Adding Potentials to Superintegrable Systems with Symmetry
aa r X i v : . [ n li n . S I] S e p Adding Potentials to Superintegrable Systems with Symmetry
Allan P. Fordy ∗ and Qing Huang † September 25, 2020
Abstract
In previous work, we have considered Hamiltonians associated with 3 dimensional conformally flatspaces, possessing 2, 3 and 4 dimensional isometry algebras. Previously our Hamiltonians have repre-sented free motion, but here we consider the problem of adding potential functions in the presence ofsymmetry.Separable potentials in the 3 dimensional space reduce to 3 or 4 parameter potentials for Darboux-Koenigs Hamiltonians. Other 3D coordinate systems reveal connections between Darboux-Koenigs andother well known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.
Keywords : Darboux-Koenigs metric, Hamiltonian system, super-integrability, Poisson algebra, conformalalgebra, Kepler problem.MSC: 17B63, 37J15, 37J35,70G45, 70G65, 70H06
In our previous papers [10, 11] we used the method introduced in [7] to construct kinetic energies , relatedto 3 dimensional conformally flat spaces and having 4 or 5 independent first integrals. In both these papers,it is evident that an isometry group for the corresponding metric simplified the construction of a closedPoisson algebra of integrals.When adding potentials to these kinetic energies, we have the choice of whether or not to impose invarianceunder some or all of the isometries. Clearly, imposing more symmetries restricts the potential. In [11], weconsidered isometry algebras of dimension 2, 3 and 4, since there is a “gap phenomenon” [14] which tells usthat 5 symmetries imply 6 symmetries, in which case the space has constant curvature, which we wished toavoid.In [11], we were particularly interested in using the isometries to reduce from 3 to 2 dimensions, andshowed that for each symmetry algebra, there was a universal reduction , achieved by adapting coordinates toparticular isometries. In each case, the general kinetic energy takes separable form, so is certainly Liouvilleintegrable. Furthermore, the 2 dimensional metric always has a Killing vector. In the super-integrable cases,we can reduce any additional integral which commutes with the special isometry, being used in the reduction.The super-integrable reductions are therefore, either constant curvature (possibly flat) or of Darboux-Koenigstype.For each such reduction we can add a general separable potential and investigate the restrictions imposedby requiring additional integrals. In the Darboux-Koenigs reductions, the potentials we derive just belongto the classification given in [12, 13]. However, since our 2 dimensional reductions are, in fact, embeddedin a 3 dimensional system, we can choose other coordinate systems, which relate these potentials to otherknown potentials of super-integrable systems, which can be found in the reviews [2, 4, 16]. ∗ School of Mathematics, University of Leeds, Leeds LS2 9JT, UK. E-mail: [email protected] † School of Mathematics, Center for Nonlinear Studies, Northwest University, Xian 710069, Peoples Republic of China E-mail: [email protected]
1n particular, curved space resonant oscillators (12) arise in relation to D (using the notation of [12]).Curved space generalisations of the Kepler problem arise in (23) and (31), in relation to D and in (51), inrelation to D . In the cases of (23) and (51), one of the Darboux-Koenigs first integrals is a generalisationof a component of the Runge-Lenz vector.In Section 2 we give our general notation regarding the 10 dimensional conformal algebra, used through-out. Section 3 gives a summary of important results from [11] and an outline of the general approach used inthis paper. Following [11], we then present our results for individual isometry algebras, in sequence. Sections4 and 5 are concerned with the 2D algebras h e , h i and h h , h i , while Sections 6 to 8 are concerned withthe 3D algebras h e , h , f i , h e , e , h i and h h , h , h i . In Section 9, we make some general remarks andconclusions. In [11], we considered metrics which are conformally related to the standard Euclidean metric in 3dimensions, with Cartesian coordinates ( q , q , q ). The corresponding kinetic energy takes the form H = ϕ ( q , q , q ) (cid:0) p + p + p (cid:1) . (1)A conformal invariant X , linear in momenta, will satisfy { X, H } = λ ( X ) H , for some function λ ( X ). Theconformal invariants form a Poisson algebra, which we call the conformal algebra . In flat spaces, of dimension n , the infinitesimal generators consist of n translations , n ( n − rotations , 1 scaling and n inversions ,totalling ( n + 1)( n + 2). This algebra is isomorphic to so ( n + 1 ,
1) (see Volume 1, p143, of [3]). For specialcases of ϕ ( q , q , q ) there will be a subalgebra for which { X, H } = 0, thus forming true invariants of H .These correspond to infinitesimal isometries (Killing vectors) of the metric. Constant curvature metricspossess n ( n + 1) = 6 Killing vectors (when n = 3).The conformal algebra of this 3D metric has dimension ( n + 1)( n + 2) = 10 (when n = 3). A convenientbasis is as follows e = p , h = − q p + q p + q p ) , f = ( q + q − q ) p − q q p − q q p , (2a) e = p , h = 2( q p − q p ) , f = − q q p − q − q − q ) p − q q p , (2b) e = p , h = 2( q p − q p ) , f = − q q p − q q p − q − q − q ) p , (2c) h = 4 q p − q p . (2d)The Poisson relations of the ten elements in the conformal algebra (2) are given in Table 1. Note that this isan example of the conformal algebra given in [9] (Table 3), corresponding to the case a = a = 2 , a = 0.The subalgebra g , with basis (2a) is just a copy of sl (2). We then make the vector space decomposition ofthe full algebra g into invariant subspaces under the action of g : g = g + g + g + g . The basis elements for g i have the same subscript and are given in the rows of (2).The algebra (2) possesses a number of involutive automorphisms: ι : ( q , q , q ) ( q , q , q ) , ι : ( q , q , q ) ( q , q , q ) , ι : ( q , q , q ) ( q , q , q ) ,ι ef : ( q , q , q ) (cid:18) − q q + q + q , − q q + q + q , − q q + q + q (cid:19) , whose action is given in Table 2. 