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yfl K 517.925.42 , 521.13 1

AM S MS C 58F22 , 58F0 5 E . A . KUDRYAVTSEV A
Facult y of Mechanic s an d Mathematic s Departmen t of Differentia l Geometr y an d Application s Mosco w Stat e Universit y Vorob'iev y Gor y Moscow , 119899 , Russi a E-mail : ekudr@beta.math.msu.s u

GENERALIZATION OF GEOMETRIC POINCARE THEOREM FOR SMALL PERTURBATIONS.
Received July 4, 1998

We consider th e dynamica l system , which phas e space containe s a closed submanifold filled by periodi c orbits . Th e following proble m is analysed . Let us consider a small perturbation s of th e system . Wha t we can say abou t th e numbe r of periodi c orbits , survived unde r perturbation , an d abou t thei r locatio n in th e neighborhoo d of th e submanifold unde r consideration ? We obtai n th e solutio n of thi s proble m for th e perturbation s of general typ e in term s of avereged perturbation . Th e mai n resul t of th e pape r is as follows. Theorem : Let us consider th e Hamiltonia n syste m wit h Hamiltonia n function H on a symplecti c manifold (M^",!^) . Let A C H'^(h) b e th e closed nondegenerat e submanifold filled by periodi c orbit s of thi s system . The n for th e arbitrar y perturbe d function H, which is C^{close t o th e initia l function H, th e syste m wit h th e Hamiltonia n H ha s a t least two periodi c orbit s on th e isoenergy surface H~^(h). Moreover, if eithe r th e fibration of A by closed orbit s is trivial , or th e base B = A=S^ of thi s fibration is locally fiat, the n th e numbe r of such orbit s is no t less tha n th e minima l numbe r of critical point s of smoot h function on th e quotien t manifold B.

1. Introductio n
Thi s paper s contain s tw o mai n results . Th e first on e ca n b e considere d a s th e ne w an d mor e simpl e proo f of partia l result s of A . Weinstei n [16]. Th e secon d on e is th e generalizatio n of th e mai n resul t of th e pape r [8] obtaine d b y J . Moser , an d i t wa s obtaine d b y Weinstei n i n th e pape r [9]. Ou r result s als o confir m th e well-know n V . I . Arnol d conjectur e tha t th e geometri c H . Poincar e theore m [1], [12], [27] ca n b e generalize d o n th e cas e of arbitrar y symplecti c manifold s an d arbitrar y perturbations . Le t u s not e tha t ou r resul t d o no t giv e ye t th e complet e generalizatio n of Poincar e theore m for th e arbitrar y perturbations , becaus e we hav e prove d thi s genera l resul t onl y for smal l perturbations . I t is possibl e tha t thi s resul t ca n b e extende d wit h th e hel p of techniqu e of paper s [8], [16]{[26]. Besides , th e first ou r resul t (theorem s 1, 3) generalize s th e resul t of [24] b y P . L. Ginzburg . Th e diflerenc e betwee n ou r wor k (see als o work s [9], [10], [15], [16] b y Weinstein ) an d th e pape r [24] is a s follows. Ginzbur g i n [24] ha d considere d th e periodi c system s an d obtaine d th e estimatio n for th e numbe r of periodi c orbit s for perturbe d systems . I t turn s ou t (see [9]) tha t th e sam e estimatio n tak e plac e als o i n genera l case . Namely , whe n th e syste m ha s submanifol d filled b y th e periodi c solution s of th e unperturbe d system .

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Th e secon d ou r resul t (theorem s 2, 4) is formulate d a s th e averagin g principl e o n a submanifol d [9]. Thi s resul t is th e generalizatio n of th e averagin g principl e o n a manifold , obtaine d b y Mose r i n th e pape r [8]. I n thi s pape r Mose r ha d considere d periodi c systems . Further , we formulat e tw o statement s (proposition s 2, 3) generalizin g theore m 1 i n th e followin g cases . 1) Whe n th e submanifold , filled b y periodi c orbits , contain s equilibriu m point s of th e system . 2) Whe n th e symplecti c structur e is perturbe d analogousl y t o Hamiltonian . I n th e first of thi s statement s we conside r th e situatio n whic h partia l case s ha d bee n studie d b y Weinstei n [9], [10] an d Mose r [13]. I n thes e work s th e submanifol d considere d b y u s coincide s wit h a n equilibriu m point , an d som e spetia l estimatio n is prove d for th e numbe r of periodi c orbit s nea r a n equilibrium . Th e secon d ou r generalizatio n is illustrate d i n separat e chapter , whe n we discus s som e author' s resul t o n celestia l mechanics . I n thi s situatio n th e initia l symplecti c structur e is degenerate , whil e th e generalise d Poincar e theore m remain s vali d unde r admissibl e perturbation s of th e symplecti c structure . Then , we give th e formulation s of bot h ou r theorem s 1 an d 2 (i. e. generalize d geometri c Poincar e theore m an d th e averagin g principl e o n th e submanifold ) for dynamica l system s of genera l type , i. e. no t necessaril y Hamiltonia n ones . Th e clos e result s hav e bee n obtaine d als o b y F.B . Fulle r [6] an d Mose r [8]. Bot h result s of th e presen t pape r ar e applie d no t onl y for th e cas e of standard , regula r fibrations , bu t als o for th e fibration s wit h singula r fiber s (i.e . singula r fibrations) . Namely , for th e multidimensiona l Seifert fibrations . Suc h fibration s appear , for example , i n th e analysi s of periodi c solution s far fro m th e equilibriu m point s of th e syste m (see work s [9], [10], [13] an d [15]). I n ou r pape r we introduc e th e notio n of th e circula r function s o n Seifer t fibration . W e give a n estimatio n for th e numbe r of periodi c solution s of th e perturbe d syste m i n term s of suc h functions . Le t u s not e tha t for arbitrar y fibration s ou r estimatio n seem s t o b e weake r tha n Weinstein' s on e [16], becaus e circula r function s no t alway s ca n b e projecte d ont o th e bas e of Seifer t fibration . I n addition , i n th e pape r [16] Weinstei n ha d considere d mor e wid e clas s of perturbations . Namely , h e assume d tha t C^{nor m of th e perturbatio n is small . I n additio n t o thi s assumption , we assum e tha t C^{nor m of th e perturbatio n is als o small . Bu t ou r proo f is mor e simpl e an d mor e geometric . Developin g Poincare' s ideas , we ar e base d o n th e classica l implici t functio n theorem . I n particular , i n ou r presen t pape r we d o no t conside r th e infinite-dimensiona l spac e of loops , i n contras t wit h th e pape r [16]. Le t u s note , that , b y ou r technique , for th e wid e clas s of fibration s we succeede d i n provin g th e Weinstein' s estimation . Namely , thi s is th e clas s of fibrations , whic h base s ar e locall y flat , for example , tori .

2. Generalize d geometri c Poincar e theore m
I n thi s sectio n we formulat e th e mai n resul t of th e paper . I n th e followin g sectio n we wil l formulat e th e averagin g principl e an d giv e proofs . Le t u s conside r th e Hamiltonia n syste m wit h Hamiltonia n functio n H o n symplecti c manifol d (M^n,!^) . Le t A C H~^(h) b e a close d submanifol d filled b y periodi c orbit s of thi s system . W e assum e tha t th e surfac e H~^(h) is regular , i.e. withou t singula r points . Le t u s conside r th e orientabl e smoot h fibratio n S^^Ap B (1)

wit h fibe r circle . Al l thes e fiber s ar e assume d t o b e periodi c orbit s o n A, an d th e bas e B is th e quotien t manifol d wit h natura l quotien t topology . We'l l assum e tha t A is no t a n isolate d periodi c orbit , i. e. it s dimensio n is greate r tha n 1. Le t u s assum e tha t ther e is th e continuou s (an d consequentl y smooth ) functio n o n A, whic h is th e perio d functio n T : A --?· M for th e periodi c orbit s o n thi s submanifold . Thi s mean s tha t o n A ther e is th e smoot h actio n of th e circle . Thi s actio n foliate s A int o th e regula r an d singula r fibers , whic h ar e homeomorphi c t o th e circle . Th e orbit s if thi s actio n coincid e wit h periodi c orbit s of th e give n

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syste m o n A. Th e tim e of motio n alon g thes e orbit s is prorortiona l t o th e natura l paramete r o n th e circl e wit h th e facto r ^ . I n particular , thi s actio n is locall y free (i. e. ha s n o fixed points) , becaus e we hav e assume d th e regularit y of th e isoenerg y surface . Suc h fibratio n of A b y circle s we cal l periodic fibration, o r Seifert fibration. Th e functio n T is no t necessaril y minima l perio d for som e orbits , whic h ar e calle d singular fibers. I n thi s cas e th e functio n T ha s th e for m T = (minima l period ) · k, wher e k is som e intege r numbe r dependin g o n th e fiber . I n th e cas e of locall y trivia l fibratio n th e functio n T is equa l t o th e minima l perio d everywher e o n A, an d th e quotien t manifol d B = A=S^ is smooth . Le t th e periodi c fibratio n (1) ha s singula r fibers . The n th e quotien t B wil l n o longe r b e smoot h manifold , bu t i t ha s th e structur e of so calle d generalised manifold, o r V{manifold [4], [5], [15]. Namely , th e functio n f o n B is calle d smoot h if th e invers e imag e p* f : A --)· M of thi s functio n is smoot h functio n o n A. Thus , critica l point s of th e tunctio n f coincid e wit h projection s of critica l orbit s for th e functio n p*f. W e cal l critica l poin t of functio n f nondegenerate , o r Mors e point , if th e correspondin g critica l orbi t of functio n p* f is Bot t critica l circle . Le t u s not e tha t ther e is natura l well-define d close d 2{for m o n th e considere d V{manifold . Namely , i t is th e "projection " o n B of th e restrictio n of th e symplecti c structur e t o th e submanifol d A. Thi s 2{for m ca n als o b e a symplecti c dtructur e o n B, see below . Le t u s conside r for arbitrar y orbi t 7 C A th e smal l transversa l cross-sectio n a C M^ calle d Poincare section. The n we conside r th e Poincar e mappin g A : a ^ a fro m thi s cross-sectio n int o itself. Thi s mappin g is determine d b y th e flow correspondin g t o th e initia l system . Th e tim e of th e motio n alon g th e orbi t fro m th e initia l poin t o n th e cross-sectio n t o th e imag e of thi s poin t unde r Poincar e mappin g is approximatel y equa l t o th e perio d T\^ . I t is clea r tha t th e intersectio n poin t m = 7 n a of initia l orbi t 7 wit h th e cross-sectio n cr is a fixed poin t of th e Poincar e mappin g A. Le t u s conside r th e linearizatio n dA(m) of thi s mappin g i n th e fixed poin t m. Definitio n 1 . Th e submanifol d A is calle d nondegenerate if th e fibration (1) is periodic , an d for an y orbi t 7 C A th e kerne l of th e operato r dA(m) -- I coincide s wit h th e tangen t spac e T^(^ n cr) t o th e submanifol d A n cr: ker(dA(m) - I ) = T,,( A n a) , wher e I is th e identit y operato r i n th e tangen t spac e T^c r t o th e Poincar e sectio n a. Le t u s conside r o n (M^",!^ ) th e perturbe d syste m wit h th e Hamiltonia n H, whic h is close t o H wit h respec t t o th e nor m C^ . Definitio n 2 . I n th e presen t paper , speakin g abou t a closed orbit of th e perturbe d system , we alway s assum e tha t thi s close d orbi t satisfie s a n additiona l conditio n of "almos t T{periodicity" . Thi s mean s tha t thi s orbi t is clos e t o th e submanifol d A, an d it s perio d is clos e t o th e perio d T\^ of som e unperturbe d orbi t 7 C A, whic h is close t o th e orbi t 7 . W e cal l th e manifol d (or generalize d manifold ) locally flat, if thi s manifol d ha s locall y fiat affine connectedness . W e conside r a t first th e cas e of periodi c fibrations of specia l type . Namely , th e cas e of trivia l fibration an d th e cas e of fibration wit h locall y fiat base . Theore m 1 . Let the submanifold A C M^ be filled by the closed orbits of the unperturbed system. Let this submanifold be closed (i. e. compact and without boundary), and is nondegen-rate. Let the fibration (1) of the submanifold A by periodic orbits be periodic and has special type, namely: A) either this fibration is trivial, B) or the base B = k=S^ of this fibration is locally flat (for example, is diffeomorphic to the multidimensional torus). Then the number of geometrically different closed orbits of perturbed system on the isoenergy surface H~^(h) is not less than the minimal number of critical points of the smooth function on the quotient manifold B. Besides, the number of such orbits with their multiplicities is not less than the minimal number of critical points of Morse functions on B.

