Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems

Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems

The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerat...

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Veröffentlicht in:Symmetry, Integrability and Geometry: Methods and Applications
Datum:2012
Hauptverfasser: Kalnins, E.G., Miller Jr., W.
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Veröffentlicht: Інститут математики НАН України 2012
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Zitieren:Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-148418
record_format dspace
spelling Kalnins, E.G.
Miller Jr., W.
2019-02-18T12:09:46Z
2019-02-18T12:09:46Z
2012
Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ.
1815-0659
2010 Mathematics Subject Classification: 20C35; 22E70; 37J35; 81R12
DOI: http://dx.doi.org/10.3842/SIGMA.2012.034
https://nasplib.isofts.kiev.ua/handle/123456789/148418
The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler-Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k₁,k₂) and reducing to the usual systems when k₁=k₂=1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.
This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html. This work was partially supported by a grant from the Simons Foundation (# 208754 to Willard Miller, Jr.).
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
spellingShingle Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
Kalnins, E.G.
Miller Jr., W.
title_short Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
title_full Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
title_fullStr Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
title_full_unstemmed Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems
title_sort structure theory for extended kepler-coulomb 3d classical superintegrable systems
author Kalnins, E.G.
Miller Jr., W.
author_facet Kalnins, E.G.
Miller Jr., W.
publishDate 2012
language English
container_title Symmetry, Integrability and Geometry: Methods and Applications
publisher Інститут математики НАН України
format Article
description The classical Kepler-Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler-Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler-Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler-Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k₁,k₂) and reducing to the usual systems when k₁=k₂=1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.
issn 1815-0659
url https://nasplib.isofts.kiev.ua/handle/123456789/148418
citation_txt Structure Theory for Extended Kepler-Coulomb 3D Classical Superintegrable Systems / E.G. Kalnins, W. Miller Jr. // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 28 назв. — англ.
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last_indexed 2025-11-24T04:39:42Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 034, 25 pages Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems? Ernie G. KALNINS † and Willard MILLER Jr. ‡ † Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: math0236@math.waikato.ac.nz ‡ School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA E-mail: miller@ima.umn.edu URL: http://www.ima.umn.edu/~miller/ Received March 14, 2012, in final form June 04, 2012; Published online June 07, 2012 http://dx.doi.org/10.3842/SIGMA.2012.034 Abstract. The classical Kepler–Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler–Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn’t close polynomially. The 3D 4-parameter potential for the extended Kepler–Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler–Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k1, k2) and reducing to the usual systems when k1 = k2 = 1. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras deter- mined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations. Key words: superintegrability; Kepler–Coulomb system 2010 Mathematics Subject Classification: 20C35; 22E70; 37J35; 81R12 1 Introduction A quantum superintegrable system is an integrable n-dimensional Hamiltonian system with Schrödinger operator H = ∆n + V (x), where ∆n = 1√ g ∑n j,k=1 ∂xj ( √ ggjk)∂xk is the Laplace–Beltrami operator on a Riemannian mani- fold, in local coordinates xj , n ≥ 2. The system is required to admit 2n − 1 algebraically independent globally defined partial differential symmetry operators Sj , j = 1, . . . , 2n− 1, n ≥ 2, ?This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html mailto:math0236@math.waikato.ac.nz mailto:miller@ima.umn.edu http://www.ima.umn.edu/~miller/ http://dx.doi.org/10.3842/SIGMA.2012.034 http://www.emis.de/journals/SIGMA/SESSF2012.html 2 E.G. Kalnins and W. Miller Jr. with S1 = H and [H,Sj ] ≡ HSj − SjH = 0, apparently the maximum number possible. The system is of order ` if the maximum order of the symmetry operators, other than the Schrödinger operator, is `. Similarly, a classical superintegrable system with Hamiltonian H = ∑ gjkpjpk + V (x) on phase space with local coordinates xj , pj , where ds2 = ∑ gjkdxjdxk is an integrable system such that there are 2n − 1 functionally independent functions polynomial in momenta, (easily provable to be the maximum number possible): Sj(p,x), j = 1, . . . , 2n− 1, with S1 = H, and globally defined such that {Sj ,H} = 0, where {F ,G} = n∑ j=1 ( ∂F ∂xj ∂G ∂pj − ∂F ∂pj ∂G ∂xj ) is the Poisson bracket. The system is of order ` if the maximum order of the generating constants of the motion is `. As has been pointed out many times [2, 22] such systems are of enormous historic and present day practical importance. In essence, superintegrable systems are those Hamiltonian systems that can be “solved” exactly, analytically and algebraically, without re- quiring numerical approximation. Superintegrable systems are used as the basis for exact and perturbation methods that underlie planetary motion determination, orbital maneuvering, the periodic table of the elements, the boson calculus and much of special function theory, [14, 15]. The key property that makes a system “superintegrable” is that, in contrast to merely inte- grable systems, the symmetry algebra generated by the basis symmetries is nonabelian. This nonabelian structure can be analyzed and used to deduce properties of the system. Thus in the quantum case the irreducible representations of the symmetry algebra determine