The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion
TThe classical Taub-Nut System: factorization, Spectrum GeneratingAlgebra and solution to the equations of motion D anilo L atini , O rlando R agnisco Department of Mathematics and Physics and INFN, Roma Tre University, Via della Vasca Navale , I- Rome, Italy
Abstract T he formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way tobe suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System,the so-called Taub-Nut system, associated with the Hamiltonian: H η ( q , p ) = T η ( q , p ) + U η ( q ) = | q | p m ( η + | q | ) − k η + | q | ( k > η > ) .In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we showthat the classical Taub-Nut system shares a number of essential features with the Kepler system, that is justits Euclidean version arising in the limit η →
0, and for which a “SUSYQM” approach has been recentlyintroduced by S. Kuru and J. Negro. In particular, for positive η and negative energy the motion is alwaysperiodic; it turns out that the period depends upon η and goes to the Euclidean value as η →
0. Moreover,the maximal superintegrability is preserved by the η -deformation, due to the existence of a larger symmetrygroup related to an η -deformed Runge-Lenz vector, which ensures that in R closed orbits are again ellipses.In this context, a deformed version of the third Kepler’s law is also recovered. The closing section is devotedto a discussion of the η < We consider the classical Hamiltonian in R N given by: H η ( q , p ) = T η ( q , p ) + U η ( q ) = | q | p m ( η + | q | ) − k η + | q | , ( )where k and η are real parameters, q = ( q , . . . , q N ) , p = ( p , . . . , p N ) ∈ R N are conjugate coordinates andmomenta, and q ≡ | q | = ∑ Ni = q i . We recall that H η has been proven to be a maximally superintegrableHamiltonian by making use of symmetry techniques [ ]. This means that H η is endowed with the maximumpossible number ( N − ) of functionally independent constants of motion (including H η itself). In fact,besides the integrals of motion provided by the so ( N ) symmetry, H η is endowed with an η − deformed N DLaplace–Runge–Lenz vector R implying the existence of N additional constants of motion coming from thecomponents of R , which are given by: R i = m N ∑ j = p j ( q j p i − q i p j ) + q i | q | ( η H η + k ) , i =
1, . . . , N . ( ) E-mail: latini@fis.uniroma .it E-mail: ragnisco@fis.uniroma .it a r X i v : . [ m a t h - ph ] J a n he squared modulus of R is radially symmetric, and turns out to be expressible in terms of H η and L : R = N ∑ i = R i = L m H η + ( η H η + k ) . ( )As a matter of fact the system associated with H η ( ) under the canonical symplectic structure can beconsidered as a genuine (maximally superintegrable) η -deformation of the N D usual Kepler-Coulomb (KC)system, since the limit η → H η ( ) yields: H = p m − k | q | . ( )Moreover H η can be naturally related to the Taub-NUT system [ , , , , , , , , , , , ] since M N can be regarded as the (Riemannian) N D Taub-NUT space [ ]. It is also known that according to thePerlick classification [ , , , ] the system ( ) pertains to the class II, and thus it has to be regarded as an“intrinsic oscillator”. In the sequel it will be clear that with respect to the Euclidean KC it plays an analogousrole to the Darboux III (D-III in the following) in comparison with the standard harmonic oscillator [ ].Actually it turns out ??mutatis mutandis”, the solution to the classical equations of motion for D-III can befound on the same footing, through the factorisation of the corresponding classical Hamiltonian. The resultsconcerning D-III will be published later in a larger more general paper [ ], where the quantum cases willbe also investigated, mostly through the Shape Invariant Potentials approach.
