Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrands theorem to curved manifolds

Hamiltonian systems admitting a Runge–Lenz v ector and an optimal extension of Bertrand’s theorem to curv ed manifolds ´ Angel Ballesteros a ∗ , Alb erto Enciso b † , F rancisco J . Herranz a ‡ , Orlando Ragnisco c § a Depto. de F ´ ısica, Univ ersidad de Burgos, 09001 Burgos, Spain b Depto. de F ´ ısica T e´ orica I I, Univ ersida d Complutense, 2804 0 Madrid, Spain c Dip. di Fisica, Universit` a di Roma 3, and Istituto Nazionale di Fisica Nucleare, 00146 Rome, Ita ly Abstract Bertrand’s theorem asserts that any spherically symmetric na tural Hamiltonian sys- tem in Euclidean 3-space which p ossesses s table circular orbits and whose b ounded tra- jectories are all p erio dic is either a harmo nic oscillator or a Kepler system. In this pap er we ex tend this classical r esult to c ur v ed spaces b y pro ving that a n y Hamiltonian on a spherically symmetric Riemannian 3 -manifold which satisfies the sa me conditions a s in Bertrand’s theo rem is sup erintegrable a nd given by an intrinsic oscilla tor o r Kepler sys- tem. As a byproduct we obtain a wide pa noply of new sup e rin tegra ble Hamiltonian systems. The demonstr ation r elies o n Perlick’s classification of Bertrand spacetimes and on the cons tr uction of a suitable, globally defined gener alization of the Runge–Lenz vec- tor. P ACS: 02 .30.Ik, 02 .40.Ky , 45.20.J j Keywords: Int egr able Hamiltonia n systems, Runge–Le nz vector, spherical symmetry , static spacetimes, p erio dic or bits, Kepler pro blem. 1 In tro duction and pr eliminary definitions The Kepler problem an d the harmonic oscillator are probably the m ost thoroughly studied systems in classical mec hanics. The r easons for this are tw ofold. First, these p oten tials play a prep ond eran t role in Ph ysics due their connection with planetary motio n and oscillat ions around a nond egenerat e equilibrium. Second, these p oten tials are o f particular mathemati- cal interest due to the existence of additional (or “hidden”) symmetries yieldin g additional constan ts of motion. In fact, b oth the Kepler and the h arm onic oscillato r Hamiltonians are (maximally) sup erinte gr able in the sense that they ha ve the maximum num b er (four) of functionally indep end en t first in tegrals other than the Hamiltonian. 1 Bertrand’s theorem [6] is a landmark result whic h charac terizes the K ep ler and h arm onic oscillato r Hamiltonians in terms of their qualitativ e dynamics. A pr ecise statemen t of this ∗ angelb@ubu.es † aenciso@fis.ucm.es ‡ fjherranz@ubu.es § ragnisco@fis.uniroma3.i t 1 As usual, by functional ind epend ence of th e in tegrals I 1 , . . . , I k w e mean that the ( k + 1)-form d H ∧ d I 1 ∧ · · · ∧ d I k is nonzero in an open and dense sub set of phase space, H b eing the Hamiltonian fun ction. 1 theorem is giv en b elo w. W e recall [18] that the first condition, which is o ccasionally f orgott en, is nece ssary in ord er to exclude p oten tials of th e form V ( q ) = − K k q k − s , with K > 0 and s = 2 , 3 , . . . Theorem 1 (B ertrand ) . L et H = 1 2 k p k 2 + V ( q ) b e a natur al, spheric al ly symmetric Hamil- tonian system in a domain of R 3 . L et us supp ose th at: (i) Ther e exist stable cir cu lar or bits. (ii) Al l the b ounde d tr aje ctories ar e close d. Then the p otential is either a Kepler ( V ( q ) = A/ k q k + B ) or a harmonic oscil lator p otential ( V ( q ) = A k q k 2 + B ) . In p articular, H is sup erinte gr able. Analogues of the Kep ler and harmonic oscillator systems in curv ed spaces hav e b een of in terest since the disco v ery of non-Eu clidean geometry . In fact [49], the “intrinsic” Kepler an d harmonic oscillator p roblems on spaces of constan t curv atur e were studied by Lipsc hitz and Killing already in the 19th cen tury , and later r edisco vered by Sc hr¨ odinger [48] and Higgs [25]. In b oth cases it was established th at these s ystems are su p erin tegrable and s atisfy Prop erties (i) and (ii) ab o v e. A consid er ab ly more ambiti ous devel opment was Perlic k’s introd uction and classification of Bertrand spacetimes [47], which w as based on the follo w ing observ ation. Let ( M , g ) b e a Riemann ian 3-manifold and consider the sp ace M = M × R endo we d with th e w arp ed Loren tzian metric η = g − 1 V d t 2 , with V a smo oth p ositiv e function on M . Th en the tr aje c- tories in ( M , η ), that is, the pr o jections of inextendible timelik e geo desics to a constan t time leaf M × { t 0 } , corresp ond to in tegral curv es of the Hamiltonian H = 1 2 k p k 2 g + V ( q ) in (t he cotangen t bun dle of ) M . Thus Perlic k in tro duced the follo wing Definition 2. A Lorent zian 4-manifold ( M × R , η ) is a Bertr and sp ac etime if : (i) It is sph ericall y symm etric and static in the sense that η = g − 1 V d t 2 and M is dif- feomorphic to ( r 1 , r 2 ) × S 2 , where the smo oth function V d ep ends only on r and the Riemannian metric g on M tak es the form g = h ( r ) 2 d r 2 + r 2  d θ 2 + sin 2 θ d ϕ 2  (1) in the adapted co ordinate system ( r , θ , ϕ ). Here r 1 , r 2 ∈ R + ∪ { + ∞} . (ii) Th ere is a circular ( r = const.) tra jectory passing through eac h p oin t of M . (iii) Th e ab o ve circular tra jectories are stable, that