Poisson Algebras and 3D Superintegrable Hamiltonian Systems
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2018), 022, 37 pages Poisson Algebras and 3D SuperintegrableHamiltonian Systems
Allan P. FORDY † and Qing HUANG ‡† School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
E-mail:
[email protected] ‡ School of Mathematics, Northwest University, Xi’an 710069, People’s Republic of China
E-mail: [email protected]
Received August 24, 2017, in final form March 06, 2018; Published online March 16, 2018https://doi.org/10.3842/SIGMA.2018.022
Abstract.
Using a Poisson bracket representation, in 3D, of the Lie algebra sl (2), we firstuse highest weight representations to embed this into larger Lie algebras. These are theninterpreted as symmetry and conformal symmetry algebras of the “kinetic energy”, related tothe quadratic Casimir function. We then consider the potentials which can be added, whilstremaining integrable, leading to families of separable systems, depending upon arbitraryfunctions of a single variable. Adding further integrals, in the superintegrable case, restrictsthese functions to specific forms, depending upon a finite number of arbitrary parameters.The Poisson algebras of these superintegrable systems are studied. The automorphisms ofthe symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enablingus to build the full set of Poisson relations. Key words:
Hamiltonian system; super-integrability; Poisson algebra; conformal algebra;constant curvature
This paper is in two parts. Sections 2–4 are mainly algebraic, building Lie algebras with a givencopy of sl (2) as a subalgebra. Since the second part of the paper (Sections 5 and 6) is aboutcompletely integrable Hamiltonian systems (and their super-integrable restrictions), the Lie al-gebraic part is presented in a Poisson bracket representation (with 3 degrees of freedom), so weare constructing Poisson algebras with linear relations. Our emphasis is on building a Poissonalgebra with a desired Lie algebraic structure.We extend the 3-dimensional algebra sl (2) to 6- and 10-dimensional algebras. The quadraticCasimir of the 6-dimensional algebra can be written in the form (with n = 3) H = 12 n (cid:88) i,j =1 g ij ( q ) p i p j . When the matrix of coefficients g ij is nonsingular, it may be considered as the inverse of a met-ric tensor g ij and the function H represents the kinetic energy of a freely moving particle onthe corresponding manifold (geodesic motion). For a metric with isometries, the infinitesimalgenerators (Killing vectors) correspond to functions which are linear in momenta and whichPoisson commute with the kinetic energy H (the corresponding Noether integrals). When thespace is either flat or constant curvature, it possesses the maximal group of isometries, which isof dimension n ( n + 1). In this case, H is actually the second order Casimir function of thesymmetry algebra (see [10]). This is exactly the case we have, with n = 3 and a 6-dimensional a r X i v : . [ n li n . S I] M a r A.P. Fordy and Q. Huangisometry algebra. The maximal number of conformal symmetries (including isometries as a sub-algebra) is of dimension ( n + 1)( n + 2) = 10, when n = 3. Our 10-dimensional extensions arejust the corresponding conformal algebras. When g ij is singular , the Poisson algebras have thesame structure, but without the geometric interpretation.Our main application of the algebraic structures we construct is to build some superintegrablesystems with nontrivial, nonlinear Poisson algebras, which generalise the Lie algebraic
Poissonalgebras of Sections 2–4. Below we give a brief reminder of the meaning of complete and super-integrability.A Hamiltonian system of n degrees of freedom, Hamiltonian H , is said to be completelyintegrable in the Liouville sense if we have n independent functions I n , which are in involution (mutually Poisson commuting), with H being a function of these and typically just one ofthem. Whilst n is the maximal number of independent functions which can be in involution ,it is possible to have further integrals of the Hamiltonian H , which necessarily generate a non-Abelian algebra of integrals of H . The maximal number of additional independent integralsis n −
1, since the “level surface” of 2 n − I k = c k , k = 1 , . . . , n −
1. Maximally superintegrable systems have a number of interestingproperties: they can be separable in more than one coordinate system; all bounded orbits areclosed; they give rise to interesting Poisson algebras with polynomial Poisson relations. Theidea can be extended to quantum integrable systems , with first integrals replaced by commutingdifferential operators. For some examples of superintegrable quantum systems it is possible to usethe additional commuting operators to build sequences of eigenfunctions [7, 9]. There is a largeliterature on the classification and analysis of superintegrable systems (see the review [14]) andthey naturally occur in many applications in physics (additional integrals being referred to as“hidden symmetries” [1]).Clearly our geodesic flow, with Hamiltonian H is super-integrable. There are 6 Noetherintegrals, but only 5 are functionally independent, since there is a quadratic constraint on the6-dimensional algebra (see equation (3.3)). Furthermore, each element of the algebra commuteswith at least one other element (see Table 1), so the Hamiltonian H belongs to several involutivetriples, each of which renders it completely integrable.In Section 5 we show how to use the symmetry algebra of the kinetic energy H to buildquadratic (in momenta) integrals, and to add potential functions to build completely integrablesystems, which are, in fact, separable. Explicitly, we extend the Hamiltonian functions H toinclude a potential function: H = H + h ( q ) , and use the symmetry algebra of H to construct two functions F i = K i + g i ( q ) , with { H, F i } = { F , F } = 0 , i = 1 , , where the functions K i are quadratic forms of the Noether constants of H . For some particularexamples, we derive the explicit form of the functions h , g i , which depend upon three arbitraryfunctions of a single variable (the three separation variables). Some examples are related througha Lie algebra automorphism, a property that would not be easy to see without the relation tothe symmetry algebra of H .In Section 6 we consider the superintegrable restrictions of these separable systems, by addingtwo further integrals. These now typically depend upon a small number of arbitrary parameters oisson Algebras and 3D Superintegrable Hamiltonian Systems 3instead of arbitrary functions. Whilst some of these functions still Poisson commute, not all ofthem can and the non-zero Poisson relations are no longer linear , but polynomial. In 3 degreesof freedom, these are considerably more complicated than in the 2 degrees of freedom case of[3, 7, 8], and it’s not clear that we can always close the algebra in a finite way. However, since ourkinetic energy has a 6-dimensional symmetry algebra with automorphisms which can be realisedas canonical transformations, these act on the nonlinear Poisson algebra of our superintegrablesystems, enabling us to obtain the full set of Poisson relations. We consider a 6-dimensional space M , with (local) canonical coordinates q i , p i , i = 1 , , { q i , q j } = { p i , p j } = 0, { q i , p j } = δ ij , for all i, j =1 , , f, g are any functions on M , then the Hamiltonian vector field of f is X f = (cid:88) i =1 ( { q i , f } ∂ q i + { p i , f } ∂ p i ) and [ X f , X g ] = − X { f,g } . Functions which are linear in momenta define vector fields on configuration space, with coordi-nates ( q , q , q ). For any function on configuration space, f ( q , q , q ), we have h ( q , p ) = (cid:88) i =1 a i ( q ) p i ⇒ { f, h } = (cid:88) i =1 a i ( q ) ∂f∂q i . (2.1)Later, we use this to represent a Killing vector by its Noether constant, which is linear inmomenta. g (cid:39) sl (2) We start with a Poisson representation of the Lie algebra sl (2), e = p , h = − q p + q p + q p ) ,f = − q q p + (cid:0) q − q − q (cid:1) p − q q p , (2.2a)satisfying { e , h } = 2 e , { f , e } = h , { f , h } = − f . (2.2b)With this choice, the corresponding Hamiltonian vector fields will satisfy the standard commu-tation rules for sl (2).We can calculate the most general function on this phase space which commutes with thewhole algebra g , which is a function of 3 variables: Proposition 2.1 (general invariant of g ) . The most general function, I , on our phase space,satisfying { e , I } = { h , I } = { f , I } = 0 , is given by I = F ( r , r , r ) , (2.3) with r = q q , r = q p + q p , r = − (cid:0) q + q (cid:1) p + q p − q q p p , where F is an arbitrary function of variables. A.P. Fordy and Q. HuangIn particular, the Casimir function is given by C = e f + 14 h = r r + (cid:0) r − (cid:1) r . (2.4) Remark 2.2.
In 2-dimensions, all invariants would be just functions of the quadratic Casimir,but in this larger space the general invariant includes all the Casimirs of larger algebras con-taining g as a subalgebra.The most general quadratic (in momenta) function of the form (2.3) is given by I = ψ ( r ) r − ϕ ( r ) r = ϕ (cid:18) q q (cid:19) q (cid:0) p − p − p (cid:1) + (cid:18) ϕ (cid:18) q q (cid:19) + ψ (cid:18) q q (cid:19)(cid:19) (cid:0) q p + q p (cid:1) . (2.5)The determinant of the matrix of coefficients, G ij , is det G = ϕ q (cid:0) ϕr + (cid:0) r − (cid:1) ψ (cid:1) . When thisis nonzero, G defines a conformally flat metric, but the Ricci scalar is generally a complicateddifferential expression in the functions ϕ and ψ , even in the diagonal case, for which ψ = − ϕ .In the diagonal case there are two interesting cases:1) ϕ = ( c r + c ) , which is a constant curvature space with R = 6 (cid:0) c − c (cid:1) ,2) ϕ = c (cid:0) r − (cid:1) , which is not a constant curvature space, but does have constant scalarcurvature R = 2 c .In Section 5.3 we consider the involutive system H = H + h, F = C + g , F = K + g , where H = I is conformally flat (but not constant curvature) and K is some element of g , todetermine separable choices of potential function. We can build the standard Lie algebra automorphisms of sl (2) as canonical transformations. Wedenote by ι and ι , the involutive automorphisms ι : ( e , h , f ) (cid:55)→ ( f , − h , e ) , (2.6a) ι : ( e , h , f ) (cid:55)→ ( − e , h , − f ) , (2.6b)which can be realised by canonical transformations, generated by S = q P − q P + q P q − q − q , S = q P − q P + q P . (2.6c)Each of the functions r , r , r of (2.3) is invariant under these automorphisms. sl (2) The calculations of this paper could be carried out for any choice of representation of sl (2).Perhaps the most natural choice would be the linear representation e = 2( Q P + Q P ) , h = 2( Q P − Q P ) , f = Q P + Q P , (2.7)which is related to the representation (2.2a) through the point transformation Q = 1 q , Q = 2 q q , Q = 2 (cid:0) q + q − q (cid:1) q . oisson Algebras and 3D Superintegrable Hamiltonian Systems 5However, the first step in our calculation of Section 2.3 is to seek functions which commute with e , so it is natural to transform e to p i for some i . The invariants of e are Q and2 Q Q − Q , and we have (cid:8) Q Q , e (cid:9) = 1, so we initially choose q = ρ ( Q ) , q = Q Q , q = Θ (cid:0) Q , Q Q − Q (cid:1) , which imply { q , h } = 2 Q ρ (cid:48) ( Q ) , { q , h } = − q , { q , h } = 2 Q Θ (cid:0) Q , Q Q − Q (cid:1) , where Θ ( y , y ) is the partial derivative of Θ( y , y ) with respect to y .If we choose to have a symmetric formula for h , so that { q i , h } = − q i , then q = 1 Q , q = σ (cid:0) Q Q − Q (cid:1) Q , with q defined up to an arbitrary function of one variable only. The inverse of this transforma-tion is just Q = 1 q , Q = 2 q q , Q = 12 q (cid:18) q q + σ − (cid:18) q q (cid:19)(cid:19) , and the corresponding canonical transformation, with (2.7), gives e = p , h = − q p + q p + q p ) ,f = − q q p + (cid:18) σ − (cid:18) q q (cid:19) q − q (cid:19) p − q q p , (2.8a)which is exactly (2.2a) when σ − ( r ) = 4 (cid:0) r − (cid:1) , so σ ( y ) = √ y + 4.In fact, given this choice of e , h , the most general form of f is determined only up to 3 arbitrary functions : f = (cid:0) uq − q q (cid:1) p + (cid:0) vq − q (cid:1) p + (cid:0) wq − q q (cid:1) p , (2.8b) u , v , w being arbitrary functions of q q . Our transformed elements (2.8a) just correspond to u = w = 0 , v = σ − . Remark 2.3.