In the following we drastically simplify our setting, and limit our considerations to the physical (i.e. 3-dimensional) case. As mentioned in the abstract we adapt and in a sense generalise the construction and theresults derived in [ ].We will study the Hamiltonian H = T ( r , p ) + V e f f ( r ) = rp m ( r + η ) + l mr ( r + η ) − kr + η = K ( r ) H , ( )where m , k and l are positive constants, η is the deformation parameter, p ≡ p r is the radial momentum, H is the “undeformed” Kepler-Coulomb Hamiltonian and K ( r ) . = rr + η . In ( ) we introduced the radialcoordinate r : = | q | , canonically conjugated to p .The main idea is to use the framework of SUSYQM in the context of classical mechanics to derive alge-braically the classical trajectories (see Ref. [ ]). Let us then consider the Hamiltonian ( ) written in a slightlydifferent form, namely: H = rr + η (cid:18) p m + l mr − kr (cid:19) . ( )Multiplying both sides of ( ) by r ( r + η ) we get: r ( r + η ) H = r (cid:18) p m + l mr − kr (cid:19) = m ( r p + l − mkr ) . ( )Now, as it has been done in the undeformed case by Kuru and Negro ([ ]), at any r we can factorize ( ) asfollows: r p − mr ( k + η H ) − mr H = A + A − + γ ( H ) = − l , ( )where for the time being A + , A − are unknown functions of r , p . Paraphrasing [ ] we make the following ansatz for A + , A − A ± = (cid:18) ∓ irp + ar √− H + b ( H ) √− H (cid:19) e ± f ( r , p ) . ( ) he “arbitrary function” f ( r , p ) will be determined by requiring the closure of the Poisson algebra generatedby H and A ± . More precisely, we impose: { H , A ± } = ∓ i α ( H ) A ± ( ) { A + , A − } = i β ( H ) , ( )where the functions α , β wait to be determined. Inserting A ± in ( ) we get a = √ m , b ( H ) = − (cid:114) m ( k + η H ) , γ ( H ) = m ( k + η H ) H , ( )and requiring that A ± obey the proper Poisson brackets we arrive at f ( r , p ) = − i (cid:114) m rp √− H ( k − η H ) , α ( H ) = − (cid:114) m H √− H ( k − η H ) , β ( H ) = √ m ( k + η H ) √− H , ( )and finally: A ± = (cid:18) ∓ irp + r √− mH − (cid:114) m ( k + η H ) √− H (cid:19) e ∓ i (cid:113) m rp √− H ( k − η H ) ( ) { H , A ± } = ± i (cid:114) m H √− H ( k − η H ) A ± , { A + , A − } = i √ m ( k + η H ) √− H . ( )A mandatory requirement is that in the limit η → m = k =
1, the result found in [ ]. To make the identification even more perspicuous we canintroduce A . = (cid:113) m ( k + η H ) √− H entailing the following su (
1, 1 ) algebra relations: { A , A ± } = ∓ iA ± , { A + , A − } = iA . ( )Now we can define the “time-dependent constants of the motion” Q ± = A ± e ∓ i α ( H ) t , ( )such that dQ ± dt = { Q ± , H } + ∂ t Q ± =
0. Those dynamical variables take complex values admitting the polardecomposition Q ± = q e ± i θ and allowing in fact to determine the motion, which turns out to be boundedfor E = −| E | <
0. Indeed we have: (cid:18) ∓ irp + r (cid:113) m | E | − (cid:114) m ( k − η | E | ) (cid:112) | E | (cid:19) e ∓ i (cid:0) (cid:113) m rp √ | E | ( k + η | E | ) + (cid:113) m | E | √ | E | ( k + η | E | ) t (cid:1) = q e ± i θ , ( )or else − irp + r (cid:112) m | E | − (cid:113) m ( k − η | E | ) √ | E | = q e i (cid:0) (cid:113) m rp √ | E | ( k + η | E | ) + (cid:113) m | E | √ | E | ( k + η | E | ) t + θ (cid:1) + irp + r (cid:112) m | E | − (cid:113) m ( k − η | E | ) √ | E | = q e − i (cid:0) (cid:113) m rp √ | E | ( k + η | E | ) + (cid:113) m | E | √ | E | ( k + η | E | ) t + θ (cid:1) , ( )where q = (cid:114) − l + m ( k − η | E | ) | E | (following from A + A − + γ ( H ) = q + γ ( H ) = − l ). Summing and subtract-ing ( ) we obtain: r (cid:112) m | E | − √ m ( k − η | E | ) √ | E | = q cos (cid:18)(cid:113) m rp √ | E | ( k + η | E | ) + (cid:113) m | E | √ | E | ( k + η | E | ) t + θ (cid:19) rp = − q sin (cid:18)(cid:113) m rp √ | E | ( k + η | E | ) + (cid:113) m | E | √ | E | ( k + η | E | ) t + θ (cid:19) . ( )It is immediate to verify that taking the sum of the square of these two equations we obtain the equation ( )restricted to the level surface H = −| E | . Finally, thanks to the above relations, we are able to obtain t as afunction of r : t ( r ) = Ω ( η ) ( E ) (cid:20) arccos (cid:18) − (cid:114) m (cid:0) ( k − η | E | ) − | E | r (cid:1) q (cid:112) | E | (cid:19) − (cid:114) m (cid:112) | E | ( k + η | E | ) (cid:113) mr ( k − η | E | ) − m | E | r − l − θ (cid:21) , ( ) here Ω ( η ) ( E ) = (cid:113) m | E | √ | E | ( k + η | E | ) ≡ α ( E ) is the angular frequency of the motion. Concerning ( ) it is evidentthat, due to the presence of the “inverse cosine” function, t is a multivalued function of r defined mod2 π / Ω . To recover univaluedness, we have to introduce a “uniformization map” which is trivially given bythe periodic function cos ( Ω t ) . In the limit η →
0, the results for the flat Kepler-Coulomb are recovered (see[ ]). We can say that the motion has been algebraically determined .A number of plots are reported, showing the behavior of V e f f ( r ) . = l mr ( r + η ) − kr + η as a function of r , andthe orbits on the phase plane ( r , p ) for different values of the deformation parameter (for 2 l = m = k = E = −
1, in appropriate units). - - r p r - - - - V eff ( r ) Figure : Phase plane ( r , p ) and effective potential V e f f ( r ) for η = - - r p Figure : Phase plane ( r , p ) for η =
0, 0.01, 0.05, 0.1, 0.15, 0.2, 0.25 .
As we have shown in the introduction, our system is maximally superintegrable and this maximal superin-tegrability is strictly related to the existence of the Runge-Lenz vector: then, as it happens for the standardKepler-Coulomb system, we expect that this extra symmetry will play a crucial role in determining theshape of the orbits. As is well known, in the undeformed case the orbits are conic sections, namely ellipsesfor bounded trajectories. To identify the analytic form of the orbits when η (cid:54) =
0, we will first consider thesimplest and more physical case, corresponding to η >
0. To this end, we will closely follow Refs. [ , ]. n R the Runge-Lenz vector R , when evaluated on-shell , can be written as: R = m [ p ( p · q ) − q ( p · p )] + q | q | ( k − η | E | ) . ( )Again, we see that its expression is formally identical to the one holding in the flat case and is obtained byletting k → k − η | E | : = K . For its square we can write (again on-shell ): R = K − l | E | m , ( )and then − l m (cid:18) | E | − Kr (cid:19) = R + K + K | R | cos ( θ − θ ) . ( )At this point, by elementary algebraic manipulations, it is easy to write the equation for the orbits in termsof r : = | q | and θ , getting: r ( θ ) = p ( η ) + (cid:101) ( η ) cos ( θ − θ ) , ( )( p ( η ) being the parameter and (cid:101) ( η ) the eccentricity of the ellipses) which is formally the same expressionholding in the flat case. But now we have: (cid:40) p ( η ) ≡ p ( E , η ) = l mK (cid:101) ( η ) ≡ (cid:101) ( E , η ) = | R | K , ( )so that (cid:101) ( η ) = − | E | l / mK . In the above expression θ and θ are the angles that the vectors q and R formwith the half-line θ = θ is a constant of the motion).To check whether the third Kepler’s law holds in the deformed case as well, we have to compute the ratio τ / a = π / a Ω where Ω ( η ) ( E ) = | E | (cid:112) | E |√ m ( k + η | E | ) , ( )and a is the larger semi-axis defined as a = r + + r − . The inversion points r ± (where p r + = p r − =
