Dynamics on the cone: closed orbits and superintegrability
aa r X i v : . [ m a t h - ph ] O c t Dynamics on the cone:closed orbits and superintegrability
Y. Brihaye
Department of Theoretical and Mathematical Physics,University of Mons,20, Place du Parc, B7000 Mons, Belgium
P. Kosi´nski, P. Ma´slanka ∗ Department of Theoretical Physics and Computer Science,University of L´od´z,Pomorska 149/153, 90-236 L´od´z, Poland
Abstract
The generalization of Bertrand’s theorem to the case of the motionof point particle on the surface of a cone is presented. The superin-tegrability of such models is discussed. The additional integrals ofmotion are analyzed for the case of Kepler and harmonic oscillatorpotentials.
The cone geometry appears in a number of physical contexts such as 2 +1-dimensional counterpart of Schwardschild solution [1], cosmic strings [2],defects in condensed matter physics [3], [4] and other. ∗ e-mail: [email protected]
1n view of possible applications the motion of a particle on the surfaceof a cone has been studied both classicaly and quantum mechanically [5]-[16]. In particular, the motion under the influence of central potential (i.e.depending on the distance from the tip of a cone) has attracted some at-tention. In a very interesting paper Al-Hashimi and Wiese [17] consideredthe case of Kepler and harmonic oscillator potentials. They showed that allbounded orbits are then closed provided the scale factor s , defined bellow byeq. (2), is rational. Moreover, they constructed the generators of accidentalsymmetries which appeared to be the straightforward generalizations of thestandard (i.e. corresponding to s=1) ones. The situation becomes even moreinteresting in the quantum domain. One can still define the accidental sym-metry generators obeying proper commutation rules. However, since one isdealing with unbounded operators, the problem of proper definition of theirdomains becomes urgent. It appears that, contrary to the standard case, forgeneral s being a rational multiple of π , the degenerate eigenstates of theHamiltonian do not form the complete multiplets of symmetry algebra.The present paper is devoted to the study of symmetry aspects of classicalmotion of a particle on a cone under the influence of a central potential. Weshow first that the Bertrand’s theorem [18] generalizes to the case of particlemotion on the surface of a cone. Namely, the following statement holds true: All bounded trajectories of a nonrelativistic particle moving on the surface ofthe cone under the influence of a central potential V ( r ) are closed if and onlyif V ( r ) = − κr or V ( r ) = mω r and the scale factor s is rational. Noting further that our system:( i ) is integrable because there are two globallydefined integrals of motion, H and J ; ( ii ) can be obtained from the specialcase s = 1 by a (locally defined) cannonical transformation, we show that theproperty that all bounded orbits are closed is equivalent to the superintegra-bility. Moreover, the additional integral of motion can be found by applyingthe above mentioned canonical transformation to the integral of motion forthe s = 1 case. The resulting expression is, for s noninteger, defined onlylocally. However, for s rational one immediately finds the global counterpartby taking an appropriate function of it.Viewing our system as an example of twodimensional integrable dynamicswe find that all conclusions are next to obvious.2 Dynamics on the cone
