Central force problem in space with SU(2) Poisson structure
CCentral force problem in space withSU(2) Poisson structure
Taraneh Andalib and Amir H. Fatollahi Department of Physics, Alzahra University, Tehran 19938-93973, Iran
Abstract
The central force problem is considered in a three dimensional space inwhich the Poisson bracket among the spatial coordinates is the one by theSU(2) Lie algebra. It is shown that among attractive power-law potentialsit is only the Kepler one that all of its bound-states make closed orbits.The analytic solution of the path equation under the Kepler potentialis presented. It is shown that except the Kepler potential, in contrast toordinary space, all of the potentials for which all of the almost circular or-bits are closed are non-power-law ones. For the non-power-law potentialsexamples of the numerical solutions of the path equations are presented.
PACS numbers:
Keywords:
Noncommutative geometry; Formalisms in classical mechanics;Orbital and rotational dynamics e-mail: [email protected] e-mail: [email protected] a r X i v : . [ h e p - t h ] D ec Introduction
During recent years much attention has been paid to the formulation and studythe physics living on noncommutative spaces. The motivation, among others[1, 2], partly is the natural appearance of noncommutative spaces in some areasof physics, for example in the string theory. In particular it has been understoodthat the canonical relation [ˆ x a , ˆ x b ] = i θ a b , (1)with θ is an antisymmetric constant tensor, describes the longitudinal directionsof D-branes in the presence of a constant B-field background as seen by theends of open strings [3–6]. The theoretical and phenomenological implicationsof possible noncommutative coordinates have been extensively studied [7].One direction to extend studies on noncommutative spaces is to considerspaces where the commutators of the coordinates are not constants. Examplesof this kind are the noncommutative cylinder and the q -deformed plane [8],the so-called κ -Poincar´e algebra [9–12], and linear noncommutativity of the Liealgebra type [13, 14]. In the latter the dimensionless spatial position operatorssatisfy the commutation relations of a Lie algebra:[ˆ x a , ˆ x b ] = f ca b ˆ x c , (2)where f ca b ’s are structure constants of a Lie algebra. One example of this kindis the algebra SO(3), or SU(2). A special case of this is the so called fuzzysphere [15, 16], where an irreducible representation of the position operators isused which makes the Casimir of the algebra, (ˆ x ) + (ˆ x ) + (ˆ x ) , a multipleof the identity operator (a constant, hence the name sphere). One can considerthe square root of this Casimir as the radius of the fuzzy sphere. This is,however, a noncommutative version of a two-dimensional space (sphere). In [17],a four dimensional differential calculus has been developed corresponding to thealgebra SU(2), based on which the equations of motion corresponding to spin 0,1/2, and 1 particles living in that four dimensional space have been investigated.In [18–20] a model was introduced in which the representation was notrestricted to an irreducible one, instead the whole group was employed; seealso [21]. In particular the regular representation of the group was considered,which contains all representations. As a consequence in such models one isdealing with the whole space, rather than a sub-space like the case of fuzzysphere as a 2-dimensional surface. In [18, 19] basic ingredients for calculus ona linear fuzzy space, as well as the basic notions for a field theory on such aspace, were introduced. Models based on the regular representation of SU(2)were treated in more detail, giving explicit forms of the tools and notions intro-duced in their general forms [18, 19]. In [19, 20, 22] the tree and loop diagramsfor self-interacting scalar and spinor fields discussed. It is observed that modelsbased on Lie algebra type noncommutativity enjoy three features: • They are free from any ultraviolet divergences if the group is compact. • The momentum conservation is modified, in the sense that the