Bifurcation diagrams and critical subsystems of the Kowalevski gyrostat in two constant fields

Bifurcation diagrams and critical subsystems of the Ko w alevs ki gyrostat in t w o constan t fields Mikhail P . Kharlamo v Gagarin Str e et 8, V olgo gr ad, 400131 Russia e-mail: mharlamov@vags .ru Abstract The Kow alevski gyrostat in tw o constant fields is known as the u nique example of an integrable Hamiltonian system with th ree degrees of freedom not reducible to a f am- ily of systems in few er d imensions and still having the clear mechanical interpretation. The practica l explicit in tegration of this system can hardly b e obtained by the existing techniques. Then t he challenging problem b ecomes to fulfil the qualitative inv estigation based on the study of t he L iouv ille foliation of the phase space. As the first app roac h to top ological analysis of this s ystem we find the stratified critical set of the momen tum map; this set consists of the tra jectories with number of frequencies less than three. W e obtain the equations of the b ifurcation dia gram in th ree- dimensional space. These equations h ave the form con venient for the classification of t h e b ifurcation sets ind uced on 5-dimensional iso-energetic levels. MSC: 7 0E17, 70G4 0, 70H06 P ACS: 45.20 .Jj, 45.40.Cc Key W ords: Kow alevski gyro s tat, t wo co nstant fields, critica l set, bifur c ation diag ram 1 In tro duction The famous integrable ca se of S. Kow a levski of the motion of a heavy r igid b o dy ab out a fixed po int [1] has received several gener alizations. Some o f them supp ose restrictions to submanifolds in the phase space (par tial cases ), others are far from mec hanics inv o lving p otential functions on the config uration space S O (3) with singular ities. The most essential generaliza tion having the cle a r mechanical sense was found by A.G. Rey man and M.A. Semenov-Tian-Shansky in the work [2]. The authors intro duce the dynamical system o n the dual space of the Lie alg ebra e ( p, q ) of the Lie group defined as the semi-dire ct pro duct of S O ( p ) and q copies o f R p . Such systems are known as the Euler equa tions on Lie (co )algebras [3]. The case p = 3 , q = 2 corres p o nds to the Euler– Poisson equations of the motio n of a g yrostat in tw o constant fields . F or a rigid b o dy without gyrosta tic momen tum, the mo del of tw o constant fields was intro- duced by O .I. Bo goy avlensky [3]. The physical o b ject can b e either a heavy electrically charged rigid b o dy rotating in g ravitational and constant electric fields, or a heavy magnet ro tating in gravitational and constant mag netic fields. The cor resp onding eq uations a re Hamiltonian on the orbit of coadjoint action on e (3 , 2) ∗ of the Lie group defined as the s e mi-direct pro duct S O (3) × ( R 3 ⊗ R 3 ). The typical orbit is diffeomorphic to T S O (3) ∼ = R 3 × S O (3). Therefore, the gyrostat in t wo constant fields is the Hamiltonian system with three degrees of freedom. Bogoya vlensky [3] s uggested the c o nditions o f the Kowalevski type and found the analo gue of the K ow alev s ki integral K for the top in tw o constant fields. H. Y ehia [4 ] generalized this int egral for the Kowalevski gyro s tat in tw o constant fields. Almost simultaneously with Y ehia, I.V. Komarov [5] and L.N. Ga v rilov [6] proved the Lio uville in tegrability o f the Kow a levski gyro- stat in the gr avit y field. But for t wo constant fields the K ow alevs ki gyrostat was not considered int egrable due to the fact that the existence of the sec ond field de s troys the axial symmetry of the p otential and, conse q uen tly , the cor resp onding cyclic integral. F inally , Reyma n a nd 1 Semenov-Tian-Shansky [2] found the Lax represe ntation with a sp ectral parameter for the fa m- ily of Euler equations on e ( p, q ) ∗ . F or e (3 , 2) this repr e sentation immediately gave ris e to the new integral fo r the Kow a levski gyros tat in tw o constant fields. F or the classical K ow alevs ki top this integral turns into the square of the cyclic integral. The Kow alevski gyrostat in t wo constant fields do es not hav e any explicit symmetry groups and, therefore, is not reducible, in a standar d wa y , to a family of sy stems with t wo degree s o f freedom. Pha se topolog y of s uch systems has not be e n studied y et. The theor y of n -dimensional int egrable systems started in [7] is not illustr ated by a n a pplication to any real irreducible ph ysical or geo metrical problem with n > 2. In the pap er [8], the author s give a detailed exp ositio n of the r esults of [2] as well a s a study of the a lg ebraic geometry o f the Lax pair for the generalized Ko walevski sy s tem. They announce the p os sibilit y of its integration by the finite-ba nd techniques and fulfil such integration for the classical top. F o r t wo constant fields the in tegration of the Kow alevski top is no t given up-to- date. The problem of the K ow alev ski gyr ostat motion in tw o c onstant fields is not studied at all. The technical difficulties here are extremely high. It is not likely that, in the genera l regular ca se, the a na lytical so lutions can be obtained having the for m useful for the qualitative top ologica l analysis or the computer sim ulation. How ever, there is a g o o d experience of studying the critical subsystems, i.e., the systems with t wo degrees of freedom induced o n 4-dimensiona l inv arian t submanifolds of the phas e space. F or the K ow alevs ki top in tw o co nstant fields we ha ve now the complete descr iption of all singular ities of the momentum map [9], [10], [11], [12], [13], [1 4] and the classifica tion of the bifurcation diagr ams for the r estriction of this map to 5-dimensional iso-energ etic surfaces [15], [16], [17]. This result is a nece ssary and hig hly co mplicated pa r t of the study of Liouville folia tion of the integrable system and shows the actual need in the generaliza tion of the Liouville inv a riants theory [18] for the dimensio ns greater than tw o. The pres en t pap er contains similar re s ults for the K ow alevs ki gyro stat in tw o constant fields. The 6-dimensiona l phase space is str atified by the ra nk of the momentum map. W e find the equations of inv a riant s ubmanifolds on which the induce d systems ar e Hamiltonian with less than three deg rees of freedom (critical manifo lds of ra nk 0,1, or 2). W e straig h tforwardly prov e that the imag e of these critical manifolds (the bifurcation