Semi-global symplectic invariants of the Euler top
SSEMI-GLOBAL SYMPLECTIC INVARIANTS OF THE EULERTOP
George Papadopoulos and Holger R. Dullin
School of Mathematics and StatisticsThe University of SydneySydney, NSW 2006, Australia
Abstract.
We compute the semi-global symplectic invariants near the hy-perbolic equilibrium points of the Euler top. The Birkhoff normal form at thehyperbolic point is computed using Lie series. The actions near the hyper-bolic point are found using Frobenius expansion of its Picard-Fuchs equation.We show that the Birkhoff normal form can also be found by inverting theregular solution of the Picard-Fuchs equation. Composition of the singularaction integral with the Birkhoff normal form gives the semi-global symplecticinvariant. Finally, we discuss the convergence of these invariants and showthat in a neighbourhood of the separatrix the pendulum is not symplecticallyequivalent to any Euler top. Introduction
The Euler top is a classical Hamiltonian system which describes the rotationalmotion of a rigid body in free space in the absence of a potential. In the centre ofmass frame this can be considered as the rotation of the rigid body about a fixedpoint. This system has been treated extensively in the literature for the last 300years, so we only give a few selected references here [13, 14, 6]. After reduction bythe SO (3) symmetry the Hamiltonian ˜ H can be written in terms of the moment ofinertia tensor M := diag (Θ , Θ , Θ ) and the angular momentum L := ( L , L , L ) in the body frame as(1) ˜ H ( L ) = 12 L T M − L = 12 (cid:88) i =1 L i Θ i . Without loss of generality, we assume the ordering < Θ < Θ < Θ . Theprincipal moments of inertia satisfy triangle inequalities Θ ≤ Θ + Θ and so forthcyclically. We exclude the degenerate cases Θ = Θ or Θ = Θ since then there isno hyperbolic equilibrium. This way of writing the Hamiltonian is not symplectic asthe angular momenta are not canonical variables. Instead the system has a Poissonstructure with Poisson structure matrix P = − L L L − L − L L Mathematics Subject Classification.
Primary: 37J35; Secondary: 37J15, 70H06, 70E40.
Key words and phrases.
Euler top, Picard-Fuchs equation, semiglobal symplectic invariants,Birkhoff normal form, Liouville integrable.HRD was supported in part by ARC grant DP110102001. a r X i v : . [ m a t h . S G ] J un YMPLECTIC INVARIANTS OF THE EULER TOP 2 so that the differential equation is written as ˙ L = P ∇ ˜ H = L × ∇ ˜ H . Due to conservation of angular momentum, the dynamics of the Euler top takesplace on spheres of constant magnitude of angular momentum (cid:107) L (cid:107) = (cid:96) . In factthe total angular momentum is a Casimir C ( L ) := (cid:107) L (cid:107) of the Poisson structure, P ∇ C = 0 . The level set of the Casimir C (cid:96) := (cid:8) L ∈ R : C ( L ) = (cid:96) (cid:9) is a sphere and the level set of the Hamiltonian, the energy surface E ˜ h := (cid:110) L ∈ R : ˜ H ( L ) = ˜ h (cid:111) is an ellipsoid. The solution curves are given by the intersection C (cid:96) ∩ E ˜ h . The (non-degenerate) intersections of the sphere with the ellipsoid give concentric ellipse-likecurves, centred about elliptic equilibria of which there are four. There are twoseparatrices, the intersections of which occur at the two hyperbolic equilibria. Intotal there are six equilibria given by ± (cid:96) ˆ L i , i ∈ { , , } where ˆ L i is defined asthe unit vector along the L i axis. We are interested in the unstable hyperbolicequilibria ( i = 2 ), corresponding to a steady rotation about the principal axis ofinertia corresponding to the middle moment of inertia Θ . The eigenvalues of thelinearisation about the hyperbolic equilibria are ± λ , where(2) λ = (cid:96) Θ (cid:115) (Θ − Θ ) (Θ − Θ )Θ Θ . In this work the main aim is to calculate the semi-global symplectic invariantsof the Euler top near the hyperbolic equilibrium point. In [11, 18], Dufour, Molinoand Toulet introduce the classification of integrable systems using their semi-globalsymplectic invariants. Their approach considers the triple ( M , ω, F ) , where M is atwo-dimensional manifold, ω is a symplectic 2-form, and F is a Morse foliation givenby the levels of a Morse function F . Then the equivalence between two integrableHamiltonian dynamical systems with one degree of freedom is introduced in [11] bythe following definition: Definition 1.1.
Two triples ( M , ω , F ) and ( M , ω , F ) are said to be equival-ent if there exists a symplectomorphism between M and M that preserves thefoliation.In formulas in the analytic case this means there is a symplectic diffeomorphism Ψ : M → M and a diffeomorphism Ξ : R → R such that(3) F ◦ Ψ = Ξ ◦ F . They introduce the semi-global symplectic invariant as the regular part of the actionintegral measuring the symplectic area near a separatrix (see Figure 1), written ina particular canonical coordinates system defined using the Birkhoff normal format the hyperbolic equilibrium point. The main theorem in [11] then is
Theorem 1.2.
In a full neighbourhood of a separatrix of the same topological typetwo systems are equivalent if and only if their semi-global symplectic invariantscoincide.
YMPLECTIC INVARIANTS OF THE EULER TOP 3
This theorem motivates us to calculate the semi-global symplectic invariant ofthe Euler top, the first step that may lead to a non-trivial equivalence between theEuler top and other Hamiltonian systems with one degree of freedom. We can,for example, ask whether a particular Euler top (it is a three parameter family) isequivalent to the pendulum, for which the invariants have been calculated in [7].In the last section we will show that the answer to this question is no.The theory of semiglobal symplectic invariants was further developed by San VuNgoc in [15] who extended it to focus-focus equilibria for system of two degrees offreedom. The first explicit computation of semi-global invariants near a focus-focuspoint was done by Dullin in [7] for the spherical pendulum. Hyperbolic-Hyperbolicpoints were treated in [8] where the C. Neumann system was used to illustrate thegeneral theory. Other notions of equivalence have been used to study the Euler topin [3, 1, 16, 2, 9, 14].