2able 1: The 10-dimensional conformal algebra (2) e h f e h f e h f h e e − h − e − h − e − h h f − e f − e f f − h − f − h − f e e − h h e h − f − h h f h f e e − h − e h − f − h f − f h Table 2: The involutions of the conformal algebra (2) e h f e h f e h f h ι e h f e − h f e − h f − h ι e h f e h f e − h f h ι e h f e h f e h f − h ι ef − f − h − e − f h − e − f h − e h g In [11], we considered subalgebras of g , having dimensions 2, 3 and 4. We showed that there were initially
15 inequivalent subalgebras of interest, but then reduced this to a list of 7 subalgebras, given in Table 3.Specifically, we enumerated all such subalgebras, with bases of the form h K , . . . , K m i , m = 2 , ,
4, where K i are chosen from the list (2). A classification of subalgebras of so ( n + 1 ,
1) is given in [18], but, since theemphasis is on Lie algebras, this is up to conjugations with respect to the matrix group O (4 , In [11] we considered the general Hamiltonian H (kinetic energy) of the form: H = ϕ ( q , q , q ) (cid:0) p + p + p (cid:1) , (3)where ϕ ( q , q , q ) is chosen to be one of those listed in Table 3, corresponding to a particular isometryalgebra. 3able 3: Invariant Hamiltonians for subalgebras of g Dimension Representative ϕ of Invariant Hamiltonian2D h e , h i ϕ = ψ (cid:0) q + q (cid:1) h h , h i ϕ = ( q + q ) ψ (cid:16) q q + q (cid:17) h e , h , f i ϕ = q ψ (cid:16) q q (cid:17) h e , e , h i ϕ = ψ ( q ) h h , h , h i ϕ = ψ ( q + q + q )4D h e , h , f i ⊕ h h i ϕ = q + q h h , h , h i ⊕ h h i ϕ = q + q + q By adapting coordinates to particular symmetries, we then constructed canonical point transformations( q , q , q ) ( Q , Q , Q ), after which H took one of two forms: H = χ ( Q ) (cid:0) P + P + ρ ( Q i ) P (cid:1) , (4a) H = χ ( Q ) (cid:0) P + P + ρ ( Q i ) P (cid:1) , (4b)having respectively, P or P as a first integral. On the level sets of these conserved momenta, the corre-sponding H represents a two dimensional kinetic energy in either 1 − − ρP and ρP being considered as “potential terms”. The function ρ is shown to depend upon only one of ( Q , Q ) in thefirst case or one of ( Q , Q ) in the second. The function ρ is, in fact, totally fixed by the procedure. Remark 3.1 (Universal Reductions)
The forms of H given in (4a) and (4b) depend only upon thespecific symmetry algebra possessed by (3) and can therefore be considered as universal for the entire classof metric. In [11], we referred to these as universal reductions . It is straightforward to add separable potentials: H = χ ( Q ) (cid:0) P + P + ρ ( Q i )( P + V ( Q )) + V ( Q ) + V ( Q ) (cid:1) , (4c) H = χ ( Q ) (cid:0) P + P + ρ ( Q i )( P + V ( Q )) + V ( Q ) + V ( Q ) (cid:1) , (4d)for three arbitrary functions, V ℓ ( Q ℓ ), of a single variable. Here it is important that ρ depends upon only onevariable. In this paper we restrict to the case when H continues to possess the same conserved momentum , settingrespectively V ( Q ) = c (in (4c)) or V ( Q ) = c (in (4d)). In each case we evidently have three involutive integrals, which are respectively:
H, P , G = P + δ i ρ ( Q i )( P + c ) + V ( Q ) , (4e) H, P , G = P + δ i ρ ( Q i )( P + c ) + V ( Q ) , (4f)where δ ij is the usual Kroneker delta. In [11] we also identified specific forms of the functions ψ , of Table 3, which allowed us to constructquadratic first integrals and thus build super-integrable systems, associated with various kinetic energies H .4his process fixes the function χ ( Q ). In this paper we deform these “kinetic” first integrals with “potential”terms and require that they Poisson commute with H .In the case of (4c), if { H , F } = 0, for some quadratic (in momenta) function F , then we require { H, G } = 0 , where G = F + σ ( Q , Q ) , (5)for some function σ ( Q , Q ), to be determined. The bracket { H, G } = 0 only contains linear terms in P i , whose coefficients give formulae for the first derivatives of σ . This is an overdetermined system, whoseintegrability conditions impose conditions on the unknown functions V ( Q ) , V ( Q ), giving explicit formsfor each super-integrable case. These calculations are of the standard type, so we omit most of the details.The case of (4d) is exactly analogous and again gives explicit forms for the unknown functions V ( Q ), V ( Q ) and σ ( Q , Q ).In these universal coordinates the super-integrable cases are all of Darboux-Koenigs type with potentialsfrom the classification of [12, 13]. In other coordinate systems, these take the form of generalisations of somewell known super-integrable systems, as mentioned in the introduction. In [11] we constructed the super-integrable cases in the original Cartesian coordinates ( q , q , q ).We found that, in the case of the two dimensional algebras, all additional integrals commuted with oneof the basis elements. As a consequence, these are essentially 2 dimensional systems, so the rank of theintegrals is only 4. These systems are therefore super-integrable, but not maximally.In the case of the three dimensional algebras, we have a much bigger Poisson