Separation of variables and integral manifolds in one problem of motion of generalized Kowalevski top

Separation of V ariables and In tegral Manifolds in One Problem of Motion of Generalized Ko w alevski T op Mic hael P . Kharlamo v and Alex ander Y. Sa vush kin ( Pr esente d by I.A. Lukovskii ) Abstract In th e phase space of the integrable Hamiltonian system with three degrees of freedom used to d escribe the motion of a Ko w alevski-type top in a double constan t force field, w e point out the f our-dimensional inv ariant manifold. It is sho wn that this manifold consists of critical motions generating a smo oth sheet of the b ifurcation diagram, and the induced dyn amic system is Hamiltonian wi th certain s ub set of p oints of degeneration of the symp lectic s tructu re. W e find the tra nsformation separating va riables in this s ystem. As a result, th e solutions can b e represented in terms of elliptic f unctions of time. The corresp on d ing phase top ology is completely describ ed . MSC 2000 . 70E17 , 70 E 40, 37J35 Key w ords. Kow alevsk i top, double field, inv ar iant manifold In tro duction The equa tions of r otation o f a rig id b o dy ab o ut a fixed p oint in a do uble constant force field hav e the form I d ω dt = I ω × ω + r 1 × α + r 2 × β , d α dt = α × ω , d β dt = β × ω , (1) where r 1 and r 2 are vectors immov able with r esp ect to the b o dy , α and β are vectors immov able in the inertial space, I is the tens or o f iner tia at the fixed p oint O , and ω is the instantaneous angular velocity (all of these ob jects are expr e ssed via their comp onents relative to certain axes strictly attached to the b o dy). It is as sumed that the vectors r 1 and r 2 hav e the origin O . The p o ints s pec ified by these vectors in the moving space are called the centers o f r ig ging. System (1) is a Hamiltonian system in the p hase space P 6 sp ecified in R 9 ( ω , α , β ) b y geo metric int egr als; P 6 is diffeomo rphic to the tangent bundle T S O (3). In [3], it is pr o p osed to use the pr oblem o f motion of a mag netized rigid b o dy in gravitational and magnetic fields and the problem o f motion of a rig id bo dy with constant distribution of e le ctric charge in gravitational and electr ic fields a s physical mo dels of Eqs. (1). T he r esults obtained for system (1) in [3] ar e presented in more details in [4] within the framework of inv estigation of the Euler equa tions on Lie algebr as. In the general case r 1 × r 2 6 = 0 and α × β 6 = 0, sys tem (1) without additional restrictions imp osed on the par ameters, unlike the classical Euler–Poisson equa tions, is not reducible to a Hamiltonian system with t wo degrees of freedom a nd do es not have known fir s t integrals o n P 6 , except for the energy integral H = 1 2 I ω · ω − r 1 · α − r 2 · β . In [3], the fo llowing assumptions are int ro duced for system (1): in the principa l axes of the inertia tensor O e 1 e 2 e 3 , (2) the momen ts of inertia satisfy the conditions I 1 = I 2 = 2 I 3 and the vectors r 1 and r 2 are parallel to the equatoria l pla ne O e 1 e 2 and mutually o rthogona l. F or β = 0, the pro blem reduces to the Kow alevski 1 int egr able ca s e of rotation of a heavy r igid b o dy [8]. Therefo re, for the sa ke of brevity , the problem prop osed in [3] is called the generalized Kow alevs ki ca se and the problem with β = 0 is ca lled the classic a l Kowalevski case. By the appropr iate choice of measurement units and axe s (2) one can o btain I = diag { 2 , 2 , 1 } ; (3) r 1 = e 1 , r 2 = e 2 . (4) In [3 ], a new genera l integral is indicated for the genera lized Kow alevski case. In virtue of r elations (3 ) and (4 ), this in tegra l a dmits a representation: K = ( ω 2 1 − ω 2 2 + α 1 − β 2 ) 2 + (2 ω 1 ω 2 + α 2 + β 1 ) 2 , (5) where ω i , α i , and β i ( i = 1 , 2 , 3) ar e the comp onents of the vectors ω , α , and β relative to the reference frame (2). In [10], in tegral (5) is generalized to the case of motion o f a gyros ta t in a linear force field by sup- plement ing a b o dy having pro pe rty (3) with an inner ro to r gener ating a co nstant moment along the axis of dyna mic symmetry . As shown, e.g, in [5], the comp onent of the moment genera ted by p otential forces int ro duced in [10] can be reduced to the same form a s in Eqs. (1) with prop erty (4 ). The complete Liouville integrability of the K ow alevski g y rostat in a double force field was prov ed in [2]. The Lax r epresentation of equatio ns o f type (1) (with gyros copic term in the moment of ex ternal forces, as in [10]) co ntaining the sp ectral parameter was obtained under co nditions (3) and (4). The sp ectral curve of this represe ntation ma de it possible to find a new first integral which is in inv olution with the corres p o nding gener a lization of integral (5) and turns into the squar e of the momentum in tegr a l for β = 0. Multi-dimensional ana logs of the Kow alevski problem w ere introduced in [2 ]. It was propo s ed to solv e these proble ms by the metho d of finite-band in tegr ation. This progra m was realized in [2] for the classical Kow alevski top and new ex pr essions for the pha se v ariables in the form of sp ecial hyper-elliptic functions of time were o btained. The explicit integration of the problem of motion of the Kow alevski top in a do uble field and its qualitative and topo logical ana ly sis hav e not b een p erformed yet (see also a survey in [5]). The top ologica l analy s is o f an integrable Ha milto nia n system includes the descriptio n (in cer tain terms ) of the foliation of its phase s pace into Liouville tori. In particular, this requires finding all separating cases. These cases corres p o nd to the p oints of the bifurc a tion diagram of the in tegra l map and, in the phase space, are for med by the tra jectorie s co mpletely co nsisting of the p o int s at which the fir st integrals ar e not independent. In a system with three degrees o f freedom, t wo-dimensional Liouville tori are , as a rule, filled with sp ecial motions corr esp onding to a po int of the smo oth tw o-dimensional s heet of the bifurcatio n dia gram. The union of these tor i ov er all p oints of the shee t is an in v ariant subse t of the pha se space . In the neig hborho o d o f a po int of general p osition, this subset is a fo ur -dimensional manifold, a nd the dyna mic system induced on this subset must b e Hamiltonian with tw o degrees of freedom (degener a tions o f v arious kinds are exp ected a t the bo undary o f this shee t or at the p oints of intersection of shee ts ). Thus, the inv ariant subsets of maximum dimension formed by the p oints of depe ndence of integrals a re sp ecified (at least, lo cally) by systems of t wo inv ar iant rela tions of the form f 1 = 0 , f 2 = 0 . (6) The k nowledge of all these systems and the analysis o f the dynamics on inv ar ia nt manifolds s pe cified by thes e systems is essential to fulfil the top o lo gical analy s is of the en tire problem. In the gener alized Kowalevski case, w e know tw o sy stems o f the form (6). The first one is obtained in [3]. Co nsider the manifold { K = 0 } ⊂ P 6 . (7) Due to the structure of function (5), this ma nifold is sp ecified by t wo indep endent equa tions Z 1 = 0 and Z 2 = 0. An additional pa rtial integral (Poisson brack et { Z 1 , Z 2 } ) is indicated. The top olo g ical analysis of the induced Ha milto nia n system with tw o deg r ees of freedom is car ried out in [1 1]. It turns out that the inv ar iant set is a four -dimensional manifold, whic h is smo oth everywhere but the res tr iction to it of the symplectic structure degener ates on the set of zeros of the a dditio na l int egr al. This case genera lizes the first Appelr ot cla ss (Delone class) [1] of mo tions of the classica l K ow alevski case. 2 The second system of the for m (6) is obtained in [7]. It is shown that, for β = 0 , the cor resp onding motions transfor m into so-ca lled esp e cial ly r emarkable motions o f the second and third App elrot class es. The present work is dev oted to the in vestigation of the dynamic sy stem on the inv a riant subse t indicated in [7]. First, we make a gener al remar k imp ortant for the sequel. The moment o f external forces r 1 × α + r 2 × β app earing in (1) is preser ved by the ch ang e  ˜ r 1 ˜ r 2  = Θ  r 1 r 2  ,  ˜ α ˜ β  = (Θ T ) − 1  α β  , (8) where Θ is an ar bitrary non-singular 2 × 2 matrix. Therefore, the a-priori assumption made in [3], [2 ] concerning the orthogo nality of the radius vectors o f the centers of rigging is redundant (it suffices to r equire that these centers lie in the equator ial plane of the b o dy ). This statement is trivial enough; the po s sibility of ortho gonalizatio n of any pair ( r 1 , r 2 ) or ( α , β ) is indicated, e.g., in [5]. Howev er, the author s of [5] also indicate that, in genera l case, the sec o nd pair is not orthog onal. Moreov er, in [3, 4, 2 , 1 1, 7 ], the ang le betw een α a nd β remains ar bitr ary . This fact makes the corresp o nding fo rmulas more complicated. Note that if the pair ( r 1 , r 2 ) is made ortho no rmal, then there remains the arbitrar y c hoice of Θ ∈ S O (2). Under such transfor mation, a new pair of radius vectors o f the c e n ters of r igging r emains orthonorma l and can b e used a s equato rial unit vectors of the principal a xes (2) to pres e rve prop erties (4). At the same time, by cho osing Θ =     cos θ sin θ − sin θ cos θ     , tan 2 θ = 2 α · β α 2 − β 2 , we g et the orthogonal pair ( ˜ α , ˜ β ). Thu s, w itho ut los s o f ge ne r ality , in addition to relations (4), we ca n assume that the force fields are orthogo nal. This simple fa c t ha s not b een indicated yet. The elimination of the redundant par ameter makes it p oss ible to sig nificantly simplify all subseq uent ca lculations and to obtain r esults in a symmetric form. 1 In v arian t S ubset and Its Prop erties In the sequel, we consider sy stem (1) under a ssumptions (3) and (4) with the phase spa ce P 6 sp ecified by the for mulas α 2 = a 2 , β 2 = b 2 , α · β = 0 . (9) The ca se a = b is singula r. Indeed, as indicated in [10], in this case, ther e exist a gr oup of symmetries generated by the transfor mations of the c o nfiguration space a nd, hence, a cyclic integral linear in the a ngular velocities. Therefore, for the sake of being definite, w e s e t a > b. (10) Denote by G the general in tegr a l of the problem o bta ined in [2] and represent it in the form G = 1 4 ( g 2 α + g 2 β ) + 1 2 ω 3 g γ − b 2 α 1 − a 2 β 2 , (11) where g α , g β , a nd g γ are the scalar pr o ducts of the kinetic momentum I ω and the vectors α , β , α × β , resp ectively . Int ro duce the function F by s etting F = (2 G − p 2 H ) 2 − r 4 K, where the parameters p and r are sp ecified as follows: p 2 = a 2 + b 2 , r 2 = a 2 − b 2 . (12) The la tter is well defined due to ineq ua lity (10). Obviously , F is a co mbined first integral of E qs. (1). Note that the ze r o level of the function F is specifie d by o ne o f the conditions 2 G − p 2 H − r 2 √ K = 0; (13) 3 2 G − p 2 H + r 2 √ K = 0 . (14) If β = 0 , then these conditions reduce to the equations of the second a nd third Appelr ot classes, resp ectively [1]. Define the subset N ⊂ P 6 as the set o f critical p o ints of the function F lying on the level F = 0. The set N is definitely non- empt y; it co nt ains, e.g ., a ll po int s of the for m ω 1 = ω 2 = 0 , α 1 − β 2 = 0, and α 2 + β 1 = 0, whic h a r e critica l for K a nd turn the expres s ion 2 G − p 2 H into zero. The se t N is inv a riant under the phase flow of (1) as the set of critical p o in ts o f the general integral. The co ndition dF = 0 means that the differentials of the functions H , K , and G are linea rly dep endent at the p oints of N . It immediately implies that the relation (2 g − p 2 h ) 2 − r 4 k = 0 (15) for the constan ts of these integrals is the equation of one of the s he e ts of the bifurcation diagram (the inv estigation of this diagr am has not b een co mpleted yet) of the genera lized K ow alevski top. W e use the following complex change of v ariables [7] (a g eneralizatio n of the K ow alevski change; see [8]): x 1 = ( α 1 − β 2 ) + i ( α 2 + β 1 ) , x 2 = x 1 , y 1 = ( α 1 + β 2 ) + i ( α 2 − β 1 ) , y 2 = y 1 , z 1 = α 3 + iβ 3 , z 2 = z 1 , w 1 = ω 1 + iω 2 , w 2 = w 1 , w 3 = ω 3 . (16) Denote the op er ation of differentiation with resp ect to the ima ginary time it by primes and r ewrite the equations o f motion in terms of the new v a riables: 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 , 2 w ′ 1 = − ( w 1 w 3 + z 1 ) , 2 w ′ 2 = w 2 w 3 + z 2 , 2 w ′ 3 = y 2 − y 1 . (17) Constraints (9) now ta ke the form z 2 1 + x 1 y 2 = r 2 , z 2 2 + x 2 y 1 = r 2 , x 1 x 2 + y 1 y 2 + 2 z 1 z 2 = 2 p 2 . (18) F urther on, instead of integral (11), it is conv enient to consider another general integral linear ly ex- pressed via G a nd H , na mely , M = 1 r 4 (2 G − p 2 H ) . (19) On the level F = 0, we have K = r 4 M 2 . (20) In terms of v ariables (16 ), r e w r ite the integrals H , K , a nd M a s follows: H = w 1 w 2 + 1 2 w 2 3 − 1 2 ( y 1 + y 2 ) , K = U 1 U 2 , M = − 1 2 r 4 F 2 1 + 1 2 r 2 ( U 1 + U 2 ) , where F 1 = √ x 1 x 2 w 3 − 1 √ x 1 x 2 ( x 2 z 1 w 1 + x 1 z 2 w 2 ) , U 1 = x 2 x 1 ( w 2 1 + x 1 ) , U 2 = x 1 x 2 ( w 2 2 + x 2 ) . Consider the function F 2 = U 1 − U 2 . 4 Prop ositi on 1. 