Remark 1.
It should be noted that for C ∞ systems the relation (3) is not true assuch, but may need to be formulated separately in each sector defined by the sep-aratrix. Furthermore, in general the symplectic invariant which is the Taylor seriesof the regular part of the action integral is not necessarily convergent. Howeverwe need not worry about these two issues, since both the Hamiltonian and Poissonstructure of the Euler top are analytic.Plan of the paper: first we will be calculating the Birkhoff normal form algorith-mically using Lie series. For one degree of freedom, the Birkhoff normal form abouta hyperbolic equilibrium is a series in powers of the regular action only. Next weshow that the action integrals satisfy a Picard-Fuchs ODE. Then we solve this ODEusing the method of Frobenius, and arrive at series expansions for the action in-tegrals. We find that if we invert the regular action integral then we recover theBirkhoff normal form. Once we have the Birkhoff normal form, we calculate thesymplectic invariant by composing the singular action integral with the Birkhoffnormal form, and extracting the regular part. Finally we discuss convergence andnon-equivalence to the pendulum.2. Calculating the Birkhoff normal form
In order to calculate the Birkhoff normal form of the Hamiltonian of the Eulertop at the unstable equilibrium, we introduce local canonical variables ( q, ˜ p ) withsymplectic structure S := (cid:18) − (cid:19) : Lemma 2.1.
The Poisson map Φ : R → R defined by q = Arg( L + i L ) (4a) ˜ p = L (4b) maps the Hamiltonian ˜ H ( L ) of (1) with value ˜ h and Poisson structure P on thesymplectic leaf C (cid:96) into the standard symplectic structure S in canonical variables ( q, ˜ p ) and Hamiltonian ˜ H ( q, ˜ p ) with value ˜ h given by (5) ˜ H ( q, ˜ p ) = 12 (cid:2) ˜ p (cid:0) Θ − − f ( q ) (cid:1) − (cid:96) (cid:0) Θ − − f ( q ) (cid:1)(cid:3) where f ( q ) := Θ − sin q + Θ − cos q . YMPLECTIC INVARIANTS OF THE EULER TOP 4
Proof.
Substituting equations (4) into (5) and using (cid:96) = L + L + L yields therequired Hamiltonian (1) up to the constant (cid:96) Θ − , so that ˜ H ( Φ ( L )) + (cid:96) Θ − =˜ H ( L ) . The hyperbolic equilibrium (cid:96) ˆ L is mapped to the origin ( q, ˜ p ) = (0 , , while − (cid:96) ˆ L is mapped to ( q, ˜ p ) = ( π, . To derive the new symplectic structure, computethe × Jacobian matrix D Φ and verify that D Φ P ( D Φ ) T = S . (cid:3) Note that this transformation to “cylindrical coordinates for the sphere” is notdefined globally on sphere, but only on the punctured sphere with the two points ± (cid:96) ˆ L (where L = L = 0 ) removed. However, the transformation is valid nearthe unstable equilibria ± (cid:96) ˆ L and in a full neighbourhood of their separatrix. Analternative transformation introduces L as momentum instead of L , and the cor-responding Hamiltonian ˜ H ( q, ˜ p ) has Θ and Θ interchanged.At this stage it is convenient to define the dimensionless real parameters ρ := (cid:115) Θ (Θ − Θ )Θ (Θ − Θ ) (6a) κ := ρ − ρ − , (6b)which will be fundamental in the upcoming analyses. Note that if ρ → ρ − ⇔ κ → − κ are exchanged then Θ → Θ are exchanged. The involution ρ → − ρ − leaves κ invariant. Furthermore, κ = ρ − ρ − is rational in the momentsof inertia. When restricting to the physical range ρ > , making ρ the subjectin (6b) yields the unique injection ρ = ( κ + √ κ + 4 ) . We are then able to re-write our Hamiltonian (originally posed with three parameters) in terms of a singledimensionless parameter by performing the following non-dimensionalisation: Lemma 2.2.
Using λ as units of time, (cid:96) as units of angular momentum, and (cid:96)λ as units of moment of inertia, the Hamiltonian in non-dimensional form is (7) H ( q, p ) = 12 (cid:0) − p ( ρ + ρ − sin q ) + ρ − sin q (cid:1) . The proof is a simple calculation. The new scaled angular momentum p = ˜ p(cid:96) isdimensionless, as is the value of the Hamiltonian h = ˜ hλ(cid:96) . Remark 2.
Each time a transformation is done, the variables change. For clarityand simplicity of notation we use the same letters ( q, p ) for old and new variables,but it should be noted that each transformation introduces different variables. Inour notation the tilde designates quantities with dimensions, while from this pointonwards we use non-dimensionalized quantities H , q , h without tilde. Sans-serif fontis used to designate quantities in the original Poisson system, so that ˜ h − (cid:96) Θ − =˜ h = hλ(cid:96) . Remark 3.
