SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2019), 083, 17 pages Collective Heavy Top Dynamics
Tomoki OHSAWADepartment of Mathematical Sciences, The University of Texas at Dallas,800 W Campbell Rd, Richardson, TX 75080-3021, USA
E-mail: [email protected]
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Received July 20, 2019, in final form October 22, 2019; Published online October 30, 2019https://doi.org/10.3842/SIGMA.2019.083
Abstract.
We construct a Poisson map M : T ∗ C → se (3) ∗ with respect to the canonicalPoisson bracket on T ∗ C ∼ = T ∗ R and the ( − )-Lie–Poisson bracket on the dual se (3) ∗ ofthe Lie algebra of the special Euclidean group SE (3). The essential part of this map isthe momentum map associated with the cotangent lift of the natural right action of thesemidirect product Lie group SU (2) (cid:110) C on C . This Poisson map gives rise to a canonicalHamiltonian system on T ∗ C whose solutions are mapped by M to solutions of the heavytop equations. We show that the Casimirs of the heavy top dynamics and the additionalconserved quantity of the Lagrange top correspond to the Noether conserved quantitiesassociated with certain symmetries of the canonical Hamiltonian system. We also constructa Lie–Poisson integrator for the heavy top dynamics by combining the Poisson map M witha simple symplectic integrator, and demonstrate that the integrator exhibits either exact ornear conservation of the conserved quantities of the Kovalevskaya top. Key words: heavy top dynamics; collectivization; momentum maps; Lie–Poisson integrator
Consider a rigid body spinning around a fixed point ( not the center of mass) in a gravitationalfield. We specify the configuration of the body by a rotation matrix around the fixed point: therotation matrix defines a transformation from the body frame to the spatial frame; hence theoriginal configuration space is SO (3). Let t (cid:55)→ R ( t ) ∈ SO (3) be the rotational dynamics of thebody and define the body angular velocity by ˆΩ( t ) := R − ( t ) ˙ R ( t ) ∈ so (3). One may identifyit with a vector Ω ( t ) ∈ R via the standard identification between so (3) and R summarizedin Appendix A. Let I := diag( I , I , I ) be the moment of inertia of the body with respect tothe principal axes, and define the body angular momentum Π ( t ) := I Ω ( t ). Let e be the unitvector pointing upward in the spatial frame and set Γ ( t ) := R ( t ) − e ; this vector signifies thevertically upward direction – essentially the direction of gravity – seen from the body frame.Then, the equations of motion are given by˙ Π = Π × (cid:0) I − Π (cid:1) + mgl Γ × c , ˙ Γ = Γ × (cid:0) I − Π (cid:1) , (1.1)where m is the mass of the body, c ∈ R is the unit vector from the point of support to thedirection of the body’s center of mass in the body frame, and l is the length of the line segmentbetween these two points. The above set of equations is often called the heavy top equations . a r X i v : . [ m a t h - ph ] O c t T. Ohsawa
The heavy top equations are known to be a Hamiltonian system. Let us define the heavy topbracket for the space of smooth functions on the space R × R = (cid:8) ( Π , Γ ) | Π ∈ R , Γ ∈ R (cid:9) asfollows { f, h } ( Π , Γ ) := − Π · (cid:18) ∂f∂ Π × ∂h∂ Π (cid:19) − Γ · (cid:18) ∂f∂ Π × ∂h∂ Γ − ∂h∂ Π × ∂f∂ Γ (cid:19) . (1.2)The Hamiltonian h : R × R → R of the heavy top is given by h ( Π , Γ ) := 12 Π · (cid:0) I − Π (cid:1) + mgl Γ · c . (1.3)Then the Hamiltonian system defined with respect to the above Poisson bracket and this Hamil-tonian, i.e.,˙ Π = { Π , h } , ˙ Γ = { Γ , h } gives the heavy top equations (1.1).The heavy top bracket (1.2) turns out to be the ( − )-Lie–Poisson bracket on the dual se (3) ∗ ofthe Lie algebra se (3) of the special Euclidean group SE (3) := SO (3) (cid:110) R ; note that we identify se (3) = so (3) (cid:110) R – and hence se (3) ∗ as well – with R × R in the standard manner (seeAppendix A for details). In fact, [12, 13] showed that the heavy top bracket is a special case ofthe semidirect product theory for reduction of Hamiltonian systems with broken symmetry; seealso [7] for its Lagrangian counterpart. Specifically, the heavy top is originally a Hamiltoniansystem on T ∗ SO (3) where the presence of gravity breaks the SO (3)-symmetry the system wouldotherwise possess. As a result, the system retains only the SO (2)-symmetry with respect torotations about the vertical axis. The semidirect product theory effectively recovers the fullsymmetry by considering the extended configuration space SE (3) as opposed to SO (3). Asa result, one obtains a Hamiltonian dynamics in se (3) ∗ defined in terms of the Hamiltonian (1.3)and the ( − )-Lie–Poisson bracket (1.2) on se (3) ∗ ∼ = R × R . Given a Poisson manifold P and an equivariant momentum map M : P → g ∗ associated witha right action of a Lie group G on P , one can show that M is a Poisson map with respect to thePoisson bracket on P and the ( − )-Lie–Poisson bracket on g ∗ ; see, e.g., [11, Theorem 12.4.1].Now, given a Hamiltonian system on P with Hamiltonian H : P → R , suppose that thereexists a function h : g ∗ → R such that h ◦ M = H ; such a function h is called a collective Hamilto-nian . If t (cid:55)→ z ( t ) is a solution of the Hamiltonian system on P defined by H , then t (cid:55)→ M ( z ( t )) isa solution of the Lie–Poisson equation on g ∗ defined in terms of the ( − )-Lie–Poisson bracket andthe collective Hamiltonian h . This is the basic idea of collective motions/dynamics originallydue to [4]. Their motivating examples are aggregate motions of a number of particles “as if itwere a rigid body or liquid drop”, such as the liquid drop model in nuclear physics; hence theterm “collective”. See also [8] and [5, Section 28] for details.More recently, [14, 15] used this idea to develop geometric integrators for Lie–Poisson equa-tions. In one of their examples, they considered a natural right action of SU (2) on C ∼ = T ∗ R and used its associated momentum map M : C → su (2) ∗ to construct a natural Poisson mapfrom C ∼ = T ∗ R to so (3) ∗ where su (2) ∗ ∼ = su (2) and so (3) ∗ ∼ = so (3) are identified in a naturalmanner, as summarized in Appendix A. As a result, they obtained a canonical Hamiltoniansystem on T ∗ R whose solutions are mapped by M to solutions of the free rigid body equationin so (3) ∗ . By applying a symplectic integrator to the canonical Hamiltonian system on T ∗ R ollective Heavy Top Dynamics 3and taking the image by M of its numerical solutions, they obtained a Lie–Poisson integratorfor the free rigid body equation and also for the point vortices equations on S – a coadjointorbit in su (2) ∗ .We also note that [3] extended the above work to Hamiltonian systems on the direct product (cid:0) S (cid:1) n × T ∗ R m motivated by an application to the spin-lattice-electron equations. However,the symplectic/Poisson structure there is the standard one on the direct product (interactionsbetween the (cid:0) S (cid:1) n part and the T ∗ R m part come from the Hamiltonian), whereas our Poissonstructure is on a semidirect product and hence is geometrically more involved. We present a collective formulation of the heavy top equations by constructing a Poisson mapfrom T ∗ C ∼ = T ∗ R to se (3) ∗ . More specifically, we first consider, in Section 2, the cotangentlift of the natural right action of the semidirect product SU (2) (cid:110) C on C ∼ = R and find theassociated momentum map L : T ∗ C → (cid:0) su (2) (cid:110) C (cid:1) ∗ .Next, in Section 3, we construct a Poisson map (cid:36) : (cid:0) su (2) (cid:110) C (cid:1) ∗ → se (3) ∗ with respect to the( − )-Lie–Poisson brackets on both duals. Then it follows that the composition M : T ∗ C → se (3) ∗ defined as M := (cid:36) ◦ L yields a Poisson map that gives a collective formulation of the heavy topdynamics. We note that a more natural and straightforward Poisson map T ∗ SE (3) → se (3) ∗ is not appropriate for collective dynamics; see Remark 2.1.In Section 4, we state our main result, Theorem 4.1. It essentially states that we can realizeany heavy top dynamics as a collective dynamics of a canonical Hamiltonian system in T ∗ C ∼ = T ∗ R . We also look into the symmetries possessed by the system in the general case as well asfor the Lagrange top. It turns out that those momentum maps associated with these symmetriesare essentially the well-known Casimirs (cid:107) Γ (cid:107) and Π · Γ of the general heavy top as well as theadditional conserved quantity Π for the Lagrange top.In Section 5, we develop a collective Lie–Poisson integrator for the heavy top dynamics bycombining our result and the idea of [14, 15]. Our test case is the Kovalevskaya top, which alsopossesses an additional conserved quantity – the Kovalevskaya invariant – in addition to theCasimirs. We apply both symplectic (implicit midpoint) and non-symplectic (explicit midpoint)integrators to the collective dynamics. We refer to the former as the collective Lie–Poissonintegrator . We also apply these integrators directly to the heavy top equations (1.1); note thatneither is a Poisson integrator for the heavy top equations although the implicit midpoint rulehas favorable properties; see [2]. We numerically demonstrate that the non-symplectic integra-tor, applied to either the collective or direct formulation, exhibits drifts in all the conservedquantities – the Hamiltonian, the Casimirs, and the Kovalevskaya invariant. On the other hand,the collective Lie–Poisson integrator preserves the Casimirs exactly (or more practically to ma-chine precision) and exhibits near-conservation – small oscillations without drifts – of the exactvalues of the Hamiltonian and the Kovalevskaya invariant. T ∗ C → (cid:0) su (2) (cid:110) C (cid:1) ∗ T ∗ C and T ∗ R Let χ ∈ C be arbitrary and identify it with q ∈ R as follows C → R , χ = (cid:20) χ χ (cid:21) = (cid:20) q + i q q + i q (cid:21) (cid:55)→ q = ( q , q , q , q ) . We equip C with the inner product (cid:104) · , · (cid:105) C : C × C → R , ( ψ, φ ) (cid:55)→ (cid:104) ψ, φ (cid:105) = Re( ψ ∗ φ ) . (2.1) T. OhsawaThen it is easy to see that this inner product is compatible with the dot product in R . Setting ψ = (cid:20) ψ ψ (cid:21) = (cid:20) p + i p p + i p (cid:21) ↔ p = ( p , p , p , p ) , we have (cid:104) ψ, χ (cid:105) C = p · q. Therefore, we may identify the dual (cid:0) C (cid:1) ∗ ∼ = (cid:0) R (cid:1) ∗ with C ∼ = R via this inner product. Asa result, we may identify the cotangent bundles T ∗ C and T ∗ R as follows T ∗ C → T ∗ R , ( χ, ψ ) = (cid:18)(cid:20) q + i q q + i q (cid:21) , (cid:20) p + i p p + i p (cid:21)(cid:19) (cid:55)→ ( q, p ) = ( q , . . . , q , p , . . . , p ) . Then, for any χ ∈ C , the natural dual pairing between T ∗ χ C ∼ = (cid:0) C (cid:1) ∗ and T χ C ∼ = C isidentical