1. In the domain x 1 x 2 6 = 0 , the invariant set N is sp e cifie d by t he following system of functional ly indep endent e quations: F 1 = 0 , F 2 = 0 . (21) Pr o of. Repr esent re la tion (20) in the form  F 2 1 − r 2 ( p U 1 − p U 2 ) 2  F 2 1 − r 2 ( p U 1 + p U 2 ) 2  = 0 , (22) where p U 1 and p U 2 are chosen to b e complex c o njugates o f each other. On the level F = 0, the functions F 1 , p U 1 , and p U 2 are indep endent everywhere except for the set w 1 w 2 = 0 , x 1 = x 2 . (23) Therefore, the co ndition that the left-hand side of Eq . (2 2 ) ha s a critical p oint leads to E q s. (21). It is clear that p oints (23 ) als o satisfy these equations. Thus, it remains to notice that the functions F 1 and F 2 are indep endent o n the le vel F = 0 everywhere in their domain of definition including p oints (23). The s ystem of inv ar iant rela tio ns (21) is obtained in [7] without using the first integrals. In virtue of the above definition a nd Pro po sition 1.1, the indicated system describ es a cer ta in smo o th four -dimensional (non-closed) ma nifo ld N 4 = N ∩ { x 1 x 2 6 = 0 } , and N is the le a st inv ariant subset of P 6 containing N 4 . Remark 1.1. It is eas y to see that the in v ariant set N , a s a whole, is stra tified, namely , N = 4 ∪ i =1 N i , dim N i = i, ∂ N i ⊂ i − 1 ∪ j =1 N j . Moreov er, in virtue o f Pr op osition 1.1, a ll N i with i < 4 b elo ng to a s ubset of the phase s pa ce s pe cified by the equa tion x 1 x 2 = 0 (24) (e.g., N 1 = { w 1 w 2 = 0 , w 3 = 0 , x 1 x 2 = 0 } is diffeomo rphic to 2 S 1 ). Therefore, the existence of singula rity (24) in the ex pr essions for F 1 and F 2 is in no case a ccidental. If, in re la tions (21 ), we remove the denomina- tors, then the set o f solutio ns of the obtained s ystem co nt ains the entire four -dimensional manifo ld (2 4 ). This manifold is not everywhere critical for the function F (how ev er, F is identically zero on it). In pa rticular, all tr a jectories starting from this manifold fill a set in P 6 , which is almos t everywhere five-dimensional. The follo wing statement demonstra tes that if w e restrict ourselves to relations (21 ), i.e., study the dynamics only on N 4 , then we do no t lo s e any tra jectory of the dynamic sy stem on N . Prop ositi on 1 .2. Set (24) do es not c ontain subset s invariant under the phase flow of system (1) . T o prov e this prop osition, it is necess ary to compute the deriv atives of x 1 x 2 in virtue of Eqs. (17) up to the fourth order , inclusively , and show that they cannot v anish simult aneo usly on the set s pe c ified by relation (2 4). It is worth noting that the indica ted str o ng degenera tio n o f this subset a lso takes place for motions of the heavy Kowalevski top. In that case, condition (24) means that the axis of dynamic sy mmetry of the top is vertical. Specia l a ttent ion is given to this phenomenon b o th in the cla ssical pap ers (see, e .g ., [1]) and in r ecent inv estigations dealing with the computer animation o f mo tio n (see [9 ] , where one can also find an extensive bibliogr aphy o f works in this field devoted to the investigation of heavy Kow alevski tops). Prop ositi on 1.3. The differ ential 2-form induc e d on the manifold N 4 by t he symple ctic structur e of the sp ac e P 6 pr oviding the Hamiltonian pr op erty of Eqs. (1) is non-de gener ate everywher e exc ept for the su bset sp e cifie d by the e qu ation L = 0 , wher e L = 1 √ x 1 x 2 h w 1 w 2 + x 1 x 2 + z 1 z 2 2 r 2 ( U 1 + U 2 ) i . 5 Pr o of. T he Poisson brack ets of the functions on R 9 ( ω , α , β ) sp ecifying the indicated symplectic structur e are deter mined acc ording to the following rules [3]: { g i , g j } = ε ij k g k , { g i , α j } = ε ij k α k , { g i , β j } = ε ij k β k , { α i , α j } = { β i , β j } = { α i , β j } = 0 , (25) where g 1 = 2 ω 1 , g 2 = 2 ω 2 , and g 3 = ω 3 are the comp onents o f the kinetic mo men tum. In relations (25), we now pass to v ar iables (16 ) and co mpute the Poisso n brack et for the functions F 1 and F 2 . In view of relations (21), this gives { F 1 , F 2 } = − r 2 L. The tangent space T q N 4 is a s kew-orthog onal complemen t of the span o f vectors included in the Hamiltonian fields with Hamiltonians F 1 and F 2 . By the Ca rtan for mula (see, e.g ., [6], p. 231), the restriction of the symplectic structure to T q N 4 is non-dege nerate provided that { F 1 , F 2 } ( q ) 6 = 0. Prop ositi on 1.4. The fun ct ion L is the first inte gr al of the dynamic syst em induc e d on N 4 . Mor e over, t his inte gr al is in involution with the