algebra, which has rank 5,so these systems are maximally super-integrable. However, in the universal coordinates, we can only reducethose integrals which commute with the conserved momentum (respectively P and P ). The resulting systemis again of rank 4 (including the conserved momentum).Indeed, in the case of reduction (4c), we have exactly H, P , G , G and G = { G , G } (which is cubic in momenta), with one constraint of the form G = P ( H, P , G , G ), which is polynomial of degree 6 inmomenta. The case of reduction (4d) is similar, but with P in place of P . We only explicitly present suchan algebra once, in (9).In all these cases (resulting from either 2 or 3 dimensional symmetry algebras) the resulting 2 dimensionalsystem is maximally super-integrable (rank 3). h e , h i Since this algebra is commutative, we can simultaneously adapt coordinates to both basis elements: h e , h i = h P , P i . The general Hamiltonian in this class is: H = H + U ( q , q , q ) = ψ (cid:0) q + q (cid:1) (cid:0) p + p + p (cid:1) + U ( q , q , q ) . (6)We give our universal reductions (4c) and (4d) for this case, and use (5) to determine the functions V i ( Q i )and σ for each of our super-integrable cases. h The generating function S = q P + q q + q P + 14 arctan (cid:18) q q (cid:19) P , (7a)5ives rise to a Hamiltonian of type (4c): H = ψ (cid:0) Q (cid:1) (cid:18) P + P + P + c Q + V ( Q ) + V ( Q ) (cid:19) , (7b)where we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 1 − P = const., with P + c Q (times the conformal factor) corresponding to a potential term. Since the conformal factor is a function ofonly Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = P , T h = − Q P + Q P ) , T f = ( Q − Q ) P − Q Q P , (7c) T e = P , T h = 2( Q P − Q P ) , T f = 2( Q − Q ) P − Q Q P , which satisfy the relations of g + g in Table 1, together with the algebraic constraints T e T f + 14 T h + 12 (cid:18) T e T f − T h (cid:19) = 0 , T e T f − T e T f + T h T h = 0 . (7d)The Hamiltonian (7b) has the involutive integrals H, P and G = P + V ( Q ). For the super-integrablecases, the additional integrals can be written in terms of the conformal elements (7c). ψ ( z ) = zαz + β With this ψ , the Hamiltonian (7b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the two functions V ( Q ) , V ( Q ). Each of our previousquadratic functions gives a different class of potential.Adapting our previous function F , we consider { H, G } = 0, where G = T h − α ( Q + Q ) H + σ ( Q , Q ) , with T h given in the list (7c). As discussed for (5), we obtain equations for the derivatives σ Q i , whoseintegrability condition implies restrictions on the functions V i , giving V ( Q ) = c Q + c Q , V ( Q ) = c Q , σ ( Q , Q ) = 4 c ( Q + Q ) , (8a)giving H = Q αQ + β (cid:18) P + P + P + c Q + c ( Q + Q ) + c Q (cid:19) , (8b)which is the D kinetic energy with a potential of “type B” in the classification of [12].Adding the cubic integral G = { G , G } , we find the closed Poisson algebra { G , G } = 8 αH ( P + 4 G − βH ) − G G + 8 α ( c − c ) H, (9a) { G , G } = − H ( αG − βc ) + 32 G − c P − c G − c + 16 c ) c , (9b) G = 256 αβH G − αH ( P G + 4 G G ) + 64 G G − α c + β c ) H +16 H (cid:0) βc P + 8 βc G + α (16 c − c ) G (cid:1) − c P ( P + 8 G ) − c G − c ( c − c )(2 P − βH + c − c ) − c + 16 c ) c G . (9c)6dapting our previous function F , we consider { H, G } = 0, where G = P T h + 2 αQ H + σ ( Q , Q ) , with T h given in the list (7c). Again this leads to restrictions on the functions V i , giving V ( Q ) = c Q + c Q , V ( Q ) = 14 c Q , σ ( Q , Q ) = − c Q (2 Q + Q ) − c Q + Q ) , (10a)so H = Q αQ + β (cid:18) P + P + P + c Q + 14 c (cid:0) Q + Q (cid:1) + c Q (cid:19) , (10b)which is the D kinetic energy with a potential of “type A” in the classification of [12]. Again, adding thecubic integral G = { G , G } leads to a closed Poisson algebra. Other Coordinate Systems
Using the canonical transformation (7a), we can return to the original Cartesian coordinates. TheHamiltonians (8b) and (10b) then take the forms H = q + q α ( q + q ) + β (cid:18) p + p + p + c q + q ) + c ( q + q + q ) + c q (cid:19) , (11a) H = q + q α ( q + q ) + β (cid:18) p + p + p + c q + q ) + c q + q + q ) + c q (cid:19) , (11b)respectively. These are extensions of harmonic oscillators, with h symmetry, reducing to the flat case when β = 0, being cases 5 and 6 of Table II in [4].In polar coordinates , with ( q , q , q ) = ( r cos θ, r sin θ cos ϕ, r sin θ sin ϕ ), the case (11a) is separable: H = r sin θαr sin θ + β p r + p θ r + p ϕ r sin θ + c r sin θ + c r + c r cos θ ! . (12a) Remark 4.1 (Jacobi’s Theorem)
In the initial coordinate system, ( Q i , P i ) , H and G were simultane-ously diagonalised quadratic forms (in the momenta), since G is related to separation in those coordinates.The existence of such quadratic integrals is guaranteed by Jacobi’s Theorem. On the other hand, G was notrelated to the separation of variables and is not diagonalised. In polar coordinates, G is no longer diagonal,but G is: G = 4 βr αr sin θ + β p r − α sin θβ p θ − αp ϕ β − α β ( c + 16 c tan θ ) + c r ! . Indeed, this can be derived independently, through the separation of variable process. In cylindrical polar coordinates , with ( q , q , q ) = ( z, r cos θ, r sin θ ), the case (11b) is separable: H = r αr + β (cid:18) p r + p θ r + p z + c r + 14 c r + c z + c z (cid:19) . (12b)Since Q = z , the integral G remains diagonal. 