vectoraddition is replaced by some non-Abelian operation [18, 23]. • In the transition amplitudes only the so-called planar graphs contribute.1n [24] the quantum mechanics on a space with SU(2) fuzziness was examined. Inparticular, the commutation relations of the position and momentum operatorscorresponding to spaces with Lie-algebra noncommutativity in the configura-tion space, as well as the eigen-value problem for the SU(2)-invariant systemswere studied. The consequences of the Lie type noncommutativity of space onthermodynamical properties have been explored in [25, 26].The classical motion on noncommutative space has attracted interests aswell [27, 28]. In particular, the central force problems on space-times withcanonical and linear noncommutativity and their observational consequenceshave been the subject of different research works [29–34]. In [35] the classicalmechanics defined on a space with SU(2) fuzziness was studied. In particular,the Poisson structure induced by noncommutativity of SU(2) type was investi-gated, for either the Cartesian or Euler parameterization of SU(2) group. Theconsequences of SU(2)-symmetry in such spaces on integrability, were also stud-ied in [35].The purpose of the present work is to examine in more detail the classi-cal central force problem in a space with linear SU(2) fuzziness. In particular,using the path equation, the conditions are obtained under which the slightlyperturbed circular orbits are stable or closed. The differential equation is ob-tained that the potentials should satisfy to guarantee that all of the perturbedcircular orbits of them would be closed. The numerical solutions of the men-tioned differential equation as well as the path equation under the obtainedpotentials are presented. It is shown that among attractive power-law poten-tials it is only the Kepler one that all of its bound-states make closed orbits.The analytic solution of the path equation under the Kepler potential is given.The scheme of the rest of this paper is the following. In section 2, a shortreview of the construction in [35] is presented. In section 3 the circular orbitsand various aspects of their perturbations are considered. In section 4 theKepler problem and its analytic solution is presented. Appendix A is devotedto the derivation of the differential equation by which the potentials with closedperturbed circular orbits are obtained.
The classical dynamics on a space whose Poisson structure is originated from aLie algebra is given in [35]. To make this presentation self contained, below ashort review of the construction is presented.
Denote the members of a basis for the left-invariant vector fields correspondingto the group G by ˆ x a ’s. These fields satisfy the commutation relation (2), withthe structure constants of the Lie algebra corresponding to G . The members ofthis basis would contain the quantum mechanical (operator form) counterpartsof the classical spatial coordinates of the system (for the moment they aredimensionless). The group elements of G are parametrized by the coordinates k a ’s as: U ( k ) := exp( k a ˆ x a ) U ( ) . (3)2hese coordinates would play the role of the conjugate momenta of ˆ x a ’s. Thecoordinates and the momenta in their operator forms (denoted by hat) wouldsatisfy the following relations [ˆ k a , ˆ k b ] = 0 , (4)[ˆ x a , ˆ k b ] = ˆ x ab , (5)with x ab ’s as scalar functions of ˆ k a ’s, having the property x ab ( k = ) = δ ba . (6)Accordingly, the operator forms of ˆ x a ’s in the k -basis areˆ x a → x ab ∂∂k b . (7)The explicit forms of the ˆ x ab scalars for the SU(2) group will be given later [35].There are also the right-invariant vector fields whose basis ˆ x R a satisfy[ˆ x R a , ˆ x R b ] = − f ca b ˆ x R c , (8)[ˆ x R a , ˆ x b ] = 0 . (9)Using these, one defines the vector field ˆ J a throughˆ J a := ˆ x a − ˆ x R a . (10)Using (9) and the definitions of the left- and right-actions, it is found that J a ’sare the generators of similarity transformation (adjoint action):exp( α a ˆ J a ) U ( k ) = U ( − α ) U ( k ) U ( α ) . By all these above the following