Phase topology of one system with separated variables and singularities of the symplectic structure
aa r X i v : . [ n li n . S I] J u l Phase topology of one system with separated variablesand singularities of the symplectic structure ∗ M.P. Kharlamov
Russian Academy of National Economy and Public AdministrationVolgograd Branch, Volgograd, Russia
Abstract
We consider an example of a system with two degrees of freedom admitting separation ofvariables but having a subset of codimension 1 on which the 2-form defining the symplecticstructure degenerates. We show how to use separation of variables to calculate the exacttopological invariant of non-degenerate singularities and singularities appearing due to thesymplectic structure degeneration. New types of non-orientable 3-atoms are found.
Keywords : phase topology, separated variables, Kowalevski top, double field, loopmolecule, non-orientable atoms
MSC 2000 : 70E17, 70G40
The Lie co-algebra e (3 , ∗ with coordinate functions g i , α j , β k has the Lie – Poisson bracket { g i , g j } = − ε ijk g k , { g i , α j } = − ε ijk α k , { g i , β j } = − ε ijk β k , { α i , α j } = { β i , β j } = { α i , β j } = 0 ( i, j, k = 1 , , . For a given function H : e (3 , ∗ → R the system of equations written as˙ x = { x, H } (1.1)is called Euler equations on e (3 , ∗ with the Hamilton function H [1].Suppose we have a rigid body rotating about a fixed point O and relate all vector andtensor objects to a reference frame moving with the body. Let α , β be some vectors fixed inthe inertial space and g the kinetic momentum of the body. The equations of motion havethe form (1.1) with H = g · I − g + W ( α , β ). The constant symmetric matrix I is theinertia tensor and ω = I − g is the angular velocity of the body. The function W is treatedas the potential energy. In what follows we use the coordinates ω i of ω in the moving frameinstead of g i for convenience. In the generic case α × β = 0, common levels P of the Casimirfunctions α , β , α · β are 6-dimensional symplectic leaves of the Lie – Poisson bracket. System(1.1) restricted to P becomes a Hamiltonian system with three degrees of freedom.The Hamilton function H = ω + ω + 12 ω − α − β (1.2)defines an integrable generalization of the Kowalevski top [2] to a double force field. Additionalintegrals in involution are [1, 3]: K = ( ω − ω + α − β ) + (2 ω ω + α + β ) ,G = ( ω α + ω α + ω α ) + ( ω β + ω β + ω β ) + ω [ ω ( α β − α β ) + ω ( α β − α β ) + ω ( α β − α β )] − α β − β α + ( α + β )( α · β ) . ∗ J. of Geometry and Physics, On-line July 2014, DOI: 10.1016/j.geomphys.2014.07.004
1t is known that for a wide class of Hamilton functions including (1.2) without loss ofgenerality one can choose the following constants of the Casimir functions [4] α = a , β = b , α · β = 0 ( a > b > . (1.3)For the function (1.2) the cases b = 0 and b = a correspond to the classical Kowalevski case [2]and the case of Yehia [5]. Both of these cases have symmetries, globally reduce to systems withtwo degrees of freedom and are not considered here. For irreducible cases (1.3) the system (1.1),(1.2) admits a Lax representation given by A.G. Reyman and M.A. Semenov-Tian-Shansky [3]but has not been yet reduced to quadratures. Let us call this system the RS-system .The study of irreducible integrable 3D-systems begins with detecting the so-called criticalsubsystems. These are even-dimensional invariant submanifolds of the phase space with theinduced Hamiltonian systems having less than three degrees of freedom. All critical subsystemsfor the RS-system were found in [1, 6, 4]. Separation of variables is known for two of them [7, 8].Consider a critical subsystem on a four-dimensional invariant submanifold. The 2-form definingthe Hamiltonian type of the induced dynamics is obtained as the restriction of the symplecticstructure of P . It appeared that, in the RS-system, all critical subsystems with two degrees offreedom have 3-dimensional subsets on which this 2-form degenerates. Such systems are nowcalled almost Hamiltonian. For the first subsystem [1], as shown in [9], the Fomenko – Zieschanginvariant [10] can be applied. Here we study the second subsystem found in [6]. It is denotedby N . This notation stands for the dynamical system and therefore includes both the phasespace and the induced dynamics. For the sake of brevity we also call N the phase space of thesubsystem meaning the corresponding subset of P . The rough phase topology of the system N was described in [7]. Nevertheless, some properties of N has not been completely establishedand the character of some “strange” bifurcations has not been explained. In this paper we fulfilthe complete topological investigation of the system N in terms of the topological invariants[10, 11, 12] calculated using the global separation of variables. New topological effects arerevealed due to non-orientable bifurcations, which are possible in almost Hamiltonian systems. We use the definition of N given in [7]. Consider the first integral of the RS-system F = (2 G − p H ) − r K, where p = √ a + b , r = √ a − b ( p > r > F = 0 appliedto the integral constants gives a leaf of the bifurcation diagram of the global integral map H × K × G : P → R . Therefore, we define a non-empty invariant set as follows N = { x ∈ P : F ( x ) = 0 , d F ( x ) = 0 } . The invariant submanifold
N ⊂ P was first found in [6] in terms of invariant relations on P having singularities on the set { Λ = 0 } , whereΛ = ( α − β ) + ( α + β ) . The Lie – Poisson bracket L of these relations is defined as L = 1 √ Λ (cid:8) ω + ω + [ a + b − α β − α β )] M (cid:9) , M = 1 r (2 G − p H ) . It is proved in [7] that in the domain { Λ = 0 } the set N is a smooth 4-dimensional manifold, L is a partial integral on N and the restriction to N of the symplectic structure is non-degenerateeverywhere except for the subset { L = 0 } . This subset is non-empty. Moreover, it does notcontain critical points of L and therefore is a 3-dimensional submanifold in N . Thus, the inducedsystem on N is almost Hamiltonian, i.e., the 2-form defining the Hamiltonian field degenerates2n a subset of codimension 1. Below we give some simple explanation of the fact that N iseverywhere a smooth 4-dimensional manifold. Moreover, it appears to be non-orientable. Inthe sequel, we call N a manifold without further comments on the fact.From now on having a first integral denoted by an upper case letter we denote its arbitraryconstant by the corresponding lower case letter. On N , the following identities hold [7] g = 12 ( p h + r m ) , k = r m , (2.1) ℓ = 1 + 2 mh + 2 p h . (2.2)These relations show that it is convenient to take M, H for the functionally independent pairof the first integrals on N , while the pairs L, H and
L, M are not in one-to-one correspondencewith the triple (
H, K, G ). Thus, we define the integral map J = M × H : N → R and study the bifurcations of the integral manifolds J m,h = { x ∈ N : M ( x ) = m, H ( x ) = h } . For almost all integral manifolds J m,h we can take two values of ℓ with different signs accordingto (2.2).Introduce the new variables u , u as u = a √ Λ a − ( α β − α β ) , u = b − ( α β − α β ) b √ Λ . It readily follows from (1.3) that | u | , | u | . (2.3)These are the so-called natural restrictions. Let τ , = 2 amℓ ∓ , σ , = ℓ ∓ bm , Θ = 1 a − b u u ,h ∗ = −
12 ( h + p m ) , ψ = ab m u + h ∗ u − ( a + b u u ) . Consider the two-valued (algebraic) radicals Q = p − u , Q = p − u ,P = p h ∗ ( u − τ )( u − τ ) , P = b p m ( u − σ )( u − σ ) . (2.4) Proposition 1.