7 .2 Reduction with respect to e The generating function S = q P + 18 log (cid:0) q + q (cid:1) P + 14 arctan (cid:18) q q (cid:19) P , (13a)gives rise to a Hamiltonian of type (4d): H = 116 e − Q ψ (cid:0) e Q (cid:1) (cid:0) P + P + 16 e Q ( P + c ) + V ( Q ) + V ( Q ) (cid:1) , (13b)where we have set V ( Q ) = c to retain P = e as a first integral.This case corresponds to a conformally flat metric in the 2 − P = const., with16 e Q ( P + c ) (times the conformal factor) corresponding to a potential. Since the conformal factor is afunction of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = 14 e Q ( P sin 4 Q − P cos 4 Q ) , T h = 12 P , T f = − e − Q ( P sin 4 Q + P cos 4 Q ) , (13c) T e = 14 e Q ( P cos 4 Q + P sin 4 Q ) , T h = − P , T f = 12 e − Q ( P sin 4 Q − P cos 4 Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (13b) has the involutive integrals H, P and G = P + V ( Q ). For the super-integrablecases, the additional integrals can be written in terms of the conformal elements (13c). ψ ( z ) = αz + β With this ψ , the Hamiltonian (13b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).Adapting our previous function F , we obtain H = e − Q αe Q + β ) (cid:18) P + P + 16 e Q ( P + c ) + c + c sin (8 Q )cos (8 Q ) (cid:19) , (14a) G = − T f T f + α e Q sin (8 Q ) H + e − Q ( c sin (8 Q ) + c )32 cos (8 Q ) , (14b)which is the D kinetic energy with a potential of “type B” in the classification of [12].Adapting our previous function F = T f − T f − αe Q cos (8 Q ) H leads to an equivalent Hamiltonian. Cartesian coordinates
Using (13a) to return to the original Cartesian coordinates, (14a) takes the form H = 1 α ( q + q ) + β (cid:18) p + p + p + c + c + c q − q ) + c − c q + q ) (cid:19) , (15)which can be further simplified by a rotation of π in the q − q space.8 .2.2 The Case ψ ( z ) = √ zα √ z + β With this ψ , the Hamiltonian (13b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).Adapting our previous function F , leads to H = e − Q α e Q + β ) (cid:18) P + P + 16e Q ( P + c ) + c sin (4 Q ) + c cos (4 Q ) (cid:19) , (16a) G = − P T f + 2 β sin (4 Q ) H + c e − Q − e − Q ( c + c sin (4 Q ))4 cos (4 Q ) , (16b)which is the D kinetic energy with a potential of “type B” in the classification of [12].Adapting our previous function F = P T f + 2 β cos (4 Q ) H , a similar calculation leads to an equivalentHamiltonian. Other Coordinate Systems
Using (13a) to return to the original Cartesian coordinates, (16a) takes the form H = p q + q α p q + q + β p + p + p + c + c q q p q + q + c q ! . (17)When β = 0, this can be found in Table II (case 4) of [4].In cylindrical coordinates ( q , q , q ) = ( z, r cos θ, r sin θ ), the Hamiltonian (17) takes the form H = rαr + β (cid:18) p r + p θ r + p z + c + c cos θ r sin θ + c r sin θ (cid:19) . (18)In these coordinates, ( e , h ) = ( p z , − p θ ). h h , h i Since this algebra is commutative, we can simultaneously adapt coordinates to both basis elements: h h , h i = h P , P i . The general Hamiltonian in this class is: H = H + U ( q , q , q ) = ( q + q ) ψ (cid:18) q q + q (cid:19) (cid:0) p + p + p (cid:1) + U ( q , q , q ) . (19)We give our universal reductions (4c) and (4d) for this case, and use (5) to determine the functions V i ( Q i )and σ for each of our super-integrable cases. h The generating function S = −
14 log (cid:0) q + q + q (cid:1) P + 12 arctan q p q + q ! P −
12 arctan (cid:18) q q (cid:19) P , (20a)gives a Hamiltonian of type (4c): H = 14 cos (2 Q ) ψ (cid:0) tan Q (cid:1) (cid:0) P + P + sec (2 Q )( P + c ) + V ( Q ) + V ( Q ) (cid:1) , (20b)9here we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 1 − P = const., withsec (2 Q )( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformalfactor is a function of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = 12 e Q ( P cos 2 Q − P sin 2 Q ) , T h = P , T f = 12 e − Q ( P cos 2 Q + P sin 2 Q ) , (20c) T e = 12 e Q ( P sin 2 Q + P cos 2 Q ) , T h = P , T f = e − Q ( P sin 2 Q − P cos 2 Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (20b) has the involutive integrals H, P and G = P + V ( Q ). For the super-integrablecases, the additional integrals can be written in terms of the conformal elements (20c). ψ ( z ) = √ zα √ z + β √ z With this ψ , the Hamiltonian (20b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).Adapting our previous function F , we consider { H, G } = 0, and obtain H = cos (2 Q )4( α + β sin (2 Q )) (cid:0) P + P + sec (2 Q )( P + c ) + c e − Q + c e − Q (cid:1) , (21a) G = P T e − βe Q H − (cid:0) c e − Q + c (cid:1) sin (2 Q ) , (21b)which is the D kinetic energy with a potential of “type A” in the classification of [12].Adapting our previous function F = 2 P T f + 2 β e − Q H , we find an equivalent Hamiltonian, given by Q
7→ − Q . Other Coordinate Systems
Using (20a) to return to the original Cartesian coordinates, (21a) takes the form H = p q + q + q ( q + q ) α p q + q + q + βq p + p + p + c q + q ) + c c p q + q + q ! . (22a)This is a conformally flat extension of case 7 in Table II of [4], constant scalar curvature when β = 0. Inthese coordinates G = h + c ( q + q + q ) + c q q + q + q , (22b) G = e h − βH p q + q + q − q c + c p q + q + q ! . (22c)In polar coordinates , with ( q , q , q ) = ( r sin θ cos ϕ, r sin θ sin ϕ, r cos θ ), we have H = r sin θα + β cos θ p r + p θ r + p ϕ r sin θ + c r sin θ + c c r ! , (23a)10ith ( h , h ) = ( − rp r , p ϕ ) and G = 4 r p r + c r + c r, (23b) G = 2 p r ( p θ sin θ − rp r cos θ ) − βHr −