0) areobtained by taking the roots of r − ( k − η | E | ) | E | r + l m | E | = ⇒ r ( η ) ± = k − η | E | | E | ± (cid:115) ( k − η | E | ) | E | − l m | E | , ( )entailing a ( η ) = r ( η )+ + r ( η ) − = k − η | E | | E | . ( )In the limit η → τ a = π mk . ( )We remind that the so-called third Kepler’s law is obtained by assuming that the ratio mM between the massof the planet and the mass of the sun be very small, so that the reduced mass can be identified with the massof the planet, entailing k = GMm and thus τ a = π mk = π GM . In the deformed case the analogous formulareads: τ ( η ) a ( η ) = π m ( k + η | E | ) ( k − η | E | ) . ( )The Kepler’s third law is then violated as the r.h.s of ( ), againassuming k = GMm , keeps its dependenceupon m and E : τ ( η ) a ( η ) = π m ( k + η | E | ) ( k − η | E | ) = π GM (cid:0) + η | E | GMm + O ( η ) (cid:1) . ( ) ome comments are in order, as the formulas we have derived seem to imply a sort of difference betweenthe classical and the quantum case. Namely, according to the results obtained in [ ] in the quantum case,for η > coupling constant metamorphosis , amounting just to replace k by k + η E . Thissubstitution holds both for the spectrum and for the eigenfunctions. We have already seen in [ ] that asimilar simple substitution applies for the quantum D-III as well. However in the classical Taub-Nut case,in order to close the Poisson algebra, one has to cope both with k − η E and with k + η E . An analogousbehaviour is exhibited by the classical D-III [ ]. For the sake of completeness we present here the explicit derivation of the trajectory t ( r ) using the standardanalytic method [ ]. A comparison with the results obtained through the Spectrum Generating Algebrawill provide a definite proof of the correctness of the algebraic approach. The starting point is the usualHamiltonian ( ): H = rr + η (cid:20) p m + l mr − kr (cid:21) . ( )The radial momentum p is related to the radial component of the velocity through the Hamilton’s equation:˙ r = ∂ p H = rr + η pm ⇒ p = r + η r m ˙ r . ( )Inserting in the Hamiltonian the expression of p in terms of r and ˙ r we obtain: H = rr + η (cid:20) ( r + η ) r m r + l mr − kr (cid:21) . ( )By solving the above expression with respect to ˙ r ( t ) and setting H = E we get:˙ r ( t ) = ± (cid:114) m rr + η (cid:114) E + k + η Er − l mr . ( )Comparing with the Euclidean case ( η = t ( r ) bytaking the positive branch of the square root: t ( r ) − t = (cid:114) m (cid:90) rr dr r + η r (cid:113) E + k + η Er − l mr = (cid:114) m (cid:90) rr dr (cid:113) E + k + η Er − l mr + (cid:114) m η (cid:90) rr drr (cid:113) E + k + η Er − l mr . ( ) The two integrals involved in the above formula can be conveniently calculated by introducing the so-called eccentric anomaly Ψ ( η ) through the relation [ ]: r = a ( η ) ( − (cid:101) ( η ) cos Ψ ( η ) ) . ( )In the previous section we have already shown that the semi-major axis is given by a ( η ) = − k + η E E and theeccentricity reads (cid:101) ( η ) = (cid:113) + l Em ( k + η E ) . Let us now pass to the explicit calculation of the two integralscontained in ( ), setting there E = −| E | <