Lie classified the 2-dimensional realisations of sl (2 , C ) and sl (2 , R ). There are 5inequivalent realisations of sl (2 , R ) (see [2, Section 2]). No such classification exists for 3-dimensional realisations, but not all choices of u , v , w in (2.8b) lead to equivalent realisations.For example, the determinant of the matrix of coefficients in (2.4) (for general u , v , w ) vanisheswhen q w (cid:0) q q (cid:1) − q u (cid:0) q q (cid:1) = 0 (as in our case) and this cannot be equivalent to a non-degeneratecase. g We now build highest weight representations of g , starting with Z = A ( q , q , q ) p + B ( q , q , q ) p + C ( q , q , q ) p and requiring (2.9b) below, which leads immediately to Z = q − λ (cid:18) A (cid:18) q q (cid:19) p + B (cid:18) q q (cid:19) p + C (cid:18) q q (cid:19) p (cid:19) , (2.9a) A.P. Fordy and Q. Huangsatisfying { Z , e } = 0 , { Z , h } = λZ . (2.9b)Defining Z i +1 = { Z i , f } , i = 1 , , . . . , (2.9c)the Poisson relations (2.2b) then imply { Z i , h } = ( λ − i + 2) Z i and { Z i , e } = ( i − λ − i + 2) Z i − . (2.9d)From this point, A , B and C are functions of r = q q .For general λ , we have an infinite-dimensional representation, but when λ = 2 m ( m a positiveinteger), it is finite, with dimension 2 m + 1. We are particularly interested in the cases m = 0 , The case m = 0 : It is easy to see that the general formula for Z is given by Z = q − λ (cid:0) ( λq A + 2 q B ) p + ( λq B + 2 q A − q C ) p + ( λq C + 2 q B ) p (cid:1) , so that, for Z = 0, we require λ = 0, A = q q C and B = 0. This leads to Z = C (cid:18) q q (cid:19) ( q p + q p ) and { Z , e } = { Z , f } = { Z , h } = 0 . (2.10)In the notation of (2.3), Z = C ( r ) r .2. The case m = 1 : For m ≥ λ = 2 m , we automatically have Z m +2 = 0, without anyrestrictions on the functions A , B , C . When m = 1, we have Z = Ap + Bp + Cp ,Z = 2( Aq + Bq ) p + 2( Aq + Bq − Cq ) p + 2( Cq + Bq ) p , (2.11) Z = 2 (cid:0) A (cid:0) q + q + q (cid:1) + 2 Bq q − Cq q (cid:1) p + 2 (cid:0) B (cid:0) q + q − q (cid:1) + 2 Aq q − Cq q (cid:1) p + 2 (cid:0) C (cid:0) q − q − q (cid:1) + 2 Aq q + 2 Bq q (cid:1) p . The Poisson bracket relations (2.9c) and (2.9d) take the explicit form { Z , h } = 2 Z , { Z , f } = Z , { Z , e } = 2 Z , { Z , f } = Z , { Z , e } = 2 Z , { Z , h } = − Z , { Z , e } = { Z , h } = { Z , f } = 0 . (2.12) We have a Lie algebra g and its action (through the Poisson bracket) on the representationspace { Z i } m +1 i =1 . We may consider the linear space spanned by g and g : g = g + g , where g = { Z i } m +1 i =1 and { g , g } ⊂ g , { g , g } ⊂ g , but need to specify the possible forms of { g , g } if we wish to consider g as a Lie algebra.Using this approach, we now build 6-dimensional algebras. The quadratic Casimir functionof the algebra g defines a matrix, which can be interpreted as an upper-index metric, when it isnon-singular. In this case its inverse defines a metric with Killing vectors corresponding to theelements of g .The calculation splits into two parts. First of all we use the Jacobi identity to derive abstractrelations. Then, in Section 3.2 we use these relations to restrict the functions in the concreterealisation of (2.11).In Section 4, we further extend to 10-dimensional algebras, which can be interpreted asconformal symmetry algebras of these metrics.oisson Algebras and 3D Superintegrable Hamiltonian Systems 7 If we consider g to have the basis Z , Z , Z , defined by (2.11), then it follows from the Poissonbracket relations (2.12), that { Z , h } = 2 Z , { Z , h } = 0 , { Z , h } = − Z , so, for this 3-dimensional invariant space we introduce the notation e = Z , h = Z , f = Z . For g to form a Lie algebra, we must have { g , g } ⊂ g + g , including the special case when g is Abelian .Noting that {{ e , h } , h } = 2 { e , h } , we have { e , h } = αe + βe , for arbitrary constants α, β. The action of f leads to { e , f } = − αh + βh , and { h , f } = − αf + βf . In fact, we may choose β = 0 without loss of generality, as shown by the following: Proposition 3.1 ( { g , g } ⊂ g ) . If the vector space g = g + g forms a Lie algebra, satisfyingthe Poisson bracket relations (2.2b) and (2.12) , then a basis can be chosen for g , satisfying { g , g } ⊂ g . Specifically, there exists a parameter a , such that { e , h } = ae , { e , f } = − ah , { h , f } = − af . (3.1) The possibility of a = 0 is included, in which case g forms an Abelian subalgebra. Proof .
Definingˆ e = e + γe ⇒ ˆ h = h − γh and ˆ f = f − γf , for arbitrary parameter γ , then { ˆ e , ˆ h } = (cid:0) α − γ (cid:1) e + ( β − γ ) e . Choosing γ = β , we have (cid:8) ˆ e , ˆ h (cid:9) = ˆ αe , where ˆ α = α − β . The action of f then leads to (cid:8) ˆ e , ˆ f (cid:9) = − ˆ αh , and (cid:8) ˆ h , ˆ f (cid:9) = − αf , giving (3.1) after dropping “hats” and setting ˆ α = a . (cid:4) Casimir functions.
This 6-dimensional algebra has a quadratic Casimir function C = 2 a (cid:18) e f + 14 h (cid:19) + 2 e f − h , (3.2)which will play an important role in what follows.As an abstract (rank 2) algebra, there is a second independent (fourth order) Casimir element C = (cid:0) e f + f e + h h + h h − f e + e f ) (cid:1) . In the 6 × e f + h h − f e = 0 . (3.3) A.P. Fordy and Q. Huang { g , g } ⊂ g The relations (3.1) impose conditions on the functions A , B , C , giving 3 subcases:1) A ( r ) = (cid:113) a r − , B ( r ) = 0, C ( r ) = r A ( r ),2) A ( r ) = 0, B ( r ) = (cid:113) − a , C ( r ) = 0, for a < B ( r ) = 0, with A ( r ) and C ( r ) satisfying the equation A (cid:48) ( r ) − r C (cid:48) ( r ) = 2 A ( r ) − C ( r ) + a r A ( r ) − C ( r ) , with C ( r ) (cid:54) = r A ( r ) . (3.4) Here we have the explicit solution (given here for a = 2): e = q p + q p (cid:112) q − q , h = 2 q ( q p + q p ) − (cid:0) q − q (cid:1) p (cid:112) q − q ,f = 2 (cid:0) q + q − q (cid:1) ( q p + q p ) − q (cid:0) q − q (cid:1) p (cid:112) q − q . In this case2 e f − h = − (cid:18) e f + 14 h (cid:19) , so the Casimir (3.2) vanishes, corresponding to a quadratic constraint between the basis ele-ments.The most general invariant of this 6-dimensional algebra is a restriction of (2.3), given by I = F ( r , r ), with the most general quadratic invariant being H = ψ ( r ) r = ψ (cid:18) q q (cid:19) ( q p + q p ) , with arbitrary function ψ. This just leads to the trivial case g = g . This is the most interesting case, depending on two arbitrary functions, subject to one differentialconstraint (3.4). The explicit form of the Casimir (3.2) is H = 2 (cid:0) q A − q C ) + a (cid:0) q − q (cid:1)(cid:1)(cid:0) p − p − p (cid:1) + 2 (cid:0) a + 2 (cid:0) A − C (cid:1)(cid:1) ( q p + q p ) , (3.5)which is a specific example of the general quadratic integral (of g ), given in (2.5). Remark 3.2 (constant curvature) . When C ( r ) (cid:54) = r A ( r ) and 2( A ( r ) − r C ( r )) (cid:54) = a (cid:0) r − (cid:1) , then the matrix of coefficients is invertible and defines a metric with constant curvature, satis-fying R ij = 1 n Rg ij , (3.6)where, in our case n = 3 and R = − a .oisson Algebras and 3D Superintegrable Hamiltonian Systems 9The six first degree (in momenta) Hamiltonian functions generate six Killing vectors (by theformula (2.1)) of the metric corresponding to the Hamiltonian (3.5). The Poisson algebra isgiven by Table 1. Table 1.