commutation relations hold[ ˆ J a , ˆ J b ] = f ca b ˆ J c , (11)[ ˆ J a , ˆ x b ] = f ca b ˆ x c , (12)[ˆ k c , ˆ J a ] = f ca b ˆ k b . (13)According to the above commutation relations, in the case of the group SU(2)( f ca b = (cid:15) ca b ), J a ’s simply satisfy the commutation relation of the rotation gen-erators, and thus represent the components of the angular momentum.To construct the phase space, all that is needed is to transform the commuta-tion relations to Poisson brackets. This can be done through the correspondence[ . , . ] / (i (cid:126) ) → { . , . } . However, one should also take care of the dimension of thequantities, and their reality ( i.e. Hermitian in operator form). To do so, let usdefine the following quantities p a := ( (cid:126) /(cid:96) ) ˆ k a , (14) x a := i (cid:96) ˆ x a , (15) x ab ( p ) := ˆ x ab [( (cid:96)/ (cid:126) ) p ] , (16) J a := i (cid:126) ˆ J a , (17)3here (cid:96) is a constant of dimension length. One then arrives at the followingPoisson brackets { p a , p b } = 0 , (18) { x a , p b } = x ab , (19) { x a , x b } = λ f ca b x c , (20) { J a , x b } = f ca b x c , (21) { p c , J a } = f ca b p b , (22) { J a , J b } = f ca b J c , (23)where the dimension of λ is that of inverse momentum: λ := (cid:96) (cid:126) . (24)One notes that x a ’s and p b ’s are independent variables, and other variables canbe expressed in terms of these. So that among the Poisson brackets (18) to (23),only (18), (19), and (20) are independent. All others can be derived from these.Using (6) it is seen that in the limit λ → (cid:96) → For the special case of group SU(2), the independent Poisson brackets (18), (19),and (20) are in fact the Poisson structure of a rigid rotator, in which the angularmomentum and the rotation vector have been replaced by x and p , respectively,that is, the roles of position and momenta have been interchanged. It would beconvenient to use the Euler parameters, defined throughexp( φ T ) exp( θ T ) exp( ψ T ) := exp( k a T a ) , (25)where T a ’s are the generators of SU(2) satisfying the commutation relation[ T a , T b ] = (cid:15) ca b T c . (26)It should be emphasized that in this setting the Euler angular parameters, inopposite to their ordinary definition, are parametrizing the momentum space,and their canonical conjugate components X φ , X θ , and X ψ play the role ofthe parametrization of the (real) configuration space. One also mention thatin this setup, φ , θ , and ψ are dimensionless, while X φ , X θ , and X ψ have thedimension of action. One advantage of using the Euler parameters over theordinary rotational generator θ · L is that the Poisson brackets of φ , θ , ψ , X φ , X θ , and X ψ are the standard canonical ones [36], namely the only nonzeroPoisson brackets are { X φ , φ } = 1 , (27) { X θ , θ } = 1 , (28) { X ψ , ψ } = 1 . (29)4sing this set of parameters one can find the basis for the generators of leftaction, whose non-operator forms as spatial coordinates are found to be [35] x = λ ï − cos ψ sin θ X φ + sin ψ X θ + cos ψ cos θ sin θ X ψ ò , (30) x = λ ï sin ψ sin θ X φ + cos ψ X θ − sin ψ cos θ sin θ X ψ ò , (31) x = λ X ψ . (32)Similarly the angular momentum components are found as follow [35] J = cos φ cos θ − cos ψ sin θ X φ + (sin φ + sin ψ ) X θ + − cos φ + cos ψ cos θ sin θ X ψ , (33) J = sin φ cos θ + sin ψ sin θ X φ + ( − cos φ + cos ψ ) X θ + − sin φ − sin ψ cos θ sin θ X ψ , (34) J = − X φ + X ψ . (35)It is mentioned that the above components have the correct dimension ( i.e. length for x a ’s and action for J a ’s). One also has [35, 36]cos k λ p θ φ + ψ , (36)where k := √ k · k and p := √ p · p .In the case with motion under a central force, the Poisson brackets of Hamil-tonian H with J a ’s vanish. A Hamiltonian which is a function of only ( p · p ) and( x · x ) is clearly so. For such a system, H , J · J and one of the components of J (say J ) are involutive constants of motion, hence any SU(2)-invariant classicalsystem is integrable. As J is a constant vector, one can choose the axes so thatthe third axis is parallel to this vector: J = J = 0 , (37)by which, after subtracting J cos φ + J sin φ and J cos ψ − J sin ψ , one has:(1 + cos θ )sin θ (1 − cos( φ + ψ ))( X φ − X ψ ) = 0 . Assuming J = − X φ + X ψ (cid:54) = 0, leads to φ + ψ = 0 ⇒ X φ + X ψ = 0 . (38)Defining the new variables χ := ψ − φ ,J := − X φ + X ψ , (39)5or which { J, χ } = 1. So only ( J, χ ) and ( X θ , θ ) are left as the canonicallyconjugate variables. Applying (38), one arrives at x = λ Å J θ sin θ cos χ + X θ sin χ ã ,x = λ Å − J θ sin θ sin χ + X θ cos χ ã ,x = λ Å J ã , x · x = λ ï X θ + J Å θ ãò , cos k θ . (40)It is seen that the motion is not in the plane x = 0, but in a plane parallel tothat, as x does not vanish but is a constant.By defining the polar coordinates ( ρ, α ) in the ( x , x ) plane: x =: ρ cos α,x =: ρ sin α, (41)and x · x = x + x + x , one has x · x = ρ + λ J ,ρ = λ ( X θ + J u ) ,α = − χ + tan − X θ J u , (42)where u := 12 cot θ . (43)One then has x · x = λ ï X θ + J Å
14 + u ãò . (44)In the line similar to the motion on ordinary space, one can proceed to derivethe equation for the path, in terms of ρ and α . The first-order path equation isfound to be [35]: 1 u = λ J ñ ρ + 1 ρ Å d ρ d α ã ô . (45)We mention that, by (40) and (43), u can be expressed as the kinetic energy.In the case of our interest, the k -space as angle variable in (25) is a compactone, and so the momenta is taken to be bounded, that is 0 ≤ p ≤ π/λ .Following [18–20, 35], the kinetic term is taken to be K = 4 λ m Å − cos λ p ã (46)6or a particle of mass m . By this choice, the kinetic term is a SU(2)-invariant(rotationally-invariant), and hence a class-function of SU(2), which is also mono-tonically increasing in the above mentioned interval 0 ≤ p ≤ π/λ , just as in thecase we have in ordinary classical mechanics, with p / (2 m ). Also we mentionin the limit λ →
0, the expression (46) coincides with the ordinary kinetic term,as it should. One can express this kinetic term in terms of u : K = 4 λ m Å − u √ u ã . (47)Using above one easily find1 u = 4 ñÅ − λ m K ã − − ô . (48)Assuming the usual form for the Hamiltonian as H = K + V ( r ) , r := √ x · x (49)with V ( r ) as the potential term, the path equation (45) comes to the form1 ρ + 1 ρ Å d ρ d α ã = 4 λ J ®ï − λ m (cid:0) E − V ( r ) (cid:1) ò − − ´ , (50)in which r = (cid:112) ρ + x = (cid:112) ρ + λ J /
4, by first of (42). It is easy to checkthat in the limit λ →
0, the above equation is reduced to the path equation onordinary space. For the case of free particle, it is easy to see that the solutionof (50) is of the form ρ cos( α − α ) = c , representing a straight line [35]. In this section the circular orbits under a central force are considered. In par-ticular, the existence condition for circular orbits, as well as the condition forstable and closed slightly perturbed circular orbits are obtained. Similar to themotion on ordinary space, it is useful to define the new variable: w := 1 ρ (51)by which the path equation (50) takes the form w (cid:48) + w = 4 λ J ®ï − λ m (cid:0) E − V ( r ) (cid:1) ò − − ´ (52)with w (cid:48) = d w/ d α , and r = 1 w … λ J w . (53)Differentiating once again with respect to α from (52) we find w (cid:48)(cid:48) + w = − mJ ï − λ m (cid:0) E − V ( r ) (cid:1) ò − d V ( r )d w (54)7 .1 Existence condition of circular orbits The circular orbit is given by the conditions w (cid:48) = 0 ,w (cid:48)(cid:48) ≡ . (55)By using (52) the first of above gives w = 4 λ J ®ï − λ m (cid:0) E − V ( r ) (cid:1) ò − − ´ (56)with r given by (53) at w . Above can be converted to1 + λ J w ï − λ m (cid:0) E − V ( r ) (cid:1) ò − . (57)By using (54) the second condition of (55) leads to w = − mJ ï − λ m (cid:0) E − V ( r ) (cid:1) ò − d V ( r )d w (cid:12)(cid:12)(cid:12)(cid:12) r (58)By combination (57) and (58), one finds the following as the condition for exis-tence of the circular orbit: w = − mJ Å λ J w ã d V ( r )d w (cid:12)(cid:12)(cid:12)(cid:12) r . (59)The above relation is interpreted as an equation for w , whose positive solutions( w >