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A s th e corollary , we obtain , tha t if th e fibratio n p is of specia l typ e (eithe r trivia l fibratio n o r th e fibratio n wit h locall y flat base) , the n th e numbe r of geometricall y different orbit s of th e perturbe d syste m is no t less tha n th e Lusternik-Schnirelma n categor y ca t B of th e quotien t manifol d B. Becides , if suc h fibratio n is locall y trivial , the n th e numbe r of suc h orbit s is no t less tha n th e su m P A(B ) of Bett i number s of th e manifol d B. Remar k 1. Locally flat manifolds, different from the tori, really exist, and their number is sufficiently large, see Wolf's book [29]. Let us remark also that the periodic fibration over locally flat manifold is not necessarily trivial. The restrictions on the topology of the fibration (1) are indeed not so important, i. e. the theorem 1 is valid also for arbitrary periodic fibrations, see [16]. No w we conside r i n mor e detail s th e cas e whe n Seifert fibratio n p is no t necessaril y trivial . W e introduc e th e notio n of circula r function s o n th e manifol d A. Le t u s denot e b y v th e vertica l vecto r field o n A, tangen t t o fiber s of th e fibratio n p, i. e. v is th e restrictio n of th e give n unperturbe d syste m o n th e submanifol d A. Le t v~ b e som e vecto r field o n A, whic h is clos e t o th e vertica l field v. Speakin g abou t th e nearnes s of vecto r fields , we alway s mea n thei r nearnes s wit h respec t t o th e C^{norm . Th e definitio n of almos t T{periodi c orbit s of th e field v is analogou s t o th e definitio n 2. Le t F b e a smoot h functio n o n A, whic h is th e first integra l of th e field v~, i. e. th e functio n F is constan t alon g eac h orbi t of thi s field. Definitio n 3 . Suc h functio n F o n A is calle d circular function for th e field v, if an y orbi t of thi s field, whic h is critica l for F , is close d orbit . I n othe r words , thi s orbi t is close d an d almos t T{periodi c orbi t of th e field v. W e cal l suc h a n orbi t critical circle of th e circula r functio n F. If, i n addition , al l critica l circle s of th e functio n F ar e Bot t critica l submanifold s for thi s function , the n we cal l th e functio n F Bot t circula r function . Further , we wil l cal l a functio n F o n A circular , if ther e exist s som e vecto r field v, clos e t o vertica l field v, an d satisfyin g t o al l propertie s liste d above . I n othe r words : 1) th e functio n F is constan t alon g al l orbit s of th e field v, an d 2) th e functio n F if th e circula r functio n for thi s field. Le t u s not e som e propertie s of circula r function s o n th e spac e A of Seifer t fibration . a ) Th e clas s of circula r function s include s th e clas s of al l smoot h function s o n th e quotien t manifol d B, becaus e an y functio n o n B is "th e image " of som e circula r functio n o n A for vertica l field v. b ) Critica l submanifold s of arbitrar y circula r functio n wit h finit e numbe r of critica l circles , ar e smoot h close d curve s (circles ) whic h ar e close t o th e fiber s of th e fibratio n p. c) I t is eas y t o see , tha t for an y nondegenerat e singula r fibe r 7 of Seifert fibratio n ther e exist s th e clos e orbi t ~ of th e field v. An d suc h a orbi t is critica l for circula r functio n F. Her e we cal l th e fibe r 7 of Seifert fibratio n nondegenerate , if th e following condition s ar e valid . Le t u s conside r th e smal l cross-sectio n i n A for 7 , an d conside r th e Poincar e mappin g of thi s cross-sectio n ont o itself, whic h is determine d b y th e flow of field v o n A. If th e spectru m of th e differentia l of Poincar e mappin g doe s no t contai n 1, the n suc h a n orbi t is calle d nondegenerat e for th e fibration (1) . I n analogou s way, we ca n defin e th e notio n of nondegenerat e submanifol d i n A, filled b y singula r fibers. Theore m 1* . Let A C M^ be the submanifold, filled by closed orbits of the system with Hamiltonian H . Let this submanifold be nondegenerate, closed (i. e. compact, without boundary) submanifold as in the theorem 1. But, in contrast with the theorem 1, the periodic fibration (1) of this submanifold is not necessarily trivial. Then: A) There are at least two closed orbits of the perturbed system on the surface H ~^(h) . B) Moreover, the number of such orbits is not less that the minimal number of the critical circles of the circular function on the manifold A . Also, the number of such orbits, counted with their multiplicities, is not less than the minimal number of critical circles of Bott circular function on A . A s th e corollary , we obtai n th e following roug h estimatio n for th e numbe r of close d orbit s of th e perturbe d syste m o n th e surfac e ~ ~^(h) . Namely , th e numbe r of geometricall y different close d orbit s of perturbe d syste m o n th e surfac e H~^(h) is no t less tha n ^ ca t A. If th e fibers of th e fibration p ca n

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b e contracte d int o th e point , the n th e numbe r of suc h orbit s is no t les s tha n ca t A. If th e fibration p is locall y trivial , the n th e numbe r of suc h orbits , counte d wit h thei r multiplicities , is no t less tha n 1 P / i (A) . Le t u s note , tha t th e estimatio n i n term s of th e bas e B, analogou s t o th e estimatio n obtaine d fro m th e theore m 1, doe s no t follows fro m thi s result , becaus e th e followin g inequalitie s ar e ca t A valid : 1 i (A) ^ Pi (B), -- j -- ^ < ca t B , wher e S 1 is th e fiber of th e fibration (1) . Ther e ar e catAS 1 fibrations, for whic h thes e inequalitie s becom e strong . Actually , a s i t wa s mentione d above , th e estimatio n fro m th e theore m 1 is vali d for arbitrar y fibrations, becaus e th e estimatio n i n term s of critica l point s of smoot h function s o n th e bas e B is vali d i n genera l case . See [14]. Le t u s not e tha t th e ite m A ) of th e theore m 1 is th e simpl e corollar y of th e theore m 1*. Le t u s underlin e tha t we d o no t restric t th e structur e of th e quotien t manifol d B wit h natura l 2{for m o n it , an d d o no t restric t it s dimensio n an d it s topology , excep t of th e cas e B ) of theore m 1. I n particular , ou r manifol d B is no t necessaril y symplectic , no t necessaril y isotropic , an d no t necessaril y orientable . Bu t i n som e specia l case s th e theore m 1* give s u s mor e convenien t an d mor e concret e estimation . Fo r example , Mose r [13] an d Weinstei n [15] note d tha t ca t B > 1 di m B + 1 = 1 (dim A +1 ) i n cas e whe n eithe r th e quotien t manifol d B is symplectic , o r (whic h is th e same ) th e multiplicit y of 1 i n th e spectru m of th e operato r dA(m) equal s dimB . Mor e surprisin g an d interestin g fact is that , th e previou s estimatio n is vali d als o i n th e cas e whe n A is homotopicall y equivalen t t o th e odd-dimensiona l sphere . See th e paper s of Krasnosel'ski i [3] an d Weinstei n [10]. Th e secon d of thes e paper s prove s th e correspondin g estimatio n for th e numbe r of close d orbit s of perturbe d system , withou t th e complicate d techniqu e use d i n th e pape r [16]. Her e we cal l th e quotien t manifol d B symplecti c wit h respec t t o th e natura l 2{for m o n it , if th e restrictio n of th e initia l symplecti c structur e ! ^ t o th e intersectio n of th e cross-sectio n a an d A is nondegenerat e 2{form . Remar k 2 . Let the fibration (1) be not locally trivial, i. e. it has singular fibers. It turns out that many closed orbits of perturbed system has minimal period close to the number T=k, but not to T, where k is an integer number greater than 1. Indeed, let us consider all orbits on A, which periods are factors of T=k. Let us assume that the periodic fibration is nondegenerate, i. e. for any k each connected component of union Ak of all such singular fibers is smooth submanifold in A. Besides, let this union be nondegenerate in the fibration (1) with respect to natural sense. Then periods of corresponding closed orbits of perturbed system be close to T=k. Indeed, we can apply theorem 1 to each of this components with period function T=k on Ak . Thus, for nondegenerate periodic fibrations minimal periods of many closed orbits of perturbed system will be close to T=k, but not to T, where k > 1. In particular, each isolated nondegenerate fiber remains under perturbation. In other words, this orbit generates the closed orbit of perturbed system with period closed to minimal period of the unperturbed orbit.