Consider the nonrelativistic particle of mass m confined to the surface of acone and bound to its tip by a potential V ( r ) , r being the distance from thetip. Let δ be the deficit angle and χ the polar angle (cf. Fig. 1). r δχ Fig. 1Then 0 ≤ χ < π − δ (1)Let us rescale the polar angle χ as to extend it to the range < , π ): ϕ ≡ χs , ≤ ϕ < πs ≡ − δ π (2)3he scaling parameter s is related to the angle α between the symmetry axisand the generating of the cone (see Fig. 2) by the formula s = sinα (3) q q q α Fig. 2The relevant Lagrangian describing the dynamics of our particle reads L = m ~q − V ( r ) , r = | ~q | (4)Using ˙ ~q = ˙ r + r ˙ χ = ˙ r + r s ˙ ϕ (5)one can write L = m r + mr s ˙ ϕ − V ( r ) (6)4t is convenient to introduce the Cartesian coordinates parametrizing theplane x = rcosϕx = rsinϕ (7)Then L takes the form L = m ˙ ~x m ( s − r ( ~x × ˙ ~x ) − V ( r ) == ms ˙ ~x − m ( s − r ( ~x ˙ ~x ) − V ( r ) (8)Moreover ~p ≡ ∂L∂ ˙ ~x = ms ˙ ~x − m ( s − r ( ~x ˙ ~x ) ~x. (9)or ˙ ~x = ~pms + ( s − ms r ( ~x~p ) ~x. (10)The Hamiltonian reads H = ~p ˙ ~x − L = ~p m + 1 − s ms r J + V ( r ) , (11)where J ≡| ~x × ~p | = mr s ˙ ϕ is the angular momentum. Defining p r = ~x~pr . (12)we find finally H = p r m + J ms r + V ( r ) . (13)The same results can be obtained by adding to the unconstrained La-grangian in q -space the constraint multiplied by Lagrange multiplier (thisyields what is called in classical mechanics the Lagrangian equations of thefirst kind) and applying the Dirac analysis of constrained systems.5 Bertrand’s theorem and superintegrability
We shall show that the analogue of Bertrand’s theorem holds in the formpresented in Sec. 1. Namely, all bounded orbits are closed only provided V ( r ) corresponds to the Kepler problem V ( r ) = − κr (14)or harmonic oscillator V ( r ) = βr s is rational. The proof goes, with small modifications, along the samelines as in the standard case [19]. We shall assume that either V ( r ) → ∞ or V ( r ) → c as r → ∞ ; in the latter case one can safely put c =0. Usingthe fact that J and E are constant of motion we easily find the equationdetermining the trajectory ϕ = ± Z Jms r dr q m ( E − J ms r − V ( r )) (16)In particular, denoting by △ ϕ the angle between the directions of adjacentperigeum and apogeum we get s △ ϕ = Z r max r min λmr dr q m ( E − λ mr − V ( r )) (17)where λ ≡ Js . The right-hand side of eq. (17) has the same form as in thecase of plane (i.e. s = 1) motion. Therefore, one can follow the same line ofreasoning as in the standard case. In order to have a closed orbit ∆ ϕπ mustbe rational. Now, the right hand side of eq. (17) is (at least in a certaindomain) a continuous function of λ and E . Therefore, it must be a constant.Making the change of variables x = λmr we finally conclude that the integral Z x max x min dx r m (cid:16) E − mx − V ( λmx ) (cid:17) (18)6ust be independent on E and λ . It represents (half of) the period ofonedimensional motion in the potential U ( x ; λ ) ≡ mx V (cid:18) λmx (cid:19) (19)It is easy to see that, for the period to be energy independent, U ( x ; λ ) musthave unique minimum x for any fixed λ . Now, assume E to be close tothe minimum value of U ( x ; λ ). Then one can use the harmonic oscillatorapproximation. Within this approximation, T πωω = 1 m d U ( x ; λ ) dx | x = x (20)Eqs. (19) - (20) imply mx − λmx V ′ (cid:16) λmx (cid:17) = 0 ω = 1 + 2 λm x V ′ (cid:16) λmx (cid:17) + λ m x V ′′ (cid:16) λmx (cid:17) = 0 (21)or ω = 3 + λmx V ′′ ( λmx ) V ′ ( λmx ) (22)As it has been mentioned above x is a unique function of λ ; moreover, firsteq. (21) implies that λx varies as λ varies. Therefore, the independence ofthe integral (18) on E and λ leads to the equation rV ′′ ( r ) V ′ ( r ) = C − , C ≡ ω > V ( r ) = Ar α , α > − V ( r ) = Bln (cid:16) rr (cid:17) (25)7ow, in oder to select the potentials U ( x ; λ ) leading to energy independentperiod beyond the harmonic oscillator approximation one can use the methoddescribed in Ref. [20]. Namely, U ( x ; λ ) should obey x ( U ) − x ( U ) = a p U − U (26)where a is some constant and the notation is explained on Fig. 3. UUU X ( U ) X ( U )Fig. 3It is not difficult to check that the condition (26) excludes (25) and im-plies α = − α = 2 in the case (24).Finally, let us compute the righ-hand side of eq. (17); for α = 2 we get π while α = − π . Going back to the eq. (17) we see that s must berational