Completely Integrable Systems — A Generalization

Completely integrable systems: a generalization arXiv:math/9906202v2 [math.SG] 30 Jun 1999 D. V. Alekseevsky1 , J. Grabowski2 , G. Marmo3 , and P. W. Michor4 Erwin Schr¨odinger International Institute of Mathematical Physics, Wien,Austria Abstract We present a slight generalization of the notion of completely integrable sys- tems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups. 1 Introduction A Hamiltonian system on a 2n-dimensional symplectic manifold M is said to be completely integrable if it has n first integrals in involution, which are functionally independent on some open dense submanifold of M. This definition of a completely integrable system is usually found, with some minor variants, in any modern text on symplectic mechanics [AM, Ar, LM, MSS, Th]. Starting with this definition, one uses the so called Liouville-Arnold the- orem to introduce action-angle variables and write the Hamiltonian system 1 Gen. Antonova 2 kv 99, 117279 Moscow B-279, Russia 2 Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland; and ´ Mathematical Institute, Polish Academy of Sciences, ul. Sniadeckich 8, P.O. Box 137, 00-950 Warszawa, Poland e-mail: [email protected] 3 Dipartimento di Scienze Fisiche, Universit`a di Napoli, Mostra d’Oltremare, Pad. 19, 80125 Napoli, Italy e-mail: [email protected] 4 Institut f¨ ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, 1090 Wien, Austria; and Erwin Schr¨ odinger International Institute of Mathematical Physics, Bolzmanngasse 9, 1090 Wien, Austria e-mail: [email protected] 1 in the form I˙k = 0, (1.1) ∂H φ˙ k = = νk (I), (1.2) ∂Ik where k ∈ {1, . . . , n}. The corresponding flow is given by I k (t) = I k (0), (1.3) φk (t) = φk (0) + νk t. (1.4) The main interest in completely integrable systems relies on the fact that they can be integrated by quadratures. It is clear, however, that even if νk dI k is not an exact (or even a closed) 1- form, as long as ν˙ k = 0, the system can always be integrated by quadratures. In this letter, we would like to take up this remark from the view point of Lie groups and their cotangent bundles, as well as double Lie groups, plying the role of deformed cotangent bundles, to show how the notion of a completely integrable system can be generalized by relaxing the property of being Hamiltonian and retaining only that it has enough constants of the motion (first integrals) to warrant it to be integrable by quadratures. More precisely, we define some class of dynamical systems on a Poisson manifold (M, Λ), defined by a vector field Γ = Λ(η), where η is a 1-form, which are integrable by quadratures. Here M is the cotangent bundle of a Lie group or its appropriate deformation. 2 A universal model for completely integrable system If we consider the abelian Lie group Rn , we can construct a Hamiltonian action of Rn on T ∗ Rn induced by the group addition: Rn × T ∗ Rn −→ T ∗ Rn . (2.1) This can be generalized to the Hamiltonian action Rn × T ∗ (Rk × T n−k ) −→ T ∗ (Rk × T n−k ), (2.2) 2 of Rn , where T m stands for the m-dimensional torus, and reduces to Rn × T ∗ T n or T n × T ∗ T n , when k = 0. By using the standard symplectic structure on T ∗ Rn , we find the momen- tum map µ : T ∗ Rn −→ (Rn )∗ , (q, p) 7→ p, induced by the natural action of Rn on itself via translations, which is a Poisson map if (Rn )∗ is endowed with the (trivial) natural Poisson structure of the dual of a Lie algebra. It is now clear that any function on (Rn )∗ , when pulled back to T ∗ Rn or T ∗ T n , gives rise to a Hamiltonian system which is completely integrable (in the