is, an y initial condition su fficien tly close to that of a circular tra j ectory giv es a p erio dic tra jectory . P erlic k’s main result wa s the classification of all Bertrand spacetimes, r eco ve ring the classical Bertrand theo rem as a su b case. Ho we ver, t wo main r elat ed questions remained to b e settled. On the one hand , the p otentials V in Pe rlick’ s classificatio n lac k ed an y physica l in terpretation, and this was in strong con trast with the Euclidean case. This dra wb ac k w as circum v en ted in Ref. [3], w here we show ed that the t wo families of Perlic k’s p otenti als corresp ond to either the “intrinsic” Kepler or harmonic oscilla tor p oten tials in the underlying 3-manifold ( M , g ). On the other hand, the issue of whether the corresp ond ing Hamiltonian systems we re sup erintegrable in some reasonable sense was left wid e op en. In fact, Perlic k’s 2 only remark in this direction w as th at, b y virtue of a theorem of Hauser and Malhiot [24], only t w o concrete mo dels among the f amily of Bertrand spacetimes adm itted a quadratic additional in tegral coming from a second rank Killing tens or. A careful analysis of the literature revea ls that man y particular cases of Bertrand m etrics ha v e b een thoroughly analyze d and sho wn to b e su p erin tegrable [21, 22, 27, 28], and that in man y cases they hav e b een shown to admit a generalization of the classical Runge–Lenz ve c- tor as an additional first in tegral. T he physica l and mathematica l interest of these models (and th us of Bertrand space times) is fostered by their connections with the th eory of mag- netic monop oles, with differentia l and algebraic geomet ry , and with lo w -dimensional mani- fold th eory [9, 10, 33– 35, 43, 45, 46, 50]. The relation b et w een Bertrand sp aces and monop ole motion is not to tally incidental. Indeed, an ample sub class of Bertrand sp acet imes admit- ting some kin d of generalized Runge–Lenz v ectors (the so-called m ultifold Kepler systems) w ere in tro duced by Iwa i an d Kata ya ma [27, 28] as generalizations of the T aub–NUT met- ric, whose geo desics asymptotic ally describ e the relativ e motio n of t wo monopoles (se e, for instance, [1, 7, 11, 29, 39, 40]). Interesti ngly , su p erin tegrable Hamiltonian systems on curve d spaces ha ve recen tly attracted considerable atten tion also w ith in the integ rable systems com- m unity , esp ecially in low dim en sions (cf. [2, 4, 5, 30– 32] and r eferences th erein). The main result of this article is that all Bertrand spacetimes are indeed su p erin tegrable, their sup erin tegrabilit y b eing linked to the existence of a generalized Runge–Lenz vecto r. This enables us to presen t an optimal v ersion of Bertrand’s theorem (Theorem 16) on spheri- cally symmetric manifolds which includes th e classificatio n of the natur al Hamiltonians whose b ounded orbits are all p er io dic [47], the ph ysical in terpretation of the corresp onding p otent ials as Kepler or harmon ic oscillator p oten tials, in eac h case, and the pr oof of the sup erinte gra- bilit y of these mo dels. This settles in a quite satisfactory w ay a p roblem with a large b o dy of p revious partial resu lts scattered in the literature. It is standard that the sup erin tegrabilit y of the Kepler system stems f rom th e existence a conserv ed Runge–Lenz vect or, whose geometric s ignifi cance is describ ed f r om a m od ern p ersp ectiv e in [23]. On the other hand, the su p erin tegrabilit y of the harm onic oscillator is usually established either u s ing exp licit (scalar) first inte grals or the conserved ran k 2 tensor C = 2 ω 2 q ⊗ q + p ⊗ p , whic h is s ometimes preferable for algebraic reasons [1 9]. That the latter approac h is closely related to a (m ultiv alued) analogue of th e Ru nge–Lenz v ector w as firmly established in [26]. Motiv ated by this connection, we ha ve b ased our appr oac h to the in tegrabilit y of the Bertrand systems on the construction of a ge neralized Runge–Lenz v ector, globally defined on a finite co ver of M . Th is construction relies on a detailed analysis of the inte gral curv es of the app ropriate Hamiltonians. The literature on generaliz ations of the Runge–Lenz v ector for cent ral p otent ials on Eu clidean space is v ast (see the survey [36] and r eferences th erein), but u nfortunately s ev eral in teresting pap ers are severely fl aw ed by the lac k of distinction b etw een local, semi-global and global existence. The article is organized as follo ws. In Sectio n 2 w e recall the t w o families of Bertrand spacetimes en tering Pe rlic k’s classification, whic h are labeled b y t wo coprime positive inte - gers n an d m . W e also include the c haracterizatio n of P erlic k’s p otentia ls as th e in trinsic Kepler or harmonic oscillator p oten tials of the corresp ondin g Riemann ian 3-manifolds ( M , g ) and b riefly discuss sev eral physic ally relev ant examples. In Sectio n 3 we consider the asso- ciated natur al Hamiltonian sys tems on ( M , g ) and compute th eir in tegral curv es in closed form (Prop osition 7). Usin g this r esult we easily derive that th e latter Hamiltonians are geo- metrically su p erin tegrable (cf. Definition 9 and Prop osition 10) in the region of phase space foliated b y b ound ed orbits, as happ ens with the harmonic oscillator and Kepler p oten tials in 3 R 3 . Our cent ral result is a stronger s u p erin tegrabilit y theorem (Theorem 12) that w e presen t in Section 4 , w here w e constru ct a generalized Ru nge–Lenz vecto r globally defin ed on an n -fold co v er of M . As a corollary of this constr u ction w e also obtain a global rank n tensor field in M inv arian t under the flo w and a wide panoply of new sup erinteg rable Hamiltonian systems. Lastly , in Section 5 we com bine the results established in the pr evious sections to obtain an optimal extension of Bertrand’s theorem to curved spaces (Th eorem 16). 