0) determine the radii of the allowed circular orbits. On ordinary spacethe above condition takes the form [36] w = − mJ d V ( r )d w (cid:12)(cid:12)(cid:12)(cid:12) ρ (ordinary space) , (60)which is simply retrieved as the limit λ → r d w = − w Å λ J w ã − / (61)by which one can replace for the derivative d V ( r ) / d w appearing in (59). Theabove condition for the power law potentials of the form V ( r ) = ± g r n , g > , n > w = ± mJ n g w − n − Å λ J w ã ( n +1) / (63)or in a more compact form J mρ = ± n g r n +10 (64)8howing that for the minus sign, for which the force is repulsive, the circularorbit is not allowed. It is instructive to compare the above with the conditionon ordinary space: J mρ = n g ρ n +10 (ordinary space) . (65)For the power law potentials of the form V ( r ) = ± gr n , g > , n > w = ∓ mJ n g w n − Å λ J w ã − n/ / (67)or J mρ = ∓ n gr n − (68)Again for the repulsive force (positive sign in (66)) the circular orbit is notpossible. In this case the condition on ordinary space is J mρ = n gρ n − (ordinary space) . (69) It is a matter of interest to look for the condition by which the circular orbitsstay bounded after a slight perturbation. To obtain the condition, we assume aslight change in the parameter w as w ( α ) = w + w ( α ) (70)for which | w ( α ) | (cid:28) w . (71)The purpose is to find the conditions under which the above remains true notonly as an initial condition, but also for all α . By inserting (70) in (54), we findthe following as the equation that w ( α ) should satisfy: w (cid:48)(cid:48) ( α ) + w + w ( α ) = − mJ ®ï − λ m (cid:0) E − V ( r ) (cid:1) ò − d V ( r )d w ´ (cid:12)(cid:12)(cid:12)(cid:12) r + r . (72)By expanding the right hand side of above in the first order of w , and using(57) and (59), one finds the following for w w (cid:48)(cid:48) ( α ) + β w ( α ) = 0 , (73)in which β := 1 − w dd w ln Å − d V ( r )d w ã (cid:12)(cid:12)(cid:12)(cid:12) r − λ J r . (74)9he condition for having a stable circular orbit is that (73) does not developgrowing solutions in α , which is guaranteed if β >
0. On ordinary space β is the same as above but in the limit λ → β > V ( r ) = g r n , (75)for which the circular orbit is allowed, we find β = 1 + 1 + nr w , (76)which is always positive. The expression on ordinary space is gained in the limit λ → i.e. r w → V ( r ) = − gr n (77)we then find β = 1 + 1 − nr w , (78)by which the β > n < r w = 2 + λ J w / . (79)This result is less constrained than the condition on the ordinary ( i.e. n < n = 1, the Kepler potential.In this case β = 1 and is independent of the initial condition. So, having (76)and (78), one should expect that the only power law force for which not onlythe perturbed circular orbits but also all of the bound-states’ orbits are closedcould be the Kepler one. Later we will see that this is indeed the case.To have a closed perturbed circular orbit, β should be a rational number [36].So whenever β by (74) would be a rational number, the perturbed circular orbitis closed. On the ordinary space, for power law potentials the condition on n for having closed orbits is free from the initial conditions, and only depends on n [36]. Here, except for the case with n = 1 in (78), due to existence of r w factor in (76) and (78), both the value of n and the initial conditions, by which w is fixed, are important.One can ask about the condition by which all the perturbed circular orbitsare closed. Interpreting (74) as a differential equation that potential V ( r ) shouldsatisfy for a fixed rational β rat . , one would find the form of the potentials forwhich, no matter what the initial conditions are, all of the perturbed circularorbits are closed [36]. To do so one should write the condition (74) in theway that the constant of motion J would not appear in it. In particular, theparameter w , which is related to 3-dimensional radial variable r by (53), shouldbe replaced by r . This can be done, and the result comes to the form (seeAppendix A) β . = 3 − λ J ( r ) + r Å − λ J ( r ) ã dd r ln Å − d V ( r )d r ã , (80)10 .0 0.5 1.0 1.5 2.0 2.5 3.001234 (cid:45) (cid:45) Figure 1:
Left: The plot of d V ( r ) / d r for β rat . = 3 / m = 2, λ = 0 .