which concludes the proof.Let us now discuss the notion of superintegrability. Assume that the Hamil-tonian admits bounded motion in some region of phase space. Our system isintegrable because it admits two Poisson commuting integrals of motion, H and J . Using standard methods one can construct the Liouville-Arnold tori8nd the corresponding action-angle variables; the Hamiltonian is a functionof the action variables I and I only H = H ( I , I ) (27)Let ω k ( I ) ≡ ∂H∂I k , k = 1 , , be the frequencies corresponding to the anglevariables ϕ k . It is well known that the trajectories corresponding to thefixed values of H and J (i.e. lying on some Liouville-Arnold torus) are closedif and only if ω ( I ) ω ( I ) is a rational number. If all bounded orbits (i.e. lying onall L − A tori) are closed then, as can be easily shown, the frequencies havethe form ω k ( I ) = n k ω ( I ) , k = 1 , n k are some integers. In such acase there exists simple way to construct third integral of motion [19]-[21](for a recent review of superintegrability see [22]). Namely, the above formof frequencies implies H ( I ) = H ( n I + n I ) (28)Then n ϕ − n ϕ is an integral of motion and, due to the fact that n , are integers, any trigonometric function of it is globally well-defined. So weconclude that the property of all bounded orbits being closed implies super-integrability.On the other hand, if the ratio ω ( I ) ω ( I ) is an irrational number the trajectoriescover densely a given torus. Therefore, they cannot result from intersectionof L − A torus with the level surface of the third integral of motion; thesystem is not superintegrable.Let us point out that the conclusion concerning nonsuperintegrability isstrongly related to the existence of bounded orbits. If all orbits are un-bounded the invariant hypersurfaces in phase space obtained by fixing thevalues of commuting integrals of motion are no longer tori. We are no longerdealing with angle variables. However, it is the 2 π -periodicity which playsa decisive role in the above considerations. It sets the normalization of anglevariables and, consequently, the frequencies which makes it sensible to posethe question concerning their ratio. Therefore, the superintegrability is muchless restrictive condition in this case. As we have discussed in the previous section superintegrability is equivalentto the property that all bounded orbits are closed. Therefore, in the attractive9ase central motion on the surface of a cone is superintegrable if and onlyif the potential describes the Kepler system or harmonic oscillator and s isrational.In this section we consider the Kepler problem. The Hamiltonian reads H = p r m + J ms r − κr (29)For s rational all orbits are closed so the system is superintegrable. Theaction variables read (for H < I = 12 π I J dϕ = J (30) I = 12 π I p r dr = 1 π Z r max r min r m (cid:16) H − J ms r + κr (cid:17) dr (31)This yields H = − mκ (cid:16) I s + I (cid:17) (32)For rational s , s = n n one finds H = − mn κ n I + n I ) (33)in accordance with eq. (28) of previous section. One can now constructthe third integral of motion following the method presented above. However,the derivation is simplified by noting that the (local) canonical transforma-tion J → sJ , ϕ → ϕs reduces the problem to the standard two-dimensionalKepler one. Therefore, we can obtain our integral from the correspondingRunge-Lenz one. This results in the following expression C ≡ ( A − iB ) e isϕ (34) A = J ms r − κ (35) B = J prms (36)10ne easily checks that A + B = 2 HJ ms + κ (37) C is not single-valued unless s is integer. For s irrational nothing can bedone. On the other hand, for s = kn (38)we find that Z ≡ C n = ( A − iB ) n e ikϕ (39)is a single valued integral of motion. Now H, J, Z and ¯ Z form a finite W -algebra [23] with respect to the Poisson bracket. The nontrivial bracketsread { J, Z } = kZ, { J, ¯ Z } = − k ¯ Z (40) { Z, ¯ Z } = 4 in mk J H (cid:16) n J mk + κ (cid:17) n − (41)We see that only for n ≡