2 Harmonic oscillators and Kepler p oten tials in B ertrand space- times In this section w e shall define th e “int rins ic” K epler and harmonic oscillator p oten tials in a spherically sym m etric 3-manifold and sho w h ow Bertrand spacetimes are related to the Kepler and harmonic oscillato r p oten tials of an y of its constant time lea ves. M ost of the material includ ed here is essenti ally take n f rom Ref. [3]; for the sak e of completeness, let us men tion that fur ther information on geometric prop erties of Gr een fun ctio ns can b e consulted e.g. in [13– 15, 37, 38]. W e start by letting ( M , g ) b e a Riemannian 3-manifold as in Definition 2 . In p articular, the m etric g ta ke s the form d s 2 = h ( r ) 2 d r 2 + r 2  d θ 2 + sin 2 θ d ϕ 2  . (2) It is stand ard that if u ( r ) is fu n ction whic h d ep ends only on the radial co ordinate, then its Laplacian is also radial and reads as ∆ g u ( r ) = 1 r 2 h ( r ) d d r  r 2 h ( r ) d u d r  . As the Kepler p oten tial in Euclidean three-dimensional space is simply th e radial Green function of the Laplacian and the harmonic oscillator is its inv erse square, it is natural to mak e the follo wing Definition 3. Th e (in trinsic) Ke pler and the harmonic oscil lator p otentials in ( M , g ) are resp ectiv ely giv en by the r adial functions V K ( r ) = A 1  Z r a r ′− 2 h ( r ′ ) d r ′ + B 1  , V H ( r ) = A 2  Z r a r ′− 2 h ( r ′ ) d r ′ + B 2  − 2 , (3) where a, A j , B j are constan ts. Example 4. Let ( M , g ) b e the s imple connected, thr ee-dimensional space form of sectional curv ature κ . In th is case the metric has the form (2 ) w ith h ( r ) 2 = 1 1 − κr 2 . The corresp onding K epler and h arm onic oscillator p otenti als are therefore V K = p r − 2 − κ , V H = 1 r − 2 − κ (4) 4 up to additiv e and m ultiplicativ e constan ts. In terms of the distance function ρ κ to the p oin t r = 0 th is can b e rewritten as V K = √ κ cot  √ κ ρ κ  , V H = tan 2 ( √ κ ρ κ ) κ , th us repro ducing the known p rescriptions for the sphere and the h yp erb olic space [5, 49]. Th e Euclidean case is reco v ered b y letting κ → 0. No w let us consider th e spherically symmetric spaces ( M , g j ) ( j = I, I I) defined b y the metrics T yp e I : d s 2 = m 2 d r 2 n 2 (1 + K r 2 ) + r 2 (d θ 2 + sin 2 θ d ϕ 2 ) (5a) T yp e I I : d s 2 = 2 m 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  n 2 ((1 − D r 2 ) 2 − K r 4 ) d r 2 + r 2 (d θ 2 + sin 2 θ d ϕ 2 ) (5b) where D and K are real constan ts and m an d n are copr im e p ositiv e int egers. The maximal in terv al ( r 1 , r 2 ) can b e easily found f r om these expressions. These Riemannian 3-manifolds, whic h fir st ap p eared in [47] (wh ere the quotien t n/m w as called β ), will b e h enceforth called Bertr and sp ac es . A short computation shows that, u p to a multi plicativ e constan t, the Kep ler p oten tial of a Bertrand space of t yp e I is V I = p r − 2 + K + G , (6a) whereas the harmonic oscillat or p otentia l of one of t yp e I I can b e wr itten in the con v enient form V II = G ∓ r 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  − 1 . (6b) Here G is an arb itrary constant. By comparing with Ref. [47], the ab o ve digression imm ed iate ly yields the follo wing Prop osition 5. ( M , η ) is a Bertr and sp ac etime if and only if it is isometric to the warp e d pr o duct ( M × R , g j − d t 2 V j ) , with ( M , g j ) a Bertr and sp ac e of typ e j ( j = I, II, cf. (5) ) and V j given by (6) . In p articular, this sho ws that Perlic k’s obten tion of t wo different kinds of Bertrand space- times has a natural interpretatio n [3]: they a re asso ciat ed to either Kepler (t yp e I) or har- monic oscillato r (t yp e I I) p oten tials. The m ultiplicativ e constan t of the p oten tials is inessen- tial and can b e eliminated b y rescaling the time v ariable. Example 6. W e conclude this section with a brief discussion of a few examples of physically relev an t spaces that are Bertrand . T his intends b oth to serve as motiv ation and to help the reader gain some insigh t on Bertrand sp aces. A more detailed discu s sion can b e foun d in [3]. (i) Sp ac es of c onstant curvatur e . The m etric of the simply connected Riemannian 3- manifold of constan t sectional cu rv ature κ is usually written as d s 2 = d r 2 1 − κr 2 + r 2  d θ 2 + sin 2 θ d ϕ 2  . 