The 6-dimensional symmetry algebra g , when { g , g } ⊂ g . e h f e h f e e − h − e − h h − e f − e f f h − f − h − f e e h ae − ah h e f − ae − af f h − f ah af The Lie algebra automorphisms ι and ι The automorphisms of g , given by (2.6) also act on this extended algebra: ι : ( e , h , f , e , h , f ) (cid:55)→ (cid:18) f , − h , e , − f , − h , − e (cid:19) ,ι : ( e , h , f , e , h , f ) (cid:55)→ ( − e , h , − f , e , − h , f ) , with the Casimir function (3.5) being invariant (it being a function of the invariants r , r and r ). g is Abelian When g is an Abelian algebra, we have { e , h } = 0 , { e , f } = 0 , { h , f } = 0 , so we have the Poisson algebra of Table 1, but with a = 0, giving a 3 × g is degenerate, but the Casimircan be obtained by taking the limit of (3.2) as a →
0, giving H = 2 e f − h . (3.7)As with Case 3, above, we have B ( r ) = 0 and the functions A ( r ) and C ( r ) satisfy thedifferential constraint A (cid:48) ( r ) − r C (cid:48) ( r ) = A ( r ) − C ( r ) r A ( r ) − C ( r ) , (3.8)which is just (3.4) with a = 0.The explicit form of the Casimir (3.7) is given by H = 4( q A − q C ) (cid:0) p − p − p (cid:1) + 4 (cid:0) A − C (cid:1) ( q p + q p ) , (3.9)which is just (3.5), with a = 0, and non-degenerate when( A − r C )( C − r A ) (cid:54) = 0 , (3.10)in which case it corresponds to a flat metric when the functions A and C satisfy (3.8).0 A.P. Fordy and Q. Huang In Sections 3.2 and 3.3, we gave two classes of Poisson algebra g = g + g , with Casimirfunctions (3.5) and (3.9), corresponding (when non-degenerate) to constant curvature and flatspaces, respectively. These depend on 2 functions A ( r ) and C ( r ), which must satisfy thedifferential relations (3.4) or (3.8) respectively. In this section we consider the general solutionof these equations and some particular cases of interest.The general solution is constructed in two steps. First we reduce the problem to finding onlyone function A ( r ), with C ( r ) = 1 or C ( r ) = 0. The second step reintroduces the secondfunction. First, we note that Z i of (2.11) are only defined up to an overall multiple of a function of r ,since this is an invariant of the algebra g . Therefore, to satisfy (2.11), we have two cases1) C ( r ) (cid:54) = 0, in which case we may set C ( r ) = 1 and then determine the one function A ( r ),2) C ( r ) = 0 ⇒ A ( r ) = (cid:112) c r − a .For the case C ( r ) = 1, (3.4) takes the form A (cid:48) ( r ) = 2 A ( r ) − a r A ( r ) − , with r A ( r ) (cid:54) = 1 . We then have a number of subcases.When A (cid:48) ( r ) (cid:54) = 0, we have the general solution A = ( a − (cid:0) r + c (cid:112) a − (cid:112) − ( a − c + ( a − r (cid:1) c ( a − − , when a (cid:54) = 2 , (3.11a) A = 1 r ± (cid:112) r − c = r ∓ (cid:112) r − c c , when a = 2 . (3.11b)When A (cid:48) ( r ) = 0, then A ( r ) = (cid:114) − a . (3.11c)Clearly, when ( a − c − c by c = ( a − c − (cid:0) A + a − (cid:1) c = ( a − (cid:0) r A + ( a − r − (cid:1) ⇒ A = (2 − a ) r + 24 r , when c = 0 . (3.12a)On the other hand, when c = 0, we have the simple solution A = (2 − a ) r . (3.12b)The first form of (3.11b) allows us to set c = 0 (with the “+” sign) to obtain the special solution A = 12 r . (3.12c)oisson Algebras and 3D Superintegrable Hamiltonian Systems 11 The solution of (3.8) is just a reduction of those of (3.4), but with a = 0, giving A = r + c (cid:112) r + c − − c , (3.13a) A = r + 12 r , (3.13b) A = r , (3.13c) A = 1 , (3.13d)which are respectively reductions of (3.11a), (3.12a), (3.12b) and (3.11c). We can now reinstate the second function by writing Z = σ ( r )( A ( r ) p + p ) , (3.14)where A ( r ) is one of the solutions (3.11) or (3.12). The conditions (3.1) then imply2 σσ (cid:48) σ − a ( r − A )( r A − , (3.15)which can be directly integrated for a given solution A ( r ).We see from (3.15) that when a = 0, we generically have σ (cid:48) = 0, so can just multiply thesolutions (3.13) by an arbitrary constant. There is a singular solution of (3.15): A = r and σ arbitrary . The general formulae for the Casimirs (3.5) and (3.9) depend upon the specific functions A and C . For any solution given in Section 3.4, we can calculate the specific form of H (thecorresponding kinetic energy). Each one corresponds to a constant curvature or flat manifold,so will not all be independent. In fact, all constant curvature metrics with the same dimension,signature and scalar curvature R are isometrically related (see [5, p. 84]). Since, in our case, wehave R = − a , any two cases with the same value of a should be isometric, even though thetransformation may be difficult to find. For this choice, (3.5) takes the form H = 2 (cid:0) √ q − √ − aq (cid:1) (cid:0) p − p − p (cid:1) (3.16a)= 4 q (cid:0) p − p − p (cid:1) , when a = 2 . (3.16b)This restriction of a = 2 corresponds to A = 0, and gives the 6-dimensional isometry algebra e = p , h = − q p + q p + q p ) ,f = − q q p + (cid:0) q − q − q (cid:1) p − q q p ,e = p , h = 2( q p − q p ) ,f = − q ( q p + q p ) − (cid:0) q − q + q (cid:1) p , (3.16c)which satisfies the relations of Table 1 for a = 2. This will be embedded into the 10-dimensionalalgebra (4.3) in Section 4.3.2 and will be one of our main examples in the context of super-integrability in Section 6.2 A.P. Fordy and Q. Huang For this choice, (3.5) takes the form H = (cid:0) q + ( a − q (cid:1) q q (cid:0)(cid:0) q + q (cid:1) p − q p + 2 q q p p (cid:1) , which simplifies with the reduction a = 2 and also reduces to the flat case, with a = 0. For this choice, (3.5) takes the form H = (cid:0) q + ( a − q (cid:1) (cid:18) a (cid:0) p − p − p (cid:1) + (cid:18) a − q (cid:19) (cid:0) q p + q p (cid:1) (cid:19) = 4 q (cid:0) p − p − p (cid:1) , when a = 2 . This restriction to a = 2 is identical to (3.16b), so corresponds to the same algebra (3.16c). The flat case (3.13d) is just the case (3.11c), with a = 0, so A = 1. However, we saw that when a = 0, equation (3.15) has a constant solution, so we make the choice σ = , in which case, theCasimir (3.9) takes the form H = ( q − q ) (cid:0) p − p − p (cid:1) . (3.17a)The 6-dimensional isometry algebra now takes the form e = p , h = − q p + q p + q p ) ,f = − q q p + (cid:0) q − q − q (cid:1) p − q q p ,e = 12 ( p + p ) , h = q ( p + p ) + ( q − q ) p , (3.17b) f = (cid:0) q + ( q − q ) (cid:1) p + 2 q ( q − q ) p + (cid:0) q − ( q − q ) (cid:1) p , which satisfies the relations of Table 1 for a = 0. This will be embedded into the 10-dimensionalalgebra (4.7) in Section 4.4.2 and will be one of our main examples in the context of super-integrability in Section 6. Flat coordinates.
Since e , h , f are in involution, we can consider them as new momenta , P = e , P = h , P = f , and find new coordinates Q i , which are canonically conjugate. Thisis just Lie’s theorem on complete integrability in the Poisson case. The equations { Q i , P j } = δ ij give us a system of equations for Q i , which in the current case are easy to solve: Q = q − q − q q − q , Q = q q − q , Q = − q − q ) . (3.18)With generating function S = ( q − q − q ) P + q P − P q − q , we then have e = − Q P + Q P ) , h = 2( Q P − Q P ) , f = − Q P − Q P ,e = P , h = P , f = P , leading to H = 2 P P − P . (3.19)The form of this is dictated by the form of the Casimir (3.7). It can, of course, be diagonalisedto H = 2 P − P − P by using Q ± Q .oisson Algebras and 3D Superintegrable Hamiltonian Systems 13 In Section 3 we built 6-dimensional Poisson algebras which included g as a subalgebra. Thequadratic Casimir function was interpreted as a Hamiltonian function (the kinetic energy), withthe algebra g = g + g being its symmetry algebra . When the matrix of coefficients was non-degenerate, this defined a metric, and the symmetry algebra corresponded to its Killing vectors .In this section we further extend the algebra g to include conformal symmetries, which, in themetric case, correspond to conformal Killing vectors . In fact, we will first construct an extensionwith the appropriate Poisson bracket relations and then prove directly that these are conformalsymmetries of the above Hamiltonian. In 2 dimensions, as is well known, the conformal group is infinite . For n ≥ finite and has maximal dimension ( n + 1)( n + 2), which is achieved for conformally flat spaces (whichincludes flat and constant curvature spaces). We are particularly interested in the case n = 3,so will be looking for a 10-dimensional algebra.In flat spaces, 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 , X s , and “con-formal symmetries”, which we label X c . The “true symmetries” form a subalgebra of the con-formal symmetry algebra. Here we discuss the general structure of the conformal algebra.Suppose X s is a symmetry and X c , X c are conformal symmetries of H , satisfying { X s , H } = 0 , { X ci , H } = w i H, where w i are functions of the coordinates q , q , q . Then the Jacobi identity implies: {{ X s , X ci } , H } = −{ w i , X s } H, {{ X c , X c } , H } = ( { w , X c } − { w , X c } ) H. The symmetry X s is, of course, just a special conformal symmetry, with w = 0. Whilst it maybe that { w i , X s } = 0 for some particular choices of X s or X ci and that ( { w , X c } − { w , X c } )may or may not be zero, these relations show that conformal symmetries form an invariant spaceunder the action of the “true” symmetries and that the set of conformal symmetries (includingthe “true” symmetries) form a Lie algebra. In particular, the conformal symmetries must forman invariant space under the action of g . We start with the 6-dimensional Lie algebra g = g + g , where g is either Case 3 of Section 3.2or the Abelian case of Section 3.3. The respective Casimir functions H correspond to a spacewith non-zero, constant curvature and a space of zero curvature.We first algebraically extend g by adding a further 4 basis elements, so that, as a vector space,we haveˆ g = g + g + g + g , where g is another 3-dimensional invariant space in the form of either Case 3 of Section 3.2 orthe Abelian case of Section 3.3, and g is a 1-dimensional representation of the form (2.10). Wealready know the bracket relations { g , g } , { g , g } , { g , g } , { g , g } , { g , g } and { g , g } , { g , g } , { g , g } , { g , g } . In fact, once we have determined the first of these, the remaining pair follow by the Jacobiidentity.We introduce the following notation for the basis elements of ˆg : g k = { e k , h k , f k } , for k = 1 , , , and g = { h } , with g + g satisfying the relations given by Table 1 (with a = a , possibly zero) and g + g satisfying the relations given by Table 1 (with a = a , possibly zero). We also have that h commutes with g .For ˆg to be a Lie algebra, we must have { g , g } ⊂ g + g + g + g . Noting that {{ e , h } , h } = 2 { e , h } , we have { e , h } = αe + βe + γe , for arbitrary constants α, β, γ. We can repeat the argument of Proposition 3.1 to show that, without loss of generality, we maychoose β = γ = 0. Definingˆ e = e + µe ⇒ ˆ h = h − µh and ˆ f = f − µf , ˆ e = e + νe ⇒ ˆ h = h − νh and ˆ f = f − νf , for arbitrary parameters µ , ν , then { ˆ e , ˆ h } = ( α − µν ) e + ( β − ν ) e + ( γ − µ ) e . Choosing µ = γ , ν = β , we have { ˆ e , ˆ h } = ˆ αe , where ˆ α = α − βγ. Dropping “hats”, we have shown that { e , h } = a e , for some parameter a . The next propo-sition extends this to the whole of { g , g } , as shown in Table 2. Proposition 4.1.