diagram) lies in the discriminant set of the a lgebraic cur ve o f the Lax repr e sentation given in [8]. Mor eov er , the sp ectral para meter on the Lax c urve is explicitly expres s ed in terms o f the c o nstant s of the additional partial integral arising on the critical submanifolds. It then follows that the equations of the surfaces con taining the bifurcatio n diagram ar e written in the para metric form such that the parameters ar e the energy constant h a nd the co ns tant s o f the partia l integral. Fixing the v alue of h we come to explicit e q uations o f the bifurcation dia g rams induced on iso-energ etic le vels. The proble m of classification of these diag rams seems quite complicated due to the existence o f several physical parameters . Nevertheless, it is certainly solv a ble with the help of cont emp orary c omputer progra ms of ana ly tical calculations. First w e sho w that the n umber of physical pa r ameters for the gyr ostat in tw o constant fields can b e reduced by a simple pro cedure, which may b e called the o r thogonaliza tion of the fields. More pr ecisely , for the problems of gyro stat motion there ex ists a gr oup of diffeomorphisms of the phase spac es (men tioned ab ove o rbits of the coa djoin t action) that is an equiv alence g r oup for the corr esp onding dyna mical s ystems. It app e ars that each equiv alence class contains a problem with an o rthonormal pair of r adius vectors of the centers of forces a pplication and with an orthogona l pair of the intensit y vectors. Such for ce field is c har acterized b y only one essential parameter—the ratio of the mo dules of the intensit y vectors. F or a dynamically symmetric gyrosta t having the centers of forces applicatio n in the equatoria l pla ne, the or thogonaliza tion pro cedure along with the a ppropriate choice of the measur e units lea ve, in addition to the forces ra tio, only tw o physical para meters of the b o dy itself, namely , the ra tio of the equato rial and axial inertia moments a nd the non-zero axial comp onent of the gyrosta tic mo mentum. In the generalized Kow a le vski case the first o f them equals 2. Thus, the who le problem ha s, in fact, t wo essential parameter s . In par ticular, each o f the critical four -dimensional submanifolds found b elow provides a tw o-par ametric family of completely integrable Hamiltonian systems with tw o degr ees of freedom. 2 2 Gyrostat equations and parametrical reduction Consider a rig id b o dy B rotating around a fixed p oint O . Cho ose a tr ihedral at O moving along with the b o dy and refer to it all vector and tensor ob jects. Denote b y e 1 e 2 e 3 the canonical unit basis in R 3 ; then the moving trihedr al itse lf is re presented as O e 1 e 2 e 3 . Let ω b e the vector of the angula r velocity o f B . Suppose that B is b earing an a xially symmetric rigid r otor B ′ rotating freely a round its symmetr y axis fixed in B . Such sy s tem o f tw o b o dies is the simples t mo del of a gyro stat. The notion of a gyr ostat was in tro duced by N.E. Zh uko vsky [19] for a b o dy having cavities totally filled with ho mo geneous fluid. Both mo dels have the common feature usually taken as the definition of a gyro stat: the total a ngular momentum of such system is M = I ω + λ , where the inertia tensor I and the vector λ (called the gyro static momentum) are constant with r esp ect to the moving trihedral. Using the term ”gyro stat” we alwa ys supp ose λ 6 = 0. In the cas e λ = 0 we us e the terms ”rigid bo dy” or ”top” instead. The top is usually suppo sed to have a dyna mical symmetry a xis. Let M F denote the mo ment of external forces with r esp ect to O (the rota ting mo men t). Constant field is a force field inducing the rotating moment of the for m r × α with constant vector r and with α corr esp onding to some physical vector fixed in inertial space; r p oints from O to the center of a pplication of the field, α is the field intensit y . F or two c onstant fields the rota ting moment is M F = r 1 × α + r 2 × β with r 1 , r 2 constant in the bo dy and α , β corresp onding to the vectors fixed in iner tial space . O b viously , M F can be r epresented as the moment of one c onstant field if either r 1 × r 2 = 0 or α × β = 0. Supp ose that r 1 × r 2 6 = 0 , α × β 6 = 0 . (2.1) Two co nstant fields satisfying (2.1) are sa id to b e indep endent . The e q uations defining the resp ective evolution of M , α , β in tw o constant fie lds a re d M dt = M × ω + r 1 × α + r 2 × β , d α dt = α × ω , d β dt = β × ω . (2.2) These equations are Euler equations in the space R 9 ( M , α , β ) considered as the dual space to the semi-direct sum so (3) + ( R 3 ⊗ R 3 ). The Lie–Poisson bracket applied to the c o ordinate functions y ields { M i , M j } = ε ij k M k , { M i , α j } = ε ij k α k , { M i , β j } = ε ij k β k , { α i , α j } = 0 , { α i , β j } = 0 , { β i , β j } = 0 . (2.3) Such bra ck et is non-deg enerate on ea ch or bit of the co adjoint ac tio n. The or bits are defined by the g eometric integrals (common level of the Ca simir functions) α · α = c 11 , β · β = c 22 , α · β = c 12 . If c 11 > 0 , c 22 > 0 , c 2 12 < c 11 c 22 , then the o rbit in R 9 is diffeo morphic to R 3 × S O (3), and the induced Hamiltonian system has three degr ees o f freedom (see [3], [8] for the details ). F rom ph ysical p oint o f view the constants c 11 , c 22 , c 12 characterize the fo r ce fields in tensities. Along with the co or dinates of r 1 , r 2 in the moving frame, we have 9 parameters of the int eraction of the b o dy with the external forces. W e now show how to reduce this num b er. Int ro duce s ome notation. Let L ( n, k ) b e the space o f n × k -ma trices. Put L ( k ) = L ( k , k ). Ident ify R 6 = R 3 × R 3 with L (3 , 2) by the iso morphism j that joins tw o columns A = j ( a 1 , a 2 ) = k a 1 a 2 k ∈ L (3 , 2) , a 1 , a 2 ∈ R 3 . F or the inv er s e map, we write j − 1 ( A ) = ( c 1 ( A ) , c 2 ( A )) ∈ R 3 × R 3 , A ∈ L (3 , 2) . 