0. It is not too difficult to arrive at the following results: (cid:114) m (cid:90) rr dr (cid:113) −| E | + k − η | E | r − l mr = (cid:118)(cid:117)(cid:117)(cid:116) ma ( η ) k − η | E | (cid:90) Ψ ( η ) d Ψ (cid:48) ( η ) ( − (cid:101) ( η ) cos Ψ (cid:48) ( η ) )= (cid:118)(cid:117)(cid:117)(cid:116) ma ( η ) k − η | E | ( Ψ ( η ) − (cid:101) ( η ) sin Ψ ( η ) ) , ( ) m η (cid:90) rr drr (cid:113) −| E | + k − η | E | r − l mr = (cid:115) ma ( η ) k − η | E | η (cid:90) Ψ ( η ) d Ψ (cid:48) ( η ) = (cid:115) ma ( η ) k − η | E | η Ψ ( η ) . ( )Hence, dividing and multiplying the output of the second integral by the same quantity a ( η ) and rearrangingthe two integrals in a single expression, we get the trajectory (with the initial condition t = t ( r ) = (cid:118)(cid:117)(cid:117)(cid:116) ma ( η ) k − η | E | (cid:18) η + a ( η ) a ( η ) (cid:19) Ψ ( η ) − (cid:118)(cid:117)(cid:117)(cid:116) ma ( η ) k − η | E | (cid:101) ( η ) sin Ψ ( η ) , ( )namely: t ( r ) = (cid:118)(cid:117)(cid:117)(cid:116) ma ( η ) k − η | E | (cid:18) η + a ( η ) a ( η ) (cid:19)(cid:20) Ψ ( η ) − a ( η ) η + a ( η ) (cid:101) ( η ) sin Ψ ( η ) (cid:21) = Ω ( η ) ( E ) (cid:20) Ψ ( η ) − a ( η ) η + a ( η ) (cid:101) ( η ) sin Ψ ( η ) (cid:21) , ( )which is the deformed Kepler equation : Ω ( η ) ( E ) t ( r ) = Ψ ( η ) − a ( η ) η + a ( η ) (cid:101) ( η ) sin Ψ ( η ) . ( )The frequency of the motion is given by Ω ( η ) ( E ) = (cid:118)(cid:117)(cid:117)(cid:116) k − η | E | ma ( η ) a ( η ) η + a ( η ) = (cid:114) m | E | (cid:112) | E | k + η | E | , ( )which is nothing but the same frequency obtained through the Spectrum Generating Algebra. Now we havejust to plug in the equation ( ) the explicit form of Ψ ( η ) and check whether it coincides with the one derivedvia the algebraic method. By solving for Ψ ( η ) one gets Ψ ( η ) = arccos (cid:20) (cid:101) ( η ) (cid:18) − ra ( η ) (cid:19)(cid:21) , ( )whence, owing to the well known relation sin ( arccos ( x )) = √ − x , it follows Ω ( η ) ( E ) t ( r ) = arccos (cid:20) (cid:101) ( η ) (cid:18) − ra ( η ) (cid:19)(cid:21) − a ( η ) η + a ( η ) (cid:118)(cid:117)(cid:117)(cid:116) (cid:101) ( η ) − (cid:18) − ra ( η ) (cid:19) . ( )Equation ( ) represents the trajectory calculated through the standard analytic method.On the other hand, the equation ( ) for the trajectory derived by means of the algebraic method yields (inthe case θ = Ω ( η ) ( E ) t ( r ) = arccos (cid:18) − (cid:114) m (cid:0) ( k − η | E | ) − | E | r (cid:1) q (cid:112) | E | (cid:19) − (cid:114) m (cid:112) | E | ( k + η | E | ) (cid:113) mr ( k − η | E | ) − m | E | r − l , ( ) where q = (cid:114) − l + m ( k − η | E | ) | E | . After easy algebraic manipulations equation ( ) acquires the form: Ω ( η ) ( E ) t ( r ) = arccos (cid:20) − (cid:101) ( η ) (cid:18) − ra ( η ) (cid:19)(cid:21) − a ( η ) η + a ( η ) (cid:118)(cid:117)(cid:117)(cid:116) (cid:101) ( η ) − (cid:18) − ra ( η ) (cid:19) . ( )In other words, by the algebraic method we get t ( r ) evaluated for − (cid:101) ( η ) . As we expected, this result is justthe η − deformation of the Kuru-Negro result [ ]. t h e c a s e η < : n e w f e a t u r e s This section is devoted to a terse investigation of the main features arising in the case η <