Bases can be chosen for g and g , satisfying { g , g } ⊂ g + g . Specifically,there exist parameters a , γ , such that the relations shown in Table are satisfied. Table 2.
The relations for { g , g } . e h f e a e γh − a h h − a e − γh − a f f γh + a h a f Proof .
First, we note that since {{ e , e } , h } = 4 { e , e } , we have { e , e } = 0. Similarly, wefind { f , f } = 0. We then have {{ e , e } , f } = 0 ⇒ { h , e } = −{ e , h } = − a e , {{ f , f } , e } = 0 ⇒ { h , f } + { f , h } = 0 . oisson Algebras and 3D Superintegrable Hamiltonian Systems 15Further action of f leads to { h , h } + { e , f } = − a h and { f , e } + { h , h } = a h . Since {{ h , h } , h } = 0, we have { h , h } = − γh + δh , so, bracketing this with f gives { h , f } + { f , h } = 2 δf ⇒ δ = 0 . Piecing these results together, we obtain Table 2. (cid:4)
Now that we have { g , g } , we calculate { g , g } and { g , g } by using the Jacobi identity.We require γ (cid:54) = 0 if g is to enter our calculations, so, without loss of generality, we may take γ = 1, but leave a arbitrary.Since { h , h } = − h , we have { e , h } = −{ e , { h , h }} = { h , { h , e }} + { h , { e , h }} = 2( a e − a e ) , using the relations we already have. Similarly, we can derive the remaining brackets to completeTable 3. The lower part of the table is, of course, determined by skew symmetry. Table 3.
The 10-dimensional conformal algebra when { g i , g i } ⊂ g . 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 a e − a h a e h − a h a e − a e ) h − a f − a e − h − a f a h − a h ) f h + a h a f a f − a f ) e a e − a h a e − a e ) h − a f a h − a h ) f a f − a f ) h The cases for which { g , g } = and/or { g , g } = are obtained by setting a = 0 and/or a = 0. The Lie algebra automorphisms ι and ι The automorphisms of g , given by (2.6) also act on this 10-dimensional algebra: e h f e h f e h f h ι : f − h e − f − h − e − f − h − e h ι : − e h − f e − h f e − h f h Note that the four spaces g , g , g and g are each invariant. Table 3 was obtained from Table 2 by requiring algebraic consistency as an abstract Poissonalgebra. However, these Poisson relations impose additional differential relations on the functionsused to define the basis elements. We will solve the resulting equations for ( A , C ) in termsof ( A , C ), which will be arbitrary solutions of equations (3.4) or (3.8).6 A.P. Fordy and Q. HuangFrom Case 3 of Section 3.2 or the Abelian case of Section 3.3, we have e i = A i p + C i p ,h i = 2 A i q p + 2( A i q − C i q ) p + 2 C i q p ,f i = 2 (cid:0) A i (cid:0) q + q + q (cid:1) − C i q q (cid:1) p + 4 q ( A i q − C i q ) p + 2 (cid:0) C i (cid:0) q − q − q (cid:1) + 2 A i q q (cid:1) p , where i = 2 , A i ( r ), C i ( r ) satisfy either (3.4) (with parameter a → a i ) or (3.8), as well as h = C ( r )( q p + q p ) , as the basis of g .We must solve the two equations { e , h } = a e , (4.1a) { h , h } = − h , (4.1b)each of which has 3 components (the coefficients of p i ).Equations (3.4) (or (3.8)), together with the p component of (4.1a), can be used to eliminatethe derivatives A (cid:48) i ( r ) and C (cid:48) i ( r ) (for i = 2 , p component of (4.1b) gives theformula C = 4( A C − A C ) . It is then possible to solve the remaining parts of (4.1) for A ( r ) and C ( r ), but the solutiondepends upon whether or not a a = 0. { g , g } ⊂ g and { g , g } ⊂ g When a a (cid:54) = 0, we obtain A ( r ) = (cid:112) a a − a (2 r A C − A − a ) a (cid:113) (cid:0) a (cid:0) r − (cid:1) − r C − A ) (cid:1) + a A a ,C ( r ) = (cid:112) a a − a (2 r C − A C − a r ) a (cid:113) (cid:0) a (cid:0) r − (cid:1) − r C − A ) (cid:1) + a C a , (4.2) C ( r ) = 4( A C − A C ) = 4 (cid:112) a a − a ( C − r A ) (cid:113) (cid:0) a (cid:0) r − (cid:1) − r C − A ) (cid:1) , where A ( r ), C ( r ) are arbitrary solutions of equation (3.4) with a = a . Table 3 can be rearranged by re-ordering the 4 subspaces of ˆg . We can take • g + g as Killing vectors of a Casimir H , with conformal Killing vectors in the space g + g . • g + g as Killing vectors of a Casimir H , with conformal Killing vectors in the space g + g . • g + g as Killing vectors of a Casimir H , with conformal Killing vectors in the space g + g .oisson Algebras and 3D Superintegrable Hamiltonian Systems 17Since they all have the same 10-dimensional conformal algebra, they are conformally equivalentto one another. The Hamiltonian H will denote the Casimir corresponding to the sub-algebra g + g , andis given by (3.5), but with ( a, A, C ) = ( a , A , C ). This corresponds to a metric of constantcurvature , with R = − a . The 6-dimensional algebra g + g is just the symmetry algebraand g + g correspond to conformal symmetries, satisfying { e , H } = w H , { h , H } = w H , { f , H } = w H , { h , H } = w H . We need to calculate w directly, obtaining w = a a − a a ( q A − q C ) − a ( q A − q C ) , but the remaining (infinitesimal) conformal factors can be derived by using the Poisson bracketrelations of Table 3: w = { w , f } = 2 q w , w = { w , f } = 2 (cid:0) q + q − q (cid:1) w , and { h , H } = {{ h , h } , H } = {{ h , H } , h } = { w , h } H ⇒ w = 4( q A − q C ) w . Remark 4.2.
The 3 functions w i form a representation space for our algebra g . Under theaction of the Poisson bracket, we have f : ( w , w , w ) (cid:55)→ ( w , w , , h : ( w , w , w ) (cid:55)→ (2 w , , − w ) ,e : ( w , w , w ) (cid:55)→ (0 , w , w ) . The function w is invariant with respect to g . The Hamiltonian H corresponds to the sub-algebra g + g and is again of the form (3.5),but now with ( a, A, C ) = ( a , A , C ), so corresponds to a metric of constant curvature , with R = − a . The 6-dimensional algebra g + g is now the symmetry algebra and g + g correspond to conformal symmetries.Since H has the same conformal algebra as H , the corresponding metrics must be confor-mally related. To see this (on the level of the inverse metric) we use formulae (4.2) to replace A , C in H to obtain H = φ H , where φ = φ + 2 a (cid:113) (cid:0) a a − a (cid:1) a φ + a a φ , where φ = 2 a ( q A − q C ) a (cid:0) a (cid:0) q − q (cid:1) − q A − q C ) (cid:1) ,φ = q C − q A (cid:113) a (cid:0) a (cid:0) q − q (cid:1) − q A − q C ) (cid:1) ,φ = a (cid:0) q − q (cid:1) − q A − q C ) a (cid:0) q − q (cid:1) − q A − q C ) . { e , H } = 0, we have { e , H } = { e , log( φ ) } H , giving w = { e , log( φ ) } = a (cid:113) (cid:0) a a − a (cid:1)(cid:113) (cid:0) a a − a (cid:1) ( q A − q C ) − a (cid:113) a (cid:0) q − q (cid:1) − q A − q C ) . Again, with the notation { h , H } = w H , { f , H } = w H , we use the action of f tofind w = 2 q w , w = 2 (cid:0) q + q − q (cid:1) w . We can then use h = { h , h } to obtain w = − a (cid:16) a ( q A − q C ) + (cid:113) (cid:0) a a − a (cid:1)(cid:113) a (cid:0) q − q (cid:1) − q A − q C ) (cid:17) w , where { h , H } = w H . The Hamiltonian H corresponds to the sub-algebra g + g . Since g contains the singleelement h , defined by Z of (2.10), with C given by (4.2), and since { g , g } = { g , g } = ,this is an algebraically trivial extension, since it is just a direct sum . However, the Casimir, H = e f + 14 h + αh = (cid:0) r + αC ( r ) (cid:1) r + (cid:0) r − (cid:1) r = (cid:0) q + αq C (cid:1) p + 2 q q (cid:0) αC (cid:1) p p + (cid:0) q − q (cid:1) p + (cid:0) q + αq C (cid:1) p , defines a non-degenerate upper-index metric whenever α (cid:54) = 0, which is conformally equivalentto H when α = a a − a ) , satisfying H = φ H , where φ = q − q a (cid:0) q − q (cid:1) − q A − q C ) . The metric, corresponding to H , has constant scalar curvature R = −
2, but is not actuallya constant curvature metric, since it does not satisfy (3.6) and, indeed, only has a 4-dimensionalsymmetry algebra.The elements of g + g correspond to conformal symmetries of H . Defining z ki by { e k , H } = z k H , { h k , H } = z k H , { f k , H } = z k H , for k = 2 , , we again have z = { e , log( φ ) } and use the action of f to find z = 2( q A − q C ) q − q , z = 2 q z , z = 2 (cid:0) q + q − q (cid:1) z . Noting that { e , H } = { e , log( φ ) } H + φ { e , H } ⇒ z = { e , log( φ ) } + w , we find that z i are given by the same formulae as z i , but with ( A , C ) replaced by ( A , C ). Remark 4.3.