7, d V ( r ) / d r | r =1 =1; dashed-line: 1 /r − β = 1 /r . . Right: The perturbed circular path is closed; m = 2, J = 1, β = 3 / r (0) = 1 / . r (cid:48) (0) = 0. r circ . (cid:39) / . in which J ( r ) := m r V ( r )d r Ñ (cid:115)
64 + m λ r Å d V ( r )d r ã − m λ r d V ( r )d r é . (81)It is easy to see that the Kepler potential V ( r ) = − g/r satisfies (80) with β rat . = 1. In fact, by direct insertion, as expected by (78), it can be checked that,except the Kepler potential, the other kinds of attractive power law potentials(75) and (77) do not satisfy the above equation. This observation is in deepcontrast with the case on ordinary space in which it can be shown that allthe perturbed circular orbits of the power law potentials in the form V ( r ) ∝ /r − β . are closed [36].For a given β rat . , one still may look for the non-power law solutions of (80).The potentials by the numerical solutions of (80) can be easily obtained andsubsequently would be used in the path equation (54). In Figs. 1 & 2 twoexamples of such potentials and their perturbed circular orbit solutions arepresented ( β rat . = 3 / / − g/r ) and the harmonic oscillator one ( g r ) [36].By the above considerations the latter one is excluded here (see also Fig. 3),but the Kepler one is still a candidate for such a kind of potentials on SU(2)space. In the next section it is shown explicitly that this is indeed the case.So on SU(2) space, among the power law potentials, it is the Kepler problemthat all the orbits of bound-state solutions are closed. Other potentials withthe mentioned property, if any, should be looked for among the non-power lawsolutions of (80). 11 .5 1.0 1.52468 (cid:45) (cid:45) Figure 2:
Left: The plot of d V ( r ) / d r for β rat . = 5 / m = 2, λ = 0 .
4, d V ( r ) / d r | r =1 =1; dashed-line: 1 /r − β = r . . Right: The perturbed circular path is closed; m = 2, J = 1, β = 5 / r (0) = 1 / . r (cid:48) (0) = 0. r circ . (cid:39) / . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 3:
The path for harmonic oscillator potential V ( r ) = g r , with m = 2, J = 1, g = 1, λ = 0 . r (0) = 1, r (cid:48) (0) = 0. The plot is for three revolutions and the path isnot closed. Here the solution to the equation (52) for the Kepler potential is presented. Thepotential in terms of the parameter w comes to the form V ( r ) = − gr = − g w Å λ J w ã − / . (82)In the following, without lose of generality, we set g = 1. By above (52) can bewritten as 1 + a ( w (cid:48) + w ) = (cid:104) η − b w (cid:0) a w (cid:1) − / (cid:105) − . (83)in which a := λJ , b := λ m g, η := 1 − λ m E. (84)12sing the new variable v defined by a w ( α ) =: sinh v ( α ) , a w (cid:48) ( α ) = v (cid:48) cosh v ( α ) , (85)(83) comes to the form 1 + v (cid:48) = 1( η cosh v − γ sinh v ) (86)in which γ := b/a . So the dependence of the new variable v on the polar angle α can be obtained by (cid:90) η cosh v − γ sinh v (cid:112) − ( η cosh v − γ sinh v ) d v = ± ( α − α ) , (87)with α appearing as the constant of integration. The above integral can beevaluated in two regimes: η > γ and η < γ . For the case with η > γ , introducing v := tanh − ( γ/η ) and µ := 1 / (cid:112) η − γ , one has η cosh v − γ sinh v = cosh( v − v ) /µ, (88)by which the above integral takes the form (cid:90) cosh( v − v ) » µ − cosh ( v − v ) d v = ± ( α − α ) . (89)The above integral is known [37], leading tosin − sinh( v − v ) (cid:112) µ − ± ( α − α ) , (90)or sinh( v − v ) (cid:112) µ − ± sin( α − α ) . (91)The last expression indicates that, through the intermediate variables v and w ,the polar coordinate ρ depends on sin( α − α ). As the consequence, the pathafter every revolution around the force center repeats itself, and hence is closed.For the other case η < γ , again by introducing ˜ v := tanh − η/γ and ˜ µ :=1 / (cid:112) γ − η , one has η cosh v − γ sinh v = sinh(˜ v − v ) / ˜ µ, (92)by which (87) takes the form (cid:90) sinh(˜ v − v ) » ˜ µ − sinh (˜ v − v ) d v = ± ( α − α ) . (93)The above integral is known [37], leading to − sin − cosh( v − ˜ v ) (cid:112) ˜ µ + 1 = ± ( α − α ) , (94)13 (cid:45) (cid:45) (cid:45) Figure 4:
The sample plots of paths under Kepler potential; solid-line: SU(2) space;dashed-line: ordinary space. For both paths: m = 2, J = 1, g = 1, r (0) = 1, r (cid:48) (0) = 0.For SU(2) path: λ = 2 . or cosh( v − ˜ v ) (cid:112) ˜ µ + 1 = ∓ sin( α − α ) . (95)The last expression indicates that the path is periodic and closed. As a demon-stration and to compare the path in the present case with the one on ordinaryspace the orbits with equivalent parameters are plotted in Fig. 4.It would be instructive to find the turning radii, in which the radial velocityvanishes ( w (cid:48) = 0); the analogy of apsides of the elliptical path on ordinary space.By (83) one has the equation for the apsides1 + a w . = (cid:104) η − b w aps . (cid:0) a w . (cid:1) − / (cid:105) − (96)or (1 + b w aps . ) = η (1 + a w . ) , (97)leading to w aps . = b ± | η | (cid:112) a + b − a η a η − b . (98)Restoring the original values the solutions come to the form w aps . = 16 m g ± | − mλ E | (cid:112) m g + mJ E (8 − mλ E ) J (4 − mλ E ) − m λ g . (99)The condition to have the real number roots is4 m g + J E (8 − mλ E ) ≥ , (100)by which, for fixed J , we have4 mλ Ç − … m g λ J å ≤ E ≤ mλ Ç … m g λ J å . (101)14he condition − mg / (2 J ) ≤ E on ordinary space can be retrieved as the limit λ → E is bounded from both below and above.In the allowed region for E only positive roots can be accepted as turningpoints. In the case where both roots are positive the problem is in fact abound-state one, and the roots represent the apoapsis (smaller root) and theperiapsis (larger root) of the path (remind w = 1 /ρ ). In the cases where onlyone acceptable root exists, the problem in fact is an unbounded motion, withthe root representing the least distance to the center of force.The circular path, if exists, is obtained by the condition w apoa . = w peri . ,which comes from (100) by the equal-sign. Note the condition 4 − mλ E = 0leads to the unaccepted solution w apoa . = w peri . <
0. On ordinary space thecondition for circular path is E = − mg / (2 J ), which is obtained by λ → J , the bound-state with the farthest path from circle is obtained. On theordinary space this comes by the condition E → − , by which the apoapsis goesto ∞ ( w apoa . →
0) [36]. In the present case, setting w (cid:48) = 0 and w → E → − . Inserting E → − in (99), we find w peri ./ apoa . → mg ± mg J − m λ g , (102)which indicates bound-state paths with extremely large apoapsis are possibleonly for 2 J > mλg . Otherwise, the apoapsis takes place in a finite value.
A Derivation of (80) and (81)
The purpose is to replace J ’s and w ’s in (74) by 3-dimensional r . The staringpoint is (74) β := 1 − w dd w ln Å − d V ( r )d w ã (cid:12)(cid:12)(cid:12)(cid:12) r − λ J r . (103)First of all, using (53) and (61), the above can be brought to (80) β = 3 − λ J r + r Å − λ J r ã dd r ln Å − d V ( r )d r ã (cid:12)(cid:12)(cid:12)(cid:12) r . (104)Combining the relations (53) and (61), together with the circular orbit condition(59), one finds the following1 m r (d V ( r ) / d r | r ) Å J r ã + λ J r − J r = m r V ( r )d r Ñ (cid:115)
64 + m λ r Å d V ( r )d r ã − m λ r d V ( r )d r é (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r . (106) Acknowledgement : This work is supported by the Research Council of theAlzahra University. 15 eferences [1] S. Doplicher, K. Fredenhagen, & J. E. Roberts, Commun. Math. Phys. (1995) 187;Phys. Lett.
