The quasi-bi-Hamiltonian formulation of the Lagrange top 2 0 Carlo Morosi1, Giorgio Tondo2 0 2 n 1 Dipartimento di Matematica, Politecnico di Milano, a J P.za L. da Vinci 32, I-20133 Milano, Italy 5 e–mail: [email protected] 1 2 Dipartimento di Scienze Matematiche, Universit`a di Trieste, ] via A. Valerio 12/1, I-34127 Trieste, Italy I S e–mail: [email protected] . n i l n [ 1 Abstract v 8 Startingfromthetri-HamiltonianformulationoftheLagrangetopinasix-dimensional 2 phase space, we discuss the possible reductions of the Poisson tensors, the vector 0 1 field and its Hamiltonian functions on a four-dimensional space. We show that the 0 vector field of the Lagrange top possesses, on the reduced phase space, a quasi- 2 0 bi-Hamiltonian formulation, which provides a set of separation variables for the / corresponding Hamilton-Jacobi equation. n i l n : v i X Keywords: Lagrange top, Hamiltonian formulation, Hamilton-Jacobi separability. r AMS 2000 Subject classifications: 37K10, 37J35, 53D17, 70E40, 70H06. a 1 Introduction The classical theory of separation of variables for the Hamilton-Jacobi equation provides the most effective tool to solve the equations of motion of a given Hamiltonian system. In this framework, the main problem is to have an efficient (possibly algorithmic) way to produce a set of separation variables. To this purpose, two new approaches, stemming fromsolitontheory,havebeenrecentlyintroduced: the“magicSklyanin’srecipe”[1],based on the Lax representation of the equations of the motion, and the bi-Hamiltonian (bH) approach to separation of variables [2, 3, 4, 5], based on the bi-Hamiltonian structures associated with the equations of motion. A remarkable feature of the latter approach is that if the Hamiltonian system admits a quasi-bi-Hamiltonian (qbH) formulation, then a set of separation variables can be algorithmically computed [3]; moreover, the qbH property is independent of the coordinate system in which the bH structure is written down. The aim of this paper is to apply the approach based on the qbH property to the classical Lagrangetop (LT); in particular, we show how the (complex) separation variables for LT, introduced in [6] in an algebraic-geometric setting, arise quite naturally as distinguished functions for its tri-Hamiltonian structure. The starting point of our analysis is the fact that, on a six-dimensional phase space M, the LT vector field X admits a tri-Hamiltonian formulation X = P dh (throughout L L α α the paper, the index α takes values 0,1,2), each one of the three compatible Poisson tensors P possessing two independent Casimir functions. α When one tries to eliminate the Casimirs by fixing their values, one is faced with a typical situation, occurring also for other bH finite-dimensional integrable systems [7, 8, 5]: to each one of the symplectic leaves S one can restrict only the vector field X and the α L corresponding pair (P ,h ), but not the entire triple of the Poisson tensors, so that the α α tri-Hamiltonian formulation of X is lost under restriction. Nevertheless, using a more L general reduction process `a la Marsden-Ratiu, we will show that the symplectic leaf S of 0 the Poisson tensor P can be endowed with a Poisson-Nijhenuis structure [9, 10] (hence a 0 bH structure) and that X can be given a qbH formulation. So, the separability of LT is L obtained from its Hamiltonian structures as a natural outcome of the reduction process. The paper is organised as follows. In Section 2 the tri-Hamiltonian structure of LT is shortly reviewed; in