The function φ satisfies φ = 18 (cid:0) a a − a (cid:1) (cid:18) w w − w (cid:19) , which is an invariant of the representation mentioned in Remark 4.2.oisson Algebras and 3D Superintegrable Hamiltonian Systems 19 The Lie algebra automorphisms ι and ι Under the action of ι of (2.6a), we have q q (cid:55)→ q q , q − q − q (cid:55)→ q − q − q , ( H , H , H ) (cid:55)→ ( H , H , H ) . For k = 2 ,
3, the functions( w k , w k , w k , w k ) (cid:55)→ (cid:18) − w k , − w k , − w k , w k (cid:19) and ( φ , φ ) (cid:55)→ ( φ , φ ) , and similarly for z ki . The action of ι is even simpler.These automorphisms will be very important in later sections, when we discuss super-integrable systems associated with some of our Casimir functions. Consider the Hamiltonian H , which is of the form (3.5), but with ( a, A, C ) = ( a , A , C ). Theonly off-diagonal term is the coefficient of p p , which vanishes when 2 A − C + a = 0, whichthen implies that the right-hand side of (3.4) also vanishes, so we have A (cid:48) − r C (cid:48) = 0 , A A (cid:48) − C C (cid:48) = 0 ⇒ ( r A − C ) C (cid:48) = 0 . Since we require that r A − C (cid:54) = 0, we have C = c (a constant) ⇒ A = (cid:114) c − a , so H = 2 (cid:16) √ c q − (cid:113) c − a q (cid:17) (cid:0) p − p − p (cid:1) = 2 a q (cid:0) p − p − p (cid:1) , when c is chosen so that 2 c = a . This is just the case of equations (3.16). For the choice a = − a = 2 , a = 0, the conformal algebra has the explicit form: e = p , h = − q p + q p + q p ) ,f = − q q p + (cid:0) q − q − q (cid:1) p − q q p ,e = p , h = 2( q p − q p ) , f = − q ( q p + q p ) − (cid:0) q − q + q (cid:1) p , (4.3) e = p , h = 2( q p + q p ) , f = 2 (cid:0) q + q + q (cid:1) p + 4 q ( q p + q p ) ,h = − q p + q p ) , which is a 10-dimensional extension of the algebra (3.16c). In this case we have H = 4 q (cid:0) p − p − p (cid:1) , H = 4 q (cid:0) p − p − p (cid:1) ,H = (cid:0) q − q (cid:1)(cid:0) p − p − p (cid:1) . (4.4) Remark 4.4 (further automorphism) . As can be seen, the Casimir H is invariant under theinterchange 2 ↔
3, which induces the following involution ι of the 10-dimensional algebra( e , h , f , e , h , f , e , h , f , h ) (cid:55)→ (cid:18) e , h , f , e , − h , f , e , − h , f , − h (cid:19) . (4.5)This is no longer an automorphism of g , so its representation spaces are not individuallypreserved, but it is an automorphism of the symmetry algebra g = g + g and of the conformalelements g + g .0 A.P. Fordy and Q. Huang H of (4.4) as a reduction from flat space in 4-dimensions Starting with g of (4.3), we can build a 3-dimensional, highest weight representation in thespace of functions of q , q , q . We obtain y = 1 q , y = 2 q q , y = 2 (cid:0) q + q − q (cid:1) q , satisfying { ( y , y , y ) , e } = (0 , y , y ), { ( y , y , y ) , h } = (2 y , , − y ), { ( y , y , y ) , f } =( y , y , g , we need to add the function y = q q , which Poissoncommutes with the whole of g (it is just 2 r (see (2.3))). The action of g is given by { ( y , y , y , y ) , e } = (0 , , y , y ) , { ( y , y , y , y ) , h } = (0 , − y , , y ) , { ( y , y , y , y ) , f } = (2 y , , , y ) . These clearly define a linear action of g + g on { y i } i =1 , given by T x f = { f, x } , satisfying [ T x , T y ] f = − T { x,y } f. The four variables y i satisfy the quadratic constraint 2 y y − y − y = −
4, which definesa quadratic form with matrix S = − − , and the matrices T x for x ∈ g + g are “infinitesimally orthogonal” with respect to this “metric”,satisfying T x S + ST tx = 0. This means that our symmetry algebra g + g is just so (1 , S to define the corresponding Lorentzian metric, we findd s = 2d y d y − d y − d y = 4 q (cid:0) d q − d q − d q (cid:1) , corresponding to the Hamiltonian H of (4.4). { g , g } = 0 and { g , g } ⊂ g Here we must solve equations (4.1) with a = 0, so A ( r ), C ( r ) satisfy equation (3.8), while A ( r ), C ( r ) satisfy equation (3.4), with a = a , which can be either zero or non-zero. Thecalculation soon gives the choice of • a = 0, leading to A = r C , which means that the determinant condition (3.10) is notsatisfied, so the Casimir function (3.7) cannot be associated with a flat metric. • a (cid:54) = 0, which leads to a non-degenerate flat metric, but has a singular limit as a → a (cid:54) = 0 we find A = a A a + a (cid:0) q q C − (cid:0) q + q (cid:1) A (cid:1) q A − q C ) , C = a C a + a (cid:0)(cid:0) q + q (cid:1) C − q q A (cid:1) q A − q C ) ,C = 4( A C − A C ) = 2 a ( q C − q A ) q A − q C , where A ( r ), C ( r ) are arbitrary solutions of equation (3.8).oisson Algebras and 3D Superintegrable Hamiltonian Systems 21 We now have Table 3, with a = 0 and again consider various 6-dimensional subalgebras andtheir respective Casimir functions. The automorphisms (2.6) are still valid in this case. The Hamiltonian H will again denote the Casimir corresponding to the sub-algebra g + g ,and is given by (3.9), but with ( A, C ) = ( A , C ). This corresponds to a flat metric. The6-dimensional algebra g + g is just the symmetry algebra and g + g correspond to conformalsymmetries, satisfying { e , H } = w H , { h , H } = w H , { f , H } = w H , { h , H } = w H . The coefficients are calculated in the same way to give w = a q A − q C , w = 2 q w , w = 2 (cid:0) q + q − q (cid:1) w , w = 4 a . The Hamiltonian H corresponds to the sub-algebra g + g and is of the form (3.5), with( a, A, C ) = ( a , A , C ), so corresponds to a metric of constant curvature , with R = − a . The6-dimensional algebra g + g is now the symmetry algebra and g + g correspond to conformalsymmetries.Again H is conformally related to H , with H = φ H , where φ = (cid:0) a ( q A − q C ) − a (cid:0) q − q (cid:1)(cid:1) a ( q A − q C ) Defining w k by { e , H } = w H , { h , H } = w H , { f , H } = w H , { h , H } = w H , we have w = 4 a ( q A − q C ) a (cid:0) q − q (cid:1) − a ( q A − q C ) , w = 2 q w ,w = 2 (cid:0) q + q − q (cid:1) w , w = − a (cid:0) q − q (cid:1) + 2 a ( q A − q C ) a ( q A − q C ) w . The Hamiltonian H corresponds to the sub-algebra g + g , and is given by H = e f + 14 h + αh = (cid:0) r + αC ( r ) (cid:1) r + (cid:0) r − (cid:1) r = (cid:0) q + αq C (cid:1) p + 2 q q (cid:0) αC (cid:1) p p + (cid:0) q − q (cid:1) p + (cid:0) q + αq C (cid:1) p , which is non-degenerate whenever α (cid:54) = 0, and is conformally equivalent to H when α = − a ,satisfying H = φ H , where φ = − q − q q A − q C ) . As before, the metric, corresponding to H , has constant scalar curvature R = −
2, but isnot actually a constant curvature metric, since it does not satisfy (3.6) and, indeed, only hasa 4-dimensional symmetry algebra.2 A.P. Fordy and Q. HuangThe elements of g + g correspond to conformal symmetries of H . Defining z ki by { e k , H } = z k H , { h k , H } = z k H , { f k , H } = z k H , for k = 2 , , we use the action of f to find z = 2( q A − q C ) q − q , z = 2 q z , z = 2 (cid:0) q + q − q (cid:1) z , with z i being given by the same formulae as z i , but with ( A , C ) replaced by ( A , C ). The diagonalisation of the Hamiltonian H is simpler in the flat case. The only off-diagonal termis the coefficient of p p , which now vanishes when A − C = 0, so C = ± A , corresponding to H = 4( q ∓ q ) A (cid:0) p − p − p (cid:1) . For simplicity, we choose C = A = , after which we find H = ( q − q ) (cid:0) p − p − p (cid:1) , H = (cid:18) a ( q + q ) + a a ( q − q ) (cid:19) (cid:0) p − p − p (cid:1) ,H = (cid:0) q − q (cid:1)(cid:0) p − p − p (cid:1) . (4.6)The conformal algebra now has the explicit form: e = p , h = − q p + q p + q p ) , f = − q q p + (cid:0) q − q − q (cid:1) p − q q p ,e = 12 ( p + p ) , h = q ( p + p ) + ( q − q ) p ,f = (cid:0) q + ( q − q ) (cid:1) p + 2 q ( q − q ) p + (cid:0) q − ( q − q ) (cid:1) p , (4.7) e = − a ( p − p ) + a a e , h = − a ( q ( p − p ) + ( q + q ) p ) + a a h ,f = − a (cid:0)(cid:0) ( q + q ) + q (cid:1) p + 2 q ( q + q ) p + (cid:0) ( q + q ) − q (cid:1) p (cid:1) + a a f ,h = 2 a ( q p + q p ) . This is an extension of the algebra given in (3.17).
Remark 4.5.