to the inner product (2.1), i.e., with an abuse of notation, (cid:104) · , · (cid:105) C : T ∗ χ C × T χ C → R , ( ψ, ˙ χ ) (cid:55)→ Re( ψ ∗ ˙ χ ) . We define a canonical 1-form on T ∗ C asΘ := Re( ψ ∗ d χ ) = Re (cid:0) ¯ ψ i d χ i (cid:1) , and define a symplectic form on T ∗ C as followsΩ := − d Θ = Re (cid:0) d χ i ∧ d ¯ ψ i (cid:1) . Note that we use Einstein’s summation convention unless otherwise stated. One easily seesthat, under the above identification of T ∗ C and T ∗ R , Θ and Ω become the canonical 1-form p i d q i and the canonical symplectic form d q i ∧ d p i on T ∗ R , respectively. Therefore, T ∗ C is alsoequipped with the following canonical Poisson bracket. For any smooth F, H : T ∗ C ∼ = T ∗ R → R , { F, H } T ∗ C ( q, p ) := ∂F∂q i ∂H∂p i − ∂F∂p i ∂H∂q i . (2.2)Alternatively, we may define the one-formΘ := 12 Re( ψ ∗ d χ − χ ∗ d ψ ) . (2.3)Then, it is ( p i d q i − q i d p i ) / T ∗ R , and satisfies Ω = − d Θ as well. SU (2) (cid:110) C Let us define the semidirect product SU (2) (cid:110) C using the natural action of SU (2) on C , i.e.,for any ( U, ρ ) , ( W, ϑ ) ∈ SU (2) × C , we define a binary operation( U, ρ ) · ( W, ϑ ) := (
U W, U ϑ + ρ ) . This renders SU (2) (cid:110) C a Lie group. Alternatively, one may think of the resulting Lie group SU (2) (cid:110) C as the matrix group (cid:26)(cid:20) U ρ (cid:21) ∈ GL (3 , C ) | U ∈ SU (2) , ρ ∈ C (cid:27) under the standard matrix multiplication.ollective Heavy Top Dynamics 5Now consider the (right) action of SU (2) (cid:110) C on C defined as followsΨ : (cid:0) SU (2) (cid:110) C (cid:1) × C → C , (( U, ρ ) , χ ) (cid:55)→ Ψ ( U,ρ ) ( χ ) := U ∗ ( χ − ρ ) . In terms of matrices, one may write this action as follows (cid:18)(cid:20)
U ρ (cid:21) , (cid:20) χ (cid:21)(cid:19) (cid:55)→ (cid:20) U ρ (cid:21) − (cid:20) χ (cid:21) = (cid:20) U ∗ − U ∗ ρ (cid:21) (cid:20) χ (cid:21) = (cid:20) U ∗ ( χ − ρ )1 (cid:21) . Its cotangent lift is then T ∗ Ψ : (cid:0) SU (2) (cid:110) C (cid:1) × T ∗ C → T ∗ C , (( U, ρ ) , ( χ, ψ )) (cid:55)→ T ∗ Ψ ( U,ρ ) − ( χ, ψ ) := ( U ∗ ( χ − ρ ) , U ∗ ψ ) . It is clear that the canonical 1-form Θ is invariant under the cotangent lift, i.e., ( T ∗ Ψ ( U,ρ ) − ) ∗ Θ = Θ for any ( U, ρ ) ∈ SU (2) (cid:110) C . Let us find the momentum map associated with the above action. Let ( ξ, φ ) be an arbitraryelement in the Lie algebra su (2) (cid:110) C = T ( I, (cid:0) SU (2) (cid:110) C (cid:1) . Its infinitesimal generator on C isdefined as( ξ, φ ) C ( χ ) := dd s Ψ exp( s ( ξ,φ )) ( χ ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 , where the exponential map takes the formexp( s ( ξ, φ )) = (cid:18) exp( sξ ) , (cid:90) s exp( σξ ) φ d σ (cid:19) . Therefore, we obtain( ξ, φ ) C ( χ ) = dd s exp( − sξ ) (cid:18) χ − (cid:90) s exp( σξ ) φ d σ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s =0 = − ξχ − φ. As a result, the associated momentum map L : T ∗ C → (cid:0) su (2) (cid:110) C (cid:1) ∗ satisfies (cid:104) L ( χ, ψ ) , ( ξ, φ ) (cid:105) = Θ ( χ, ψ ) · ( ξ, φ ) C ( χ, ψ ) = − Re ( ψ ∗ ( ξχ + φ ))= − Re(tr( χψ ∗ ξ )) − Re( ψ ∗ φ )= −
12 (tr( χψ ∗ ξ ) + tr( ξ ∗ ψχ ∗ )) − (cid:104) ψ, φ (cid:105) C = 2 tr (cid:18)
14 ( ψχ ∗ − χψ ∗ ) ξ (cid:19) + (cid:104)− ψ, φ (cid:105) C = 2 tr (cid:18)
14 ( χψ ∗ − ψχ ∗ − i Im( ψ ∗ χ ) I ) ∗ ξ (cid:19) + (cid:104)− ψ, φ (cid:105) C = (cid:28)
14 ( χψ ∗ − ψχ ∗ − i Im( ψ ∗ χ ) I ) , ξ (cid:29) su (2) + (cid:104)− ψ, φ (cid:105) C , where we used the inner products on su (2) and C defined in (A.1) and (2.1), respectively.Note also that, in the second last line, we added an extra term to render the matrix inside theconjugate transpose traceless to come up with an element in su (2). Using the inner products, T. Ohsawawe may identify su (2) ∗ with su (2) and (cid:0) C (cid:1) ∗ with C so that (cid:0) su (2) (cid:110) C (cid:1) ∗ ∼ = su (2) × C . Underthis identification, we obtain L ( χ, ψ ) = (cid:18)
14 ( χψ ∗ − ψχ ∗ − i Im( ψ ∗ χ ) I ) , − ψ (cid:19) . Since this momentum map is associated with the cotangent lift action of the right action Ψon T ∗ C , the momentum map L is equivariant (see [11, Theorem 12.1.4]), i.e., for any ( U, ρ ) ∈ SU (2) (cid:110) C , L ◦ T ∗ Ψ ( U,ρ ) − = Ad ∗ ( U,ρ ) ◦ L . Remark 2.1 (why not SE (3) action on R ?) . Perhaps the most natural way to construct an se (3) ∗ -valued equivariant momentum map – under a right action – on a cotangent bundle wouldbe the following. Consider the right SE (3)-action on R defined as SE (3) × R → R , (( R, a ) , x ) (cid:55)→ R T ( x − a ) , or using matrices, (cid:18)(cid:20) R a (cid:21) , (cid:20) x (cid:21)(cid:19) (cid:55)→ (cid:20) R a (cid:21) − (cid:20) x (cid:21) = (cid:20) R T ( x − a )1 (cid:21) , that is, this is the right action one obtains from the natural left action of SE (3) on R . Usingits cotangent lift action on T ∗ R , one obtains the momentum map˜ M : T ∗ R → se (3) ∗ , ˜ M ( q , p ) = − ( q × p , p )with the standard identification se (3) ∗ ∼ = so (3) × R ∼ = R × R . However, this momentum map istoo restrictive for the collective heavy top dynamics in se (3) ∗ because setting ( Π , Γ ) = ˜ M ( q , p )implies that the angular momentum Π is always perpendicular to Γ . This is clearly not alwaysthe case. For example, even the very simple case of the top spinning in the upright positiondoes not satisfy this condition: Π and Γ are both parallel to (0 , , Proposition 2.2.