3 . Th e averagin g metho d o n a submanifold . Proo f of mai n result s
Le t u s assum e that , unde r th e hypothesi s of theore m 1*, th e initia l Hamiltonia n syste m is include d int o one-paramete r famil y of perturbe d Hamiltonia n system s dependin g o n smal l paramete r e wit h Hamiltonia n H = H + eH 1 +o(e), e ^ 0: (2) I t turn s ou t [9], tha t i n th e cas e of genera l perturbatio n unde r smal l |e| we ca n estimat e th e numbe r of close d orbit s of perturbe d syste m o n th e surfac e ~ ~ 1 (h) mor e exactly . Namely , we'l l "average " th e perturbatio n H 1 over th e periodi c orbit s o n A an d loo k for critica l submanifold s of th e obtaine d functio n o n A. Thes e critica l orbit s generat e close d orbit s of perturbe d system . Thi s is essentiall y th e conten t of th e averangin g metho d o n a submanifol d whic h wa s give n i n thi s formulatio n b y Weinstei n i n [9]. I n th e presen t pape r we giv e th e proo f different fro m Weinstein' s one . Ou r techniqu e goe s bac k

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t o Poincar e [1]. Als o i t ha s bee n exploite d b y G.Ree b [2], Mose r [8] an d G . A . Krasinski i [11]. Le t u s not e tha t thi s metho d generalise s th e averagin g metho d i n Euclidea n spac e [2], o n a manifol d [8] an d o n Liouvill e tor i [1], [11]. W e pas s no w t o mor e exac t formulations . Le t u s conside r th e restrictio n H = H 1\\ of th e perturbatio n t o A. Th e followin g function :
T(m)

H( m) = Z
0

H(j(m,t))dt,

m G A,

(3)

we'l l cal l averaged perturbation wit h initia l valu e 7(m , 0) = m, tha t th e obtaine d functio n H is well-define d "projection " of Bot t critica l orbit s of functio n

o n A. Her e 7(m , t) meen s th e periodi c solutio n of unperturbe d syste m an d T : A --)· M is th e continuou s functio n of period s o n A. I t is clea r o n A is constan t alon g eac h fibe r of fibratio n (1) . Consequently , ther e thi s functio n ont o th e quotien t B. Th e followin g theore m show s tha t H give ris e t o close d orbit s of perturbe d system .

Theore m 2 . Let A C M^ be a submanifold, filled by closed orbits of the unperturbed system. Let this submanifold be nondegenerate, but not necessarily compact. Let the perturbed Hamiltonian ~ depends smoothly on a small parameter e, i. e. it has the form (2). And let 70 C A be Bott critical orbit of the averaged perturbation H, see (3). Then there exists one-parameter family of closed orbits 7e C H ~ 1 (h) of the perturbed system. This family depends smoothly on the parameter of perturbation e under small e, and 7^ coincides with 70 under e = 0. I n case s whe n th e functio n H is no t constan t an d no t necessaril y Bott , on e ca n appl y th e followin g statements . Statemen t 1-A . Let, under the hypothesis of theorem 2, j C A be some orbit of the unperturbed system. Let this orbit be not critical for the averaged perturbation H, see (3). Then there exists such a neighborhood U of this orbit in M 2" , that for sufficiently small e on the surface U fl H ~ 1 (h) there are no closed orbits of the perturbed system with Hamiltonian H. I n orde r t o formulat e th e secon d th e statement , we introduc e th e followin g notion . Definitio n 4 . Le t m b e a n isolate d critica l poin t of th e smoot h functio n F : El^ --)· ffi, F(m) = 0. Le t u s conside r th e subcritica l se t M = { x G W!^\F(x) < 0 } of thi s function . Th e followin g intege r numbers : Pi (F,m) = rank H i (M,M\{m}), 0 < i < k, we cal l Bett i number s of th e functio n F i n th e poin t m. I t ca n b e show n su m ind m (xjF) = P on e of Bett i number s Hessia n d2 F(m), an d tha t th e inde x of gradien t of thi s functio n i n th e poin t m equal s th e alternate d (--1)Pi (F,m) of Bett i numbers . Besides , if m is Mors e critica l point , the n onl y is nonzero . Mor e exactly , i n thi s cas e Pi (F,m) = 1 if i is equa l t o th e inde x of Pi (F,m) = 0 for al l othe r i.

Statemen t 1-B . Let, under the hypothesis of theorem 2, ^ C A be (not necessarily Bott) isolated critical orbit for the function H on A. Let us consider the restriction F = H\^ of this function onto the cross-siction A to j in A. Let Pi (F,m) be Betti numbers of the obtained function in the intersection point m = 7 n A of 7 with this cross-section. Let us suppose, that at least one of this numbers is nonzero, and consequently their sum /? = P i (F? m ) i s positive. Then for any preassigned neighborhood U of the orbit 7 in M 2" , for sufficiently small e, on the surface U fl H ~ 1 (h) there is at least one closed orbit of system with Hamiltonian H of the form (2). In addition, the number of such orbits in U \H~1 (h), counted with their multiplicities, is at least (3. Proof of theorems 1-A, 1*, 2 and statements 1-A, 1-B. No w we'l l formulat e th e las t statement , whic h implie s al l statement s liste d here . Le t u s assume , tha t Hamiltonia n H of th e pertutbe d syste m is clos e t o H wit h respec t t o C^{norm , wher e r > 2. Le t is denot e e = \\H -- H\\cr.

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Statemen t 1-C . Let A C M h be nondegenerate submanifold, filled by closed orbits of the unperturbed system. Then there exists such a neighborhood U of this orbit in M^ n , depending only on the unperturbed system, that for any sufficiently small e > 0 there exists an embedding i : A ^^ U and a circular function S on the submanifold A, which have the following properties: 1) The image ~ = i(A ) of the submanifold A under the embedding i lies on the surface H~^(h). 2) Images under the embedding i of all critical circles of function S coincide with closed orbits of the perturbed system on U \ H ~^(h) . In particular, the submanifold ~ containes all closed orbits of perturbed system on U\H~^(h). 3) The embedding i is close to identity mapping, and the function --S is close to function H, obtained by averaging (3) of the perturbation Hi = (H -- H)=e over the closed orbits of the unperturbed system on A. 4) If the perturbed Hamiltonian H smoothly depends on the small parameter, then both embedding i : A M- M and function S on A together with the corresponding to it vector field v, close to the vertical, depend smoothly on this parameter. Her e we cal l tw o mapping s o r function s clos e t o eac h othe r if the y ar e e{close wit h respec t t o C r~^{metrics , an d th e nearnes s of tw o vecto r fields we conside r wit h respec t t o C r~^{norm .

Proof. W e spli t th e proo f of statemen t 3 int o severa l steps . Ste p 1. Le t u s fix som e afiine connectednes s o n th e isoenerg y surfac e H~^(h). Le t am C H~^(h) b e a smal l cross-sectio n i n th e isoenerg y surfac e t o th e orbi t passin g throug h th e poin t m G A. Le t u s conside r (dependin g o n m) Poincar e mappin g A of th e cross-sectio n am int o itself. Further , le t u s construc t th e co-kerne l of th e operato r dA(m) -- I. Mor e exactly , le t u s denot e b y D m = Im(dA(m) -- I) th e imag e of thi s operator . Le t surfac e 9m C Om b e transversa l t o thi s th e imag e D m i n (m an d ha s th e sam e dimentio n a s A \ Om . We'l l assum e tha t bot h constructe d surface s am an d 6m C am ar e compose d b y geodesic s passin g throug h th e poin t m an d depen d smoothl y o n th e poin t m G A. No w le t u s fix a smal l neigborhoo d U of th e submanifol d A i n M an d denot e b y U h th e intersectio n of thi s neighborhoo d wit h th e isoenerg y surfac e H~^(h). We'l l assum e tha t bot h of thes e surface s am an d 6m C am ar e determine d for al l point s m G U h , compose d b y geodesic s an d smoothl y depen d o n th e point . Le t u s transfe r th e affine connectednes s an d bot h constructe d fields of surface s ont o th e "perturbed " isoenerg y surfac e H~^(h), usin g som e diffeomorphis m H~^(h) --)· H ~^(h ) clos e t o th e identit y mapping . Le t u s denot e U h = U \ H ~^(h). Ste p 2. Le t u s conside r i n th e neigborhoo d U th e restrictio n of th e perturbe d syste m ont o th e isoenerg y surfac e H~^(h). Fro m eac h poin t m G U h = U \ H ~^ (h) of thi s surfac e we'l l mov e alon g th e integra l curv e m 9 m of th e perturbe d syste m durin g th e time , close t o th e perio d T\^ of th e orbi t 7 C A clos e t o 7~O. Le t u s denot e b y A(m) th e intersectio n poin t of thi s orbi t wit h th e cross-sectio n am . Th e obtaine d mappin g A : U h ^ U h we cal l th e perturbe d Poincar e mapping . I t is clear , tha t for th e unperturbe d syste m thi s mappin g coincide s wit h unperturbe d Poincar e mappin g A : U h ^ U h . Le t u s not e severa l propertie s of Poincar e mapping : A ) Fixe d point s of Poincar e mappin g A coincid e wit h point s i n U h , lyin g o n th e close d orbit s of th e perturbe d system . B ) Poincar e mappin g is symplectic , i. e. A preserve s th e restrictio n of th e symplecti c structur e ont o th e isoenerg y surfac e H~^(h). C) Poincar e mappin g "preserve s th e cente r of mass" . Mor e exactly , for an y close d curv e ^ C U h , th e integra l of 2{for m ! = !"^ over a chai n F , @F = 7 -- A(7) , "relating " thi s curv e an d it s imag e ~ (7 ) unde r Poincar e mapping , vanishes : / / r ! = 0: Her e 2{chai n F is homologou s t o th e "cylinder" , whic h is compose d b y th e geodesi c segment s relatin g eac h poin t m G 7 wit h it s imag e A(m) i n H~^(h). T o prov e th e las t propert y of Poincar e mappin g (i. e. preservin g of th e cente r of mass) , le t u s conside r th e difi"erential 2{for m ~ = ! -- dH A dt i n th e extende d phas e spac e M^ n xR. Asfa r a s th e whol e 2{chai n F C M^ n x R^ lies intirel y i n th e isoenerg y surfac e H~^(h) i n th e extende d phas e space ,