Section 3 the main properties of the qbH model are discussed in view of application to LT. In Sections 4 and 5, respectively, the reduction of the Poisson tensors P and of the vector field X with its Hamiltonian functions are considered; α L the qbH formulation for X is explicitly constructed, together with a solution of the L corresponding Hamilton-Jacobi equation. Our results are summarised in Section 6, where some potential extensions of this work are pointed out. 1 2 The multi-Hamiltonian structure of the Lagrange top A modern formulationof LT can be found in[11, 12]; as usual in this framework, the com- ponents of vectors and covectors and the entries of matrices are referred to the comoving frame, whose axes are the principal inertia axes of the top, with fixed point O. The phase space M of LT isparametrised bythe pairm = (ω,γ), where ω = (ω ,ω ,ω )T 1 2 3 and γ = (γ ,γ ,γ )T are the angular velocity and the vertical unit vector, respectively. 1 2 3 The following notations are introduced: µ is the mass of the top, g the acceleration of gravity, J = diag(A,A,cA) the principal inertia matrix (c = 1), G = (0,0,a)T is the 6 center of mass; at last, normalisations are chosen so that µag/A = 1. The Euler-Poisson equations are dL /dt = M (change of the angular momentum) and o o dγ/dt = 0 (invariance of the vertical unit vector); with the above notations and normali- sations, these equations take the well-known form (1 c)ω ω γ 2 3 2 − −  (1 c)ω ω +γ  3 1 1 − − dm 0 = X (m) , X (m) =   . (2.1) dt L L  γ ω γ ω  2 3 3 2  −   γ ω γ ω  3 1 1 3  −   γ ω γ ω  1 2 2 1  −  The LT vector field X can be given a tri-Hamiltonian formulation L X = P dh = P dh = P dh ; (2.2) L 0 0 1 1 2 2 the compatible Poisson tensors P , written in matrix block form, are α 0 B B 0 T R P = , P = − , P = , (2.3) 0 (cid:18) B C (cid:19) 1 (cid:18) 0 Γ (cid:19) 2 (cid:18) RT 0 (cid:19) − where B, C, Γ, T and R are 3 3 matrices × 0 1 0 0 c ω ω 0 γ γ 3 2 3 2 − − − B =  1 0 0 , C =  c ω 0 ω , Γ =  γ 0 γ , 3 1 3 1 − − 0 0 0 ω ω 0 γ γ 0 2 1 2 1    −   −  0 c ω ω /c 0 γ γ 3 2 3 2 − − T =  c ω 0 ω /c  , R =  γ 0 γ  . (2.4) 3 1 3 1 − − ω /c ω /c 0 γ /c γ /c 0 2 1 2 1  −   −  The Hamiltonian functions h can be written as α 1 h = F +2σcF F , h = σc2F3 F 2σcF F , h = F , (2.5) 0 2 4 1 3 1 1 − 3 − 1 2 2 2 2 where σ = c−1 and 2c 1 F = ω , F = (ω2 +ω2 +c ω2) γ , (2.6) 1 3 2 2 1 2 3 − 3 F = ω γ +ω γ +c ω γ , F = γ2 +γ2 +γ2 . 3 1 1 2 2 3 3 4 1 2 3 As it is known, the functions F (i = 1,...,4) are integrals of motion for Eq.(2.1); they i are independent and in involution w.r.t. each one of the three Poisson tensors. Moreover, (F ,F ) are Casimir functions of P , (F ,F ) of P and (F ,F ) of P . 1 2 0 1 4 1 3 4 2 The vector field X can be immersed in two different bH chains, starting and ending with L the Casimirs of the Poisson tensors P : α P dF = 0, P dF = P dh = X , P dh = P d( σF2), P d( σF2) = 0 ; (2.7) 0 2 2 2 0 0 L 2 0 0 − 3 2 − 3 P dF = 0, P dF = P dh , P dh = P dh = X , P dh = P d( σcF F ), 0 2 1 2 0 1 1 1 0 0 L 1 0 0 1 4 − P d( σcF F ) = 0 . 1 1 4 − 2.1 Remark. The Hamiltonian formulation of LT w.r.t. P is classical (see, e.g.,[12]). 