Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces
The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidde…
Authors: Angel Ballesteros, Francisco J. Herranz
Maximal sup erin tegrabilit y of the generali zed Kepler–Coulom b system on N -dimensional curv ed spaces ´ Angel Ballesteros a and F rancisco J. Herranz b a Departamen to de F ´ ısica, F acultad de Ciencias, Univ ersidad de Burgos, 09001 Burgos, Spain E-mail: angelb@ubu.es b Departamen to de F ´ ısica, Escuela Polit ´ ecnica Sup erior, Univ ersidad de Burgos, 09001 Burgos, Spain E-mail: fjherranz@ubu.es Abstract The s up erp osition of the Kepler–Coulomb p oten tial on the 3D Euclidean space with thr e e centrifugal terms has recen tly b een sho wn to b e maximally sup erin te- grable [V errier P E and Ev ans N W 2008 J. Math. Ph ys. 49 022902] by fin ding an additional (hidden) integral of motion wh ic h is quartic in the momen ta. In this pap er we present th e generalizati on of this result to the N D spherical, hy- p erb olic and Eu clidean spaces by making u se of a unified symmetry app roac h that make s use of th e curv ature parameter. The r esulting Hamiltonian, formed b y the (curve d ) Kep ler–Coulom b p otenti al together with N cen tr if u gal terms , is shown to b e end o we d with 2 N − 1 fun ctionally indep endent integ r als of the motion: one of th em is quartic and the remaining ones are quadr atic. T he tran- sition from the p rop er K epler–Coulom b p oten tial, with its asso ciated quadr atic Laplace–Runge–Lenz N -v ector, to the generalized system is fully describ ed. The role of sp h erical, n on lin ear (cubic), and coalgebra sym metries in all th ese systems is h ighligh ted. P AC S: 02.30 .Ik 02.40.Ky KEYW ORDS: In tegr a ble systems , curv ature, coalgebras, Lie algebras, nonlinear sym- metry 1 In tro ducti on The Kepler–Coulom b ( KC) p oten tial on R iemannian spaces of constan t curv ature w a s already studied b y Lipsc hitz a nd Killing in the 19th cen tury , and later redisco v ered b y Sc hr¨ odinger [1] (see [2] f o r a detailed discussion). In terms of a geo desic radial distance r b etw een the particle a nd the origin of the space, the K C p o ten tial on the N -dime nsional ( N D) spherical S N , Euclidean E N and h yp erb olic H N spaces reads (see, e.g., [3, 4 , 5] and references t herein) − K tan r on S N ; − K r on E N ; − K tanh r on H N . (1.1) In this pap er w e shall deal with the integrabilit y prop erties of the so-called ND gen- er alize d KC system , whic h is defined as the sup erp osition of the (curv ed) K C p ot en tial with N ‘cen trifugal’ terms. In the N D Euclidean space E N suc h a system reads H = 1 2 p 2 − K p q 2 + N X i =1 b i q 2 i (1.2) where K and b i ( i = 1 , . . . , N ) are real constan ts. This system was kno wn to b e quasi- maximal ly sup erinte gr able [3] in the Liouville sense [6], since a set of 2 N − 2 functionally indep enden t quadr atic integrals o f motion ( including the Hamiltonia n) w ere explicitly kno wn. In fact, in the remark a ble classification on sup erin tegrable systems on E 3 b y Ev ans [7], this Hamiltonian w a s called ‘w eakly’ or minim al ly sup erinte gr a b le since it had one in t egr a l of motion more (fo ur) than the necessary num b er to b e completely in tegrable (three), but o ne less than the maximum possible num b er of indep enden t in tegrals for a 3D system (five). In con trast, it w as also w ell kno wn that when at least one of the cen trifugal terms v a nishes (w e shall call this case the quasi-gener alize d KC system ), the resulting Hamil- tonian turns out to be maxim