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th e result s of integratio n over thi s chai n of form s ~ an d ! wil l b e equa l t o eac h other . Le t u s close th e chai n F b y th e tub e compose d b y orbit s passin g throug h th e initia l curv e 7 . Th e obtaine d 2{cycl e we denot e b y [T]. B y virtu e of Hamilton' s principl e of th e leas t action , th e restrictio n of 2{for m ~ o n th e adde d tub e is zero . Therefore , we actuall y mus t prov e tha t th e integra l of thi s 2{for m over th e obtaine d 2{cycl e [T] vanishes . Fo r provin g thi s fact le t u s notice , tha t 2{for m ~ is co-homologou s t o unperturbe d 2{for m Q of th e sam e type , an d tha t homologou s curve s 7 give ris e t o homologou s 2{cycle s [F] i n M^ n x t . Fro m thes e fact s we conclud e tha t i t is enoug h t o verify th e "preservatio n of th e cente r of mass " onl y for th e unperturbe d system . An d moreover , we ma y restric t oursel f onl y t o curve s 7 lyin g i n A, becaus e th e valu e of th e integra l unde r consideratio n depend s onl y o n th e homolog y clas s of th e curv e 7 C M^ n . I t remain s t o notic e tha t for an y suc h curv e 7 C A th e chai n F is "degenerate" , sinc e i t turne s int o th e curv e 7 . Thus , th e integra l of th e symplecti c structur e over suc h a chai n vanishes . Ste p 3 . Le t u s determin e th e perturbe d submanifol d ~ a s th e se t of al l point s m G U h , suc h tha t A(m) G ~ m . I t is easil y t o see tha t for th e unperturbe d syste m thi s se t coincide s wit h A. B y th e implici t functio n theorem , for sufficiently smal l e, thi s se t ha s th e for m ~ = i(A) , wher e i is a n embeddin g clos e t o th e identit y mapping . Indeed , th e implici t functio n theore m is applicable , because , i n accordin g t o th e constructio n of th e surface s 6m , m G A, thei r tangen t space s T mdm ar e tranversa l i n T mCTm t o image s of operator s dA(m) -- I, wher e A is th e unperturbe d Poincar e mapping , I is th e identit y operato r i n T mUm . Le t u s not e that , b y definio n of th e se t ~ C ~/j , th e embeddin g i satisfie s th e propert y 1) fro m statemen t 1-C, an d al l close d orbit s of th e perturbe d syste m i n U h ar e automaticall y containe d i n th e constructe d submanifol d ~ = i(A) . Ste p 4 . Th e followin g ou r construction s ar e completel y analogou s t o Ginzburg' s ones , se e [16], wher e onl y periodi c system s wer e studied . Le t u s construc t i n U h th e smoot h functio n ^ b y th e followin g way. Thi s functio n is determine d u p t o a n additiv e constant . Th e difference of it s value s i n an y tw o point s m 0,mi G U h is equa l t o th e integra l of 2{for m ! over th e chai n r(m 0,mi), compose d b y geodesi c segment s relatin g eac h poin t m t wit h it s imag e A(m t ) unde r Poincar e mapping , wher e m t ^0 < t < 1, mean s an y pat h i n U h relatin g point s m 0 an d m i : "^(m 0 ) - 'l'(mi) = /
r{mo,mi)

!:

Th e las t propert y C ) of Poincar e mappin g (preservatio n of th e cente r of mass ) implie s tha t th e smoot h functio n 'I ' is globa l single-value d functio n i n U h , althoug h i t is determine d u p t o a n additiv e constant . Fo r definiteness , le t u s assume , tha t thi s functio n vanishe s i n th e poin t i(m 0 ) , wher e m 0 is som e fixed poin t i n A. I t is clea r tha t th e determine d i n suc h a wa y unperturbe d functio n 'I ' identicall y vanishe s o n A. Le t u s denot e b y eS th e functio n ^ o i o n th e submanifol d A, wher e i is th e embeddin g constructe d o n th e previou s step . Ste p 5. O n thi s ste p we give a n explici t representatio n for differentia l of th e functio n S, an d sho w tha t thi s functio n is circular . Le t u s denot e b y g(m,t),0 < t < 1, th e geodesi c segmen t relatin g th e poin t m = g(m,0) G U h wit h it s imag e A(m) = g(m, 1) . Fo r an y curv e g(t), 0 < t < 1, in U h we denot e b y ^t G T gi^t) U h , 0 < t < 1, th e paralle l carr y of th e tangen t vecto r ^ 0 ё T g^0) U h alon g thi s curve . I t follows fro m th e definitio n tha t th e differentia l of th e functio n ^ i n a n arbitrar y poin t m G U h is equa l t o f^ @g(mt ) @g(m,t) d^(m)-n = 0 !(^^^^,-^^@mr/)dt , W e conclud e tha t thi s differentia l ha s th e for m d'^(m)'q = Q(~@ t bilinea r for m o n U h : Q ( 0,^) = 0 ! ( t,^^@mr?)dt : ~ 'n^T m U h : |t=0?^) , wher e Q is th e followin g

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W e see, tha t i n an y fixed poin t m of th e mappin g A we hav e Q((,0,ri) = 1 !(^ 0, dA(m)'ri+r]), d"^(m) = 0, d '^(m)r i = Q(dA(m)ri -- rj^rj) = ! (dA(m)ri,rj) . I n particular , for unperturbe d system , for ever y poin t m G A an d tangen t vecto r r] G T mA, we hav e Q(*,1]) = ! (*,ri) . Fro m th e las t equalit y we mak e th e followin g importan t conclusion , whic h we prov e below . Namely , we conclud e tha t th e "orthogonality " of a vecto r ^ 0 G T m ~ m t o al l tangen t vector s rj G T m ~ (wit h respec t t o th e bilinea r for m Q) implie s ^ 0 = 0. Usin g th e dimentiona l reason , we obtai n tha t th e bilinea r for m Q give s ris e t o som e non-degenerat e couplin g betwee n tangen t space s of th e surfac e ~ m an d th e surfac e ~ n dm , for an y poin t m G ~ . Thus , o n ~ ther e exist s single-value d field of direction s f~ G T m ~ , m G ~ , clos e t o vertica l field, an d orthogona l (wit h respes t t o th e for m Q) t o al l vector s W e see tha t th e functio n S = 'I'| ^ is indee d circula r for th e uni t vecto r field v, tangen t t o th e field f~ of directions . I t remaine s t o prov e th e conclusio n fro m th e formul a Q(* , rj) = ! (* , rj), rj G T mA, mentione d above . I t is enoug h t o give th e proo f onl y for th e unperturbe d system . Le t u s notic e tha t th e imag e of th e operato r dA(m) -- I is skew-orthogona l t o it s kerne l (b y virtu e of symplecticit y of mappin g A). Consequently , b y th e nondegenerac y conditio n o n A, thi s imag e coincide s wit h th e skew-orthogona l complemen t t o T m (^ n am ) i n CTm . Fro m thi s fact an d b y constructio n of th e surfac e 6m , m G A, we see tha t an y tangen t vecto r t o thi s surface , whic h is skew-orthogona l t o th e subspac e T m (A n am ) , is equa l t o zero . Fro m th e implici t functio n theore m we obtai n th e propert y 4) fro m statemen t 1-C abou t th e smoot h dependenc e o n th e smal l paramete r of th e embeddin g i, th e vecto r field v an d th e functio n eS. Thi s implie s th e smoot h dependenc e o n paramete r of th e functio n S. Ste p 6. No w le t m G ~ b e an y critica l poin t of th e functio n ^| ~ ^ . We'l l sho w tha t thi s poin t is fixed unde r Pioncar e mappin g A, i. e. th e vecto r of "translation " 0= 77-^--lt=0 vanishes . Indeed ,
@t

thi s vecto r is tangen t t o th e surfac e 6m i n accordin g t o th e constructio n of th e submanifol d ~ . Fro m th e othe r hand , we hav e reall y assume d tha t vecto r ^ 0 is orthogona l t o al l tangen t vector s t o ~ (wit h respec t t o th e firm Q). Hence , i n agreemen t t o th e conclusio n o n th e previou s step , ^ 0 = 0. W e hav e prove d th e propert y 2) fro m statemen t 1-C and , consequently , theore m 1{A . Ste p 7. Le t u s prov e th e res t propert y 3) of statemen t 1-C, whic h implie s th e averagin g metho d o n th e submanifold . Le t u s conside r th e direc t produc t of th e extende d phas e spac e M 2n x Rt o n th e rea l axi s R wit h th e coordinat e e'. O n th e isoenerg y surfac e of eac h syste m wit h Hamiltonia n H ^' = H + e'H 1 , 0 < e' < e, le t u s conside r th e smoot h functio n ^ , constructe d above . B y analog y wit h th e secon d step , we'l l "close " 2{chai n r(m 0,m 1 ) b y th e chain , compose d b y orbit s of perturbe d syste m o n th e isoenerg y surface , an d integrat e 2{for m ~ o n th e obtaine d chain . Le t u s subtrac t fro m th e functio n S, determine d b y thi s way, th e unperturbe d (identica l zero ) function , an d le t e --)· 0. T o sho w tha t th e limi t equal s --H, we'l l appl y Stokes ' theore m t o th e integra l of th e for m O^/ = ! --d H ^'Adt over th e boundar y of th e infinitesima l 3{chain , compose d b y infinitesima l 2{chain s correspondin g t o system s wit h Hamiltonian s H ^i = H + e'H 1 , 0 < e' < e. Accordin g t o Hamilton' s principl e of th e leas t action , th e par t of thi s integra l correspondin g t o th e bot h infinitesima l "sides " of suc h 3{chai n vanishes . Consequently , th e require d limi t of th e functio n S is equa l t o th e limi t of th e integra l of th e differentia l -de' A dH 1 A dt of th e for m Q^i over th e 3{chai n unde r consideration , devide d b y e. Bu t th e las t limi t coincide s wit h th e functio n --H. Thi s implie s propert y 3) fro m statemen t 1-C. Statemen t 1-C is completel y proved . Fro m thi s statemen t on e ca n obtai n b y standar d techniqu e th e averagin g principl e (theore m 2) an d statement s 1-A, B . Theorem s 1{A , 1*, 2 an d statement s ar e proved . I t remaine s t o give a proo f of theore m 1{B . · Proof of theorem 1{B. Le t u s assume , tha t th e quotien t manifol d B = A=S 1 is locall y flat. W e clai m tha t i n thi s cas e th e embeddin g i an d th e functio n S (see statemen t 1-C) ca n b e constructe d i n suc h a wa y tha t th e functio n S is constan t alon g ever y orbi t of th e unperturbe d syste m o n A. Th e proo f wil l b e analogou s