3 If A, B ∈ L (3 , 2), a ∈ R 3 , by definition, put A × B = 2 P i =1 c i ( A ) × c i ( B ) ∈ R 3 ; a × A = j ( a × c 1 ( A ) , a × c 2 ( A )) ∈ L (3 , 2) . (2.4) Lemma 1. L et Λ ∈ S O (3) , D ∈ GL (2 , R ) , a ∈ R 3 , A, B ∈ L (3 , 2) . Then Λ( A × B ) = (Λ A ) × (Λ B ); ( AD − 1 ) × ( B D T ) = A × B ; Λ( a × A ) = (Λ a ) × (Λ A ); a × ( AD ) = ( a × A ) D . The pr o of is by direct ca lculation. In no tation (2.4) we write E qs. (2.2) in the form I d ω dt = ( I ω + λ ) × ω + A × U, dU dt = − ω × U . (2.5) Here A = j ( r 1 , r 2 ) is a constant matrix, U = j ( α , β ). The pha se space of (2.5) is { ( ω , U ) } = R 3 × L (3 , 2). In fact, U in (2.5) is restr icted by the geometric integrals; i.e., for so me constant symmetric matrix C ∈ L (2) U T U = C. (2.6) Let O b e the s et defined by Eq. (2.6) in L (3 , 2). In orde r to empha size the C -dep endence, we write O = O ( C ). Let P = ( I , λ , A, C ) denote the complete set of constan t para meters of the problem. Deno te by X P the vector field on R 3 × O ( C ) induced by (2.5). Given the s et P , the problem o f motion of the gyro stat in tw o consta n t fields descr ibe d by the dynamical system X P will b e called, for short, the DG-pr oblem . Asso ciate to Λ ∈ S O (3), D ∈ GL (2 , R ) the linear a utomorphisms Ψ(Λ , D ) and ψ (Λ , D ) of R 3 × L (3 , 2) and L (3 ) × R 3 × L (3 , 2) × L (2) Ψ(Λ , D )( ω , U ) = (Λ ω , Λ U D T ) , ψ (Λ , D )( I , λ , A, C ) = (Λ I Λ T , Λ λ , Λ AD − 1 , DC D T ) . (2.7) Eqs. (2.6) and (2.7) imply Ψ(Λ , D )( R 3 × O ( C )) = R 3 × O ( D C D T ). Using Lemma 1 we obtain the following statement. Lemma 2. F or e ach (Λ , D ) ∈ S O (3) × GL (2 , R ) , we have Ψ(Λ , D ) ∗ ( X P ( v )) = X ψ (Λ ,D )( P ) (Ψ(Λ , D )( v )) , v ∈ R 3 × O ( C ) . Thu s, any t wo DG-problems determined by the sets of parameter s P and ψ (Λ , D )( P ) are completely equiv a lent . Let us call a DG-problem c anonic al if the centers of application o f forces lie on the fir st t wo axes of the moving trihedral at unit distance from O and the intensities of the forces are orthogo nal to each other. Theorem 1. F or e ach DG-pr oblem with indep endent for c es t her e exists an e qu ivalent c anonic al pr oblem. Mor e over, in b oth e quivalent pr oblems the c enters of applic ation of for c es b elong to the same plane in the b o dy c ontaining the fixe d p oint. Pr o of. Let the DG-pr oblem with the set of parameter s P = ( I , λ , A, C ) satisfy (2.1). This means that the symmetric ma trices A ∗ = ( A T A ) − 1 and C ar e p ositively definite. Acco rding to the well-known fact from linea r alg ebra, A ∗ and C can be r educed, resp ectively , to the identit y matrix a nd to a dia gonal matrix via the same co njugation o p er ator D A ∗ D T = E , D C D T = dia g { a 2 , b 2 } , D ∈ GL (2 , R ) , a, b ∈ R + . 4 Then c 1 ( AD − 1 ) a nd c 2 ( AD − 1 ) form a n o rthonormal pair in R 3 . There exists Λ ∈ S O (3) such that Λ c i ( AD − 1 ) = e i ( i = 1 , 2). The first statement is obtained by a pplying Lemma 2 with the previously chosen Λ , D to the initia l vector field X P . T o finish the pro o f, no tice that the tra nsformation A 7→ AD − 1 preserves the span of c 1 ( A ), c 2 ( A ). The matrix Λ in (2.7) sta nds for the change of the mo v ing trihedral. Ther efore, if a ∈ R 3 represents some physical vector in the initial pro ble m, then Λ a is the s ame vector with res pec t to the b o dy in the eq uiv alent pr oblem. Remark 1. The fact that a ny DG-problem c a n b e reduced to the pro ble m with one of the pairs r 1 , r 2 or α , β orthono rmal is obvious. Sim ultaneo us orthog onalization of b oth pairs was first esta blis hed in [11] for a r igid b o dy and crucially simplifies a ll calculatio ns. It follows fro m Theo rem 1 that, without loss of gener ality , for indep endent for ces we may suppo se r 1 = e 1 , r 2 = e 2 , (2.8) α · α = a 2 , β · β = b 2 , α · β = 0 . (2.9) Change, if necess a ry , the order of e 1 , e 2 (with simultaneous change of the dir ection of e 3 ) to obtain a > b > 0. Consider a dynamically symmetric top in tw o constant fields with the centers of application of forces in the equatoria l plane of its inertia ellips oid. Cho ose a moving trihedral suc h that O e 3 is the symmetry ax is. Then the inertia tenso r I b ecomes diag onal. Let a = b . F or any Θ ∈ S O (2) deno te by ˆ Θ ∈ S O (3) the corr esp onding ro tation of R 3 ab out O e 3 . T a ke in (2.7) Λ = ˆ Θ, D = Θ. Under the conditions (2.8), (2 .9), ψ = Id and Ψ b eco mes the symmetry group. The s ystem (2.5) has the cyc lic integral I ω · ( a 2 e 3 − α × β ). Therefo r e it is p ossible to reduce such a DG-problem to a family of systems with t wo degr ees of freedo m. F or the a nalogue of the K ow alevs ki case this system be c omes integrable [4]. Let us ca ll a DG-problem irr e ducible if, in its cano nical re pr esentation, a > b > 0 . (2.10) The following sta temen ts are needed in the future; they also r e veal so me fea tures of a wide class o f DG-problems. Lemma 3. In an irr e ducible DG-pr oblem, the b o dy has exactly four e quilibria. Pr o of. The set of singular p oints of (2.5 ) is defined by ω = 0 , A × U = 0. F or the equiv alent canonical pro blem with (2.8) we have e 1 × α + e 2 × β = 0 . Then the four vectors e 1 , α , e 2 , β are para llel to the same plane a nd | e 1 × α | = | e 2 × β | . Given (2.10), this equa lit y yields α = ± a e 1 , β = ± b e 2 . (2.11) Thu s, in the c a nonical irreducible system, an eq uilibrium ta kes place only if the radius vectors of the centers of application are par allel to the corres po nding fields int ensities. Note that the e xistence of the gyros tatic momen tum do es not change the equilibria. Ther e- fore, the res ult here is the s a me as in the ca se of a r igid b o dy in tw o constant fields [15]. Lemma 4. L et an irr e ducible DG-pr oblem in its c anonic al form have t he diagonal inertia tensor I = diag { I 1 , I 2 , I 3 } and λ = 0 . Then t he b o dy has the fol lowing families of p erio dic motions of 5 p endulum t yp e P 1 :  ω = ϕ · e 1 , α ≡ ± a e 1 , β = b ( e 2 cos ϕ − e 3 sin ϕ ) , I 1 ϕ ·· = − b sin ϕ ; (2.12) P 2 :  ω = ϕ · e 2 , β ≡ ± b e 2 , α = a ( e 1 cos ϕ + e 3 sin ϕ ) , I 2 ϕ ·· = − a sin ϕ ; (2.13) P 3 :    ω = ϕ · e 3 , α × β ≡ ± ab e 3 , α = a ( e 1 cos ϕ − e 2 sin ϕ ) , β = ± b ( e 1 sin ϕ + e 2 cos ϕ ) , I 3 ϕ ·· = − ( a ± b ) sin ϕ. (2.14) If λ 6 = 0 bu t λ = λ e i for some i = 1 , 2 , 3 , then the only family r emaine d is P i with the c orr esp onding index. The pro of is obvious. The fa milies (2.12)–(2.1 4) were first found in [11] (the c a se λ = 0). Note that for tw o constant fields these families ar e the only mo tio ns with a fixe d direction of the angular velo c ity . In pa rticular, the b o dy in tw o indep endent co nstant fields do es not have any unifor m rotatio ns. 3 Critical set of the Ko w alevski gyrostat Suppo se that the irreducible DG-pr oblem ha s the diagona l inertia tenso r with the principal moments of inertia s a tisfying the ratio 