Energy and Angular Momentum Dependent Potentials with Closed Orbits
EEnergy and Angular Momentum DependentPotentials with Closed Orbits
M. H. Al-Hashimi ∗ April 3, 2019
Abstract
The Bertrand theorem concluded that; the Kepler potential, and theisotropic harmonic oscillator potential are the only systems under whichall the orbits are closed. It was never stressed enough in the physicalor mathematical literature that this is only true when the potentials areindependent of the initial conditions of motion, which, as we know, de-termine the values of the constants of motion E and L . In other words,the Bertrand theorem is correct only when V ≡ V ( r ) (cid:54) = V ( r, E, L ). Ithas been derived in this work an alternative orbit equation, which is asubstitution to the Newton’s orbit equation. Through this equation, itwas proved that there are infinitely many energy angular momentum de-pendent potentials V ( r, E, L ) that lead to closed orbits. The study wasdone by generalizing the well known substitution r = 1 /u in Newton’sorbit equation to the substitution r = 1 /s ( u, E, L ) in the equation ofmotion. The new derived equation obtains the same results that can beobtained from Bertrand theorem. The equation was used to study differ-ent orbits with different periodicity like second order linear differentialequation periodicity orbits and Weierstrasse periodicity orbits, where in-terestingly it has been shown that the energy must be discrete so that theorbits can be closed. Furthermore, possible applications of the alterna-tive orbit equation were discussed, like applications in Bohr-Sommerfeldquantization, and in stellar kinematics. ∗ Contact information: M. H. Al-Hashimi: [email protected], Bern, Switzerland, +41 31534 8948. a r X i v : . [ phy s i c s . c l a ss - ph ] A p r Introduction
The notion of closed orbit is a fundamental concept that shaped the conceptualdevelopment of astronomy, and the very first cosmological models created by theancient world thinkers, and later developed by medieval thinkers. Take for examplethe idea of celestial spheres, where all the astronomical objects circulating earth[1], the model based on observing repletion of the position of the celestial objectswhich takes place after a certain period of time. This essentially a feature of aclosed orbit. The significant development made by Kepler’s three laws [2, 3] cannotbe comprehended without the notion of closed orbit explicitly or implicitly. A wellknown historical fact, that Newton developed his gravitational theory, in order tofind a mathematical discerption to the planetary motion which match, or explainKepler’s three laws. Of course, in addition to explain the free fall of objects. Thebrilliant concept of potential function was invented by Newton in order to explainthe behavior of a particle moving under the action of gravity. Newton found thatthe successful potential for this purpose is V ( r ) = − GmMr , (1.1)where m and M are two masses separated by the distant r , and G is the gravitationconstant, a universal constant that does not depend on the initial conditions, likeenergy and angular momentum. This is a property that we have to keep in mind tounderstand the rest of this work. It is very important here to remember that thechoice of V ( r ) is totally an empirical choice [4]. It has been made to account for theproperties of gravity as they were discovered by all the thinkers prior to Newton.Their thoughts was driven by observation, or logical reasoning, or by both.From the Newton’s second law, the equation of motion for a single particle withmass m under the action of a arbitrary force (cid:126)F ( (cid:126)r ) takes the following form m d (cid:126)rdt = (cid:126)F ( (cid:126)r ) = − (cid:126) ∇ V ( (cid:126)r ) , (1.2)By integrating the above equation with respect to time we get the total energy E as a constant of motion. Then eq.