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t o on e give n above . Ste p 1. Le t Um C H~^(h), b y analog y wit h th e previou s proof, b e a smal l cross-sectio n i n th e isoenerg y surfac e t o th e orbi t passin g throug h th e poin t m G A. Le t u s conside r (dependin g o n m) Poincar e mappin g A of th e cross-sectio n am int o itself. Further , le t u s construc t th e co-kerne l of th e operato r dA(m) -- I. Mor e exactly , le t u s denot e b y D m = Im(dA(m) -- I) th e imag e of thi s operator . I n contras t t o th e previou s proof, le t u s denot e b y Om C T mUm a subspac e (bu t no t a surface) , havin g th e sam e dimentio n a s A \ m an d transversa l t o th e subspac e D m i n T mCm . We'l l assum e tha t bot h conctructe d transversals : th e surfac e Um an d th e subspac e Om C T mOm depen d smoothl y o n th e poin t m G A. Le t u s conside r th e obtaine d fibratio n over A wit h fiber s Om , m G A. Le t u s notice , tha t th e subspac e 6m is natural y isomorphi c t o th e co-kerne l of th e operato r dA(m) -- I , i. e. t o th e quotien t spac e E m = (T mO'm )=D m , m G A. Le t u s not e som e propertie s of thi s quotien t fibratio n E --)· A wit h fiber E m,m G A. A ) Th e ran k of th e fibratio n E --?· A is equa l t o th e dimentio n dim B = di m A -- 1 of th e quotient manifol d B. B ) Th e natura l actio n of th e circl e o n A ca n b e extende d t o th e whol e fibratio n E, wit h us e of th e tangen t flow of th e give n system . I n othe r words , th e field of subspace s D m,m G A, is invarian t unde r th e tangen t flow, so tha t th e actio n of th e flow o n th e fibratio n E is well-defined . Besides , th e mappin g "ove r th e period " is th e identit y operato r o n th e fibratio n E. C ) Ther e exist s natura l isomorphis m (relate d wit h couplin g mentione d above ) betwee n th e fibratio n E an d th e subfibratio n of th e cotangenc e fibratio n T*A , consistin g of al l covector s vanishin g o n th e tangen t vecto r t o th e fibe r o n A. Namelly , unde r thi s isomorphis m an y vecto r ^ G E m is mappe d t o th e covector , whic h valu e o n th e arbitrar y vecto r r] G T mA equal s !^(^,77) . I t is clea r tha t suc h a n isomorphis m of fibration s is well-define d an d commute s wit h natura l actio n of a circl e S^ o n thes e fibrations . Thes e propertie s permi t u s t o realis e th e followin g construction . Le t u s conside r som e locall y flat (afhne) connectednes s o n th e quotien t manifol d B. Le t u s conside r th e correspondin g connectednes s o n th e cotangen t fibratio n T*B o n B, an d "lift " th e las t connectednes s o n A. A s a resul t we obtai n th e locall y flat connectednes s o n th e subfibratio n of th e cotangen t fibratio n T* A describe d i n propert y C) . Finally , usin g th e isomorphis m betwee n thi s subfibratio n an d th e fibratio n E, see C) , an d th e natura l isomorphis m betwee n th e fibratio n E an d th e field of subspace s 6m , m G A, we obtai n th e correspondin g connectednes s o n th e las t field of subspace s 6m , m G A. B y construction , we hav e tha t thi s connectednes s is flat , an d tha t th e actio n of th e circle , see B) , o n th e fibratio n is th e paralle l carr y wit h respec t t o thi s connectedness . I t implies , tha t for an y close d curv e i n A, homotopi c t o th e fibe r S^ , th e holonom y operato r is identical . Now , b y analog y wit h th e previou s proof, le t u s prolong e th e describe d constructio n o n th e whol e neighborhoo d of A i n M. Namelly , le t u s fix a smal l tubula r neigborhoo d U of th e submanifol d A i n M an d denot e b y U h th e intersectio n of thi s neighborhoo d wit h th e isoenerg y surfac e H~^(h). We'l l assum e tha t bot h transversals : th e surfac e am an d th e subspac e 6m C T mm ar e determine d for al l point s m G U h an d smoothl y dependen t o n th e point . Le t u s denot e b y Q th e fibratio n over th e neighborhoo d U h wit h fiber s 6m, m G U h . Le t u s conside r som e retractio n U/j --)· A of th e tubula r neighborhoo d ont o A. Usin g thi s retraction , on e ca n easel y construc t som e locall y flat connectednes s o n th e fibratio n Q over U h , suc h tha t it s restrictio n o n A coincide s wit h th e connectednes s constructe d above . Le t u s underlin e one s mor e th e following specia l propert y of th e constructe d connectednes s o n th e fibratio n Q: th e paralle l carr y alon g an y close d curv e i n U h , homotopi c t o th e fiber s o n A, is th e identit y operator . Finally , similarl y t o th e previou s proof, le t u s transfe r bot h constructe d fields of transversal s (am an d 6m C T mcrm ) an d th e connectednes s (o n th e fibratio n Q) ont o th e "perturbed " isoenerg y surfac e U h = U \H~^(h), wit h us e of som e diffeomorphis m H~^(h) --)· H ~^(h ) close t o th e identit y mapping .

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Ste p 2. Fo r th e bot h system s (i. e. unperturbe d an d perturbe d ones ) we her e construc t i n th e spac e @ D U h som e "extended " dynamica l syste m an d "extended " Poincar e mappin g A : 0 --)· 0 . Fo r convenienc e of notation s we'l l describ e th e constructio n onl y for th e unperturbe d system . Le t V b e th e vecto r field i n U h , correspondin g t o th e initia l dynamica l system . We'l l define th e dynamica l syste m o n th e "extended " spac e 0 b y presentin g th e integra l orbit s of thi s system . Namelly , a smoot h curv e m(t) G U h,^(t) G 0m(t) i n th e spac e of th e fibratio n 0 we cal l integra l orbit , if: 1) Th e "projection " m(t) of thi s curv e i n U h ha s th e velocit y dm(t)=dt = V,,(t ) + ^(t). 2) Th e field ^(t ) is paralle l alon g thi s curv e m(t). On e ca n sho w tha t thes e condition s giv e u s th e well-define d dynamica l syste m o n th e spac e of th e fibratio n 0 . I t is clea r tha t th e zer o sectio n of 0 is invarian t unde r thi s system , an d th e restrictio n of thi s dynamica l syste m t o th e zer o sectio n coincide s wit h th e initia l syste m V on U h . No w we ca n define th e extende d Poincar e mapping . Fro m eac h "point " m G U h,^ G 6m of th e spac e 0 we'l l mov e alon g th e integra l curv e m(t),^(t) of th e describe d syste m (m(0 ) = m , ^(0 ) = ^) durin g a tim e T(m,^) , close t o th e perio d Tj of th e orbi t 7 C A clos e t o m(t). Le t th e curv e m(t) intersect s th e cross-sectio n am over th e tim e T(m , ^) . The n we denot e b y A th e mappin g fro m th e spac e 0 int o itself, whic h move s th e "point " (m , ^) t o th e valu e of th e orbi t (m(t),^(t)) i n th e momen t t = T(m,^). Th e obtaine d mappin g A : 0 --)· 0 we cal l th e extended Poincare mapping. I t is clear , tha t th e restrictio n of thi s mappin g t o th e zer o sectio n of th e fibratio n 0 coincide s wit h th e usua l Poincar e mappin g A : U h ^ U h . I n analogou s way, on e define s th e extende d Poincar e mappin g A for th e perturbe d system . Le t u s ad d t o th e propertie s A){C ) of Poincar e mappin g (see ste p 2 of th e previou s proof ) som e propert y of th e extende d Poincar e mappin g A : D ) Le t u s conside r th e linearizatio n ^ A of th e extende d Pioncar e mappin g A i n an y fixed "point " (m , ^) , m G A, ^ = 0. Le t u s introduc e o n 0 loca l "coordinates " (m , ^) an d conside r th e representatio n of th e operato r 5A i n term s of "variations " (5m, S^) of thes e coordinate s i n th e poin t (m , 0) . Then , unde r thi s operator , an y vecto r ((5m, 6^) is move d t o som e vecto r havin g th e for m (5m + e T5^ + d, 5^). Her e T = T(m) > 0 denote s th e valu e of perio d functio n T i n th e poin t m G A, d G D m . Le t u s conside r th e imag e of th e submanifol d A i n 0 unde r th e natura l embeddin g a s th e zer o section . We'l l denot e thi s imag e als o b y A. I t is clear , tha t th e submanifol d A C 0 is filled b y close d orbit s of th e constructe d dynamica l syste m o n 0 . Moreover , on e ca n sho w wit h us e of th e propert y D) , tha t thi s submanifol d is nondegenerat e i n th e sens e of th e definitio n 1. Ste p 3 . Le t u s defin e th e perturbe d submanifol d ~ C 0 a s th e se t of al l "points " (m,^) G 0 , whic h image s unde r th e mappin g A hav e th e for m (m , *) . I t turne s ou t tha t thi s se t is filled b y close d orbit s of th e perturbe d dynamica l syste m i n 0 (whic h ha s bee n constructe d i n th e previou s step) . I n fact , it' s enoug h t o sho w tha t th e se t ~ coincide s wit h th e fixed poin t se t of th e extende d Poincar e mappin g A . Bu t thi s follows immediatel y fro m th e definitio n of ~ an d th e specia l propert y of th e connectednes s o n ~ , see ste p 1. B y th e implici t functio n theorem , for sufficiently smal l e, th e se t ~ is a smoot h submanifol d an d ha s th e for m ~ = i(A) , wher e i is a n embeddin g clos e t o th e identit y mapping . Indeed , th e implici t functio n theore m is applicabl e du e t o th e propert y D ) of th e extende d Poincar e mapping , see ste p 2. Sinc e th e submanifol d ~ is filled b y close d orbit s of th e perturbe d dynamica l syste m o n 0 , on e ca n choos e suc h a n embeddin g i, tha t image s of fiber s o n A unde r i coincid e wit h th e orbit s of th e dynamica l syste m o n 0 . Besides , th e tim e of th e motio n alon g thes e orbit s is proportiona l t o th e natura l paramete r o n th e circl e wit h facto r T , wher e T is th e perturbe d functio n of period s o n ~ . Further , we'l l denot e b y ~ als o th e submanifol d 7r( ~ ) i n U h , wher e TT : ~ --)· ~/j is th e natura l projection . I t is evident , tha t suc h ~ is close t o A. Ste p 4 . Her e we'l l define th e smoot h functio n ip o n th e quotien t manifol d B = A=S^ b y a n explici t formula . Le t 7 = {bt} C B b e a n arbitrar y smoot h curv e i n B. Le t u s conside r th e 2{chai n p"^(7 ) i n A, whic h is th e invers e imag e of th e curv e 7 , an d integrat e th e symplecti c structur e o n th e imag e of thi s chai n unde r th e embeddin g i. Th e obtaine d rea l numbe r is th e difference of value s of th e functio n

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·0 o n th e end s of th e curv e 7 : iP(b 0 ) - i>(b)=
jop-i(7 )

!:

Th e propert y C ) of Poincar e mappin g (see ste p 2 of th e previou s proof ) implie s tha t th e smoot h functio n tp is globa l single-value d functio n o n B. Fo r definiteness , le t u s assume , tha t thi s functio n vanishe s i n som e fixed poin t b 0 o n B. Sinc e unde r e = 0 th e functio n tp identicall y vanishe s o n B, we'l l denot e th e constructe d functio n b y eS. Ste p 5. Her e we give a n explici t representatio n for th e differentia l of th e functio n tp, an d we'l l conside r thi s functio n a s determine d o n th e whol e A. On e ca n sho w tha t th e differentia l of th e functio n tp i n a n arbitrar y poin t m G A ha s th e followin g form : dip(m)rj = Q m'(i-,T^ ° i*f]), wher e (m',^ ) = i(m) G 0 , ^ is th e analo g of th e vecto r of "translation" , Q mi is th e "coupling " betwee n subspace s ~ ' an d T m ( ~ n am') . Th e bilinea r for m Q is S 1 {invarian t unde r th e natura l actio n of th e circle , sinc e b y definitio n i t is obtaine d b y "averaging " of th e symplecti c structure . Fo r th e unperturbe d syste m we have : Q m (^.r])=T(m)!(^,7]), C G ^ m, ^ G T m (A n m ) : Fro m th e las t equality , usin g th e skew-orthogonalit y of th e considere d subspaces , we conclud e tha t th e equalit y Q m'(^iV) = 0 unde r an y 7 G T m ^ implie s ^ = 0. 7 Ste p 6. Finally , le t m G A b e an y critica l poin t of th e functio n tp. We'l l sho w tha t it s imag e i(m) is fixed unde r Pioncar e mappin g A, i. e. th e vecto r of "translation " ^ vanishes . Indeed , we hav e reall y assume d tha t vecto r ^ is "orthogonal " t o al l tangen t vector s t o ~ (wit h respec t t o th e fir m Q), see ste p 5. Hence , i n agreemen t t o th e conclusio n o n th e previou s step , ^ = 0. Thi s immediatel y implie s th e ite m B ) of theore m 1. Thus , theore m 1 is completel y proved . ·