the multiplic- ities of the degenerate energy eigenspaces and permit algebraic computation of the eigenvalues. The classical Kepler–Coulomb system in 3 dimensions is well known to be 2nd order super- integrable, with a symmetry algebra that closes polynomially in an so(4)-like structure, e.g. [1]. This polynomial closure (though not usually a Lie algebra) is typical for 2nd order superinte- grable systems in 2D [8, 11] and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler–Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn’t close polynomially [7]. We write it in the form H = p2x + p2y + p2z + α r + β x2 + γ y2 , (1.1) where x, y, z are the usual Cartesian coordinates with conjugate momenta px, py, pz in phase space and r = √ x2 + y2 + z2. The 3D 4-parameter extended Kepler–Coulomb system is not even 2nd order superintegrable. We write it in the form H = p2x + p2y + p2z + α r + β x2 + γ y2 + δ z2 . (1.2) However, Verrier and Evans [28] showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis [21] showed in the quantum case that, if a second 4th order symmetry is added to the generators, the symmetry algebra closes polynomially in the sense that all second commu- tators of the generators can be expresses as symmetrized polynomials in the generators. (Note Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 3 that the 3-parameter potential is not just a restriction of the 4-parameter case, because it admits symmetries that are not inherited from the 4-parameter symmetries.) Here we introduce an analog of the TTW construction [23, 24] and consider an infinite class of classical extended Kepler–Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers (k1, k2) and reducing to the usual systems when k1 = k2 = 1. We construct explicitly a set of generators, show these systems to be superintegrable of arbitrarily high order and determine the structure of the generated symmetry algebras. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Much of the paper is quite technical but, as this is the first work devoted to uncovering the structure of the symmetry algebras of 3D superintegrable systems of arbitrary order, we think it is important to expose the details of computations and concepts that later may prove to be routine. For the 4-parameter system in the case k1 = k2 = 1, where discrete symmetry is present and a 6th generator is needed to obtain polynomial closure, we work out the 12th order functional relationship between the 6 generators. In Section 2 we review the action angle construction that we employ to show that our systems are superintegrable and to enable the determination of the structure of the symmetry algebra. In Section 3 we use the fact that the 3-parameter Kepler–Coulomb system (1.1) separates in spherical coordinates r, θ1, θ2 and, by replacing the angles by k1θ1, k2θ2 for k1, k2 rational, define an infinite family of extended Kepler–Coulomb systems, no longer restricted to flat space. We demonstrate that each of these systems is superintegrable, but of arbitrarily high order. We use our method of raising and lowering symmetries to determine the structure of the symmetry algebras generated by these systems. The general construction does not yield generators of minimum order and we show in Section 3.2 how a limit argument exposes the minimum order generators. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations. We show that the symmetry algebra closes rationally, not polynomially. In Section 4 we apply our method to the 4-parameter Kepler–Coulomb system (1.2) and use its separation in spherical coordinates r, θ1, θ2. By replacing the angles by k1θ1, k2θ2 for k1, k2 rational, we define an infinite family of extended Kepler–Coulomb systems, again not restricted to flat space. We demonstrate that each of these systems is superintegrable, but of arbitrarily high order. We use raising and lowering symmetries to determine the structure of the symmetry algebras generated by these systems. Again the general construction does not yield generators of minimum order and we show in Section 5 how a limit argument exposes the minimum order generators. In general, the symmetry algebra closes rationally, but not polynomially. However in two cases k1 = k2 = 1 and k1 = k2 = 1/2 the system admits additional symmetry: the 6-element permutation group S3. The general construction shows that these systems admit 5 generators of orders 2, 2, 2, 2, 4, but in Section 5.2 we show that the permutation symmetry implies the existence of a 6th symmetry of order 4 such that the 6 symmetries are linearly independent. Then we demonstrate that the algebra generated by these 6 symmetries closes polynomially, in analogy with the computation for the quantum case in [21]. We go further and work out the 12th order functional relation between the 6 generators. In Section 6 we present our conclusions and prospects for additional research. 2 Review of the action-angle construction In [6, 9, 10, 12, 27] it was described how to determine a complete set of 2n−1 functionally inde- pendent constants of the motion for a classical Hamiltonian on an n-dimensional Riemannian or pseudo-Riemannian manifold whose Hamilton–Jacobi equation separates in an orthogonal sub- 4 E.G. Kalnins and W. Miller Jr. group coordinate system. In the special case n = 3 the defining equations for the HamiltonianH, expressed in the separable coordinates q1, q2, q3, take the form H = L1 = p21 + V1(q1) + f1(q1)L2, L2 = p22 + V2(q2) + f2(q2)L3, L3 = p23 + V3(q3). The additional constants of the motion can be constructed as L′1 = N1(q2, p2)−M1(q1, p1), L′2 = N2(q3, p3)−M2(q2, p2). Here, Mj = 1 2 ∫ fj(qj) dqj√ Lj − Vj(qj)− fj(qj)Lj+1 , Nj = 1 2 ∫ dqj+1√ Lj+1 − Vj+1(qj+1)− fj+1(qj+1)Lj+2 , j = 1, 2, and L4 ≡ 0. With this construction the functions L1, L2, L3, L′1, L′2 are functionally independent constants of the motion. The functions L2, L3 are second order polynomials in the momenta and determine the separation of variables in coordinates q1, q2, q3. In general the functions L′1, L′2 are only locally defined and are not polynomials. The system will be superintegrable only if we can supplement L1, L2, L3 with two more polynomial functions such that the full set is functionally independent. This will be possible only for very special systems In the following we will look at several candidate systems for superintegrability, show how to construct polynomial constants of the motion from L′1, L′2 and work out the structure of the symmetry algebra generated by these constants. 