(1.2) gives E = m (cid:126)r · ˙ (cid:126)r + V ( (cid:126)r ) . (1.3)For a central force, V ( (cid:126)r ) = V ( r ) , (cid:126)F ( (cid:126)r ) = (cid:126)rr F ( r ) , (1.4)as a result of the above equation, we get (cid:126)r × (cid:126)F ( (cid:126)r ) = ddt ( (cid:126)r × m(cid:126)v ) = d(cid:126)Ldt = 0 , (1.5)2hich gives the angular momentum vector (cid:126)L as another constant of motion. Thedirection of (cid:126)L as well as the magnitude L are constant with respect to time, andcan written in terms of the angular velocity ˙ ϕ as L = mr ˙ ϕ. (1.6)In general, solving the equation of motion eq.(1.2) for most potentials is not astraightforward process, and even when r ( t ) can be obtained, the expression is toocomplicated to be representable. Therefore, a substitution like r ( ϕ ) = 1 u ( ϕ ) , (1.7)would lead to a solvable equation of motion. For a central force, eq.(1.2) can bewritten as [3] d udϕ + u = − mF ( u − ) L u . (1.8)The above equation known as Newton’s orbit equation. For the Kepler problem, F ( r ) = − κr = − κu , (1.9)the solution of eq.(1.8) is u ( ϕ ) = 1 r = κmL (1 + e cos( ϕ − ϕ )) , (1.10)where e is the eccentricity of the elliptical orbit, which is given by the followingrelation e = (cid:114) EL mκ , (1.11)and ϕ is the initial value of the angle variable ϕ . From now on, we will take ϕ = 0.For the case of the isotropic harmonic oscillator the force is F ( r ) = mω r = mω u , (1.12)the solution of eq.(1.8) is 1 r = EmL (1 + f cos(2 ϕ )) , (1.13)where f = (cid:114) − ω L E . (1.14)The above two examples are particularly important because they lead to a closedorbit. In fact Bertrand’s theorem [5] proves that the Kepler force, and the isotropic3armonic oscillator force are the only two systems where all the orbits are closed.This is correct regardless of the initial conditions (which include initial position,and initial velocity that fixed the value of E and L ). In his theorem, Bertrand onlyconsidered forces of the form F ( r ) = cr n , (1.15)where c is a constant that does not depend on constants of motion like E and L .In the previous examples, n = − c = − κ for the Kepler system, and n = 1, c = mω for the isotropic harmonic oscillator system. The proof of the theorembased on guarding the stability of a circular orbit that could be perturbed by asmall oscillation. Bertrand have shown that, these two systems are the only onesthat do not lead to an orbit that can spiral down to the origin [3]. I have to mentionhere, that the event of a particle going through the origin, seems to catastrophic tosome physicists, especially before the development of the new quantum mechanics,where a particle can have a zero angular momentum for the s -state. However, aparticle passing through the center of force is not forbidden mathematically.Aside from solutions for the Newton’s orbit equation that leads to closed obits,there are other studies that aiming at finding potentials for which eq.(1.8) has so-lutions in terms of known functions. For example, Whittaker have shown that for n = 5 , , , − , − , −
7, then r ( ϕ ) can be expressed in terms of elliptical functions [6](p83). His work is a result of collecting many studies that go back to the nineteencentury. A more generalized approach by Broucke [7] have shown that potentialswith multiple terms of different powers of r can lead to solution in terms of knownfunctions. Systems with potentials that depends on the initial conditions can be found in manyexamples in the physics literature. One of the prominent example is the generalrelativity correction to the Newton gravitational potential. Its importance is due toits success in accounting for the perihelion shift for mercury’s orbit. According togeneral relativity, for weak gravitational field, slow particle, and an orbit with smalleccentricity e , the gravitational potential can be approximated to the form [8, 9] V ( r ) = − GM mr − GM L mr c . (1.16)This means that the corrected gravitational potential does depend on the initialconditions of motion through L .Angular