B331 (1994) 39.[2] J. Madore, Rept. Math. Phys. (1999) 231;M. Buric, J. Madore, & G. Zoupanos, 0709.3159 [hep-th];SIGMA (2007) 125.[3] N. Seiberg & E. Witten, JHEP (1999) 032.[4] A. Connes, M. R. Douglas, & A. Schwarz, JHEP (1998) 003.[5] M. R. Douglas & C. Hull, JHEP (1998) 008.[6] H. Arfaei & M. M. Sheikh-Jabbari, Nucl. Phys. B526 (1998) 278.[7] M. R. Douglas & N. A. Nekrasov, Rev. Mod. Phys. (2001) 977;R. J. Szabo, Phys. Rept. (2003) 207.[8] M. Chaichian, A. Demichev, & P. Presnajder, Nucl. Phys. B567 (2000)360;J. Math. Phys. (2000) 1647.[9] S. Majid & H. Ruegg, Phys. Lett. B334 (1994) 348.[10] J. Lukierski, H. Ruegg, & W. J. Zakrzewski, Annals Phys. (1995) 90;J. Lukierski & H. Ruegg, Phys. Lett.
B329 (1994) 189;G. Amelino-Camelia, Phys. Lett.
B392 (1997) 283.[11] G. Amelino-Camelia & M. Arzano, Phys. Rev.
D65 (2002) 084044;G. Amelino-Camelia, M. Arzano, & L. Doplicher, in “25th Johns HopkinsWorkshop on Current Problems in Particle Theory,” hep-th/0205047.[12] P. Kosinski, J. Lukierski, & P. Maslanka, Phys. Rev.
D62 (2000) 025004;D. Robbins & S. Sethi, JHEP (2003) 034;H. Grosse & M. Wohlgenannt, Nucl. Phys. B748 (2006) 473.[13] J. Madore, S. Schraml, P. Schupp, & J. Wess, Eur. Phys. J.
C16 (2000)161.[14] N. Sasakura, JHEP (2000) 015;S. Imai & N. Sasakura, JHEP (2000) 032;Y. Sasai & N. Sasakura, 0711.3059 [hep-th].[15] J. Madore, Class. Quant. Grav. (1992) 69.[16] P. Presnajder, Mod. Phys. Lett. A18 (2003) 2431;H. Grosse & P. Presnajder, Lett. Math. Phys. (1998) 61;Lett. Math. Phys. (1995) 171.[17] E. Batista & S. Majid, J. Math. Phys. (2003) 107.[18] A. H. Fatollahi & M. Khorrami, Europhys. Lett. (2007) 20003.1619] H. Komaie-Moghaddam, A. H. Fatollahi, & M. Khorrami, Eur. Phys. J. C53 (2008) 679.[20] H. Komaie-Moghaddam, M. Khorrami, & A. H. Fatollahi, Phys. Lett.
B661 (2008) 226.[21] A. B. Hammou, M. Lagraa, & M. M. Sheikh-Jabbari, Phys. Rev. D (2002) 025025.[22] A. Shariati, M. Khorrami, & A. H. Fatollahi, Int. J. Mod. Phys. A (20)(2012) 1250105.[23] S. Ghosh & P. Pal, Phys. Rev. D75 (2007) 105021.[24] A. H. Fatollahi, A. Shariati, & M. Khorrami, Eur. Phys. J.
C60 (2009)489.[25] H. Shin & K. Yoshida, Nucl. Phys.
B701 (2004) 380;W.-H. Huang, JHEP (2009) 102.[26] A. Shariati, M. Khorrami, & A. H. Fatollahi, J. Phys. A: Math. Theor. (2010) 285001.[27] Y.-G. Miao, X.-D. Wang, S.-J. Yu, Ann. Phys. (2011) 2091.[28] F. J. Vanhecke, C. Sigaud, A. R. da Silva, Braz. J. Phys. (2006) 194.[29] C. Leiva, J. Saavedra, J. R. Villanueva, arxiv: 1211.6785.[30] P. M. Zhang, P. A. Horvathy, J.-P. Ngome, Phys. Lett. A (2010) 4275.[31] E. Harikumar, A. K. Kapoor, Mod. Phys. Lett. A (2010) 2991.[32] D. Khetselius, Mod. Phys. Lett. A (2005) 263.[33] J. M. Romero, J. D. Vergara, Mod. Phys. Lett. A (2003) 1673.[34] B. Mirza, M. Dehghani, Commun. Theor. Phys. (2004) 183.[35] M. Khorrami, A. H. Fatollahi, & A. Shariati, J. Math. Phys.50