14 ( c + 2 c r ) cos θ. (23c)It can be seen that this is an extension of the Kepler problem (but neither flat nor fully rotationally invariant).The first integral G is an extension of one component of the Runge-Lenz vector, which is most easily seenin the Cartesian form (22c). The Hamiltonian (23a) is separable in these coordinates. ψ ( z ) = zα + βz With this ψ , the Hamiltonian (20b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).Adapting our previous function F , we consider { H, G } = 0, and find H = sin (4 Q )8(( α − β ) cos (4 Q ) + α + β ) (cid:0) P + P + sec (2 Q )( P + c ) + c e − Q + c e − Q (cid:1) , (24a) G = T e − αe Q sin (2 Q ) H + c e − Q sin (2 Q ) , (24b)which is the D kinetic energy with a potential of “type A” in the classification of [12].Adapting our previous function F = 4 T f − α e − Q sin (2 Q ) H , leads to an equivalent Hamiltonian, correspondingto Q
7→ − Q . Remark 5.1 (Variations on the “type”)
The potential of (24a) differs from the true “type A” in theterm sec (2 Q ) , which replaces the term csc (4 Q ) in the classification of [12] (taking into account a rescal-ing). In fact the classification of [12] is very dependent on the choice of integrals being used. In (21b), weuse P T e as the leading term in G (coinciding with the choice in [12]), so obtain exactly their potential.In (24b), we have used T e , which leads to this small variation. In fact, we are not choosing these integralsat this stage, since they just arise through the reduction process from some integrals, which were previouslychosen in [11]. Cartesian coordinates
In the original Cartesian coordinates, (24a) is of the form H = ( q + q ) q α ( q + q ) + βq (cid:18) p + p + p + c q + q ) + c q + q + q ) + c (cid:19) . (25)This is a conformally flat extension of case 5 in Table II of [4], constant scalar curvature when β = 0. h The generating function S = −
14 log (cid:0) q + q + q (cid:1) P + 12 log q + p q + q + q p q + q ! P −
12 arctan (cid:18) q q (cid:19) P , (26a)gives a Hamiltonian of type (4d): H = 14 ψ (cid:0) sinh Q (cid:1) (cid:0) P + P + sech (2 Q ) ( P + c ) + V ( Q ) + V ( Q ) (cid:1) , (26b)11here we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 2 − P = const., withsech (2 Q ) ( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformalfactor is a function of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = 12 e Q ( P sin 2 Q − P cos 2 Q ) , T h = P , T f = − e − Q ( P sin 2 Q + P cos 2 Q ) , (26c) T e = 12 e Q ( P cos 2 Q + P sin 2 Q ) , T h = − P , T f = e − Q ( P sin 2 Q − P cos 2 Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (26b) has the involutive integrals H, P and G = P + V ( Q ). For the super-integrablecases, the additional integrals can be written in terms of the conformal elements (26c). ψ ( z ) = (1+ z ) zα + βz With this ψ , the Hamiltonian (26b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).Adapting our previous function F , we find H = sinh (4 Q )8( β cosh (4 Q ) + 2 α − β ) (cid:0) P + P + sech (2 Q ) ( P + c )+ c (4 Q ) + c cos (4 Q ) + c sin (4 Q ) (cid:19) , (27a) G = 4( T e − T f ) − (2 T e − T f ) + 16 α cos (4 Q )sinh (2 Q ) H − c cos (4 Q )4 sinh (2 Q ) + 4 c sinh (2 Q )sin (4 Q ) , (27b)which is the D kinetic energy with a potential of “type B” in the classification of [12]. Apart from theswitch of trigonometric and hyperbolic functions, this also has a slight variation of the type described inRemark 5.1.Adapting our previous function F = ( T e − T f )(2 T e − T f ) − α sin (4 Q )sinh (2 Q ) H , we obtain an equivalentHamiltonian. Cartesian coordinates
In the original Cartesian coordinates, (27a) is of the form H = ( q + q + q ) q α ( q + q ) + βq (cid:18) p + p + p + 16 c − c q + q + q ) + 116 (cid:18) c + c q + c − c q + c q (cid:19)(cid:19) . (28)This is a conformally flat extension of case 1 in Table II of [4], constant scalar curvature when α = 0. ψ ( z ) = zα + β √ z With this ψ , the Hamiltonian (26b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integral G = P + V ( Q ), from the separation of variables. We can adda modification of one of our previous integrals to fix the functions V ( Q ) , V ( Q ) and σ ( Q , Q ) (of (5)).12dapting our previous function F , we find H = cosh (2 Q )4( α + β sinh (2 Q )) (cid:18) P + P + sech (2 Q ) ( P + c ) + c + c cos (2 Q )sin (2 Q ) (cid:19) , (29a) G = P ( T f − T e ) − β cos (2 Q ) H + (cid:18) c cos (2 Q ) + c sin (2 Q ) − c (cid:19) sinh (2 Q ) , (29b)which is the D kinetic energy with a potential of “type B” in the classification of [12]. Adding the cubicintegral G = { G , G } , we again find a closed Poisson algebra.Adapting our previous function F = P ( T f − T e )+4 β sin (2 Q ) H , leads to an equivalent Hamiltonian. Other Coordinate Systems
Using (26a) to return to the original Cartesian coordinates, (29a) takes the form H = p q + q ( q + q + q ) α p q + q + βq p + p + p + c q + q + q ) + c q + c q q p q + q ! , (30a)and G = h + c (cid:18) q + q q (cid:19) + c q p q + q q , G = h h − βq p q + q H + c q q + c ( q + 2 q )2 q p q + q . (30b)In polar coordinates , with ( q , q , q ) = ( r sin θ cos ϕ, r sin θ sin ϕ, r cos θ ), we have H = r sin θα sin θ + β cos θ p r + p θ r + p ϕ r sin θ + c r + sec ϕ r sin θ ( c + c sin ϕ ) ! , (31a)with ( h , h ) = ( − rp r , p ϕ ) and G = 4 p ϕ + sec ϕ ( c + c sin ϕ ) , (31b) G = 4 p ϕ ( p ϕ cot θ sin ϕ − p θ cos ϕ ) − β sin ϕH + 2 c sec ϕ sin ϕ + c (1 + 2 tan ϕ ) . (31c)It can be seen that this is an extension of the Kepler problem (but neither flat nor fully rotationally invariant).When β = c = c = 0, the Hamiltonian is fully rotationally invariant, with G and G reducing to productsof angular momenta. The Hamiltonian (31a) is separable in these coordinates. h e , h , f i In this (non-commutative) case we adapt coordinates to either e → P or h → P . The case of f isrelated to that of e through the involution ι ef . The general Hamiltonian in this class is: H = H + U ( q , q , q ) = q ψ (cid:18) q q (cid:19) (cid:0) p + p + p (cid:1) + U ( q , q , q ) . (32) h P The generating function S = 12 log p q + q + q + q p q + q + q − q ! P + arctan (cid:18) q q (cid:19) P −