The area form on the original sphere in these variables is (cid:96) d q ∧ d˜ p = (cid:96) d q ∧ d p so that the scaled symplectic area d q ∧ d p differs from the true area onthe Casimir sphere C (cid:96) by a factor of (cid:96) .The original Hamiltonian has a group of discrete symmetries generated by L i →− L i , i = 1 , , . In the canonical variables these correspond to q → − q , q → π − q , p → − p , respectively. The global analysis we are going to present later is simplestif there is only a single hyperbolic equilibrium on the separatrix in question, andtherefore we are going to consider the Euler top modulo its discrete symmetry group. YMPLECTIC INVARIANTS OF THE EULER TOP 5
Any two pairs of the three discrete symmetries generate the group of symplecticdiscrete symmetries of the Euler top, which is isomorphic to Klein’s Vierergruppe V = Z ⊗ Z . In the canonical variables a possible choice of generators is S ( q, p ) =( − q, − p ) and S ( q, p ) = ( π + q, p ) which are both involutions. A fundamental regionfor the quotient of the cylindrical ( q, p ) phase space [ − π, π ) × ( − , by the group V generated by S and S can be chosen as the positive quadrant ( q, p ) ∈ [0 , π ) × [0 , .This corresponds to a quarter of the original sphere V ( C (cid:96) ) := { L ∈ C (cid:96) : 0 < L < (cid:96), − (cid:96) < L < (cid:96), < L < (cid:96) } . We Taylor expand the non-dimensional Hamiltonian H ( q, p ) about the origin ( q, p ) = (0 , for analysis near the equilibrium L = (cid:96) ˆ L . The quadratic terms are ( ρ − q − ρp ) . The Williamson (linear) normal form of the hyperbolic equilibriais found after a symplectic linear transformation (e.g. as outlined in [3]). Althoughthe Williamson normal form is unique up to the overall sign of the qp term which wechose to be positive, the transformation is not; we chose to perform a symplecticscaling q → √ ρ q, p → p √ ρ followed by a rotation by − π , so that the positivequadrant in the new coordinates corresponds to positive Hamiltonian. With thisconvention the Williamson normal form becomes unique. Lemma 2.3.
The symplectic linear transformation (cid:18) qp (cid:19) (cid:55)→ (cid:18) √ ρ √ ρ (cid:19) √ (cid:18) − (cid:19) (cid:18) qp (cid:19) puts the quadratic part of the Hamiltonian H into Williamson’s normal form, andthe new Hamiltonian is H ∗ = qp − ρ ( q − p ) − ρ
24 ( q + p ) + O (6) In order to arrive at the Birkhoff normal form, we use the method of Lie trans-forms to remove terms that are not powers of qp . A recursive algorithm for thisis given in, e.g. [12]. The algorithm is implemented in Mathematica; we state themain results in the following Theorem: Theorem 2.4.
The Birkhoff normal form of the Euler top at the hyperbolic equi-librium point is given by H ∗ ( J ) = J − κ J − κ + 416 J − κ (cid:0) κ + 4 (cid:1) J − (cid:0) κ + 4 (cid:1) (cid:0) κ + 12 (cid:1) J − κ (cid:0) κ + 4 (cid:1) (cid:0) κ + 20 (cid:1) J (8) − (cid:0) κ + 4 (cid:1) (cid:0) κ + 1776 κ + 720 (cid:1) J + O (cid:0) J (cid:1) where J = qp in the new variables. Note that the parameter dependence on the right hand side is only throughthe dimensionless parameter κ = ρ − ρ − and the power series is in terms of thedimensionless action J . The normal form coefficients are displayed up to order 14in the canonical variables q and p . YMPLECTIC INVARIANTS OF THE EULER TOP 6
Casimir sphere Canonical phase plane
Figure 1.
Areas enclosed by the same closed orbits after discretesymmetry reduction (left) on the quarter Casimir sphere V ( C (cid:96) ) and (right) on the scaled and discrete symmetry reduced canonicalphase plane ( q, p ) ∈ [0 , π ) × [0 , . The parameter is κ = 0 . . Thethick black line is the separatrix with h = 0 . The lighter shade(blue) below the separatrix shows the action of an orbit with h > and the darker shade (red) above the separatrix shows the actionof an orbit with h < . The symplectic invariant is calculated fromthese areas in the singular limit h → .3. Action integrals via Picard-Fuchs equation
The essential step in the calculation of the semi-global symplectic invariants is thecomputation of the action integrals, which are given by complete elliptic integralsin the case of the Euler top. Since we are interested in the series expansions of theseintegrals the most natural approach is not through the integral itself, but insteadthrough the so called Picard-Fuchs ODE that the integral satisfies. The derivationof the Picard-Fuchs equation proceeds in a way similar to [10]. Frobenius expansionsof this linear ODE then gives the desired series. This gives a basis for the vectorspace of solutions of the linear ODE, and in a second step the particular solutionscorresponding to the action integrals of the Euler top are found.
Lemma 3.1.
The scaled and discrete symmetry reduced action of the Euler top withscaled energy h normalised to 0 at the unstable equilibrium is a complete ellipticintegral of the third kind on the curve Γ := { ( z, u ) ∈ C | u = (2 h − z ) w ( z ) , w ( z ) = z ( z + κz − } over the Abelian differential ζ given by I β ± ( h ) = 14 π ˛ β ± ζ ( h ) , ζ ( h ) := √ h − zw ( z ) d z along the cycles β ± for ± h > as specified in Fig. 2. YMPLECTIC INVARIANTS OF THE EULER TOP 7
Figure 2.
The α and β cycles, choices of branch cuts and branchpoints of Γ in the z ∈ C plane for h > (top) and h < (bottom). Proof.