The differential equations induced on N by the RS-system separate in variables u , u , ˙ u = Q P , ˙ u = Q P , (2.5) and on the integral manifolds J m,h the phase variables have the following expressions in termsof u , u α = − a Θ [( au − bu ) ψ + bQ Q P P ] ,α = 2 a Θ [( au − bu ) P P − bQ Q ψ ] ,β = 2 b Θ [ aQ Q ψ − ( bu − au ) P P ] ,β = 2 b Θ [ aQ Q P P − ( bu − au ) ψ ] ,ω = r ( ℓu − am )Θ P , ω = r (2 bmu − ℓ )Θ P ,α = ar Θ Q , β = br Θ u Q , ω = 2Θ( bQ P − aQ P ) . (2.6)The result easily follows from [7]. The variables of separation s , s found in [7] canoscillate on infinite segments. Therefore here we use dimensionless variables u = as − , u = b − s , always restricted to the segment [ − , a − bu u > Geometrical representation of the integral manifolds
Fixing the values m, h and choosing ℓ according to (2.2), let us treat (2.6) as a map π : V = R ( u , Q , P ) × R ( u , Q , P ) → N . Then the integral manifold J m,h is the π -image of the direct product of two curves Γ i ⊂ R ( u i , Q i , P i ) ( i = 1 , : (cid:26) Q + u = 1 ,P − f ( u ) = 0; Γ : (cid:26) Q + u = 1 ,P − f ( u ) = 0 , (3.1)where f ( u ) = h ∗ u + aℓu − a m = h ∗ ( u − τ )( u − τ ) f ( u ) = b mu − bℓu − h ∗ = b m ( u − σ )( u − σ ) . Introduce the enhanced space V = R ( ℓ, m, h ) × R ( u , Q , P ) × R ( u , Q , P ), and define M ⊂ V by equations (2.2) and (3.1). Let ˆ π : M → N be the map given by (2.6).
Lemma 1.
The set M is a connected smooth 4-dimensional manifold in V .Proof. It is easy to check that the system of five equations (2.2), (3.1) always has rank 5.Therefore, M is a smooth 4-dimensional manifold.Let us call a point ( ℓ, m, h ) ∈ R admissible if the corresponding set∆ = ∆( ℓ, m, h ) = Γ × Γ is not empty. For a given ∆, we call the projection of it onto the ( u , u )-plane a region ofpossible motions. If not empty, this region is a rectangle in the square (2.3) cut out by thesystem of inequalities { f ( u ) > , f ( u ) > } . This yields that the image of the integral map J in the ( m, h )-plane isIm J = { h > min[ r m − a, − r m − b ] , p m + 2 hm + 1 > } . The set D of all admissible points is a two-sheet covering of Im J . The sheets with oppositesigns of ℓ are glued together along the curve L : 2 p m + 2 hm + 1 = 0 , ℓ = 0 , m < D is connected. Take any two points in M and connect their images in D by a path γ with Int γ ⊂ Int D . At the points of Int γ the curves Γ i never degenerate to one point (weomit technical details, since this fact follows from the rough topological analysis given below).Therefore the initial two points in M can also be connected by a path. Lemma 2.
Consider the involution χ : V → V ( χ = Id) χ : ( ℓ, m, h, u , Q , P , u , Q , P ) ( − ℓ, m, h, − u , Q , − P , − u , − Q , P ) . The manifold M and the map ˆ π are χ -invariant and χ changes the orientation on M . Indeed, for x ∈ M let µ x ∈ L (5 ,
9) denote the Jacobi matrix of the system (2.2), (3.1). Wereadily obtain that µ χ ( x ) ≡ µ x · A χ , where A χ is the matrix of χ . So χ changes the orientation of V but preserves the orientation of the space normal to M . Therefore, it changes the orientationof M .Let us emphasize that the regions of possible motions for ( ℓ, m, h ) and ( − ℓ, m, h ) arecentrally symmetric to each other and do not coincide except for ℓ = 0. In the latter case χ becomes a Z -symmetry of ∆. In particular, χ : M → M is a diffeomorphism and N = M /χ .Summarizing the above statements we come to the following theorem.4 QP Q P V Æ { } · S S uQ P Q P V S S u QP Q P W S S Q Q P P Figure 1: Possible transformations of Γ.
Theorem 1.
The set N is a smooth connected 4-dimensional non-orientable submanifold in P . For any point ( m, h ) ∈ (Im J ) \ L the integral manifold J m,h is diffeomorphic to Γ × Γ .On L we have J m,h ∼ = (Γ × Γ ) / Z . As we can see from (3.1), the set Γ i in R ( u i , Q i , P i ) is an intersection of the round cylindergenerated by the unit circle in the ( u i , Q i )-plane and a cylinder with elliptical directrix in the( u i , P i )-plane. The form of Γ i depends on the position of the roots of f i ( u i ) with respect to ± i are shown in Fig. 1:1) ∅ → { · } → S → V → S ;2) S → V → S ;3) 2 S → W → S .Here V and W stand respectively for the eight-curve S ˙ ∪ S with transversal self-intersectionand for the curve S ¨ ∪ S formed by two circles transversally intersecting at two points. Thedifference between cases 1 and 2 is that the two circles of 1 differ by the sign of Q i , and thetwo circles of 2 differ by the sign of P i . In the first case there is an exit to ∅ when the commonroot of P i and Q i turns out to be an isolated point. In the second case both exits would be ofhyperbolic type because Q i never has a multiple root in this system. To accomplish the rough topological analysis of an integrable system one has to do the following:1) find the admissible region (the image of the integral map);2) for each regular value of the integral map find the number of Liouville tori in the pre-image of this value, i.e., establish the topology of regular integral manifolds;4) for each critical value of the integral map find the topological type of its pre-image, i.e.,establish the topology of irregular integral manifolds usually called (critical) integral surfaces;5) show a collection of paths in the integral constants space with complete description ofthe topological type of bifurcations taking place along this path; this set of paths should besufficient to find out the character of any bifurcation occurring in the phase space.For the classical integrable systems in the rigid body dynamics, this program was fulfilledin [13, 14, 15, 16, 17]. In this section, we find out the rough topology of the system N , thusdescribing its rough Liouville equivalence class. Further, to describe the exact topology of thesystem, i.e., to establish its Liouville equivalence class (for detailed definitions see [12]), weneed to use general classifications of critical points and their neighborhoods [18, 12] and exact topological invariants. By this term we mean the invariants that completely define Liouvillefoliations on some sufficient collection of 3-dimensional integral manifolds, e.g. Fomenko –Zieschang invariants [10] or marked loop molecules [11, 19]. This will be done in the followingsections.According to Lemma 2, the integral manifolds undergo topological transformations only intwo cases, namely, when ( m, h ) crosses the discriminant set of one of the product polynomials P i Q i ( i = 1 ,
2) or when ( m, h ) reaches the curve L . In the first case, for ℓ = 0, the map χ able 1. Regular cases Chamber Position of θ , u ∈ u ∈ Γ Γ J m,h I + − b < θ < b < a < θ [ τ , − , σ ] S S T I − θ < − a < − b < θ < b < a II + b < θ < a < θ [ τ , − , S S T II − θ < − a < − b < b < θ < a III − a < θ < − b < θ < b < a [ − , − , σ ] 2 S S T IV − a < − b < θ < θ < b < a [ − , σ , σ ] 2 S S T V − a < θ < − b < b < θ < a [ − , − , S S T VI + a < θ < θ [ τ , τ ] [ − , S S T VI − θ < − a < a < θ identifies in N two sets ∆( ℓ, m, h ) and ∆( − ℓ, m, h ) different in M . Therefore, in the sequel weby default suppose that ℓ >
0. In the second case the set ∆(0 , m, h ) is factorized by the actionon Γ × Γ diagonal with respect to the induced actions on R ( u i , Q i , P i ) χ : ( u , Q , P ) ( − u , Q , − P ) ,χ : ( u , Q , P ) ( − u , − Q , P ) . (4.1)In what follows if S stands for some set and n is a positive whole number, then nS denotes n isolated copies of S . h mp p p p q q IIIV VIV IIII q q p VIII III
11 13 h m
Figure 2: Bifurcation diagram, chambers and segments.