Under the identification (cid:0) su (2) (cid:110) C (cid:1) ∗ ∼ = su (2) × C , the image of L contains su (2) × (cid:0) C \{ } (cid:1) , i.e., su (2) × (cid:0) C \{ } (cid:1) ⊂ L (cid:0) T ∗ C (cid:1) . Remark 2.3.
As we shall see later in Section 3, we will construct a map (cid:36) : (cid:0) su (2) (cid:110) C (cid:1) ∗ → se (3) ∗ , and consider the composition M := (cid:36) ◦ L : T ∗ C → se (3) ∗ . It then turns out that theorigin ψ = 0 in C corresponds to the origin Γ = 0 in R (see (4.1) below) and hence has nopractical importance in the heavy top dynamics because Γ is a unit vector. This propositionmakes sure that the image of M contains any possible values of Π ∈ R and Γ ∈ R \{ } , and sosetting ( Π , Γ ) = M ( χ, ψ ) does not impose any practical restriction in the collective heavy topdynamics. Proof .
Let ψ ∈ C \{ } be arbitrary and consider the map C → su (2) ∗ , χ (cid:55)→ (cid:0) χψ ∗ − ψχ ∗ − i Im( ψ ∗ χ ) I (cid:1) . Then it suffices to show that this map is surjective. In fact, one may rewrite the map with theidentification C ∼ = R defined by ( χ, ψ ) (cid:55)→ ( q, p ) as shown above as well as the identification su (2) ∗ ∼ = R (see Appendix A below) to obtain the following linear map R → R , q (cid:55)→ Aq with A := 12 p − p p − p − p − p p p p − p − p p . Then AA T = (cid:107) p (cid:107) I , whereas p (cid:54) = 0 because ψ (cid:54) = 0. Hence this linear map is surjective. (cid:4) ollective Heavy Top Dynamics 7 (cid:0) su (2) (cid:110) C (cid:1) ∗ Let us equip (cid:0) su (2) (cid:110) C (cid:1) ∗ with the ( − )-Lie–Poisson bracket. For any smooth f, h : (cid:0) su (2) (cid:110)C (cid:1) ∗ → R , we define { f, h } ( su (2) (cid:110)C ) ∗ ( µ, α ) := − (cid:28) µ, (cid:20) δfδµ , δhδµ (cid:21)(cid:29) su (2) − (cid:28) α, δfδµ δhδα − δhδµ δfδα (cid:29) C , (2.4)where ( δf /δµ, δf /δα ) ∈ su (2) × C (evaluated at ( µ, α )) is defined so that, for any ( δµ, δα ) ∈ (cid:0) su (2) (cid:110) C (cid:1) ∗ , (cid:28) δµ, δfδµ (cid:29) su (2) + (cid:28) δα, δfδα (cid:29) C = dd s f ( µ + sδµ, α + sδα ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 . (2.5)Then the equivariance of L implies that it is Poisson with respect to the canonical Poissonbracket (2.2) and the above Lie–Poisson bracket (2.4) (see, e.g., [4], [5, Section 28], and [11,Theorem 12.4.1]): { f ◦ L , h ◦ L } T ∗ C = { f, h } ( su (2) (cid:110)C ) ∗ ◦ L . (cid:36) : (cid:0) su (2) (cid:110) C (cid:1) ∗ → se (3) ∗ (cid:0) su (2) (cid:110) C (cid:1) ∗ to se (3) ∗ Just like we identified (cid:0) su (2) (cid:110) C (cid:1) ∗ with su (2) × C , we also identify se (3) ∗ with so (3) × R viathe inner product (A.2) on so (3) and the dot product on R .Under this identification, let us define (cid:36) : (cid:0) su (2) (cid:110) C (cid:1) ∗ → se (3) ∗ , ( µ, α ) (cid:55)→ (ˆ µ, (cid:36) ( α )) (3.1a)with (cid:36) : C → R , α (cid:55)→ (cid:0) α α ) , α α ) , | α | − | α | (cid:1) . (3.1b)The first part of the map is the well-known identification of su (2) with so (3) summarized inAppendix A below. The second part of the map is also well known in the context of the Hopffibration. Particularly, (cid:36) maps the three-sphere with radius √ R centered at the origin in C ,i.e., S √ R := (cid:8) α ∈ C | (cid:107) α (cid:107) = √ R (cid:9) , to the two-sphere with radius R centered at the origin in R .Now let us define M : T ∗ C → se (3) ∗ , M := (cid:36) ◦ L , (3.2)i.e., so that the diagram T ∗ C ( su (2) (cid:110) C ) ∗ se (3) ∗ LM (cid:36) commutes. T. Ohsawa (cid:36) is a Poisson map Let us equip se (3) ∗ with the ( − )-Lie–Poisson bracket as well. For any smooth f, h : se (3) ∗ → R ,we define { f, h } se (3) ∗ ( ˆΠ , Γ ) := − (cid:28) ˆΠ , (cid:20) δfδ ˆΠ , δhδ ˆΠ (cid:21)(cid:29) so (3) − Γ · (cid:18) δfδ ˆΠ ∂h∂ Γ − δhδ ˆΠ ∂f∂ Γ (cid:19) , (3.3)where δf /δ ˆΠ ∈ so (3) (evaluated at ( ˆΠ , Γ )) is defined in a similar manner as in (2.5). With theidentification so (3) ∼ = R , this Poisson bracket is nothing but the heavy top bracket (1.2). Thenwe have the following: Proposition 3.1.
The map (cid:36) : (cid:0) su (2) (cid:110) C (cid:1) ∗ → se (3) ∗ is Poisson with respect to the ( − ) -Lie–Poisson brackets (2.4) and (3.3) , i.e., for any smooth f, h : se (3) ∗ → R , { f ◦ (cid:36), h ◦ (cid:36) } ( su (2) (cid:110)C ) ∗ = { f, h } se (3) ∗ ◦ (cid:36). Proof .
See Appendix B. (cid:4)
Corollary 3.2.