inte gr al M . Pr o of. As shown in [7], in virtue o f (17) we can write F ′ 1 = µ 1 F 2 , F ′ 2 = µ 2 F 1 . Here µ 1 and µ 2 are functions smo oth in the neighbor ho o d of N 4 . In v ie w of these equalities, apply the Jacobi ident ity to the functions H , F 1 , a nd F 2 and obtain that the double Poisson bra ck et { H , { F 1 , F 2 }} is a line a r combination of the functions F 1 and F 2 . Therefo re, L ′ ≡ 0 on the s et sp ecified by r elation (21 ). It is shown by direct ca lculation that the following rela tion is tr ue under conditions (21 ): L 2 = 2 p 2 M 2 + 2 H M + 1 (26) and, ther efore, L { L, M } = M { H, M } ≡ 0 . It means that { L , M } = 0 for L 6 = 0. Hence, by contin uity , { L, M } = 0 everywhere on N 4 . Thu s, in the smo oth par t N 4 of the inv ar iant s ubset N completely sp ecifying the entire dyna mics on N , the equations of motion o f the gener alized Kowalevski top define the Hamiltonian system with t wo deg rees of free dom with a closed s ubset o f p oints of degenera tion of the s ymplectic s tr ucture nowhere dense in N 4 . 2 Analytic Solution By Pr op osition 1.4, to integrate the eq ua tions of motion in the set N , we can use the integrals M and L . The orig inal gener al in tegra ls H , K , and G a re expressed via these in tegra ls b y using relations (19), (20), and (2 6 ). Theorem 2.1 . On an arbitr ary inte gr al manifold J m,ℓ = { M = m, L = ℓ } ⊂ N , (27) the e quations of motion ar e sep ar ate d in the variables s 1 = x 1 x 2 + z 1 z 2 + r 2 2 √ x 1 x 2 , s 2 = x 1 x 2 + z 1 z 2 − r 2 2 √ x 1 x 2 (28) and take t he form s ′ 1 = q s 2 1 − a 2 r ms 2 1 − ℓs 1 + 1 4 m ( ℓ 2 − 1) , s ′ 2 = q s 2 2 − b 2 r ms 2 2 − ℓs 2 + 1 4 m ( ℓ 2 − 1) . (29) 6 Pr o of. In virtue o f the fir st eq uation in (21 ), the function M tak es the following form on N : M = 1 2 r 2 ( U 1 + U 2 ) . In virtue of the sec o nd eq uation in (21), we get U 1 = U 2 . There fore, the integral equation M = m yields U 1 = r 2 m and U 2 = r 2 m. (30) Determine w 3 from the first equation in (21) and w 1 and w 2 from Eqs . (30). W e o btain w 3 = z 1 w 1 x 1 + z 2 w 2 x 2 , w 2 = r x 2 x 1 R 1 , w 1 = r x 1 x 2 R 2 , (31) where R 1 = p r 2 m − x 1 and R 2 = p r 2 m − x 2 . (32) Substituting these quantities in the equation of the integral L , w e o btain m ( x 2 + z 2 ) − ℓx + p r 4 m 2 − 2 r 2 mx co s σ + x 2 = 0 . (33) The v ariables x, z , and σ are defined as follows x 2 = x 1 x 2 , z 2 = z 1 z 2 , x 1 + x 2 = 2 x cos σ , (34) and the radical in (33 ) corresp o nds to w 1 w 2 , and therefore is no n-negative. The o ther radicals use d ab ov e, including R 1 and R 2 , are algebraic. Equation (3 3) now yields R 1 R 2 = ℓx − m ( x 2 + z 2 ) , R 2 1 + R 2 2 = 1 r 2 m { [ ℓx − m ( x 2 + z 2 )] 2 − x 2 } + r 2 m, Int ro ducing the p olyno mial Φ( s ) = 4 ms 2 − 4 ℓ s + 1 m ( ℓ 2 − 1) , we c a n write in terms o f v ariables (28) R 1 + R 2 = r s 1 − s 2 p Φ( s 2 ) and R 1 − R 2 = r s 1 − s 2 p Φ( s 1 ) . (35) Using c o nstraints (18) and rela tions (34), we obtain ( z 1 ± z 2 ) 2 = 1 r 2 [( x 2 + z 2 ± r 2 ) 2 − 2 x 2 ( p 2 ± r 2 )] . Hence, in terms of v a riables (28), z 1 + z 2 = 2 r s 1 − s 2 q s 2 1 − a 2 , z 1 − z 2 = 2 r s 1 − s 2 q s 2 2 − b 2 . (36) W e now differentiate rela tions (28) in virtue of system (17). In v iew of (31), we g et s ′ 1 = r 2 4 x 2 ( z 1 + z 2 )( R 1 − R 2 ) and s ′ 2 = r 2 4 x 2 ( z 1 − z 2 )( R 1 + R 2 ) . Substituting expressions (35 ) and (36 ) in these equalities, we ar rive a t sy s tem (29). Remark 2.1. It is clear that the deduced equatio ns ca n b e integrated in elliptic functions o f time. B y using the standar d pr o cedure, the solutions are express e d in terms of Jaco bi functions. Their specific for m depe nds on the lo ca tion of the ro ots of the p o lynomials under the r adicals on the right-hand s ides. The bifurcation solutions of systems o f this type co rresp ond to stationar y p o int s of o ne of the equatio ns, i.e., to the cas es for whic h the p o lynomial ( s 2 − a 2 )( s 2 − b 2 )Φ( s ) (37) 7 po ssesses a multiple ro ot. F or dimension rea s ons, the or iginal phase v ariables on manifold (27) a re expressed via s 1 and s 2 , though, in genera l, these expressions mig ht b e multi-v alued functions . W e now show that the latter hav e a fairly simple a lgebraic form. Int ro duce the following notation: S 1 = q s 2 1 − a 2 , ϕ 1 = p − Φ( s 1 ) , S 2 = q b 2 − s 2 2 , ϕ 2 = p Φ( s 2 ); (38) ψ = 4 ms 1 s 2 − 2 ℓ ( s 1 + s 2 ) + 1 m ( ℓ 2 − 1) . (39) Theorem 