14 log (cid:0) q + q + q (cid:1) P , (33a)13ives a Hamiltonian of type (4c): H = cos Q ψ (tan Q ) (cid:18) P + P + 14 cosh Q ( P + c ) + V ( Q ) + V ( Q ) (cid:19) , (33b)where we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 1 − P = const., with
14 cosh Q ( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformalfactor is a function of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = e Q ( P sin Q + P cos Q ) , T h = 2 P , T f = e − Q ( P sin Q − P cos Q ) , (33c) T e = e Q ( P cos Q − P sin Q ) , T h = − P , T f = 2 e − Q ( P cos Q + P sin Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (33b) has the involutive integrals H, P and G = P +
14 cosh Q ( P + c ) + V ( Q ).For the super-integrable cases, the additional integrals can be written in terms of the conformal elements(33c). ψ ( z ) = z α + βz With this ψ , the Hamiltonian (33b) takes the form given below. Apart from H and P , we have thequadratic integrals G , G : H = sin (2 Q )2( α + β ) + 2( α − β ) cos(2 Q ) (cid:18) P + P + 14 sech Q (cid:0) P + c (cid:1) + c sin (2 Q ) + c sinh Q (cid:19) ,G = P + 14 sech Q (cid:0) P + c (cid:1) + c sinh Q ,G = 12 ( T e + T f ) + β (cos 2 Q − cosh 2 Q ) sec Q H −
12 ( P + c ) cos Q sech Q + c cosh Q Q . This Hamiltonian is the D kinetic energy with a potential of “type B” in the classification of [12]. In Cartesian Coordinates
In the original Cartesian coordinates, the Hamiltonian is of the form H = q q αq + βq (cid:18) p + p + p + c q + q + q ) + c (cid:18) q + 1 q (cid:19) + c q (cid:19) . (34)This is a conformally flat extension of case 1 in Table II of [4], constant curvature when either α = 0 or β = 0. ψ ( z ) = √ z α √ z + βz With this ψ , the Hamiltonian (33b) takes the form of (35a). Apart from H and P , we have the quadraticintegrals G , G , given below. H = cos Q α + β sin Q (cid:18) P + P + 14 sech Q (cid:0) P + c (cid:1) + c sinh Q Q + c sec Q (cid:19) , (35a) G = P + 14 sech Q (cid:0) P + c (cid:1) + c sinh Q Q , (35b) G = 4 P ( T e − T f ) + 2 sec Q sinh Q ( β cos (2 Q ) − β − α sin Q ) H + sin Q (2 c + 8 c sec Q sinh Q ) . (35c)14his Hamiltonian is the D kinetic energy with a potential of “type B” in the classification of [12]. Other Coordinate Systems
Using (33a) to return to the original Cartesian coordinates, the Hamiltonian (35a) takes the form H = q p q + q α p q + q + βq p + p + p + c q + q + q ) + c q p q + q ( q + q + q ) + c q ! . (36a)When β = 0, this reduces to a constant curvature case.This is separable in polar coordinates ( q , q , q ) = ( r cos θ, r sin θ cos ϕ, r sin θ sin ϕ ): H = r sin θ cos ϕα + β sin ϕ p r + p θ r + p ϕ r sin θ + c r + c cot θ r + c sec ϕr sin θ ! . (36b) e P The generating function S = q P + arctan (cid:18) q q (cid:19) P + 12 log (cid:0) q + q (cid:1) P , (37a)gives a Hamiltonian of type (4d): H = cos Q ψ (tan Q ) (cid:0) P + P + e Q ( P + c ) + V ( Q ) + V ( Q ) (cid:1) , (37b)where we have set V ( Q ) = c to retain P = e as a first integral.This corresponds to a conformally flat metric in the 2 − P = const, with e Q ( P + c )(times the conformal factor) corresponding to a potential term. Since the conformal factor is a function ofonly Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = e Q ( P sin Q + P cos Q ) , T h = 2 P , T f = e − Q ( P sin Q − P cos Q ) , (37c) T e = e Q ( P cos Q − P sin Q ) , T h = − P , T f = 2 e − Q ( P cos Q + P sin Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (37b) has the involutive integrals H, P and G = P + e Q ( P + c ) + V ( Q ). Forthe super-integrable cases, the additional integrals can be written in terms of the conformal elements (37c). ψ ( z ) = z α + βz With this ψ , the Hamiltonian (37b) takes the form of (38a). Apart from H and P , we have the quadraticintegrals G , G , given below. H = sin (2 Q )2( α + β ) + 2( α − β ) cos(2 Q ) (cid:0) P + P + e Q ( P + c ) + c csc Q + c e Q (cid:1) , (38a) G = P + e Q ( P + c ) + c e Q , (38b) G = T f − βe − Q sec Q H + c e Q cos Q . (38c)This Hamiltonian is the D kinetic energy with a potential of “type A” in the classification of [12], but witha slight variation of the type described in Remark 5.1.15 n Cartesian Coordinates In the original Cartesian coordinates, the Hamiltonian is of the form H = q q αq + βq (cid:18) p + p + p + c + c q + c ( q + q ) (cid:19) . (39)This is a conformally flat extension of case 2 in Table II of [4], constant curvature when either α = 0 or β = 0. ψ ( z ) = √ z α √ z + βz With this ψ , the Hamiltonian (37b) takes the form of (40a). Apart from H and P , we have the quadraticintegrals G , G , given below. H = cos Q α + β sin Q (cid:16) P + P + e Q (cid:0) P + c (cid:1) + c sec Q + c e Q (cid:17) , (40a) G = P + e Q ( P + c ) + c e Q , (40b) G = 2 P T f − e − Q (3 β − β cos(2 Q ) + 4 α sin Q ) sec Q H + sin Q (2 c e − Q sec Q + c ) . (40c) This Hamiltonian is the D kinetic energy with a potential of “type A” in the classification of [12]. Other Coordinate Systems