As can be seen, this algebra is no longer invariant under the involution ι , givenby (4.5). In this section we consider Hamiltonian systems of the form H = H + h ( q ) , with the kinetic energy H being one of our diagonal cases of H , given by (4.4) or (4.6), I (of (2.5)) or H , given by (4.4).For complete integrability (in the Liouville sense) we need two functions F and F , such that H , F , F are in involution: { H, F } = { H, F } = { F , F } = 0 . (5.1)oisson Algebras and 3D Superintegrable Hamiltonian Systems 23We restrict attention to functions F i , whose dependence on momenta is at most quadratic. Suchfunctions will be the sum of two homogeneous parts, F i = F (2) i + F (0) i , and { H, F i } = 0 ⇒ (cid:8) H , F (2) i (cid:9) = 0 and (cid:8) H , F (0) i (cid:9) + (cid:8) h, F (2) i (cid:9) = 0 . The first of these means that the coefficients of p i p j in F (2) i define a second order Killing tensorof the metric corresponding to H . When this metric is constant curvature, all Killing tensorsare built as tensor products of Killing vectors (see [10]). In the Poisson representation, thisjust means that F (2) i is some quadratic form of the elements of g = g + g (of Section 4.3.2or Section 4.4.2). Since, in each case, this algebra is of rank 2, any K ∈ g will commute withexactly one other element ¯ K . Since we require (cid:8) F (2)1 , F (2)2 (cid:9) = 0, we must choose these quadraticparts to be independent quadratic form of some pair K , ¯ K . For simplicity, we will choose ourpairs to be one of e , e , or h , h or f , f . In the conformally flat case of H = I , we havea smaller symmetry algebra, but since I is no longer the Casimir function, we can use H , C and a choice of K to generate our integrals.The choice of quadratic integrals means that our systems will be separable. The calculation ofseparable potentials is standard and it is well known that in the standard orthogonal coordinatesystems, with separable kinetic energies, we can add potentials which depend upon a numberof arbitrary functions of a single variable [12]. If a complete (possessing n parameters) solutionof the Hamilton–Jacobi equation is found, then, by Jacobi’s theorem, these parameters, whenwritten in terms of the canonical variables, are quadratic (in momenta) first integrals of H . Theproblem has also been posed in the “opposite” direction: given a pair of Poisson commuting,homogeneously quadratic integrals (in two degrees of freedom) what sort of potentials can beadded, whilst maintaining commutativity? This is a classical problem (see Whittaker [16, Chap-ter 12, Section 152]) and leads to the Bertrand–Darboux equation for the potential [13, 15]. Thisapproach will be used in this section. The calculations are very similar, so the details will beomitted (with a few more included in the first case). Here we consider the Hamiltonian H = H + h ( q ) , where H = q (cid:0) p − p − p (cid:1) , with the kinetic energy H = H of (4.4) and the specific conformal algebra (4.3). h , h Consider the case of F = 14 h + g ( q ) = ( q p + q p + q p ) + g ( q ) ,F = 14 h + g ( q ) = ( q p − q p ) + g ( q ) . Each of the equations (5.1) is linear in momenta, so give us 9 equations in all. This is anoverdetermined system for the 3 functions h , g , g , which can be solved explicitly in terms of 3functions, each of a single variable. We find ∂ g = 0 , ( q ∂ + q ∂ ) g = 0 ⇒ g = ϕ (cid:18) q q (cid:19) , q ( q ∂ + q ∂ ) g = ( q ∂ + q ∂ + q ∂ ) g = 0 ⇒ g = Φ (cid:0) q , q + q (cid:1) ,q ( q ∂ + q ∂ ) h = q ∂ g ⇒ h = ψ (cid:0) q , q + q (cid:1) − q ϕ (cid:0) q q (cid:1) q + q . There are 2 more independent equations, leading to( q ∂ + q ∂ )Φ = 0 ⇒ Φ (cid:0) q , q + q (cid:1) = ϕ (cid:0) q − q − q (cid:1) , ( q ∂ + q ∂ + q ∂ ) h = q ∂ Φ ⇒ ψ (cid:0) q , q + q (cid:1) = q ϕ (cid:0) q − q − q (cid:1) q − q − q + ϕ (cid:18) q + q q (cid:19) . In summary, we have h = q ϕ (cid:0) q − q − q (cid:1) q − q − q − q ϕ (cid:0) q q (cid:1) q + q + ϕ (cid:18) q + q q (cid:19) ,g = ϕ (cid:0) q − q − q (cid:1) , g = ϕ (cid:18) q q (cid:19) . This solution immediately gives us the separation variables u = q − q − q , v = q q , w = q + q q ⇒ h = ϕ ( u )1 − w − ϕ ( v ) w + ϕ ( w ) , g = ϕ ( u ) , g = ϕ ( v ) . Remark 5.1 (action of automorphism) . Under the action of the automorphism ι of (2.6a), wehave ( u, v, w ) (cid:55)→ (cid:18) u , − v, w (cid:19) , so this solution is invariant up to redefining some arbitrary functions. e , e and f , f These two cases are connected by the action of the automorphism ι , of (2.6a). The simplestcase to calculate is with the pair e , e : F = e + g ( q ) = p + g ( q ) , F = e + g ( q ) = p + g ( q ) . (5.2a)The simple form of e and e means that we are already in separation coordinates, leading to h = − q ( ϕ ( q ) + ϕ ( q )) + ϕ ( q ) , g = ϕ ( q ) , g = ϕ ( q ) . (5.2b)The much more difficult case to calculate, involving f and f , is simply obtained by usingthe automorphism ι , which preserves H , whilst mapping e (cid:55)→ f and e (cid:55)→ f . This gives F = f + g ( q ) = (cid:0) − q q p + (cid:0) q − q − q (cid:1) p − q q p (cid:1) + g ( q ) ,F = 14 f + g ( q ) = (cid:0) q ( q p + q p ) + (cid:0) q − q + q (cid:1) p (cid:1) + g ( q ) , where g = ϕ (cid:18) − q q − q − q (cid:19) , g = ϕ (cid:18) q q − q − q (cid:19) ,h = ϕ (cid:18) q q − q − q (cid:19) − q (cid:16) ϕ (cid:16) − q q − q − q (cid:17) + ϕ (cid:16) q q − q − q (cid:17)(cid:17)(cid:0) q − q − q (cid:1) . oisson Algebras and 3D Superintegrable Hamiltonian Systems 25 Here we consider the Hamiltonian H = H + h ( q ) , where H = ( q − q ) (cid:0) p − p − p (cid:1) , (5.3)with the kinetic energy H = H of (4.6) and the specific conformal algebra (4.7). h , h Consider the case of F = 14 h + g ( q ) = ( q p + q p + q p ) + g ( q ) ,F = h + g ( q ) = ( q ( p + p ) + ( q − q ) p ) + g ( q ) . The relations { H, F } = { H, F } = { F , F } = 0 lead to h = ( q − q ) ϕ (cid:0) q − q − q (cid:1) q − q − q − ϕ (cid:18) q − q q (cid:19) + ϕ (cid:18) q − q q − q ( q − q ) (cid:19) ,g = ϕ (cid:0) q − q − q (cid:1) , g = ϕ (cid:18) q − q q (cid:19) . Again, we have the separation variables u = q − q − q , v = q − q q , w = 2 q − q q − q ( q − q ) , with h = ϕ ( u ) w − − ϕ ( v ) + ϕ ( w ) , g = ϕ ( u ) , g = ϕ ( v ) , and again we have ( u, v, w ) (cid:55)→ (cid:0) u , − v, w (cid:1) , under the automorphism ι , so the solution is invariantup to redefining some arbitrary functions. e , e and f , f These two cases are again connected by the action of the automorphism ι . The simplest caseto calculate is with the pair e , e : F = e + g ( q ) = p + g ( q ) , F = 4 e + g ( q ) = ( p + p ) + g ( q ) . The relations { H, F } = { H, F } = { F , F } = 0 lead to h = − ( q − q ) ϕ ( q ) + q ( q − q ) ϕ (cid:48) ( q − q ) + ϕ ( q − q ) ,g = ϕ ( q ) , g = ϕ ( q − q ) . Again, the case involving f and f is simply obtained by using the automorphism ι , whichpreserves H , and still maps e (cid:55)→ f and e (cid:55)→ f . This gives F = f + g ( q ) = (cid:0) − q q p + (cid:0) q − q − q (cid:1) p − q q p (cid:1) + g ( q ) ,F = f + g ( q ) = (cid:0)(cid:0) q + ( q − q ) (cid:1) p + 2 q ( q − q ) p + (cid:0) q − ( q − q ) (cid:1) p (cid:1) + g ( q ) , where g = ϕ (cid:18) q q − q − q (cid:19) , g = ϕ (cid:18) q − q q − q − q (cid:19) ,h = − ( q − q ) ϕ (cid:16) q q − q − q (cid:17)(cid:0) q − q − q (cid:1) + q ( q − q ) ϕ (cid:48) (cid:16) q − q q − q − q (cid:17)(cid:0) q − q − q (cid:1) + ϕ (cid:18) q − q q − q − q (cid:19) . In this section we consider H to be the diagonal case of I (see (2.5)) H = H + h ( q ) , where H = ϕ (cid:18) q q (cid:19) q (cid:0) p − p − p (cid:1) . (5.4)We exclude the case ϕ ( r ) = ( c r + c ) , since this corresponds to the constant curvature caseof Section 4.3.2. Generally, this H has only 3 symmetries ( g ), but in the case ϕ = c ( r −
1) ithas a fourth symmetry and corresponds to the case H = H of (4.4), with symmetry algebra g + g .For generic ϕ ( r ), the kinetic energy H has the symmetry algebra g . It is easy to checkthat H , C (of (2.4)) and K (for any element K of g ) are functionally independent, so we usethese to construct some associated involutive systems, with H = H + h, F = C + g , F = K + g , (5.5)where h , g , g are arbitrary functions of q , q , q .We just present the results. The calculations are straightforward. The Case K = e Involutivity of (5.5) leads to h = ϕ ( r ) + q ϕ ( r ) q − q (cid:0) ϕ (cid:0) q − q (cid:1) + (cid:0) q − q (cid:1) ϕ ( q ) (cid:1) ,g = ϕ (cid:0) q − q (cid:1) + (cid:0) q − q (cid:1) ϕ ( q ) , g = ϕ ( q ) , (5.6)which gives the separation variables u = q q , v = q − q , w = q , in terms of which H = (cid:0) u − (cid:1) ϕ ( u ) p u + ϕ ( u ) − ϕ ( u ) u − F ,F = 4 v p v + ϕ ( v ) + vF , F = p w + ϕ ( w ) . Remark 5.2 (the involution ι ) . We can use the involution ι to transform this system to anequivalent one for which K = f . The case K = h Involutivity of (5.5) leads to h = ϕ ( r ) + q ϕ ( r ) q − q (cid:32) ϕ (cid:18) q − q q (cid:19) + q − q (cid:0) q − q − q (cid:1) ϕ (cid:0) q − q − q (cid:1)(cid:33) ,g = ϕ (cid:18) q − q q (cid:19) + q − q (cid:0) q − q − q (cid:1) ϕ (cid:0) q − q − q (cid:1) , g = ϕ (cid:0) q − q − q (cid:1) , (5.7)which gives the separation variables u = q q , v = q − q q , w = q − q − q , in terms of which H = ( u − ϕ ( u ) p u + ϕ ( u ) − ϕ ( u ) u − F ,F = 4 v ( v + 1) p v + ϕ ( v ) + v v + 1) F , F = 16 w p w + ϕ ( w ) . Remark 5.3 (the involution ι ) . This system is invariant under the action of the involution ι ,up to a relabelling of ϕ , since q − q − q (cid:55)→ q − q − q .oisson Algebras and 3D Superintegrable Hamiltonian Systems 27 H of (4.4) The kinetic energy H = H of (4.4) is a specific example of H of (5.4), corresponding to ϕ ( r ) = 1 − r , and has the 4-dimensional symmetry algebra g + g , with basis e , h , f , h ,with h commuting with the whole of g . Consequently the cases (5.6) and (5.7) simply reduceto this choice of ϕ ( r ). However, there are additional possibilities involving the element h .Since H = C − h , we cannot use (5.5), with K = h . We can, however, use H , h and anyelement K of g . The commuting pair h , e With the choice F = 116 h + g ( q ) , F = e + g ( q ) , a simple calculation leads to h = ϕ (cid:0) q − q (cid:1) + (cid:0) q − q (cid:1) ϕ ( q ) − ϕ (cid:18) q q (cid:19) , depending upon 3 arbitrary functions, with g = ϕ (cid:16) q q (cid:17) and g = ϕ ( q ).We can use ι to derive