We derive the action using the Hamiltonian H ( q, p ) in canonical variables ( q, p ) . Solve H ( q, p ) = h for p and integrating with respect to q in order to find thelighter (blue) area in Fig. 1 gives I β + ( h ) = 12 π ˆ q q (cid:115) − h − ρ − sin qρ + ρ − sin q d q where q = π/ − q and q = π/ q are the roots of the numerator of theintegrand with q = cos − ( √ hρ ) . Upon performing the change of integrationvariable z = ρ − sin ( q ) we arrive at the required integral over the differential 1-form ζ ( h ) as defined above. The new denominator [ w ( z )] = z (cid:0) z + κz − (cid:1) = z ( z + ρ )( z − ρ − ) depends only upon the single parameter κ = ρ − ρ − . Note thaton Γ we find that ζ = uw d z is meromorphic, so that we can equate the real integralover z ∈ [ q , q ] ⊂ R to an equivalent complex contour integral over z ∈ β + ⊂ C ,namely ˆ π − q q ζ = 12 ˛ β + ζ . which can be evaluated by continuously shrinking β + so that it sits entirely uponthe real axis. We find that β + gives two equal contributions as z traverses eachside of the branch cut. Then, q traversing the interval [ q , q ] once corresponds to z traversing the whole closed (shrunken) path β + once. The complex closed loopintegral around β + gives twice the real integral along the interval (cid:2) h, ρ − (cid:3) , and soaltogether I β + = π ¸ β + ζ .A slightly more complicated argument applies in the case h < . To get thelighter shaded (red) area in Figure 1 the integrand is (1 − p ) d q instead of just p d q and the real integration interval is q ∈ [0 , π ] . Complexifying this integral gives the YMPLECTIC INVARIANTS OF THE EULER TOP 8 loop β ∗ so that I β − = − π ¸ β ∗ ζ . On the Riemann sphere we have β ∗ + β + + γ ∞ = 0 where γ ∞ is a loop around the pole at infinity. Due to the non-zero residue of ζ atinfinity the constant term cancels and we get I β − = π ¸ β − ζ . (cid:3) Remark 4.
The unscaled action ˜ I as a function of the unscaled and unshiftedenergy ˜ h can be rewritten in the symmetric form ˜ I β ± (˜ h , (cid:96) ) = 12 π ˛ β ± (cid:115) − h − (cid:96) ˜ z (cid:0) ˜ z − Θ − (cid:1) (cid:0) ˜ z − Θ − (cid:1) (cid:0) ˜ z − Θ − (cid:1) d˜ z = 2 (cid:96)I β ± ( h ) . The scaled action I depends on ˜ h and (cid:96) only through h = ˜ h − Θ − (cid:96) λ(cid:96) , and on Θ , Θ , Θ only through κ . In the transformed variable the roots ˜ z =Θ − , Θ − , Θ − correspond to the roots z = − ρ, , ρ − of w , respectively.3.1. Derivation of the Picard-Fuchs equation.
We now derive the Picard-Fuchs ODE of I ( h ) . The Abelian differential ζ lives on the complex manifold Γ .By de Rham cohomology theory, there must exist a relationship between derivativesof ζ on Γ . In fact, we follow the route of Clemens in [5] and find that there exists alinear combination of the h derivatives of ζ that equals a total differential. However,we cannot exploit the same simplifications as Clemens does since our independentvariable is fixed to be the energy h , because otherwise we would lose the connectionto the Birkhoff normal form (see below). A similar approach was taken in [10]. Therelation between the differentials is given in the following Lemma: Lemma 3.2.
There exists a function v ( z ) meromorphic on Γ and coefficients c i such that (9) (cid:88) i =0 c i d i ζ d h i = d v. Proof.
Observe that w ( z )(2 h − z )
52 3 (cid:88) i =0 c i d i ζ d h i = (cid:2) − c z + (6 c h + c ) z + ( − c h − c h + c ) z +(8 c h + 4 c h − c h + 3 c ) (cid:3) d z . Now choose v ( z ) := u (2 h − z ) = w ( z )(2 h − z ) , which is meromorphic on Γ , and hasdifferential w ( z )(2 h − z ) d v = (cid:2) (3 h + κ ) z + (2 κh − z − h (cid:3) d z. Equating the coefficients of the polynomials in z and solving for c i yields the uniquesolution c = 0 c = 3 h + κc = 12 h + 4 κh − c = h (4 h + 2 hκ −
1) = 12 [ w (2 h )] YMPLECTIC INVARIANTS OF THE EULER TOP 9 and thus by construction we have proven the Lemma. (cid:3)
Now we are ready to derive the linear and homogeneous Picard-Fuchs equationfor the action I ( h ) : Theorem 3.3.
The scaled action I ( h ) satisfies the Picard-Fuchs equation (10) [ w (2 h )] I (cid:48)(cid:48)(cid:48) ( h ) + 2(12 h + 4 κh − I (cid:48)(cid:48) ( h ) + (6 h + κ ) I (cid:48) ( h ) = 0 with the scaled energy h as the independent variable.Proof. To obtain the Picard-Fuchs ODE, perform a closed complex contour integralto both sides of equation (9). By definition we have ¸ ζ = 4 πI , and given v ismeromorphic on Γ the residues of d v are vanishing, so that the right hand sidegives ¸ d v = 0 for any closed integration path. (cid:3) The fact that the Picard-Fuchs equation is of third order is related to the factthe curve is elliptic (genus g = 1 ) and that the differential ζ ( h ) is of third kind witha single pole. Moreover, the derivative of the residue of the pole with respect to h vanishes, so that the first h -derivative of ζ is a differential of 2nd kind on a genus g = 1 curve. Now w ( z ) is independent of h and further derivatives do not createadditional poles. So the order of the Picard-Fuchs equation is g + 1 = 3 .3.2. Solving the Picard-Fuchs equation.
Clearly (10) is an ODE in I (cid:48) thus I = k is a constant solution. To lower the order we introduce the scaled period T ( h ) = 2 πI (cid:48) ( h ) , which has the first kind differential d zu on Γ .The scaled period T ( h ) hence satisfies the second order linear homogeneous ODE(11) T (cid:48)(cid:48) ( h ) + 2 12 h + 4 κh − w (2 h )] T (cid:48) ( h ) + 6 h + κ [ w (2 h )] T ( h ) = 0 . It is interesting to observe that the leading coefficient c is proportional to [ w (2 h )] , and thus in normalising the ODE the roots of w (2 h ) given by h ∈ (cid:8) , − ρ, ρ − (cid:9) become the (regular) singular points of the Picard-Fuchs ODE. Thusthe partial fraction decomposition of the coefficient of T (cid:48) simply is (cid:18) h + 12 h + ρ + 12 h − ρ − (cid:19) . We are interested in series solutions at the singular point corresponding to theunstable equilibrium, namely h = 0 . The general theory and procedure for solving(11) via the method of Frobenius can, e.g., be found in [4]. We seek series solutionsof the form ∞ (cid:88) n =0 a n ( (cid:37) ) h n + (cid:37) where (cid:37) is a root of the indicial equation. At the finite singular points the indicialequation is (cid:37) = 0 . Remark 5.