2:2:1 , the gyrosta tic momentum is dir ected along the dynamical s ymmetry a x is λ = λ e 3 and the centers of the fields applicatio n lie in the equa torial plane r 1 ⊥ e 3 , r 2 ⊥ e 3 . These are the conditions of the in tegrable case [2] of the Kowalevski gyrosta t in tw o constant fields . The or thogonaliza tio n pro cedure in this cas e do es not change the e 3 -axis a nd w e obtain (2.8), (2.9). Cho osing the appropriate units o f measurement, represent Eqs. (2.5) in the form 2 ˙ ω 1 = ω 2 ( ω 3 − λ ) + β 3 , 2 ˙ ω 2 = − ω 1 ( ω 3 − λ ) − α 3 , ˙ ω 3 = α 2 − β 1 , ˙ α 1 = α 2 ω 3 − α 3 ω 2 , ˙ β 1 = β 2 ω 3 − β 3 ω 2 , ˙ α 2 = α 3 ω 1 − α 1 ω 3 , ˙ β 2 = β 3 ω 1 − β 1 ω 3 , ˙ α 3 = α 1 ω 2 − α 2 ω 1 , ˙ β 3 = β 1 ω 2 − β 2 ω 1 . (3.15) The phase space is P 6 = R 3 × O , wher e O ⊂ R 3 × R 3 is defined by (2.9); O is diffeomor phic to S O (3). The c o mplete s et o f the first integrals in inv olution o n P 6 includes the ener gy integral H , generalized Ko walevski integral K [3], [4], and the int egral G found in [2]. After the parametric a l reduction, these integrals ar e H = ω 2 1 + ω 2 2 + 1 2 ω 2 3 − α 1 − β 2 , K = ( ω 2 1 − ω 2 2 + α 1 − β 2 ) 2 + (2 ω 1 ω 2 + α 2 + β 1 ) 2 + + 2 λ [( ω 3 − λ )( ω 2 1 + ω 2 2 ) + 2 ω 1 α 3 + 2 ω 2 β 3 ] , G = 1 4 ( M 2 α + M 2 β ) + 1 2 ( ω 3 − λ ) M γ − b 2 α 1 − a 2 β 2 . Here M α = ( I ω + λ ) · α , M β = ( I ω + λ ) · β , M γ = ( I ω + λ ) · ( α × β ). Int ro duce the momentum map J = G × K × H : P 6 → R 3 (3.16) and denote by C ⊂ P 6 the se t o f cr itica l p oints o f J . By definition, the bifurcatio n dia g ram of J is the set Σ ⊂ R 3 ov er which J fails to b e lo cally trivia l; Σ defines the cases whe n the integral manifolds J c = J − 1 ( c ) , c = ( g , k , h ) ∈ R 3 6 change its to p olo gical (and s mo oth) type. T o find C and Σ is the necessary part of the glo bal top ological a nalysis of the problem. It follows fro m Liouville–Arno ld theorem that for c / ∈ Σ the manifold J c , if not empt y , is the union of three-dimensional tor i. The consider ed Hamiltonian system on P 6 is non-degene r ate a t least fo r small enough v alues of b . Therefo r e the tra jector ies on such tor i are almost everywhere quasi-p erio dic with three indep endent frequencies. The critical se t C is preser ved by the phase flow and consists of the tra jectories ha ving less than three frequencies. W e call these tra jectories the critic al motions . The se t C is stratified by the rank of J . Let C j = { ζ ∈ C : r ank J ( ζ ) = j } ( j = 0 , 1 , 2). It is natural to exp ect that C j consists of the Liouville to ri of dimension j and the image J ( C j ), as a subset of Σ, is a smo o th sur face Σ j of dimens io n j . More pr ecisely , for each j 6 2 we have to take Σ j = J ( C j ) \ j − 1 [ i =0 J ( C i ) . Then, as a who le , we may consider Σ as a tw o- dimensional cell complex , Σ j as its j -skeleton. F or j = 1 , 2 we will have ∂ Σ j ⊂ Σ j − 1 . F or c ∈ Σ 2 the set J c ∩ C consists of tw o-dimensiona l tor i. T ake the unio n of such tori ov er the v a lues c fro m so me op en subset in Σ 2 . The dynamical system induced on this union will b e Hamiltonia n with tw o degree s o f freedom. Vice versa, let M b e a submanifold in P 6 , dim M = 4 , and supp ose that the induced system on M is Hamiltonian. Then obviously M ⊂ C . This sp eculation g ives a useful to ol to find out whether a common level of functions consists of critical p oints of J . Lemma 5. Consider a system of e quations f 1 = 0 , . . . , f 2 k = 0 (3.17) on a domain W op en in the phase sp ac e P 2 n of the int e gr able Hamiltonian system X . L et M ⊂ W b e the set define d by ( 3.17 ) . Su pp ose ( i ) f 1 , . . . , f 2 k ar e sm o oth funct ions indep endent on M ; ( ii ) X f 1 = 0 , . . . , X f 2 k = 0 on M ; ( iii ) t he matrix of the Poisson br ackets k { f i , f j }k is non-de gener ate almost everywher e on M . Then M c onsists of critic al p oints of the momentu m map. Pr o of. Conditions (i), (ii) imply that M is a smo oth (2 n − 2 k )-dimensional manifold inv a riant under the restriction of the phase flow to the o pen s et W . Co ndition (iii) means that the close d 2-form induced on M by the symplectic str ucture on P 2 n is a lmo st everywhere non-degener ate. Thu s the flow on M is almost everywhere Hamiltonian with n − k degr e es of freedom. It inher its the prop erty of complete integrability . The n almost a ll its integral ma nifo lds consis t o f ( n − k )- dimensional tori and therefore lie in the critical set o f the mo ment um map. Since M is closed in W and the c ritical set is closed in P 2 n , we conclude that M totally consists o f the critical po int s of the momentum map. Remark 2. In our case n = 3 a nd the ab ov e lemma is applied in the situations when k = 1 or k = 2 . The critical se t and the bifurcation diag ram o f the ma p (3 .1 6) in the ca s e λ = 0 are known. The critical set is describ ed by o ne system of the type (3.17) with k = 2 and three systems of the type (3.1 7) with k = 1. The complete pr esentation of these results and the list o f publications are given in [12], [17]. Except for the pa rtial integrable case of Bogoya vlensky [3] (case K = 0), all of the cr itical subsy stems hav e b een either explicitly integrated o r reduced to separated sy s tems of equations [13], [14], [16 ]. Int ro duce the change of v ariables [10] based on the change given by S. Kowalevski and on the La x repr esentation [2] ( i 2 = − 1) x 1 = ( α 1 − β 2 ) + i ( α 2 + β 1 ) , x 2 = ( α 1 − β 2 ) − i ( α 2 + β 1 ) , y 1 = ( α 1 + β 2 ) + i ( α 2 − β 1 ) , y 2 = ( α 1 + β 2 ) − i ( α 2 − β 1 ) , z 1 = α 3 + iβ 3 , z 2 = α 3 − iβ 3 , w 1 = ω 1 + iω 2 , w 2 = ω 1 − iω 2 , w 3 = ω 3 . (3.18) 7 Then E qs. (3.15) yield 2 w ′ 1 = − w 1 ( w 3 − λ ) − z 1 , 2 w ′ 2 = w 2 ( w 3 − λ ) + z 2 , 2 w ′ 3 = y 2 − y 1 , x ′ 1 = − x 1 w 3 + z 1 w 1 , x ′ 2 = x 2 w 3 − z 2 w 2 , y ′ 1 = − y 1 w 3 + z 2 w 1 , y ′ 2 = y 2 w 3 − z 1 w 2 , 2 z ′ 1 = x 1 w 2 − y 2 w 1 , 2 z ′ 2 = − x 2 w 1 + y 1 w 2 . (3.19) Here prime sta nds for d/d ( it ). Consider (3.18) a s the map R 9 → C 9 and denote its ima ge by V 9 . E qs. (2.9) o f the phase space P 6 in V 9 take the form z 2 1 + x 1 y 2 = r 2 , z 2 2 + x 2 y 1 = r 2 , (3.20) x 1 x 2 + y 1 y 2 + 2 z 1 z 2 = 2 p 2 . (3.21) Here we introduce the p ositive constants p = p a 2 + b 2 , r = p a 2 − b 2 . Using (3.20) and (3.21), expres s the fir st integrals in new co ordinates, H = w 1 w 2 + 1 2 w 2 3 − 1 2 ( y 1 + y 2 ) , K = ( w 2 1 + x 1 )( w 2 2 + x 2 ) + 2 λ ( w 1 w 2 w 3 + z 2 w 1 + z 1 w 2 ) − 2 λ 2 w 1 w 2 , G = 1 4 ( p 2 − x 1 x 2 ) w 2 3 + 1 2 ( x 2 z 1 w 1 + x 1 z 2 w 2 ) w 3 + + 1 4 ( x 2 w 1 + y 1 w 2 )( y 2 w 1 + x 1 w 2 ) − 1 4 p 2 ( y 1 + y 2 )+ + 1 4 r 2 ( x 1 + x 2 ) + 1 2 λ ( z 1 z 2 w 3 + y 2 z 2 w 1 + y 1 z 1 w 2 )+ + 1 4 λ 2 ( p 2 − y 1 y 2 ) . (3.22) Let f be an arbitrar y function on V 9 . F or brevity , the term ”critical po in t of f ” will alwa ys mean a critical p oint o f the restrictio n of f to P 6 . Similarly , d f means the restrictio n of the differential of f to the s et o f vectors tange nt to P 6 . While calculating critical p oints of v a rious functions, it is conv enient to av oid in tro ducing Lag range’s multipliers for the res tr ictions (3 .20) and (3.21). Lemma 6. Critic al p oints of a function f on V 9 , in the ab ove sense, ar e define d by t he system of e quations X i f = 0 ( i = 1 , . . . 