an equivalent system with e replaced by f . The commuting pair h , h With the choice F = 116 h + g ( q ) , F = 14 h + g ( q ) , a simple calculation leads to h = ϕ (cid:18) q − q q (cid:19) + (cid:0) q − q (cid:1) ϕ (cid:0) q + q − q (cid:1) q + q − q − ϕ (cid:18) q q (cid:19) , depending upon 3 arbitrary functions, with g = ϕ (cid:0) q q (cid:1) and g = ϕ (cid:0) q + q − q (cid:1) .This system is invariant (up to a simple redefinition of ϕ ) under the action of ι . In this section we consider the possibility of adding further integrals, F , F , to separable casesof Section 5, which can no longer be in involution with H , F , F , but should Poisson commutewith H itself: { H, F } = { H, F } = 0. Having any additional integrals, the system is referredto as super-integrable . The functions should be chosen to be functionally independent , so theJacobian matrix ∂ ( H, F i ) ∂ x , where x = ( q , . . . , p ) , has maximal rank . Whilst the maximal rank for a set of functions in this space is 6, the maximalrank for a set of first integrals is 5, since in this case, the level surface S = { x : H = c , F i = c i } i =1 , one , so represents an (unparameterised) trajectory of the dynamical system.A super-integrable system with the maximal number of functionally independent integrals iscalled maximally super-integrable . “Solving” the system of equations defining S , gives thesolution, but this cannot in general be determined explicitly. Being only 5 equation in a 6-dimensional space, this solution will depend upon a single additional parameter (as well as theparameters c i ), which will be some function of t , but not necessarily t itself.If we start with a separable system of Section 5, depending upon 3 arbitrary, single-variablefunctions, then each additional integral imposes differential constraints on these arbitrary func-tions. Our maximally super-integrable systems depend on a finite number of arbitrary parame-ters , whose coefficients are specific functions (rational in our examples). The set of functions, { H, F i } i =1 , will then generate a non-Abelian Poisson algebra, which may or may not be finite-dimensional.To simplify all of these calculations, we choose F , F to be a pair of functions whose leadingorder parts (in momenta) commute , but allow for the case { F , F } (cid:54) = 0, in which case { F , F } = (cid:88) i =1 X i ( q ) p i is a first order integral.By choosing the leading order terms of each integral { F i } i =1 to be just K , for some ele-ment K of the symmetry algebra g , the automorphisms ι and ι (where appropriate) of g induce corresponding automorphisms of the Poisson algebra generated by { F i } i =1 . This will beimportant when deriving the Poisson relations on the full Poisson algebra. Here we have the symmetry algebra (4.3), with Casimir H = q (cid:0) p − p − p (cid:1) . We start withthe involutive system given in (5.2), with integrals H = q (cid:0) p − p − p (cid:1) + h ( q ) , F = e + g ( q ) = p + g ( q ) ,F = e + g ( q ) = p + g ( q ) , where the potential functions, h ( q ), g ( q ), g ( q ), are given by (5.2b). We then add two furtherfunctions F = f + g ( q ) and F = 14 f + g ( q ) , where f , f are defined in the list (4.3). When we impose the conditions { H, F } = { H, F } = 0,it is a simple calculation to derive the following solution: h = q (cid:18) k q + k q (cid:19) , g = − k q , g = − k q ,g = − k (cid:0) q + q − q (cid:1) q , g = − k (cid:0) q + q − q (cid:1) q . (6.1)In this case, we also find that h is a first integral.We therefore have 6 first integrals ( H, F , F , F , F , h ), but the rank of the Jacobian is 5, sothere should be an algebraic relation between them. Nevertheless, we consider these 6 functionsas generators of our Poisson algebra. Under the action of the involutions (automorphisms of thesymmetry algebra) ι and ι , we have ι : ( H, F , F , F , F , h , k , k ) (cid:55)→ ( H, F , F , F , F , − h , k , k ) ,ι : ( H, F , F , F , F , h , k , k ) (cid:55)→ ( H, F , F , F , F , h , k , k ) , oisson Algebras and 3D Superintegrable Hamiltonian Systems 29so the entire Poisson algebra should obey such symmetry rules. We can use this in the derivationof the Poisson algebra. For example, if we know the formula for { F , F } , then we can use ι to deduce the formula for { F , F } . Whenever we introduce a new element of our algebra, weshould simultaneously introduce any new elements which are derived through the action of theseinvolutions. In this way we add a further 5 elements, which, by construction, satisfy { H, F i } = 0,for all i .Some Poisson relations are very simple to derive { F , F } = { F , F } = 0 and { F i , h } = λ i F i , i = 1 , . . . , , where λ = (4 , , − , − , , , , , − { F , F } , { F , F } , { F , F } , { F , F } , are all cubic in momenta andcould be linear combinations of { h F i , h H, h } i =1 , but this is not the case. However, note that { F , F } and { F , F } are related through the involutions, as are { F , F } and { F , F } . We candefine two new quadratic elements F , F through the relations { F , F } = h (cid:0) h − H − F − k (cid:1) , { F , F } = h (cid:0) h − H − F − k (cid:1) , with F ↔ F under ι . These functions can be written F = q F + q F + q p ( q p + 2 q p ) , F = q F + q F + q p ( q p + 2 q p ) . We define F , F by the equations { F , F } = 2 h F + 4 F , { F , F } = − h F + 4 F , related by F ↔ F under ι .The function F is defined by the second of the following equations { F , F } + { F , F } = 8 h (cid:18) H + F + F − h (cid:19) , { F , F } − { F , F } = 16 F , after which, we find that { F , F } = 4 F .The action of the two involutions is then given by: H F F F F F F F F F h k k ι : H F F F F F F F F F − h k k ι : H F F F F F F − F − F − F h k k The action of ι on { F , F } and { F , F } then gives { F , F } = 2 h F − F , { F , F } = − h F − F . This phenomenon of connecting four different Poisson relations through the involutions is de-picted in Fig. 1(a), where we define P ij = { F i , F j } (see Table 4). Sometimes only two relationsare connected, such as with P and P (Fig. 1(b)), or even just one , such as with P , becauseof invariance properties. Remark 6.1 (commutativity) . The actions of ι and ι commute on these functions.After inputting these known brackets, it is possible to use the Jacobi identity to derive allthe others. The full set of Poisson relations is given in Table 4, with the array P = ( P ij ), usingthe order ( F , F , F , F , F , F , F , F , F , F = h ). The lower part of the matrix is given by0 A.P. Fordy and Q. Huang P P P P ι ι ι ι (a) Four connected bracket relations. P P (cid:45) ι (b) Two connected bracket relations. Figure 1.
Bracket relations connected through ι and ι . skew-symmetry. In this context, H is just a parameter, since it commutes with all 10 elements.We find P = h (cid:0) h − H + F + k ) (cid:1) ,P = h (cid:0) H + F + F ) − h (cid:1) + 8 F ,P = P = 12 (( F + F ) (cid:0) H + 4 F − h (cid:1) + 4( F F + k F )) ,P = 2(2( F + H )(2( F + H ) + F ) − F F ) − h (3 F + 2 F + 4 H − k ) + 12 h − k ( F + H + k ) + F k ) ,P = H (4 F − h + 4 H ) + 12 F (cid:0) F − h + 12 H (cid:1) + 2 F − k + k ) F + 12 k (cid:0) h − H (cid:1) − k k ,P = 14 h ( F + F ) (cid:0) h − H + F + F ) (cid:1) . From the list given above, we can derive all except P by using the involutions (as in Fig. 1).The relevant groupings are( P , P ) , ( P , P ) , ( P , P , P , P ) , ( P , P , P , P ) , ( P , P , P , P ) , ( P , P ) , ( P , P ) . For example, applying ι to the formula for P , we get P = h ( h − H + F + k )). Theequality P = P follows from the Jacobi identity for the elements F , F , F . The mostcomplicated entry in matrix P is P , which is not obtainable in this way, since it is invariant(up to a sign) under both involutions: P = 2( F F − F F ) + 14 h (cid:0) H + F + F ) − h (cid:1)(cid:0) h − H − F − F (cid:1) + 12 k h (cid:0) H + F + F ) − h (cid:1) + k h (cid:0) H + F ) − h (cid:1) + 4 h k k . Under ι , P (cid:55)→ − P , as it should. Under ι , we should have P (cid:55)→ P , but, in fact, P (cid:55)→ P + I , where I = 2( F + F ) F − F + F ) F + 12 h ( F − F ) (cid:0) H + F + F − k − k ) − h (cid:1) + 12 ( k − k ) h (cid:0) H − h (cid:1) , which satisfies I (cid:55)→ − I under both involutions. However, this does not pose a contradiction,since in the explicit form of the Poisson algebra, I = 0.oisson Algebras and 3D Superintegrable Hamiltonian Systems 31 Table 4.
The 10-dimensional Poisson algebra { F i , F j } = P ij . P = P P F + 2 F h F F P P F P P F h − F − F F P P F F − F h P P F F − F − F − F h P P − F F − F F P P P P P P P P F P − F , The functions (
H, F , F , F , F ) are functionally independent integrals of the Hamiltoniansystem, with Hamiltonian H . The functions ( F , F , F , F , F , h ) must therefore satisfy sixfunctionally independent relations. These can be obtained by looking at the full set of Jacobiidentity relations, some of which have non-trivial entries, all of which vanish when the functionalforms of F i are inserted. We label such functions by the Jacobi identity which gave rise to them,so J ijk is (up to an overall constant multiple) the entry corresponding to F i , F j , F k . These alsobelong to “families”, which are related through the action of the involutions. Six such functionsare J = 2 F F − F F + 2 F F − F H + F h + 4 k F ,J = 2 F F − F F + 2 F F − F H + F h + 4 k F ,J = 2 F (cid:0) h + 4 H − k + 4 F (cid:1) − F F + 4 h ( F H + F k + F k + ( F + F ) F ) − F h ,J = 2 F (cid:0) h + 4 H − k + 4 F (cid:1) − F F − h ( F H + F k + F k + ( F + F ) F ) + F h ,J = − F (cid:0) h + 4 H − k + 4 F (cid:1) + 8 F F + 4 h ( F H + F k + F k + ( F + F ) F ) − F h ,J = − F (cid:0) h + 4 H − k + 4 F (cid:1) + 8 F F − h ( F H + F k + F k + ( F + F ) F ) + F h . These functions satisfy J ijk = 0, and their Jacobian (with respect to the functions H, F i ) hasrank 6, thus giving us the necessary six relations on our algebra. Remark 6.2 (action of the involutions) . J and J are invariant under ι and transforminto one-another (up to sign) under ι . J , J , J and J are also connected, as depictedin Fig. 2.The very simple relation J = F F − F F + 2 F h , also exists and is invariant (up to sign) under the action of both involutions. Remark 6.3 (comparison with the literature) . The Hamiltonian H , with potential h givenin (6.1), can be written H = q (cid:18) p − p − p + k q + k q (cid:19) , J J J J ι ι ι ι Figure 2.