2.2. O n the c ommon level of the first inte gr als (27) , by using notation (38) , (39) , t he phase variables of t he gener alize d K owalevski c ase c an b e ex pr esse d, in t erms of variables (28) , as fol lows: α 1 = 1 2( s 1 − s 2 ) 2 [( s 1 s 2 − a 2 ) ψ + S 1 S 2 ϕ 1 ϕ 2 ] , α 2 = 1 2( s 1 − s 2 ) 2 [( s 1 s 2 − a 2 ) ϕ 1 ϕ 2 − S 1 S 2 ψ ] , β 1 = − 1 2( s 1 − s 2 ) 2 [( s 1 s 2 − b 2 ) ϕ 1 ϕ 2 − S 1 S 2 ψ ] , β 2 = 1 2( s 1 − s 2 ) 2 [( s 1 s 2 − b 2 ) ψ + S 1 S 2 ϕ 1 ϕ 2 ] , (40) α 3 = r s 1 − s 2 S 1 , β 3 = r s 1 − s 2 S 2 , ω 1 = r 2( s 1 − s 2 ) ( ℓ − 2 ms 1 ) ϕ 2 , ω 2 = r 2( s 1 − s 2 ) ( ℓ − 2 ms 2 ) ϕ 1 , ω 3 = 1 s 1 − s 2 ( S 2 ϕ 1 − S 1 ϕ 2 ) . Pr o of. B y using notation (12 ), we r epresent the compatibility conditions of constra ints (18) in the v ariables x and z as follows: x 2 + z 2 + r 2 > 2 a | x | ,   x 2 + z 2 − r 2   6 2 b | x | , whence we get the natura l ra nges for v ariable s (2 8 ): s 2 1 > a 2 and s 2 2 6 b 2 . (41) Hence, r ewriting Eqs . (29 ) in the rea l for m, we conclude that, for given m and ℓ , the domain of p ossible motions in the plane ( s 1 , s 2 ) is determined, along with ineq ua lities (41), by the inequalities Φ( s 1 ) 6 0 and Φ( s 2 ) > 0 . (42 ) Thu s, in par ticular, a ll v alues (38) ar e rea l on the tra jectories of the analyzed system. The expressio ns for the complex v ariables x 1 , x 2 , z 1 , z 2 , w 1 , w 2 , a nd w 3 in ter ms of s 1 and s 2 are obtained by a pplica tion, in sequence, of relations (35) with r e gard for (32), then (36 ), and, finally , (31 ). After this, the v ar iables y 1 and y 2 are determined from the fir s t tw o rela tions in (18). By the change of v ariables inverse to (16 ), we arrive a t the required dep endencies (40). Note that the v a lue s 1 = ∞ is or dinary for the firs t equation in (29 ) (be c ause the deg ree of the po lynomial under the ra dical on the right-hand s ide is even). More ov er, if this v a lue b elo ngs to the domain of p os sible motions, then it is reached dur ing a finite p erio d o f time. Similar ly , rela tions (40 ) also do not hav e singula rities in this case. This can b e proved by the change of v ariables s 1 7→ 1 /s 1 . Thus, in particula r, we hav e deduced a nalytic expr essions for all c a ses in which the tr a jector ie s pass the set sp ecified by r elation (24). It means that we have cons tr ucted the complete analytic solution o f the proble m on the inv ariant set N . 8 3 Phase T op ology In the regula r case, the integral manifold J m,ℓ consists of tw o-dimensional Liouville tori. The ca s es when they top olog ically rea r range gener a te the bifurc ation diagra m of the sy stem o n N . It is na tural to s tudy this diagram in the plane of cons tants of the used in tegra ls, i.e., a s the set of critica l v alues of the map J = M × L : N → R 2 . (43) Theorem 3.1 . The bifur c ation diagr am of m ap (43) is a p art of the syst em of str aight lines ℓ = − 2 ma ± 1 , ℓ = 2 ma ± 1 , ℓ = − 2 mb ± 1 , ℓ = 2 mb ± 1 , (44) and the c o or dinate ax es of t he plane ( ℓ, m ) lying in the half-plane ℓ > 0 and sp e cifie d by the c onditions of existenc e of r e al s olut ions ℓ > max (2 ma − 1 , − 2 mb + 1) , m > 0; ℓ 6 − 2 mb + 1 , m < 0; ℓ = 1 , m = 0 . (45) Pr o of. Acco rding to Remar k 2.1, the diagr am co n tains the discriminant set of p oly nomial (37) for med by straight lines (44) (in the part cor resp onding to the existence of motions). The p oints of the co ordinate axes in the plane ( m, ℓ ) b elonging to J ( N ) m ust b e included in the diagram; indeed, it can b e shown that the v alues m = 0 a nd ℓ = 0 a re attained, in par ticular, on the subsets N i ( i < 4) , where N fails to b e smo o th (see Remark 1.1). The analytica l foundation for this inclusion is as follows. In Eq s. (2 9), we can pass to the limit as m → 0. As a result, by using relations (26 ), we conclude that | ℓ | → 1 and ( ℓ 2 − 1) / 2 m → h . At the same time, the de g ree o f the p olynomials under the r adicals decreases to three; the form of so lutions changes. Moreov er, it is clear that K equals zero o n the set N ∩ { M = 0 } , i.e., the corr esp onding motions a lso b elong to the cla ss (7). [It is worth noting that, a s shown in [1 1], the restriction o f symplec tic structure to manifold (7 ) deg enerates just at the p oints o f corr esp onding tra jectories.] Therefor e, the v alue m = 0 should b e rega rded as corresp o nding to bifurcation. On the other ha nd, the integral s urface J m,ℓ is preser ved