, t h e n there is only a single bounded solution. I f k indices s satisfy (A) but n - k satisfy (B), t h e n the linear homogeneous equations (1) with ~ - 0 admit just k linearly independent bounded solutions. For k = 0 there is no bounded solution other t h a n the trivial solution $~ = 0.
Kinematic Similarity Let Mn be the set of all n x n matrices, whose elements are complex-valued functmns of a real variable t, continuous and bounded for 0 __< t < oo. For A, B e Mn we say A ,~ B, i.e. A is kinematically similar to B if there exists a matrix P with P, p - I and P e M , , such t h a t _ p - x [ p _ AP] -= B (3) for all 0 _< $ < ~ . The idea has been introduced b y Perron (~7) and extended b y Jacobsen, (~) and b y Cameron, (2°) Lille, (~) and Markus ¢8x) to the case of almost-periodic coefficients. Consider ~(=AX, AeM,. For each non-zero solution X~($) we define a characteristic number
lira t-~ log Ix,(t)] = ~,. t,..~ oO
Perron showed t h a t 2~ are bounded and t h a t there arc at most n distinct characteristic numbers of.4, and t h a t every matrix in M n is kinematically equivalent to a triangular matrix. For such a solution vector X(t) with characteristic n u m b e r ~ we define a type of A as
1,()t, X(t)) = l,(;t) --- lira [log IX(t) e-at]/log t]. !'--* oO
The set of all solution vectors of X = A X with characteristic n u m b e r ~ =< ~ , i = 1,..., k, forms a linear space E(~). E ( ~ H ) is properly included in E ( ~ ) . We make the convention E(~0) = 0. We define the type of a coset in the factor space E(~)/E(~_x) , taking the idcntity-coset to be of zero type. The collection of cosets of E(2~)/E(2~_l), whose t y p e =< rtj, is a linear subspace of E(,~t)[E(2~_I) called T ( ~ , r~j) where we set T(2t, rt0) -- 0. Let hi < ~ < ... < 4, be the distinct characteristic numbers of .4 e M , , with corresponding types ~x < ~ < .... ~ < ~ < .... ~ < ~ < ... The multiplicity of 2~, i = 1, ..., k, is m(2~) = dim ~(2~)/(E(2H). The multiphcity of ~ is m ( ~ ) = dim T(2~, r~)/T(2~, ~.~-1). The characteristic numbers 2~, the types r~, and the multiplicities m(2~), m ( ~ ) are the invariants of kinematic similarity. B y a non-singular coordinate transformation Y = P - i X , P e M , , we get ~" = B Y , B = - P - ~ [ P - .4P] This is kinematic similarity. The solution space for X = A X is isomorphic to the solution for 1~ = B Y and the corresponding solutions X(t), Y(t) are of the same type with the same characteristic numbers.
Strong Stability Consider a canonical system dY dt
-
IH(t)
Y,
Y =
l(yl,.
,y.),
(4)
44 where
Celestial Mechanics m the Soviet Umon
,:(
--I n
)
2n '
l,j=l
and hu(t ) = h jr(t) are piecewise continuous real functions with the common period eo A canomcal system is called strongly stable if it is stable and if there exists an e > 0, such t h a t for any real symmetric matrix H(t) wath pieeewise continuous elements satisfying
R(t + co) =//(t),
[IB(t) - H(t)lJ <
all solutions are bounded for t -~ ~ , where [[ I[ denotes the norm. Strongly stable canonical systems form an open set in the set of all canonical systems. We find the conditions under which there exists a sequence of symmetric matrices H(t, ~) subject to H ( t + w, v) = H(t, v), and plecewise continuous in t and ~ such t h a t Y= IH(t,v)Yis strongly stable for all 0 < ~ < 1, and such t h a t H ( t , O ) = H l ( t ), H(t, 1) = H~(t). Such systems are said to belong to one stability domain The necessary and sufficient condition for the matrices of solutions of canomcal systems to belong to one stabihty domain, accordingly the topological structure of the stability domain, has been studied by Krein, (a2) ¥abuboviS, (aa) Gelfand and Lidskd. (34) We normahze the solution such t h a t Y(0) = E2,. We have Y(t + neo) = Y(t) Y"(w), n = 1, 2, ..., ff the elements of Y(t) are bounded in 0 _< t _< w. For stability all elements of Yn(w) are necessarily bounded for n -* oo ; or all elementary divisors of Y(eo) should be linear with unit moduli, t h a t is, Y(eo) should be reducible to a diagonal matrix. Y(co) is a monodromy matrix for the monodromy group. The roots of the characteristic equation are called multipliers. Krasnoselski (aS) has given a method for computing the multipliers. Each stability domain is determined by one of the 2 n possible types of relative distributions of multipliers on the unit circle, and b y the serial number of the stability domain AT, -- ~ < N < ~ , which is called the index of rotation of the system. A matrix Y(t) satisfying Y*(t) I Y ( t ) = I for all t ]s called symplectw. E v e r y curve Y(t) in the group of real symplectic matrices m 0 < t _< co possessing a piecewase continuous derivative satisfies a matrix equation of the form (4) in [0, o~]. The symplectic matrix Y is a matrix of stable type ff and only if it can be represented in the form Y = G R G -1, where G and R are real symplectic matrices and R=(