Constraints connected through ι and ι . which is in the form of a St¨ackel transform to a flat metric, so can be compared with theclassification given in [6]. We can compare with the list of non-degenerate potentials (albeit inthe quantum case with H replaced by a Laplacian) given in [6, Section 5]. Allowing for thefact that this classification is for the Euclidean case and involves some complex coordinates, theabove potential can be considered as a reduction of the 5 parameter potential V [2 , , , of [6]; itis, in fact a reduced case, with our parameters ( k , k ) corresponding to their ( a , a ). The simplest flat case, with H given by (5.3), has integrals of the form F = e + g , F = 4 e + g , F = f + g , F = f + g , since this is invariant under the action of the involution ι , so the Poisson algebra possesses thisautomorphism. This is the only case we consider here.The functions ϕ , ϕ of Section 5.2.2 are constrained by the additional integrals, giving h = k ( q − q ) q + 2 k q q − q , g = − k q , g = − k ( q − q ) ,g = − k (cid:0) q − q − q (cid:1) q , g = − k (cid:0) q − q − q (cid:1) ( q − q ) . (6.2)In this case, we also find that h is a first integral.We therefore have 6 first integrals ( H, F , F , F , F , h ), but the rank of the Jacobian is 5, sothere should be an algebraic relation between them. Nevertheless, we consider these 6 functionsas generators of our Poisson algebra. Under the action of the involution (automorphism of thesymmetry algebra) ι , we have ι : ( H, F , F , F , F , h , k , k ) (cid:55)→ ( H, F , F , F , F , − h , k , k ) , so the entire Poisson algebra should obey such symmetry rules, induced by the definitions below.We have { F , F } = { F , F } = 0 and the cubic expression { F , F } = 4 h F factorises ,giving us a new quadratic integral F , which is invariant under the action of ι . Two new cubic integrals (related through ι ) are defined by { F , F } = 8 F and { F , F } = 8 F , whilst { F , F } is just linear : { F , F } = − k h . The first four brackets with F are { F , F } = 2 h F , { F , F } = 4 F , { F , F } = − h F , { F , F } = 4 F , giving us two new cubic integrals. We find the following factorisation: F + F = h F , oisson Algebras and 3D Superintegrable Hamiltonian Systems 33which defines another quadratic integral. All remaining brackets can be determined in terms ofthe above F i and h . In these relations, H acts as a parameter, since it (by definition) commuteswith all F i . The action of the involution ι is then given by: H F F F F F F F F F F h ι : H F F F F F − F − F F F F − h The integral h acts diagonally: { F i , h } = λ i F i , i = 1 , . . . , , where λ = (4 , , − , − , , , , , − , . Defining P ij = { F i , F j } , the remaining independent entries in the Poisson matrix are P = 2 F ( H + 3 F − k ) + 4 k F , P = 2( F F + F h ) , P = 2 F F ,P = − F ( H + 2 F − k ) − F F − k F , P = 4 F , P = − k F ,P = 4( F F + k F ) , P = 0 , P = 4 F ( k − H − F ) + 8 k ( F + 2 k ) ,P = 0 , P = 2( F ( H + 2 F − k ) − F h + 2 k F ) ,P = F F − F h − k F − F ( H + 3 F − k ) , P = 2( F − F ) ,P = F F − F F + h ( F ( H − k ) − k (cid:0) F + 2 k ) + F (cid:1) ,P = 2( F F + F ( − F − H + k )) , P = 2 F (2 F + H − k ) − F F ,P = 2 (cid:0) F ( H − k ) − k ( F + 2 k ) + F (cid:1) ,P = − h (cid:0) F H − F k + 2 F − k k (cid:1) , P = 4 F k + 2 F F . The remaining entries can be obtained by using the involution ι .The functions ( H, F , F , F , F ) are functionally independent integrals of the Hamiltoniansystem, with Hamiltonian H . The functions ( F , F , F , F , F , F , h ) must therefore satisfyseven functionally independent relations. Again, these can be obtained by looking at the full setof Jacobi identity relations, some of which have non-trivial entries, all of which vanish when thefunctional forms of F i are inserted. We label such functions by the Jacobi identity which gaverise to them, so J ijk is (up to an overall constant multiple) the entry corresponding to F i , F j , F k .These also arise in “pairs”, which are related through the action of the involution ι . The first,in fact, comes from the definition of F (which satisfies I (cid:55)→ − I under the action of ι ): I = F + F − h F , J = F F − F F + ( F − F ) h ,J = F ( F + H − k ) + F ( F + 2 k ) − F h ,J = F ( F + H − k ) + F ( F + 2 k ) + F h ,J = F F − F ( F + H − k ) + k h ( F + 2 k ) ,J = F F + F ( F + H − k ) − k h ( F + 2 k ) ,J = F F − F ( F + H − k ) + k h F . We saw that the flat coordinates (3.18) reduce this H to the form (3.19). In these coordinates,the first 5 (functionally independent) integrals of our Poisson algebra take the form H = 2 P P − P + k Q + k (cid:0) Q − Q Q (cid:1) ,F = 4 (cid:18) Q P + Q (cid:18) P − k Q (cid:19) + 2 Q Q P P (cid:19) , F = 4 (cid:0) P − k Q (cid:1) , F = ( Q P + Q P ) − k Q Q , F = P − k Q . The functions F , . . . , F can similarly be found and since the transformation is canonical, thePoisson relations do not change.In these coordinates, the involution ι is generated by S = − (cid:0) q P + q P + q P (cid:1) . Remark 6.4 (comparison with the literature) . The Hamiltonian H , with potential h givenin (6.2), can be written H = ( q − q ) (cid:18) p − p − p + k q + 2 k q ( q − q ) (cid:19) , which is in the form of a St¨ackel transform to a flat metric, so, again, can be compared with theclassification given in [6]. We can compare with the list of non-degenerate potentials (albeit inthe quantum case with H replaced by a Laplacian) given in of [6, Section 5]. Allowing for thefact that this classification is for the Euclidean case and involves some complex coordinates, theabove potential can be considered as a reduction of the 5 parameter potential V [2 , , of [6]; it is,in fact a reduced case, with our parameters ( k , k ) corresponding to their ( a , a ).In the flat coordinates (diagonalised), it again corresponds to V [2 , , , but now with ( k , k )corresponding to their ( a , a ). In this section we consider the involutive system (5.4), with K = e , giving the potential func-tions (5.6). To simplify the calculations of this section, we choose the specific metric coefficient ϕ ( r ) = r , leading to H = H + h ( q ) , where H = q q (cid:0) p − p − p (cid:1) , whose Ricci tensor is non-constant: R = (cid:0) q − q (cid:1) q q . In addition to F , F , given in (5.4) and (5.6),we require that the quadratic function F = 14 h + g ( q ) , satisfies { F , H } = 0, which restricts the component functions of h ( q ), as well as determining g ,but leaves ϕ (cid:0) q q (cid:1) arbitrary . Specifically, we find ϕ (cid:0) q − q (cid:1) = k + k (cid:0) q − q (cid:1) + k (cid:0) q − q (cid:1) , ϕ ( q ) = k q − k q ,g = − (cid:0) q − q − q (cid:1)(cid:0) k − k (cid:0) q − q − q (cid:1)(cid:1) . Setting ϕ (cid:0) q q (cid:1) = 0 (since it effectively an additive constant), we therefore have H = q q (cid:18) p − p − p + k q − q − k + k (cid:0) q − q − q (cid:1) + k q (cid:19) ,F = C − k (cid:0) q − q (cid:1) + k (cid:0) q − q (cid:1)(cid:0) q − q − q (cid:1) + k (cid:0) q − q (cid:1) q ,F = e + k q − k q , F = 14 h − (cid:0) q − q − q (cid:1)(cid:0) k − k ( q − q − q (cid:1)(cid:1) . oisson Algebras and 3D Superintegrable Hamiltonian Systems 35The algebra of these three integrals is easily calculated: { F , F } = 0 , { F , F } = 0 , { F , F } = 2 F , where F is a cubic expression F = F h − k q p + 4 k q (cid:0) q ( q p + q p ) + (cid:0) q − q (cid:1) p (cid:1) , which cannot be written as a polynomial in H and its integrals, but does satisfy the algebraicrelation F = 4 F F + 4 k F ( F − F ) − k ( F − F ) + 4 k k ( F + F )+ 2 k k F + k (2 k − k k ) . We can use this to derive the formulae for { F i , F } : { F , F } = 0 and { F , F } = 4 F + 4 k F − k ( F − F ) + 4 k k , { F , F } = − F F − k ( F − F ) − k k . Remark 6.5.
We have 4 functionally independent integrals, H , F , F , F , which form a closedalgebra (with the inclusion of the functionally dependent F ). In 3-dimensions, this system issuper-integrable, but not maximally. Since we only have a 3-dimensional symmetry algebra g ,we cannot build any further integrals out of this. However, since the metric is not constantcurvature, it is possible that other integrals exist, which are not built in this way. We started this paper by considering a specific 3-dimensional realisation of sl (2) (our algebra g )and showed how to embed this into some 6- and 10-dimensional Lie algebras, with very specificstructure. The Casimir functions (3.5) and (3.9) of the 6-dimensional algebras, represent thekinetic energy on manifolds with these symmetries, and the 10-dimensional algebra gave us thecorresponding conformal algebra . This was used in Sections 5 and 6, where we considered somespecific diagonal examples of these Casimir functions.Indeed, the main aim of this paper was to build super-integrable systems (and the associatednon-Abelian Poisson algebras) with a given kinetic energy, which itself has a high degree ofsymmetry. The approach is most suited to kinetic energies derived from constant curvature(including flat) manifolds, as mainly considered here. However, we saw that it is possible toapply the method when the isometry algebra is smaller, but, in this case, it may not be possibleto build enough independent integrals for the system to be maximally super-integrable.We just considered 3-dimensional manifolds in this paper. We used the structure of thesymmetry algebra to construct involutive triples, giving us separable systems. We further usedthe structure of the symmetry algebra to select the leading parts of additional integrals to obtainsuper-integrable restrictions. We selected our integrals in such a way that the automorphisms ofthe symmetry algebra (realised as canonical transformations) could be extended to act simplyon the non-Abelian Poisson algebra, which enabled us to find a finite closure of the Poissonalgebra.We did not consider the classification of super-integrable systems, such as can be found in theliterature (see, for example, [6, 11, 14]), but our approach can be applied to any kinetic energyassociated with a flat or constant curvature metric, which includes most physical systems.In this paper we made several choices which simplified our calculations, leaving us witha number of open problems.6 A.P. Fordy and Q. HuangNot all realisations of sl (2) are equivalent. Whilst 2-dimensional realisations were classifiedby Lie (see [2, Section 2]), no such classification exists for the 3-dimensional case. Within ourgeneral g , with f given by (2.8b), there are at least two equivalence classes (correspondingto degenerate and non-degenerate Casimir functions), but we don’t have a full classification ofinequivalent cases. Given such a choice of g , the constructions of Sections 2.3, 3 and 4 couldbe carried out: the general problem is to find all 6- and 10-dimensional, nontrivial extensionsof g , which are then to be used in the construction of separable and super-integrable systems.We saw in Section 4 that the 10-dimensional algebra contains several subalgebras, whoseCasimir functions correspond to conformally equivalent metrics. The (infinitesimal) conformalfactors formed another representation space of the algebra g . We don’t yet have the fullclassification of subalgebras of our 10-dimensional conformal algebra. For any subalgebra, theCasimir function, representing the corresponding kinetic energy, could be used in the context ofour analysis of Sections 5 and 6.In Section 5 we made the simplest choice of involutive triple. More general quadratic formswould clearly lead to more complicated calculations, but could lead to some interesting examples.In Section 6 we chose additional integrals that would minimise the complexity of our calcu-lations. Clearly there are many different choices which could lead to interesting systems andcorresponding Poisson algebras. Even for the simple choices we made, the Poisson algebraswere very complicated and we have little understanding of the general structure. The Poissonalgebras satisfy many polynomial constraints, which can be used to simplify some of the Poissonrelations, but what is the minimal set of generators of these constraints?The current paper is about classical Poisson algebras, but a similar analysis can be car-ried out for the quantum case, where super-integrability should allow us to construct expliciteigenfunctions, as was found in 2-dimensions [7]. Acknowledgements
This work was carried out while QH was visiting Leeds for one year, funded by the ChinaScholarship Council. QH would like to thank the School of Mathematics, University of Leeds,for their hospitality. Significant improvements were made after the comments of both refereesand an editor. We thank them for their input.
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