Theorem 2.
The bifurcation diagram Σ of the system N consists of the half-lines R − b : h = − r m − b, h > − a − b,R + b : h = − r m + 2 b, h > − a + b,R − a : h = r m − a, h > − a − b,R + a : h = r m + 2 a, h > a − b and of the curve L defined by (3.2) .The set Σ divides the admissible region in the ( m, h ) -plane into six chambers I − VI shownin Fig. . The corresponding regular integral manifolds are listed in the last column of Table .The integral surfaces in the pre-image of the segments forming the set Σ are listed in thelast column of Table . The proof almost obviously follows from Lemma 2. Let us make only some remarks.6 able 2. Critical cases
Seg. u ∈ u ∈ Γ Γ J m,h { ∗ } [ − , σ ] {·} S S τ , {− ∗ } S {·} S { ∗ } [ − , {·} S S − , {− ∗ } S {·} S τ , − , ∗ ] S V V× S − ∗ , − , σ ] V S V× S − ∗ , − , V S V× S − , − , ∗ ] 2 S V V× S − , − ∗ , σ ] 2 S V V× S
10 [ τ , ∗ ] [ − , V S V× S
11 [ − , − σ, σ ] 2 S S T
12 [ − , − , S S T
13 [ − τ, τ ] [ − , S S T The half-lines R ± a and R ± b correspond to the cases of a multiple root in one of the poly-nomials P i Q i ( i = 1 , τ , = ± σ , = ±
1. The corresponding criticalmotions are pendulum type motions R ± a : α = α = 0 , α = ∓ a, β = 0 ,β = b sin φ, β = b cos φ,ω = ω = 0 , ω = ˙ φ, φ = b cos φ,h = ˙ φ ± a − b sin φ, r m = ˙ φ ∓ a − b sin φ,R ± b : β = β = 0 , β = ∓ b, α = 0 ,α = a cos φ, α = a sin φ,ω = ω = 0 , ω = ˙ φ, φ = − a sin φ,h = ˙ φ ± b − b cos φ, r m = − ˙ φ ± b + a cos φ. (4.2)They include closed orbits of rank 1, singular points of rank 0 at the intersections of the half-lines and separatrices of rank 1 of unstable singular points. From here we readily obtain theinequalities for h on the half-lines.Note that the mutual position of the values τ i , σ i , ± θ , = ( ℓ ∓ / (2 m ) with respect to ± a, ± b . In Table 1, we present the inequalities for θ , and, consequently, define the accessible regions (segments of oscillation) for u , u , the topologyof Γ i and the resulting type of regular integral manifolds J m,h . Here the ± sign attached tothe notation of a chamber means the sign of m in the corresponding part of this chamber. Itaffects the values θ , but does not change the rest of information. The number of tori in J m,h for ℓ = 0 equals 2 k where k is the number of those radicals among P i , Q i which have constantsign on Γ × Γ . Agreement 1.
Suppose that the radical P i or Q i does not vanish on the connected componentof a regular integral manifold or a critical integral surface. Then we respectively denote e i = sgn P i or d i = sgn Q i . In Table 2, we collect the information on the critical cases. In the first column we givethe notation of the smooth segments of Σ according to Fig. 2. In the corresponding accessibleregions ± ∗ stand for the double root of P i Q i .More analysis is needed for the segments on the curve L . Here the roots of P i = f i ( u )become centrally symmetric and are denoted by ± τ , ± σ for i = 1 ,
2. The curve L is tangentto R + b and R + a respectively at the points q , q = (cid:18) − b , a + 3 b b (cid:19) , q = (cid:18) − a , a + b a (cid:19) . Proposition 2.
The integral manifolds in the pre-image of the curve L are as follows: T on segments , ; T on segment ; W× S at the point q ; V× S at the point q .Proof. We see from Table 2 that the four components of the set ∆(0 , m, h ) differ by the signs of P , Q on segment 11, by the signs of P , P on segment 12, and by the signs of Q , P on segment13. At the same time according to (4.1) the Z -symmetry χ of ∆(0 , m, h ) simultaneously changesthe signs of P and Q . This means that χ identifies the components having the same product e d on segment 11 as in Fig. 3, a . Here the arrows show the connected components of the directproduct Γ × Γ , the numbers stand for the resulting connected components in the quotient set.On segment 12 (Fig. 3, b ) χ identifies the components with the opposite sign e but the samesign e . Therefore, the result is (Γ × Γ ) /χ = 2 T . On segment 13 the symmetry preserves allfour components of ∆(0 , m, h ). The result is (Γ × Γ ) /χ = 4 T (Fig. 3, c ). ( ) a ( ) b ( ) c e =+1 e = -1 e =+1 e = -1 d =+1 d = -1 e =+1 e = -1 e =+1 e = -1 d =+1 d = -1
112 2 1 11 222 3 4
Figure 3: Gluing the components in J − ( L ).Let us take the points q , . The critical points in the pre-image are of rank 1 and formclosed orbits. For the accessible regions we have q : u ∈ [ − , , τ > , u ∈ [ − ∗ , ∗ ] , σ = 1; q : u ∈ [ − ∗ , ∗ ] , τ = 1 , u ∈ [ − , , σ > . The sets Γ , for q and q are shown in Fig. 4, a and b . Again, χ acts as simultaneous centralsymmetry in ( u , P )- and ( u , Q )-planes. At q , it glues together the components of Γ andpreserves Γ . The result is W× S . At q , χ preserves the components of Γ and thereforefactorizes Γ . The result is 2 V× S . This proves the statement. u Q P u Q P ´ ´ ( ) a ( ) b e =+1 e = -1 e =+1 e = -1 u Q P u Q P Figure 4: The ∆-sets at q , q .Now we describe all bifurcations in terms of the atoms according to the contemporarynotation [20, 10, 12]. Let us recall some terminology from [12].Consider a 2-surface (two-dimensional compact manifold without boundary) and a Morsefunction f on it having a critical value f . A 2-atom is a neighborhood of a connected componentof the set f − δ f f + δ for sufficiently small δ , foliated into level lines of f and considered8p to the fiber equivalence. A 2-atom is supposed to have only one connected singular fiber { f = f } . If an atom U is given, its singular fiber is denoted by L ( U ).In Fig. 5, the following atoms are shown: A ( ∂A = S , L ( A ) = {·} ); B ( ∂B = 3 S , L ( B ) = V ); C ( ∂C = 4 S , L ( C ) = W ); C ( ∂C = 2 S , L ( C ) = W ). A BC2 C1
Figure 5: Some known 2-atoms.For integrable Hamiltonian systems with two degrees of freedom, bifurcations of the Liou-ville tori are described in terms of 3-atoms. Let F × F be the integral map. Fixing a regularvalue r of F we obtain a 3-dimensional iso- F manifold I = I ( r ). Let us suppose for simplicitythat I is connected, otherwise we take one connected component of I . Let r be a critical valueof F on I and L be the connected component of ( F × F ) − ( r , r ). Then L is called a singularleaf of the Liouville foliation on I . A 3-atom is a small enough connected neighborhood of L in I containing no other singular leaves and invariant under the phase flow. In fact, 3-atomsas foliated manifolds are considered up to the fiber equivalence as defined in [12]. In the sameway as before, for a given 3-atom U we denote its singular leaf by L ( U ).We easily obtain 3-atoms from the above mentioned 2-atoms by considering their directproducts with a circle. Then, traditionally [12], we keep for them the same notation. Thus, L ( A ) = S , L ( B ) = V× S , L ( C ) = L ( C ) = W× S and the bifurcations of tori when crossingthe singular leaves are A : ∅ → T , B : T → T , C : 2 T → T , C : T → T . (4.3)Note that the 3-atom C was predicted [12] but