3 The classical 3D extended Kepler–Coulomb system with 3-parameter potential The extended Kepler–Coulomb Hamiltonian is H = p2r + α r + L2 r2 , where L2 = p2θ1 + L3 sin2(k1θ1) , L3 = p2θ2 + β cos2(k2θ2) + γ sin2(k2θ2) . Here, L2, L3 are constants of the motion that determine additive separation of the Hamilton– Jacobi equation. Further {L2,L3} = 0 so L2 and L3 are in involution. Applying our action angle construction to get two independent constants of the motion we note that q1 = r, q2 = θ1, q3 = θ2 and f1 = 1 r2 , f2 = 1 sin2(k1θ1) , f3 = 0, V1 = α r , V2 = 0, V3 = β cos2(k2θ2) + γ sin2(k2θ2) , to obtain functions Aj , Bj , j = 1, 2 such that sinhA1 = − i √ L2 cos(k1θ1)√ L2 − L3 , coshA1 = sin(k1θ1)pθ1√ L2 − L3 , Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 5 sinhA2 = i(L3 cos(2k2θ2) + γ − β)√ (β − γ − L3)2 − 4γL3 , coshA2 = − √ L3 sin(2k2θ2)pθ2√ (β − γ − L3)2 − 4γL3 , sinhB1 = − i(α+ 2L2/r)√ α2 + 4HL2 , coshB1 = 2 √ L2pr√ α2 + 4HL2 , sinhB2 = i(2L3 csc2(k1θ1)− L2 − L3) L3 − L2 , coshB2 = −2 √ L3 cot(k1θ1)pθ1 L3 − L2 . Here, k1 = p1/q1, k2 = p2/q2 where p1, q1 are relatively prime positive integers and p2, q2 are relatively prime positive integers. From our general theory, N1 = − iA1 2k1 √ L2 , M1 = − iB1 2 √ L2 , N2 = − iA2 4k2 √ L3 , M2 = − iB2 4k1 √ L3 , so p1q2A2 − p2q1B2, q1A1 − p1B1 are two constants of the motion such that the full set of five constants of the motion is func- tionally independent. We work with the exponential functions, [4], see also [25, 26]. We have, for j = 1, 2, eAj = coshAj + sinhAj = Xj/Uj , e−Aj = coshAj − sinhAj = Xj/Uj , eBj = coshBj + sinhBj = Yj/Sj , e−Bj = coshBj − sinhBj = Yj/Sj , where X1 = sin(k1θ1)pθ1 − i √ L2 cos(k1θ1), X1 = sin(k1θ1)pθ1 + i √ L2 cos(k1θ1), X2 = − √ L3 sin(2k2θ2)pθ2 + i(L3 cos(2k2θ2) + γ − β), X2 = − √ L3 sin(2k2θ2)pθ2 − i(L3 cos(2k2θ2) + γ − β), Y1 = 2 √ L2pr − i ( α+ 2 L2 r ) , Y 1 = 2 √ L2pr + i ( α+ 2 L2 r ) , Y2 = −2 √ L3 cot(k1θ1)pθ1 + i ( 2L3 csc2(k1θ1)− L2 − L3 ) , Y 2 = −2 √ L3 cot ( k1θ1)pθ1 − i(2L3 csc2(k1θ1)− L2 − L3 ) , U1 = √ L2 − L3, U2 = √ −(β − γ − L3)2 + 4γL3, S1 = √ α2 + 4HL2, S2 = L3 − L2. (Here, X, Y are, in general, not the complex conjugates of X, Y , respectively, unless all of the coordinates are real.) Now note that eq1A1−p1B1 and e−q1A1+p1B1 are constants of the motion, where eq1A1−p1B1 = ( eA1 )q1(e−B1 )p1 = Xq 1Y1 p1 U q11 S p1 1 , e−q1A1+p1B1 = ( e−A1 )q1(eB1 )p1 = X1 q1Y p1 1 U q11 S p1 1 . Moreover, the identity eq1A1−p1B1e−q1A1+p1B1 = 1 can be expressed as Xq1 1 X1 q1Y p1 1 Y1 p1 = U2q1 1 S2p1 1 = P1(H,L2,L3) = (L2 − L3)q1(α2 + 4HL2)p1 , where P1 is a polynomial in H, L2 and L3. 6 E.G. Kalnins and W. Miller Jr. Similarly, ep1q2A2−p2q1B2 and e−p1q2A2+p2q1B2 are constants of the motion, where ep1q2A2−p2q1B2 = ( eA2 )p1q2(e−B1 )p2q1 = Xp1q2 2 Y2 p2q1 Up1q22 Sp2q12 , e−p1q2A2+p2q1B2) = ( e−A1 )p1q2(eB1 )p2q1 = X2 p1q2Y p2q1 2 Up1q22 Sp2q12 . The identity ep1q2A2−p2q1B2e−p1q2A2+p2q1B2 = 1 can be written as Xp1q2 2 X2 p1q2Y p2q1 2 Y2 p2q1 = U2p1q2 2 S2p2q1 2 = P2(H,L2,L3) = (L2 − L3)2p2q1((β − γ − L3)2 − 4γL3)p1q2 , where P2 is a polynomial in H, L2 and L3. Let a, b, c, d be nonzero complex numbers and consider the binomial expansion(√ Lka+ ib )q(√Lkc+ id )p + (√ Lka− ib )q(√Lkc− id)p = ∑ 0≤`≤q,0≤s≤p ( q ` )( p s ) b`dsaq−`cp−sL(q+p−`−s)/2k [ i`+s + (−i)`+s ] . (3.1) Here, either k = 2 or k = 3 and p, q are positive integers. Suppose p + q is odd. Then it is easy to see that the sum (3.1) takes the form √ LkTodd(Lk) where Todd is a polynomial in Lk. On the other hand, if p + q is even then the sum (3.1) takes the form Teven(Lk) where Teven is a polynomial in Lk. Similarly, consider the binomial expansion 1 i [(√ Lka+ ib )q(√Lkc+ id )p − (√Lka− ib)q(√Lkc− id)p] = ∑ 0≤`≤q,0≤s≤p ( q ` )( p s ) b`dsaq−`cp−s L(q+p−`−s)/2k i [ i`+s − (−i)`+s ] . (3.2) Suppose p+ q is odd. Then the sum (3.2) takes the form Vodd(Lk) where Vodd is a polynomial in Lk. On the other hand, if p+q is even then the sum (3.2) takes the form √ LkVeven(Lk) where Veven is a polynomial in Lk. A third possibility is( a+ i √ Lkb )q(√Lkc+ id )p + ( a− √ Lkb )q(√Lkc− id)p = ∑ 0≤`≤q,0≤s≤p ( q ` )( p s ) b`dsaq−`cp−sL(`+p−s)/2k [ i`+s + (−i)`+s ] . Then we must have p even to get a polynomial in Lk. If p is odd the sum takes the form√ LkT (Lk) where T is a polynomial. A fourth possibility is( a+ i √ Lkb )q(√Lkc+ id )p − (a−√Lkb)q(√Lkc− id)p = ∑ 0≤`≤q,0≤s≤p ( q ` )( p s ) b`dsaq−`cp−sL(`+p−s)/2k [i`+s − (−i)`+s]. Then we must have p odd to get a polynomial in Lk. If p is even the sum takes the form√ LkT (Lk) where T is a polynomial. Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 7 We define basic raising and lowering symmetries J + = Xq1 1 Y1 p1 , J − = X1 q1Y p1 1 , K+ = Xp1q2 2 Y2 p2q1 , K− = X2 p1q2Y p2q1 2 . At this point we restrict to the case where each of p1, q1, p2, q2 is an odd integer. (The other cases are very similar.) Let J1 = 1√ L2 (J − + J +), J2 = 1 i (J − − J +), K1 = 1 i √ L3 (K− −K+), K2 = K− +K+. Then we see from the explicit expressions for the symmetries and the preceding parity argu- ment that J1, J2, K1, K2 are constants of the motion, polynomial in the momenta. Moreover, the identities J +J − = P1, K+K− = P2 hold. The following relations are straightforward to derive from the definition of the Poisson bracket: {L3, X1} = {L3, X1} = {L3, Y1} = {L3, Y1} = 0, {L2, Y1} = {L2, Y1} = {L3, Y2} = {L3, Y2} = 0, {L2, X1} = −2ik1 √ L2X1, {L2, X1} = 2ik1 √ L2X1, {L3, X2} = −4ik2 √ L3X2, {L3, X2} = 4ik2 √ L3X2, {L2, X2} = − 4ik2 sin2(k1θ1) √ L3X2, {L2, X2} = 4ik2 √ L3 sin2(k1θ1) X2, {L2, Y2} = − 4ik1 √ L3 sin2(k1θ1) Y2, {L2, Y2} = 4ik1 √ L3 sin2(k1θ1) Y2, {H, X1} = 4ik1 √ L2 r2 X1, {H, X1} = −4ik1 √ L2 r2 X1, {H, X2} = 4ik2 √ L3 r2 sin2(k1θ1) X2, {H, X2} = − 4ik2 √ L3 r2 sin2(k1θ1) X2, {H, Y1} = 2i √ L2 r2 Y1, {H, Y1} = −2i √ L2 r2 Y1, {H, Y2} = 4ik1 √ L3 r2 sin2(k1θ1) Y2, {H, Y2} = − 4ik1 √ L3 r2 sin2(k1θ1) Y2, From these results, we find {L3,J ±} = 0, {L2,J ±} = ∓2ip1 √ L2J ±, {L2,K±} = 0, {L3,K±} = ∓4ip1p2 √ L3K±. Thus we obtain {L2,J2} = −i ( 2ip1 √ L2J − + 2ip1 √ L2J + ) = 2p1L2J1, {L2,J1} = 1√ L2 ( 2ip1 √ L2J − − 2ip1 √ L2L+ ) = −2p1J2, {L3,J1} = {L3,J2} = 0. 