4. Th e rol e of constanc y for value s of th e energ y an d th e perio d function . Som e generalizatio n
4.1 . Le t u s poin t ou t tha t theorem s 1 an d 1* impl y th e existenc e of th e require d numbe r of close d orbit s o n an y isoenerg y surfac e H~1 (h') clos e t o th e initia l one , bu t no t onl y o n th e surfac e H ~ 1 (h). I t follows fro m th e fact tha t on e ca n conside r th e energ y valu e a s th e additiona l paramete r of perturbation . I n an y case , theorem s 1 H 1* give estimatio n for th e numbe r of close d orbit s lyin g o n th e sam e isoenerg y surface . Bu t on e ca n obtai n th e sam e estimatio n if we fix th e perio d of require d close d orbits , instea d of th e energ y value . Her e we give th e analo g of theorem s 1,1 * an d 2 for th e cas e of estimatio n for th e numbe r of close d orbit s (of perturbe d system ) wit h give n period . I n contras t t o th e situatio n above , we assum e her e tha t th e perio d functio n T o n th e submanifol d A is constan t (for som e orbit s i t ma y b e no t minima l period) , for exampl e T = 1, bu t A doe s no t necessaril y lie o n th e isoenerg y surface . Besides , i n thi s situatio n A ca n contai n critica l point s of Hamiltonian , so tha t som e orbit s o n A ma y b e equilibriu m points . Le t u s defin e th e Poincar e mappin g a s th e flow A = g 1^ of th e syste m over th e tim e 1 i n th e whol e phas e spac e (bu t no t a s th e restrictio n of thi s flow t o th e isoenerg y surface , i n contras t t o th e situatio n above) . Th e deflnitio n of th e nondegenerac y is analogou s t o th e deflnitio n 1. Theore m 3 . Let the submanifold A, filled by 1{periodic orbits of the unperturbed system, be compact (without boundary) and nondegenerate. Then: A) The number of 1{periodic orbits of the perturbed system is not less than the minimal number of critical orbits of circular function on the manifold A. Besides, the number of such orbits, counted w it h their multiplicities, is not less than the minimal number of critical orbits of Bott circular function on the manifold A .

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B) Let the fibration of the submanifold A by periodic orbits be of special type, namely: either this fibration is trivial; or the base B = k=S 1 of this fibration is locally flat (for example, is diffeomorphic to the multidimensional torus). Then the number of geometrically different 1{periodic orbits of perturbed system is not less than the minimal number of critical points of the smooth function on the quotient manifold B. Besides, the number of such orbits, counted with their multiplicities, is not less than the minimal number of critical points of Morse functions on B. Theore m 4 . Let the submanifold A, filled by 1{periodic orbits of the unperturbed degenerate. Let us assume, that the Hamiltonian of the perturbed system depends small parameter e, i. e. has the form (2). Let us consider the averaged perturbation smooth function H (m ) = 0 H('y(m,t))dt, m G A, where H = L1\\. Let 70 be Bott this function. Then there exists one-parameter family of 1{periodic orbits 7^ C H ~ 1 systems. This family depends smoothly on the parameter of perturbation e, where small, and 7^ coincides with 70 under e = 0. Le t u s sho w 3 we conside r no (3) b y th e perio d o n A C H~1 (h). system, be nonsmoothly on the on A, i. e. the critical orbit of (h) of perturbed e is sufficiently

th e differenc e betwee n th e averagin g principl e i n theorem s 3 an d 5. I n fact , i n theore m t standar d averagin g over th e time , becaus e we d o no t divid e th e valu e of th e integra l functio n T . Thi s is essentia l becaus e th e perio d functio n T is no t necessaril y constan t Bu t i n som e of th e importan t case s th e perio d functio n turne s t o b e constant .

Statemen t 2 . Let T, be a connected submanifold of the phase space (M 2n , ! 2 ) , filled by closed orbits of Hamiltonian system with Hamiltonian H. Let the differential d(H\^ ) of the restriction of the function H to T, does not vanish anywhere on E . Let T be some continuous function of period (not necessarily minimal period) of the orbits on E . Then this function is the smooth function depending on the value h of Hamiltonian H on T.. Moreover, the period function has the form T = dI(h)=dh, where value I( h ) is equal to the integral of the form u2 over any 2{chain, composed by periodic orbits of the oneparameter family ^h' C S n H~1 (h'), h 0 < h' < h, where h 0 is some fixed value of the Hamiltonian H. I n othe r words , i n thi s situatio n th e perio d functio n T equal s th e derivativ e wit h respec t t o h of th e "area " I(h) , "swept " b y periodi c orbit s of th e for m h/ C S n H~1 (h'), wher e h 0 < h' < h. Proof. Ther e is geometri c proo f of thi s statement . I n ou r situatio n thi s proo f is completel y analogou s t o th e standar d proo f [7], [8], althoug h i n thi s paper s i t ha d bee n don e onl y for periodi c systems . W e giv e her e th e anothe r proo f of th e first par t of th e statement , base d o n th e averagin g principle . We'l l impos e som e additiona l conditio n o n S , namell y tha t eac h submanifol d A = E n H~1 (h) is almos t everywher e nondegenerat e i n th e sens e of definitio n 1. Le t H = H -- e be th e perturbe d Hamiltonian . The n th e perturbe d syste m o n th e initia l energ y level H~1 (h) is simila r t o th e unperturbe d syste m o n th e close energ y level H~1 (h + e) . I t is clear , tha t th e average d perturbatio n o n A coincide s wit h th e perio d functio n T . Thus , fro m statemen t 1{A , we conclud e tha t if T is no t constan t o n th e submanifol d A C 57, the n thi s submanifol d ca n no t b e include d int o th e famil y of analogou s submanifold s A^ C H~1 (h + e) , A 0 = A. Thi s contradic t t o th e conditio n tha t th e differentia l of th e functio n H\^ doe s no t vanis h anywher e o n A. Thus , unde r th e hypothesi s of propositio n 1, th e perio d functio n T wil l b e constan t o n A. Con sequently , i n thi s cas e we ca n us e th e usua l averaging : H (m ) = -T0 H(j(m,t))dt, m G A, so tha t theore m 2 an d statement s remai n valid . 4.2 . No w we pas s t o formulat e som e generalization s of theorem s above . A t first, we conside r th e sets , filled b y close d orbits , whic h lie o n th e singula r isoenerg y surface . Further , we conside r th e cas e whe n th e symplecti c structur e als o is perturbe d unde r th e perturbatio n of th e system . An d i n th e following sectio n we describ e th e proble m fro m th e celestia l mechanic s i n whic h th e generalise d Poincar e theore m is successfull y applied , althoug h th e unperturbe d 2{for m of th e symplecti c structur e is degenerate .

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B y analog y wit h th e situatio n of theore m 1, le t A b e som e invarian t close d subse t of th e isoenerg y surfac e H~^(h), filled b y close d orbits . Le t th e fibratio n of A b y close d orbit s is periodic , i. e. ther e exist s som e continuou s positiv e functio n T o n A, suc h tha t g H (m) = m, wher e g tj is th e flow of th e syste m over th e tim e t. I n othe r words , ever y orbi t 7 o n A is "closed " over th e tim e T\^ . But , i n contras t t o theore m 1, we'l l assum e tha t som e orbit s o n A ar e equilibriu m point s of th e considere d Hamiltonia n system . I n particular , th e give n isoenerg y surfac e H~^(h) is singular . · Definitio n 5 . W e cal l suc h a singula r subse t A C H~^(h) nondegenerate if al l it s singularitie s ar e of Mors e type . Mor e exactly , A is calle d nondegenerat e if i t satisfie s th e followin g conditions : 1) Firstly , al l equilibriu m point s i n A ar e Mors e critica l point s of th e Hamiltonia n functio n H. 2) Secondly , th e complemen t t o th e se t of equilibriu m point s i n A is nondegenerat e i n th e sens e of definitio n 1, see chapte r 2. 3) Finally , for an y equilibriu m poin t m G A an d tangen t vecto r ^ G T m M t o thi s point , th e followin g equalitie s unde r e = 0: dg H (m)^ = ^ an d d H(m)^ = e,

implie s tha t vecto r ^ is tangen t t o th e subse t A. Her e d'^H(m) denote s Hessia n (i. e. "quadrati c part" ) of th e functio n H i n th e critica l poin t m. Besides , her e we cal l vecto r ^ G T m M tangent t o th e subse t A if thi s vecto r ha s th e for m ^ = dm(t)=dt\t=o for som e smoot h curv e m(t) C A, 0 < t < 1. Le t u s appl y a smal l perturbatio n t o th e Hamiltonia n H, unde r whic h th e "energ y level " become s regular . I n othe r words , th e perturbe d isoenerg y surfac e H~^(h) doe s no t contai n critica l points . Befor e th e formulatio n of th e result , we'l l construc t th e smoot h submanifol d ~ C M , obtaine d fro m A b y som e surgerie s nea r eac h equilibriu m poin t m G A, analogou s t o th e usua l Mors e surgery . T o ever y equilibriu m poin t m G A we attac h a sig n m ^ = ± 1 i n th e following natura l way. Thi s sig n depend s o n that : if th e valu e h is greate r o r smalle r tha n th e critica l valu e of th e perturbe d functio n H in a smal l neighborhoo d of th e poin t m. Th e se t {m --)· m } of thes e sign s we cal l the type of the perturbation. Th e following constructio n we'l l appl y t o eac h equilibriu m poin t m G A: 1) Firstly , we remov e fro m A th e poin t m bot h wit h a smal l bal l D m , havin g m a s th e center . 2) Secondly , we conside r th e smoot h submanifol d i n T m M, consistin g of al l tangen t vector s ^ G T m M, satisfyin g equation s abov e wit h th e correspondin g valu e e = em . 3) An d finally, we replac e th e removin g par t of A i n D m b y th e smoot h "handle " describe d jus t now . I n othe r words , we past e thi s "handle " t o A\D m b y natura l one-to-on e correspondenc e of thei r boundaries . A s th e result , we obtai n a close d smoot h submanifol d ~ C A U D m C M close t o A. W e see fro m th e constructio n of thi s manifold , tha t i t depend s onl y o n th e typ e of th e perturbation . Besides , ther e exist s a natura l fibratio n ~ --? B o n thi s submanifol d wit h fibe r th e circle . Finally , unde r th e · natura l projectio n ~ --)· A al l fiber s ar e transfere d t o orbit s of th e initia l syste m o n A. Th e obtane d submanifol d ~ we cal l Morse surgery of th e initia l se t A unde r th e give n perturbation . Propositio n 1 . Let A be a compact see def. 5, filled by closed orbits. Let given type. Then the estimation above system (see theorem 1) remain valid. with fibration on it) should be replaced nondegenerate singular subset of the isoenergy surface H~^(h), ~ be the Morse surgery of this set under a perturbation of the for the number of periodic orbits of the perturbed Hamiltonian More exactly, in the estimation of theorem 1 the set A (both by the manifold ~ with corresponding fibration on it.