The map M := (cid:36) ◦ L : T ∗ C → se (3) ∗ is Poisson with respect to the Poissonbrackets (2.2) and (3.3) , i.e., for any smooth f, h : se (3) ∗ → R , { f ◦ M , h ◦ M } T ∗ C = { f, h } se (3) ∗ ◦ M . Concrete expressions of the map M defined above in (3.2) are the following M ( χ, ψ ) = − Im (cid:0) χ ¯ ψ + χ ¯ ψ (cid:1) Re (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1) Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1) , ψ ψ )2 Im( ¯ ψ ψ ) | ψ | − | ψ | = q p − q p − q p + q p q p − q p − q p + q p q p − q p − q p + q p , p p − p p ) − p p + p p ) p + p − p − p . (4.1)Now define a Hamiltonian H : T ∗ C ∼ = T ∗ R → R as H := h ◦ M , where h : se (3) ∗ → R is the heavy top Hamiltonian defined in (1.3). Then we have H ( χ, ψ ) = 18 (cid:32) (cid:0) Im (cid:0) χ ¯ ψ + χ ¯ ψ (cid:1)(cid:1) I + (cid:0) Re (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1)(cid:1) I + (cid:0) Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1)(cid:1) I (cid:33) + mgl (cid:0) ψ ψ ) c + 2 Im( ¯ ψ ψ ) c + ( | ψ | − | ψ | ) c (cid:1) . (4.2)Define a Hamiltonian vector X H ∈ X ( T ∗ C ) by setting i X H Ω = d H, (4.3)or equivalently, using the Poisson bracket (2.2),˙ χ = { χ, H } T ∗ C , ˙ ψ = { ψ, H } T ∗ C , ollective Heavy Top Dynamics 9which yield˙ χ = 2 ∂H∂ ¯ ψ , ˙ ψ = − ∂H∂ ¯ χ . (4.4a)In terms of the real coordinate ( q, p ) for T ∗ R , the symplectic form Ω is canonical, and so wehave the canonical Hamiltonian system˙ q = ∂H∂p , ˙ p = − ∂H∂q . (4.4b)We are now ready to state our main result: Theorem 4.1 (collective heavy top dynamics) . ( i ) For any ( Π , Γ ) ∈ R × (cid:0) R \{ } (cid:1) , there exists a corresponding element ( χ, ψ ) ∈ T ∗ C suchthat M ( χ, ψ ) = ( Π , Γ ) . ( ii ) Let I ⊂ R be a time interval, and let Φ : I × T ∗ C → T ∗ C and ϕ : I × se (3) ∗ → se (3) ∗ bethe flows of the canonical Hamiltonian system (4.4) and of the heavy top equations (1.1) ,respectively. Then, for any t ∈ I , ϕ t ◦ M = M ◦ Φ t . Proof . (i) From Proposition 2.2, we have su (2) × (cid:0) C \{ } (cid:1) ⊂ L (cid:0) T ∗ C (cid:1) , and so (cid:36) (cid:0) su (2) × (cid:0) C \{ } (cid:1)(cid:1) ⊂ M (cid:0) T ∗ C (cid:1) . However, from the definition of (cid:36) in (3.1), we have (cid:36) (cid:0) su (2) × (cid:0) C \{ } (cid:1)(cid:1) = so (3) × (cid:0) R \{ } (cid:1) , because the second part (cid:36) maps S √ R ⊂ C to S R ⊂ R for any R >
0; note also thatwe identified so (3) ∗ with so (3) here. Therefore, the assertion follows upon the identificationof so (3) with R .(ii) It follows easily from Corollary 3.2: The fact that M is Poisson implies that the map M pushes the flow Φ to ϕ (see, e.g., [11, Proposition 10.3.2]). (cid:4) Remark 4.2.
The first assertion is the result alluded in Remark 2.3. It shows that any solutionof the heavy top equations (1.1) can be realized as the image by M of a corresponding solutionof the canonical Hamiltonian system (4.4). Suppose that F : T ∗ C → R is a conserved quantity of the (canonical) Hamiltonian system (4.3)or (4.4), and that F is collective, i.e., there exists f : se (3) ∗ → R such that f ◦ M = F . Thenone easily sees that f is a conserved quantity of the heavy top dynamics.There are two conserved quantities of the Hamiltonian system (4.3) associated with its sym-metries. For the first one, consider the following R -action on T ∗ C : R × T ∗ C → T ∗ C , ( b, ( χ, ψ )) (cid:55)→ ( χ + bψ, ψ ) . We easily see that the alternative one-form Θ defined in (2.3) is invariant under this action,and hence so is Ω, i.e., this action is symplectic. It is also easy to see that the Hamiltonian (4.2)0 T. Ohsawais invariant under this action as well, i.e., H ( χ + bψ, ψ ) = H ( χ, ψ ) for any b ∈ R . Let v ∈ R bean the element of the Lie algebra T R ∼ = R of R . Its infinitesimal generator is then v T ∗ C ( χ, ψ ) = v (cid:18) ψ i ∂∂χ i + ¯ ψ i ∂∂ ¯ χ i (cid:19) = v p j ∂∂q j . Let J : T ∗ C → R be the associated momentum map. Then, it satisfies J ( χ, ψ ) · v = Θ ( χ, ψ ) · v T ∗ C ( χ, ψ ) = 12 Re (cid:0) v (cid:107) ψ (cid:107) (cid:1) = 12 (cid:107) ψ (cid:107) · v. Hence we obtain J ( χ, ψ ) = (cid:107) ψ (cid:107) /
2. Therefore, F : T ∗ C → R , F ( χ, ψ ) := 4 J ( χ, ψ ) = (cid:107) ψ (cid:107) is a conserved quantity of the Hamiltonian system (4.3) by Noether’s theorem; see, e.g., [11,Theorem 11.4.1, p. 372]. One easily sees that f ◦ M = F with f : se (3) ∗ → R , f ( Π , Γ ) := (cid:107) Γ (cid:107) . In fact, this is a well-known Casimir of the heavy top bracket (1.2) or (3.3).For the second one, consider the following SO (2) ∼ = S -action on C : S × C → C , (cid:0) e i θ , χ (cid:1) (cid:55)→ e i θ χ. Its cotangent lift is S × T ∗ C → T ∗ C , (cid:0) e i θ , ( χ, ψ ) (cid:1) (cid:55)→ (cid:0) e i θ χ, e i θ ψ (cid:1) , and the Hamiltonian (4.2) is invariant under this action, i.e., H (cid:0) e i θ χ, e i θ ψ (cid:1) = H ( χ, ψ ) for anye i θ ∈ S . Its associated momentum map is J : T ∗ C → so (2) ∗ ∼ = R , J ( χ, ψ ) := − Im( ψ ∗ χ ) , and again by Noether’s theorem this is a conserved quantity of the Hamiltonian system (4.3).However, because (cid:107) ψ (cid:107) is conserved, we may define an alternative conserved quantity F : T ∗ C → so (2) ∗ ∼ = R , F ( χ, ψ ) := − (cid:107) ψ (cid:107) ψ ∗ χ ) . Then we see that f ◦ M = F with f : se (3) ∗ → R , f ( Π , Γ ) := Π · Γ . This is the other well-known Casimir of the heavy top bracket (1.2) or (3.3).