8 E.G. Kalnins and W. Miller Jr. Similarly, {L3,K2} = −4p1p2L3K1, {L3,K1} = 4p1p2K2, {L2,K2} = {L2,K1} = 0. Since J 2 1 = 1 L2 [ (J +)2 + 2J +J − + (J −)2 ] , J 2 2 = − [ (J +)2 − 2J +J − + (J −)2 ] , (3.3) we have J 2 2 + L2J 2 1 = 4J +J − = 4P1, so J 2 2 = −L2J 2 1 + 4P1(H,L2,L3). Further, {J +,J −} = { J +, P1 J + } = (J +)−1{J +,P1} = (J +)−1 ( ∂P1 ∂L2 {J +,L2}+ ∂P1 ∂L3 {J +,L3} ) , so {J +,J −} = 2ip1 √ L2 ∂P1 ∂L2 . To evaluate {J2,J1} we have {J2,J1} = 1 i {J − − J +, J + + J −√ L2 } = 1 i [ −1 2 (L2)−3/2(J − + J +){L2,J + − J −}+ (L2)−1/2{J − − J +,J − + J +} ] = − p1 L2 (J − + J +)2 − 4p1 ∂P1 ∂L2 . Then, using (3.3), we conclude that {J2,J1} = −p1J 2 1 − 4p1 ∂P1 ∂L2 . Similarly, we have the K-related identities K2 1 = − 1 L3 [ (K+)2 − 2K+K− + (K−)2 ] , K2 2 = [ (K+)2 + 2K+K− + (K−)2 ] , K2 2 = −L3K2 1 + 4P2(H,L2,L3), {K+,K−} = 4ip1p2 √ L3 ∂P2 ∂L3 , {K2,K1} = −2p1p2K2 1 + 8p1p2 ∂P2 ∂L3 . Commutators relating the J and K symmetries are somewhat more complicated to compute. We have {X1, X2} = −2k2 √ L3 cot2(k1θ1) sin(k1θ1) √ L2 X2, {X1, Y2} = 2k1 √ L3 cot(k1θ1)√ L2 sin2(k1θ1) Y2 − 2k1 √ L2X1 + k1 ( cos(k1θ1)pθ1 + i √ L2 sin(k1θ1) ) × ( 2 √ L3 cot(k1θ1) ) − k1 sin(k1θ1) ( 2 √ L3pθ1 sin2(k1θ1) + 4iL3 cos(k1θ1) sin3(k1θ1) ) , Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 9 {X2, Y1} = ( 4ipr√ L2 − 8 r ) k2 √ L3 sin2(k1θ1) X2. Now we can determine the nonpolynomial constant of the motion {J +,K+}: {J +,K+} = { Xq1 1 Y p1 1 , Xp1q2 2 Y p2q1 2 } = q1X q1−1 1 Y p1 1 { X1, X p1q2 2 Y p2q1 2 } + p1X q1 1 Y p1−1 1 { Y1, X p1q2 2 Y p2q1 2 } = q1X q1−1 1 Y p1 1 ( p1q2X p1q2−1 2 Y p2q1 2 {X1, X2}+ p2q1X p1q2 2 Y p2q1−1 2 {X1, Y2} ) + p1X q1 1 Y p1−1 1 ( p2q1X p1q2 2 Y p2q1−1 2 {Y1, Y2}+ p1q2X p1q2−1 2 Y p2q1 2 {Y1, X2} ) , where the last term in braces vanishes identically. We conclude that {J +,K+} J +K+ = 2iq1p1p2 L2 − L3 ( √ L2 + √ L3). Once we have {J +,K+} explicitly, we can obtain the remaining Poisson relations between the J and K symmetries with little additional work. We use the fact that J +J − = P1 and K+K− = P2. Then we have {J −,K−} = { P1 J + , P2 K+ } = − P1 (J +)2 { J +, P2 K+ } + 1 J + { P1, P2 K+ } = − P1 (J +)2K+ {J +,P2}+ P1P2 (J +)2(K+)2 {J +,K+} − P2 J +(K+)2 ){J +,K+} = − P1 (J +)2K+ ∂P2 ∂L2 {J +,L2}+ P1P2 (J +)2(K+)2 {J +,K+}− P2 J +(K+)2 ∂P1 ∂L3 {L3,K+} = −2ip1 √ L2P1 J +K+ ∂P2 ∂L2 + P1P2 (J +)2(K+)2 {J +,K+}+ 4ip1p2 √ L3P2 J +K+ ∂P1 ∂L3 . We can write this relation in the more compact form {J −,K−} J −K− = −4iq1p1p2 √ L2 + √ L3 L2 − L3 + {J +,K+} J +K+ . Similarly, we have {J +,K−} J +K− = 4iq1p1p2 √ L2 L2 − L3 − {J +,K+} J +K+ , {J −,K+} J −K+ = 4iq1p1p2 √ L3 L2 − L3 − {J +,K+} J +K+ . Set {J +,K+} = QJ +K+. Then {J1,K1} = −i { J − + J + √ L2 , K− −K+ √ L3 } = −i√ L2L3 {J − + J +,K− −K+} = −i√ L2L3 ( −4ip1p2q1 L2 − L3 [√ L2(J − − J +)K− + √ L3(K− +K+)J − ] + (J − − J +)(K− +K+)Q ) , where Q = 2iq1p1p2 L2 − L3 (√ L2 + √ L3 ) . 10 E.G. Kalnins and W. Miller Jr. Thus, {J1,K1} = 2q1p1p2√ L2L3(L2 − L3) × [ − (√ L2 + √ L3 )( J −K− + J +K+ ) + (√ L2 − √ L3 )( J −K+ + J +K− )] . In summary: J +J − = P1, K+K− = P2, P1(H,L2,L3) = (L2 − L3)2q1(α2 + 4HL2)p1 , P2(H,L2,L3) = (L2 − L3)2p2q1((β − γ − L3)2 − 4γL3)p1q2 , {L3,J ±} = 0, {L2,J ±} = ∓2ip1 √ L2J ±, {L2,K±} = 0, {L3,K±} = ∓4ip1p2 √ L3K±, {J +,J −} = 2ip1 √ L2 ∂P1 ∂L2 , {K+,K−} = 4ip1p2 √ L3 ∂P2 ∂L3 , {J +,K−} J +K− = −{J −,K+} J −K+ = 2iq1p1p2( √ L2 − √ L3) L2 − L3 , {J −,K−} J −K− = −{J +,K+} J +K+ = −2iq1p1p2( √ L3 + √ L2) L2 − L3 . These relations prove closure of the symmetry algebra in the space of functions polynomial in J ±, K±, rational in L2, L3, H and at most linear in √ L2, √ L3. 3.1 Structure relations for polynomial constants of the motion Since, J − = 1 2 ( √ L2J1 + iJ2), J + = 1 2 ( √ L2J1 − iJ2), K− = 1 2 (i √ L3K1 +K2), K+ = 1 2 (−i √ L3K1 +K2), we have {J1,K1} = 2q1p1p2 L2 − L3 (−J1K2 + J2K1). Similar computations yield {J2,K1} = 2q1p1p2 L2 − L3 (J2K2 + L2J1K1), {J1,K2} = 2q1p1p2 L2 − L3 (L3K1 + J2K2), {J2,K2} = 2q1p1p2 L2 − L3 (L3J2K1 − L2J1K2). Note: It can be verified that the numerators are divisible by L2−L3, so that {J1,K1}, {J2,K1} {J1,K2} and {J2,K2} are true polynomial constants of the motion, although not polynomial in the generators. 3.2 Minimal order generators The generators for the polynomial symmetry algebra that we have produced so far are not of minimal order. Note that L1, L2, L3 are of order 2 and the orders of J1, K1 are one less than the orders of J2, K2, respectively. We will construct a symmetry K0 of order one less than K1. Note that the symmetry K2 is a polynomial in L3. The constant term in this polynomial expansion Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 11 is ip1q2+p2q1Lp2q12 (γ − β)p1q2((−1)p1q2 + (−1)p2q1), itself a constant of the motion. In the case we are considering p1, q1, p2, q2 are each odd, so the constant term is D2(L2) = 2(−1)(p1q2+p2q1)/2+1Lp2q12 (γ − β)p1q2 . Thus K0 = K2 −D2 L3 is a polynomial symmetry of order two less than K2. We have the identity K2 = L3K0 +D2. (3.4) From this, {L3,K2} = L3{L3,K0}. We already know that {L3,K2} = −4p1p2L3K1 so {L3,K0} = −4p1p2K1, {L2,K0} = 0. The same construction fails for J2. It is a polynomial in L2, but the constant term in the expansion is not a constant of the motion. Indeed, in the special case k1 = k2 = 1, the symmetry J1 is of minimal order 2, so J1 cannot be realized as a commutator. Now we choose L1, L2, L3, J1, K0 as the generators of our algebra. We define the basic nonzero commutators as R1 = {L2,J1} = −2p1J2, R2 = {L3,K0} = −4p1p2K1, R3 = {J1,K0}. Then we have R2 1 4p21 = J 2 2 = −L2J 2 1 + 4P1, a polynomial in the generators. Further, R2 2 16p21p 2 2 = K2 1 = −(L3K0 +D2) 2 + 4P1 L3 which again can be verified to be a polynomial in the generators. Note, however, that R1R2 = 8p21p2J2K1, a product of Poisson brackets of the generators, is not a polynomial in the generators, although (R1R2) 2 is such a polynomial. Using the identity (3.4) and the expression for {J1,K2}, it is easy to see that R3 is rationally related to R2. It is clear that all additional commutators can be expressed as rational functions of the constants of the motion already computed. We conclude that the polynomial symmetry algebra generated by the 5 basic generators and their 3 commutators closes rationally, but not polynomially. 