by the in version ( α 3 , β 3 , ω 3 ) 7→ ( − α 3 , − β 3 , − ω 3 ) . In re lations (40 ), this inv ersion is realized either by changing the sign o f the radicals S 1 and S 2 or by the substitution ( ℓ , s 1 , s 2 ) 7→ ( − ℓ, − s 1 , − s 2 ). This means that J m,ℓ and J m, − ℓ are the same subset of the phase space. T he r efore, we need to restrict ourse lves to the v alues of ℓ of the same s ign (to b e definite, we choos e non-neg ative v alues); the axis ℓ = 0 b ecomes the outer b ounda r y of the doma in o f existence of motions in the plane of c o nstants of the integrals. In virtue of E q. (33), ℓ can equal zer o only for neg a tive v alues of m . Thu s, in a ddition to (4 4), the diagra m should als o be supplemented with the p oint { m = 0, ℓ = 1 } and the semi-a xis { ℓ = 0 , m < 0 } . (46) By ana ly zing the compatibility of conditions (41) and (42), we determine the actual regio n of existence of motio ns in the form (45). In Fig. 1 , the doma ins with num ber s 1 –9 defined by the dia g ram in the plane ( m, ℓ ) corresp o nd to different types of the integral surfaces (27 ). The motion is impo ssible in the sha ded r e gion. T o determine the num be r of tori for the r e g ular manifold, we note that relatio ns (40) give a one- v alued depe ndence o f the pha se v ariables on tw o collections of quantities ( s 1 , S 1 , ϕ 1 ) and ( s 2 , S 2 , ϕ 2 ) . (47) In this case , the signs of the ra dic a ls in (3 8) o n e ach J m,ℓ are a rbitrary . How ever, along the tra jectory , some r a dicals tur n to zero a nd then change the s ign. This means that tw o p oints that only differ by the sig n of such radical lie in the same connected co mpo nent o f J m,ℓ . Ther e fore, the num b er of co nnected comp onents of the regular in tegra l manifold is equal to 2 n , where n is the n umber of quantities (38) non-zer o along the tra jectory . The v alue of n is determined a c c ording to the lo cation o f r o ots o f p olyno mia l (37) a nd do es not exceed 2. 9 Figure 1: Bifurcation diagram and the domains of existence of motions Prop ositi on 3 .1. Assume that the analyze d domains ar e numb er e d as indic ate d in Fig. 1 . Then the inte gr al manifolds J m,ℓ c an b e describ e d as fol lows: a) T 2 , in domains 1 and 8 , b) 2 T 2 , in domains 2 , 7 , and 9 , and c) 4 T 2 , in domains 3 – 6 . T o determine the type of critica l integral surfaces, we note that, in each thre e-dimensional space of collections (47), equa lities (38) sp ecify a pa ir of cylinders (elliptic o r h yp erb olic) with mutually orthogo nal generatric es. F or the p oints of the s traight lines (44), a pair of cylinder s co rresp onding to one of the v ariables s 1 or s 2 has a tangency p oint. Hence, the line of their intersection is the eig ht curve S 1 ∨ S 1 . Thus, on segments o f the str aight lines (44) b ounded by the p oints of lines int ers ection and internal for doma in (45), the integral sur face co ns ists of the comp onents ho meomorphic to the pro duct S 1 × ( S 1 ∨ S 1 ). Crossing s uch segment, we obser ve one o f bifurcations T 2 → 2 T 2 t ypical for sy stems with tw o deg rees of free do m. The nu mber of connec ted comp onents of the form S 1 × ( S 1 ∨ S 1 ) in the critica l J m,ℓ is determined by the num b er of tori in the adjacent domains . Actually , the critica l per io dic tra jectories (the tr aces o f centers of the eight curve) are motions in which one of v ariables s 1 or s 2 remains constant and equal to the multiple ro ot of the corres p o nding p o ly nomial under the radical. In this case, either S 1 ≡ 0 and ϕ 1 ≡ 0 or S 2 ≡ 0 and ϕ 2 ≡ 0. Hence, it follows from r elations (40 ) that ω 2 = ω 3 ≡ 0 in the first case and ω 1 = ω 3 ≡ 0 in the second case. The b o dy p erfor ms p endulum motions in which the radius vector of one of the centers of r igging is per manently dir ected along the corres po nding force field. In approa ching the outer b oundar y of domain (45) with the exception of the half-line (46), the tori degenera te in to circles (p erio dic solutions of the indicated pendulum type) and the surfaces S 1 × ( S 1 ∨ S 1 ) degene r ate into eight curves. It is clear that the cr itical single-fr e quency motions do no t app ear in the half-line (46). The cor r esp ond- ing bifurcation in the seg men ts adjacent to domains 5 a nd 6 is characteriz e d by the fact that the num b er of connected comp onents o f J m, 0 is half as large as a t the close r egular p oint of the plane ( m, ℓ ). These ar e so- called