Actually , an y submanifol d A, whic h doesn' t contai n equilibriu m points , ca n b e considere d a s th e partia l cas e of subset s containin g suc h points . Fro m thi s poin t of view , propositio n 2 generalize s theore m 1.

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Remar k 3 . Partial case of the described situation had been studied by Moser [13] and Weinstein [9], [10], [15]. Namelly, if speaking in our notations, in these works set A was an equilibrium point, the submanifold ~ was diffeomorphic to sphere, and the perturbation of the Hamiltonian was the identical e.

4.3 . Le t u s not e tha t i n al l theorem s formulate d abov e we assume d tha t th e symplecti c structur e ! ^ is fixed o n th e manifol d M^ n . Sometime s i n application s th e symplecti c structur e change s unde r perturbation . I n turne s ou t tha t i n som e of suc h case s ou r theorem s remai n valid . Fo r example , i t is so if eithe r H^(A ) = 0, o r H^(B) = 0, o r th e topolog y of th e fibratio n (1) satisfie s som e mor e wea k conditio n tha n th e previou s ones . Her e H^(A ) is th e 2{dimentiona l d e Rha m cohomolog y grou p of th e manifol d A. Bu t i n man y cases , importan t t o application s (for example , whe n A is torus) , ou r theorem s wil l no t remai n vali d for arbitrar y smal l perturbation s of th e symplecti c structure . Nevertheless , on e ca n generalis e them , imposin g som e restrictio n t o th e perturbation s of ! ' , natura l for Hamiltonia n mechanics . Namelly , usin g proof s of ou r theorems , on e easil y prove s th e followin g statement . Propositio n 2 . Theorems 1, 1*, 2, 3, 4 remain valid, if under the perturbation of the system we permit C^{small perturbation of the symplectic structure ! ' , such that the cohomology class of ! does not shange under perturbation. Moreover, the last condition can be replaced by the following (more weak) condition, called "preserving of the center of mass", see chapter 3 (and also works [12], [18], [24]). Namelly, this condition means that for any closed curve 7 in B the integral of the form !~^ over the 2{chain p~^('j) vanishes, see [24]. Let us note that for the unperturbed 2{form ! this condition is always satisfied. Le t u s underlin e one s mor e tha t we don' t impos e an y restrictio n t o th e initia l symplecti c structur e !^ , sinc e al l of require d condition s wil l b e satisfie d automatically . Moreover , i t turne s ou t tha t eve n if th e unperturbe d structur e ! ^ is degenerate , bu t th e perturbe d on e is "admissible" , the n ou r theorem s remai n valid . Fo r example , suc h situatio n appear s i n stud y of relativel y periodi c motion s of planet satellit e system , whic h we discus s i n th e followin g section . Th e simila r resul t for plane t syste m wit h doubl e planet s is discusse d i n th e author' s wor k [28]. Anothe r example s illustratin g suc h a n applicatio n of th e generalize d geometri c Poincar e theore m ar e S. V . Bolotin' s an d A . I . Neishtadt' s result s o n th e existenc e of periodi c motion s of a charge d particl e i n multidimentiona l stron g magneti c field (or periodi c motio n of mechanica l syste m unde r larg e gyroscopi c forces) .

5. Periodi c motion s of a planet-satellit e syste m
Le t u s conside r th e proble m of celestia l mechanics , abou t 1 materia l point s i n Euclidea n plane , attractin g t o eac h othe r tha t on e of thes e point s M 0 (sun ) is centra l poin t of mas s 1. point s Ml,:: : , M N ar e decompose d int o tw o groups : n "planets "satellites " wit h masse s of smalle r orde r ij,u, wher e /j, an d u ar e betwee n point s b e Newtonia n wit h th e potentia l th e motion s of th e syste m of N + du e t o Newton' s law . W e ussum e Further , we ussum e tha t al l othe r " wit h masse s of orde r // , an d N -- n smal l parameters . Le t th e attractio n

U= -

E 0
mm
ij

j

Her e m i denote s th e mas s of th e poin t M i , an d r i j = \M j -- M i| , 0 < i,j < N, distance s betwee n th e points . Thus , th e motio n is describe d b y th e followin g equations : -@U=@M i , 0
ar e mutuall y m i d M i =dt'^ = an d se t u p th e respec t t o th e over th e tim e

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T > 0 th e configuration s togethe r wit h velocitie s of al l point s of th e syste m ca n b e obtaine d fro m th e initia l one s b y th e rotatio n over th e sam e angl e a(mod2-K) aroun d th e origin . Definitio n 6 . Suc h a motio n we cal l relatively periodic wit h parameters T, a. Le t u s conside r th e othe r motio n whic h ca n b e constructe d fro m th e initia l on e b y th e followin g way. A t first, on e ca n mov e th e origi n of th e tim e axis , an d the n on e ca n rotat e th e plan e aroun d th e origin e over som e angle . I t is clea r tha t al l th e motion s obtaine d b y suc h a wa y remai n relativel y periodi c wit h th e sam e parameter s T , a. Al l of thes e motion s we'l l conside r a s th e sam e relativel y periodi c motion . Le t u s not e tha t th e se t of thes e motion s is diffeomorphi c t o th e two{dimentiona l torus . Pu t e = 27r=T. We'l l assum e tha t th e positiv e number s // , u an d e ar e small , functionall y independen t parameters . Anothe r parameter s of ou r proble m wil l b e th e fixed numbe r a an d th e se t of N -- 1 intege r number s whic h order s wil l b e dependen t o n e. Le t u s describ e thes e parameter s mor e precisely . A t first, le t u s introduc e mor e convenienc e numeratio n of points . Le t u s remind , tha t we implicitl y assum e tha t eac h satellit e is associate d wit h som e planet . Fo r notatio n of suc h a correspondenc e we introduc e th e ne w numeratio n of point s wit h us e of tw o indexes : M ij , 1 < i < n , 0 < j < n i . Her e Moo is th e sun , Mio, 1 < i < n, an d M ij, 1 < j < n i, ar e planet s an d thei r satellites , respectively . Satellite s wit h thei r plane t compos e th e so-calle d i{t h satellit e system . We'l l als o designat e th e masse s of planet s a s // m iO, an d masse s of satellite s a s jjum ij , wher e m ij = const > 0, 1 < i < n , 0 < j < n i . Le t u s note , tha t we stud y specia l solution s of th e describe d N + 1 bod y problem . Namely , we assum e tha t th e mutua l distance s betwee n eac h plane t an d it s satellite s hav e th e orde r 1, whil e th e mutua l distance s betwee n th e su n an d al l planet s hav e th e mor e greate r orde r R > > 1, whic h is compute d b y th e formul a e = 1= pJiR, an d consequently , i t is automaticall y large . I n accordin d t o this , we introduc e th e "relativ e coordinates" . Namelly , fro m eac h plane t we dra w radiu s vector s y ij = M ij - M io, 1 < j < n i,

t o al l it s satellites . Besides , for eac h satellit e syste m we conside r th e normalize d radiu s vecto r xi =(C - Moo)=R, 1
drawin g fro m th e su n t o th e cente r of mas s Ci = (TOiOMO + ^ Y^ni m ij M ij ) = ( mio + '^Y^nim ij ) of thi s satellit e system . (Actually , i t shoul d b e considere d on e mor e radiu s vector : drawin g fro m th e origi n t o th e cente r of mas s C of th e whol e syste m of points . Bu t sinc e C is "cycli c variable" , on e ca n eithe r eliminat e it , o r pas s t o th e fram e wit h origin e i n C , whic h is equivalent. ) Further , we assum e tha t "months " ar e muc h smalle r tha n "years" . Mor e exactly , afte r rescalin g th e time , we assum e tha t "mea n frequencies " of rotation s of radiu s vector s y ij ar e "fast " of orde r 1 (months) , an d "mea n frequencies " of rotation s of radiu s vector s x ar e "slow " of orde r e (years) . Le t u s conside r th e se t of "mea n relativ e frequencies " of rotation s of describe d radiu s vector s unde r som e relativel y periodi c motion . I t is clea r tha t thi s se t is proportiona l t o som e se t of intege r numbers , i. e. thi s se t is maxima l resonance . Indeed , th e se t of "mea n frequencies " is of th e for m ! = ! i + k ie (1 < i < n ) ( 1 < j < n i ), (4)

Oi j = ! i + K ije

wher e ! I = ea=2-K, an d Ai , K ij ar e som e intege r numbers numbe r of rotation s of th e correspondin g radiu s vecto r over frame) . Her e we assume , withou t loss of generality , tha t ki equal s th e whol e angl e o n whic h th e first planet s rotate s

. Namely , eac h of thes e number s is th e th e tim e T (wit h respec t t o th e potatin g = 0, so tha t th e rea l numbe r a = ! I T over th e tim e T . Thus , al l th e intege r

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E.A.KUDRYAVTSEVA

number s ki = 0, k 2,:: : ,k n , correspondin g t o slow frequencie s (of planets ) ar e bounded , whil e th e intege r number s K ij correspondin g t o fast frequencie s (of satellites ) hav e th e orde r 1=e: ki =0,k 2 ^ ::: ^ k n ^ 1, K ij « 1=e, 1 < i < n, 1 < j < ni : (5)

Le t u s conside r a s th e unperturbe d syste m th e se t of N independen t Keple r problems : ( d 2 x= d t 2 =-2 x i =\iif (1
d 2 y ij =dt 2 = -m i y ij =\y ijf I t is easil y t give n se t of frequencies . space . Le t u s havin g th e a = ! iT .

o see tha t unde r a ^ 0 (mo d 27r) relativel y periodi c motion s of unperturbe d proble m wit h mea n frequencie s (4) correspon d t o circula r motion s of Keple r problem s wit h thi s se t of Consequently , orbit s of thes e motion s compos e th e N{dimentiona l toru s i n th e phas e fix th e sufficiently smal l valu e of paramete r e. Le t ! j , Qi j , b e an y se t of rea l numbers , for m (4) , (5) . Le t u s specif y value s of parameter s T^a i n definitio n 6, settin g T = 27r=e,

Propositio n 3 . A) Under any a ^ 0 (mo d 2-K), if /j,, v are sufficiently small, then there exist at least 2 N~2 (counted with their multiplisities) relatively periodic motions of planet-satellite system, with parameters T, a. Besides, among these motions there are at least N -- 1 geometrically different motions. B) Under each of these motions, the mean values of the radius vectors x = ( Ci -- MQ)=R and y ij = M ij -- M iQ are exactly equal to ! i, Qij , and the motion of each of these radius vectors is close to the circular motion with the same frequency of the corresponding Kepler problem (6). Her e th e nearnes s mean s a s follows. Th e motio n x i wit h frequenc y (!=e ) b y th e valu e of th e orde r u=R 2 . Th e motio n b y th e valu e of th e orde r e2 . Her e we impl y tha t close wit h respec t t o C^{nor m over th e "ow n time " r an d = x i (T=) differs fro m th e circula r motio n motio n y ij = y ij (t) differs fro m th e circula r th e correspondin g coordinat e function s ar e t respectively .

