Bifurcation diagrams of the Kowalevski top in two constant fields

Bifurcation diagrams of the Ko w alevski top in tw o consta n t fields ∗ Mikhail P . Kharlamo v Abstract The Kow alevski top in tw o constant fields is kno wn as the unique profo und example of an integ rable Hamil- tonian system with three degrees of fre edom not reducible to a family of systems in few er dimensions. As the first approac h to top ological analysis of th is system we fin d the critical set of the integ ral map; this set con- sists o f the tra jectories with num ber of frequencies less th an th ree. W e obtain the e quations of th e bifurcation diagram in R 3 . A correspond ence to t h e App elrot classes in the classical Kow alevski p roblem is established. The adm issible regions for the v alues of the fi rst integral s are found in the form of some inequ alities of general chara cter and b oundary conditions for the induced diagrams on energy lev els. 1 In tro duction During the las t 20 y ears th e in tegr able cas e of S. K ow a levski [1] has received several generalizations. Among them a special place is given to the c ase [2] of rota tion ab out a fixed p oint of a heavy electrically charged gy rostat in gravitational and electr ic fields of force. F or a rigid b o dy without gyro s tatic effects the co rresp onding equations were first considered in [3 ] a nd interpreted as the equations o f motion of a massive magnet sub ject to the gravity force a nd constant magnetic force fields . The mathematical mo del of sup erp osition o f such fields is r eferred to a s two c onstant fields [4]. The case [2] does not hav e any explicit g roups of symmetry and ther efore provides an illustra tion of a physically realizable system with three degrees of freedom not admitting any obvious reduction to a fa mily o f systems with t wo degr ees o f freedom. The phase top ology of irreducible sys tems has not been s tudied yet. The theory of n - dimensional integrable systems or iginated in [5 ] has not b een further dev elop ed due to the absence, at that moment, of non-trivial natural examples. The result [2] s ucceeded some previous publications dealing with rig id b o dies a nd gyr ostats satisfying the conditions of Kow alevski type: I.V. Komarov [6] has prov ed the complete integrability of the Kow alevski gy rostat in gravit y force field b y finding the firs t gener a lization of the Kow alevski integral K ; the cor resp onding integral for the rigid b o dy in tw o constant fields was p ointed out b y O.I. Bogoy a vlensky [3]; this integral w as upg r aded to the ca se o f gyr ostat by H. Y ehia [7]. Y et the a nalog of the Kowalevski case for tw o constant fields had no t bee n considered integrable un til A.G. Reyman and M.A. Semenov-Tian-Shansky [2] found the Lax repre s en tation with sp ectral parameter; this immediately led to the new in tegral generalizing the s quare of momentum integral for axially symmetric force fields. Later, in a joint publication with A.I. B o benko [4 ], the authors of [2] pr esented algebr aic foundations for the in tegrability of multidimensional Ko w alevski gyros tats and described a v iable w ay of explicit integration using finite-band technique. F or tw o constan t fields this integration w as nev er fulfilled. This paper starts the in v estigation of three-dimensional phase top ology of a rigid b ody of Ko w alevski t yp e in t wo co nstan t fields. 2 Preliminaries Consider a rig id bo dy with fixed p oin t O . Choose a trihedral at O r otating alo ng with the bo dy and refer to it all vector and t ensor ob jects. Denote b y e 1 e 2 e 3 the canonical unit basis in R 3 ; then the moving trihedral itself is represented as O e 1 e 2 e 3 . Constant field is a force field inducing the rotating momen t a bo ut O of the form r × α (2.1) ∗ REGULAR AND CHAOTIC D YNAMICS, V. 10, N 4, 2005, pp. 381 -398, DOI: 10.1070/RD2005v01 0n04ABEH000321 1 with co nstant v ector r a nd with α corr espo nding to some physical v ector fixed in iner tial space; r po in ts fro m O to the center o f application of the field, α is the field’s intensit y . F or tw o constant fields the rota ting moment is r 1 × α + r 2 × β . It can be represented a s (2.1) if either r 1 × r 2 = 0 or α × β = 0. In the sequel w e supp ose that r 1 × r 2 6 = 0 , α × β 6 = 0 . (2.2) Two co nstant fields satis fying (2.2) are said to be indep endent . Int ro duce some notation. Let L ( n, k ) be the space of n × k -matr ices. Put L ( k ) = L ( k, k ). Ident ify R 6 = R 3 × R 3 with L (3 , 2) b y the isomorphism 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 erse map, we w r ite j − 1 ( A ) = ( c 1 ( A ) , c 2 ( A )) ∈ R 3 × R 3 , A ∈ L (3 , 2) . If A, B ∈ L (3 , 2), a ∈ R 3 , by definition, put A × B = 2 X i =1 c i ( A ) × c i ( B ) ∈ R 3 ; a × A = j ( a × c 1 ( A ) , a × c 2 ( A )) ∈ L (3 , 2) . (2.3) 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 pro of is by direct calculation. Denote by I the iner tia tensor of the b o dy at O a nd by ω the angular velo city . Using the notation (2.3) we write the Euler - Poisson eq uations of motion in the for m I ω · = I ω × ω + A × U, U · = − ω × U. (2.4) Here A = j ( r 1 , r 2 ) is a co ns tan t ma tr ix, U = j ( α , β ). The phase space of (2.4) is { ( ω , U ) } = R 3 × L (3 , 2). In fa c t, U in (2.4 ) is restricted by geometric integrals; that is, for some consta nt symmetric C ∈ L (2) U T U = C . (2.5) Let O r epresent the set (2.5) in L (3 , 2). In order to emphasize the C -dep endence, we