5 W e ha ve already seen that the Kepler and harmonic oscillator p oten tials in these sp aces are give n b y (4), an d it is well known that all the b ounded inte gral curves of b oth systems are p erio dic. This result is immediately reco v ered b y noticing that the Kepler system is reco v ered from the t yp e I Bertrand spacetimes when n = m = 1 and K = − κ , whereas the harmonic oscillator is obtained as the typ e I I Bertrand sp acetime with n/m = 2, K = 0 and D = κ . (ii) Darb oux sp ac e o f typ e III. Cons ider the metric d s 2 = k 2 + 2 r 2 + k √ k 2 + 4 r 2 2( k 2 + 4 r 2 ) d r 2 + r 2  d θ 2 + sin 2 θ d ϕ 2  , whose in trinsic harm onic oscillator p otenti al is given by V II = 2 k 2 r 2 k 2 + 2 r 2 + k √ k 2 + 4 r 2 up to m ultiplicativ e and additiv e constant s. This defines a Bertrand spacetime of t yp e I I with parameters n /m = 2, K = 4 /k 4 and D = − 2 /k 2 . Let us introd uce coord inates Q = ( Q 1 , Q 2 , Q 3 ) as Q =  ( k 2 + 4 r 2 ) 1 / 2 − k 2  1 / 2  cos θ cos ϕ, cos θ sin ϕ, sin θ  . In terms of these co ordinates, the ab o v e metric and p oten tial r ead as d s 2 =  k + k Q k 2  k d Q k 2 , V II = k 2 k Q k 2 k + k Q k 2 . Th us w e reco ve r the three-dimensional Darb oux system of t yp e I I I [32]. The Dar- b oux system of typ e I I I is th e only quadratically sup erinteg rable natural Hamiltonian system in a surf ace of nonconstan t curv ature whic h is kno wn to adm it quadratically sup erinte grable N -dimensional generalizatio ns [2 ]. (iii) Mu ltifold Kepler systems. The family of multifold Kepler s ystems w as introdu ced b y Iw ai and Kata y ama [27 , 28] as Hamiltonian r eductions of the geo desic flow in a gener- alized T aub–NUT metric. These systems are giv en by the m etrics and p oten tials d s 2 = k Q k n m − 2  a + b k Q k n m  k d Q k 2 , V II = k Q k 2 − n m a + b k Q k n m  µ 2 k Q k − 2 + µ 2 c k Q k n m − 2 + µ 2 d k Q k 2 m n − 2  , with Q = ( Q 1 , Q 2 , Q 3 ), a, b, c, d, µ constan ts and n, m coprime p ositiv e in tegers. Th e substitution Q =  ( a 2 + 4 br 2 ) 1 2 − a 2 b  m n  cos θ cos ϕ, cos θ sin ϕ, sin θ  sho ws that the multifold Kepler mo dels are equiv alen t to the type I I Bertrand sy s tems with parameters K = 4 a − 4 b 2 and D = − 2 b a 2 . I t should b e notic ed that the Darb oux space of typ e I I I is a particular case of the multifo ld Kepler systems. 6 3 The orbit equation and geometric sup erin tegrabilit y Hereafter w e shall analyze the pr op erties of the Hamiltonian systems in ( M , g j ) give n by H j := 1 2 k p k 2 g j + V j ( q ) , j = I, I I , (7) where the metric g j and the p ote ntia l V j are resp ectiv ely defined b y (5) and (6). As previously discussed, the orbits of these systems corresp ond to tra jectories o f the asso ciated Be rtrand spacetimes. I t sh ou ld b e noticed that in the adapted co ordinate s ystem, th ese Hamiltonians read as H I = 1 2 "  n m  2  1 + K r 2  p 2 r + p 2 θ r 2 + p 2 ϕ r 2 sin 2 θ # + p r − 2 + K + G , (8a) H II = 1 2 " n 2  (1 − D r 2 ) 2 − K r 4  p 2 r 2 m 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  + p 2 θ r 2 + p 2 ϕ r 2 sin 2 θ # ∓ r 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  − 1 + G , (8b) where p r is th e momentum conju gate to r and p θ and p ϕ are defined analogously . In this section we shall d eriv e the simplest su p erin tegrabilit y p rop ert y of the Hamiltonian systems (7 ) (cf. Prop osition 10), whic h nonetheless seems to hav e esca p ed unn otice d so far. The pro of of this result relies on the f act that, b y definition, the orbits of (7) define an in v ariant foliation by (top ologica l) circles in an op en su bset Ω ⊂ T ∗ M of the ph ase space of the s y s tem. E.g., in the classical Kep ler p r oblem Ω =  ( q , p ) ∈ R 3 × R 3 : H ( q , p ) < 0 , q × p 6 = 0  is the set of p oints w ith nega tive energy and n onzero angular momen tum , whereas for the harmonic oscillator one can tak e Ω = ( R 3 × R 3 ) \{ (0 , 0) } , i.e., the w hole ph ase space minus the equilibrium. In Prop osition 7 b elo w we compu te the expression of the orbits in clo sed form, r ev ealing th at the ab o v e foliation is actually a lo cally trivial fibration. This allo ws u s to resort to the geometric theory of su p erin tegrable Hamiltonian systems [12], yielding the first s u p erin tegrabilit y resu lt for (7). Before discussing th e precise statemen t of Prop osition 10, let us compute the orbits of the Hamiltonian (7). In fact, the closed expression that w e shall d eriv e is not only used in the pro of of P rop osition 10, bu t it is also a k ey elemen t of Theorem 12, wh ere a stronger sup erinte grabilit y result is p r esen ted. It is con ve nient to introd u ce the r ecta ngular co ordinates q = ( q 1 , q 2 , q 3 ) asso ciated to the s pherical co ordinates ( r, θ , ϕ ) as q =  r cos θ cos ϕ, r cos θ sin ϕ, r sin θ  . (9) The conju gate moment a will b e denoted by p = ( p 1 , p 2 , p 3 ). Clearly the co ordinates ( q , p ) are globally defi n ed in T ∗ M . W e sh all use the n otation · , × and k · k f or the Eu clidean inner pro duct, cross p rod uct and n orm in R 3 and call E = H j ( p , q ) and J 2 = k q × p k 2 the energy and angular momen tum of an integ ral curv e ( q ( t ) , p ( t )) of (7) . Ob viously E an d J 2 are constan ts of motion. 