never has been met in real systems. Of course,bifurcations with non-symmetric atoms can be written the other way round depending on thedirection in which we cross the bifurcation diagram.In the case of minimal or maximal integral surfaces, symmetric atoms can be folded twice,so that the bifurcation S → U → S turns into ∅ → U → S . Let us use the notation R forthe atom of a minimal (maximal) torus. Here I is a regular level of F and on it F has theform ( g − r ) with regular function g . The 2-atom R (already not associated with any Morsefunction) is just an annulus foliated into circles, and the 3-atom R is the direct product of anannulus and a circle foliated into 2-tori. Considering the torus { F = r } as the singular leafwe have the following bifurcation R : ∅ → T .The atoms in (4.3) correspond to the critical points which are called simple [18] or non-degenerate [12]. For all points forming the motions (4.2) including those of rank 0 in thepreimage of p . . . , p (see Fig. 2) the non-degeneracy is proved in [21]. The only exceptionsare the motions in the pre-image of the points q , q , which are degenerate as critical points ofrank 1 in the RS-system (with three degrees of freedom). Let us also mention that all tori inthe pre-image of L are degenerate as critical points of rank 2 in P [21].It is now easy to describe all non-degenerate bifurcations. We have the following atoms:1) A on segments 1 , A on segments 3 , B on segments 5 ,
6; 9) 2 B on segments 7 , , , p i ( i = 1 , . . . ,
4) have h -coordinate equal to ∓ a ∓ b and are enumerated alongthe h -axis. The points c i of rank 0 in the pre-image of p i are also non-degenerate. For suchpoints the local phase topology is described in terms of the almost direct products of 2-atoms[22, 12]. Here we have only direct products. This fact follows immediately from the generalclassification of the cases with one singular point on a singular leaf [18, 12] and the atoms onthe adjacent segments. Thus, small enough invariant under the phase flow neighborhoods U i ofthe points c i (called extended or saturated neighborhoods) are U = A × A ; U , = A × B ; U = B × B. (4.4)To finish the description of the rough topology we need to point out the bifurcationsoccurring on the pre-image of the curve L . Two of them are obvious. As shown in the proofof Proposition 2, the symmetry χ on segments 11 ,
12 glues together pairwise the four tori ofchambers IV and V respectively. Therefore we have here bifurcations with two atoms R . Q P S S Q P u Q P u Q P T Figure 6: The sets Γ i in chamber VI and the 2-atom T .Let us consider a path reaching segment 13 from chamber VI . Each of the sets ∆( ± ℓ, m, h ) ⊂M along this path has four components corresponding to the pair of signs ( d , e ). Considera continuous set of these components T ( ℓ ; d , e ) marked by ℓ ∈ ( − δ, δ ). The componentof Γ ( ℓ ) defined by ( d , e ) is well projected onto the plane ( u , P ) and the correspondingcomponent of Γ ( ℓ ) is well projected onto the plane ( u , Q ) as shown in Fig. 6. We thensee that both χ i : Γ i ( ℓ ) → Γ i ( − ℓ ) are almost central symmetries on these planes and becomereal central symmetries on Γ i (0). Finally for a point on segment 13 we obtain the 3-atom T × ( − δ, δ ) /χ = M × S , where M is the M¨obius band foliated into circles in the natural waywith one singular central circle (the band’s axis) twice shorter than all close ones. The productof the axis with a circle stands for the torus in the pre-image of L . The M¨obius band itselffoliated this way gives a non-orientable 2-atom, which we denote by T . The same notation weuse for the corresponding 3-atom M × S . Its bifurcation in the direction from the border intothe chamber is T : ∅ → T , but unlike the atom A having a circle as a singular leaf, herethe singular leaf is a torus covered twice by the close regular torus as ℓ →
0. This 3-atomis impossible if we deal with a Hamiltonian system without degenerations of the symplecticstructure.To describe the topology in small enough neighborhoods of q , q we first consider thesituation arising in the covering manifold M , where all transformations are easily seen fromthe sets ∆( ℓ, m, h ).Consider a neighborhood of q and unfold the picture from the integral constants spaceonto the plane ( ℓ, m ). We obviously obtain the cross formed by the lines ℓ = − bm − ℓ = 2 bm + 1. On each of the lines the bifurcation is 2 B , along the horizontal line the bifurcationis 2 C and along the vertical line the bifurcation is 2 C . In fact, this picture reflects twoconnected components of the neighborhood of the pre-image of q in M . On each componentthe bifurcation diagram together with the atoms is shown in Fig. 7, a . The vertical iso- M graphschange as in Fig. 7, b , and the horizontal iso- L graphs change as in Fig. 7, c . This phenomenon10an be called the splitting of the atom C . The possibility of topologically unstable systems isdiscussed in [12]. The case we obtain here is described as a possible transformation of iso-energyFomenko graphs in [23]. ( ) a ( ) b ( ) c B BB B T T ℓ m BB BBC1 B BB BC2
Figure 7: The splitting of the atom C .Let us consider a path reaching the point q from chamber III . All the way the curveΓ has two components with different signs of P while the closed curve Γ ( ℓ, m, h ) covers thewhole set W as ℓ → Q P W S Q P S Q P u Q P u Q P Figure 8: The sets Γ i in chamber III while moving towards q .In M , we unfold this path to a small vertical path m = − b , ℓ ∈ ( − δ, δ ) and this processforms two atoms S × C , where C denotes the 2-atom. Representing the sets Γ = 2 S and ∪ Γ ( ℓ ) = C on the plane as in Fig. 9, we see that in this representation both χ and χ act ascentral symmetries, so χ identifies the components of Γ = 2 S and χ acts as a Z -symmetryon ∪ Γ ( ℓ ) = C . Factorizing by the diagonal action, we can write, admitting some inexactness,(2 S × C ) /χ ∼ = (2 S /χ ) × C . Therefore the pre-image of the chosen path in N gives one atom S × C , i.e., one 3-atom C . On the diagram Σ of the system N , crossing the point q from theborder into chamber III we have the bifurcation C written in the form ∅ → T . G ÈG Figure 9: The sets Γ and ∪ Γ along the path in chamber III .Let us consider a neighborhood of q and, in the same way as in the previous case, unfoldthe picture from the integral constants space onto the plane ( ℓ, m ). We obtain the cross formedby the lines ℓ = − am − ℓ = 2 am + 1. The topological picture in M again gives twocopies of the bifurcation shown in Fig. 7. The essential difference appears after applying thefactorization with respect to χ . Consider a small vertical path m = − a , ℓ ∈ ( − δ, δ ) coveringthe path reaching q transversally from chamber II . The set Γ = 2 S does not transform in11he whole neighborhood of q . Along the chosen path the union ∪ Γ ( ℓ ) fills the 2-atom C (seeFig.10). Q P W S Q P S Q P u Q P u Q P Figure 10: The sets Γ i in