wr ite O = O ( C ). Let S = ( I , A, C ). Denote by X S the vector field on R 3 × O ( C ) corr e spo nding to the sys tem (2.4). Asso ciate to Λ ∈ S O (3), D ∈ GL (2 , R ) the linea r automorphisms Ψ(Λ , D ) and ψ (Λ , D ) of R 3 × L (3 , 2) and L (3) × L (3 , 2) × L (2) Ψ(Λ , D )( ω , U ) = (Λ ω , Λ U D T ) , ψ (Λ , D )( I , A, C ) = (Λ I Λ T , Λ AD − 1 , DC D T ) . (2.6) It is easy to see that (2.5 ) and (2.6) imply Ψ(Λ , D )( R 3 × O ( C )) = R 3 × O ( D C D T ). Using Lemma 1 we obtain the following sta temen t. Lemma 2. F or e ach (Λ , D ) ∈ S O (3) × GL (2 , R ) , we have Ψ(Λ , D ) ∗ ( X S ( v )) = X ψ (Λ ,D )( S ) (Ψ(Λ , D )( v )) , v ∈ R 3 × O ( C ) . Thu s, any tw o problems of rig id b o dy dyna mics in tw o co nstant fields (for shor t, RBD-pr oblems) determined by the sets of pa r ameters S and ψ (Λ , D )( S ) ar e completely equiv a len t. Let us call an RBD-problem c anonic al if the centers of application of forces lie on the first t w o axes of the moving trihedral at unit distance fro m O and the int ensities of the for c es are orthogo nal to each other. Prop osition 1. F or e ach RBD-pr oblem with indep endent for c es ther e exists an e quivalent 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 s ame plane in t he b o dy c ontaining the fixe d p oint. 2 Pr o of. Let the RBD-pr oblem de ter mined by the s et o f pa r ameters S = ( I , A, C ) satisfy (2.2). This mea ns tha t the symmetric matrices A ∗ = ( A T A ) − 1 and C a re p ositively definite. According to the well-known fact from linear alg ebra, A ∗ and C ca n b e r educed, resp ectively , to the identit y matrix a nd to a diag o nal matrix via the sa me conjugation o per ator D A ∗ D T = E , D C D T = diag { a 2 , b 2 } , D ∈ GL (2 , R ) , a, b ∈ R + . Then c 1 ( AD − 1 ) and c 2 ( AD − 1 ) form an orthono rmal pair in R 3 . There exists Λ ∈ S O (3) s uc h that Λ c i ( AD − 1 ) = e i ( i = 1 , 2). The first sta temen t is o btained b y applying L e mma 2 with the previo usly chosen Λ , D to the initial vector field X S . T o finish the pro of, notice that the transformation A 7→ AD − 1 preserves the plane spa nning c 1 ( A ), c 2 ( A ). The matrix Λ in (2 .6) repre s en ts the change o f the moving trihedral. Therefo r e, if a ∈ R 3 represents some ph ysical vector in the initial pro blem, then Λ a is the same vector with resp ect to the b o dy in the equiv alen t problem. Remark . The fact that any RBD-problem can b e r e duced to the problem with o ne of the pairs r 1 , r 2 or α , β orthonor mal is known from [8 ]. Simultaneous orthogo nalization of b oth pairs cr ucially simplifies c a lculations below. It follows fr o m Pro pos itio n 1 that, without loss of gener alit y , for indep endent forces we may supp ose r 1 = e 1 , r 2 = e 2 , (2.7) α · α = a 2 , β · β = b 2 , α · β = 0 . (2.8) Change, if necessary , the order of e 1 , e 2 (with simultaneous change o f the direction of e 3 ) to obta in a > b > 0. Consider a dynamica lly symmetric top in t wo co ns tan t fields with the centers o f applica tion of for ces in the equatoria l plane of its inertia ellipsoid. Cho ose a moving trihedra l such tha t O e 3 is the symmetry a xis. Then the inertia tens or I b ecomes diago nal. Let a = b . F or a n y Θ ∈ S O (2) denote by ˆ Θ ∈ S O (3) the cor r esp onding rotation of R 3 ab out O e 3 . T a k e in (2.6) Λ = ˆ Θ, D = Θ. Under the co nditions (2.7), (2.8) ψ = Id and Ψ b ecomes the symmetry group. The system (2 .4) has the cyclic integral I ω · ( a 2 e 3 − α × β ) pointed o ut in [7] for the analog of the Kow alevski case. Therefore it is po ssible to reduce such an RBD-problem to a family of systems with tw o degrees of fre edom. Let us call a n RBD-problem irr e ducible if for its cano nic a l r epresentation (2.7), (2 .8) the following inequality holds a > b > 0 . (2.9) The following sta tements are needed in the future; they also reveal some featur es of a wide class of RBD- problems. Lemma 3. In an irr e ducible RBD-pr oblem, the b o dy has exactly four e quilibria. Pr o of. The se t of singular p oints of (2.4) is defined b y ω = 0 , A × U = 0. F or the equiv alen t ca nonical pro blem with (2.7) e 1 × α + e 2 × β = 0 . (2.10) Then the four vectors in (2.10) are para llel to the same plane and | e 1 × α | = | e 2 × β | . With (2.8), (2.9) this equality y ields α = ± a e 1 , β = ± b e 2 . (2.11) F ro m mechanical p oint of view, the result is absolutely clear: none of the o r thogonal forces with unequal int ensities and ”or tho normal” centers of applica tion can pro duce a non-zero moment at a n equilibrium. Lemma 4. L et an irr e ducible RBD-pr oblem in its c anonic al form have the diago nal inertia tensor I . Then the b o dy has the fol lowing families of p erio dic motions of p endulu m typ e α ≡ ± a e 1 , ω = ϕ · e 1 , β = b ( e 2 cos ϕ − e 3 sin ϕ ) , 2 ϕ ·· = − b sin ϕ ; (2.12) β ≡ ± b e 2 , ω = ϕ · e 2 , α = a ( e 1 cos ϕ + e 3 sin ϕ ) , 2 ϕ ·· = − a sin ϕ ; (2.13) α × β ≡ ± ab e 3 , ω = ϕ · e 3 , α = a ( e 1 cos ϕ − e 2 sin ϕ ) , β = ± b ( e 1 sin ϕ + e 2 cos ϕ ) , ϕ ·· = − ( a ± b ) sin ϕ. (2.14) The pro of is ob vious. Note that in the case considered the pointed o ut fa milies are the only motions with constant direction of the angular velo cit y . In particular , the b o dy in tw o indep endent constant fields do es not have any unifor m ro tations. 