2 The bH formulation w.r.t. (P ,P ) was introduced in [13] in the semidirect product 0 2 so(3) so(3), and was later recovered in [6] in an algebraic-geometric setting. The tri- × Hamiltonian formulation w.r.t. (P ,P ,P ) was constructed in [14], by a suitable reduction 0 1 2 of the Lie-Poisson pencil defined in the direct sum of three copies of so(3). (To compare the above-quoted results, let us recall that the angular momentum and the vertical unit vector are taken as dynamical variables in [12, 13, 14], whereas the angular momentum is replaced by the angular velocity ω in [6] and in the present paper.) ⋄ 3 The quasi-bi-Hamiltonian model TheqbH modelwasintroducedin[15,2]anddevelopedin[3,16](seealso[4]andreferences therein). Here we summarise some facts to be used in the rest of the paper. Let Q , Q be two compatible Poisson tensors on a manifold M; a vector field X is said 0 1 to admit a qbH formulation w.r.t. Q and Q if there are three functions ρ, H, K such 0 1 that 1 X = Q dH = Q dK . (3.1) 0 1 ρ In other words, X is Hamiltonian w.r.t. Q with Hamiltonian function H, and it is quasi- 0 Hamiltonian (qH) w.r.t. Q , with qH function K and conformal factor 1/ρ. In spite 1 of the presence of ρ, equation (3.1) implies that H and K are in involution w.r.t. both Poisson brackets corresponding to Q and Q (as well as in the particular bH case ρ = 1). 0 1 3 If dim M = 2n, the qbH formulation is said to be of maximal rank if at each point m M ∈ the Poisson tensors Q , Q are non degenerate and the associated tensor N = Q Q−1 0 1 1 0 (with vanishing Nijenhuis torsion) has nindependent eigenvalues λ (m),...,λ (m). Inthis 1 n case, one can introduce a local chart (λ ,µ ) (i = 1,2,...,n), called a Darboux-Nijenhuis i i chart [17], such that Q , Q and N take the canonical form 0 1 0 I 0 Λ Λ 0 Q = n , Q = , N = , (3.2) 0 (cid:18) I 0 (cid:19) 1 (cid:18) Λ 0 (cid:19) (cid:18) 0 Λ (cid:19) n − − with Λ = diag(λ ,...,λ ); in general, the coordinate functions µ , canonically conjugated 1 n i to λ , can be computed by quadratures. At last, the qbH formulation is said to be of i n Pfaffian type if ρ = λ . i=1 i The following resultQhas been proved in [3] for a Pfaffian qbH vector field. 3.1 Proposition. The general solution of Eq.(3.1) for the Pfaffian case is given by functions H and K which, in a Darboux-Nijenhuis chart (λ ,µ ), take the “canonical” i i form n n f ρ f i i H = , K = , ∆ = (λ λ ) , (3.3) i i j ∆ λ ∆ − Xi=1 i Xi=1 i i Yj6=i where each f is an arbitrary function, depending at most on the pair (λ ,µ ). Moreover, i i i the Hamilton-Jacobi equations for both H and K are separable. ⋄ This Proposition has a straightforward consequence. 3.2 Corollary. Let X = Q dH be a Hamiltonian vector field; if in a Q -Darboux chart 0 0 (x,y) the Hamiltonian H takes the canonical form (3.3), then X admits a Pfaffian qbH formulation w.r.t. a Poisson tensor Q and a qH function K of the form (3.2) and (3.3), 1 respectively. Viceversa, let X = 1/ρ Q dK be a qH vector field w.r.t. Q ; if, in a chart (x,y), Q and 1 1 1 n K take the canonical forms (3.2) (3.3) and ρ = x , then it is also X = Q dH with i=1 i 0 Q0 and H given by (3.2) (3.3), respectively. HencQe, the chart (x,y) is a Darboux-Nijenhuis chart for the Poisson pair Q ,Q . 0 1 ⋄ For n = 2, this Corollary can be slightly generalised, in a way that is useful for subsequent applications to LT . 3.3 Proposition. Let S be a four-dimensional manifold and Y = Q dH be a Hamil- 0 tonian vector field w.r.t. a non degenerate Poisson tensor Q . Let there is a Darboux 0 chart (x,y) such that the Hamiltonian H can be written as a linear combination of two functions Hˆ, Kˆ with the canonical form (3.3), i.e., H(x,y) = βHˆ(x,y)+Kˆ(x,y) β = const , (3.4) 1 Hˆ(x,y) = fˆ(x ,y ) fˆ(x ,y ) , 1 1 1 2 2 2 x x − 1 2 (cid:16) (cid:17) − 4 1 Kˆ(x,y) = x fˆ(x ,y ) x fˆ(x ,y ) . 