Liouville sense). Because the level sets of this function carry on the action of Rn , the completely integrable system gives rise to a one-dimensional subgroup of the action of Rn on the given level set. The specific subgroup will, however, depend on the particular level set, i.e. the ‘frequencies’ are first integrals. The property of being integrable by quadratures is captured by the fact that it is a subgroup of the Rn -action on each level set. It is now clear, how we can preserve this property, while giving up the requirement that our system is Hamiltonian. We can indeed consider any 1-form η on (Rn )∗ and pull it back to T ∗ Rn or T ∗ T n , then associated vec- tor field Γη = Λ0 (µ∗ (η)), where Λ0 is the canonical Poisson structure in the cotangent bundle, is no more Hamiltonian, but it is still integrable by quadratures. In action-angle variables, if η = νk dI k is the 1-form on (Rn )∗ , the associated equations of motion on T ∗ T n will be I˙k = 0, (2.3) φ˙ k = νk , (2.4) with ν˙ k = 0, therefore the flow will be as in (1.3), (1.4), even though ∂νk ∂νj j 6= k . (2.5) ∂I ∂I We can now generalize this construction to any Lie group G. We consider the Hamiltonian action G × T ∗ G −→ T ∗ G, (2.6) of G on the cotangent bundle, induced by the right action of G on itself. The associated momentum map µ : T ∗ G ≃ G ∗ × G −→ G ∗ (2.7) 3 It is a Poisson map with respect to the natural Poisson structure on G ∗ (see e.g. [AG1, LM]). Now, we consider any differential 1-form η on G ∗ which is annihilated by the natural Poisson structure ΛG ∗ on G ∗ associated with the Lie bracket. Such form will be called a Casimir form. We define the vector field Γη = Λ0 (µ∗ (η)). Then, the corresponding dynamical system can be written as (for the proof we refer to the general case described in Theorem 1) g −1 g˙ = η(g, p) = η(p), (2.8) p˙ = 0, (2.9) since ω0 = d(< p, g −1dg >) (cf. [AG1]). Here we interpret the covector η(p) on G ∗ as a vector of G. Again, our system can be integrated by quadratures, because on each level set, obtained by fixing p’s in G ∗ , our dynamical system coincides with a one-parameter group of the action of G on that particular level set. We give a familiar example: the rigid rotator and its generalizations. Example 1 In the case of G = SO(3) the (right) momentum map µ : T ∗ SO(3) −→ so(3)∗ (2.10) is a Poisson map onto so(3)∗ with the linear Poisson structure Λso(3)∗ = εijk pi ∂pj ⊗ ∂pk . (2.11) P Casimir 1-forms for Λso(3)∗ read η = F dH0 , where H0 = p2i /2 is the ‘free Hamiltonian’ and F = F (p) is an arbitrary function. Clearly, F dH0 is not a closed form in general, but (pi ) are first integrals for the dynamical system Γη = Λ0 (µ∗ (η)). It is easy to see that ci , Γη = F (p)Γ0 = F (p)pi X (2.12) where X ci are left-invariant vector fields on SO(3), corresponding to the basis (Xi ) of so(3) identified with (dpi ). Here we used the identification T ∗ SO(3) ≃ SO(3)×so(3)∗ given by the momentum map µ. In other words, the dynamics is given by p˙i = 0 (2.13) −1 g g˙ = F (p)piXi ∈ so(3) (2.14) 4 and it is completely integrable, since it reduces to left-invariant dynamics on SO(3) for every value of p. We recognize the usual isotropic rigid rotator, when F (p) = 1. We can generalize our construction once more, replacing the cotangent bundle T ∗ G by its deformation, namely a group double D(G, ΛG ) associated with a Lie-Poisson structure ΛG on G (see e.g. [AG2, Lu1]). This double, denoted simply by D, carry on a natural Poisson tensor Λ+ D which is non-degenerate on the open-dense subset D + = G·G∗ ∩G∗ ·G of D (here G∗ ⊂ D is the dual group of G with respect to ΛG ). We