3 3 Critical set of the Ko w alevski top in t w o constan t fields Suppo se that the irr educible RBD-problem has a diago nal inertia tensor with principa l moments o f inertia s atisfying the ratio 2 :2:1; then w e o btain the in tegrable cas e [2] o f the K ow a levski top in tw o co nstant fields. By an appro priate choice o f meas urement units, we present eq uations (2.4) in scala r form 2 ω · 1 = ω 2 ω 3 + β 3 , 2 ω · 2 = − ω 1 ω 3 − 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 . (3.1) The phase s pace is P 6 = R 3 × O , wher e O ⊂ R 3 × R 3 is defined by (2.8); O is diffeo mo rphic to S O (3). The complete se t of first in tegrals in involution on P 6 consists of the energy int egral H , the generalized Kow alevski integral K [3], a nd the integral G found in [2]: 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 , G = 1 4 (2 α 1 ω 1 + 2 α 2 ω 2 + α 3 ω 3 ) 2 + 1 4 (2 β 1 ω 1 + 2 β 2 ω 2 + β 3 ω 3 ) 2 + + 1 2 ω 3 (2 γ 1 ω 1 + 2 γ 2 ω 2 + γ 3 ω 3 ) − b 2 α 1 − a 2 β 2 . (3.2) Here we deno te by γ i the co mponents of γ = α × β relative to the moving ba sis. Int ro duce the in tegral map J = G × K × H : P 6 → R 3 . (3.3) Let σ ⊂ P 6 be the set of cr itical p o in ts of J . By definition, the bifurcation dia gram of J is the subset Σ ⊂ R 3 ov er which J fails to b e lo ca lly trivial; Σ determines the cases when the top olog ical type of the integral manifolds J c = J − 1 ( c ) , c = ( g , k , h ) ∈ R 3 (3.4) changes. Finding the cr itical set σ and the bifurca tion diagr am is the necess a ry step in the top olog ical analysis o f the pr oblem a s a whole. It follows fro m the Liouville – Ar nold theorem that for c / ∈ Σ the manifold (3.4 ), if not empty , is a union of three-dimensional tori. The c o nsidered Hamiltonian sys tem is non-deg e ne r ate (at least for sufficiently small v a lues of b ); then the tr a jectories on s uc h a to r us are q uasi-p erio dic with three almo st everywhere indep enden t frequencie s. The critica l set σ is inv ariant under the phas e flow and co nsists of tra jectories with num ber o f frequencies less than three. These tra jectories are called critic al mot ions . F or a generic v a lue c ∈ Σ the set J c ∩ σ consists of tw o - dimensional tor i. The dynamical sys tem induced on the unio n of such tori for c in some op en subset in Σ is a Hamiltonian system with tw o degrees o f freedom. Vice versa, let M be a s ubmanifold of P 6 , dim M = 4, a nd suppo se that the induced system on M is Hamiltonian. Then, o bviously , M ⊂ σ . This sp eculation gives a useful to ol to find out whether a common level of functions co nsists of critica l p oint s of J . Lemma 5. Conside r a system of e quations f 1 = 0 , f 2 = 0 (3.5) on a domain W op en in P 6 . L et X b e the ve ctor field on P 6 c orr esp onding to (3.1) and M ⊂ W define d by (3.5). Supp ose (i) f 1 and f 2 ar e smo oth functions indep endent on M ; (ii) X f 1 = 0 , X f 2 = 0 on M ; (iii) t he Poisson br acket { f 1 , f 2 } is non-zer o almost everywher e on M . Then M c onsists of critic al p oints of the map J . Pr o of. Conditions (i), (ii) imply that M is a smo oth four-dimensio nal manifold inv ariant under the re striction of the pha s e flow to the o pen set W . Condition (iii) mea ns that the closed 2 -form induced on M by the symplectic structure on P 6 is almo st everywhere non-degenerate. T hus the flow on M is almost everywhere Hamiltonian with t wo degrees o f freedom. It inherits the pro p erty o f complete in tegrability . Then almos t all its in tegral manifolds consist of tw o-dimensional tori and necess arily lie in σ . Since M is close d in W and σ is closed in P 6 , we conc lude that M ⊂ σ . Two sys tems o f the t yp e (3.5) are known. The first one was p ointed out in [3]. It is the zero level of the int egral K . T he c o ndition K = 0 leads to tw o indep endent equa tio ns defining the smo oth four -dimensional manifold 4 M ⊂ σ . It is shown in [9] that the 2-for m induced on M by the sy mplectic structur e on P 6 is degenerate o n the surface of co dimension 1 . The s e c ond critical subset N ⊂ σ was found in [10] in the for m of a system of t wo equa tions s atisfying the conditions of Lemma 5. The functions in these equa tions hav e essential singularities a t the p oints α 1 = β 2 , α 2 = − β 1 . (3.6) The set N w as inv estigated in [1 1]. It was shown tha t N is the set of critical points o f some smo oth function F on P 6 . Then N is str atified by the rank of Hesse’s matrix o f F and fails to be a smo oth fo ur-dimensional manifold at some p oints of the set (3.6). In pa rticular, it cannot b e defined by any glo bal system of t w o independent equations. In this c a se the induced 2-for m a lso has degener ate p oints even in the smo oth pa rt of N . The fo llowing result completes the descr iptio n of the critical set σ b y a dding a ne w in v ariant subset O ⊂ P 6 ; O is almost everywhere a smo o th four-dimensional manifold. Note that the sets M , N and O hav e pairwise nonempt y int ersections co rresp onding to bifurcatio ns o f cr itica l integral manifolds of the induced ” almost Hamiltonia n” sys- tems with tw o degr ees of freedom. Let us int ro duce the following notation p 2 = a 2 + b 2 , r 2 = a 2 − b 2 ; ξ 1 = α 1 − β 2 , ξ 2 = α 2 + β 1 , η 1 = α 1 + β 2 , η 2 = α 2 − β 1 . Theorem 1. The set of critic al p oints of the inte gr al map (3.3) c onsists of the fol lowing su bsets in P 6 : 1) the set M define d by t he system Z 1 = 0 , Z 2 = 0 (3.7) with Z 1 = ω 2 1 − ω 2 2 + ξ 1 , Z 2 = 2 ω 1 ω 2 + ξ 2 ; (3.8) 2) the set N define d by t he system F 1 = 0 , F 2 = 0 , ξ 2 1 + ξ 2 2 6 = 0 (3.9) with F 1 = ( ξ 2 1 + ξ 2 2 ) ω 3 − 2[( ξ 1 ω 1 + ξ 