2 1 1 1 1 2 2 2 x x − 1 2 (cid:16) (cid:17) − Then, the vector field Y admits the Pfaffian qbH formulation (3.1)-(3.3); a Darboux- Nijenhuis chart (λ,µ) is given by the following map: 1 Φ : (x,y) (λ,µ) λ = , µ = y (x +β)2 (i = 1,2) . (3.5) i i i i 7→ x +β − i Hence, H is separable in the chart (λ,µ). Moreover, H is separable also in the chart (x,y) and the corresponding Hamilton-Jacobi equation H(x ,x ,∂W/∂x ,∂W/∂x ) = h (3.6) 1 2 1 2 ˆ ˆ ˆ ˆ ˆ ˆ has the complete solution W(x ,x ;h,k) = W (x ;h,k) +W (x ;h,k) , W and W ful- 1 2 1 1 2 2 1 2 filling the Sklyanin separation equations [1] fˆ(x ,W′(x )) = x hˆ kˆ, fˆ(x ,W′(x )) = x hˆ kˆ , (3.7) 1 1 1 1 1 − 2 2 2 2 2 − ˆ ˆ with βh+ k = h . Proof. It is straightforward to check that the map Φ : (x,y) (λ,µ) is a Darboux map 7→ for Q ; moreover, since x x = (λ λ )/λ λ , the Hamiltonian H takes the canonical 0 1 2 1 2 1 2 − − − form (3.3): H x(λ,µ),y(λ,µ) = βHˆ x(λ,µ),y(λ,µ) +Kˆ x(λ,µ),y(λ,µ) = (3.8) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) 1 1 1 = λ fˆ( β, λ2µ )+λ fˆ( β, λ2µ ) = λ λ (cid:18)− 1 1 λ − − 1 1 2 2 λ − − 2 2 (cid:19) 1 2 1 2 − 1 = f (λ ,µ ) f (λ ,µ ) , 1 1 1 2 2 2 λ λ − 1 2(cid:16) (cid:17) − where 1 1 f (λ ,µ ) = λ fˆ( β, λ2µ ) , f (λ ,µ ) = λ fˆ( β, λ2µ ) . (3.9) 1 1 1 − 1 1 λ − − 1 1 2 2 2 − 2 2 λ − − 2 2 1 2 On account of Corollary 3.2, the vector field Y = Q dH admits the qH formulation 0 Y = 1/ρ Q dK and H is separable. 1 Obviously enough, H is separable also in the chart (x,y), since the map Φ is a separated map [18], i.e., it maps separated coordinates into separated ones. Indeed, taking into account the form (3.4) of the function H, it is easily checked that the Hamilton-Jacobi ˆ ˆ ˆ ˆ equation H(x,∂W/∂x) = h has a complete solution W(x ,x ;h,k) = W (x ;h,k) + 1 2 1 1 ˆ ˆ ˆ ˆ W (x ;h,k), with βh+ k = h , and that W , W fulfil the Sklyanin separation equations 2 2 1 2 (3.7) for the Hamilton-Jacobi equations Hˆ(x,∂W/∂x) = hˆ, Kˆ(x,∂W/∂x) = kˆ. ⋄ 5 4 The reduction of the tri-Hamiltonian structure of the Lagrange top If a vector field X on a manifold M is bH w.r.t. a pair of degenerate Poisson tensors (P ,P ), a preliminary step in analysing its integrability is trying to reduce the vector 0 1 field, its Hamiltonian functions and the Poisson tensors on a lower-dimensional manifold M′, where one of the two Poisson tensors, say P , be invertible. A natural way to do 0 that is to fix the values of the Casimir functions of P . Of course, both P and X can 0 0 be properly restricted to a symplectic leaf S , giving rise to a Poisson tensor P′ and to a 0 0 vector field X′ = P′ dH′, H′ being the restriction to S of the original Hamiltonian H. 0 0 However, without additional assumptions, P is not assured to restrict to S , so that X′ 1 0 loses the original bH formulation. This situation occurs also for the tri-Hamiltonian structure of LT . Each one of the three Poisson tensors P has two independent Casimir functions, and the generic symplectic α leaves S are four-dimensional submanifolds of M. On account of Eq.(2.6), they are α defined as a S = m M ω = 1, ω2 +ω2 +cω2 2γ = 2a , (4.1) 0 { ∈ | 3 2c 1 2 3 − 3 2} a S = m M ω = 1, γ2 +γ2 +γ2 = a , 1 { ∈ | 3 2c 1 2 3 4} 1 S = m M ω γ +ω γ +cω γ = a , γ2 +γ2 +γ2 = a , 2 { ∈ | 1 1 2 2 3 3 −2 3 1 2 3 4} where a , a , a and a are arbitrary constants. Each Poisson tensor P can be properly 1 2 3 4 α restricted to a corresponding symplectic leaf S , but the other two tensors do not restrict α to the same leaf. Nevertheless, a quite general reduction technique given by the Marsden-Ratiu theorem [19] can be applied; it will enables us to construct on S a Poisson-Nijhenuis