We find that the Picard-Fuchs ODE has a regular singular point at h = ∞ , confirming it is of Fuchsian type. The indicial equation for the singularpoint at infinity is (cid:37) ∞ − (cid:37) ∞ + = 0 with roots (cid:37) ∞ = , differing by an integer. YMPLECTIC INVARIANTS OF THE EULER TOP 10
By substitution of the Frobenius series into the ODE, the recursion relation for a n ( (cid:37) ) is found to be(12) a n ( (cid:37) ) = 2 n + 2 (cid:37) − n + (cid:37) ) (cid:16) κ n + 2 (cid:37) − a n − ( (cid:37) ) + (2 n + 2 (cid:37) − a n − ( (cid:37) ) (cid:17) Setting (cid:37) = 0 and a n := a n (0) yields the Frobenius expansion of the regularsolution T r ( h ) = ∞ (cid:88) n =0 a n h n . with coefficients obtained from (12) at (cid:37) = 0 as(13) a n = 2 n − n (cid:16) κ n − a n − + (2 n − a n − (cid:17) . We solve this second order recursion relation for a n . Without loss of generality,normalise the initial condition a := 1 , and we require that a − := 0 . Thus we findthat the next few coefficients are a = κ ,a = 316 (cid:0) κ + 4 (cid:1) ,a = 532 κ (cid:0) κ + 12 (cid:1) ,a = 351024 (cid:0) κ + 120 κ + 48 (cid:1) ,a = 63 κ (cid:0) κ + 280 κ + 240 (cid:1) . Theorem 3.4.
The recursion for a n is solved by a n = 14 n (cid:18) nn (cid:19) (cid:98) n (cid:99) (cid:88) k =0 (cid:18) n − kk, n − k, n − k (cid:19) (cid:16) κ (cid:17) n − k where (cid:0) ni,j,k (cid:1) = n ! i ! j ! k ! with n = i + j + k is the trinomial coefficient. The proof of this theorem will be given later as a special case of the (approximate)solution of the more general recursion for a n ( (cid:37) ) . Remark 6.
It is interesting to note that the sum a n can be summed to the hyper-geometric function F a n = 8 − n (cid:18) (2 n )!( n !) (cid:19) κ n F (cid:18) − n − , − n − n −
12 ; − κ (cid:19) which is always terminating because n is an integer. It is well known that thecomplete elliptic integral of first kind T can be expressed in terms of the hypergeo-metric function as the function of the modulus of the elliptic curve, while here thecoefficients of the Taylor series of T r ( h ) are given by the hypergeometric function.Since we have repeated indicial roots, we expect the second independent solutionto be singular. The general theory (see for example [4]) says that the singularsolution is of the form T s ( h ) := T r ( h ) log h + ∞ (cid:88) n =1 b n h n YMPLECTIC INVARIANTS OF THE EULER TOP 11 where b n := d a n ( (cid:37) )d (cid:37) (cid:12)(cid:12)(cid:12) (cid:37) =0 . The recursion relation for the b n at (cid:37) = 0 is thus given by(14) b n = (2 n − (cid:0) κa n − + κ n (2 n − b n − + n (2 n − b n − (cid:1) + (8 n − a n − n . Along with the initial conditions on the a n , we also impose that b − := 0 and b := 0 . The first few coefficients are b = κ,b = 116 (cid:0) κ + 20 (cid:1) ,b = 196 κ (cid:0) κ + 372 (cid:1) ,b = 16144 (cid:0) κ + 56760 κ + 18672 (cid:1) ,b = 120480 κ (cid:0) κ + 416360 κ + 313680 (cid:1) . We were not able to find an explicit solution for a n ( (cid:37) ) . However, since we onlyneed the derivative of a n ( (cid:37) ) at (cid:37) = 0 it is enough to find an approximate solution ˆ a n ( (cid:37) ) that is valid up to terms of O ( (cid:37) ) . Using this we obtain an explicit formulafor b n = a (cid:48) n (0) = ˆ a (cid:48) n (0) . Lemma 3.5.
The recursion for a n ( (cid:37) ) given by (12) is approximately solved by ˆ a n ( (cid:37) ) = 2 n κ n n ! (cid:0) (cid:37) + (cid:1) n ( (cid:37) + 1) n (cid:98) n (cid:99) (cid:88) k =0 (cid:0) (cid:37) + (cid:1) n − k ( n − k )! k ! κ n − k to leading order in (cid:37) , where ( x ) n is the Pochhammer symbol. Note that this formula reduces to the explicit formula for a n given earlier when (cid:37) = 0 using the identity (cid:0) (cid:1) n = (2 n − n − n − . Hence the following proof will alsoprove Theorem 3.4. Proof.