momentum dependent potentials are also discussed in the context ofquantum field theory, where the scattering amplitude be approximated to a potentialusing Born approximation. For example the Møller scattering has an amplitudecorrespond to spin-orbit interaction potential between two electrons. This potential4epends on the relative angular momentum between the two electrons explicitly[10]. This is in addition to countless, other examples where the potential dependson angular momentum, total energy, or both, which can obtained from quantumfield theories. The energy momentum dependent potentials as leading term and notas a mere correction, was also discussed by when the work of Broucke was furtherexpanded to more potentials using symmetry as tool of investigation [7]. The notion of closed or periodic orbit is central in the development of the old quan-tum mechanics. In Bohr hydrogen like atom, the electron moves in circular closedorbits around the nucleus. In Sommerfeld hydrogen like atom, the closed orbits areelliptical. For closed orbit, and as a result of mere observation, the old quantummechanics rules of quantization are (cid:73) p ϕ dϕ = hn ϕ , n ϕ = 1 , , , ... (1.17)where h is the Plank constant. Also (cid:73) p r dr = hn r , n r = 0 , , , ..., (1.18)where p ϕ , p r are the conjugate generalized momentums to the generalized coordinates ϕ and r respectively. In general, we can write (cid:73) p i dq i = hn i , (1.19)where n i = 0 , , , ... for lateral motion, and n i = 1 , , , ... for rotational motion[3, 11]. This work is based on deriving the expressions of potentials from the periodicityof the motion of a particle as an input. To achieve that, instead of the usualsubstitution that lead to Newton’s orbit equation r = 1 /u eq.(1.8) we use a moregeneral expression r ( ϕ ) = g ( u ( ϕ ) , E, L ) , (2.1)It is important to chose r = g ( u, E, L ) ≥ , g ( u, E, L ) ∈ (cid:60)∀ u ∈ [ u min , u max ] . (2.2)5everal differential equations with periodic solutions are considered. What we meanby periodic solution in this work is u ( ϕ ) = u ( ϕ + 2 nπ ) , (2.3)where n a non-zero positive integer. For second order differential equation orbits. d udϕ = γ ( u, E, L ) , (2.4)where γ ( u, E, L ) is a function that characterized the kind of periodicity of the so-lution of the differential equation, as a function of u and the initial conditions thatspecify E and L . For example, γ ( u, E, L ) = − λ u + a ( E, L ) is corresponds to asolution of the form u = a ( E, L ) λ − + b ( E, L ) cos λϕ , where λ is a rational number.Another example for γ ( u, E, L ) = 6 u − g ( E, L ) / dudϕ = ± g ( u, E, L ) Lg (cid:48) ( u, E, L ) (cid:112) g ( u, E, L ) m ( E − V ( u, E, L )) − L , (2.5)by using the chain rule, and substituting for u (cid:48)(cid:48) ( ϕ ) from eq.(2.4), γ ( u, E, L ) can beexpressed as d udϕ = dudϕ ddu dudϕ = γ ( u, E, L ) , (2.6)The above equation can be integrated with respect to u . Further more, by substi-tuting for u (cid:48) ( ϕ ) from eq.(2.5) then the above equation leads to following expressionfor the potential V ( u, E, L ) = E − L mg ( u, E, L ) − L g (cid:48) ( u, E, L ) mg ( u, E, L ) Ω( u, E, L ) , (2.7)where Ω( u, E, L ) = (cid:90) ua γ ( ζ, E, L ) dζ. (2.8)The function Ω( u, E, L ) is named here as the periodicity characterization function.It represents an input that is restricted by the periodicity of the orbit given by γ ( u, E, L ). This phrase will be more clear later on in the rest of this article. Thenumber of expressions for V ( u, E, L ) for a given Ω( u, E, L ) is infinite, that is becausethere is infinite possibilities of choosing g ( u, E, L ). The expression of V ( r, E, L ) canbe retrieved if an inverse solution g − ( r, E, L ) = u, (2.9)6xists. The same result of eq.(2.7) can be obtained using different root. By usingeq.(1.2) and eq.(1.6), for central force, it can be proved that the force expression F ( r ) is given by the following equation F ( r ) = L mr d rdϕ − L mr (cid:18) drdϕ (cid:19) − L mr . (2.10)Then using the above equation with eq.