refer to D as being complete if D + = D. Identifying D with G×G∗ if D is complete (or D + with an open submanifold of G × G∗ in general case; we assume completeness for simplicity) via the group product, we can write Λ+ ∗ D in ‘coordinates’ (g, u) ∈ G × G in the form Λ+ l r D (g, u) = ΛG (g) + ΛG∗ (u) − Xi (g) ∧ Yi (u), (2.15) where Xil and Yir are, respectively, the left- and right-invariant vector fields on G and G∗ relative to dual bases Xi and Yi in the Lie algebras G and G ∗ , and where ΛG and ΛG∗ are the corresponding Lie-Poisson tensors on G and G∗ (see [Lu1, AG2]). It is clear now that the projections µG∗ and µG of (D, Λ+ ∗ D ) onto (G, ΛG ) and (G , ΛG∗ ), respectively, are Poisson maps. Note that we get the cotangent bundle (D, Λ+ ∗ D ) = (T G, Λ0 ) if we put ΛG = 0. The group G acts on (D, Λ+ D ) by left translations which, in general are not canonical transformations. This is, however, a Poisson action with respect to the inner Poisson structure ΛG on G, which is sufficient to develop the momentum map reduction theory (see [Lu2]). For our purposes, let us take a Casimir 1-form η for ΛG∗ , i.e. ΛG∗ (η) = 0. By means of the momentum map µG∗ : D −→ G∗ , we define the vector field on D: Γ = Λ+ (µ∗ ∗ (η)). η D G (2.16) In ‘coordinates’ (g, u), due to the fact that η is a Casimir, we get Γη (g, u) =< Yir , η > (u)Xil (g), (2.17) so that Γη is associated with the ‘Legendre map’ Lη : D ≃ G × G∗ −→ T G ≃ G × G, Lη (g, u) =< Yir , η > (u)Xi , (2.18) which can be viewed also as a map Lη : G∗ −→ G. Thus we get the following. 5 Theorem 1 The dynamics Γη on the group double D(G, ΛG ), associated with a 1-form η which is a Casimir for the Lie-Poisson structure ΛG∗ on the dual group, is given by the system of equations u˙ = 0, (2.19) −1 g g˙ = < Yir , η > (u)Xi ∈ G, (2.20) and is therefore completely integrable by quadratures. Example 2 We consider now the double Lie group D = SL(2, C) with G = SU(2) and G∗ = SB(2, C) (see e.g. [AG2]. We will write the elements as follows:   z1 z2 D = SL(2, C) ∋ a = , where zi ∈ C, z1 z4 − z2 z3 = 1, z3 z4   α −¯ ν G = SU(2) ∋ g = , where α, ν ∈ C, |α|2 + |ν|2 = 1, ν α ¯   ∗ r γ G = SB(2, C) ∋ u = , where r > 0, γ ∈ C. 0 r −1 The Poisson structure ΛSL(2,C) is the restriction of the following quadratic (real) Poisson brackets on C4 : {z1 , z2 } = − 2i z1 z2 {z2 , z3 } = iz1 z4 i i {z1 , z3 } = zz 2 1 3 {z2 , z4 } =zz 2 2 4 {z1 , z4 } = 0 {z3 , z4 } − 2i z3 z4 = {z1 , z¯1 } = − 2i |z1 |2 − i|z3 |2 {z2 , z¯2 } − 2i |z2 |2 − i|z1 |2 − i|z4 |2 = {z3 , z¯3 } = − 2i |z3 |2 {z4 , z¯4 } − 2i |z4 |2 − i|z3 |2 = i {z1 , z¯2 } = −iz3 z¯4 {z2 , z¯3 } =z z¯ 2 2 3 {z1 , z¯3 } = 0 {z2 , z¯4 } = −iz1 z¯3 i {z1 , z¯4 } = z z¯ 2 1 4 {z3 , z¯4 } = 0. (2.21) The lacking commutators may be obtained from this list if we remember that zi , z¯j } = {zi , zj }. One can then check that, the Poisson bracket is real, e.g., {¯ indeed, det and det are Casimir functions, and that z1 ↔ z4 , z2 7→ −z2 , and z3 7→ −z3 defines a symmetry of the bracket associated to the inverse a 7→ a−1 in SL(2, C). 6  Our double group is complete, since we have the following unique (Iwa- sawa) decompositions: 1 SL(2, C) ≃ SU(2).SB(2, C), where s= p , |z1 |2 + |z3 |2      z1 z2 sz1 −s¯z3 1/s s(¯ z1 z2 + z¯3 z4 ) = (2.22) z3 z4 sz3 s¯ z1 0 s 1 SL(2, C) ≃ SB(2, C).SU(2), where t= p , |z3 + |z4 |2 |2      z1 z2 t t(z1 z¯3 + z2 z¯4 ) t¯ z4 −t¯z3 = . (2.23) z3 z4 0 1/t tz3 tz4 Hence, the bracket { , } is globally symplectic on SL(2, C). This bracket is projectable on the subgroups SU(2) and SB(2, C), and for the ‘left trivial- ization’ SL(2, C) = SU(2).SB(2, C) it gives us the Poisson Lie brackets