al ly s up erinte gr able in arbitrary dimens ion since a maximal set o f 2 N − 1 functionally indep endent and quad r atic in tegrals of the motion is explicitly kno wn (see [3 , 7, 8, 9, 10] and r eferences therein). Moreov er, suc h maximal sup erin- tegrabilit y of the quasi-generalized KC system has also b een pro ven for the spherical and hy p erb olic spaces [11, 12, 13] a s w ell as for the Minko wskian and (a n ti-)de Sitter spacetimes [4, 14]. Nev ertheless, in a recen t w ork V errier and Ev ans [1 5] ha ve sho wn tha t the gener- alized K C system on E 3 ( i.e. , the superp osition of the 3D K C p oten tial with three cen trifugal terms) is maximal ly suprinte gr able , but the additional in tegra l of motion is quartic in the momen ta. The aim of this pap er is to sho w tha t this result holds for an arbitrary dimension N and, moreo v er, that the generalized K C system is also maxi- mally sup erintegrable on t he N D curv ed Riemannian spaces of constan t curv ature: the spherical S N and h yp erb o lic H N spaces. In t his w ay the N D Euclidean system arises as a smo oth limiting flat case t ha t can b e in terpreted as a con traction in terms of the curv ature pa rameter κ . In order to pro ve this result w e shall explicitly construct the set of 2 N − 1 function- ally indep enden t in tegra ls of motion f o r the generalized KC Hamilto nian. In particular, 2 2 N − 2 of them will b e quadr atic and pro vided by an sl (2 , R ) P oisson coalgebra sym- metry [3, 16] (together with the Hamilto nian), while the remaining ‘hidden’ o ne is quartic in the momen ta and generalizes t he result of [15] on E 3 to these three N D classical spaces of constant curv ature. In this wa y the short list of N D maximally sup erin tegrable Hamiltonians (see [17] an references therein) is enlarged with another instance. The pap er is organized as follows . In t he next section w e introduce the geometric bac kground on which the rest of the pap er will b e based: the P oincar´ e and Beltrami phase spaces a rising, resp ectiv ely , as the stereographic a nd the cen tra l pro jection fro m a linear am bien t space R N +1 [3, 18]. The next section is dev oted to recall the description of the maximal in tegrabilit y o f the curv ed KC system in t erms o f these tw o phase spaces. In section 4 the spherical and ‘hidden’ nonlinear symmetries of the K C system are fully describ ed, th us providing a detailed explanation of the tec hniques making p ossible the ‘transition’ from the in tegrability prop erties of the curv ed K C system to the generalized one. The core of the pap er is contained in section 5, where we explicitly sho w how the sphe rical symmetry breaking induced by the cen t rifugal t erms can b e appropriately replaced by an sl (2 , R ) Poisson coalg ebra symmetry [19, 20] that allo ws us to construct the ‘additional’ quartic in tegra l of motion for the N D generalized curv ed K C systems. F ina lly , some remarks close the pap er. 2 P oincar ´ e and Beltrami p h ase spaces T o start with w e presen t the structure of the P o incar ´ e and Beltrami phase spaces [3], whic h will allo w us to deal with the three classical Riemannian spaces in a unified setting. F urthermore, as a new result, w e also presen t the explicit canonical transfor- mation relat ing b oth of them. Giv en a constant sectional curv a t ure κ , the N D spherical S N ( κ > 0), Euclidean E N ( κ = 0) and hyperb olic H N ( κ < 0) spaces can b e sim ultaneously em b edded in an ambie nt linear space R N +1 with am bien t (or W eiertrass) co ordinates ( x 0 , x ) = ( x 0 , x 1 , . . . , x N ) b y requiring them to fulfill the ‘sphere’ constraint Σ: x 2 0 + κ x 2 = 1. Hereafter for any t wo N D v ectors, sa y a = ( a 1 , . . . , a N ) and b = ( b 1 , . . . , b N ), w e denote a 2 = P N i =1 a 2 i , | a | = √ a 2 and a · b = P N i =1 a i b i . The metric on these three Riemannian spaces of constant curv ature is giv en, in am bien t co o rdinates, by [2 1]: d s 2 = 1 κ d x 2 0 + κ d x 2 Σ . (2.1) No w, if w e consider the stereographic pr o jection [18] from ( x 0 , x ) ∈ Σ ⊂ R N +1 to the Poinc ar´ e c o or dinates q ∈ R N with p ole ( − 1 , 0 ) ∈ R N +1 , that is, ( − 1 , 0 ) + λ (1 , q ) ∈ Σ, w e obtain that λ = 2 1 + κ q 2 x 0 = λ − 1 = 1 − κ q 2 1 + κ q 2 x = λ q = 2 q 1 + κ q 2 . (2.2) On the other hand, if we apply the cen tr al pro jection from ( x 0 , x ) to the Beltr ami 3 c o or dinates ˜ q ∈ R N with p ole (0 , 0 ) ∈ R N +1 , suc h tha t (0 , 0 ) + µ (1 , ˜ q ) ∈ Σ, w e find µ = 1 p 1 + κ ˜ q 2 x 0 = µ x = µ ˜ q = ˜ q p 1 + κ ˜ q 2 . (2.3) The image o f these pro jections is the subset of R N determined b y either λ > 0 (a nd then 1 + κ q 2 > 0), or b y µ ∈ R (and th us 1 + κ ˜ q 2 > 0). This means that for S N ( κ > 0) b oth pro jections lead to R N with the exception of a single p oint; for H N they giv e the op en subset q 2 < 1 / | κ | o r ˜ q 2 < 1 / | κ | (t he P oincar´ e disc in 2 D); and in b oth cases, if κ = 0, w e reco ve r E N in Cartesian co ordinates x ∈ R N ≡ E N since x = 2 q = ˜ q . Therefore, it can b e sho wn that the metric (2.1) in b oth co o r dinate systems reads d s 2 = 4 d q 2 (1 + κ q 2 ) 2 = (1 + κ ˜ q 2 )d ˜ q 2 − κ ( ˜ q · d ˜ q ) 2 (1 + κ ˜ q 2 ) 2 (2.4) so that the corresp onding geo desic flow on these spaces can b e describ ed through the free Lagrangian g iv en (up to a p o sitive multiplic a tiv e constant) by : T = 2 ˙ q 2 (1 + κ q 2 ) 2 = (1 + κ ˜ q 2 ) ˙ ˜ q 2 − κ ( ˜ q · ˙ ˜ q ) 2 2(1 + κ ˜ q 2 ) 2 . (2.5) Hence the canonical momen ta p , ˜ p conjuga te to q , ˜ q are obtained t hr o ugh a Legendre transformation yielding p = 4 ˙ q (1 + κ q 2 ) 2 ˜ p = (1 + κ ˜ q 2 ) ˙ ˜ q − κ ( ˜ q · ˙ ˜ q ) ˜ q (1 + κ ˜ q 2 ) 2 (2.6) and the geo desic flo w kinetic energy is found to b e T = 1 8 1 + κ q 2 2 p 2 = 1 2 (1 + κ ˜ q 2 ) ˜ p 2 + κ ( ˜ q · ˜ p ) 2 . (2.7) Eviden tly , T can b e written with a factor 1 / 2 instead of 1 / 8 in t he P oincar ´ e phase space but w e hav e k ept the latt er factor in order to mak e explicit the equalit y b etw een P oincar´ e and Beltrami expressions. The canonical equiv alence b etw een b o th phase spaces is c hara cterized by t he fol- lo wing statemen t, that can b e prov en thro ugh direct computations and b y taking in to accoun t the expressions (2.2 ) – (2.6). Prop osition 1. L et ( q , p ) b e the Poinc ar ´ e phase sp ac e variables such that { q i , p j } = δ ij and ( ˜ q , ˜ p ) the Beltr ami ones satisfying { ˜ q i , ˜ p j } = δ ij . Both sets of c anonic al variables ar e r elate d thr ough the c anonic al tr ansformation given by q = ˜ q 1 + p 1 + κ ˜ q 2 p = 1 + p 1 + κ ˜ q 2 ˜ p + κ ˜ q ( ˜ q · ˜ p ) ˜ q = 2 q 1 − κ q 2 ˜ p = 1 − κ q 2 2(1 + κ q 2 ) (1 + κ q 2 ) p − 2 κ q ( q · p ) . (2.8) 4 Moreo v er, from (2 .8) w e obta in the fo llowing useful relations to b e considered b elow: q i p j − q j p i = ˜ q i ˜ p j − ˜ q j ˜ p i q i q j = ˜ q i ˜ q j q i p q 2 = ˜ q i p ˜ q 2 ˜ p + κ ( ˜ q · ˜ p ) ˜ q = 1 2 (1 − κ q 2 ) p + κ ( q · p ) q ˜ q · ˜ p = 1 − κ q 2 1 + κ q 2 ( q · p ) . (2.9) 3 The Kepler– Coulom b system In order to construct the KC Hamiltonian w e recall that fo r t he three Riemannian spaces with constant curv ature, the geo