2 ω 2 ) α 3 + ( ξ 2 ω 1 − ξ 1 ω 2 ) β 3 ] , F 2 = ( ξ 2 1 − ξ 2 2 )(2 ω 1 ω 2 + ξ 2 ) − 2 ξ 1 ξ 2 ( ω 2 1 − ω 2 2 + ξ 1 ) , (3.10) and by t he system ξ 1 = ξ 2 = 0 , α 3 = ± r , β 3 = 0 , η 2 1 + η 2 2 = 2( p 2 − r 2 ) , ( ω 2 1 + ω 2 2 )( α 3 ω 3 + η 1 ω 1 + η 2 ω 2 ) + r 2 ω 1 = 0; (3.11) 3) the set O define d by the syst em R 1 = 0 , R 2 = 0 (3.12) with R 1 = ( α 3 ω 2 − β 3 ω 1 ) ω 3 + 2 ξ 1 ω 1 ω 2 − ξ 2 ( ω 2 1 − ω 2 2 ) + η 2 ( ω 2 1 + ω 2 2 ) , R 2 = ( α 3 ω 1 + β 3 ω 2 ) ω 2 3 + [ α 2 3 + β 2 3 + ξ 1 ( ω 2 1 − ω 2 2 ) + 2 ξ 2 ω 1 ω 2 + + η 1 ( ω 2 1 + ω 2 2 )] ω 3 + 2[ ξ 1 ( α 3 ω 1 − β 3 ω 2 ) + ξ 2 ( α 3 ω 2 + β 3 ω 1 )] . (3.13) Pr o of. In tro duce the change o f v ariables [10] ( i 2 = − 1) x 1 = ξ 1 + iξ 2 , x 2 = ξ 1 − iξ 2 , y 1 = η 1 + iη 2 , y 2 = η 1 − iη 2 , 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 . (3.14) The system (3.1 ) takes the form 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 . (3.15) Here the prime sta nds for d/d ( it ). 5 Denote by V 9 the s ubspace of C 9 defined by (3.14). On V 9 , e quations (2.8) of the phase space P 6 bec ome 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 . (3.16) By virtue of (3.14) and (3.1 6) the integrals (3.2) ta k e the form H = 1 2 w 2 3 + w 1 w 2 − 1 2 ( y 1 + y 2 ) , K = ( w 2 1 + x 1 )( w 2 2 + x 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 ) . (3.17) Let f be an arbitra ry function on V 9 . F or bre v it y , the ter m ”cr itical po in t o f f ” will alwa ys mean a critical po in t of the restriction o f f to P 6 . Similarly , d f means the restr iction o f the differential of f to the set of vectors tangent to P 6 . While calculating critical po in ts of v arious functions (in the a b ove s ense), it is conv enient to av oid in tr o ducing Lagra ng e multipliers for the restr ictions (3.16). Notice that the following vector fie lds X 1 = ∂ ∂ w 1 , X 2 = ∂ ∂ w 2 , X 3 = ∂ ∂ w 3 , Y 1 = z 2 ∂ ∂ x 2 + z 1 ∂ ∂ y 2 − 1 2 x 1 ∂ ∂ z 1 − 1 2 y 1 ∂ ∂ z 2 , Y 2 = z 1 ∂ ∂ x 1 + z 2 ∂ ∂ y 1 − 1 2 y 2 ∂ ∂ z 1 − 1 2 x 2 ∂ ∂ z 2 , Y 3 = x 1 ∂ ∂ x 1 − x 2 ∂ ∂ x 2 + y 1 ∂ ∂ y 1 − y 2 ∂ ∂ y 2 are tangent to P 6 ⊂ V 9 and linear ly indep endent at any p oint of P 6 . Then the set of critical p oints of f is defined by the system o f equa tions X 1 f = 0 , X 2 f = 0 , X 3 f = 0 , (3.18) Y 1 f = 0 , Y 2 f = 0 , Y 3 f = 0 . (3.19) 1. Apply (3.18) and (3.19) to f = K . Then a critica l p oint of K satisfies either w 2 1 + x 1 = 0 , w 2 2 + x 2 = 0 (3.20) or w 1 = w 2 = 0 , z 1 = z 2 = 0 . (3.21) The system (3.20) coincides with (3.7) and the only inv ariant s et genera ted by (3.21) consists of all p oints of the tr a jectories (2.14). Such p oints sa tisfy (3.12). 2. Co ns ider the regula r p oints of K at which H and K a r e dependent. Applying (3 .18) to f = H + sK with Lagra ng e m ultiplier s we immediately obtain w 3 = 0. Then from (3.1 5) we co me to solutions (2.12), (2.13). Along the co rresp onding tra jector ies bo th conditio ns (3.9), (3.12) are v alid. 3. W e now as sume that H and K ar e independent. Intro duce the function with Lagra nge multipliers τ , s L = 2 G + ( τ − p 2 ) H + sK . The m ultiplier of G is non- z ero by a ssumption. The term with p 2 is added for convenience. The set σ 0 ⊂ σ of the p oints satisfying for some τ , s the condition 2 dG + ( τ − p 2 ) dH + sdK = 0 (3.22) is preserved by the phase flow of (3.15). Applying the co rresp onding Lie deriv ative to (3.22) gives τ ′ dH + s ′ dK = 0 . Since dH and dK a re supp osed to b e linearly indep endent, on σ 0 we obtain τ ′ = 0 , s ′ = 0 . (3.23) 6 Hence τ , s ar e partial int egrals o f motion on the inv a r iant surfac e σ 0 . Equations (3.18) with f = L give x 2 z 1 w 3 + x 2 y 2 w 1 + ( τ − z 1 z 2 ) w 2 + 2 sw 1 ( w 2 2 + x 2 ) = 0 , x 1 z 2 w 3 + ( τ − z 1 z 2 ) w 1 + x 1 y 1 w 2 + 2 sw 2 ( w 2 1 + x 1 ) = 0; (3.24) ( τ − x 1 x 2 ) w 3 + x 2 z 1 w 1 + x 1 z 2 w 2 = 0 . (3.25) First consider the ca se (3.6). F rom (3.16) we come to the following v alue s x 1 = x 2 = 0 , z 2 1 = z 2 2 = r 2 , y 1 y 2 = 2( p 2 − r 2 ) . (3.26) Equations (3.24) a nd (3.25) hold if w 1 w 2 = 0 or w 3 = 0. If either of these equalities takes place on some int erv al of time (and hence identically), then we obtain o ne of the so lutions (2.12) – (2.14). Let w 1 w 2 6 = 0 , w 3 6 = 0 at so me p oint satisfying (3.26). Then (3.2 4) and (3.2 5) y ield τ = 0 , s = r 2 / (2 w 1 w 2 ) . (3.27) Since z 1 and z 2 are complex conjugates of each other, it follows from (3.26) that they are real and equal. Denote their v alue by z = ± r . With (3.26) and (3.2 7) the system (3.19) reduces to a single e q uation w 1 w 2 [2 z w 3 + ( w 2 y 1 + w 1 y 2 )] + r 2 ( w 1 + w 2 ) = 0 , (3.28) which corres ponds to (3.11). Note that (3.28) is obtained from (3.9) as ρ = q ξ 2 1 + ξ 2 2 tends to zero only after dividing b y the maximal av ailable p ow er of ρ . Thus, at the po in ts (3.6) the system (3.9), without the assumption that ρ 6 = 0, has extra solutions not b elonging to σ . Suppo se x 1 x 2 6 = 0. The determinant of (3.2 4) with resp ect to τ , s eq ua ls δ = 2 ( x 1 w 2 2 − x 2 w 2 1 ). Let δ ≡ 0 on so me interv al of time; calculating the deriv ativ es of this identit y in virtue of (3.15), we o btain o ne o f the cases (3.20), (3.21). Therefor e we may a ssume that δ 6 = 0. Then (3.2 4) implies s = 1 2( x 1 w 2 2 − x 2 w 2 1 ) [( x 2 z 1 w 1 − x 1 z 2 w 2 ) w 3 + x 2 y 2 w 2 1 − x 1 y 1 w 2 2 ] , (3.29) τ = z 1 z 2 + 1 x 1 w 2 2 − x 2 w 2 1 { [ x 1 x 2 ( z 2 w 1 − z 1 w 2 ) − w 1 w 2 ( x 