We will show that a n ( (cid:37) ) = ˆ a n ( (cid:37) ) + O ( (cid:37) ) by induction. From the explicitrecursion we find a ( (cid:37) ) = (1 + 2 (cid:37) ) (cid:37) ) κ and a ( (cid:37) ) = (1 + 2 (cid:37) )(3 + 2 (cid:37) )(2 + (cid:37) ) + (1 + 2 (cid:37) ) (3 + 2 (cid:37) ) (cid:37) ) (2 + (cid:37) ) κ . One can easily check that a ( (cid:37) ) = ˆ a ( (cid:37) ) , Furthermore, the coefficients of κ in a ( (cid:37) ) and ˆ a ( (cid:37) ) coincide and the constant and linear coefficients of a ( (cid:37) ) and ˆ a ( (cid:37) ) areboth + (cid:37) + O ( (cid:37) ) . Now assume that the identity a n ( (cid:37) ) = ˆ a n ( (cid:37) ) + O ( (cid:37) ) holds for all ≤ n ≤ m − forsome fixed ≤ m ∈ N . Now we are going to show that under these assumptions, a m ( (cid:37) ) = ˆ a m ( (cid:37) ) + O ( (cid:37) ) holds.We substitute the formulas ˆ a m − ( (cid:37) ) and ˆ a m − ( (cid:37) ) into the right hand side of (12)with n = m and need to verify that a m ( (cid:37) ) so obtained is equal to ˆ a m ( (cid:37) ) + O ( (cid:37) ) . YMPLECTIC INVARIANTS OF THE EULER TOP 12
In order to simplify the recursion define d m := a m ( (cid:37) )(2 m + 2 (cid:37) + 1) so that d m = g m ( κ d m − + d m − ) , where g m := (2 m + 2 (cid:37) + 1)(2 m + 2 (cid:37) − m + (cid:37) ) . Using the formula for ˆ a m and the above definition of d m we define(15) ˆ d m := ˆ a m ( (cid:37) )(2 m + 2 (cid:37) + 1) = G m ( (cid:37) ) (cid:98) m (cid:99) (cid:88) k =0 S m,m − k ( (cid:37) ) κ m − k , where G m ( (cid:37) ) := 2 m +1 m ! (cid:0) (cid:37) + (cid:1) m +1 ( (cid:37) + 1) m , S i,j ( (cid:37) ) := (cid:0) (cid:37) + (cid:1) i + j (cid:0) i − j (cid:1) ! j ! . The equivalent claim is that d m = ˆ d m + O ( (cid:37) ) , which implies(16) ˆ d m = g m ( κ ˆ d m − + ˆ d m − ) + O ( (cid:37) ) . Inserting the series (15) for ˆ d m and collecting powers of κ gives G m ( (cid:37) ) S m,m − k ( (cid:37) ) = g m (cid:0) G m − ( (cid:37) ) S m − ,m − k − ( (cid:37) ) + G m − ( (cid:37) ) S m − ,m − k ( (cid:37) ) (cid:1) + O ( (cid:37) ) for k = 0 , . . . , (cid:4) m (cid:5) . Dividing this equation by G m − ( (cid:37) ) S m − ,m − k ( (cid:37) ) and using theidentities G m ( (cid:37) ) G m − ( (cid:37) ) = m (2 m + 2 (cid:37) + 1)( m + (cid:37) ) = m m + 2 (cid:37) − g m and S m,m − k ( (cid:37) ) S m − ,m − k ( (cid:37) ) = 2 m − k + 2 (cid:37) − k , S m − ,m − k − ( (cid:37) ) S m − ,m − k ( (cid:37) ) = m − kk , we have after some simplification m m + 2 (cid:37) − − ( m + (cid:37) − ( m − m + 2 (cid:37) − − O (cid:0) (cid:37) (cid:1) G m − ( (cid:37) ) S m − ,m − k ( (cid:37) ) . Multiplication by ( m − m + 2 (cid:37) − gives (cid:37) = − ( m − m + 2 (cid:37) − G m − ( (cid:37) ) S m − ,m − k ( (cid:37) ) O (cid:0) (cid:37) (cid:1) . The coefficient of O (cid:0) (cid:37) (cid:1) in this equation evaluated at (cid:37) = 0 is − ( m − m − G m − (0) S m − ,m − k (0) = − (cid:0) m − m + 1 (cid:1) ( k − m − m − k )!2 m (cid:0) (cid:1) m (cid:0) (cid:1) m − k − , which is non-zero for all k = 0 , . . . , (cid:4) m (cid:5) and m ≥ , and hence the the Taylor seriesof the factor multiplying O (cid:0) (cid:37) (cid:1) has a non-zero constant term. This shows that foreach power of κ in (16) the estimation holds, and thus it holds for the whole finiteseries. Hence by mathematical induction the Lemma is proved. (cid:3) With the previous Lemma it is now straightforward to find an explicit formulafor the coefficients b n . YMPLECTIC INVARIANTS OF THE EULER TOP 13
Theorem 3.6.
The recursion relation for b n is solved by b n = 14 n (cid:18) nn (cid:19) (cid:98) n (cid:99) (cid:88) k =0 (cid:18) n − kk, n − k, n − k (cid:19) f n,k (cid:16) κ (cid:17) n − k where f n,k := 2O n + 2O n − k − n and H n is the Harmonic number and O n its odd cousin defined by H n := n (cid:88) k =1 k , O n := n (cid:88) k =1 k − . Proof.
Using the previous Lemma we can simply differentiate ˆ a n ( (cid:37) ) and evaluate at0 in order to get b n . The derivative of the Pochhammer function is given in termsof the digamma function ψ , which for integer and half-integer values n can beexpressed in terms of the Euler-Mascheroni constant γ and the Harmonic numbers, ψ ( n ) = − γ + H n − , where H n = (cid:80) nk =1 1 k . Using the recursion H n = H n − + n and H = 2 − the Harmonic number is thus also defined for half integers;explicitly H n − = 2O n + H − . Denote by ˆ a kn ( (cid:37) ) the coefficient of κ n − k in ˆ a n ( (cid:37) ) .The logarithmic derivative of ˆ a kn ( (cid:37) ) at (cid:37) = 0 can thus be found as a kn (cid:48) (0) a kn (0) = ˆ a kn (cid:48) (0)ˆ a kn (0) = H n − + H n − k − − n + 4 log 2 = 2O n + 2O n − k − n and this determines the “correction factor” f n,k for the coefficient of κ n − k in b n . (cid:3) To obtain the solutions I r ( h ) and I s ( h ) of the Picard-Fuchs equation we integrate T r ( h ) and T s ( h ) term-by-term, respectively, and get I r ( h ) := 12 π ˆ T r ( h ) d h = 12 π ∞ (cid:88) n =0 a n n + 1 h n +1 (17a) I s ( h ) := 12 π ˆ T s ( h ) d h (17b) = I r ( h ) log h + 12 π ∞ (cid:88) n =0 n + 1 (cid:26) b n − a n n + 1 (cid:27) h n +1 (17c)where the integration constants are fixed by the requirements that I r (0) = 0 and I s ( h ) → as h → .3.3. Particular action integrals.