Now consider the Lagrange top, i.e., I = I and c = (0 , , (cid:0) Im (cid:0) χ ¯ ψ + χ ¯ ψ (cid:1)(cid:1) + (cid:0) Re (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1)(cid:1) = (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) + (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) − (cid:0) Re (cid:0) χ ¯ ψ (cid:1) Re (cid:0) χ ¯ ψ (cid:1) − Im (cid:0) χ ¯ ψ (cid:1) Im (cid:0) χ ¯ ψ (cid:1)(cid:1) = (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) + (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) − (cid:0) χ ¯ ψ χ ¯ ψ (cid:1) , ollective Heavy Top Dynamics 11and so the Hamiltonian (4.2) becomes H L ( χ, ψ ) := 18 (cid:32) (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) + (cid:12)(cid:12) χ ¯ ψ (cid:12)(cid:12) − (cid:0) χ ¯ ψ χ ¯ ψ (cid:1) I + (cid:0) Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1)(cid:1) I (cid:33) + mgl (cid:0) | ψ | − | ψ | (cid:1) . (4.5)Consider the following SO (2) ∼ = S -action on C : S × C → C , (cid:18) e i θ , (cid:20) χ χ (cid:21)(cid:19) (cid:55)→ (cid:20) e i θ χ e − i θ χ (cid:21) = (cid:20) e i θ
00 e − i θ (cid:21) χ. Its cotangent lift is S × T ∗ C → T ∗ C , (cid:0) e i θ , ( χ, ψ ) (cid:1) (cid:55)→ (cid:18)(cid:20) e i θ
00 e − i θ (cid:21) χ, (cid:20) e i θ
00 e − i θ (cid:21) ψ (cid:19) , and we see that the Hamiltonian (4.5) is invariant under this action.What is the associated momentum map J : T ∗ C → so (2) ∗ ∼ = R ? Let any ω ∈ so (2) ∼ = R bearbitrary. Its infinitesimal generator is then ω C ( χ ) = dd s (cid:20) e i sω
00 e − i sω (cid:21) χ (cid:12)(cid:12)(cid:12)(cid:12) s =0 = i ω (cid:20) − (cid:21) χ. Then we have J ( χ, ψ ) · ω = Θ ( χ, ψ ) · ω C ( χ ) = Re (cid:18) ψ ∗ i ω (cid:20) − (cid:21) χ (cid:19) = − Im (cid:18) ψ ∗ (cid:20) − (cid:21) χ (cid:19) · ω = Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1) · ω. Therefore, we obtain J ( χ, ψ ) = Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1) , and hence F : T ∗ C → R , F ( χ, ψ ) := 12 J ( χ, ψ ) = 12 Im (cid:0) χ ¯ ψ − χ ¯ ψ (cid:1) is a conserved quantity of (4.3). Then we see that f ◦ M = F with f : se (3) ∗ → R , f ( Π , Γ ) := Π . Again, f is a well-known conserved quantity of the Lagrange top. Remark 4.3.
It is well known that h (with I = I and c = (0 , , f , f , and f are ininvolution with respect to the heavy top bracket (1.2). Since M : T ∗ C → se (3) ∗ is Poisson,this implies that H L , F , F , and F are in involution with respect to the canonical Poissonbracket (2.2) as well. The collective formulation suggests that any symplectic integrator for the canonical Hamiltoniansystem (4.4) on T ∗ C ∼ = T ∗ R gives rise to a Lie–Poisson integrator for the heavy top dynamics2 T. Ohsawavia the map M . Specifically, let ∆ t be the time step, and Φ d∆ t : T ∗ C → T ∗ C be the discreteflow defined by the symplectic integrator. Then, φ d∆ t := M ◦ Φ d∆ t : se (3) ∗ → se (3) ∗ is clearlyPoisson. This is the basic idea of the collective Lie–Poisson integrator of [14]; it is implementedfor the free rigid body in [15].In our case, the symplecticity of the integrator implies that the momentum maps J and J from Section 4.2 are conserved exactly along the flow Φ d∆ t . This implies that the correspondingCasimirs f and f are exactly conserved along the flow φ d∆ t of the collective integrator aswell. Furthermore, since the Hamiltonian H is nearly conserved without drifts with symplecticintegrators, the Hamiltonian h for the heavy top behaves in a similar manner as well.We note in passing that [16] showed that the spherical midpoint method [17] on (cid:0) S (cid:1) n isa collective integrator corresponding to the midpoint rule applied to a canonical Hamiltoniansystem on T ∗ R n . That is, for this special case, the collective integrator gives rise to an integratorintrinsically defined on the symplectic leaves of a Poisson manifold. It is an interesting futurework to look into an extension of this property to our setting. In our case, each non-trivialcoadjoint orbit (symplectic leaf) in se (3) ∗ is known to be diffeomorphic to either S or itstangent bundle [11, Section 14.7]. As a test case, we consider the Kovalevskaya top [9] – the case with I = I = 2 I and c = (1 , ,
0) – with the parameters m = g = l = I = 1 and initial condition Π (0) = (2 , , Γ (0) = (cid:0) / , , √ / (cid:1) . By solving M ( χ (0) , ψ (0)) = ( Π (0) , Γ (0)) with the constraintRe( χ (0)) = 1, we have χ (0) = (cid:32) − i (cid:0) √ √ (cid:1) , − √ √ − i 2 √
61 + √ (cid:33) , ψ (0) = 12 √ (cid:0) √ , √ − (cid:1) as a corresponding initial condition for (4.3) or (4.4).The system has four conserved quantities: the Hamiltonian h , the Casimirs f and f , andthe Kovalevskaya invariant K ( Π , Γ ) := (cid:12)(cid:12) (Π + iΠ ) − mglI (Γ + iΓ ) (cid:12)(cid:12) , and is known to be integrable [9]; see also [1]. Our focus here is to compare the behaviors ofthese four conserved quantities for several heavy top integrators.Fig. 1 shows the time evolutions of the four conserved quantities along the numerical solutionsobtained by applying the explicit and implicit midpoint rules to the canonical Hamiltoniansystem (4.3) as well as that obtained by applying the explicit midpoint rule directly to the heavytop equations (1.1). The time step ∆ t is 1 /
50 in all the cases. We note that the implicit midpointrule is a symplectic integrator for canonical Hamiltonian systems (see, e.g., [6, Theorem VI.3.5]and [10, Section 4.1]). So the implicit midpoint rule applied to (4.3) gives a collective Lie–Poissonintegrator for the heavy top equations (1.1) via the map M , and is our main focus here.The explicit midpoint rule applied to either (1.1) or (4.3) suffers from fairly significant driftsand/or oscillations in all the four conserved quantities. On the other hand, the collective Lie–Poisson integrator conserves the Casimirs f and f exactly because it is a symplectic integratorfor canonical Hamiltonian systems; note that we identified these Casimirs as Noether conservedquantities, i.e., the momentum maps associated with the symmetries of the canonical Hamil-tonian system (4.3). One can observe this in Fig. 1(b) and (c) as well. Furthermore, as iswell known (see, e.g., [6, Section IX.8] and [10, Section 5.2.2]), symplectic integrators exhibitnear-conservation of the Hamiltonian, and one observes this in Fig. 1(a).Interestingly, the collective Lie–Poisson integrator exhibits a similar near-conservation of theKovalevskaya invariant as well, performing better than the non-symplectic ones; see Fig. 1(d).ollective Heavy Top Dynamics 13 �� �� �� �� ���������������������������� ���������� ( �������� �������� ) ����� ��� ��� ( �������� �������� ) ���������� ( �������� �������� ) (a) Hamiltonian h �� �� �� �� ���������������������������� (b) Casimir f �� �� �� �� ��������������������������������� (c) Casimir f �� �� �� �� ��� - ������������������������������ (d) Kovalevskaya invariant K Figure 1.