2 z 1 w 1 − x 1 z 2 w 2 )] w 3 − (3.30) − w 1 w 2 ( x 2 y 2 w 2 1 − x 1 y 1 w 2 2 ) + x 1 x 2 w 1 w 2 ( y 1 − y 2 ) } . Eliminating τ fro m (3.25) and (3 .30) we obta in S 1 = 0, where S 1 = [ x 1 x 2 ( z 2 w 1 − z 1 w 2 ) − w 1 w 2 ( x 2 z 1 w 1 − x 1 z 2 w 2 )] w 2 3 + + [( x 1 x 2 − z 1 z 2 )( x 2 w 2 1 − x 1 w 2 2 ) − w 1 w 2 ( x 2 y 2 w 2 1 − x 1 y 1 w 2 2 )+ + x 1 x 2 w 1 w 2 ( y 1 − y 2 )] w 3 − ( x 2 w 2 1 − x 1 w 2 2 )( x 2 z 1 w 1 + x 1 z 2 w 2 ) . Next w e solve (3.2 5) for τ and calculate the deriv ative τ ′ in virtue o f (3 .15). According to (3.23) we must hav e S 2 = 0, where S 2 = ( x 2 z 1 w 1 − x 1 z 2 w 2 ) w 2 3 + ( x 2 y 2 w 2 1 − x 1 y 1 w 2 2 + x 2 z 2 1 − x 1 z 2 2 ) w 3 − − ( y 1 − y 2 )( x 2 z 1 w 1 + x 1 z 2 w 2 ) . Notice that S 1 + w 1 w 2 S 2 = F 1 R. Here F 1 = x 1 x 2 w 3 − ( x 2 z 1 w 1 + x 1 z 2 w 2 ) (3.31) is the first function fro m (3 .1 0). The function R = ( z 2 w 1 − z 1 w 2 ) w 3 + x 2 w 2 1 − x 1 w 2 2 + w 1 w 2 ( y 1 − y 2 ) (3.32) is a multiple o f the first function from (3.13), precisely , R = 2 iR 1 . Thus on the tra jectories consisting of critica l po in ts, we ha ve either F 1 ≡ 0 or R 1 ≡ 0. Ca lculating the deriv atives of these iden tities in virtue of (3.15) we obtain (3.9) and (3 .1 2), resp ectively . Hence (3.9) and (3.12) provide necessar y c onditions for a p oint to b elong to σ 0 . T o prov e sufficiency , it is enough to chec k (3.1 9). W e av o id this technically complicated pr o cedure and only notice that the systems (3.9) a nd (3.1 2) satisfy the assumptions of L emma 5. 7 The phase top olog y of the induced sy stem on M w as studied in [9]. The sy stem of inv ar iant rela tions (3 .9) corres p onds to that found in [10]. In the pa per [11] the equations o f mo tion on N are separated and the initial phase v ar iables ar e express e d via tw o a ux iliary v ar iables, the la tter b eing elliptic functions o f time. The motions on M gener alize those of the 1st App elrot class (Delone class) of the Kowalevski problem [12]. As b tends to zero the motions on N , as shown in [10], conv ert to the so- called esp e cial ly marvelous motions of the 2nd and 3rd classes of Appelr ot [1 2]. The set defined by the system (3.12) was no t p ointed out earlier. T o find the classic a l analog of the set O , put β = 0 in (3.13). Then ξ 1 = η 1 = α 1 , ξ 2 = η 2 = α 2 and we obtain R 1 = 2 ℓω 2 , R 2 = 2 ℓ ( ω 1 ω 3 + α 3 ) , where 2 ℓ = 2 α 1 ω 1 + 2 α 2 ω 2 + α 3 ω 3 is a cons tan t of the momentum integral existing in the case o f one force field. Therefore equations (3.12) y ield either ℓ = 0 o r ω 2 = 0 , ω 1 ω 3 + α 3 = 0 . (3.33) The condition ℓ = 0 follows from the fact that when β = 0 the integral G tak es the v alue ℓ 2 . Equations (3.33) define esp e cia l ly marvelous motio ns of the 4th c la ss of Appe lrot [12]. 4 Bifurcation diagram Since all common levels of the first integrals (3.2) are compact, the bifurcation diagram Σ coincides with the set of critical v alues of the map (3 .3 ), that is, Σ = J ( σ ). Let γ = | α × β | . Acco r ding to (2.8), γ = ab . Denote by ∆ the region of e xistence o f motions, that is, the se t of c = ( g , k , h ) ∈ R 3 for which the in tegral manifolds (3.4) are not empty . Theorem 2. The bifur c ation diagr am of the map G × K × H is the interse ction of ∆ with the union of t he surfac es Γ 1 : k = 0 ; (4.1) Γ 2 : p 2 h − 2 g + r 2 √ k = 0 ; (4.2) Γ 3 : p 2 h − 2 g − r 2 √ k = 0 ; (4.3) Γ 4 :      k = 3 s 2 − 4 hs + p 2 + h 2 − γ 2 s 2 g = − s 3 + hs 2 + γ 2 s , s ∈ R \{ 0 } (4.4) and the line se gment Γ 5 : g = γ h, k = p 2 − 2 γ , h 2 6 4 γ . (4.5 ) In the p ar ametric r epr esentation of t he surfac e Γ 4 the p ar ameter s st ands for a multiple r o ot of the p olynomial Φ( s ) = s 4 − 2 hs 3 + ( h 2 + p 2 − k ) s 2 − 2 g s + γ 2 . (4.6) Pr o of. 1. The eq uation of the sur fa ce (4.1) follows immediately from (3.7), (3.8), and the expressio n of K in (3.2). 2. Relatio ns (4.2), (4.3) are e quiv alent to ( p 2 h − 2 g ) 2 − r 4 k = 0 . (4.7) Int ro duce the function F = ( p 2 H − 2 G ) 2 − r 4 K. F or x 1 x 2 6 = 0 denote U 1 = r x 2 x 1 ( w 2 1 + x 1 ) , U 2 = r x 1 x 2 ( w 2 2 + x 2 ) ( U 2 = U 1 ) . (4.8) F ro m repr esentations (3.17) and (3.3 1) we obtain p 2 H − 2 G + r 2 √ K = 1 2 x 1 x 2 F 2 1 + 2 r 2 (Im U 1 ) 2 , p 2 H − 2 G − r 2 √ K = 1 2 x 1 x 2 F 2 1 − 2 r 2 (Re U 1 ) 2 . (4.9) 8 Here √ K is the principa l squa re ro ot o f K . The equation of the z ero level of F splits into tw o distinct e q uations F 2 1 + 4 r 2 x 1 x 2 (Im U 1 ) 2 = 0 , (4.10) F 2 1 − 4 r 2 x 1 x 2 (Re U 1 ) 2 = 0 . (4.11) F ro m (3.1 0) and (4.8) we have F 2 = x 1 x 2 2 i ( U 2 1 − U 2 2 ) = 2 x 1 x 2 Im U 1 Re U 1 . Thu s the solutions of (3.9 ) satisfy either (4.10) o r (4.11) and there fore lie on the zero level o f the function F . The corres ponding v alues of the first integrals sa tisfy (4.7). F ro m (3.17) it follows that (4.7) holds for a ll p oints of the phase s pa ce such that x 1 x 2 = 0 (regar dless of their critical or regular nature). Hence (4.7) holds for the p oin ts (3.11). 