1, i.e. integer s , the above algebra becomes a Liealgebra on the submanifold of constant energy. The relevant Hamiltonian has the form H = p r m + J ms r + mω r s is rational. Therefore, the system isthen superintegrable. The action variables read now I = J (43) I = 1 π Z r max r min r m (cid:16) H − J m s r − mω r (cid:17) dr (44)which leads to H = ω (cid:16) I s + 2 I (cid:17) (45)11or rational s , s = n n , one obtains H = ωn (cid:16) n I + 2 n I (cid:17) (46)Again, the general method discussed previously allows us to construct thethird integral of motion. As in the case of Kepler problem it is sufficient toknow the form of the integral in the standard case ( s = 1) and to apply thecanonical transformation J → sJ, ϕ → ϕs which yields the locally definedconstant of motion. C = ( A − iB ) e isϕ (47) A = J ms r − H (48) B = prJmsr (49)where A + B = H − ω J s (50)Now, for rational s = kn one constructs the globally defined integral of motionby Z = C n = ( A − iB ) n e ikϕ (51)The integrals Z, ¯ Z, J and H generate a finite W -algebra described by thefollowing nontrivial Poisson brackets { J, Z } = 2 kZ (52) { J, ¯ Z } = − k ¯ Z (53) { Z, ¯ Z } = − in k ω J (cid:16) H − ω n J k (cid:17) n − (54)For n = 1, i.e. integer s , we arrive at the Lie algebra. We considered the motion of a particle on a cone under the influence of centralpotential V ( r ). Such a system is integrable since there are two commutingintegrals of motion, H and J . Assume that V ( r ) is such that there arebounded orbits. We have shown that all such orbits are closed if and only if12 ( r ) = − κr or V ( r ) = mω r and s is rational. By the same arguments asin the standard ( s = 1) case the dynamics is then superintegrable (and onlythen provided there are bounded orbits).Our proof can be reformulated as follows. Consider the central motionon the plane. Let J and I r be the action variables. The proof of Bertrand’stheorem shows that H depends on linear combination of action variablesonly in the Kepler (cid:16) H = H ( J + I r ) (cid:17) and oscillator (cid:16) H = H ( J + 2 I r ) (cid:17) cases. Now, J → Js and ϕ → sϕ is the canonical transformation relating thedynamics on the cone and on the plane. It is defined only locally due to thecyclic character of angle variable ϕ . However, this is irrelevant as long as theaction variables are concerned. By applying this transformation we find theform of the Hamiltonian on the cone. The theorem follows.The above canonical transformation allows also to write out immediatelythe integral of motion for s = 1 provided it is known for s = 1. If thelatter does not depend on ϕ the former is globally defined. However, for ϕ -dependent integrals the problem is more involved. Namely, for s irrationalthey are only locally defined; on the other hand for rational s one can definethe global integral by taking an appropriate function of the local one.It is interesting to consider the quantum counterpart of Kepler and oscil-lator problems. One can easily show that our integrals generalize to quantumdomain and their commutators define the W -algebras which reduce to theones derived here in the limit ~ →
0. However, as it has been pointed outin the interesting paper [17] the situation is now much more subtle. Theproblem of the operator domains becomes very important. As a result thedegenerate states of the Hamiltonian do not span complete representationsof the symmetry algebra; we have an interesting example of formal symmetryalgebra which does not determine the degeneracy. For details we refer to [17].
Note added:
Just recently we have become aware of two papers dealingwith Aharonov-Bohm effect in conical space [24].
Acknowledgements
We are very grateful to Cezary Gonera, JoannaGonera and Krzysztof Andrzejewski for interesting discussion.13 eferences [1]
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