6) , (3.23) wher e X 1 = ∂ ∂ w 1 , X 2 = ∂ ∂ w 2 , X 3 = ∂ ∂ w 3 , X 4 = z 2 ∂ ∂ x 2 + z 1 ∂ ∂ y 2 − 1 2 x 1 ∂ ∂ z 1 − 1 2 y 1 ∂ ∂ z 2 , X 5 = z 1 ∂ ∂ x 1 + z 2 ∂ ∂ y 1 − 1 2 y 2 ∂ ∂ z 1 − 1 2 x 2 ∂ ∂ z 2 , X 6 = x 1 ∂ ∂ x 1 − x 2 ∂ ∂ x 2 + y 1 ∂ ∂ y 1 − y 2 ∂ ∂ y 2 . Indeed, six vector fields X i are tangent to P 6 and linearly indep endent at any p oint of P 6 . The fo llowing tw o prop ositio ns define the str a ta C 0 and C 1 of the cr itical set. Prop ositio n 1. The set C 0 c onsists exactly of the four e quilibria existing in t his pr oblem. 8 Pr o of. The condition of zero ra nk of the momentum map at a point ζ ∈ P 6 suppo ses, in particular, that dH = 0. Then ζ is the p oint of equilibrium and it follows from Lemma 3 that ζ is one o f the po int s (2.11). Using the complex v aria bles we have w 1 = w 2 = w 3 = 0 , z 1 = z 2 = 0 , x 1 = x 2 = ε 1 a − ε 2 b, y 1 = y 2 = ε 1 a + ε 2 b ( ε 1 = ± 1 , ε 2 = ± 1) . Use Eqs. (3.23) with f = K and f = G to obtain that dK ( ζ ) = 0 and dG ( ζ ) = 0 . Therefo r e, rank J ( ζ ) = 0. Note that in class ical problems of the rigid bo dy dynamics with a n axia lly sy mmetric force field, the r ank of the momentum map is everywhere not less than 1 due to the reg ula rity of the cyclic integral. In our case, all e q uilibria are non-de gener ate (in the Mor se s e nse) critica l p oints of the Hamilton function (see [1 5]). Ther efore, these p oints a re cr itical for any first integral of the sy stem. It is es s ent ial that in the s equel λ 6 = 0. Prop ositio n 2. The set C 1 is c ompletely define d by t he c ondition rank { dK, dH } = 1 and c onsists of the p oints of the fol lowing p erio dic tr aje ctories: 1) p endulu m motions ( 2.14 ) ; 2) motions define d by the e quations w 1 = q ( w ) √ w , w 2 = √ w q ( w ) , w 3 = λ σ w, (3.24) x 1 = 1 σ u [ r 2 λ 2 σ 2 − ( λ 2 + σ ) u q 2 ( w ) w ] , x 2 = 1 σ u [ r 2 λ 2 σ 2 − ( λ 2 + σ ) u w q 2 ( w ) ] , y 1 = σ (1 + σ λ 2 − r 4 λ 2 σ u 2 ) + r 2 λ 2 u q 2 ( w ) w , y 2 = σ (1 + σ λ 2 − r 4 λ 2 σ u 2 ) + r 2 λ 2 u w q 2 ( w ) , z 1 = − r 2 λ σ u √ w q ( w ) + λ 2 + σ λ q ( w ) √ w , z 2 = − r 2 λ σ u q ( w ) √ w + λ 2 + σ λ √ w q ( w ) . (3.25) Her e q ( w ) is the r o ot of the e quation q 4 − 2 Q ( w ) q 2 + 1 = 0 , wher e Q ( w ) = σ u 3 + ( λ 2 + σ )[ λ 2 w 2 + σ 2 (2 w − σ )] u 2 + r 4 λ 4 σ 4 2 r 2 λ 2 σ 2 ( λ 2 + σ ) uw ; (3.26) σ , u ar e c onstants satisfying the e quation λ 2 ( λ 2 + σ ) 2 u 5 + ( λ 2 + σ )[2 p 2 λ 4 − ( λ 2 + σ ) 3 σ ] σ u 4 + + r 4 λ 6 σ 2 u 3 + 2 r 4 λ 4 σ 4 ( λ 2 + σ ) 2 u 2 − r 8 λ 8 σ 6 = 0 . (3.27) The evolution w ( t ) is define d by t he e quation  dw dt  2 = − λ 2 4 σ 2 P + ( w ) P − ( w ) , (3.28) wher e P ± ( w ) = w 2 + 2 σ 2 u ± r 2 λ 2 λ 2 u w + σ [ u 3 − ( λ 2 + σ ) σ 2 u 2 + r 4 λ 4 σ 3 ] ( λ 2 + σ ) λ 2 u 2 . (3.29) 9 Pr o of. It follows fr om ab ov e that dH 6 = 0 at the p oints of C 1 . Then to investigate the dep en- dence o f the functions K and H it is s ufficien t to int ro duce the function with one Lagr a nge’s m ultiplier σ . W rite E q s. (3 .2 3) with f = K − 2 σ H , ( w 2 1 + x 1 ) w 2 + λ [ z 1 + w 1 ( w 3 − λ )] − σ w 1 = 0 , ( w 2 2 + x 2 ) w 1 + λ [ z 2 + w 2 ( w 3 − λ )] − σ w 2 = 0 , (3.30) λw 1 w 2 − σ w 3 = 0 , (3.31) ( w 2 1 + x 1 ) z 2 − λ ( w 2 x 1 + w 1 y 1 ) + σ z 1 = 0 , ( w 2 2 + x 2 ) z 1 − λ ( w 1 x 2 + w 2 y 2 ) + σ z 2 = 0 , (3.32) x 1 w 2 2 − x 2 w 2 1 + σ ( y 1 − y 2 ) = 0 . (3.33) First consider the critical p oints of the function K . F or this purp os e we must put σ = 0. Eq. (3.31) gives w 1 = w 2 = 0. Then Eqs. (3.30) imply z 1 = z 2 = 0. Eqs. (3.32) and (3.33) bec ome iden tities. The sa me v alues satisfy Eqs. (3.23) if we take f = 4 G + ( x 1 x 2 − y 1 y 2 ) H . Therefore, dK = 0 and 4 dG + ( x 1 x 2 − y 1 y 2 ) dH = 0. Since dH 6 = 0 , it means that r ank J = 1. The initial v ariables on the c o rresp onding tra jectories are ω 1 = ω 2 ≡ 0 , α 3 = β 3 ≡ 0 . Substitute these v alues to Eqs. (3.15) to obtain the so lutio ns (2.14). Let σ 6 = 0. The equilibria of the system are alrea dy excluded. Then it follows from (3.3 1) that w 1 w 2 6 = 0. Satisfying (3.31), in tro duce new v ariables w , q as shown in (3.24). F o ur equations (3.30), (3.32) fo rm the linear system in y 1 , y 2 , z 1 , z 2 , fro m which we obtain these v ariables a s the functions of x 1 , x 2 , w, q identically satisfying (3.33). Deno te u = ( w − σ ) 2 ( λ 2 + σ ) − σx 1 x 2 . (3.34) Then Eqs. (3.20) ar e e a sily solved for x 1 , x 2 as the functions o f w , q , u . A s a re sult we obtain the ex pr essions (3.2 5). Let Q = 1 2 ( q 2 + 1 q 2 ) . Then the substitution of x 1 , x 2 from (3.25) bac k to (3.34) giv es (3.26). The last unused equation (3.21) provides the relation (3 .27) betw een u and the co nstants λ, σ . It shows that u defined as (3.3 4) app ear s to b e a consta n t. Thu s, a ll pha se v ariables are expres sed via one v ariable w , for which from (3.19) we find the differential equa tion (3 .2 8). Note that due to (3.29) the so lutions are elliptic functions of time. T o finish the pro o f, we need to show that a t the p oints of the tra jecto r ies found we really hav e ra nk J = 1, i.e., the linear dep endence of dK and dH implies the linear dep endence of dG and dH . Indeed, E qs. (3.23) with f = 2 G − ( p 2 + λ 2 + σ λ 2 σ u ) H are satisfied b oth by (2 .14) and by (3.24), (3.25). There fore, rank { dG, dH } = 1 and, cons e- quently , ra nk { dK, dG, dH } = 1. The fo llowing s tatement describ es one o f the critica l subsystems in C 2 . Prop ositio n 3. T he system ( 3.19 ) has the four-dimensional invariant su bmanifold O ∗ define d by the e quations U 1 = 0 , U 2 = 0 , (3.35) wher e U 1 = y 2 w 1 + x 1 w 2 + z 1 ( w 3 + λ ) w 1 − x 2 w 1 + y 1 w 2 + z 2 ( w 3 + λ ) w 2 , U 2 = w 1 w 2 U ′ 1 . (3.36) The Poisson br acket { U 1 , U 2 } is non-zer o almost everywher e on this su bmanifold. 