chamber II while moving towards q . ÈG G Figure 11: The sets ∪ Γ and Γ along the path in chamber II .In this case we can show these sets in the plane as in Fig. 11, where χ and χ act as centralsymmetries. Factorizing by the diagonal action, we can write 2( C × S ) /χ ∼ = 2( C /χ ) × S andget the pre-image of the chosen path in N as a union of two atoms of the new type, whichwe denote by T × S . Here the 2-atom T = C / Z is shown in Fig. 12. Again, we keep thesame notation T for the corresponding 3-atom T × S . On the diagram Σ of the system N ,crossing the point q from the border into chamber II we have two simultaneous bifurcations T : ∅ → T . The 3-manifold with boundary T is non-orientable and is impossible in a systemwith non-degenerate symplectic structure. T1 Figure 12: The 2-atom T .Finally, we have obtained the topological description of the integral manifolds, criticalintegral surfaces and the bifurcations in N along any chosen path in the plane of the integralconstants. This completes the rough topological analysis of the system N . The exact topological analysis is a way to establish the phase topology of the considered systemup to Liouville equivalence [12]. In order to calculate main topological invariants of suchequivalence, we need to describe the families of regular Liouville tori and to find the exact rules12y which these families are glued to the boundaries of the bifurcation atoms. Let us recall somenotions from the general theory [12].Given a Liouville integrable Hamiltonian (or almost Hamiltonian) system on a 4-dimensio-nal manifold M , let us remove from the phase space all connected components of the integralsurfaces containing critical points of the integral map F : M → R , i.e., all singular leavesof the Liouville foliation. In our case, these leaves also include the whole level { L = 0 } . Theconnected component of the remaining set is called a family of Liouville tori.Consider a path γ : [ t , t ] → R which is either closed γ ( t ) = γ ( t ) or has its endsbeyond the admissible region F ( M ). The pre-image I γ = F − ( γ ([ t , t ])) is called a loopmanifold. Under some simple transversality conditions it is indeed a smooth 3-dimensionalmanifold without boundary [24]. If the path γ is a fixed level line of some first integral Φ, wecall I γ an iso-Φ manifold. Frequently, the role of Φ is played by the Hamiltonian H . Then I γ is called an iso-energy manifold. Identifying the points that belong to the same leaf of theLiouville foliation on I γ we obtain the rough Fomenko graph W γ with edges representing thefamilies of regular tori and vertices corresponding to singular leaves of the foliation. Consider anedge of this graph bounded by two vertices. On the boundary tori of the atoms pointed by thesevertices some pairs of coordinate cycles (bases of cycles) are defined called admissible coordinatesystems (or admissible bases). Shift these bases to one regular torus corresponding to an innerpoint of the graph’s edge. Two obtained bases are connected (in the one-dimensional homologygroup) with the so-called gluing matrix. It is an integer-valued matrix whose determinant isequal to ±
1. In the orientable case without minimal or maximal tori the bases are chosen insuch a way that the determinant is always equal to −
1. Endowing each edge of the graph withthe gluing matrix we obtain one of the forms of the exact topological invariant of the Liouvillefoliation on I γ . Usually, since gluing matrices are defined up to the changes of admissiblecoordinate systems, they are replaced by some sets of numerical invariants called marks. Theresulting topological invariant is called the marked molecule and is denoted by W ∗ γ (see [12] forcomplete details). The goal of the exact topological analysis of an integrable system is to findthe existing marked molecules and the corresponding loop manifolds for a reasonably full setof paths in the integral constants plane. We return to the system N and use the advantages of the separation of variables to describeformally the families of regular tori and introduce, in some universal way, the coordinate system(the pair of independent cycles) on each family. In the general case of an integrable systemwith two degrees of freedom, each family is parameterized by the value of the integral mapand the image of a family is an open connected set in R . The image of one family can coverseveral chambers and the walls between them if there are walls on which some families do notbifurcate. For the system N this is not the case. Indeed, let us collect all information aboutthe existing atoms and the number of regular tori in the chambers in Fig. 13. The arrows showthe atoms and the direction in which the number of tori increases, the number of tori itself isgiven in squares. We see that on each wall all tori of the adjacent chambers are involved inthe corresponding bifurcation. Indeed, in each chamber the accessible region for the separatedvariables in this system consists of one rectangle, therefore, all the tori projecting onto thatrectangle bifurcate simultaneously.Fix some chamber and consider the corresponding accessible regions u i ∈ [ ξ i , η i ] ( i = 1 , . Here ξ i , η i are the roots of P i Q i = (1 − u i ) f i ( u i ). Theorem 1 states that regular tori are theconnected components of Γ × Γ , where Γ i are defined by (3.1). Let us introduce the angularvariables ϕ , ϕ in such a way that u i = ξ i cos ( ϕ i ) + η i sin ( ϕ i ) , √ η i − u i = √ η i − ξ i cos( ϕ i ) , √ u i − ξ i = √ η i − ξ i sin( ϕ i ) . (5.1)13 mp p p p q q q q p
11 13
4T B2B 2BB B B2B 2B2A2A AA 2R2R
C1 2T1
Figure 13: Atoms and families.Square roots of constant values are always supposed non-negative. Substituting (5.1) into(2.4) we get the equations of Γ i containing sin ϕ i , cos ϕ i and, maybe, some radicals which haveconstant sign on each connected component of the integral manifold. According to Agreement 1this sign is either e i or d i and, for the chosen connected component of J m,h , it does not changeinside the corresponding chamber. Finally we obtain that the family of tori is defined by theset of those signs out of e i , d i which remain in the final expressions for P i , Q i on the connectedcomponents of Γ i . Agreement 2.
Having expressed P i , Q i on the connected component of Γ i in terms of sin ϕ i , cos ϕ i , e i , d i we will always consider this component oriented by the direction of increasing of the angle ϕ i . In fact, the separation of variables allows us to assign the universal orientation to the toriof each family. Indeed, the signs of the radicals in equations (2.5), though arbitrary, are strictlyconsistent with the signs in equations (2.6). Substituting (5.1) into (2.5), we obtain˙ ϕ i = ε i Φ i ( ϕ i ) ( i = 1 , , (5.2)where Φ i > ε i can equal e i , d i or ±
1. It means that the orientation of the cycle ε i Γ i onall tori is given by the phase flow of the system N . Agreement 3.