In the original Cartesian coordinates, the Hamiltonian is of the form H = q p q + q α p q + q + βq p + p + p + c + c q + c p q + q ! . (41a)This is a conformally flat extension of case 4 in Table II of [4], constant curvature when β = 0.In polar coordinates ( q , q , q ) = ( r cos θ, r sin θ cos ϕ, r sin θ sin ϕ ), the general H and G take the form: H = r sin θ cos ϕα + β sin ϕ p r + p θ r + p ϕ r sin θ + c + c sec ϕr sin θ + c r sin θ ! , (41b) G = ( r p r + p θ ) sin θ + c r sin θ + c r sin θ. (41c)In general, (41b) separates into only two components, in ( r, θ ) and ϕ . The integral G is related to thisseparability. When c = 0, the system fully separates in these coordinates. h e , e , h i In this (non-commutative) case we adapt coordinates to either e → P or h → P . The case of e isrelated to that of e through the involution ι . The general Hamiltonian in this class is: H = H + U ( q , q , q ) = ψ ( q ) (cid:0) p + p + p (cid:1) + U ( q , q , q ) . (42) h P The generating function S = q q + q P + q P −
12 arctan (cid:18) q q (cid:19) P , (43a)16ives a Hamiltonian of type (4c): H = ψ ( Q ) (cid:18) P + P + 14 Q ( P + c ) + V ( Q ) + V ( Q ) (cid:19) , (43b)where we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 1 − P = const., with Q ( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformal factor isa function of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = P , T h = − Q P + Q P ) , T f = ( Q − Q ) P − Q Q P , (43c) T e = P , T h = 2( Q P − Q P ) , T f = 2( Q − Q ) P − Q Q P , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (43b) has the involutive integrals H, P and G = P + Q ( P + c ) + V ( Q ). For thesuper-integrable cases, the additional integrals can be written in terms of the conformal elements (43c). ψ ( z ) = αz + β With this ψ , the Hamiltonian (43b) takes the form given below. Apart from H and P , we have thequadratic integrals G , G , given below. H = 1 αQ + β (cid:18) P + P + P + c Q + c ( Q + 4 Q ) + c (cid:19) ,G = P + P + c Q + c Q , G = − P T h − (cid:0) β + αQ ) Q + αQ (cid:1) H + 8 c ( Q + 2 Q ) + 4 c Q . This is the D kinetic energy with a potential of Case 1 in [13]. In Cartesian Coordinates
In the original Cartesian coordinates, the Hamiltonian is of the form H = 1 αq + β (cid:18) p + p + p + c q + q ) + c ( q + q + 4 q ) + c (cid:19) . (44)When α = 0, this is just case 6 in Table II of [4]. ψ ( z ) = z αz + β With this ψ , the Hamiltonian (43b) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integrals G , G , given below. H = Q αQ + β (cid:18) P + P + P + c Q + c ( Q + Q ) + c (cid:19) ,G = P + P + c Q + c Q , G = P T f + 2 (cid:18) αQ − βQ Q (cid:19) H − c ( Q + Q ) Q − c Q . This is the D kinetic energy with a potential of type B in [12].17 n Cartesian Coordinates In the original Cartesian coordinates, the Hamiltonian is of the form H = q αq + β (cid:18) p + p + p + c q + q ) + c ( q + q + q ) + c (cid:19) . (45)When β = 0, this is just case 5 in Table II of [4]. e P Here we have the very simple case, of type (4d): Q = q , Q = q , Q = q ⇒ ˜ H = ψ ( Q ) (cid:0) P + P + ( P + c ) + V ( Q ) + V ( Q ) (cid:1) , (46a)where we have set V ( Q ) = c to retain P = e as a first integral.This case corresponds to a conformally flat metric in the 2 − P = const., with( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformal factor is afunction of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = P , T h = − Q P + Q P ) , T f = ( Q − Q ) P − Q Q P , (46b) T e = P , T h = 2( Q P − Q P ) , T f = 2( Q − Q ) P − Q Q P , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (46a) has the involutive integrals H, P and G = P + ( P + c ) + V ( Q ). For thesuper-integrable cases, the additional integrals can be written in terms of the conformal elements (46b). ψ ( z ) = αz + β With this ψ , the Hamiltonian (46a) takes the form given below. Apart from H and P , we have thequadratic integrals G , G , given below: H = 1 αQ + β (cid:0) P + P + P + c + c (cid:0) Q + Q (cid:1) + 2 c Q (cid:1) ,G = P + c Q + 2 c Q , G = P P − αQ ˜ H + c Q Q + c Q . This is the D kinetic energy with a potential of Case 2 in [13]. ψ ( z ) = z αz + β With this ψ , the Hamiltonian (46a) has the conformal factor of Darboux-Koenigs metric D . Apart from H and P , we have the quadratic integrals G , G , given below: H = Q αQ + β (cid:0) P + P + P + c + c ( Q + 4 Q ) + c Q (cid:1) ,G = P + 4 c Q + c Q , G = − P T h + 4 βQ Q H + (4 c Q + c ) Q . This is the D kinetic energy with a potential of type A in [12].18 Systems with Isometry Algebra h h , h , h i We give one transformation, corresponding to h → P . The transformations using either h or h areequivalent under the involutions ι and ι , respectively.The general Hamiltonian in this class is: H = H + U ( q , q , q ) = ψ (cid:0) q + q + q (cid:1) (cid:0) p + p + p (cid:1) + U ( q , q , q ) . (47) h P The generating function S = arctan p q + q q ! P + 12 log ( q + q + q ) P + 14 arctan (cid:18) q q (cid:19) P , (48a)gives a Hamiltonian of type (4c): H = e − Q ψ (cid:0) e Q (cid:1) (cid:18) P + P + 116 sin Q ( P + c ) + V ( Q ) + V ( Q ) (cid:19) , (48b)where we have set V ( Q ) = c to retain P = h as a first integral.This case corresponds to a conformally flat metric in the 1 − P = const., with
116 sin Q ( P + c ) (times the conformal factor) corresponding to a potential term. Since the conformal factoris a function of only Q , the momentum P corresponds to a Killing vector (in 2D).We have the following 6 conformal elements: T e = e Q ( P sin Q + P cos Q ) , T h = 2 P , T f = e − Q ( P sin Q − P cos Q ) , (48c) T e = e Q ( P cos Q − P sin Q ) , T h = − P , T f = 2 e − Q ( P cos Q + P sin Q ) , which satisfy the relations of g + g in Table 1, together with the algebraic constraints (7d).The Hamiltonian (48b) has the involutive integrals H, P and G = P +
116 sin Q ( P + c ) + V ( Q ). Forthe super-integrable cases, the additional integrals can be written in terms of the conformal elements (48c). ψ ( z ) = √ zα √ z + β With this ψ , the Hamiltonian (48b) takes the form given below. Apart from H and P , we have thequadratic integrals G , G , given below. H = e − Q αe Q + β (cid:18) P + P + P + c