7 Prop osition 7. L et γ b e an inextendible orbit of the Hamiltonian syst em (7) which is c on- taine d in the invariant plane { θ = π 2 } . Then γ is given by cos  nϕ m − ϕ 0  = 1 + J 2 √ r − 2 + K p 1 + 2 J 2 ( E − G ) + K J 4 (10a) if j = I and by cos  nϕ m − ϕ 0  = J 2 r − 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  + D J 2 + 2 G − 2 E p (2 E − 2 G − D J 2 ) 2 ± 4 J 2 − K J 4 (10b) if j = I I . Her e ϕ 0 is a r e al c onstant. Pr o of. W e b egin with the case j = I. The crucial observ ation is that the orbit equation m 2 J 2 n 2 r 4 (1 + K r 2 )  d r d ϕ  2 = 2 E − 2 V I − J 2 r 2 simplifies dramatically with the c hange of v ariables u = p r − 2 + K , in terms of whic h the p oten tial and th e inv erse square term read as V I = u + G , r − 2 = u 2 − K . The orbit equation is then giv en by  mJ n d u d ϕ  2 = 2 E − 2 G + K J 2 − 2 u − J 2 u 2 , whic h can b e readily in tegrated to yield cos  nϕ m − ϕ 0  = 1 + J 2 u p 1 + 2 J 2 ( E − G ) + K J 4 for some constan t ϕ 0 . When j = I I the treatmen t is analogous. No w th e orbit equation reads 1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4 r 4  (1 − D r 2 ) 2 − K r 4   mJ n d r d ϕ  2 = E − V II − J 2 2 r 2 , and it is con v enien t to in tro duce the v ariable v = r − 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  . In terms of this new co ordinate the p otent ial is simply V II = G ∓ 1 v , whereas the inv erse square term is giv en by r − 2 = v 2 + 2 D v + K 2 v . Hence a straigh tforw ard computation sho ws that the orb it equation is  mJ n d v d ϕ  2 = 4( E − G ) v − J 2  v 2 + 2 D v + K  ± 4 , 8 thereb y obtaining cos  nϕ m − ϕ 0  = J 2 ( v + D ) + 2 G − 2 E p (2 E − 2 G − D J 2 ) 2 ± 4 J 2 − K J 4 . Here ϕ 0 is a real constan t. R emark 8 . Eqs. (10) are we ll defined also when J = 0. Moreo ver, it is n ot d ifficult to c hec k that r can b e readily exp ressed as a f u nction of ϕ by p erform in g some m anipulations in the righ t-hand side of (10). W e s h all no w sp ecify what is u ndersto od by geometric sup erinteg rabilit y . Let F 0 b e a smo oth Hamiltonian d efined on a 2 d d imensional symplectic manifold N admitting s ≥ d − 1 functionally indep endent first in tegrals F 1 , . . . , F s other than the Hamiltonian. Let us supp ose that F = ( F 0 , F 1 , . . . , F s ) is a s u bmersion on to its image with compact and connected fi b ers, whic h by E h resmann’s theorem (cf. e.g. [41]) implies th at its level sets define a lo cally trivial fibration F of N . If s ≥ d , not all the latter first in tegrals can Po isson-comm ute: the usu al condition to imp ose is that there exists a matrix-v alued fun ction P : F ( N ) → Mat( s + 1) of rank s − d + 1 such that { F i , F j } = P ij ◦ F , 0 ≤ i, j ≤ s . (11) In particular, when s = d − 1 this yields the usu al definition of Liouville inte grabilit y . W ell kno wn generalizations of th e Liouville–Arnold theorem [42, 44] sho w th at ev ery fi b er of F is an in v arian t (2 d − s − 1)-torus, an d that the motion on eac h of these tori is conju gate to a linear fl o w. Moreo v er, the fi bration F has symplectic lo cal trivializations. Geometrical ly , th e existence of the fu n ction P means that F has a p olar foliation [12], i.e, a foliation F ⊥ whose tangent spaces are s ymplectica lly o rthogonal to those of F . Similarly , the rank condition in Eq. (11) is tan tamount to demand that the in v arian t (2 d − s − 1)-tori of the foli ation b e isotropic. T h us the crucial elemen t in the geometric c haracterization of sup erinte grabilit y is the b ifoliat ion ( F , F ⊥ ), whic h is a typ e of d ual pair as defined in [51]. One is thus led to in tro duce the follo wing definition (cf. [12] and the surve y [17], where slightly differen t wording is used): Definition 9. A Hamiltonian system on a symplectic 2 d -dimensional manifold is ge omet- ric al ly sup erinte gr able with s ≥ d − 1 semiglob al inte gr als if the Hamiltonian v ector field is tangen t to a lo call y tr ivial fibr atio n by isotropic (2 d − s − 1)-tori wh ic h admits a p olar f olia - tion. If s take s the maxim um v alue 2 d − 2 we shall simp ly sa y that the sys tem is ge ometric aly sup erinte gr able . Of course, generally not all the phase space of a (sup er)int egrable system is fib ered b y in v ariant isotropic tori: there can b e, e.g. , singular p oints and u n b ounded orbits. But it is customary and of in terest to restrict one’s atten tion to the r egion wher e suc h fi bration is w ell defined. In the case w h en s = d − 1 (Liouville integrabilit y), the inv arian t tori are Langrangian and th er efore F ⊥ = F , explaining why the bifibration ( F , F ⊥ ) is less well kno wn than the fibrations by Lagrangian tori. (Ho wev er, a n adv anta ge of the bifi bration is that, u n der mild tec h nical assumptions, it is un iquely determined (and finer), w h ereas f or integ rable systems with additional integrals there is some arbitrarin ess in th e c hoice of inv ariant Lagrangian tori.) It should b e n otic ed that the ab o v e structur e y ields “semiglobal” (i.e., defined in a tubular neigh b orh oo d of eac h toru s) first in tegrals asso ciated to the existence of generalized 9 action-angle coordin ates; a d etailed accoun t can b e found in [8, 12, 17]. The con ten t of the follo w ing prop osition is that the Bertrand systems (7) are geometrically sup erintegrable in the r egio n foliated by p erio dic orb its. Prop osition 10. L et Ω b e th e r e gion of T ∗ M wher e al l the orbits of H j ar e p e rio dic. Then H j | Ω is ge ometric al ly sup erinte gr able. Pr o of. It easily follo ws from Prop osition 7 that the orb its of H j define a lo cally trivial fibr atio n b y (top ologica l) circle s in Ω. The fi b ers are certainly isotropic, as they are one-dimensional, and the flow of H j on eac h fi b er is conjugate to the linear one b ecause H j do es not p ossess an y critica l p oin ts in Ω. Moreo ver, it ste ms from Pr op ositio n 7 th at the function Ω → R + mapping eac h p oin t in Ω to the length of the (p erio dic) orbit p assin g through it is smo oth, whic h in turn readily imp lies that the p erio d fun ctio n is also smo oth and n on v anishing in this r egio n. Hence a th eorem of F ass` o [16] implies that H j is geometrically su p erin tegrable, pro ving the pr op osition. 