4 The classical 3D extended Kepler–Coulomb system with 4-parameter potential Now we consider the Hamiltonian H = p2r + α r + L2 r2 , where L2 = p2θ1 + L3 sin2(k1θ1) + δ cos2(k1θ1) , L3 = p2θ2 + β cos2(k2θ2) + γ sin2(k2θ2) . 12 E.G. Kalnins and W. Miller Jr. Here L2, L3 are constants of the motion, in involution. They determine additive separation in the variables r, θ, φ. Applying our usual construction to get two independent constants of the motion we note that q1 = r, q2 = θ1, q3 = θ2 and f1 = 1 r2 , f2 = 1 sin2(k1θ1) , f3 = 0, V1 = α r , V2 = δ cos2(k1θ1) , V3 = β cos2(k2θ2) + γ sin2(k2θ2) , to obtain functions Aj , Bj , j = 1, 2 such that sinhA1 = i(−L2 cos(2k1θ1) + δ − L3)√ L23 − 2L3(L2 + δ) + (L2 − δ)2 , coshA1 = √ L2 sin(2k1θ1)pθ1√ L23 − 2L3(L2 + δ) + (L2 − δ)2 , sinhA2 = i(L3 cos(2k2θ2) + γ − β)√ (β − γ − L3)2 − 4γL3 , coshA2 = − √ L3 sin(2k2θ2)pθ2√ (β − γ − L3)2 − 4γL3 , sinhB1 = − i(α+ 2L2/r)√ α2 + 4HL2 , coshB1 = 2 √ L2pr√ α2 + 4HL2 , sinhB2 = −2i √ L3 cot(k1θ1)pθ1√ L23 − 2L3(δ + L2) + (L2 − δ)2 , coshB2 = −2L3 cot2(k1θ1) + (L2 − L3 − δ)√ L23 − 2L3(δ + L2) + (L2 − δ)2 . Here, k1 = p1/q1, k2 = p2/q2 where p1, q1 are relatively prime positive integers and p2, q2 are relatively prime positive integers. From our general theory, N1 = − iA1 4k1 √ L2 , M1 = − iB1 2 √ L2 , N2 = − iA2 4k2 √ L3 , M2 = − iB2 4k1 √ L3 , so p1q2A2 − p2q1B2, q1A1 − 2p1B1 are two constants of the motion such that the full set of five constants of the motion is func- tionally independent. We have eAj = coshAj + sinhAj = Xj/Uj , e−Aj = coshAj − sinhAj = Xj/Uj , eBj = coshBj + sinhBj = Yj/Sj , e−Bj = coshBj − sinhBj = Yj/Sj , where X1 = √ L2 sin(2k1θ1)pθ1 + i(−L2 cos(2k1θ1) + δ − L3), X1 = √ L2 sin(2k1θ1)pθ1 − i(−L2 cos(2k1θ1) + δ − L3), X2 = − √ L3 sin(2k2θ2)pθ2 + i(L3 cos(2k2θ2) + γ − β), X2 = − √ L3 sin(2k2θ2)pθ2 − i(L3 cos(2k2θ2) + γ − β), Y1 = 2 √ L2pr − i ( α+ 2 L2 r ) , Y 1 = 2 √ L2pr + i ( α+ 2 L2 r ) , Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 13 Y2 = −2L3 cot2(k1θ1) + (L2 − L3 − δ)− 2i √ L3 cot(k1θ1)pθ1 , Y 2 = −2L3 cot2(k1θ1) + (L2 − L3 − δ) + 2i √ L3 cot(k1θ1)pθ1 , U1 = √ L23 − 2L3(L2 + δ) + (L2 − δ)2, U2 = √ (β − γ − L3)2 − 4γL3, S1 = √ α2 + 4HL2, S2 = √ L23 − 2L3(L2 + δ) + (L2 − δ)2. Here, eq1A1−2p1B1 and e−q1A1+p1B1 are constants of the motion, where eq1A1−2p1B1 = ( eA1 )q1(e−B1 )2p1 = Xq1 1 Y1 2p1 U q11 S 2p1 1 , e−q1A1+2p1B1 = ( e−A1 )q1(eB1 )2p1 = X1 q1Y 2p1 1 U q11 S 2p1 1 . The identity eq1A1−p1B1e−q1A1+p1B1 = 1 can be expressed as Xq1 1 X1 q1Y 2p1 1 Y1 2p1 = U2q1 1 S4p1 1 = P1(H,L2,L3) = [ L23 − 2L3(L2 + δ) + (L2 − δ)2 ]q1[α2 + 4HL2 ]2p1 , where P1 is a polynomial in H, L2 and L3. Similarly, ep1q2A2−p2q1B2 , e−p1q2A2+p2q1B2 are constants of the motion, where ep1q2A2−p2q1B2 = ( eA2 )p1q2(e−B1 )p2q1 = Xp1q2 2 Y2 p2q1 Up1q22 Sp2q12 , e−p1q2A2+p2q1B2) = ( e−A1 )p1q2(eB1 )p2q1 = X2 p1q2Y p2q1 2 Up1q22 Sp2q12 . The identity ep1q2A2−p2q1B2e−p1q2A2+p2q1B2 = 1 becomes Xp1q2 2 X2 p1q2Y p2q1 2 Y2 p2q1 = U2p1q2 2 S2p2q1 2 = P2(H,L2,L3) = [ (β − γ − L3)2 − 4γL3 ]p1q2[L23 − 2L3(L2 + δ) + (L2 − δ)2 ]p2q1 , where P2 is a polynomial in H, L2 and L3. We define basic raising and lowering symmetries J + = Xq1 1 Y1 2p1 , J − = X1 q1Y 2p1 1 , K+ = Xp1q2 2 Y2 p2q1 , K− = X2 p1q2Y p2q1 2 . Further, we restrict to the case where each of p1, q1, p2, q2 is an odd integer. The other cases are similar. Let J1 = 1√ L2 (J − + J +), J2 = 1 i (J − − J +), K1 = 1√ L3 (K− +K+), K2 = 1 i (K− −K+). Then we see from the explicit expressions for the symmetries that J1, J2, K1, K2 are constants of the motion, polynomial in the momenta. Moreover, the identities J +J − = P1, K+K− = P2 hold. Note that the definitions of the K-symmetries differ from those for the 3-parameter potential. 14 E.G. Kalnins and W. Miller Jr. The following relations are straightforward to derive from the definition of the Poisson bracket: {L3, X1} = {L3, X1} = {L3, Y1} = {L3, Y1} = 0, {L2, Y1} = {L2, Y1} = {L3, Y2} = {L3, Y2} = 0, {L2, X1} = −4ik1 √ L2X1, {L2, X1} = 4ik1 √ L2X1, {L3, X2} = −4ik2 √ L3X2, {L3, X2} = 4ik2 √ L3X2, {L2, X2} = − 4ik2 sin2(k1θ1) √ L3X2, {L2, X2} = 4ik2 √ L3 sin2(k1θ1) X2, {L2, Y2} = − 4ik1 √ L3 sin2(k1θ1) Y2, {L2, Y2} = 4ik1 √ L3 sin2(k1θ1) Y2, {H, X1} = −4ik1 √ L2 r2 X1, {H, X1} = 4ik1 √ L2 r2 X1, {H, X2} = − 4ik2 r2 sin2(k1θ1) √ L3X2, {H, X2} = 4ik2 √ L3 r2 sin2(k1θ1) X2, {H, Y1} = −2i √ L2 r2 Y1, {H, Y1} = 2i √ L2 r2 Y1, {H, Y2} = − 4ik1 √ L3 r2 sin2(k1θ1) Y2, {H, Y2} = 4ik1 √ L3 r2 sin2(k1θ1) Y2. From these results, we find {L3,J ±} = 0, {L2,J ±} = ∓4ip1 √ L2J ±, {L2,K±} = 0, {L3,K±} = ∓4ip1p2 √ L3K±. Further, {J +,J −} = { J +, P1 J + } = (J +)−1{J +,P1} = (J +)−1 ( ∂P1 ∂L2 {J +,L2}+ ∂P1 ∂L3 {J +,L3} ) , so {J +,J −} = 4ip1 √ L2 ∂P1 ∂L2 . Similarly {K+,K−} = 4ip1p2 √ L3 ∂P2 ∂L3 . Commutators relating the J and K symmetries are somewhat more complicated to compute. We have {X1, X2} = −4k2 √ L3√ L2 ( i cot(k1θ1)pθ1 + √ L2 cot2(k1θ1) ) X2, {Y1, Y2} = 4ik1 √ L3 sin2(k1θ1) ( pr√ L2 + 2i r ) Y2, {X1, Y2} = 4ik1 √ L2X1 + 4ik1 √ L3 sin2(k1θ1) ( sin(2k1θ1)pθ1 2 √ L2 − i cos(2k1θ1) ) Y2 Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 15 + 8k1 cot2(k1θ1) ( i √ L2L3 sin(2k1θ1) 2 pθ1 − √ L3L2 sin2(k1θ1)− √ L2L3 ) , {X2, Y1} = 4ik2 √ L3 sin2(k1θ1) ( pr√ L2 + 2i r ) X2. Now we can compute the nonpolynomial constant of the motion {J +,K+}: {J +,K+} = { Xq1 1 Y 2p1 1 , Xp1q2 2 Y p2q1 2 } = q1X q1−1 1 Y 2p1 1 { X1, X p1q2 2 Y p2q1 2 } + 2p1X q1 1 Y 2p1−1 1 { Y1, X p1q2 2 Y p2q1 2 } = q1X q1−1 1 Y 2p1 1 ( p1q2X p1q2−1 2 Y p2q1 2 {X1, X2}+ p2q1X p1q2 2 Y p2q1−1 2 {X1, Y2} ) + 2p1X q1 1 Y 2p1−1 1 ( p2q1X p1q2 2 Y p2q1−1 2 {Y1, Y2}+ p1q2X p1q2−1 2 Y p2q1 2 {Y1, X2} ) , where the last term in braces vanishes identically. We conclude that {J +,K+} J +K+ = 4iq1p1p2( √ L2 − √ L3)(L2 + 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 . Once we have {J +,K+} explicitly, we can obtain the remaining Poisson relations between the J and K symmetries with little additional work. We use the fact that J +J − = P1 and K+K− = P2. Then we have {J −,K−} = { P1 J + , P2 K+ } = − P1 (J +)2 { J +, P2 K+ } + 1 J + { P1, P2 K+ } = − P1 (J +)2K+ {J +,P2}+ P1P2 (J +)2(K+)2 {J +,K+} − P2 J +(K+)2 ){P1,K+} = − P1 (J +)2K+ ∂P2 ∂L2 {J +,L2}+ P1P2 (J +)2(K+)2 {J +,K+}− P2 J +(K+)2 ∂P1 ∂L3 {L3,K+} = −4ip1 √ L2P1 J +K+ ∂P2 ∂L2 + P1P2 (J +)2(K+)2 {J +,K+}+ 4ip1p2 √ L3P2 J +K+ ∂P1 ∂L3 . We can write this relation in the more compact