desic radial distance r (along the geo desic that joins the part icle a nd the orig in in the space) can b e expressed, in this order, in am bient, P oincar´ e and Beltrami co ordinates as [3]: 1 κ tan 2 ( √ κ r ) = x 2 x 2 0 = 4 q 2 (1 − κ q 2 ) 2 = ˜ q 2 . (3.1) In fact, these three expressions provide the appropriate definition for the curve d ( Higgs) oscillator p otential [22, 23]. By using t hese co ordinates, the K C system is just the Hamiltonian [3] H = 1 8 1 + κ q 2 2 p 2 − K 1 − κ q 2 2 p q 2 = 1 2 (1 + κ ˜ q 2 ) ˜ p 2 + κ ( ˜ q · ˜ p ) 2 − K p ˜ q 2 (3.2) where t he first term is the kinetic energy (2.7) and the second one is the curv ed K C p oten tia l, whic h is obtained as the square ro ot of the inv erse of (3.1). W e stress that under this fra mek ork we are able to cov er sim ultaneously the t hree cases (1.1) fo r the particular cases of the sectional curv ature κ ∈ { +1 , 0 , − 1 } . Moreo v er, in this language the limit κ → 0 corresp onds to t he (fla t) con tra ction S N → E N ← H N . The maximal sup erin tegra bility of the curv ed KC Hamiltonian (3.2) is c ha r a cterized b y Prop osition 2. [3] L et H b e the KC Hamiltonia n (3.2) and let us c onsid er the quadr atic functions in the m omenta given by C ( m ) = m X 1 ≤ i 0, hyperb olic so ( N , 1) for κ < 0, and the Euclidean Lie–P oisson alg ebra iso ( N ) for κ = 0. Moreo v er, the three Riemannian spaces of constant sectional curv at ur e κ , de- scrib ed in section 2, can b e constructed as homogeneous spaces through the quotien t h J ij , P i i / h J ij i = h P i i : S N = so ( N + 1) / so ( N ) for κ > 0; E N = iso ( N ) / so ( N ) for κ = 0; H N = so ( N , 1) / so ( N ) for κ < 0; (4.5) with the P i ’s pla ying the role of (curv ed) translations on such spaces, and t he con trac- tion κ = 0 g iv es rise to the Euclidean (comm uta tiv e) tra nslations P i = 1 2 p i = ˜ p i . The P oisson brack ets b et w een the Ha milto nian H (3.2 ) and the so κ ( N + 1) generators read {H , J ij } = 0 {H , P i } = K q i (1 + κ q 2 ) 2 4 | q | 3 = K ˜ q i (1 + κ ˜ q 2 ) | ˜ q | 3 ≡ K x i x 3 . (4.6) W e stress that from the viewpoint of suc h so κ ( N + 1 ) - symmetry , the constan ts of the motion (3 .4) can b e expressed in a v ery natura l and simple form, as [14] L i = N X l =1 P l J li + K q i p q 2 (4.7) suc h that J ii ≡ 0 and J li = − J il if l > i . This, in turn, directly sho ws the functionally indep endence of a g iven L i with resp ect to the inte g rals (3.3), a s stated in prop osition 2, since the la t t er are only fo rmed by r o tation generatos J ij of so ( N ). 4.2 Nonlinear angular momen tum symmetry By taking in to account (4.7), it is a matter of straightforw ard computations to sho w that the N constants of the motion (3 .4) are transfor med as an N - v ector under the so ( N ) generators (4.1), { J ij , L k } = δ ik L j − δ j k L i . (4.8) 7 The N functions L i are, in fa ct, the comp onen ts of the L aplac e–R unge –L enz N -ve ctor on S N , H N and E N , whic h corresp o nd to the ‘hidden’ symmetries of the K C Hamilto- nian. Moreo v er, the Lie–Poiss o n brack ets inv olving t he L i comp onen ts read {L i , L j } = 2 κ J 2 − H J ij ≡ Λ J ij . (4.9) This expression is w orth to b e compared with (4 .4 ). Since H P oisson-comm utes with all the functions J ij and L i (in alg ebraic terms w e w ould sa y that H b ehav es as a cen tral extension), w e find that the set o f N ( N + 1) / 2 functions h J ij , L i i ( i < j ; i, j = 1 , . . . , N ) span a nonl i n e ar ( cubic) Poiss on a lgebra that w e sall denote as so (3) κ ( N + 1). Only in E N ( κ = 0) this