4 The generalized Runge–Lenz v ector In this section we sh all prov e a stronger sup erintegrabilit y r esult for the Bertrand Hamil- tonians (7). More pr ecisel y , we shall p ro vid e a semi-explicit construction of an additional v ector first int egral w hic h we shall call the generalized Run ge–Le nz ve ctor. This vec tor fi eld is defin ed on an n -fold cov er f M of the original sp ace M , and it is inv ariant un der the flo w generated by th e lift of the Bertrand Hamiltonian to the co v ering space f M . In M , this vecto r field induces a global tensor fi eld of rank n which is preserv ed und er the fl o w of H . As b efore, n is the p ositiv e in teger w hic h app ears in Eq. (7). As r egards the s up erin tegrabilit y prop erties of the Hamilto nian systems (7), the spherical symmetry of these systems readily yields three fir s t in tegrals other than the Hamiltonian, whic h can b e iden tified w ith the comp onents of the angular m omen tu m. The idea of looking for generalizations of the Run ge–Lenz v ector in ord er to find an additional in tegral of motion is not new: an up dated and rather complete r eview of the related literature can b e fou n d in [36 ]. Here w e shall u s e our information ab out the in tegral curv es of (7) and some ideas already present in the w ork of F rad k in [20] and Holas and Marc h [26]. Let us start by recalling F radkin’s construction [20] of a lo c al v ector fi rst in tegral for the Hamiltonian s ystem H 0 = 1 2 k p k 2 + U ( k q k ) , where U ( k q k ) is an arbitrary cent ral p oten tial and ( q , p ) ∈ R 3 × R 3 . The starting p oin t is the follo wing trivial remark. Consider an integral curv e q ( t ) of H 0 con tained in the plane { θ = π 2 } ⊂ R 3 , w here ( r , θ , ϕ ) are the usual spherical coordin ates. W e can assu me without loss of generalit y that we ha v e tak en th e in itial condition ϕ (0) = 0 and u se the notation r = k q k , J = p ϕ = r 2 ˙ ϕ . A simple compu tatio n sh o w s that the deriv ativ e along this int egral curv e of the unit v ector fi eld a = cos ϕ r q + sin ϕ r J q × ( q × p ) (12) is identica lly zero, as in f act a ( t ) is the constant vec tor (1 , 0 , 0). F radkin’s observ ation was that if cos ϕ and J − 1 sin ϕ can b e expressed in terms of q and p in a domain Ω ⊂ R 3 \{ 0 } , then the resulting v ector field is a first inte gral of H 0 in Ω. When H 0 is the Kepler Hamiltonian, 10 the generalized Ru nge–Lenz v ector field is w ell defi ned globally and essenti ally coincides with the cla ssical Ru n ge–Lenz vec tor d ivided by its norm. When H 0 is the h armonic oscillator, the generalized Ru nge–Lenz v ector is multiv alued (this can b e neatly understo o d by considering the turning p oin ts of the orbits), but can b e used to reco v er the conserv ed tensor field C = 2 ω 2 q ⊗ q + p ⊗ p asso ciated to the SU(3) symmetry [26]. Definition 11. L et H b e a Hamilt onian system d efined on (the cotangen t bundle of ) a 3- manifold N . W e sa y that H admits a gene r alize d Runge–L enz ve ctor if there exists a nontrivia l horizon tal vec tor field A in T ∗ N w hic h is constant along th e flo w of H . Ob viously the conserv ed v ector A is nontrivia l if it is not constant and cannot b e written in terms of the energy and the angular momentum inte grals, and w e recall that a horizon tal v ector (resp. tensor) field in T ∗ N can b e simply u ndersto o d as a vect or (resp. tensor) field in N whic h d ep ends on b oth the p ositions and the momen ta. The main problem with F radkin ’s approac h is that, of course, it is not at all ob vious ho w to obtain sufficien t conditions en suring that these lo cal integ rals are w ell defined globally , while lo cal sup erin tegrabilit y is trivial in a neigh b orho o d of an y regular p oint of th e Hamiltonian flo w. Ho w ev er, we shall see b elo w that F radkin’s appr oac h w orks w ell for the kind of Hamiltonian systems that w e are co nsid ering in th is p ap er, and that one can construct a globally defined generalized Runge–Lenz v ector (cf. Eq. (16) b elo w) whic h is roughly analogous to (12 ). Theorem 12. Consider a Hamiltonian of the f orm (7) , with m, n c oprime p ositive inte ge rs. Then ther e exists an n -fold c over f M of M su c h that the lift of this Hamiltonian to f M admits a gener alize d R u nge–L enz ve ctor. Pr o of. W e shall call H j , j = I, II, the Hamiltonian (7). Let γ b e an inextendible orbit of H j , whic h ca n b e assumed to lie in the inv arian t plane { θ = π 2 } . By Pr op ositio n 7, and taking ϕ 0 = 0 in Eq. (10) without loss of generalit y , γ is the self-intersec ting curve give n b y cos nϕ m = χ ( r 2 , J 2 , E ) , (13) where χ is the function χ ( r 2 , J 2 , E ) =            1 + J 2 √ r − 2 + K p 1 + 2 J 2 ( E − G ) + K J 4 if j = I , J 2 r − 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  + D J 2 + 2 G − 2 E p (2 E − 2 G − D J 2 ) 2 ± 4 J 2 − K J 4 if j = I I . Moreo ver, the c hain rule immediately yields sin nϕ m = − m n d d ϕ  cos nϕ m  = − m ˙ r n ˙ ϕ ∂ ∂ r χ ( r 2 , J 2 , E ) = Θ( r ˙ r, r 2 , J, E ) , (14) where Θ( r ˙ r , r 2 , J, E ) = − 2 r ˙ r mr 2 nJ ( D 1 χ )( r 2 , J 2 , E ) and D 1 χ stand s for the d eriv ativ e of the fun ction χ with resp ect to its fi r st argumen t. It should b e n oted that these expressions are well d efi ned also for J = 0. 