Since (10) is a linear third order equation, theremust be three linearly independent solutions. They are: the regular, singular, andconstant solutions. Thus the general solution is an arbitrary linear combination ofthese, namely(18) I ( h ) = k I r ( h ) + k I s ( h ) + k , and upon differentiating T ( h ) = k T r ( h ) + k T s ( h ) . We seek to find the k i that give the particular solutions corresponding to the closedloops integrals along the paths β ± as specified in Figure 2. The expansions obtainedare normalised such that T r = 1+ O ( h ) , πI r = h + O ( h ) , T s = log h + O ( h ) , πI s = h log h + O ( h ) , and so the leading terms are πI ( h ) = 2 πk + k h log h + k h + O ( h ) YMPLECTIC INVARIANTS OF THE EULER TOP 14 and T ( h ) = k + k log h + O ( h ) . Thus the constant k is given by I (0) , while both k and k are given by the leading order behaviour of T ( h ) . When T ( h ) is finitefor h → then k = 0 and k is determined by T (0) . Otherwise the leading orderlogarithmically diverging term and the constant term of T ( h ) for small h determine k and k .The four particular solutions to be found are given by the integrals I β ± and T β ± .In order to find the correct linear combinations we need to evaluate these integralsin the limit h → .The β integrals at h = 0 are computed as real integrals. The β cycles are shrunkdown, so that we are integrating along the real intervals Π [ β + ] := (cid:0) h, ρ − (cid:1) and Π [ β − ] := ( − ρ, h ) . I β ± (0) is an elementary and finite real integral that gives I β ± (0) = 14 π ˛ β ± d z √ − κz − z = 12 π ˆ Π[ β ± ] d x (cid:112) ( x + ρ )( ρ − − x ) = 1 π tan − (cid:0) ρ ∓ (cid:1) . For the singular integral T β ± ( h ) , the asymptotic behaviour for small h is T β ± ( h ) = ˛ β ± d z w ( z ) √ h − z = ˆ Π[ β ± ] (cid:112) x ( x − h ) 1 √ − κx − x d x = ± log ( ± h ) ∓
12 log (cid:18) κ + 4 (cid:19) + O ( h ) . This can be shown as follows. Define ϕ ( x ) := [1 − κx − x ] − = [( x + ρ )( ρ − − x )] − and split the integrals up as T β ± ( h ) = ˆ Π[ β ± ] ϕ ( x ) − ϕ (0) (cid:112) x ( x − h ) d x + ˆ Π[ β ± ] ϕ (0) (cid:112) x ( x − h ) d x. The first integral is a convergent elliptic integral, and when h = 0 it becomeselementary. For β + it gives − log (cid:16) ρ ρ +1 (cid:17) , and for β − it gives log (cid:16) ρ +1 (cid:17) . Thesecond integral is divergent when h → but elementary and can be integrated usinghyperbolic trigonometric substitutions. For β + it gives − log (cid:0) ρ − (cid:1) + log h + O ( h ) ,and for β − it gives log (2 ρ ) − log( − h ) + O ( h ) . Adding the two integrals gives thestated result.From these 4 integrals the coefficients k i can be determined as described aboveand the result is(19) I β ± = ∓
12 log (cid:18) κ + 4 (cid:19) I r ± I s + 1 π tan − (cid:0) ρ ∓ (cid:1) . Remark 7.
Notice that the actions related to approaching the separatrix fromeither side satisfy I β + (0) + I β − (0) = , corresponding to the fact that the total areaof the symmetry reduced scaled phase space is π . This area can also be seen as theresidue at infinity of I ( h )2 π (cid:0) I β + (0) + I β − (0) (cid:1) = 12 · π i Res ∞ { ζ ( h ) } = π. As mentioned earlier the actual area (without discrete symmetry reduction) en-closed by a single connected contour of ˜ H is twice as large, and each connected YMPLECTIC INVARIANTS OF THE EULER TOP 15 component appears twice. Undoing the scaling then gives π(cid:96) which is the area ofthe sphere C (cid:96) which is the ( SO (3) reduced) phase space of the Euler top.4. The symplectic invariants
Equipped with the Frobenius series expansions of the action integrals obtainedfrom the Picard-Fuchs ODE, we can now calculate the semi-global symplectic in-variants of the Euler top.4.1.
Revisiting the Birkhoff normal form.
The Birkhoff normal form is a seriesfor h ( J ) . From (17a) we instead have a series for the regular action in terms of theenergy πI r ( h ) = h + κ h + 116 (cid:0) κ + 4 (cid:1) h + . . . The regular action can be obtained by integrating ζ over the α cycles(20) I α ( h ) := i2 π ˛ α ζ ( h ) = h + κ h + 116 (cid:0) κ + 4 (cid:1) h + . . . where the series can be obtained by Taylor expansion and taking residues. We canomit the subscript ± for α since the two cases yield the same series expansion. Therelation between the action integral and the regular Frobenius series thus is I α ( h ) = 2 πI r ( h ) . Thus by inverting this series we recover the Birkhoff normal form, so that we canidentify I α = J . The fact that the Birkhoff normal form at a hyperbolic point isgiven by the integral of the α cycles is a general phenomenon, see [7] for a generalproof. The idea is that this works in a way similar to an elliptic point. Near sucha point the action is given by a closed loop integral over a periodic orbit whichdepends on the energy. Thus the action is obtained as a function of the energyand inverting this function gives the energy as a function of the action. Now theBirkhoff normal form is a form of the Hamiltonian that depends on a single variableonly and by uniqueness of this function we can identify it with the inverse of theaction function. A similar type of argument works near a hyperbolic point, see [7]for the details.4.2. The semi-global symplectic invariant.