on SU(2): ¯ } = −i|ν|2 {α, α {ν, ν¯} = 0 {α, ν} = 2i αν ¯ ν¯} = − 2i α¯ {α, ¯ν (2.24) i i {α, ν¯} = 2 α¯ν {α, ¯ ν} = − 2 αν,¯ and on SB(2, C): i {γ, r} = γr, γ , γ} = i(r 2 − r −2 ). {¯ (2.25) 2 The ‘interaction’ between SU(2) and SB(2, C) is described by αr −1 {ν, γ} = − 4i νγ − i¯ ν r −1 {α, γ} = − 4i αγ + i¯ i i {¯ ν , γ} = 4 ν¯γ {α, ¯ γ} = 4 αγ ¯ (2.26) {ν, r} = − 4i νr {α, r} = − 4i αr, where the lacking commutations can be derived, due to the fact that the bracket is real. One can check that the Casimir 1-forms for ΛSB(2,C) read η = F dH0 , where F = F (γ, r) is an arbitrary function on SB(2, C) and 1 1X 2 1 H0 = T r(aa∗ ) = |zi | = (|γ|2 + r 2 + r −2 ) 2 2 2 7 is the ‘free’ Hamiltonian. For the dynamics Γη on SL(2, C) induced by η, we calculate (using 2.21) that z˙1 = − 2i F (H0z1 − z¯4 ) z˙2 = − 2i F (H0 z2 + z¯3 ) (2.27) z˙3 = − 2i F (H0 z3 + z¯2 ) z˙4 = − 2i F (H0 z4 − z¯1 ). Since F and H0 are constants of the motion, it is clear that the system is completely integrable. For F = 1 this system was considered in [MSS, Za]. In variables g ∈ SU(2) and u ∈ SB(2, C), we have   r˙ γ˙ u˙ = (2.28) 0 −rr˙ −2 and i g −1g˙ = − F (H0 I + J u¯Ju−1 ) = const ∈ su(2), (2.29) 2   0 −1 where J = . In a more transparent form 1 0  2  −1 i r − r −2 + |γ|2 2γr −1 g g˙ = − F ∈ su(2). (2.30) 4 γ r −1 2¯ r −2 − r 2 − |γ|2 It follows that we get a ‘free motion’ on SU(2) along trajectories of left- invariant vector fields corresponding to i Lη = − F (u)(H0(u)I + J u¯Ju−1 ) ∈ su(2) (2.31) 2 (cf. [MSS, Za]). The mapping i SB(2, C) ∋ u 7→ Lη (u) = − F (u)(H0 (u)I + J u¯Ju−1 ) ∈ su(2) (2.32) 2 is a sort of a ‘Legendre transform’, transforming momenta from SB(2, C) into velocities from su(2). It is easy to see that, e.g. in the case F = 1, Lη is invertible and the momenta corresponding to the velocity   i s w − ∈ su(2), s ∈ R, w ∈ C, (2.33) 2 w¯ −s are p r =s+ s2 + |w|2 + 1, γ = rw. (2.34) 8 Example 3 Since the dual subgroups in the group double play entirely sym- metric role, let us consider now SU(2) to be the set of momenta and SB(2, C) to be the configuration space. The Lie-Poisson structure ΛSU (2) admits, however no global Casimir func- tion, so that we cannot produce a globally Hamiltonian system on SL(2, C) by means of the momentum map µSU (2) : SL(2, C) −→ SU(2), though we easily find out that η = iF (α, ν)(νd¯ ν − ν¯dν) (2.35) is a general Casimir 1-form (which is real for real functions F ). The equa- tions of the dynamical system Γη read α˙ = 0 (2.36) ν˙ = 0 (2.37) 1 r˙ = − F (α, ν)|ν|2γ (2.38) 2   1 2 i γ˙ = −F (α, ν) |ν| γ + α(ℜ(ν) − ℑ(ν)) . (2.39) 2 r In other words, α and ν are constant of the motion and   −1 1 |ν|2 2iα(ℜ(ν) − ℑ(ν)) uu ˙ = Lη (α, ν) = − F (α, ν) ∈ sb(2, C) 2 0 −|ν|2 (2.40) is time independent. The ‘Legendre map’ Lη : SU(2) −→ sb(2, C) is never bijective, since our set of momenta SU(2) is a compact manifold, so that ‘admissible’ velocities form a compact subset of sb(2, C). 3 A further generalization We have seen that if we concentrate on the possibility of integrating our system by quadratures, then we can do without the requirement that the system is Hamiltonian. By considering again the equations of motion in action-angle variables, we have, classically, I˙k = 0, (3.1) ϕ˙ k = ν k (I). (3.2) 9 Clearly, if we have I˙k = Fk (I), ϕ˙ = Ajk (I)ϕj , (3.3) and we are able to integrate the first equation by quadratures, we again have the possibility to integrate by quadratures the system (3.3), if only the matrices (Ajk (I(t))) commute: Z t  ϕ(t) = exp A(I(s))ds ϕ0 . (3.4) 0 Of course, because ϕk are discontinues functions on the torus, we have to be more