11 Using the p rop erties of the Cheb yshev p olynomials it is trivial to express cos nϕ and sin n ϕ in terms of r, ˙ r, J and E as cos nϕ = T m  cos nϕ m  = T m  χ ( r 2 , J 2 , E )  , sin n ϕ = sin nϕ m U m − 1  cos nϕ m  = Θ( r ˙ r , r 2 , J, E ) U m − 1  χ ( r 2 , J 2 , E )  . Here T m and U m resp ectiv ely stand for the Chebyshev p olynomials of the first and seco nd kind an d degree m . S etting S 1 =  z ∈ C : | z | = 1  , w e fin d it con v enient to define the analytic S 1 -v alued map E n ( r ˙ r, r 2 , J, E ) = T m  χ ( r 2 , J 2 , E )  + iΘ( r ˙ r , r 2 , J, E ) U m − 1  χ ( r 2 , J 2 , E )  , in terms of whic h the orbit γ is c haracterized as e i nϕ = E n ( r ˙ r, r 2 , J, E ) . (15) It stems from F radkin ’s argumen t th at (12) yields a ve ctor first inte gral of (8) in any region where e i ϕ can b e un ambiguously expressed in terms of the coord inates ( q , p ). How ever, Eq. (15) do es n ot determine the angle ϕ u niv o cally mo dulo 2 π b ecause the map z 7→ z n of the unit circle on to itself has degree n , so that F r ad k in ’s construction is, a pr iori, not global. As a matter of fact, it is obvious that Eq. (15) only d efines ϕ mo dulo 2 π /n , thus yielding an n -v alued additional inte gral. The aforementi oned problem is a consequence of the fact th at the orbit γ has self- in tersections. It is stand ard that this difficult y can b e circum v ente d by means of an app ropri- ate co v ering sp ace of our initial manifold. The construction whic h w e shall next outline is in fact analog ous to that of the R iemann su r face of the function z 7→ z n . W e denote b y γ ( t ) the p erio dic integ ral curve of (7) d efined by th e orbit γ ⊂ M and tak e an n -fold co v er Π : f M → M of M such that the lift e γ ( t ) of γ ( t ) to f M is a smo oth path without self-in tersections. Notice that e γ ( t ) is actually an int egral cur v e of the lifted Hamiltonian e H j = 1 2   e p   2 Π ∗ g j + ( V j ◦ Π)( e q ) , j = I, I I , where ( e q , e p ) ∈ T ∗ f M . f M is a fib er bund le o v er M with typical fib er Z n , and for eac h k ∈ Z n w e denote by Λ k : M → f M the section of f M with fib er v alue k . Ob viously Λ k is an injectiv e map, and an isometry from an op en and dense subset M k ⊂ M onto its image in ( f M , Π ∗ g j ). One ob viously has that Π ◦ Λ k = id and Π − 1 ( q ) = [ k ∈ Z n Λ k ( q ) for all q ∈ M . By constru ction, in eac h section Λ k ( M ) there exists a determination of the (complex) n -th ro ot whic h allo ws to solv e e i ϕ in terms o f e i nϕ univocally along Λ k ( e γ ). Therefore, for eac h k ∈ Z n there exist real f u nctions S k and C k (namely , determinations of arcsin and arccos) suc h that e i ϕ ( t ) = C k (cos nϕ ( t )) + i S k (sin nϕ ( t )) 12 whenev er the p oin t ( r ( t ) , θ = π 2 , ϕ ( t )) lies in Λ k ( e γ ). Moreo ver, an easy computation sho ws that the functions C k ( r 2 , J 2 , E ) = C k  T m ( χ ( r 2 , J 2 , E ))  , S k ( r ˙ r, r 2 , J 2 , E ) = J − 1 S k  Θ( r ˙ r , r 2 , J, E ) U m − 1 ( χ ( r 2 , J 2 , E ))  are analytic in th eir d omains. In order to express C k ( r 2 , J 2 , E ) and S k ( r ˙ r , r 2 , J 2 , E ) in a m ore con v enient wa y , w e con- sider the lift of the co ordinates q to eac h space Λ k ( M ). With a sligh t abuse of notation, w e shall s till denote these co ordinates by q . An im m ediate computation shows that Π ∗ g j | Λ k ( M ) reads as d s 2 = k d q k 2 +  h ( k q k ) 2 − 1  ( q · d q ) 2 k q k 2 , where the function h is defined as in Section 2, namely , h ( r ) 2 =            m 2 n 2 (1 + K r 2 ) if j = I , 2 m 2  1 − D r 2 ± p (1 − D r 2 ) 2 − K r 4  n 2 ((1 − D r 2 ) 2 − K r 4 ) if j = I I . By differen tiation it stems f r om this formula that the conjugate momentum p to q is give n b y p = ˙ q +  h ( k q k ) 2 − 1  q · ˙ q k q k 2 q , yielding ˙ q = v ( q , p ) with v ( q , p ) = p +  h ( k q k ) − 2 − 1  q · p k q k 2 q . As r ˙ r = q · ˙ q = q · v ( q , p ), we no w ha v e all the ingredien ts to in vo ke F radkin’s argument (cf. Eq. (12), with whic h (16) should b e compared) and derive that eac h comp onen t of the horizon tal vec tor field A k in T ∗ Λ k ( M ) d efined by A k = 1 r h C k  k q k 2 , k q × p k 2 , H j ( q , p )  q + S k  q · v ( q , p ) , k q k 2 , k q × p k 2 , H j ( q , p )  q × ( q × p ) i (16) is a constant of motion in Λ k ( M ). By constru ctio n, the v ector fields A k (with k ∈ Z n ) define an analytic global horizon tal v ector field A in T ∗ f M whose Lie deriv ativ e along the flo w of e H j is zero, thereb y obtaining the d esired unit Runge–Lenz vec tor. R emark 13 . The particular form of the orbits (10) and the fact that f M is a finite cov er of M ensu re that all the lifted orbits whic h are b oun ded are al so perio dic, a nd that th e lifted orbits do not ha v e an y self-int ersections. Note that if f M is endo we d w ith the pulled bac k metric e g j = Π ∗ g j , e H j is a natural Hamiltonian system and Π : ( f M , e g j ) → ( M , g j ) b ecomes a Riemannian co v er. Corollary 14. Consider a Hamiltonian of the form (7) with n = 1 . Then the gener alize d R unge–L enz ve ctor is wel l define d in a l l M . Pr o of. It trivially follo ws from Theorem 12. 