We call the orientation of a regular torus positive if it is defined by the pair ofcycles ( ε Γ , ε Γ ) , where ε i = ± and the orientation of ε i Γ i is induced by the phase flow. Since ε , ε in (5.2) are the same for the whole family, by the described universal algorithmwe fix a positive orientation of the family.Consider an arbitrary atom U together with some direction in which it is crossed alonga chosen path γ in R ( m, h ). The boundary ∂U consists of regular tori divided in a naturalway along γ into incoming and outgoing ones. The admissible bases ( λ, µ ) on all boundary torican be composed from the curves Γ i in one of the following ways ( ± Γ , ± Γ ) or ( ± Γ , ± Γ )depending on the type of U . Some of the atoms may not have incoming or outgoing tori. Agreement 4.
Choosing admissible bases on the boundary tori of an atom we always supposethat the orientation of these bases is positive (in the sense of Agreement 3) for all outgoing toriand negative for all incoming ones.
Let us give the explicit formulas for the parameterized cycles Γ i and the positive bases onthe families. Here we use the information given in Table 1.For the curve Γ we have three different cases. In chambers I , II the variable u oscillates14n [ τ ,
1] and therefore we putΓ : u = τ cos ϕ + sin ϕ ,Q = √ − τ p u ( ϕ ) cos ϕ ,P = √ − τ p h ∗ [ u ( ϕ ) − τ ] sin ϕ , sgn ˙ ϕ = +1 . Let us once give remarks to the sign of ˙ ϕ i applied in all similar representations of Γ i . Makingthe formal substitution of the introduced expressions for u , Q , P into the first equation (2.5)we get ˙ ϕ = 12 p u ( ϕ ) p h ∗ [ u ( ϕ ) − τ ] . In all such cases the remaining non-constant square roots will have constant sign along thecorresponding trajectory. We suppose them to be positive. Here there are two of them. Tochange both signs, it is enough to substitute ϕ → ϕ + π without changing the sign of ˙ ϕ . Tochange only one of the signs we may substitute ϕ → − ϕ or ϕ → π − ϕ . This will changeboth the default orientation of Γ and the sign of ˙ ϕ all over the chamber and, of course, willnot affect the class of equivalent gluing matrices on the corresponding families.Using the same method of the formal substitution with square roots considered positive,we obtain for chambers III − V Γ : u = − cos 2 ϕ ,Q = sin 2 ϕ ,P = e p h ∗ [ u ( ϕ ) − τ ][ u ( ϕ ) − τ ] , sgn ˙ ϕ = e , (5.3)and for chamber VI Γ : u = τ cos ϕ + τ sin ϕ ,Q = d p − u ( ϕ ) ,P = √− h ∗ ( τ − τ ) sin ϕ cos ϕ , sgn ˙ ϕ = d . (5.4)Analogously, for the curve Γ we obtain in chambers I , III Γ : u = − cos ϕ + σ sin ϕ ,Q = √ σ p − u ( ϕ ) sin ϕ ,P = b √ σ p m [ u ( ϕ ) − σ ] cos ϕ , sgn ˙ ϕ = +1 , (5.5)in chambers II , V , VI Γ : u = − cos 2 ϕ ,Q = sin 2 ϕ ,P = e b p m [ u ( ϕ ) − σ ][ u ( ϕ ) − σ ] , sgn ˙ ϕ = e , (5.6)and in chamber IV Γ : u = σ cos ϕ + σ sin ϕ ,Q = d p − u ( ϕ ) ,P = b √− m ( σ − σ ) sin ϕ cos ϕ , sgn ˙ ϕ = d . (5.7)Finally we can say that for a given chamber the number of families is equal to the number ofdifferent combinations of sgn ˙ ϕ , sgn ˙ ϕ and on each family the positive orientation is defined bythe basis (sgn ˙ ϕ Γ , sgn ˙ ϕ Γ ). The complete information on the families and the correspondingbases is given in Table 3. 15 able 3. Families and bases for the chambers Chamber
I II III IV V VI
Number offamilies 1 2 2 4 4 4Bases (cid:18) Γ Γ (cid:19) (cid:18) Γ e Γ (cid:19) (cid:18) e Γ Γ (cid:19) (cid:18) e Γ d Γ (cid:19) (cid:18) e Γ e Γ (cid:19) (cid:18) d Γ e Γ (cid:19) Let us denote by a i the atoms arising in the pre-image of segments 1 , . . . ,
13 of the bi-furcation diagram, where i is the number assigned to the corresponding segment. If there aretwo atoms in the pre-image of a point on segment i , we denote them by a ε j i using for ε j = ± a having four components. To each of them we assign the pair of sings. Obviously, in thisnotation we have the following atoms a , a , a e , a e , a , a , a e , a e , a e , a e , a e d , a e , a ( e ,d )13 . For a non-degenerate 3-atom, we can say that its sign is defined by the sign of the connectedcomponent of the curve out of Γ , Γ that does not bifurcate at this moment. The signs of theatoms in the pre-image of a point on the curve L (segments 11, 12, and 13) separate those torifamilies which are not identified with each other upon reaching L .To establish for each atom (excluding for a while the points of L ) the uniquely definedadmissible cycles ( µ -cycles for A and λ -cycles for hyperbolic atoms), we use Table 2 and equa-tions (5.3)–(5.7) giving the orientation on Γ i induced by the phase flow. The general rule ofchoosing this cycle is as follows. First, we take the curve Γ i for which the accessible region doesnot contain a double root (i.e., ± ∗ ). Then we multiply it by the sign of ˙ ϕ i from the adjacentchamber. This sign, of course, coincides with the sign of the corresponding cycle in the basistaken for the positive orientation of the family in this chamber in Table 3. For example, onsegment 4 with two atoms A we take µ = ± Γ from Table 2 and choose e for the sign fromthe basis of chamber III in Table 3. On segment 10 with two atoms B we take λ = ± Γ fromTable 2 and then choose e for the sign from the basis of any of chambers II or VI in Table 3.The second cycle in the pair ( λ, µ ) defining an admissible coordinate system is chosen accordingto Agreements 3 and 4.It is convenient to collect in one table all admissible coordinate systems for the atomson the segments of the bifurcation diagram written out for some globally fixed direction ofcrossing these atoms. Let us take the direction of the increasing h -coordinate and denote thecorresponding pairs of cycles for an atom a i by B in i and B out i . While constructing the loopmolecules we may meet with the necessity to cross some atoms in the inverse direction. Thenwe denote such admissible bases for an atom a i by C in i and C out i . The connection between thesebases is obvious C in i = (cid:18) − (cid:19) B out i , C out i = (cid:18) − (cid:19) B in i . (5.8)The result is given in Table 4. For the curve L the general rule does not work since thetori in the pre-image are in fact regular manifolds. For the new