Behaviors of the Hamiltonian and the Casimirs of the Kovalevskaya top with m = g = l = 1, I = 2, I = 2, I = 1, c = (1 , ,
0) for three integrators. Blue: explicit midpoint rule (non-symplectic)applied to (4.4). Green: explicit midpoint rule (non-Poisson) applied to (1.1). Red: implicit midpointrule (symplectic) applied to (4.4). ∆ t = 1 / Π (0) = (2 , , Γ (0) = (cid:0) / , , √ / (cid:1) . It is not clear if the Kovalevskaya invariant K corresponds to any symmetry of the canonicalHamiltonian system (4.3), but the oscillation in K with amplitudes way beyond the machineprecision suggests that K is not a Noether conserved quantity of (4.3).We also applied the implicit midpoint rule directly to the heavy top equations (1.1) andcompared the results with those of the collective Lie–Poisson integrator as well. The formerintegrator is known to conserve the Hamiltonian h and the Casimirs f and f exactly but is not Poisson; see [2]. Fig. 2 shows the behaviors of h and K for the implicit midpoint rule appliedto (1.1) or (4.3).As mentioned above, the former conserves h exactly, whereas the collective Lie–Poisson in-tegrator exhibits a near-conservation of it with oscillations of very small amplitudes. On theother hand, one observes near-conservation of K in both integrators, but the non-Poisson inte-grator exhibits an oscillation with a larger amplitude than the collective Lie–Poisson integratordoes.4 T. Ohsawa ����� ��� ��� ( �������� �������� ) ���������� ( �������� �������� ) �� �� �� �� ��� - ������� - ����������������������������������� (a) Hamiltonian h �� �� �� �� ��������������������� (b) Kovalevskaya invariant K Figure 2.
Behaviors of the Hamiltonian h and the Kovalevskaya invariant K . Gray: implicit midpointrule (non-Poisson) applied to (1.1); see [2]. Red: implicit midpoint rule (symplectic) applied to (4.4).The parameters and the time step are the same as in Fig. 1. A Identification of su (2) with so (3) and R This section gives a brief description of the well-known identification of su (2) with so (3) and R .The purpose is to introduce the notation used throughout the paper. A.1 Isomorphisms between su (2), so (3), and R Let { e i } i =1 be the standard basis for R , and define a basis { e i } i =1 for su (2) as e := − i2 (cid:20) (cid:21) , e := − i2 (cid:20) − ii 0 (cid:21) , e := − i2 (cid:20) − (cid:21) , as well as a basis { ˆ e i } i =1 for so (3) asˆ e := −
10 1 0 , ˆ e := − , ˆ e := − . So we have the isomorphisms su (2) ←→ so (3) ←→ R defined by e i ←→ ˆ e i ←→ e i . It is well known that these are Lie algebra isomorphisms in the sense that [ e i , e j ], [ˆ e i , ˆ e j ], and e i × e j with i, j ∈ { , , } are all the same under the identification.As a result, we may identify su (2) with R via the map R → su (2) , ξ = ( ξ , ξ , ξ ) (cid:55)→ ξ := (cid:88) j =1 ξ j e j = − i2 (cid:20) ξ ξ − i ξ ξ + i ξ − ξ (cid:21) , and also so (3) with R via the following “hat map”ˆ( · ) : R → so (3) , ξ = ( ξ , ξ , ξ ) (cid:55)→ ˆ ξ := (cid:88) j =1 ξ j ˆ e j = − ξ ξ ξ − ξ − ξ ξ . ollective Heavy Top Dynamics 15 A.2 Inner products
Let us define inner products on su (2) and so (3) as follows. For any ξ , η ∈ R , using the notationfrom above, (cid:104) ξ, η (cid:105) su (2) := 2 tr( ξ ∗ η ) (A.1)as well as (cid:10) ˆ ξ, ˆ η (cid:11) so (3) := 12 tr (cid:0) ˆ ξ T ˆ η (cid:1) . (A.2)It is easy to check that these are compatible with the standard dot product in R under theabove identification, i.e., (cid:104) ξ, η (cid:105) su (2) = (cid:10) ˆ ξ, ˆ η (cid:11) so (3) = ξ · η . Using these inner products we may identify the duals su (2) ∗ and so (3) ∗ with su (2) and so (3)respectively, and in turn, with R as well. B Proof of Proposition 3.1