3. Co ns ider the s ystem (3 .1 2). In ter ms of the v ar iables (3.14) it is eq uiv alent to the following equatio ns : R = 0 , R ∗ = 0 . (4.12) Here R is defined by (3 .32) and R ∗ = ( z 2 w 1 + z 1 w 2 ) w 2 3 + [ x 2 w 2 1 + x 1 w 2 2 + w 1 w 2 ( y 1 + y 2 ) + 2 z 1 z 2 ] w 3 + + 2( x 2 z 1 w 1 + x 1 z 2 w 2 ) . (4.13) Notice that, after s everal differen tiations in virtue of the system (3 .15), the po ssibility z 2 2 w 2 1 − z 2 1 w 2 2 ≡ 0 leads to the conditions (3 .21), that is, to the critical motions (2.14). Assuming (3.21) we obtain from (3.16) x 1 x 2 = p 2 − 2 q , y 1 y 2 = p 2 + 2 q , ( x 1 + x 2 ) y 1 y 2 = r 2 ( y 1 + y 2 ) ( q = ± γ ) . The corres ponding v alues of the integrals (3.17) ar e h = 1 2 w 2 3 − 1 2 ( y 1 + y 2 ) > − p p 2 + 2 q , k = p 2 − 2 q , g = q h. (4.14) If q = − γ , then all o f these v alues satisfy (4.4) w ith s = 1 2 [ h ± p h 2 + 4 γ ] . (4.15) Let q = γ . T hen the v alues (4.14) satisfy (4.4) with s = 1 2 [ h ± p h 2 − 4 γ ] , (4.16) that is , only for the energy ra ng e h 2 > 4 γ . F or q = γ and h 2 6 4 γ the v alues (4.14) fill the segment (4.5). Consider the tra jectories for which the equalities (3.2 1) do not hold identically . Express w 3 from the firs t equation (4 .12): w 3 = − 1 z 2 w 1 − z 1 w 2 [ x 2 w 2 1 − x 1 w 2 2 + w 1 w 2 ( y 1 − y 2 )] . (4.17) Replacing w 3 in ( z 2 w 1 − z 1 w 2 ) 2 R ∗ by (4 .17), we obtain the express ion 2 w 1 w 2 Q (the re s ultan t of (3.32), (4.13) as p olynomials in w 3 ), where Q is a non-homog eneous p olynomia l of third degree in w 1 and w 2 whose co efficients are p o lynomials in x i , y i and z i of degree no t greater than four. Since (3.21) is already e x cluded, the sy stem (3.12) is replaced by (4 .1 7) and the equation Q = 0 . (4.18) W e claim that in vir tue of (4.1 7) and (4.18), the v a lues (3.29) and (3.30) satisfy the identit ies τ − p 2 − 2 s ( s − H ) = 0 , ( τ − p 2 ) 2 + 4( p 2 − K ) s 2 − 8 Gs + ( p 4 − r 4 ) = 0 , ( τ − p 2 )(2 s − H ) + 2( p 2 − K ) s − 2 G = 0 . (4.19) Here the calculation se q uence is as follows. 9 W e substitute (3.1 7), (3.29), (3.30), (4.17) in the left-hand side of each equatio n (4.19 ) and m ultiply the result by the denomina to r, which is a lready supp osed to b e non-zer o. The expres sion thereby obtained app ears to b e the pro duct o f so me p olynomial in v a riables (3 .1 4) and the po lynomial Q , which equals zero due to (4.18). Replace in (4.19) the functions G, K, H by their cons tan t v a lues g , k , h and exclude τ with the help of the first relation. The remaining tw o reduce to the for m Φ( s ) = 0 , d Φ( s ) /ds = 0 , ( 4.20) where Φ is the p olynomia l (4.6). Equa tions (4.4) are eq uiv alent to (4 .2 0). Remark. It is ea s y to se e now that the re la tions (4.1) – (4.3) turn into corr espo nding relatio ns of the 1st, 2nd, a nd 3rd class e s o f Appelrot as β tends to zero. Simultaneously , the p olynomia l (4.6) turns to sϕ ( s ), where ϕ ( s ) is the E uler reso lv ent of the seco nd p olynomial of K owalevski. This provides an alter native insight into the connection of the set Γ 4 with the 4 th Appelr ot c lass of motions. The part o f the segment Γ 5 defined by the inequality h 2 < 4 γ for the classica l ca se ( γ = 0) disa ppea r s. 5 The region of existence of motions The results of the prev ious section ar e not complete until we find some conditions that g ive a criter ion to establish whether a p oint of ˜ Σ = Γ 1 ∪ Γ 2 ∪ Γ 3 ∪ Γ 4 ∪ Γ 5 belo ngs to the r e g ion of existence of motions ∆ = J ( P 6 ) ⊂ R 3 . Three ineq ua lities o f genera l character can b e obtained immediately from (3.2) and (4.9): k > 0; (5.1) h > − ( a + b ); (5.2) p 2 h > 2 g − r 2 √ k. (5.3) In case of the Kowalevski top in the gravit y field ( p 2 = r 2 ) the inequality o btained from (5.3) was established by Appelr ot [1 2]. T o ge t mor e precise e stimations for ( g , k, h ) ∈ ∆, restr ic t the problem to iso- energetic sur faces E h = { v ∈ P 6 : H ( v ) = h } . Denote J h = G × K   E h : E h → R 2 and let ˜ Σ h , Σ h , ∆ h be the cross- s ections of ˜ Σ , Σ , ∆ b y the plane parallel to and heig h t h ab ov e the ( g , k )-plane. F or any h the se t Σ h is a bifurca tion diagr am of the ma p J h . Notice that a ll sets E h are compact. As prov ed in [1 3], they are co nnected as well. Ther efore the v alues o f any co n tin uous function on E h fill a b ounded and connected segment. Let k ∗ ( h ) = min K | E h , k ∗ ( h ) = max K | E h ; g ∗ ( h ) = min G | E h , g ∗ ( h ) = max G | E h . (5.4) Then the r ectangle Π( h ) = { ( k , g ) : k ∗ ( h ) 6 k 6 k ∗ ( h ) , g ∗ ( h ) 6 g 6 g ∗ ( h ) } cuts Σ h out of ˜ Σ h and we hop e that this op eration is not ambiguous. The following statements a llow us to find explicitly the v a lues (5.4) and give so me more info r mation ab out the s ets Γ i ∩ ∆. W e ar e g oing to inv estigate v a rious ma ps co nstructed of comb inations o f the firs t integrals G, K , H a nd, po ssibly , r estricted to inv ariant submanifolds in P 6 . F or each map I : C → R k of this t yp e we call a po in t c ∈ R k an admissible value if I − 1 ( c ) 6 = ∅ . In Sections 3, 4 we o ften referred to the motions (2.12) – (2 .14). They will b e also imp ortant in the s e q uel. Calculating the related v alues of G, K, H we o btain the sets λ i in ( h, g )-plane and µ i in ( h, k )-plane ( i = 1 , ..., 6): λ 1 : g = a 2 h + a ( a 2 − b 2 ) , µ 1 : k = ( h + 2 a ) 2 , h > − ( a + b ); λ 2 : g = a 2 h − a ( a 2 − b 2 ) , µ 2 : k = ( h − 2 a ) 2 , h > a − b ; λ 3 : g = b 2 h − b ( a 2 − b 2 ) , µ 3 : k = ( h + 2 b ) 2 , h > − ( a + b ); λ 4 : g = b 2 h + b ( a 2 − b 2 ) , µ 4 : k = ( h − 2 b ) 2 , h > − a + b ; λ 5 : g = abh , µ 5 : k = ( a − b ) 2 , h > − ( a + b ); λ 6 : g = − a bh, µ 6 : k = ( a + b ) 2 , h > − a + b. The existing pa irwise intersections of the first four sets in either g roup co rresp ond to the eq uilibr ia (2.11). 