10 Pr o of. The deriv a tive U ′ 2 in virtue of (3.1 9) is pro p or tional to U 1 , i.e., (3.35) implies U ′ 2 = 0. Therefore the set (3.35) is inv ariant. Consider the function S = − 1 4  y 2 w 1 + x 1 w 2 + z 1 ( w 3 + λ ) w 1 + x 2 w 1 + y 1 w 2 + z 2 ( w 3 + λ ) w 2  . On O ∗ we obtain S ′ = − w 1 z 2 + w 2 z 1 + w 1 w 2 ( w 3 − λ ) 8 w 1 w 2 U 1 ≡ 0 . Therefore, S is a partial integral o f the induced system. Eliminate y 1 , y 2 with the help of Eqs. (3.35) a nd present S in a more s imple for m S = x 2 z 1 w 1 + x 1 z 2 w 2 + z 1 z 2 ( w 3 + λ ) 2 w 1 w 2 ( w 3 − λ ) . (3.37) Now the Poisson bracket of U 1 and U 2 calculated under the rules defined by (2.3) is expressed in ter ms of the energ y consta n t h and the constant s of the integral (3.37) in the following way { U 1 , U 2 } = − 4 s  3 s 4 − 2 s 3 ( h − λ 2 2 ) + p 4 − r 4 4  . Obviously , the rig ht part of it is a r a tio of p olynomials not identically zero on O ∗ . Ther efore the set { U 1 , U 2 } = 0 ha s co dimension 1 in O ∗ . In particula r this set is of zero measure in O ∗ . Remark 3. If λ = 0, then the manifold O ∗ turns into the phase space of the Hamiltonian system with tw o deg r ees o f freedo m studied in [14]. The geometr ic al characteristic of the motions in this system is the condition M · α M · e 1 = M · β M · e 2 = co nst , where M = I ω is the ang ula r momentum vector. The s ystem (3 .35), (3.3 6) is fo und from the same co ndition given that here M = I ω + λ . The following theorem completes the descriptio n o f the critica l s et of the momentum map for the gyr ostat. Theorem 2. The set of critic al p oints of the momentum map ( 3.16 ) c onsist s of the fol lowing subsets in P 6 : 1) the set L define d by the syst em w 1 = 0 , w 2 = 0 , z 1 = 0 , z 2 = 0; (3.38) 2) the set N define d by the s yst em F 1 = 0 , F 2 = 0 , (3.39) wher e F 1 = ( w 1 w 2 + λw 3 )( w 2 x 1 + λz 1 ) λy 1 − − w 2 ( w 2 1 + x 1 )( x 2 z 1 w 1 + x 1 z 2 w 2 − x 1 x 2 w 3 + 2 z 1 z 2 λ ) − − x 2 ( w 1 w 3 + z 1 )( w 1 z 1 − x 1 w 3 ) λ + ( x 1 w 2 3 − 2 z 1 w 1 w 3 − z 2 1 ) z 2 λ 2 , F 2 = ( w 1 w 2 + λw 3 )( w 1 x 2 + λz 2 ) λy 2 − − w 1 ( w 2 2 + x 2 )( x 2 z 1 w 1 + x 1 z 2 w 2 − x 1 x 2 w 3 + 2 z 1 z 2 λ ) − − x 1 ( w 2 w 3 + z 2 )( w 2 z 2 − x 2 w 3 ) λ + ( x 2 w 2 3 − 2 z 2 w 2 w 3 − z 2 2 ) z 1 λ 2 ; 3) the set O define d by t he system R 1 = 0 , R 2 = 0 , (3.40 ) 11 wher e R 1 = [ y 1 w 2 + x 2 w 1 + z 2 ( w 3 + λ )] w 1 ( w 3 − λ )+ + x 2 z 1 w 1 + x 1 z 2 w 2 + z 1 z 2 ( w 3 + λ ) , R 2 = [ y 2 w 1 + x 1 w 2 + z 1 ( w 3 + λ )] w 2 ( w 3 − λ )+ + x 2 z 1 w 1 + x 1 z 2 w 2 + z 1 z 2 ( w 3 + λ ) . Pr o of. W e need to prove that L ∪ N ∪ O = C . (3.41) It follows from Prop ositions 1 and 2 that L ⊂ C 0 ∪ C 1 . Indeed on L we ha ve dK ≡ 0, dG ≡ ± ab dH . Note also that the sy stem of relations (3.38) satisfies the co nditions of Lemma 5 with n = 3 , k = 2. Therefore, L is a smo oth tw o-dimensiona l ma nifold with the induced Hamiltonian sy s tem with one degree of freedo m. According to Pr op osition 3 we hav e O ∗ ⊂ C 2 . Hence, O = O ∗ ∪ L ⊂ C . Consider the set N defined by Eqs. (3.3 9). Fir st, inv estigate the ca ses when these equations cannot b e solved with resp ect to y 1 , y 2 . Supp ose that w 1 w 2 + λw 3 ≡ 0 . (3.42) Then, a fter several differe ntiations in virtue of (3.19), w e come to Eqs. (3.30) – (3.33) with σ = − λ 2 . The cor resp onding p oints belo ng to C 1 . Let ( w 2 x 1 + λz 1 )( w 1 x 2 + λz 2 ) ≡ 0 . (3.43) Then the same pro cedur e leads to the s y stem of equa tions having the only solutions of the for m (3.38), i.e., to the set L . Denote N ∗ = N \ ( C 0 ∪ C 1 ). On this set fr om (3.39) we obtain y 1 = 1 ( w 1 w 2 + λw 3 )( w 2 x 1 + λz 1 ) λ [ w 2 ( w 2 1 + x 1 )( x 2 z 1 w 1 + x 1 z 2 w 2 − − x 1 x 2 w 3 + 2 z 1 z 2 λ ) + x 2 ( w 1 w 3 + z 1 )( w 1 z 1 − x 1 w 3 ) λ − − ( x 1 w 2 3 − 2 z 1 w 1 w 3 − z 2 1 ) z 2 λ 2 ] , y 2 = 1 ( w 1 w 2 + λw 3 )( w 1 x 2 + λz 2 ) λ [ w 1 ( w 2 2 + x 2 )( x 2 z 1 w 1 + x 1 z 2 w 2 − − x 1 x 2 w 3 + 2 z 1 z 2 λ ) + x 1 ( w 2 w 3 + z 2 )( w 2 z 2 − x 2 w 3 ) λ − − ( x 2 w 2 3 − 2 z 2 w 2 w 3 − z 2 2 ) z 1 λ 2 ] . (3.44) The der iv atives of F 1 and F 2 in vir tue o f (3.1 9) v anish identically after the substitution o f the expressions (3.44). This fact prov e s that N ∗ is an inv ariant set. The Poisso n bracket { F 1 , F 2 } with (3.4 4) takes the form { F 1 , F 2 } = √ 2 λ ( w 1 w 2 + λw 3 ) 3 / 2 p ( w 2 x 1 + λz 1 )( w 1 x 2 + λz 2 ) C. (3.45) Here C = 1 s (8 s 3 λ 2 − r 4 ) p 2 s 2 − (2 h + λ 2 ) s + p 2 (3.46) depe nds o n the energy constant h and the consta n t s o f the par tial int egral S = x 1 x 2 w 3 − x 2 z 1 w 1 − x 1 z 2 w 2 − λz 1 z 2 2 λ ( w 1 w 2 + λw 3 ) . (3.47) This int egral is similar to (3.37) and indep endent of H almo st everywhere on N ∗ . The ca ses when the mu ltipliers (3.42) or (3.43) in (3.45) turn to zero a re alr eady studied ab ov e. Zer os o f the function (3 .46) hav e co dimension 1. Hence, (3.45) is non-zero almost everywhere on N ∗ . It follows from Lemma 5 that N ∗ ⊂ C 2 . Th us , L ∪ N ∪ O ⊂ C . T o pr ov e the eq ua lit y (3.4 1), we mu st show that C ⊂ L ∪ N ∪ O . 12 The p oints of the set C 0 describ ed in Prop os itio n 1 satisfy (3.38). According to Pr op osi- tion 2 the set C 1 can b e r epresented as C 11 ∪ C 12 , wher e C 11 consists of the tra jectorie s (2.14) and C 12 is defined by the system (3.2 4)–(3 .2 7). O n the tra jectories (2.14) w e have (3.38). It is e asily chec ked that the p oints given by Eqs. (3.24)–(3.26) under the condition (3.27) satisfy bo th sy stems (3.39) and (3 .40). Ther efore, C 0 ∪ C 11 ⊂ L and C 12 ⊂ N ∩ O . Consider now the set C 2 . T o in vestigate the dependence of G, H , K , intro duce the function with Lag r ange’s m ultipliers. It follows fro m Prop os itions 1 a nd 2 tha t on C 2 the differentials dK and dH are linearly indep endent. Then the multiplier at the function G is a lwa ys non- zero and can b e chosen e q ual to any no n-zero consta nt. It is conv enient to take the function 2 G + S K + ( T − p 2 ) H , where S and T are La g range’s undefined multipliers. The condition 2 dG + S dK + ( T − p 2 ) dH = 0 (3.48) is prese rved by the phase flow. Applying the Lie deriv a tive we obtain ˙ S dK + ˙ T dH = 0 . Since rank { dG, dK , dH } = 2, this linea r combination of the differentials is prop ortiona l to the left par t of (3.48). It means that, at the p oints of C 2 , ˙ S ≡ 0 , ˙ T ≡ 0 . Thu s, the functions S and T a re the par tia l integrals on the s ubma nifold C 2 . According to Lemma 6 re write Eq. (3.48) a s the system x 2 ( y 2 + 2 S ) w 1 + 2 S ( w 1 w 2 + λw 3 ) w 2 + +( T − z 1 z 2 − 2 S λ 2 ) w 2 + x 2 z 1 w 3 + ( y 2 + 2 S ) z 2 λ = 0 , x 1 ( y 1 + 2 S ) w 2 + 2 S ( w 1 w 2 + λw 3 ) w 1 + +( T − z 1 z 2 − 2 S λ 2 ) w 1 + x 1 z 2 w 3 + ( y 1 + 2 S ) z 1 λ = 0 , (3.49) ( T − x 1 x 2 ) w 3 + x 2 z 1 w 1 + x 1 z 2 w 2 + (2 S w 1 w 2 + z 1 z 2 ) λ = 0 , T z 1 + x 1 z 2 w 2 3 + [( x 1 x 2 − 2 z 1 z 2 ) w 1 + ( y 1 z 1 + x 1 z 2 ) λ + x 1 y 1 w 2 ] w 3 − − ( y 1 z 1 + x 1 z 2 ) w 1 w 2 + x 1 ( y 1 + 2 S ) w 2 λ − − [ x 2 z 1 + ( y 2 + 2 S ) z 2 ] w 2 1 + [( y 2 + 2 S ) y 1 − 2 z 1 z 2 ] w 1 λ + y 1 z 1 λ 2 − − [( y 2 + 2 S ) x 1 + z 2 1 ] z 2 = 0 , T z 2 + x 2 z 1 w 2 3 + [( x 1 x 2 − 2 z 1 z 2 ) w 2 + ( y 2 z 2 + x 2 z 1 ) λ + x 2 y 2 w 1 ] w 3 − − ( y 2 z 2 + x 2 z 1 ) w 1 w 2 + x 2 ( y 2 + 2 S ) w 1 λ − − [ x 1 z 2 + ( y 1 + 2 S ) z 1 ] w 2 2 + +[( y 1 + 2 S ) y 2 − 2 z 1 z 2 ] w 2 λ + y 2 z 2 λ 2 − − [( y 1 + 2 S ) x 2 + z 2 2 ] z 1 = 0 , ( T − x 1 x 2 )( y 1 − y 2 ) + 2 ( y 2 + S ) x 2 w 2 1 − 2( y 1 + S ) x 1 w 2 2 + +2( x 2 z 1 w 1 − x 1 z 2 w 2 ) w 3 + x 2 z 2 1 − x 1 z 2 2 + +2( y 2 z 2 w 1 − y 1 z 1 w 2 ) λ = 0 . (3.50) It follows from P rop osition 3 that this sys tem is v a lid a t the p o int s of the set O ∗ . T o find all other cases supp ose U 1 U 2 6 = 0 (3.51) and expre ss y 1 , y 2 from (3.36): y 1 = 1 2 w 1 w 2 ( w 3 − λ ) { 2 U 2 − [ w 1 w 2 ( w 3 − λ ) + w 2 z 1 − w 1 z 2 ] U 1 − − 2[ w 1 z 2 ( w 3 − λ ) 2 + ( x 2 w 2 1 + z 1 z 2 + 2 λw 1 z 2 )( w 3 − λ )+ + x 2 z 1 w 1 + x 1 z 2 w 2 + 2 λz 1 z 2 ] } , y 2 = 1 2 w 1 w 2 ( w 3 − λ ) { 2 U 2 + [ w 1 w 2 ( w 3 − λ ) + w 1 z 2 − w 2 z 1 ] U 1 − − 2[ w 2 z 1 ( w 3 − λ ) 2 + ( x 1 w 2 2 + z 1 z 2 + 2 λw 2 z 1 )( w 3 − λ )+ + x 2 z 1 w 1 + x 1 z 2 w 2 + 2 λz 1 z 2 ] } . (3.52) 13 The deter minant of the system (3.49) with re s pec t to T , 2 S is eq ual to ∆ = x 1 w 2 2 − x 2 w 2 1 − ( z 2 w 1 − z 1 w 2 ) λ. If we supp ose that ∆ ≡ 0 on some time interv al, then the sequence of the der iv atives o f this ident ity in v irtue of (3.19) le a ds to (3.38), i.e., to the p oints o f C 0 ∪ C 1 . Consider then ∆ 6 = 0 (3.53) and find from Eqs . (3.49) S = 1 ∆ [ x 2 y 2 w 2 1 − x 1 y 1 w 2 2 + ( x 2 z 1 w 1 − x 1 z 2 w 2 ) w 3 + +( y 2 z 2 w 1 − y 1 z 1 w 2 ) λ ] , T = 1 ∆ [ A 1 B 1 − A 2 B 2 ] . (3.54) Here A 1 = ( x 1 w 2 + λz 1 ) y 1 + ( x 1 w 3 − z 1 w 1 ) z 2 , B 1 = ( w 2 2 + x 2 ) w 1 + λw 2 ( w 3 − λ ) + λz 2 , A 2 = ( x 2 w 1 + λz 2 ) y 2 + ( x 2 w 3 − z 2 w 2 ) z 1 , B 2 = ( w 2 1 + x 1 ) w 2 + λw 1 ( w 3 − λ ) + λz 1 . Substitute (3.52) and (3.54) int o (3.5 0) to obtain the system o f four e q uations of the type E i = 0 ( i = 1 , ..., 4), where E i = a i 2 U 2 1 + a i 1 U 1 + a i 0 U 2 with some p olynomia ls a ij . F or the pro of of the theorem, there is no need to use all equations of this system. It is enough to co nsider, for example, the zero p oints o f the resultant o f t wo simplest functions E 1 , E 4 with resp ect to U 2 . W e obtain 4 w 1 w 2 U 1 R ∆ = 0, where R = λw 1 w 2 ( x 1 w 2 + λz 1 )( x 2 w 1 + λz 2 ) U 1 −  w 1 w 2 [ x 2 z 1 w 1 + x 1 z 2 w 2 − − x 1 x 2 ( w 3 − λ ) + 2 λz 1 z 2 ] + ( z 1 z 2 w 3 + x 2 z 1 w 1 + x 1 z 2 w 2 ) λ 2 + z 1 z 2 λ 3  ∆ . Due to (3.51), U 1 6 = 0 . A t the p o int s of C \ L the pro duct w 1 w 2 is not iden tically zer o. The set C \ ( L ∪ O ) is, o bviously , prese r ved by the phase flow. Therefo r e, (3.53) and (3.19) imply R = 0 , R ′ = 0. These equa tions ar e linea r in y 1 , y 2 and yield the expressions (3 .4 4) satisfying Eqs. (3.39). Thus, C \ ( L ∪ O ) ⊂ N . Remark 4. The system (3.40) follows from Eqs. (3.35) and (3.36). In particular, O ∗ = O \ L . Then fr o m Pro po sition 3 a nd Lemma 5 we obta in that this manifold lies co mpletely in C 2 . Remark 5. The integral S given by (3.5 4), when restr ic ted to the set N has the form (3.4 7). Indeed, it is enough to substitute (3 .44) int o the first formula (3 .54) to obtain (3.47). On the set O the sa me function S in the substitution of (3.35) a nd (3 .5 2) takes the form (3.37). Therefore, the use of the same nota tion in (3.37) a nd (3 .47) is cor rect. The ex pressions for T can also be simplified. On the set N we hav e T = 2 λ 2 S , i.e., this function do es not give rise to a new partial in tegral indep endent o f S . On the contrary , at the p oints of the s et O we hav e T = x 1 x 2 + z 1 z 2 − 2 w 1 w 2 S. In the case λ = 0 the s ame expressio n with the cor resp onding function S is the par tial int egral independent of S . The equa tions of the integral manifold defined by the pair S, T lea d to the separatio n o f v ar iables on O [14 ]. 4 The bifurcation diagram The L a x repr esentation for the cons ider ed pro ble m found in [2] can b e written in the for m L ′ = LM − M L, (4.55) 14 where L =              2 λ x 2 κ − 2 w 1 z 2 κ − x 1 κ − 2 λ − z 1 κ 2 w 1 − 2 w 1 z 2 κ − 2 w 3 − y 1 κ − 4 κ − z 1 κ 2 w 2 y 2 κ + 4 κ 2 w 3              , M =              − w 3 2 0 w 2 2 0 0 w 3 2 0 − w 1 2 w 1 2 0 w 2 2 κ 0 − w 2 2 − κ − w 3 2              . Here κ stands for the s pec tral parameter , the der iv ative in (4.55) is calcula ted in vir tue o f the system (3.19). The e quation for the eigenv a lues µ of the matr ix L defines the a lgebraic cur ve asso ciated with this repr esentation [8]. Let s = 2 κ 2 and let h, k , g b e the arbitr ary consta nts of the integrals (3.22). The equatio n of the algebra ic curve takes the form µ 4 − 4 µ 2 [ p 2 s − (2 h + λ 2 ) + 2 s ] + 4[ r 4 s 2 + 2 s (4 g − 2 p 2 h − p 2 λ 2 )+ +4( k + 2 λ 2 h ) − 8 λ 2 s ] = 0 . (4.56) It is natural to supp os e that the bifurcation diagr am o f the momentum ma p (3.16) is included in the s et o f v alues ( g , k , h ) such that the curve (4.56) either hav e singula r p oints or is r educible, i.e., the left part of E q . (4.5 6) splits into the pro duct of some ra tional non-trivia l expressions . In this wa y we can gues s the result o f the following s tatement . Nevertheless, to obtain the complete pr o of of it, we m ust fulfil the calculations on the ab ov e found critical manifolds. Theorem 3. The bifur c ation diagr am of the momentu m map G × K × H is include d in the union of the fol lowing (interse cting) subsets of R 3 ( g , k , h ) : 1) the p air of str aight lines Γ + :    k = ( a + b ) 2 , g = − ab ( h − λ 2 2 ); Γ − :    k = ( a − b ) 2 , g = ab ( h − λ 2 2 ); (4.57) 2) the surfac e Γ 1 :        k = 4 λ 2 s − 2 λ 2 h + r 4 4 s 2 , g = − λ 2 s 2 + 1 2 p 2 ( h + λ 2 2 ) − r 4 4 s , s ∈ R \ { 0 } ; (4.58) 3) the surfac e Γ 2 :        k = 3 s 2 − 4( h − λ 2 2 ) s + p 2 + ( h − λ 2 2 ) 2 − p 4 − r 4 4 s 2 , g = − s 3 + ( h − λ 2 2 ) s 2 + p 4 − r 4 4 s , s ∈ R \ { 0 } . (4.59) Pr o of. Let ζ ∈ L . Substitution o f the v alues z 1 = z 2 = 0 into (3.2 0) and (3.21) yields x 1 x 2 = ( a ± b ) 2 , y 1 y 2 = ( a ∓ b ) 2 . Then from (3 .22), (3.38) we obtain the equa tions defining the lines (4.57). Let ζ ∈ N \ L . T a ke the constant of the pa rtial in tegral (3.47) for the par ameter s in (4.58), substitute the express ions (3.2 2) for the co rresp onding constants, and fulfil the change (3.44). Then b oth Eq s. (4.5 8) b ecome the ident ities. Ther efore, J ( N \ L ) ⊂ Γ 1 . The inclus ion J ( O \ L ) ⊂ Γ 2 is proved in a simila r wa y . W e take the co nstant of the partial in tegral (3.37) for the parameter s in (4.59) and fulfil the substitution (3.52) with U 1 = U 2 = 0. 15 Remark 6. Note that the shift of the energy level ˜ h = h − λ 2 / 2 makes the equatio ns o f the lines Γ ± and the surface Γ 2 independent of λ . Thereby obtained e quations a re iden tical with the cor resp onding equations of the c ase λ = 0 [12]. The surface Γ 1 is obtained a s a p erturba tion (with resp ect to λ ) o f tw o tangent to e ach other sheets of the bifurcatio n diagr a m o f the case λ = 0, i.e., the plane k = 0 and the slanted para bo lic c y linder ( p 2 h − 2 g ) 2 − r 4 k = 0. Thus, it is ea sy to view the evolution of the App elrot class es [20] of the S. Kowalevski cas e in the pro cess of tw o-wa y generaliz a tions—adding the sec ond force field and, afterwards, the non-ze ro gyrosta tic mo men tum. The eq ua tions given in Theo rem 3 ar e in the following sense co n venien t. Let us fix the energy co nstant h . Then we obtain the parametr ic eq ua tions of a one- dimensional set in the plane ( g , k ) (with the finite n umber of singular p oints). This set is the bifurcation dia gram Σ h of the restr ic tion of the pair of in tegrals G, K onto the iso-e nergetic surface { H = h } ⊂ P 6 , which is alw ays compac t. In particular, all diagrams Σ h lie in the restr ic ted are a of the ( g , k )-plane and are easily drawn numerically . The analytica l inv es tigation of the t yp es of the diagra ms Σ h with resp ect to the essential par ameters ( b/a, λ/ √ a, h/a ) is a nece ssary but technically complicated problem. Nev ertheles s, it must b e solv able. Indee d, the set of double p oints and cusps of the curves Γ 1 , 2 in the ( g , k )-pla ne is easily defined and inv estigated analytica lly . Moreover, the v alues of the firs t int egrals on the p erio dic motions (3 .2 4)–(3.27) define the p oints of transversal int ersections Γ 1 ∩ Γ 2 . This fact, at least, guar antees tha t the num erical alg orithm can b e built for effective calculation of kno ts of one-dimensio nal cell complex Σ h for a ny h . In turn, it should b e p os sible to find a ll ca ses of bifurca tions of the set of these knots with resp ect to the parameters de fining the ab ov e set o f p erio dic motions. References [1] S. Kow alevski, Sur le probleme de la rota tion d’un co rps solide autour d’un p oint fixe, A cta Math. , 12 (188 9), 177-2 32. [2] A.G. Reyman, M.A. Semenov-Tian-Shans ky , Lax representation with a sp ectra l parameter for the Kowalewski top and its ge ner alizations, L ett. Math. Phys. , 14 , 1 (1 987), 55- 61. [3] O.I. Bog oy avlensky , E uler equa tions on finite-dimension Lie algebras a rising in physical problems, Commun . Math. Phys. , 95 (19 84), 307- 315. [4] H. Y ehia, New integrable cases in the dynamics of rigid b o dies, Me ch. R es. Commun. , 13 , 3 (19 8 6), 169- 1 72. [5] I.V. Ko ma rov, A generaliza tion o f the K ov a levsk ay a top, Phys. L etters , 1 23 , 1 (1987 ), 14-15 . [6] L.N. Gavrilov, O n the geo metry of Gor jatchev-Tc ha ply gin top, C.R. A c ad. Bulg. Sci. , 40 (1987), 33 -36. [7] A.T. F omenko, Symple ctic Ge ometry. Metho ds and Applic ations , Gor don and Brea ch (1988). [8] A.I. Bo b e nko, A.G. Reyman, M.A. Semenov-Tian-Shansk y , The Kowalewski top 99 years later: a Lax pair, generaliza tions a nd explicit solutions, Commun. Math. Phys. , 12 2 , 2 (1989), 32 1-354 . [9] D.B. Zo tev , F omenko-Zieschang in v ar ia nt in the Bogoya vlens k yi cas e, R e gular and Chaotic Dynamics , 5 , 4 (2000 ), 4 37-45 8. [10] M.P . Kharlamov, O ne c la ss of solutions with tw o inv ariant r e lations in the pr oblem of motion of the Kow alevsky to p in double constant field, Mekh. tver d. tela , 32 (2002 ), 32-38 . (In Russia n) 16 [11] M.P . Kha rlamov, Critical set and bifurcation diag ram in the problem of motion of the Kow a levsky top in double field, Mekh. tver d. tela , 34 (2004), 47 -58. (In Russian) [12] M.P . Kharlamov, Bifurcation diagr ams of the Ko walevski top in t wo constant fields, R e gular and Chaotic Dynamics , 10 , 4 (2005 ), 381 -398. [13] M.P . Khar lamov, A.Y. Savushkin, Separation of v a r iables and integral manifolds in one partial pro blem of mo tion of the gener a lized Kow alevski top, Ukr. Math. Bul l. , 1 , 4 (2004), 548-5 65. [14] M.P . Kharla mov, Separation of v ariables in the genera liz ed 4th App elrot clas s, R e gular and Chaotic Dynamics , 12 , 3 (2007 ), 267 -280. [15] M.P . Khar lamov, D.B. Zotev, No n-degenerate energy s ur faces of rigid b o dy in tw o constant fields, R e gular and Chaotic Dynamics , 1 0 , 1 (2005), 15-1 9. [16] M.P . Kharla mov, Sp ecial p erio dic motions of the generaliz e d Delone case, Mekh. tver d. tela , 36 (200 6), 23-3 3 . (In Russian) [17] M.P . Kharlamov, Regions of exis tence of motions o f the genera liz ed Kov alevsk ay a top and bifurcation dia grams, Mekh. tver d. tela , 3 6 (2006), 13-2 2. (In Russian) [18] A.V. Bols inov, A.T. F o menko, Int e gr able Hamiltonian systems: ge ometry, top olo gy, classi- fic ation , Chapman & Hall/CRC (2004 ). [19] N.E. Zhoukovsky , On the motio n of a rigid b o dy with holes filled with a homogene o us fluid, In: Col le cte d Works , Gos tek hizdat, Mo scow (194 9), 1 , 31-15 2. (In Russian) [20] G.G. App elro t, Non-completely symmetric heavy gyr oscop es. In: Motion of a rigid b o dy ab out a fixe d p oint , Moscow-Leningrad (19 40), 6 1-156 . (In Russian) 17