16 sin Q + c cos Q sin Q + c e Q (cid:19) , (49a) G = P + P + c
16 sin Q + c cos Q sin Q , (49b) G = 2 P T f + (cid:0) αe Q + β (cid:1) cos Q H + c e − Q − c cos Q . (49c)This is the D kinetic energy with the potential of “type B” in [12]. Remark 8.1 (Spherically symmetric case)
When c = c = 0 , this Hamiltonian is invariant under the3D isometry algebra h Q ) P + 12 cos (4 Q ) cot Q P , Q ) P −
12 sin (4 Q ) cot Q P , P i , which is just the original isometry algebra h h , h , h i , expressed in these coordinates. In this case, G reducesto the Casimir function; G = P + P
16 sin Q = 14 (cid:18) h + h + 14 h (cid:19) . elationship with the Kepler Problem The Hamiltonian (49a) is related to the Kepler problem, which is better seen in polar coordinates, whichwe approach via the original Cartesian coordinates, given by the inverse canonical transformation, generatedby (48a): H = p q + q + q α p q + q + q + β p + p + p + 1 q + q c
16 + c q p q + q + q ! + c p q + q + q ! , (50a) G = L + 1 q + q (cid:18) c ( q + q + q ) + c q q q + q + q (cid:19) , (50b) G = e h + q β + 2 α p q + q + q p q + q + q ! H + c − q c p q + q + q , (50c) where L = (cid:0) h + h + h (cid:1) is the usual Casimir of the rotation algebra, and G is just an extension ofthe first component of Runge-Lenz vector. When c = c = β = 0 , α = 1, we have exactly (twice) the firstcomponent of K = p × L + c q p q + q + q , where L = (cid:26) − h , − h , h (cid:27) . In this rotationally invariant case, the remaining components of the Runge-Lenz vector arise via { h , G } and { h , G } .Since we have reduced with respect to h , we must choose polar coordinates which make h proportionalto p ϕ , so use the coordinate transformation; ( q , q , q ) = ( r cos θ, r sin θ cos ϕ, r sin θ sin ϕ ), leading to H = rαr + β p r + p θ r + p ϕ r sin θ + c r sin θ + c cos θr sin θ + c r ! , (51a) G = p θ + p ϕ sin θ + c + c cos θ sin θ , (51b) G = 2 p r ( p θ sin θ − rp r cos θ ) + (2 αr + β ) cos θ H + c r − c cos θ, (51c)which is clearly an extension of the Kepler problem and is spherically symmetric when c = c = 0. Remark 8.2
As a consequence, we see that the combined transformation, which is just ( Q , Q , Q ) = (cid:0) θ, log r, ( π − ϕ ) (cid:1) gives a direct relation between the Darboux-Koenigs Hamiltonian (49a) and the gener-alised Kepler problem (51a). ψ ( z ) = αz + β With this ψ , the Hamiltonian (48b) takes the form given below. Apart from H and P , we have thequadratic integrals G , G , given below. H = e − Q αe Q + β (cid:18) P + P + P + c
16 sin Q + c sin (2 Q ) + c e Q (cid:19) , (52a) G = P + P + c
16 sin Q + c sin (2 Q ) , (52b) G = T f − αe Q cos Q H + c e − Q Q . (52c)This is the D kinetic energy with the potential of “type B” in [12].20 ther Coordinate Systems We again return to the original Cartesian coordinates, given by the inverse canonical transformation,generated by (48a): H = 1 α ( q + q + q ) + β (cid:18) p + p + p + c + 4 c q + q ) + c q + c (cid:19) , (53a) G = L + ( c + 4 c ) q q + q ) + 4 c ( q + q ) q , (53b) G = p − αq H + c q , (53c)where, again, L = (cid:0) h + h + h (cid:1) is the usual Casimir of the rotation algebra.The transformation ( q , q , q ) = ( r cos θ, r sin θ cos ϕ, r sin θ sin ϕ ) gives the system in polar coordinates: H = 1 αr + β p r + p θ r + p ϕ r sin θ + c r sin θ + c r sin θ + c ! , (54a) G = p θ + p ϕ sin θ + c
16 sin θ + c sin θ , (54b) G = (cid:18) cos θ p r − sin θr p θ (cid:19) − α r cos θ H + c r cos θ , (54c)in which H is separable. It can be seen that G is related to separability in these coordinates. In this paper we have extended the results of [11] to include potentials. In fact, we initially worked withinthe universal coordinates , described in Section 3.1, in which the kinetic energy separates. Therefore, weinitially added a general separable potential, subject only to being invariant with respect to the “reducingisometry”. In these universal coordinates, in which the kinetic energies take Darboux-Koenigs form, wenaturally reproduced the potentials found in [12, 13]. However, since our 2D reductions are actually 3DHamiltonians, we can change coordinates to find some remarkable connections with other well known super-integrable systems. Clearly, the most interesting of these is the relation to the Kepler problem. Not onlydo we obtain some general, non-flat versions of the Kepler Hamiltonian, but also generalised Runge-Lenzintegrals, which are simply related to the integrals of some Darboux-Koenigs Hamiltonians.An obvious question is whether, by breaking this connection with separability, we can obtain interestingpotentials related to higher order first integrals. Some third order integrals were considered in [7], where oneof the cases of [15] was constructed, but even in 2 degrees of freedom this was complicated. This generalproblem is discussed in [19], for systems in 2 degrees of freedom, with one Killing vector.The quantum case in 2 degrees of freedom was presented in [7], but the approach has not yet been adaptedto 3 and higher degrees of freedom. Super-integrability in the quantum case is closely related to degeneracyof eigenvalues [2, 5, 6, 8], with commuting operators being used to construct eigenfunctions with the sameeigenvalue.The general classical problem in higher degrees of freedom is quite open, although there certainly existsome interesting classes in the literature [1], including extensions of some of the examples presented in thispaper.
Acknowledgements
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