careful here. We show, however, how this idea works for double groups. In the case when the 1-form η on G∗ is not a Casimir 1-form for the Lie-Poisson structure ΛG∗ , we get, in view of (2.15), Γη (g, u) =< Yir , η > (u)Xil (g) + ΛG∗ (η)(u). (3.5) Now, the momenta evolve according to the dynamics ΛG∗ (η) on G∗ (which can be interpreted, as we will see later, as being associated with an interaction of the system with an external field) and ‘control’ the evolution of the field of velocities on G (being left-invariant for a fixed time) by a ‘variation of constants’. Let us summarize our observations in the following. Theorem 2 The vector field Γη on the double group D(G, ΛG ), associated with a 1-form η on G∗ , defines the following dynamics u˙ = ΛG∗ (η)(u), (3.6) −1 g g˙ = < Yir , α > (u)Xi ∈ G, (3.7) and is therefore completely integrable, if only we are able to integrate the equation 3.6 and < Yir , η > (u(t))Xi lie in a commutative subalgebra of G for all t. Example 4 Let us take now SU(2) for momenta and consider the Hamilto- nian H = 21 |ν|2 . The 1-form η = dH is exact, but not a Casimir 1-form for 10 ΛSU (2) . The dynamical system Γη on SL(2, C) induces the following dynamics of momenta ν˙ = 0, (3.8) i 2 α˙ = |ν| α. (3.9) 2 This is the dynamics described in [LMS]. Additionally, we get from 2.26: r˙ = 0, (3.10) i γ˙ = − α(ℜ(ν) ¯ + ℑ(ν))r −1 . (3.11) 2 Hence, ν(t) = ν0 , α(t) = α0 exp( 2i |ν|2 t), and   −1 i 0 1 uu ˙ = − α(ℜ(ν) ¯ + ℑ(ν)) ∈ sb(2, C). (3.12) 2 0 0 Here, the velocity of a particle is rotating around 0 with the radius propor- tional to the momentum ℜ(ν0 ) + ℑ(ν0 ) and the frequency proportional to the energy H = 12 |ν0 |2 . The velocities stay, however, in a commutative subalgebra of the unipotent Lie algebra sb(2, C), so that ! 1 α0 ℜ(ν0|ν)+ℑ(ν 0| 2 0) (exp(− i |ν 2 0 | 2 t) − 1) u(t) = u0 . (3.13) 0 1 Let us end up with an example which shows that we can actually weaken the assumptions of Theorem 2. In fact, it is sufficient to assume that g −1 g(t) ˙ = exp(tX)A(t) exp(−tX) (3.14) for some A(t), X ∈ G, such that X+A(t) lie in a commutative subalgebra of G for all t (e.g. A(t) = const), to assure that (3.7) is integrable by quadratures. Indeed, in the new variable g1 (t) = exp(−tX)g(t) exp(tX) (3.15) the equation (3.7) reads g˙ 1 (t) = g1 (t)(X + A(t)) − Xg1 (t) (3.16) 11 and, since the right- and the left-multiplications commute, we easily find that  Z t  g(t) = g0 exp tX + A(s)ds exp(−tX). (3.17) 0 This procedure is similar to what is known as the Dirac interaction picture in the quantum evolution. Example 5 For our group double SL(2, C), let us take the 1-form η = F (r)dH0 + λdr, r ∈ R on SB(2, C) which is a perturbation of 2.30. For the dynamics of momenta, we get r˙ = 0 (3.18) i γ˙ = − λrγ (3.19) 2 which can be easily integrated: r(t) = r0 (3.20) i γ(t) = γ0 exp(− λrt). (3.21) 2 In particular, r = const and |γ| = const. We have also   −1 i F (r)(r 2 − r −2 + |γ|2 ) − λr 2F (r)γr −1 g g˙ = − . 4 γ r −1 2F (r)¯ F (r)(r −2 − r 2 − |γ|2 ) + λr (3.22) The velocities are no longer constant and rotate around the vector  i  2 −2 2 −4 0 C0 = (F (r0)(r0 − r0 + |γ0 | ) − λr0 ) ∈ su(2). (3.23) 0 4i It can be interpreted as an effect of an interaction of moving charged particle with an external magnetic field, corresponding to the perturbation of the ‘free system’. Since, as easily seen, g −1 g(t) ˙ = exp(tX)A0 exp(−tX), (3.24) with   − 4i λr0 0 X= i (3.25) 0 4 λr 0 12 and   i F (r0 )(r02 − r0−2 + |γ0 |2 ) − λr0 2F (r0 )γ0 r0−1 A0 = − −1 −2 , 4 2F (r0 )¯ γ 0 r0 F (r0 )(r0 − r02 − |γ0 |2 ) + λr0 we can easily integrate (3.22):   2  −iF (r0 ) r0 − r0−2 + |γ0 |2 2γ0r0−1 g(t) = g0 exp t × 4 γ0 r0−1 2¯ r0−2 − r02 − |γ0 |2 