structure [9, α 10] induced by the tri-Hamiltonian structure on M, and on S a qbH formulation for 0 the vector field X′ . Essentially, one considers a Poisson manifold (M,P), a submanifold L S ֒ M and a distribution D TM such that E := D TS is a regular foliation with a → ⊂ |S ∩ good quotient = S/E. Then, the theorem states that the Poisson tensor P is reducible N to if the following conditions hold: N i) the functions on M which are invariant along D form a Poisson subalgebra of C∞(M); ii) P(D◦) TS +D (D◦ being the annihilator of D in T∗M). ⊂ Analogously to previous applications of this procedure to bH structures [8, 5, 20], let us choose as the submanifold S a generic symplectic leaf S of the Poisson tensor P and a α α distribution D such that at each point s S the following decomposition holds: α α ∈ T M = T S D , (4.2) sα sα β ⊕ sα S being the symplectic leaf of P (β = 0,1,2) passing through s . β β α This assumption assures that ii) is trivially fulfilled and that E = 0, so that the reduction procedure becomes a submersion Π : M S onto the manifold S ; then, it allows us α α → 6 to endow S with a non degenerate tri-Hamiltonian structure, since the kernels of the α reduced Poisson tensors P′ vanish. Indeed, if Π∗ denotes the (injective) pull-back of the β submersion Π, it is Ker P′ = (Π∗)−1 Im Π∗ P−1(D T S ) (4.3) sα β sα ∩ β sα ∩ sα β (cid:16) (cid:17) (4=.2) (Π∗)−1(Im Π∗ Ker P ) = 0, sα ∩ sα β where we have taken into account that Im Π∗ D◦, D◦ Ker P = D◦ (Im P )◦ = D◦ (T S )◦ (4=.2) 0. (4.4) sα ⊂ ∩ sα β ∩ sα β ∩ sα β In the LT case, the distribution is as follows. 4.1 Lemma. Let D be the distribution given by the vector fields ∂ ∂ ∂ ∂ Z = ic + , Z = i (4.5) 1 2 − ∂ω ∂ω ∂γ − ∂γ 2 3 2 3 (i = √ 1). Moreover, let ϕ ,ϕ be two generic functions. Then, for each Poisson tensor 1 2 − P there are two vector fields W and W (depending on ϕ and ϕ ) such that α 1α 2α 1 2 L (P ) = Z W +Z W (4.6) ϕ1Z1+ϕ2Z2 α 1 ∧ 1α 2 ∧ 2α (L and denoting the Lie derivative along the flow of the vector field Z and the exterior Z ∧ product of vector fields, respectively). Proof. Itiseasytocheck thatL P = Z Y +Z Y (j = 1,2),withsuitablevector Zj α 1∧ 1jα 2∧ 2jα fields Y ,Y . This result, together with the identity L (P) = fL (P)+ X P df, 1jα 2jα fX X ∧ implies (4.6), the vector fields W being W = ϕ Y +ϕ Y +P dϕ . jα jα 1 j1α 2 j2α α j ⋄ Eq.(4.6) implies the assumption i), since if f and g are invariant functions along D and (4.6) Z D, then L f,g =< df,L (P)dg > = 0 foreach function ϕ. Moreover, condition ϕZ ϕZ ∈ { } (4.2)isgenerically satisfiedasitcanbeeasilyverified. Hence, conditionsi),ii) arefulfilled and the Marsden-Ratiu reduction technique can be applied on each symplectic leaf S . α In conclusion, we have proved the following. 4.2 Proposition. The tri-Hamiltonian structure P is reducible to a non degenerate β tri-Hamiltonian structure P′ on each one of the symplectic leaves S . β α ⋄ To express the reduced tensors in a particularly simple and useful form, it is convenient to adaptthecoordinatesonM tothedistributionD,introducingaparametrisationincluding coordinate functions which span the subalgebra of the functions invariant along D. Let us choose the chart (u,v,w), related to (ω,γ) by the map Ψ : M M : (ω,γ) (u,v,w) → 7→ u = cω iω , u = iγ γ , (4.7) 1 3 2 2 2 3 − − 7 v = ω , v = γ , w = iω +cω , w = iγ γ . 