10 Recall that M = { K = 0 } ⊂ P 6 and J ( M ) = Γ 1 ∩ ∆. Denote M = p 2 H − 2 G : P 6 → R and let H (1) = H | M , M (1) = M | M . The following result b elongs to D.B. Zo tev [9]. Prop osition 2. (i) The function H (1) has thr e e critic al values h 1 = − 2 b , h 2 = 2 b and h 3 = 2 a . In p art icular, min M H = − 2 b. (ii) The bifur c ation diagr am of J (1) = H (1) × M (1) : M → R 2 c onsists of the half-line m = 0 , h > − 2 b (5.5) and the s et of solutions of t he e qu ation 27 m 4 + 4 h ( h 2 − 18 p 2 ) m 3 − 2[4 p 2 h 4 − (16 p 4 + 15 r 4 ) h 2 + 2 p 2 (8 p 4 − 9 r 4 )] m 2 + +4 r 4 h [ h 4 − 4 p 2 h 2 + 2(2 p 4 − 3 r 4 )] m − r 8 [( h 2 − 2 p 2 ) 2 − 4 r 4 ] = 0 (5.6) in t he quadr ant { m > 0 , h > − 2 b } . (iii) The set of admissible values of J (1) is 0 6 m 6 m 0 ( h ) , h > − 2 b, wher e m 0 ( h ) st ands for the gr e atest p ositive r o ot of (5.6), which is c onsider e d as an e qu ation in m . The bifurcation dia gram of J (1) is shown in Fig. 1. The a dmissible v a lues fill the shaded region. Figure 1 : The bifurc a tion diagr am of H (1) × M (1) The pr o of given in [9] is based on an ingenious c hange of v ariables o n M . Let us p oint o ut the relation b et w een this r esult a nd Theorem 2. Let m = p 2 h − 2 g . (5.7) It follows from (5.3) tha t m > 0 o n M . This inequality explains (5.5). More over, the line m = 0 in the pla ne k = 0 is the intersection Γ 1 ∩ (Γ 2 ∪ Γ 3 ) (in fact, along this line Γ 1 and Γ 2 ∪ Γ 3 are tangent to each other). The int ersection Γ 1 ∩ Γ 4 is defined by the sys tem obtained from (4.1), (4.4) 3 s 4 − 4 h s 3 + ( p 2 + h 2 ) s 2 − γ 2 = 0 , s 4 − hs 3 + g s − γ 2 = 0 . (5.8) By virtue of the notation (5.7 ) the left-hand s ide of (5.6) b ecomes the r esultant o f the p olynomials in (5.8) with resp ect to s . Th us the set (5.6) corre s ponds to Γ 1 ∩ Γ 4 . Recall that N ⊂ P 6 is defined by (3.9), (3.11) and J ( N ) = (Γ 2 ∪ Γ 3 ) ∩ ∆. Let H (2) = H | N , G (2) = G | N . Int ro duce the map J (2) = H (2) × G (2) : N → R 2 . The following statement is pr ov ed in [1 4]. 11 Prop osition 3. (i) The bifur c ation diagr am of J (2) c onsists of the half-lines λ 1 , λ 2 , λ 3 , λ 4 , the half-line g = 1 2 p 2 h, h > − 2 b, and the cu rve 2 p 2 ( p 2 h − 2 g ) 2 − 2 r 4 h ( p 2 h − 2 g ) + r 8 = 0 , p 2 h > 2 g . (ii) The admissible values of J (2) fil l the r e gion define d by the system of ine qualities  b 2 h − b ( a 2 − b 2 ) 6 g 6 a 2 h + a ( a 2 − b 2 ) , h > − ( a + b ) , 2 p 2 ( p 2 h − 2 g ) 2 − 2 r 4 h ( p 2 h − 2 g ) + r 8 > 0 . The bifurcation dia gram of J (2) is shown in Fig. 2. The a dmissible v a lues fill the shaded region. Figure 2 : The bifurc a tion diagr am of H (2) × G (2) Prop ositions 2 , 3 completely define those parts of Γ 1 , Γ 2 and Γ 3 which c o rresp ond to real critica l motions, that is , the sets Γ 1 ∩ ∆ and (Γ 2 ∪ Γ 3 ) ∩ ∆. Consider the map J (3) = H × K : P 6 → R 2 . The critica l set of J (3) is alrea dy found (see steps 1, 2 in the pr oo f of Theorem 1). In a ddition to the manifold M it co n tains all p endulum motions (2.12) – (2.1 4). Prop osition 4. (i) The bifur c ation diagr am of the map J (3) c onsists of the p ar ab olic curves µ 1 , µ 3 , µ 2 , µ 4 , the half-lines µ 5 , µ 6 , and the half-line k = 0 , h > − 2 b . (5.9) (ii) L et k ∗ ( h ) =  ( h + 2 b ) 2 , − ( a + b ) 6 h 6 − 2 b 0 , h > − 2 b , k ∗ ( h ) = ( h + 2 a ) 2 . (5.10) The admissible values of J (3) fil l the r e gion k ∗ ( h ) 6 k 6 k ∗ ( h ) , h > − ( a + b ) . (5.11) The inequality in (5.9) follows from Prop osition 2. The rela tionship k ∗ ( h ) in (5.10) is built in accordance with (5.1). The r ange of h in (5.11) is defined by (5.2). The regio n of admissible v a lues (shaded in Fig. 3 ) is found using the mentioned ab ov e fa ct that for each h > − ( a + b ) the imag e of E h under K is a bo unded co nnec ted segment. 12 Figure 3 : The bifurc a tion diagr am of H × K Finally , co nsider the map J (4) = H × G : P 6 → R 2 . F or 0 < | s | 6 b and | s | > a , let φ ( s ) = r ( s 2 − a 2 )( s 2 − b 2 ) s 2 > 0 . Prop osition 5. (i) The bifur c ation diagr am of the map J (4) c onsists of the half-lines λ 1 , λ 2 , λ 3 , λ 4 , λ 5 , λ 6 and the cu rves C 1 :    h = 2 s + φ ( s ) g = γ 2 s + s 3 + s 2 φ ( s ) , s ∈ [ − b, 0) , (5.12 ) C 2 :    h = 2 s + φ ( s ) g = γ 2 s + s 3 + s 2 φ ( s ) , s ∈ (0 , b ] , (5.13) C 3 :    h = 2 s − φ ( s ) g = γ 2 s + s 3 − s 2 φ ( s ) , s ∈ [ a, + ∞ ) . (5.14) (ii) Denote by g 0 ( h ) the one-value d funct ion define d by (5.12 ) when h > − 2 b . L et g ∗ ( h ) =  b 2 h − b ( a 2 − b 2 ) , − ( a + b ) 6 h 6 − 2 b g 0 ( h ) , h > − 2 b , g ∗ ( h ) = a 2 h + a ( a 2 − b 2 ) . (5.15) Then the r e gion of admissible values of J (4) is define d by the ine qualities g ∗ ( h ) 6 g 6 g ∗ ( h ) , h > − ( a + b ) . The bifurcation dia gram of J (4) with the sha ded regio n of a dmis s ible v alues is shown in Fig. 4. A straig h tforward pro of of Pr o po s ition 5 can be o btained using the same technique as in the pro of of Theore m 1. Here we just p oint out some genera l idea s that explain this result from the p oint of view of the geometry of Σ. Let v ∈ P 6 be the critical p oint of J (4) . If dH ( v ) = 0, then v is an equilibrium, that is, a singular p oint of the system (3.1). Such a p oint is a critical p oint for each first int egral of (3.1). In particular , dG ( v ) = 0. The v alues ( h, g ) o f J (4) at equilibria (2.11) are the p oints of pairwise intersection of the lines λ 1 − λ 4 . Let rank { dG ( v ) , dH ( v ) } = 1 , (5.16) and c = ( g , k , h ) = J ( v ). Then c ∈ Σ. It follows fr