Let us first decompose the Poisson bracket (2.4) into two to define, for any smooth ˜ f , ˜ h : (cid:0) su (2) (cid:110)C (cid:1) ∗ → R , (cid:8) ˜ f , ˜ h (cid:9) su (2) ∗ ( µ, α ) := − (cid:42) µ, (cid:34) δ ˜ fδµ , δ ˜ hδµ (cid:35)(cid:43) su (2) , (cid:8) ˜ f , ˜ h (cid:9) ( C ) ∗ ( µ, α ) := − (cid:42) α, δ ˜ fδµ δ ˜ hδα − δ ˜ hδµ δ ˜ fδα (cid:43) C . Likewise, we decompose (3.3) to define, for any smooth f, h : se (3) ∗ → R , { f, h } so (3) ∗ := − (cid:28) ˆΠ , (cid:20) δfδ ˆΠ , δhδ ˆΠ (cid:21)(cid:29) so (3) , { f, h } ( R ) ∗ := − Γ · (cid:18) δfδ ˆΠ ∂h∂ Γ − δhδ ˆΠ ∂f∂ Γ (cid:19) . Our goal is to show that { f ◦ (cid:36), h ◦ (cid:36) } su (2) ∗ = { f, h } so (3) ∗ ◦ (cid:36) (B.1)as well as { f ◦ (cid:36), h ◦ (cid:36) } ( C ) ∗ = { f, h } ( R ) ∗ ◦ (cid:36). (B.2)Let us first show (B.1). Let ( µ, α ) ∈ ( su (2) (cid:110) C ) ∗ and δµ ∈ su (2) ∗ be arbitrary. Then,setting a := (cid:36) ( α ), we have (cid:28) δµ, δ ( f ◦ (cid:36) ) δµ (cid:29) su (2) = dd s f ◦ (cid:36) ( µ + sδµ, α ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = dd s f (cid:16) ˆ µ + s (cid:99) δµ, a (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) s =0 = (cid:28)(cid:99) δµ, δfδ ˆ µ (cid:29) so (3) . δ ( f ◦ (cid:36) ) /δµ ∈ su (2) ∗ and δf /δ ˆ µ ∈ so (3) ∗ ; the same goeswith the derivatives of h . The compatibility of the inner products in su (2) and so (3) (seeAppendix A.2) then yields { f ◦ (cid:36), h ◦ (cid:36) } su (2) ∗ ( µ, α ) = − (cid:28) µ, (cid:20) δ ( f ◦ (cid:36) ) δµ , δ ( h ◦ (cid:36) ) δµ (cid:21)(cid:29) su (2) = − (cid:28) ˆ µ, (cid:20) δfδ ˆ µ , δhδ ˆ µ (cid:21)(cid:29) so (3) = { f, h } so (3) ∗ ◦ (cid:36) ( µ, α ) . So it remains to show (B.2). Writing ξ := δ ( f ◦ (cid:36) ) /δµ , η := δ ( h ◦ (cid:36) ) /δµ , β := δ ( f ◦ (cid:36) ) /δα ,and γ := δ ( h ◦ (cid:36) ) /δα for short, { f ◦ (cid:36), h ◦ (cid:36) } ( C ) ∗ ( µ, α ) = − Re ( α ∗ ( ξγ − ηβ )) = −
12 ( α ∗ ξγ + γ ∗ ξ ∗ α − α ∗ ηβ − β ∗ η ∗ α )= −
12 tr (( γα ∗ − αγ ∗ ) ξ ) + 12 tr (( βα ∗ − αβ ∗ ) η )= − (cid:28)
14 ( αγ ∗ − γα ∗ ) , ξ (cid:29) su (2) + (cid:28)
14 ( αβ ∗ − βα ∗ ) , η (cid:29) su (2) . Let us find an expression for β = δ ( f ◦ (cid:36) ) /δα ∈ C . For any δα ∈ (cid:0) C (cid:1) ∗ , (cid:28) δα, δ ( f ◦ (cid:36) ) δα (cid:29) C = dd s f ◦ (cid:36) ( µ, α + sδα ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = dd s f (ˆ µ, (cid:36) ( α + sδα )) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ∇ a f (ˆ µ, a ) · T α (cid:36) ( δα ) , where ∇ a stands for the gradient with respect to a . However, using the expression (3.1b) for (cid:36) ,we obtain the tangent map T α (cid:36) as follows T α (cid:36) ( δα ) = 2 Re (cid:0) δα α (cid:1) + Re ( α δα )Im (cid:0) δα α (cid:1) − Im ( α δα )Re (cid:0) α δα (cid:1) − Re (cid:0) α δα (cid:1) = 2 Re (cid:0) α δα (cid:1) + Re (cid:0) α δα (cid:1) − Re (cid:0) i α δα (cid:1) + Re (cid:0) i α δα (cid:1) Re (cid:0) α δα (cid:1) − Re (cid:0) α δα (cid:1) . Therefore, (cid:28) δα, δ ( f ◦ (cid:36) ) δα (cid:29) C = 2 Re (cid:18) δα (cid:18) α ∂f∂a − i α ∂f∂a + α ∂f∂a (cid:19)(cid:19) + 2 Re (cid:18) δα (cid:18) α ∂f∂a + i α ∂f∂a − α ∂f∂a (cid:19)(cid:19) , and so we have δ ( f ◦ (cid:36) ) δα = 2 α ∂f∂a − i α ∂f∂a + α ∂f∂a α ∂f∂a + i α ∂f∂a − α ∂f∂a . Then a tedious but straightforward calculation yields β = 14 ( αβ ∗ − βα ∗ ) = (cid:18) a ∂f∂a − a ∂f∂a , a ∂f∂a − a ∂f∂a , a ∂f∂a − a ∂f∂a (cid:19) = ∇ a f × a under the identification su (2) ∗ ∼ = R . Similarly,14 ( αγ ∗ − γα ∗ ) = ∇ a h × a . ollective Heavy Top Dynamics 17Therefore, we obtain { f ◦ (cid:36), h ◦ (cid:36) } ( C ) ∗ ( µ, α ) = − ( ∇ a h × a ) · ξ + ( ∇ a f × a ) · η = − a · ( ξ × ∇ a h − η × ∇ a f )= − a · (cid:0) ˆ ξ ∇ a h − ˆ η ∇ a f (cid:1) = − a · (cid:18) δfδ ˆ µ δhδ a − δhδ ˆ µ δfδ a (cid:19) = { f, h } ( R ) ∗ ◦ (cid:36) ( µ, α ) . Acknowledgments
I would like to thank the referees for their helpful comments and suggestions. This work waspartially supported by NSF grant CMMI-1824798.
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