The semi-global symplectic invari-ant σ ( J ) is the power series given by the regular part of the composition of thesingular action integral with the inverse of the regular action integral. More pre-cisely there are two cases, depending on the sign of h :(21) π (cid:0) I β ± ◦ I − α (cid:1) ( J ) = A ± ± J log ( ± J ) ∓ J ∓ σ ( J ) . Here A ± is the area enclosed by the separatrix after discrete symmetry reduction, A ± = 2 πI β ± (0) = 2 tan − (cid:0) ρ ∓ (cid:1) so that A + + A − = π . Thus we obtain thesemi-global symplectic invariant σ : Theorem 4.1.
The semi-global symplectic invariant of the Euler top with distinctmoments of inertia reduced by discrete symmetry near the hyperbolic equilibrium is
YMPLECTIC INVARIANTS OF THE EULER TOP 16 defined via (21) and is given by σ ( J ) = 12 log (cid:18) κ + 4 (cid:19) J − κ J − κ + 3296 J − κ (cid:0) κ + 36 (cid:1) J − κ + 4200 κ + 267210240 J − κ (cid:0) κ + 2960 κ + 3600 (cid:1) J − κ + 446040 κ + 801360 κ + 241664688128 J + O (cid:0) J (cid:1) . Proof.
The action I β ( h ) is given as the series expansions (19), (17) whose coefficientswhere obtained from the Frobenius expansion of the Picard-Fuchs equation. Theaction I α ( h ) is similarly given by (20), and the inverse of this series is the Birkhoffnormal form H ∗ ( J ) . Composing I β ( h ) with H ∗ ( J ) gives the series of the action I β in terms of the normal form action J , from which the symplectic invariant σ ( J ) can be read off using the definition in (21). (cid:3) Notice that for κ = 0 (i.e. ρ = 1 , i.e. Θ − − Θ − = Θ − − Θ − ) the invariantis an even function of h , so that both sides of the separatrix, for positive andnegative h or J are the same. The linear term has maximal value of log 4 at κ = 0 .For positive κ all higher order terms are negative. The series expansion of σ hasbeen numerically verified and agrees well with the values obtained from a directnumerical computation of I β ± ◦ I − α .5. Convergence of the symplectic invariant
Convergence of the Birkhoff normal form for analytic integrable systems hasrecently been proven by Zung [19], also the references therein for earlier less generalresults. Convergence of the series expansions of the actions themselves are classical,and can also be obtained from the Picard-Fuchs equation. However, we are notaware of results about the convergence of the symplectic invariant. In general it isconsidered to be a formal series only. However, in the analytic case one may expectthat the symplectic invariant has non-zero radius of convergence. Here we brieflyreport that numerically we find that the symplectic invariant has the same non-zeroradius of convergence as the Birkhoff normal form; the reason for this remarkableobservation is unclear.To analyse the asymptotics of a n define the ratio r n = (2 n +1) a n (2 n − a n − and the re-cursion becomes r n = − n − κ r − n − . The leading order iteration r n = 2 κ + r n − has twofixed points at r = − ρ − and r = 2 ρ . The positive fixed point r = 2 ρ is stablefor positive κ (or ρ > ), while the negative fixed point r = − ρ − is stable fornegative κ (or ρ < ). The radius of convergence in h is given by | r ∞ | − whichgives min( ρ, ρ − ) , and is thus controlled by the roots of w closest to zero. For b n a similar argument works after explicitly controlling the size of a n − b n − and a n − b n − .Thus both series with coefficients a n and b n have the same radius of convergence min( ρ, ρ − ) .The Birkhoff normal form is given by the inverse of the series with coefficients a n n +1 which converges, and thus this series converges as well, as expected from thegeneral theory. However, since inverting a series is a highly non-linear processexplicit formulas cannot easily be obtained. Similarly, the symplectic invariant isbased on the convergent series with coefficients b n n +1 − a n ( n +1) composed with the YMPLECTIC INVARIANTS OF THE EULER TOP 17
Birkhoff normal form, so again we expect convergence. Numerically computing theratios of successive coefficients indicates that the radii of convergence of both theBirkhoff normal form and the symplectic invariant are equal. This is a surprisingobservation. Furthermore, observing the morphology and rates of decay of ratiosof coefficients for varying κ and increasing N leads us to conclude that the radiusof convergence of the Birkhoff normal form and the symplectic invariant is at least25% larger than the radius of convergence of I α and the regular part of I β . Formore details on these results see [17].6. Non-equivalence with the pendulum
To compare the Euler top and the pendulum the topologies of their separatricesneed to be made the same by discrete symmetry reduction, as is required by The-orem 1.2. Initially they are not the same, two joined circles intersecting twice versusa “figure-eight”. The discrete reduction for the top was described in section 2 andillustrated in Figure 1. For the pendulum a discrete symmetry reduction reduces itto a similar phase portrait. The semi-global symplectic invariant near the unstablehyperbolic equilibrium of the pendulum is given in [7] to leading order as σ P ( J ) = ln 32 J + O (cid:0) J (cid:1) . For the pendulum to be equivalent to a particular Euler top the leading term in σ E ( J ) = 12 ln (cid:18) κ + 4 (cid:19) J + O (cid:0) J (cid:1) would need to coincide. However, the maximum value of the leading order termwhich is attained for κ = 0 is ln 4 , so that the leading order term for the pendulumis always bigger than that of any Euler top. Hence there is no Euler top that issemi-globally equivalent to the pendulum.This may seem to contradict a theorem in [9, 14], where it is shown that rigid-body motion reduces to pendulum motion when using a different Poisson structurefor the rigid body. However, only the Poisson structure P comes from the originalphysical system, and in our notion of equivalence we are not allowed to change thisPoisson structure. References [1] A. V. Bolsinov and H. R. Dullin. On the Euler case in rigid body dynamics and the Jacobiproblem (in Russian).
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