0. In this case theconformal factor rr + η can be more conveniently written as rr −| η | which emphasizes the singularity at r = | η | .One relevant question is whether the singularity can be overcome or not. In the first case there might betrajectories intersecting the line r = | η | . In the second case the phase plane ( r , ˙ r ) will consist of two nonoverlapping domains. In particular, for closed orbits one may ask under what conditions the following(mutually excluding) inequalities for the inversion points hold: r ( η ) − > | η | , r ( η )+ < | η | . ( )A careful analysis of ( ) shows that to characterise the corresponding regions of this plane one has to lookat both parameters η and λ , a characteristic lenght scale defined as λ : = l mk , or better at their ratio α : = | η | λ ,and at the behaviour of the effective potential V e f f ( r ) = l mr ( r −| η | ) − kr −| η | = − kr [ r − λ r − αλ ] . α < α <
1, where one has indeed two non-overlapping regionsseparated by the straight-line r = | η | . r - - - V eff ( r ) Figure : Potential V e f f ( r ) for α = . The straight lines represent the Energies associated to the critical points. • In the right domain r > | η | the conformal factor is positive. We have a Riemannian manifold with nonconstant curvature and there will be closed trajectories whenever the energy belongs to the (negative)open interval (cid:0) V e f f ( r + ) (cid:1) , where V e f f ( r + ) = − k λ ( + √ − α ) is the value of the effective potential at thecritical point r + = λ ( + √ − α ) . • In the left domain r < | η | the conformal factor is negative entailing that the kinetic energy is alsonegative. In order to get a physically significant system we are naturally led to define in this region anew Hamiltonian (cid:101) H : = − H = r | η |− r p m + (cid:101) V e f f ( r ) with (cid:101) V e f f ( r ) : = l mr ( | η |− r ) − k | η |− r , namely to look atthe system obtained by time-reversal . As it is clearly shown by Figure after that transformation in theregion 0 < r < | η | the effective potential acquires a typical ”confining” shape . There will be closed orbits forany positive energy higher than (cid:101) V e f f ( r − ) , where r − = λ ( − √ − α ) . We point out that the minimumof the potential is a monotonically decreasing function of | η | , so that it goes to infinity as | η | goes tozero. .00 0.02 0.04 0.06 0.08 0.10 r V eff ( r ) Figure : Potential V e f f ( r ) after the time-reversal transformation, i.e. (cid:101) V e f f ( r ) : = − V e f f ( r ) calculated for α = .The latter is contained into the segment 0 < r < | η | . α > r - - V eff ( r ) Figure : Potential V e f f ( r ) calculated for α = In the region r > | η | the effective potential will be proportional to − ( r − | η | ) − while in the boundedregion 0 < r < | η | its image is the full real line and furthermore it exhibits an inflection point for the value¯ r = λ (cid:0) − ( α − ) + ( α − ) (cid:1) . α = α = α =
1, i.e. V e f f ( r ) = − kr , shows that the distinctionbetween the two regions ( r > | η | , r < | η | ) disappears, in the sense that we have a single continuous line witha monotonically increasing behaviour and of course no closed orbits are allowed. However, at the same time( ) implies that at r = | η | the (absolute value of) the velocity diverges. .5 1.0 1.5 2.0 2.5 3.0 r - - - - V eff ( r ) Figure : Potential V e f f ( r ) calculated for α =