minimal tori. The transition from domain 4 to a s egment of the b oundar y set (46) is not accompa nied by the decrease in the num b e r of comp onents of J m,ℓ and all cycles homo to pic to a cer tain ma rked cy cle are folded so that each comp onent cov ers the limiting compo nent twice. In a sufficiently smo oth case (e.g., in the cas e when L is a Bott integral on the corr esp onding smo oth level of the int egr al M ), a Klein b ottle should b e obtained as a result (see, e.g., [6]). How ever, accor ding to the explic it equa tions (40), this is not true in o ur case. Most likely , this phenomeno n is connected w ith the degener ation of the induced symplectic structure. Finally , consider the no des denoted b y P 1 – P 4 in Fig. 1. F or these v alues o f the c o nstants of integrals, each s urface J m,ℓ contains one singular po int. These p oints corres po nd to the e q uilibria o f the b o dy in which bo th centers of rig ging lie o n the co r resp onding axes of force fields and, henc e , the moment of forces is e q ual to ze r o. O ne o f these p oints is stable: at P 1 , the int egr al surfa ce consists of a single p oint. The other three po ints ar e unstable. As indicated ab ove, a t the no des P 2 and P 3 the integral s urface is homeomor phic to an eight cur ve. At the no de P 4 the integral surface can b e describ ed a s follows. T ake a rectang le and identify its vertices w ith one po int ; it then can b e filled with tr a jectorie s double-asymptotic to the singular p oint. The bo undary of this set is a bunch of four circ le s. This b o unda ry represe n ts t wo pair s of p endulum mo tions; each pair is asy mptotic to the highest p osition of one of the tw o centers o f rigging. T ake four copies of the 10 obtained set and attach the bounda r y of ea ch to the same bunch of four cir c les. All indicated phenomena a re readily established by analyzing r elations (40) and the mutual lo c ation of the cylinder s formed in spaces (47). Conclusions In the present work, we p er form the complete inv estiga tio n of motions of the genera lized Kowalevski top playing the ro le of cr itical motions for the entire problem and generating bifurcations o f three-dimensiona l Liouville tori along paths crossing the sheet sp ecified by Eq. (15 ) o f the bifurcation diag ram Σ ⊂ R 3 of the general int egr als of the pr oblem. Inequalities (45) are used to deduce the eq uations of the b ounda r y of a part of this s hee t cor resp onding to the existence of actual critical motions , i.e., cont ained in Σ . Consider rela tion (22). It plays the r ole of the equa tion of the entire in tegr al sur face in the phas e space P 6 for the co llection of constants of integrals sa tisfying relation (15). It then follows, similar to the case of the heavy Kow alevski top (the s econd and third Appelro t classes), that the straight line { k = 0 , 2 g = p 2 h } splits sheet (15) into tw o cla sses. In the fir s t class sp ecified by relatio n (13) and cor resp onding to the first non-negatively definite factor in (22), the obtained integral manifolds, being cr itical for the orig inal system, exhaust the en tire cor r esp onding integral surface in P 6 as the limit o f a concentric fa mily of three-dimensiona l tori and are, in this sens e , stable in P 6 . In the second class sp ecified by relatio n (14) and corr esp onding to the second (hyperb olic ) factor in (22), all obtained critical sur faces in P 6 are hyper b o lically unstable: on the sa me level of the first three integrals, one can find tra jectories consisting o f regular po int s a nd double-asymptotic to the corre s po nding tw o-dimensional tor i of the system on the investigated inv aria nt set. References [1] G. G. App elrot, Inc ompletely S ymmet ric He avy Gyr osc op es , in: Mo tion of a Rigid Bo dy Ab out a Fixe d Poin t [in Russian], Izd. Ak ad. Nauk SSSR, Mo scow, 194 0, pp. 6 1–156 . [2] A. I. 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Kow alevsk i, Sur le pr obl ` eme de la r otation d’un c orps solide autour d’un p oint fixe / / Acta Math, 12 (188 9), 177 –232 . [9] P . H. Ric hter, and H. R. Dullin, A. Wittek, Kowalevski T op. Film C1 961 // T echn. Wiss./Naturw. (1997), No. 13, 3 3 –96. [10] H. Y ehia, New inte gr able c ases in the dynamics of rigid b o dies // Mech. Res . Commun. 13 (1 986), No. 3, 169–1 72. [11] D. B. Zo tev, F omenko–Zieschang invariant in the Bo goyavl enskyi c ase // Reg. Chaot. Dynam., 5 (200 0), No. 4 , 437– 458. 11