cos0 s i n 0 ~
-sin0
cos0]'
where 0 is a real diagonal matrix of order n, whose diagonal elements 01, ..., 0n satisfy [02[ <:~, s = 1, ..-, n, and 0~, ~:0s,,, 1 < s ' , s " < n . The group of real symplectic matrices is homeomorphic to the topological product of the circumference of a circle and a simply-connected topological space. Two continuous curves Yl(z) and Y2(z), 0 _< z < 1, with common end-points, lying entirely in a connected component of the matrices of stable type, can always be continuously deformed one to the other The topology has been studied by Segel (3e) m a dtfferent manner (B) Control I n order to maintain the stability of the origin x = 0 we use a compensating mechanism caned a control. (aT-ag) Let ~ be a scalar parameter and ~ --- 0 be a state of no control. The new equations of motion, corresponding to the lineanzed system [t = A x and obtained from y = Y(y, t), are x = ~) = A y + ~b, ~ = /(a), a = c r y -- r~,
Y. HAOmAaA
45
where b, c are n-vectors and r a scalar, while a is introduced as an intermediary and ](a) is the characteristic, assumed to be continuous, of the servo-mechanism. Assume t h a t ±ao
,~l(a) > 0 for ~ 0¢ O, l(O) = O, f l(~) da = + oo. OV IAapounov's funetional 12" -- ~ + Y • grad V is t a k e n of the form
V ( x , c/) = X T" B x + f ](~)d(r,
B > 0,
B r -- B,
0
where V is positive-definite over the whole space (x, a), such t h a t V(0, 0) = 0 and V and its first-order derivatives are continuous in the whole space. Geiss and Abbate (4°) computed the optimal quadratic estimate of the domain of attraction of an equilibrium solution of a quasi-linear differential equation
= A x + [(x),
1(0) --'- O,
where A is a stable matrix. I f ~ = X, X(0) = 0, whereby the X satisfy the relation lim &(Xo, to, t) = 0, for x(xo, to, to) e/1, t __> t o, and for all x o in G~, when F e G is given in t-*oO
a closed bounded subregion G~ ~ G; and if the non-perturbed motion is asymptotically stable, t h e n G~ is called an attractive domain around x -- 0 (Krasovskh).
Stability in A~stract Space Fomin (aI) considered an operator differential equation with distributed p a r a m e t e r in a separable Hilbert space. 1)
dt
= [z +
x,
= 0t,
where F is a self-adjoint operator, ~ is a symmetric operator with period 2~, F-loEf(T) is supposed to be uniformly continuous in [0, 2~] with uniformly continuous derivatives, and 0 characterizes the perturbation frequency. The vector solution X(t), specified in terms of the solution x(t) b y x(t) = X(t) x(0), can be represented b y X(t) = U(t) + V(t), where the operator U(t) is unitary for any t and V(t) is completely continuous for any t. Almkvist ¢~) discussed the stability of the solution of du
dt
- A(t) u(t)
(t > O)
m a separable Hflbert space, where A(t) is symmetric and periodic with period unity. K a t o (4a) studied a linear differential equation in Banach space. Ura and K i m u r a (44) discussed the flow of an incompressible fluid in Gottsehalk and Hedlund's topological dynamics.(45) Arnold (46) showed t h a t if two vector fields a and b in M of divergence zero commute, i.e. (a, b) -- 0, and are not eollinear anywhere, i e. a • b ~= 0, then almost all trajectories of the field a are closed or everywhere dense on the analytic tori T 2 immersed in M. U r a (47) further discussed the stability of trajectories in abstract spaces.
Y. HAGmA~
47
37. R. B ~ , Introduction to the Mathematical Theory of Contrd Proc~aes, Academxc Press, 1967. 38. D. SwoP.v~B, Optimal Adaptive Control Systems, Academic Press, 1966. 39. G. L E ~ (Ed.), Twpic~ ,n Optimiz~ion, Academic Press, 1967. 40. G. It. G~Iss and J. W. AB~x~, Grummer Res. Dep. Rep RE-282, Aero-Astrodynamics Lab., Marshall Space Flight Center, Hunts, 1967. 41. U. N. I~OM]~, Do/d. Akad. Na~d¢, 168 (1965) 830. 42. G. hv.uwvisT, Pacific J. Math. 16 (1966) 383. 43. T. KATo, Comm. Pure Ap~pl. Math 9 (1956) 479. 44. T. U~A and I. Ku~m~, Proc. Ja~an Acad. 40 (1964) 703. 45. W. Go~sc~Ar.w and G. H~DLV~D, Topologlc~d Dynamics, Amer. Math. Soc., 1955. 46. V. I. AJt~OLD, Doklady A ~ . Nauk, 162 (1965) No. 5; C. R. Acad Sei. Paris, 260 (1965) 5668; 261 (1965) 3719, 17. 47. T.U~,A, ~unkciala~ Ekvacio?, 2 (1959) 143; 9 (1966) 171, Contrtbutton to Di~erential Equations, 8 (1965) 249. 48. K. M. M o u s ~ ' A ~ C ~ v , B~uU..Inst. Teor. Astr. 11 (1968) 453.
No~ In addition to the above-mentioned advances, the ergodic theory has further been amplified in the Soviet Union through the introduction of relevant concepts in the fields of entropy, spectra and KSystems, by Kolmogorov, Sinai, and Arnold; see also the new volume V (MIT Press) of Y. Hag~hara's
Celesttal Mech~nic~.