form {J −,K−} J −K− = 8iq1p1p2 ( √ L3 − √ L2)(L2 + 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 + {J +,K+} J +K+ . Similarly, we have {J +,K−} J +K− = 8iq1p1p2 √ L2(−L3 + L2 − δ) (L3 − L2 − δ)2 − 4δL2 − {J +,K+} J +K+ , {J −,K+} J −K+ = 8iq1p1p2 √ L3(−L3 + L2 + δ) (L3 − L2 − δ)2 − 4δL2 − {J +,K+} J +K+ . Thus, {J +,K+} J +K+ = −{J −,K−} J −K− = 4iq1p1p2( √ L2 − √ L3)(L2 + 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 , {J +,K−} J +K− = −{J −,K+} J −K+ = 4iq1p1p2( √ L2 + √ L3)(L2 − 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 . In summary: J +J − = P1, K+K− = P2, 16 E.G. Kalnins and W. Miller Jr. P1(H,L2,L3) = [ L23 − 2L3(L2 + δ) + (L2 − δ)2 ]q1[α2 + 4HL2 ]2p1 , P2(H,L2,L3) = [ (β − γ − L3)2 − 4γL3 ]p1q2[L23 − 2L3(L2 + δ) + (L2 − δ)2 ]p2q1 , {L3,J ±} = 0, {L2,J ±} = ∓4ip1 √ L2J ±, {L2,K±} = 0, {L3,K±} = ∓4ip1p2 √ L3K±, {J +,J −} = 4ip1 √ L2 ∂P1 ∂L2 , {K+,K−} = 4ip1p2 √ L3 ∂P2 ∂L3 , {J +,K+} J +K+ = −{J −,K−} J −K− = 4iq1p1p2( √ L2 − √ L3)(L2 + 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 , {J +,K−} J +K− = −{J −,K+} J −K+ = 4iq1p1p2( √ L2 + √ L3)(L2 − 2 √ L2L3 + L3 − δ) (L3 − L2 − δ)2 − 4δL2 . These relations prove closure of the symmetry algebra in the space of functions polynomial in J ±, K±, rational in L2, L3, H and at most linear in √ L2, √ L3. 4.1 Structure relations for polynomial symmetries of the 4-parameter potential Note that J − = 1 2 (√ L2J1 + iJ2 ) , J + = 1 2 (√ L2J1 − iJ2 ) , K− = 1 2 ( iK2 + √ L3K1 ) , K+ = 1 2 ( −iK2 + √ L3K1 ) . Thus we have {J1,K1} = { J − + J + √ L2 , K− +K+ √ L3 } = 1√ L2L3 {J − + J +,K− +K+} = 4q1p1p2 (L3 − L2 − δ)2 − 4δL2 [J1K2(L2 − L3 + δ) + J2K1(L2 − L3 − δ)] . (4.1) Similarly, {J1,K2} = − 4q1p1p2 (L3− L2− δ)2− 4δL2 [J1K1L3(L2 − L3 + δ) + J2K2(−L2 + L3 + δ)] , (4.2) {J2,K2} = − 4q1p1p2 (L3− L2− δ)2− 4δL2 [J1K2L2(L2 − L3 − δ) + J2K1L3(L2 − L3 + δ)] , (4.3) {J2,K1} = − 4q1p1p2 (L3− L2− δ)2− 4δL2 [J1K1L2(L2 − L3 − δ) + J2K2(−L2 + L3 − δ)] . (4.4) Straightforward computations yield {L2,J2} = 4p1L2J1, {L2,J1} = −4p1J2, {L3,J1} = {L3,J2} = 0, {L3,K2} = 4p1p2L3K1, {L3,K1} = −4p1p2K2, {L2,K2} = {L2,K1} = 0, J 2 2 = −L2J 2 1 + 4P1(H,L2,L3), K2 2 = −L3K2 1 + 4P2(H,L2,L3), {J1,J2} = −2p1J 2 1 + 8p1 ∂P1 ∂L2 , {K1,K2} = −2p1p2K2 1 + 8p1p2 ∂P2 ∂L3 . 5 Minimal order generators The generators for the polynomial symmetry algebra that we have produced so far are not of minimal order. Here L1, L2, L3 are of order 2 and the orders of J1, K1 are one less than the Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 17 orders of J2, K2, respectively. We will construct symmetries J0, K0 of order one less than J1, K1, respectively. (In the standard case k1 = k2 = 1 it is easy to see that J1 is of order 5 and J2 is of order 6, whereas K1 is of order 3 and K2 is of order 4. Then J0, K0 will be of orders 4 and 2, respectively, which we know corresponds to the minimal generators of the symmetry algebra in this case, [21, 28].) The symmetry J2 is a polynomial in L2 with constant term D1 = 2(−1)(q1−1)/2(δ − L3)q1α2p1 , itself a constant of the motion. Thus J0 = J2 −D1 L2 is a polynomial symmetry of order two less than J2. We have the identity J2 = L2J0 +D1. From this, {L2,J2} = L2{L2,J0}. We already know that {L2,J2} = 4p1L2J1 so {L2,J0} = 4p1J1, {L3,J0} = 0. The same construction works for K2. It is a polynomial in L3, with constant term D2 = 2(−1)(p1q2+1)/2(γ − β)p1q2(L2 − δ)p2q1 . Thus K0 = K2 −D2 L3 is a polynomial symmetry of order two less than K2 and we have the identity K2 = L3K0 +D2. Further, {L3,K0} = 4p1p2K1, {L2,K0} = 0. Now we choose L1, L2, L3, J0, K0 as the generators of our algebra. We define the basic nonzero commutators as R1 = {L2,J0} = 4p1J1, R2 = {L3,K0} = 4p1p2K1, R3 = {J0,K0}. Then we have R2 1 16p21 = J 2 1 = −L2J 2 0 − 2D1J0 + 4P1 −D2 1 L2 , (5.1) where the last term on the right is a polynomial in the generators H, L2, L3. Further, R2 2 16p21p 2 2 = K2 1 = −L3K2 0 − 2D2K0 + 4P2 −D2 2 L3 , (5.2) which again can be verified to be a polynomial in the generators. Note that this symmetry algebra cannot close polynomially in the usual sense. If it did close then the product R1R2 would be expressible as a polynomial in the generators. The preceding two equations show that R2 1R2 2 is so expressible, but that the resulting polynomial is not a perfect square. Thus the only 18 E.G. Kalnins and W. Miller Jr. possibility to obtain closure is to add new generators to the algebra, necessarily functionally dependent on the original set. We see that[ (L3 − L2 − δ)2 − 4δL2 ] R3 = −4q1p1p2 L2L3 [J1(L3K0 +D2)L2(L2 − L3 − δ) + (L2J0 +D1)K1L3(L2 − L3 + δ)] + 4p1 L2L3 [(L3 − L2 − δ)2 − 4δL2] [ L2J1 ∂D2 ∂L2 − p2L3K1 ∂D1 ∂L3 ] = AJ1 +BK1, (5.3) where A and B are polynomial in the generators, so[ (L3 − L2 − δ)2 − 4δL2 ] {L2,R3} = 4p1A(L2J0 +D1)− 16q1p 2 1p2(L2 − L3 + δ)J1K1, (5.4)[ (L3 − L2 − δ)2 − 4δL2 ] {L3,R3} = −4p1p2B(L3K0 +D2)− 16q1p 2 1p 2 2(L2 − L3 − δ)J1K1. (5.5) Similarly 1 4p1 {L2,R1} = {L2,J1} = −4p1J2 = 4p1(L2J0 +D1), 1 4p1 {L3,R1} = 0, (5.6) and 1 4p1 {J0,R1} = {J0,J1} = 1 L2 ( 2p1J 2 1 − 8p1 ∂P1 ∂L2 ) + 2p1 ∂ ∂L2 ( D1 L2 ) J2 + 4p1 J 2 2 L22 , (5.7) a polynomial in the generators, 1 4p1 {K0,R1} = {K0,J1} = 4q1p1p2 L3[(L3 − L2 − δ)2 − 4δL2] × (J1K1L3(L2 − L3 + δ) + J2K2(−L2 + L3 + δ)) + 4p1 J2 L3 ∂D2 ∂L2 . (5.8) Continuing in this way, it is straightforward to show that L1, L2, L3, K0, J0 generate a sym- metry algebra that closes rationally. In particular, each of the commutators R1, R2, R3 satisfies an explicit polynomial equation in the generators. 