13 Corollary 15. Consider a Hamiltonian H j of the form (7) , with m, n c oprime p ositive inte gers. Th en ther e exists a horizonta l symmetric tensor field in M of r ank n which is invariant under the flow of H j . Pr o of. Let us use the same notation as in the pr oof of Th eorem 12. In particular, w e consider the in tegral cur v e γ ( t ) an d the maps A k used in the pr o of of Th eorem 12. F or eac h k ∈ Z n , let us denote by A k ( t ) the restriction of th e h orizonta l v ector field A k : T ∗ Λ k ( M ) → R 3 to the p ro jection of the in tegral curv e γ ( t ) to T ∗ Λ k ( M ). Th e only observ ation w e need in order to pr o ve Corollary 15 is that, by the expr ession for γ foun d in Prop osition 7 and the definitions of the co v ering space f M and of the horizonta l vec tor fields A k , it easily follo ws that A k  t + ℓ n T γ  = A k + ℓ ( t ) for all k ∈ Z n , ℓ ∈ Z , t ∈ R s u c h that γ ( t ) ∈ M k + ℓ and γ ( t + ℓ n T γ ) ∈ M k . Here T γ stands for the p erio d of th e integ ral cu rv e γ ( t ) and the su m k + ℓ is to be considered mo du lo n . This p erio dicit y p rop ert y readily implies that the symmetric tensor pro du ct C of A 1 , . . . , A n , with comp onen ts C i 1 ,...,i n ( q , p ) = A ( i 1 1 ( q , p ) · · · A i n ) n ( q , p ) , is a we ll defined, analytic tensor fi eld in M of rank n . As usual, symmetrization of th e sup erscripts delimited b y curv ed brac k ets is understo o d. T o complete the pro of of the corol- lary , it suffi ces to notice that C is trivially inv ariant under the fl o w of H j as eac h A k is a (m ultiv alued) fi rst integ ral. Some commen ts ma y b e in order. First, on e sh ould observe the dep endance of th e additional inte grals (16) on the momen ta is generally complicate d (and in p articular not quadratic), whic h explains why they are usually s o hard to sp ot [30]. Second, it sh ould b e noticed that Corollaries 14 and 15 yield the usual Ru nge–Lenz v ector and second rank con- serv ed tensor (up to a normalization constant) when the Bertrand Hamiltonian we consider is the Kepler or h armonic oscillator system in Euclidean sp ace [26, 47]. Note, how ever, th at giv en an arbitrary Bertrand Hamiltonian it is usu ally hard to compute th e conserv ed tens or C or the Runge–Lenz v ector A in closed form . In th is d irection, it should b e mentioned that an additional in tegral h as b een explicitly obtained for some of the Bertrand Hamiltonians discussed in Example 6 (cf. e.g. [2, 4, 21, 27] a nd r eferences therein). 5 Bertrand’s theorem on cu rv ed spaces In the p revious sections we hav e thoroughly analyzed the sup er integrabilit y prop erties of the spherically symmetric n atural Hamilto nian systems whose b ound ed orbits are all p eriodic. When com b ined with the discussion of harmonic oscillators and Kepler p otentia ls on Bertrand spacetimes p resen ted in Section 2, this giv es all the ingredien ts w e n eed to state a fully satisfactory analogue of Bertrand’s theorem on spherically symm etric s p aces: Theorem 16. L et H b e the H amilto nian function asso ciate d to a Bertr and sp ac etime, i.e., an autonomo us, spheric al ly symmetric natur al Hamiltonian system on a R iemannian 3-manifold ( M , g ) sat isfying Pr op erties (i) and (ii ) in Bertr and’s The or em 1. Then the fol lowing sta te- ments hold: (i) H is o f the form (7) for some c oprime p ositive inte gers n, m . 14 (ii) The p otential V is the intrinsic Kepler or oscil lato r p otential i n ( M , g ) . (iii) H is sup erinte gr able. Mor e pr e cisely, (a) H is ge ometric al ly sup erinte gr able in the r e gion of T ∗ M foliate d by b ounde d orbits. (b) Ther e exists an n -fold c over f M of M such that the lift of H to f M admits a gener alize d R unge–L enz ve ctor. (c) Ther e exists a nontrivial horizontal tensor field in M of r ank n which i s i nv ariant under the flow of H . As mentio ned in th e in tro duction, this resu lt is of in terest b oth in itself and b ecause of the abu ndan t literature dev oted to th e stu dy of p articular cases of this problem in d ifferen t con texts and fr om v arious p oin ts of view. Ac kno wledgemen ts This work was partially sup p orted b y the Spanish Ministerio de Educaci´ on un der gran t no. MTM2007- 67389 (with EU-FEDER sup p ort) (A.B. and F.J.H.), b y the Sp anish DGI and CAM–Complutense Univ ersit y under gran ts no. FIS2008-0 0209 and CCG07-2779 (A.E.), and b y the INFN–CICyT (O.R.). References [1] Atiy ah, M.F., Hitchin, N.J.: Lo w-energy s cattering o f non- Abelian mag netic monop oles. Phys. Lett. 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