1. The centrifugal and gravitational contributions add up to givesa behaviour equals to − kr − . In this case the singularity in the effective potential disappears. One of the main results obtained in our paper is the constructive proof that the Spectrum Generating Algebratechnique can be successfully employed to attack and solve maximally superintegrable systems on spaceswith variable curvature . As already mentioned throughout the article, in the next future we will provide theanalogous results for the classical D-III system. Moreover, for positive values of the deformation parameters,we will exhibit the exact solution to the corresponding quantum problems based on the
Shape InvariantPotentials techniques [ ], making a comparison with different approaches proposed in the literature [ , ].As a further interesting result, it is worth to stress that we have shown the existence of closed orbits evenfor negative values of the deformation parameter. As a matter of fact, the behaviour of the classical effectivepotential strongly suggests that in the quantum case there will be bound states also in the region 0 < r < | η | ,while in the limit | η | → . we will focus future investigations on the quantum systems exactly on the case where the deformationparameters take negative values. There, due to the confining nature of the potentials, we expect themost interesting results from a physical point of view. . we will try to solve all the classical problems belonging to the Perlick’s families I and II [ ] by meansof the Spectrum Generating Algebra approach. We are encouraged to proceed further in this directioninasmuch as we have seen that in the classical deformed versions of Taub-Nut and D-III the couplingconstant metamorphosis , hardly applicable to the full Perlick’s families, does not seem to be the essentialfeature. Acknowledgments
This work was partially supported by the grant AIC-D-2011-0711 (MINECO-INFN) (O. R.), by the italianMIUR under the project PRIN 2010-11 (Analytical and geometrical aspects of finite and infinite-dimensionalhamiltonian systems, prot. n. 2010JJ4KPA 004). The authors acknowledge with pleasure enlightening dis-cussions with the spanish and italian colleagues A. Ballesteros and F. J. Herranz (Departamento de Fisica,UBU, Spain), D. Riglioni (CRM, Montreal, Canada), F. Zullo (Math-Phys. Dept. Roma Tre, Italy) . e f e r e n c e s [ ] Ballesteros A, Enciso A,Herranz F J, Ragnisco O, Riglioni D 2011 S IGMA [ ] Manton N S 1982 Phys. Lett. B [ ] Atiyah M F and N.J. Hitchin N J 1985 Phys. Lett. A [ ] Gibbons G W and Ruback P BCJM 1988 Comm. Math. Phys. [ ] Feh´er L G and Horv´athy P A 1987 Phys. Lett. B [ ] Gibbons G W and Ruback P J 1988 Comm. Math. Phys. [ ] Iwai T and Katayama N 1994 J.. Phys. A: Math. Gen. [ ] Iwai T and Katayama N 1995 J.Math.Phys. [ ] Iwai T, Uwano Y and Katayama N 1996 J. Math. Phys. [ ] Bini D, Cherubini C and Jantzen R T 2002 Class. Quantum Grav. [ ] Bini D, Cherubini C, Jantzen R T and Mashhoon B 2003 Class. Quantum Grav. [ ] Gibbons G W and Warnick C M 2007 J. Geom. Phys. [ ] Jezierski J and Lukasik M 2007 Class. Quantum Grav. [ ] Ballesteros A, Enciso A, Herranz F J and Ragnisco O 2009 Ann. Phys. [ ] Perlick V 1992 Class. Quantum Grav. [ ] Ballesteros A, Enciso A, Herranz F J and Ragnisco O 2009 Ann. Phys. [ ] Bertrand J 1873 C.R. Acad. Sci. Paris [ ] Ballesteros A, Enciso A, Herranz F J and Ragnisco O 2009 Commun. Math. Phys. [ ] Ballesteros A, Enciso A, Herranz F J and Ragnisco O 2008 Physica D [ ] Latini D and Ragnisco O 2015 (in preparation).[ ] Kuru S and Negro J 2012 J . Phys.: Conf. Ser. [ ] Kuru S and Negro J 2008 A nn. Phys. [ ] Goldstein H, Poole C and Safko J Classical Mechanics
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