atoms R , T , C and T we needto establish some other rules.For the atoms R we take admissible bases with the λ -cycle coming from the adjacent atom B in the molecule. Thus, the notation 12 stands for the case of the edge coming from segment8 (the part of the loop molecule of the point q ) and 12 is used for the bases obtained alongthe edge coming from segment 7 (the part of the loop molecule of the point q ).Consider the 3-atom T = M × S on segment 13. Obviously, S here stands for the global λ -cycle e Γ which came from segment 10. So let it be the first cycle of the admissible basis on ∂T = T . For the µ -cycle let us take the circle in the M¨obius band that covers twice the middleline of the band oriented according to Agreement 4. On the family in chamber VI correspondingto the signs ( e , d ), the basis B in13 is negatively oriented. Then from Table 3, its orientationcoincides with that of ( d Γ , − e Γ ), which is the same as of ( e Γ , d Γ ).16 able 4. Admissible bases for the atoms Seg. 1 2 3 4 5 6 7 B in − − − − (cid:18) Γ − Γ (cid:19) (cid:18) Γ Γ (cid:19) (cid:18) e Γ Γ (cid:19) B out (cid:18) Γ Γ (cid:19) (cid:18) − Γ Γ (cid:19) (cid:18) − Γ − e Γ (cid:19) (cid:18) Γ − e Γ (cid:19) (cid:18) Γ e Γ (cid:19) (cid:18) Γ − e Γ (cid:19) (cid:18) e Γ − e Γ (cid:19) Seg. 8 9 10 11 12 B in (cid:18) e Γ − Γ (cid:19) (cid:18) e Γ − Γ (cid:19) (cid:18) e Γ Γ (cid:19) (cid:18) e Γ − d Γ (cid:19) (cid:18) e Γ − e Γ (cid:19) (cid:18) e Γ e Γ (cid:19) (cid:18) e Γ d (Γ +Γ ) (cid:19) B out (cid:18) e Γ e Γ (cid:19) (cid:18) e Γ d Γ (cid:19) (cid:18) e Γ − d Γ (cid:19) − − − − Let us write out equations (5.4) and (5.6) on the curve L (0 < τ < , σ > : u = − τ cos 2 ϕ ,Q = d p − u ( ϕ ) ,P = √− h ∗ τ sin 2 ϕ , sgn ˙ ϕ = d , Γ : u = − cos 2 ϕ ,Q = sin 2 ϕ ,P = e b p − m [ σ − u ( ϕ )] , sgn ˙ ϕ = e . It is easy to see that χ acts on Γ × Γ as the simultaneous shift ϕ → ϕ + π and ϕ → ϕ + π .Therefore the circle folding twice is Γ + Γ and we come to the basis for segment 13 as inTable 4.Since the 3-atom T is non-orientable and ∂T consists of only one torus, the solution hereis standard, i.e., for an outgoing torus we take the positive orientation of the family, but for anincoming one we take the negative orientation of the family. On the contrary, the 3-atom C isorientable and the choice of its orientation defines the orientations on two boundary tori. In ourcase at the point q both of them are incoming (supposing that one of the coordinates h or m increases while another is fixed). Two families come to the point q from chamber II . Choosingthe negative orientation from the families as given by the pair (Γ , − e Γ ) (see Table 3), wehave one of them different from the atom’s orientation. We mark this situation by the notation C ∗ . Note that we may choose the orientation on ∂C as (Γ , Γ ). Then it is consistent withsome orientation of C . Obviously, in this case one of the gluing matrices on the incoming edgeswill have the determinant equal to +1. If the topology of the molecule makes it possible tochange the direction of this edge and the following ones without general contradiction, then wecan change orientations of the corresponding families of tori and obtain an orientable molecule.But if this edge is a part of a loop, such a change may be impossible. Then the molecule definesa non-orientable loop manifold. As it was mentioned above, in the system N we have exactly four critical points of rank 0,all four are non-degenerate and their neighborhoods have representations (4.4). Therefore thecorresponding loop molecules are well-known and completely classified by the Bolsinov theorem[12]. Nevertheless, in this section we demonstrate the use of separated variables for the processof constructing these loop molecules.Let us take small closed paths around the points p , . . . , p in the clockwise direction. Then17rom Tables 3 and 4 we have the following chains: p : ( a ) B out2 → C in1 ( a ); p : ( a ) B out4 → C in6 ( a ) C out6 → C in2 ( a ); p : ( a ) B out1 → B in5 ( a ) B out5 → C in3 ( a ); p : ( a ) C out5 → B in6 ( a ) B out6 → B in8 ( a ) B out8 →→ C in7 ( a ) C out7 → C in5 ( a ) . (5.9)Substituting the bases from Table 4 and (5.8) we readily obtain not only the gluing matrices butalso the rules by which families choose their bounding atoms. Indeed, let us take for examplethe edge connecting a e and a e . We have B out8 = (cid:18) e Γ e Γ (cid:19) , C in7 = (cid:18) e Γ e Γ (cid:19) = (cid:18) (cid:19) B out8 . We see that the two families started from the B -atom a +8 ( e = +1) differ by the sign e andtherefore come to different B -atoms on segment 7.For the sake of brevity we denote some (2 × E ± = (cid:18) ± ∓ (cid:19) , D ± = (cid:18) ± ± (cid:19) , C ± = (cid:18) ± − (cid:19) . As usual, E stands for the identity matrix. The loop molecules of non-degenerate singularities c i ( J ( c i ) = p i ) generated by the chains (5.9) are shown in Fig. 14: ( a ) for c , ( b ) for c , c , and( c ) for c . a = A a = A D+ ( ) a ( ) b a + ( = A a + ) a ( = A a ) E+(E-) a ( = B a ) a - ( = A a - ) E-(E+) E-(E+) a a a a e >0 e <0 e >0 e <0 e <0 e >0 e >0 e >0 e <0 e <0 e >0 e <0 a a -++- D+D- D+D+D- D-D- ( ) c D+D+
Figure 14: Loop molecules of non-degenerate points.Of course, our method of global choice of orientation affects the gluing matrices, but sincethe possible changes (e.g. changing the order of the separation variables) are applied to all torisimultaneously, the gluing matrices of the molecules considered in this section simultaneouslychange their signs ( E ± → E ∓ , D ± → D ∓ ). This only leads to another representatives of thesame exact topological invariants. In the system N the set of degenerate closed orbits consists of the motions (4.2) in the pre-imageof the points q , q . Theorem 3.
The loop molecules of the points q , q can be represented in the form shownin Fig. 15. The first one is connected, while the second consists of two equivalent connectedcomponents differing by the sign e . +EE E EE+ a = B ℓ= a = B a = B a = B ++ -- ℓ= E ( +) e E+ d =-1 d =+1 e a = B e a = B a =T ( ) e - a =T ℓ= ℓ= q q e C- e C+ Figure 15: Loop molecules of the points q , q . Proof.