om (4.1) – (4.5) that (5.16) necessar ily implies rank { dG ( v ) , dH ( v ) , dK ( v ) } = 1 . (5.17) 13 Figure 4 : The bifurc a tion diagr am of H × G If the tangent plane to Σ a t the point c is well defined, then the se t of zero linear co mbinations of dG ( v ) , dH ( v ) and dK ( v ) is o ne-dimensional. This fact co n tradicts to (5.17). The r efore, c b elongs either to the segment Γ 5 or to the s et of tra nsversal intersections of tw o smo oth le a ves of Σ. The int ersection of Γ 1 and Γ 2 ∪ Γ 3 is nowhere tra nsversal. T ra nsversal intersections of Γ 1 and Γ 4 are given by the system (5.8 ). Solving it w ith resp ect to g and h and taking into account the admissible r egion established in Prop ositio n 2, we arrive at the curves (5.1 2) – (5.14). Consider intersections of Γ 2 ∪ Γ 3 and Γ 4 . Substitute (4.4) for k , g in (4.7): ( s 2 − a 2 )( s 2 − b 2 )[2 s 2 − 2 h s + p 2 ] 2 = 0 . (5.18) T ra nsversal in tersections corre spo nd to the v a lues s = ± a, s = ± b (the las t multiplier in (5.18) is re s pons ible for the tangency p oints of Γ 2 ∪ Γ 3 and Γ 4 ). This implies the equations of λ 1 – λ 4 . As sho wn in [11] the corresp onding motions on N are the p endulums (2.12), (2.13). F rom this fact the inequalities fo r h are obtained. Suppo se that Γ 4 has a point of self-intersection. T he n for some h the curve defined by (4.4) in ( g , k )-plane has a double p oint. Let s 1 , s 2 be the co rresp onding v a lues of s . It follows from (4.4) that s 1 + s 2 = h, s 2 1 s 2 2 = γ 2 . Hence s 1 , s 2 form one of the pair s (4 .15), (4.1 6). Substituting these pairs in (4 .4) gives (4.14). The obtained set o f po in ts in ( h, g )-pla ne united with the pr o jection of the s egment Γ 5 forms the ha lf-line s λ 5 and λ 6 . The admissible r egion for ( h, g ) is established in the same way as in the previous c a se. Prop ositions 4 and 5 g iv e the explicit formu lae (5.10), (5.15) for the v alues (5.4). Then for each h we ca n compute the limits for the parameter s in (4.4) corr e s ponding to Γ 4 ∩ ∆. Thus the set Σ h is completely determined. Finally , ∆ h is obtaine d a s the span of the curves Γ i ∩ ∆ h . 6 Conclusion A t this p oint we can draw a ll bifurca tio n diagra ms of the induced momentum maps on iso- energetic surfaces , which ar e typically five-dimensional. A lot of infor mation on the stability of the critical integral manifolds may b e immediately obtained for the tori in M and N . The inv e stigation of the new critical set O waits to b e fulfilled. Since ea c h E h is a foliatio n in to three-dimensiona l tor i with some degenera tions, we can constr uct the base B h for such a folia tion just b y factorizing E h , more exactly , by iden tifying po in ts of the same connected comp onent of J g,k ,h . Then B h is a tw o-dimensional analog o f F o menko’s gra ph [5] for an iso -energetic manifold of int egrable system with tw o degrees o f freedo m. In its turn B h is a bundle ov er ∆ h whose fibr es are finite s ets; the num ber of elements in any fibre is equal to the num ber of connected comp onents of the cor r esp onding in tegral ma nifold. The 14 problem of finding this num ber for all p ossible situa tions seems solv able. Then we o btain a co mplete descr iption of the ” cov e rings” B h → ∆ h and, cons equen tly , establish the top ology o f B h . Naturally , the next s tep requires new mathema tical ideas o n how the tori in E h glue tog ether along the pa ths in the a dmissible reg io ns. If we consider B h as a tw o-dimensional cell complex, then, for regular levels of e ner gy , 0-cells corr espo nd to clo sed orbits, a p oint of ea ch 1-cell repr esents a two-dimensional tor us, and a p oint of ea ch 2-cell repr esents a three-dimensional tor us. The union of the cells of dimensio ns 0 and 1 for ms a graph, to whic h the metho d o f marked molecules [15] ca n b e applied without any mo dification. The question is what kind of a n umeric mark should b e attached to each tw o-dimensional cell to o btain fro m B h the complete inv ariant of Liouv ille ’s folia tion of the is o-energetic sur face? Another appr oach is to consider the s e t Σ 0 h of singula r p oints of Σ h (self-intersections, tang ency p oints, and cusps), which is easy to obtain from the ab ov e results, and asso ciate to each c ∈ Σ 0 h the marked lo op molecule [16]. In this c a se, of course, the notion of a ma r k should b e changed to suit increa sed dimensions of the to r i. W e see tha t the Kow alevski top in tw o consta n t fields provides a highly non-trivia l example of int egrable Hamiltonian system and a c o mplete description of its phase top ology is r eally a challenging problem. Receiv ed 09 .04.05 References [1] S. Kowalevski. Sur le pr o bleme de la r otation d’un co rps so lide a utour d’un p oint fixe. Acta Ma th. 18 8 9. V. 12. P . 1 77–23 2. [2] A. G. R eyman, M. A. Semenov-Tian-Shansky. Lax repr esentation with a sp ectral par ameter for the Kow alewski top a nd its generaliz ations. Lett. Math. Phys. 19 87. V. 14. N 1. P . 5 5 –61. [3] O. I. Bo goyav lensky. Euler equatio ns on finite-dimensio n Lie a lgebras ar ising in physical problems. Co mm un. Math. Phys. 19 84. V. 95. P . 307 –315. [4] A. I. Bob enko, A. G. R eyman, M. A. Semen ov-Tian-Shansky. The Kow alewski top 99 years later: a La x pair, generaliza tions and explicit so lutions. Commun. Math. Phys. 1989 . V. 122. N 2. P . 321 –354. [5] A. T. F omenko. Symplectic ge ometry . 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