1 1 2 1 1 2 3 2 2 3 − − − Taking into account the tri-Hamiltonian structure P given by (2.3) and the definition α (4.1) of S , a straightforward (though lengthy) calculation allows one to verify that the α chart (u,v) gives a parametrisation on each one of the symplectic leaves S ; the reduced α Poisson tensors P′ and the tensor N take the form β 0 0 0 1 0 0 1 0  0 0 1 u   0 0 0 u  P′ = i 1 , P′ = i − 2 , (4.8) 0 0 1 0 0 1 1 0 0 0  −   −   1 u 0 0   0 u 0 0  1 2  − −    0 0 u u 1 2 − −  0 0 u 0  P′ = i − 2 . 2 u u 0 0 1 2    u 0 0 0  2   4.3 Remark. By a direct inspection, one easily concludes that the tensor N′ := P′P′−1 1 0 (with vanishing Nijenhuis torsion) is such that P′ = N′P′ and P′ = N′P′. 1 0 2 1 The matrix representation of P′ and of the adjoint tensor N′∗ of N′ are formed by Hankel 0 and Frobenius blocks, respectively, so that (u,v) are Hankel-Frobenius coordinates, in the terminology of [8]. ⋄ 4.4 Proposition. Let us consider the map Ψ : S S : (u,v) (x,y) α α → 7→ 1 1 x = ( u u2 4u ) , x = ( u + u2 4u ) , (4.9) 1 2 − 1 −q 1 − 2 2 2 − 1 q 1 − 2 1 1 y = (2v u v v u2 4u ) , y = (2v u v +v u2 4u ) . 1 2 2 − 1 1 − 1q 1 − 2 2 2 2 − 1 1 1q 1 − 2 The chart (x,y) is a Darboux-Nijenhuis chart for the tri-Hamiltonian structure on S ; α the reduced Poisson tensors P′ have the matrix block form α 0 I 0 0 2 P′ = i P′ = i X P′ = i X , (4.10) 0 − (cid:18) I 0 (cid:19) 1 − (cid:18) 0 (cid:19) 2 − (cid:18) 2 0 (cid:19) − −X −X where = diag(x ,x ) . 1 2 X Proof. A straightforward computation, taking into account Eq.s (4.8) and (4.9). ⋄ (To be more precise, in order to have the Darboux-Nijenhuis chart defined in Section 3 one should eliminate the factor ( i) in Eq.(4.10), via the map x ix, y y). − 7→ 7→ 8 5 The reduction of the vector field and the Hamilto- nians of the Lagrange top. Having established the projection of the tri-Hamiltonian structure on each one of the symplectic leaves S , the next step is to consider the reduction of the vector field X and α L of the corresponding Hamiltonian functions h . α Unfortunately, they do not project onto S , since X does not preserve the distribution D α L andtheHamiltoniansh arenotinvariantalongD;hence, thetri-Hamiltonianformulation α ofX islost onS . Nevertheless, eachpair (X ,h ) canberestrictedto thecorresponding L α L α symplectic leaf S , so that Eq.(2.1), restricted to S , keeps a Hamiltonian formulation. α α Furthermore, if we consider the reduction on a symplectic leaf S , we can recover, as 0 a reminder of the original tri-Hamiltonian formulation, a qbH formulation for X ; this L suffices to provide a set of separation variables. Indeed, the following holds. 5.1 Proposition. The vector field X , restricted to S , takes the form L 0 X = P′dH = i Q dH . (5.1) L 0 − 0 Its Hamiltonian H = h takes the form 0|S0 ˆ ˆ H(x,y) = σa H(x,y)+K(x,y) , (5.2) 1 where 1 Hˆ(x,y) = fˆ(x ,y ) fˆ(x ,y ) , (5.3) 1 1 2 2 x x − 1 2 (cid:16) (cid:17) − 1 Kˆ(x,y) = x fˆ(x ,y ) x fˆ(x ,y ) , 2 1 1 1 2 2 x x − 1 2 (cid:16) (cid:17) − 1 1 1 a2 fˆ(ξ,η) = η2 + ξ4 + a ξ3 +(a +σ 1)ξ2 . 1 2 −2 2 2 4 Proof. A straightforward computation. ⋄ On account of this result, we are just in the situation considered in Proposition 3.3, with ˆ ˆ ˆ β = σa , f = f = f . (5.4) 1 1 2 So, X admits a qbH formulation; the Darboux-Nijenhuis coordinates (λ,µ) are obtained L from (x,y) via the map (3.5): λ = (x +σa )−1, µ = y (x +σa )2 (i = 1,2). (5.5) i i 1 i i i 1 − As it follows from the general results of Propositions 3.1, 3.3, H and K are separable both in the Darboux-Nijenhuis chart (λ,µ) and in the chart (x,y). Using the latter, let us compute a solution W of the Hamilton-Jacobi equations for H and K ∂W ∂W ∂W ∂W H(x ,x , , ) = h, K(x ,x , , ) = k; (5.6) 1 2 1 2 ∂x ∂x ∂x ∂x 1 2 1 2 9