 exp − 4i λr0 t 0
. (3.26) 0 exp 4i λr0 t 4 Final comments Our identification of systems which can be integrated by quadratures with one-parameter subgroups of some Lie group G acting on the carrier space of the system gives us the possibility to dispose of the requirement that the system is Hamiltonian. We have to notice however that in some cases our system will turn out to be Hamiltonian with respect to a different symplectic structure (still invariant under the action of G) on the phase space. Indeed, if we consider η = νi dI i and make the assumption that dν1 ∧ . . . ∧ dνn 6= 0 (4.27) on some open-dense P submanifold N, we can define on N the symplectic structure ωN = dνi ∧ dφi and Γ = νi ∂φi will be associated with the iP 1 Hamiltonian H = 2 i (νi )2 . In the general situation this procedure does not apply any more. In any case, the importance of the Liouville-Arnold theorem relies on the fact that, in the hypothesis of the theorem, we can find the group G and its action on the manifold, and then show that our starting system is conjugated to the one written in the introduction (1.1,1.2) in terms of the action-angle variables. Our generalization is much more in the spirit of the Lie-Sheffers theorem [LS] and it consists of splitting our system along the orbits of an action of a Lie group and a ‘transverse’ component (which is either zero, or linear), so that the integration can be achieved easily. What seems to us relevant is that completely integrable systems (in the Liouville sense) are a part of this more general scheme. In particular, we have 13 shown that we can use the action of the group by non-canonical transforma- tions, so that systems on group doubles can be cast in this generalization. We are confident that this approach may be useful to quantize group doubles in the geometric quantization setting. Many of these questions are currently being investigated. References [AM] Abraham, R.; Marsden, J.E., Foundations of Mechanics, Benjamin, Reading, Mass. 1978. [AG1] Alekseevsky, D.V. ; Grabowski, J.; Marmo, G.; Michor, P.W., Poisson structures on the cotangent bundle of a Lie group or a principle bundle and their reductions, J. Math. Physics 35 (1994), 4909–4928. [AG2] Alekseevsky, D.V. ; Grabowski, J.; Marmo, G.; Michor, P.W., Poisson structures on double Lie groups, (to appear). [Ar] Arnol’d, V.I., Mathematical Methods of Classical Mechanics, Springer, New York 1978. [LS] Lie, S.; Sheffers, G., Vorlesungen u ¨ber Kontinuerlichen Gruppen, (1893) Teubner, Leipzig. [LM] Libermann, P.; Marle, C.M., Symplectic Geometry and Analytical Me- chanics, Reidel, Dordrecht 1987. [LMS] Lizzi, F.; Marmo, G; Sparano, G.; Vitale, P., Dynamical aspects of Lie-Poisson structures, Mod. Phys. Lett. A 8 (1993), 2973–2987. [Lu1] Lu, J.-H., Multiplicative and affine Poisson structures on Lie groups, Thesis, Berkeley 1990. [Lu2] Lu, J.-H., Momentum mappings and reduction of Poisson actions, in ‘Symplectic Geometry, Groupoids, and Integrable Systems’, eds.: P. Dazord and A. Weinstein, Springer, New York 1991, 209–225. [MSV] Marmo, G.; Saletan, E.J.; Simoni, A.; Vitale, B., Dynamical Systems. A Differential Geometry Approach to Symmetry and Reduction, John Wiley and Sons, Chichester 1985. 14 [MSS] Marmo, G.; Simoni, A.; Stern, A., Poisson Lie group symmetries for the isotropic rotator, Int. J. Mod. Phys. A 10 (1995), 99–114. [Th] Thirring, W., A Course in Mathematical Physics. Classical Dynamical Systems, Springer, New York 1978. [Za] Zakrzewski, S., Classical mechanical systems based on Poisson geome- try, Preprint. 15