nonlinear symmetry alg ebra reduces to Lie–P oisson algebras with Λ = − 2 H b eing a constan t, and w e get so ( N + 1) for H < 0 or so ( N , 1) for H > 0. Therefore, from a geometrical viewp oin t w e can sa y that the ro le of the translations P i on the homogeneous spaces (4.5), with Lie–P oisson symmetry so κ ( N + 1), is replaced b y t he Laplace–Runge–Lenz components L i , with nonlinear symme t ry so (3) κ ( N + 1), whereas t he role of the f o rmer constan t curv ature κ is no w pla y ed by the quadr atic function Λ (4.9). In this resp ect, no t ice that although J 2 and H ar e b o t h constan ts of the motion for the KC system, J 2 do es not P oisson-commute with L i . As a consequenc e, Λ is not a central function within so (3) κ ( N + 1). Ho we ver, in the case N = 2 we hav e that J 2 ≡ J 2 12 , so that so (3) κ (3) = h J 12 , L 1 , L 2 i giv es rise to the cubic P oisson algebra { J 12 , L 1 } = L 2 { J 12 , L 2 } = −L 1 {L 1 , L 2 } = 2 κJ 3 12 − 2 H J 12 . (4.10) Next we can define ‘n um b er’ L and ‘ladder’ op erators L ± as L = i J 12 L ± = L 1 ± i L 2 fulfilling the cubic comm utation relations {L , L ± } = ±L ± {L + , L − } = 4 κ L 3 + 4 HL (4.11) whic h repro duce the Poiss on algebra analogue of the Hig gs sl (3) (2 , R ) algebra [22] when- ev er κ 6 = 0 . In t his case the Poiss on brack ets are asso ciated to S 2 and H 2 , whereas the con traction κ = 0 giv es sl (2 , R ) (or gl (2) if H is considered as an actual generator) for E 2 . Note that t he Higg s algebra is endow ed with a quartic Casimir function given by C s l (3) (2 , R ) = L + L − + κ L 4 + 2 HL 2 . (4.12) Moreo v er, if w e realize this Casimir in terms o f P oincar´ e or Beltrami co ordinates, we obtain that C s l (3) (2 , R ) = K 2 , whic h is just the sq ua r e of the coupling constan t of the K C p otential. It is worth to stress that the Higgs algebra has b een deeply studied and applied to differen t quan tum ph ysical mo dels (b ey ond in tegrable systems) with an underlying nonlinear angular momen tum symmetry ( see [24, 25, 26, 2 7, 28, 2 9, 30] and references therein). 8 5 The gene ralized Kepler– Coulom b sys t em The generalized KC Hamiltonian H g is obtained by adding N cen trifugal terms (with non-v anishing parameters b i ∈ R ) to the K C system H (3.2). In the curv ed cases, the w ay to define a ppropriately such cen trifug al terms w as presen ted and fully explained in [3]. Explicitly , in P oincar´ e co ordinates the generalized KC system is giv en b y H g = H + 1 8 1 + κ q 2 2 N X i =1 b i q 2 i = 1 8 1 + κ q 2 2 p 2 − K 1 − κ q 2 2 p q 2 + 1 8 1 + κ q 2 2 N X i =1 b i q 2 i (5.1) while in terms of Beltrami v ariables this reads H g = H + 1 2 (1 + κ ˜ q 2 ) N X i =1 b i ˜ q 2 i = 1 2 (1 + κ ˜ q 2 ) ˜ p 2 + κ ( ˜ q · ˜ p ) 2 − K p ˜ q 2 + 1 2 (1 + κ ˜ q 2 ) N X i =1 b i ˜ q 2 i . (5.2) This is the curv ed κ -analogue of the generalized KC system o n E N (1.2). W e recall that the cen trifugal terms are prop er cen trifugal barriers on b oth E N ( κ = 0 ) and H N ( κ < 0 ). Moreo v er, only on S N ( κ > 0 ) all these terms can b e alternativ ely in terpreted as non- cen tral harmonic oscillators (see [4, 2 1, 31, 32] for a full discussion on the sub ject). The maximal sup erin tegrability of this Hamiltonian, whic h constitute the main result o f this pap er, can no w b e stated a nd pro ven as follows. Theorem. L et H g b e the gener alize d KC Hamiltonian (5.1)–(5.2) with al l b i 6 = 0 . L et us c onsider the quadr atic and q uartic f unction s in the m o menta giv e n by C ( m ) g = m X 1 ≤ i
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