Consider a closed path surrounding the point q clockwise. Recall that the atoms B both in a and a differ by the sign e . At the same time on L we have to identify the familiesfrom chamber IV having the same product e d and from chamber V having the same sign e . Then we obtain the picture of the molecular edges printed over a piece of the bifurcationdiagram as shown in Fig. 16. e =+1 e = 1 - d =+1 q IIIIV Vℓ = 0 h r m b = + 2 - X X Y Y Y Z Z Z a + a + a - a - d = 1 - e =+1 e =+1 Figure 16: Constructing the loop molecule of the point q .In the whole neighborhood of q all λ -cycles on the regular tori are induced by the periodiccritical trajectories of the atoms a , a . Then from Table 2 we readily obtain that both λ and λ (no matter, incoming or outgoing) are defined as e Γ . Taking the chains marked with thepoints Z i , X i , Y i we get( Z i ) C out12 → C in8 ( a ) C out8 → ( X i ) → B in9 ( a ) B out9 → B in11 ( Y i ) . From Table 4, the gluing matrix is E + at both points X i .Let us find the gluing matrices on the arcs connecting the atoms a ± . To be definite, let ustake the points Y , on the arc with e d = +1 and shift the corresponding bases B out9 to thetorus J − ( Y ). We get two bases (Γ , Γ ) and ( − Γ , − Γ ). On segment 11 with | τ i | = τ > χ (Γ ) : u = cos 2 ϕ ,Q = sin 2 ϕ ,P = − e P, sgn ˙ ϕ = e . − Γ : u = − cos 2 ϕ ,Q = sin 2 ϕ ,P = − e P, sgn ˙ ϕ = − e , where P = p − h ∗ [ τ − cos ϕ ] >
0. The obvious substitution ϕ → π − ϕ turns χ (Γ ) into − Γ . A similar reasoning leads to the equality χ (Γ ) = − Γ on J − ( Y ). Finally we have thatthe map χ identifies two incoming bases at Y , so they give the same basis on the image torus19n N . Therefore the gluing matrix on the arc is the identity matrix E . Another arc e d = − V . Shifting the bases C in8 from the points Z , tothe point Z (i.e., to the torus J − ( Z )) we get two bases (Γ , − Γ ) and ( − Γ , − Γ ). In thesame way as above, (5.3) yields χ (Γ ) = − Γ , but from (5.6) we obviously have χ (Γ ) = Γ .So again χ identifies two bases at Z and they give the same basis on the image torus in N .The gluing matrix is E . This proves the statement on the loop molecule of q .Consider a closed path surrounding the point q clockwise. Recall that the atoms B bothin a and a differ by the sign e . At the same time on L we have to identify the families fromchamber V having the same sign e . The families in chamber VI are not identified with eachother and end with the atoms T . This leads to two rough invariants homeomorphic to thatshown in Fig. 15. The picture of the molecular edges together with a piece of the bifurcationdiagram is shown in Fig. 17. In this case the globally defined λ -cycles are e Γ . Since theconnected components of the loop molecule differ exactly by the sign e , this cycle is fixed forthe whole component and is not affected by any identifications in the pre-image of L . q e =+1 e = 1 - e = 1 - II VIV ℓ = 0 h r m a = + 2 d =+1 a + a - a + a - e =+1 d = 1 - Figure 17: Constructing the loop molecule of the point q .The edges in chamber II give C out10 → B in7 . The gluing matrix is E + . In chamber V let us fix e and shift the bases B out7 to the point on L . For e = ±
1, we get two bases ( e Γ , − Γ ) and( e Γ , Γ ). Previously we stated that, in chamber V , χ (Γ ) = − Γ and χ (Γ ) = Γ , thereforethe obtained bases at the point of L give the same basis on the image torus in N . Thus, thegluing matrix on the loops is the identity matrix.Finally, comparing the bases for segments 10 and 13 from Table 4 we see that the gluingmatrix is C s where s = sgn( − e d ). The theorem is proved.Note that the structure of the molecule of q in Fig. 15 allows us to pass the whole moleculein one direction. Instead of going globally right-to-left, let us change the direction in the upperhalf. Then we have to change the admissible bases on the boundary tori of the atoms a + i . Thematrix E + remains unchanged but all the identity matrices turn into E − . So, in this case wecan avoid gluing matrices with the determinant equal to +1 and the loop manifold is orientable.Obviously, this cannot be done for the molecules of the point q and for some similar iso-integralmolecules having loops rather than arcs.Let us denote by P + k g + s m the result of gluing k handles and s M¨obius bands to aclosed 2-surface P (of course, it supposes cutting out small discs first). From Theorem 3 wefind the topology of the loop manifolds of the degenerate points. Corollary 1.
The loop manifold of the point q is homeomorphic to the direct product ( S +3 g ) × S , i.e., to the loop manifold of the non-degenerate 3-atom C . The loop manifold of thepoint q is homeomorphic to two copies of the direct product ( S + 4 m ) × S . Indeed, all r -marks in the molecules of q , q are equal to ∞ and the loop manifolds areeasily restored from the topology of the graph. This fact corresponds to the mentioned aboveproperty of globally defined λ -cycles (at the point q two λ -cycles e Γ are identified by the Z -symmetry χ in the pre-image of L ). 20 The collection of iso-integral molecules
In this section, we present the result of constructing all possible iso- M and iso-energy markedmolecules which occur in the system N , except for the critical values of the integrals at foursingular points h = ± a ± b , m = ± / ( a ± b ) and, for the restriction of H to N , the minimalvalue h ∗ = p a + b ) of the h -coordinate on the curve L . Table 5. Molecules and their codes
Code Molecule Code Molecule Code Molecule OM D+A A OM B E-D+A AAE- OM B E-D+ AAE-BAA E-E- OM BD-A B AD+BA BD- D-D-D+D+D+ A OM BD-A BD- E EA NM BD+A E NM B TTC+C-D+A NM B BD-E-A BD- E E NM B BD+D+A BD+ EE NM B B TD+ TC+C-D+A B TD+ TC+C- UM D-A D-A C1 * UM B D-A D- C1D- * UM BD+A T1T1D+D+ UM D-A T1 LM C BD+ BD+ ¥¥¥¥¥¥ IM B BD+D+A BD+ ¥¥¥¥ IM B BD+E+A BD+ ¥¥¥¥ IM BD-A ¥¥ In Table 5, the notation OM i stands for orientable molecules, NM i for non-orientable ones, UM i for splitting ones. We also include the non-compact iso- L molecule LM for the level { L = 0 } where the symplectic structure degenerates. For non-negative values of m the iso- M manifolds are non-compact. Thus, crossing the zero value, these manifolds bifurcate without anycritical points of M appearing on the zero level. Three corresponding non-compact moleculesare denoted by IM i . Note that the molecule IM was found in [9] as describing the zero level ofthe Bogoyavlensky integral. This level, due to (2.1), coincides with the manifold { M = 0 } ⊂ N .21he molecules with the gluing matrices are representatives of their Liouville equivalenceclasses obtained by our way of choosing the coordinate systems and orientations on the familiesof tori. In [12], one can find the description of possible changes that occur due to the changes ofdirections and orientations. Finally, in Table 6 we show the correspondence between the valuesof m and h and the molecules found. Table 6. Iso-integral molecules m -value Code h -value Code m < min { m ( q ) , m ( p ) } OM h ( p ) < h < h ( p ) OM m ( q ) < m < m ( p ) ( a > b ) OM h ( p ) < h < h ( p ) OM m ( p ) < m < m ( q ) ( a < b ) NM h ( p ) < h < h ( p ) OM m = m ( q ) < m ( p ) ( a > b ) UM h ( p ) < h < h ∗ OM m = m ( q ) > m ( p ) ( a < b ) UM h ∗ < h < h ( q ) OM + 2 NM max { m ( p ) , m ( q ) } < m < m ( p ) NM h = h ( q ) OM + 2 UM m ( p ) < m < m ( q ) NM h ( q ) < h < h ( q ) OM + 2 NM m = m ( q ) UM h = h ( q ) UM + 2 NM m ( q ) < m < NM h > h ( q ) OM + 2 NM m < m ( p ) IM m ( p ) < m < m ( p ) IM m > m ( p ) 2 IM This completes the exact topological analysis of the system.
Acknowledgements
The author is grateful to A.V. Bolsinov for extremely valuable discussions and advices, toV.N. Roubtsov, the University of Angers and the Organizers of FDIS-2013 for hospitality andsupport.
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