Proof.
Le t u s fix som e valu e e > 0, an d conside r th e unperturbe d problem , correspondin g to /j, = u = 0. On e show s tha t suc h a n unperturbe d proble m is decompose d int o n Keple r problems , correspondin g t o planets , an d N -- n problem s correspondin g t o satellite s an d coincidin g wit h th e unperturbe d Hil l problem . Mor e exactly , th e unperturbe d syste m ha s th e for m d2 x=d t 2 = - 2 x i =lx i p d2 y ij =dt 2 = -m i y ij =\y ijf + 2 m i =m ij @P=@y(xi,y ij ) wher e th e functio n P(x , y ) = ^-------^ (1 < i < n ) (1 < j < n i
)->

, multiplie d b y 2 is th e unperturbe d "potentia l of

th e actio n of su n o n a satellite" . W e obtain , that , unde r smal l e, variable s x an d y ar e automaticall y slow an d fast , respectively . Th e res t of th e proo f is base d o n th e geometri c Poincar e theorem , namelly , o n theore m 3{B . Th e difficulty lies i n th e fact tha t thi s theore m ca n no t b e applie d directly , sinc e th e unperturbe d syste m is no t Hamiltonian . I n particular , we mus t verify th e conditio n "preservatio n of th e cente r of mass" . Bu t i n ou r cas e i t is evident , becaus e th e perturbe d symplecti c structur e is th e standar d on e o n th e cotangen t spac e of th e configuratio n space , an d automaticall y is exact . Mor e detaile d proo f th e autho r hope s t o publi c i n th e neares t future . · Remar k 4 . In the case of the planet system without satellites (n i = 0) the analogous result had been obtained by Krasinskii in the work [11]. In this work Krasinskii not only proved the existence of the

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relatively periodic solutions, but at first he also found their "generatives" (namely, 2^~ ^ solutions, for which there exists such a moment when all points lie on the same line). But here he used the additional reason about the "reversibility" of the N + 1 body problem. And, following Poincare technique [1], Krasinskii partially proved the stability of one of these solutions. Let us note also, that in the case of the system with double planets, the smallness condition on the parameter v is unessential (see author's work [28]). If, in addition, the number N + 1 of points equals 3, i. e. the system is of the type Sun{ Earth{Moon, then the statement remains valid without smallness condition on parameters /j, and u. The corresponding solutions are well known and called generalized Hill solutions.

6. Non-Hamiltonia n cas e
I n conclusion , we giv e formulation s of ou r mai n result s for dynamica l system s of genera l typ e (no t necessaril y Hamiltonian) . Le t u s conside r o n a n arbitrar y smoot h manifol d M " th e dynamica l system , define d b y som e vecto r field V. W e assume , tha t i n th e phas e spac e M " ther e exist s a smoot h submanifol d A, whic h doe s no t contai n singula r point s an d is filled b y periodi c orbit s of th e system . W e als o assume , tha t th e fibratio n (1) of thi s submanifol d b y periodi c orbit s is periodic , i. e. o n A ther e is a continuou s functio n T , whic h is equa l t o th e (no t necessaril y minimal ) perio d of th e orbits . B y analog y wit h Hamiltonia n case , for a n arbitrar y poin t m G A we conside r a cross-sectio n CTm 3 m an d defin e th e Poincar e mappin g A : am --^ CTm of thi s cross-sectio n ont o itself. Namely , thi s mappin g is define d b y th e flow of th e vecto r field V over th e tim e clos e t o th e valu e T(m) of perio d functio n i n poin t m. Withou t loss of generality , we'l l assum e late r tha t th e perio d functio n T o n A is th e identica l 1. W e assume , tha t th e submanifol d A is nondegenerat e i n th e sens e of definitio n 1, i. e. i n eac h poin t m G A th e spac e of al l tangen t vectors , fixed unde r th e tangen t mappin g dA(m), coincide s wit h th e tangen t spac e t o th e surfac e A n am i n th e poin t m. A s i n proo f of theore m 1{B , we conside r o n A a field of th e subspace s E m = (T mam)=Im(dA(m) -- I), m G A, whic h reall y ar e co-kernel s of th e operator s dA(m) -- I , m G A. Th e obtaine d field E of th e quotien t space s is som e linea r fibration E --)· A over th e submanifol d A. I t is easil y t o see tha t thi s fibration doe s no t depen d o n th e surface s am . Mor e exactly , th e fibration E is well-deflne d u p t o a natura l fiber isomorphis m of fibrations. A s i t wa s notice d above , see sectio n 3 , th e natura l actio n of th e circl e o n A ca n b e extende d t o th e whol e spac e E b y th e eviden t way. Namely , thi s actio n is define d wit h us e of th e tangen t flow of th e give n dynamica l system . I n particular , th e mappin g "ove r th e period " is th e identit y operato r o n th e fibration E. Thi s implie s tha t th e fibration E ca n b e "projected " ont o som e fibration over th e quotien t B = A=S^. Th e resultin g fibration we'l l denot e b y pE. Fro m th e nondegenerac y conditio n o n A we obtain , flrstly, tha t th e ran k of th e linea r flbration pE over B equal s th e dimentio n dim B = di m A -- 1 of thi s quotien t manifold , and , secondly , tha t th e tota l spac e pE of thi s flbration is orientable . Consequently , th e Eule r clas s e(pE) of thi s fibration is a n intege r number , whic h is calle d Eule r numbe r of th e fibration pE . Le t u s remin d th e "geometric " sens e of th e Eule r number . I t is equa l t o th e algebrai c numbe r of zero s for an y genera l sectio n S : B ^ pE of th e linea r flbration pE over B. Le t u s conside r o n th e phas e spac e th e perturbe d vecto r fleld V, whic h is close t o V wit h respec t t o th e nor m C^. Th e following theore m is analogou s t o theore m 1. Theore m 5 . Let the submanifold A C M this submanifold be closed (i. e. compact that: A) either the periodic fibration (1) of B) or the constructed above fibration connectedness. be filled by the closed orbits of the unperturbed system. Let and without boundary), and nondegenerate. Let us ussume this submanifold by periodic orbits is trivial, pE over the quotient manifold B = A=S^ possesses

locally

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Then the number of geometrically different closed orbits of perturbed system is not less than the minimal number of zeros of section S : B ^ pE of the linear fibration pE over B. Besides, the number of such orbits, counted with their multiplicities, is not less than the minimal number of zeros of the general section of the fibration pE. In particular , if th e Euler number e(pE) of th e fibration p E does not vanish, the n th e perturbe d system has a t least one closed orbit . Besides, th e number of such orbits wit h thei r multiplicities is a t least equal t o th e absolute value \e(pE)\ of th e Euler number of thi s fibration. Let us now describe th e averaging metho d on a submanifold for arbitrar y dynamical systems. Let th e perturbe d vector field has th e form V = V + eVi+o(), (7)

where e is a small parameter . Let us consider th e restrictio n V = VI|A of th e perturbatio n Vi t o A, and define th e averaging V of th e obtaine d vector field by th e formula V= /
0

g tlB Vdt,

where

g t B V = dg-t V(g t\A),

where g t denotes th e fiow along th e vector field V over th e tim e t. Let us remind, tha t we assume withou t loss of generality, tha t th e period function T on A is th e identical 1. Th e obtained vector field V on A we project onto th e quotient fibration E = Um<^\(T mm )=Im(dA(m) -- I) wit h use of th e natura l projections im : T mO --)· Em , m G A. One can show tha t th e obtained section TTV of th e fibration E is invariant unde r th e natura l action of th e circle S^ on E. Hence, thi s section can b e "projected" onto some section pnV of th e linear fibration pE. Let us call thi s section the averaged perturbation, since it is analogous t o th e averaged perturbatio n V. in Hamiltonian case. One proves tha t thi s definition of th e averaged perturbatio n is well-defined, i. e. it does not depend on th e surfaces Th e following theorem is analogous t o theorem 2 for dynamical systems of general typ e (not necessarily Hamiltonian) . Theore m 6. Let the submanifold A C M, filled by closed orbits of the unperturmed system, is nondegenerate, but not necessarily compact. Let us assume, that the perturbed vector vield V depends smoothly on a small parameter e, i. e. it has the form (7). And let b 0 G B be a nondegenerate zero of the averaged perturbation pTr(V). Let 70 = p~^(b 0 ) C A be the orbit of the unperturbed system, corresponding to the point b 0 . Then there exists one-parameter family of closed orbits 7^ of the perturbed system. This family depends smoothly on the parameter of perturbation e under small e, and 7^ coincides with 70 under e = 0. We see that in the case of dynamical systems of general type, i. e. not necessarily Hamiltonian, one must know the topology of some linear fibration pE over quotient manifold B. This fibration depends only on the unperturbed system. It turnes out that in many important cases this fibration is fiber-isomorphic to the tangent fibration T^^B of the manifold B. Statemen t 3 . In the following cases the linear fibration pE over the manifold B = A=S^ constructed above is fiber-isomorphic to the tangent fibration T^^B of this manifold B: 1) when the unperturbed system is either Hamiltonian, or it is the restriction of some Hamiltonian system to the regular isoenergy surface M h = H~^(h); 2) when the unperturbed system is gradient, i. e. V = \/F, where F is some smooth function on M; 3) when for each point m G A the multiplicity of 1 in the spectrum of the operator dA(m) is exactly equals to the dimentional of the quotient manifold B.

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In particular , in all of these cases Euler number e(pE) of th e fibrationp E coincides wit h th e Euler characteristic x(B) of th e quotient manifold B. Thus , if th e Euler characteristic of manifold B for such a system does not vanish, then , according t o theorem 5, th e perturbe d system has at least one closed orbit .

Acknowlegmen t
Th e autho r thank s N. N. Nekhoroshev for setting of a problem, V. V. Kozov, S. V. Bolotin, A. D. Bruno , A. I. Neishtadt , V. M. Zakalukin an d V. V. Trofimov for usefull discussion, an d also A. T . Fomenko for th e attentio n t o th e reseach. Thi s reseach was supporte d by G. Soros fund (ISSEP, grant a98-1768).

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i n Hamiltonia n Mechanics . Berlin . Springer . 1996 . [28] E. A. Kudryavtseva.

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