5.1 Stäckel equivalence of Kepler–Coulomb and caged isotropic oscillator systems Consider the Hamiltonian for the caged isotropic oscillator H′ = p2R + α′R2 + L′2 R2 , (5.9) where L′2 = p2φ1 + L′3 sin2(j1φ1) + δ′ cos2(j1φ1) , L′3 = p2φ2 + β′ cos2(j2φ2) + γ′ sin2(j2φ2) . Here L′2, L′3 are constants of the motion, in involution. They determine additive separation in the spherical coordinates R, φ1, φ2. Also, j1, j2 are nonzero rational numbers. If j1 = j2 = 1, then in terms of Cartesian coordinates we have H′ = p2x+p2y+p2z +α′R2 +β′/x2 +γ′/y2 +δ′/z2. Note Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 19 that system (5.9) can be considered as the 3-variable analog of the TTW system [23, 24]. (Note, however, that this is a flat space system only if j1 = 1.) Now consider the Hamilton–Jacobi equation H′ = E′ and take the Stäckel transform that corresponds to dividing by R2. Then, making the change of variables r = R2, 2φ1 = θ1, 2φ2 = θ2, we obtain the new Hamilton–Jacobi equation H = E where H = p2r + α r + L2 r2 with L2 = p2θ1 + L3 sin2(k1θ1) + δ cos2(k1θ1) , L3 = p2θ2 + β cos2(k2θ2) + γ sin2(k2θ2) , E = −α′/4, α = −E′/4, β = β′/4, γ = γ′/4, δ = δ′/4, k1 = j1/2, k2 = j2/2. In other words, we obtain the extended Kepler–Coulomb system. Since the Stäckel transform preserves the structure of the symmetry algebra of a superintegrable system [3, 13, 19, 20], all of our structure results apply to the caged isotropic oscillator. Note, however, that the standard case k1 = k2 = 1 for Kepler–Coulomb corresponds to j1 = j2 = 2 for the oscillator. Further, only for the cases k1 = 1 and j1 = 1 is the manifold flat. The similar analysis in two dimensions [19] is always restricted to flat space, but here the manifolds depend on k1 and j1. 5.2 The special case k1 = k2 = 1 In the case k1 = k2 = 1 we are in Euclidean space and our system has additional symmetry. In terms of Cartesian coordinates x = r sin θ1 cos θ2, y = r sin θ1 sin θ2, z = r cos θ1, the Hamiltonian is H = p2x + p2y + p2z + α r + β x2 + γ y2 + δ z2 . Note that any permutation of the ordered pairs (x, β), (y, γ), (z, δ) leaves the Hamiltonian unchanged. This leads to additional structure in the symmetry algebra. The basic symmetries are L2 = (xpy − ypx)2 + (ypz − zpy)2 + (zpx − xpz)2 + β(x2 + y2 + z2) x2 + γ(x2 + y2 + z2) y2 + δ(x2 + y2 + z2) z2 , L3 = Ixy = (xpy − ypx)2 + β(x2 + y2) x2 + γ(x2 + y2) y2 . Note that the permutation symmetry of the Hamiltonian shows that Ixz, Iyz are also constants of the motion, and that L2 = Ixy + Ixz + Iyz − (β + γ + δ). The constant of the motion K0 is 2nd order: K0 = 4Iyz + 2L3 − 2(L2 + β + γ + δ) = 2(Iyz − Ixz), and J0 is 4th order: J0 = −16 ( M2 3 + δ(xpx + ypy + zpz) 2 z2 ) + 8H(Ixz + Iyz − β − γ − δ) + 2α2, 20 E.G. Kalnins and W. Miller Jr. where M3 = (ypz − zpy)py − (zpx − xpz)px − z ( α 2r + β x2 + γ y2 + δ z2 ) . If δ = 0 thenM3 is the analog of the 3rd component of the Laplace vector and is itself a constant of the motion. The symmetries H, L2, L3, J0, K0 form a generating (rational) basis for the constants of the motion. Under the transposition (x, β) ↔ (z, δ) this basis is mapped to an alternate basis H, L′2, L′3, J ′0, K′0 where L′2 = L2, L′3 = 1 4 K0 + 1 2 L2 − 1 2 L3 + β + γ + δ 2 , K′0 = 1 2 K0 − L2 + 3L3 − (β + γ + δ), R′1 = {L2,J ′0}, R′2 = {L′3,K′0} = −5 4 R2, R′3 = {J ′0,K′0} = 2R′1 − 2{L3,J ′0}, (5.10) since {L3,J ′0} = 0. All of the identities in Section 4.1 hold for the primed symmetries. It is easy to see that the K′ symmetries are simple polynomials in the L, K symmetries already constructed, e.g., K′1 = 1 4{L ′ 3,K′0} = −5 4K1. However, the J ′ symmetries are new. In particular, J ′0 = −16 ( M2 1 + β(xpx + ypy + zpz) 2 x2 ) + 8H(Ixy + Ixz − β − γ − δ) + 2α2, where M1 = (ypx − xpy)py − (xpz − zpx)pz − x ( α 2r + β x2 + γ y2 + δ z2 ) . Note that the transposition (y, γ)↔ (z, δ) does not lead to anything new. Indeed, under the symmetry we would obtain a constant of the motion J ′′0 = −16 ( M2 2 + γ(xpx + ypy + zpz) 2 y2 ) + 8H(Ixy + Iyz − β − γ − δ) + 2α2, where M2 = (zpy − ypz)pz − (ypx − xpy)px − y ( α 2r + β x2 + γ y2 + δ z2 ) . but it is straightforward to check that J0 + J ′0 + J ′′0 = 2α2, (5.11) so that the new constant depends linearly on the previous constants. For further use, note that under the transposition symmetry (x, β)↔ (y, γ) the constants of the motion L2, L3, J0, J1 are invariant, whereas K0, K1 change sign. The action on J ′0 is more complicated. Under the transposition J ′0 and J ′′0 switch places. Thus from expression (5.11) we see that J ′0 −→ J ′′0 = 2α2 − J0 − J ′0. In the paper [21], Tanoudis and Daskaloyannis show that the quantum symmetry algebra generated by the 6 functionally dependent symmetries H, L2, L3, J0, K0 and J ′0 closes poly- nomially, in the sense that all double commutators of the generators are again expressible as Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 21 polynomials in the generators, very strong evidence that the classical analog also closes polyno- mially. (However, they did not address the issue of determining the functional indenpendence explicitly.) Keys to understanding the polynomial closure are the rational structure equations (5.3)–(5.8) and the terms J1K1 and Q = (L3 − L2 − δ)2 − 4δL2. If J1K1 can be expressed as a polynomial in the generators then, with this result substituted in the rational structure equations, we will obtain polynomial structure equations. The requirement that the rational structure equations become polynomial is a strong restriction on the polynomial J1K1 = P (H,L2,L3,J0,K0,J ′0). Let’s focus on equation (5.8). Note that the left hand side of this equation is of order 6 in the momenta and J1K1 is of order 8. What can we say about the polynomial P in order that its substitution into (5.8) turns the right hand side into a polynomial structure equation? First note that the polynomial is not determined uniquely by this requirement. Indeed, if P1, P2 are two solutions their difference is of the form SQ where S is a polynomial in the generators of order ≤ 4 in the momenta. Similarly, we can add such a SQ to any solution and get another solution. A straightforward computation using the polynomial and degree conditions alone yields the result J1K1 = 1 2 [L2 + L3 − δ]J0K0 + α2[L2 − 3L3 − δ]K0 + (β − γ)[3L2 − L3 + δ]J0 + 2α2(γ − β)[L2 + L3 − 5δ] + SQ, (5.12) {K0,J1} = 1 4 {K0,R1} = −1 4 {L2,R3} = −2[2(γ − β) +K0] [ J0 − 2α2 ] + 4(L2 − L3 + δ)S, (5.13) where S = c1J0 + c2J ′0 + S1, c1 and c2 6= 0 are parameters, and S1(H,L2,L3,K0) is a polynomial of order≤ 2 in its arguments. Under the transposition (x, β)↔ (y, γ) the left hand sides of equations (5.12) and (5.13) change sign. Since Q is invariant, S must change sign. With all of these hints it is fairly easy to compute the exact result. It is S = −J0 − 2J ′0 + 2α2, in agreement with all our conditions. To determine the functional relationship between the 6 generators we can use the identity F ≡ J 2 1 K2 1 − (J1K1) 2 = 0, where the first term is obtained from (5.1) and (5.2) and the second term from (5.12). The result is a polynomial in the momenta of order 16, and of the simple form F ≡ Q ( A1(J ′0)2 +A2J ′0J0 +A3(J0)2 +A4J ′0 +A5J0 +A6 ) = 0, where Aj = Aj(H,L2,L3,K0). Since Q factors out of the equation, the basic identity is of order 12: A1(J ′0)2 +A2J ′0J0 +A3(J0)2 +A4J ′0 +A5J0 +A6 = 0. The leading coefficient is A1 = −4Q, and A2 = 8L2L3 + 2L2K0 + 2K0L3 − 4L22 − 4L23 + 4(−b+ c+ 2d)L3 + (12b− 12c+ 8d)L2 22 E.G. Kalnins and W. Miller Jr. − 2dK0 − 4cd+ 4bd− 4d2, A3 = −2L2L3 + L2K0 − L23 − L22 +K0L3 − 1 4 K2 0 + 2(−b+ c+ d)L3 + 2(7b+ c+ d)L2 + (b− c− d)K0 − b2 − c2 − d2 + 2bd+ 2bc− 2cd, A4 = 8a2L22 − 12a2K0L3 + 8a2L23 − 16a2L2L3 + 4a2K0L2 + 8a2(−b+ c− 2d)L2 − 4a2dK0 + 8a2(−b+ c− 2d)L3 + 8a2d2 − 40a2cd+ 40a2bd, A5 = +4a2L22 + 20a2L23 − a2K2 0 − 8a2L2L3 − 8a2K0L3 + 8a2(−2b+ 2c− d)L2 − 8a2(4b+ 4c+ 3d)L3 − 4a2(b− c)K0 + 4a2d2 + 12a2b2 + 16a2cd− 24a2bc + 12a2c2 + 48a2bd, A6 = −4a4L22 − 36a4L23 + 128a2L22L3H− 256a2L2L23H− 512dL22L3H2 − 512L22L23H2 + 256L32L3H2 − 256a2dL2L3H− 4a4d2 + 8a4dL2 − 24a4dL3 + 24a4L2L3 − 36a4d2 − 36a4c2 − 24a4bL2 − 40a4cL2 − 256a2(b+ c)L22H− 512(b+ c)L32H2 + 72a4bL3 + 56a4cL3 + 72a4bc− 512(b2 + c2)L22H2 + 128a2L33H+ 512a2(b+ c)L2L3H + 1024(b+ c)L22L3H2 − 256a2(b2 + c2)L2H+ 512a2bcL2H+ 1024bcL22H2 − 256a2(+c)bL23H+ 256L2L33H22− 512(b+ c)L2L23H2 + 128a2(b− c)2L3H + 256(b− c)2L2L3H2 − a4K2 0 + 24a4bd+ 104a4cd+ 12a4(c− b)K0 − 4a4L2K0 + 512a2bdL2H+ 128a2cL2K0H+ 512a2bcdH+ 128a2bdK0H− 128a2bL2K0H − 128a2cdK0H+ 1024bcdL2H2 + 256d(b− c)L2K0H2 + 512a2cL2H + 1024d(b+ c)L22H2 − 256a2d(b2 + c2 + bc+ cd)H+ 256cL2K0H2 − 512d(b2 + c2 + bd+ cd)L2H2 − 256bL22K0H2 + 4a4dK0 − 256La2dL23H − 512dL2L23H2 + 128a2d2L3H+ 256d2L2L3H2 − 32a2L3K2 0H− 64L2L3K2 0H2 + 12a4L3K0 + 512a2d(b+ c)L3H+ 1024d(b+ c)L2L3H2. By substituting (5.13) into expressions (4.1)–(4.4) and (5.4)–(5.7) for k1 = k2 = 1, and simplifying, we can recast each of these relations into polynomial structure equations in the generators. Further by squaring (5.3), substituting the structure equations for R2 1, R2 2 and R1R2 into the result and simplifying, we obtain the polynomial structure equations for R2 3. Multiplying (5.3) byR1 substituting the structure equations forR2 1 andR1R2 into the result and simplifying, we obtain the polynomial structure equations for R1R3. Similarly, multiplying (5.3) by R2 we can obtain the polynomial structure equations for R2R3. This entire construction carries over easily for the primed symmetries and their basic com- mutators (5.10). The only gap remaining to prove polynomial closure is consideration of the basic commutator R0 = {J0,J ′0} and double commutators involving J0, J ′0 simultaneously. An important observation here is that under any variable-parameter transposition R0 changes sign and R2 0 is invariant. Using this observation, we have verified that R2 0 = 65536H4 [ IxyIxzIyz − βIyz(Ixy + Ixz)− γIxz(Ixy + Iyz) − δIxy(Ixz + Iyz)− βI2yz − γI2xz − δI2xy + (β(β + 3γ + 3δ) + γδ)Iyz + (γ(γ + 3δ + 3β) + δβ)Ixz + (δ(δ + 3β + 3γ) + βγ)Ixy − 2(βγ2 + β2γ + βδ2 + β2δ + γδ2 + γ2δ) ] . In terms of our standard basis this is R2 0 = 4096H4 ( −K2 0L3 − 4βK0L2 + 4βδK0 + 4L33 − 4γδK0 + 4γK0L2 − 8βL22 − 8γL22 Structure Theory for Extended Kepler–Coulomb 3D Classical Superintegrable Systems 23 + 4L3L22 − 8L23L2 + 4β2L3 − 8βL23 + 4γ2L3 − 8γL23 − 8δL23 + 4δ2L3 + 16βγδ + 16γδL2 + 16βδL2 + 16βγL2 − 8β2L2 + 16βL3L2 − 8γ2L2 + 16γL3L2 − 8δL3L2 − 8βγL3 − 8β62δ + 16βδL3 − 8γ2δ + 16γδL3 − 8βδ2 − 8γδ2 ) . A simple consequence is {L2,R0} = 0. The expression K2 1R2 0 is a perfect square in the 6 gene- rators and gives K1R0 = −K0 2L3 − 4βK0L2 + 4βδK0 + 4L33 − 4γδK0 + 4K0 γL2 − 8L22b− 8L22γ + 4L3L22 − 8L32L2 + 4L3β2 − 8L32β + 4L3γ2 − 8L32γ − 8L32δ + 4L3δ2 + 16δβγ + 16L2δγ + 16L2δβ + 16L2βγ − 8L2β2 + 16L3L2β − 8L2γ2 + 16L3L2γ − 8L3L2 δ − 8L3βγ − 8δβ2 + 16L3δβ − 8δγ2 + 16L3δγ − 8δ2β − 8δ2γ, where the sign is determined by comparing the highest order terms. The expression J 2 1R2 0 is not a perfect square, but we can make it so by adding an appropriate, uniquely determined, multiple of the functional relation: (J1R0) 2 = J 2 1R2 0 − 642H4F . We then obtain J1R0 = 32H2 [( −2(L2 − L3)2 + (L2 + L3)K0 ) J0 − 4(L2 − L3)2J ′0 + ((−6γ + 4δ)L2 + 2(−β + γ + 2δ)L3 − δK0)J0 + ((6β + 4δ)L2 + 8δL3)J ′0 + 2δ(β − γ − δ)J0 − 4δ2J ′0 + α2 ( 4(L2 − L3)2 + (2L2 − 6L3)K0 ) + α2(−4(β − γ + 2δ)(L2 + L3)− 2δK0) + 4α2δ(5β − 5γ + δ) ] , where the sign is determined by comparing highest order terms. Note that {J1,J0} = −2J 2 0 + 128H2 ( 3L22 + L23 − 4δL2 − 2δL3 − 4L2L3 + δ2 ) + 128α2H(L2 − L3 − δ) + 8α4. The problem of computing the double commutator {J0,R0} is greatly simplified by noting that it changes sign under the transposition symmetry (x, β)↔ (y, γ). We find {J0,R0} = 512H2 [ (J ′0Iyz − J ′′0 Ixz) + δ(J ′′0 − J ′0) − γJ ′0 + βJ ′′0 + 2α2(Ixz − Iyz)− 2α2(β − γ) ] . All other commutators and products can be obtained from the preceding results by use of the discrete transposition symmetries. Thus the symmetry algebra closes polynomially, and the 6 generators obey a functional identity of order 12. 6 Final comments We have studied families of Hamiltonians that generalize the 3- and 4-parameter extended Kepler–Coulomb systems, found explicit generators for the symmetries that show these systems to be superintegrable, and worked out the explicit structure equations for the symmetry algebras determined by taking repeated Poisson brackets of the generators. We found it amazing that the structures could be computed exactly! This analysis strongly suggests that for higher order superintegrable systems in n ≥ 3 dimensions, polynomial closure of the symmetry algebra is relatively rare and dependent on additional discrete symmetry. Rational closure seems to be 24 E.G. Kalnins and W. Miller Jr. common. The structure analysis shows the fundamental importance of raising and lowering sym- metries for these systems, [5, 18]. These are nonpolynomial constants of the motion. However, all of the polynomial constants can be formed from them. We have no proof that the generators found by us are of minimal order in all cases. However, it is clear that all such generators must be expressible in terms of the basic rational raising and lowering symmetries. An obvious issue is that of quantum analogs of the classical constructions. How is the problem to be quantized? What are the operator analogs of raising and lowering symmetries and of rational closure? 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