(2.2) leads to the same expression in eq.(2.7).Another form of eq.(2.7) can be obtained by substituting s ( u, E, L ) = 1 g ( u, E, L ) , (2.11)then eq.(2.5) can be written as u (cid:48) ( ϕ ) = (cid:115) Em − L s ( u ( ϕ ) , E, L ) − mV ( u ( ϕ ) , E, L ) L s (cid:48) ( u ( ϕ ) , E, L ) , (2.12)and V ( u, E, L ) can be written as V ( u, E, L ) = E − L m (cid:0) s (cid:48) ( u, E, L ) Ω( u, E, L ) + s ( u, E, L ) (cid:1) . (2.13)Again here the following condition must hold1 r = s ( u, E, L ) ≥ , s ( u, E, L ) ∈ (cid:60)∀ u ∈ [ u min , u max ] . (2.14)The expression given by eq.(2.7) is suitable for certain problems, while the expressiongiven by eq.(2.13)is more for others, however, both of them will be called alternativeorbit equation.It is important to note at this stage that Bertrand’s theorem is applicable only onpotentials of the form V ( r, E, L ) = V ( r ) which will be called ”Bertrand potentials”. For second order linear differential equations with periodic solution, the eq.(2.4) hasthe following form d udϕ + uλ = a ( E, L ) , (3.1)then the solution is u ( ϕ ) = a ( E, L ) /λ + b ( E, L ) cos λϕ. (3.2)7rom the above equation and eq.(2.8) we getΩ( u, E, L ) = − u λ a ( E, L ) u − c ( E, L ) , (3.3)where c ( E, L ) is an arbitrary constant. Therefore eq.(2.13) for this case can bewritten as V ( u, E, L ) = E + L m (cid:18) s (cid:48) ( u, E, L ) (cid:18) c ( E, L ) + u λ − a ( E, L ) u (cid:19) − s ( u, E, L ) (cid:19) . (3.4)From eq.(1.3), and by using eq.(3.2) and eq.(3.4), we get b ( E, L ) = ± (cid:112) − λ c ( E, L ) + a ( E, L ) λ . (3.5)As eccentricity is only associated with conic section orbits, we define a more suitableparameter as a generalize conic section ˜ e ( E, L ) which can be expressed from eq.(3.2).It is defined by the following relation˜ e ( E, L ) = λ b ( E, L ) a ( E, L ) (3.6)There are countless cases for V ( u, E, L ), let us first consider the Bertrand po-tentials case V ( u, E, L ) = V ( u ) = V ( r ) , (3.7)where the expression of the potential is independent of the constants of motion E an L . A solution for this case is s ( u, E, L ) = s ( u ) = u n , (3.8)by substituting for s ( u ) from eq.(3.8)into eq.(3.4), the result is V ( u ) = E + L m (cid:0) n λ − (cid:1) u n + n L m (cid:0) u n − c ( E, L ) − u n − a ( E, L ) (cid:1) , (3.9)The above equation can only be satisfied if λ = 1 /n . Moreover, the third or thefourth terms can cancel E only if n = 1 or 2 n = 1 /
2. For n = 1, then λ = 1 and theKepler potential V ( r ) = − κ/r case is retrieved with s ( u ) = u = 1 r , a ( E, L ) = − EmL , c ( E, L ) = mκL , (3.10)the value of b ( E, L ) can be calculated from eq.(3.5), which gives b ( E, L ) = mκL (cid:114) EL mκ , (3.11)8hich is the same expression that can be obtained from eq.(3.38). For n = 1 / λ = 2 and the isotropic harmonic oscillator potential V ( r ) = mω r / s ( u ) = √ u = 1 √ r , c ( E, L ) = 2 mω L , a ( E, L ) = 4
EmL ,b ( E, L ) = 4
EmL (cid:114) − L ω E (3.12)which is in agreement with eq.(1.13). As we see here, the results comply withBertrand’s theorem, as there are no other potentials of the form V ( u ) = V ( r )except the Kepler and the isotropic harmonic oscillator potentials. However, itmust be kept in mind that this treatment only applicable on second order lineardifferential equation orbits.The other class of potentials of the form V ( u, E, L ) = V ( u, E ) which shall benamed as rescaling potentials. The following substation can lead to such potentials s ( u, E, L ) = 1 L (cid:101) s ( u, E ) , c ( E, L ) = c ( E ) , a ( E, L ) = a ( E ) , (3.13)then eq.(3.4) can be written as V ( u, E ) = E + 12 m (cid:18)(cid:101) s (cid:48) ( u, E ) (cid:18) c ( E ) + uλ − a ( E ) u (cid:19) − (cid:101) s ( u, E ) (cid:19) , (3.14)where c ( E ) and a ( E ) are arbitrary functions of the total energy. It must be men-tioned here that for such substitution V ( u, E ) = V ( rL − , E ) . (3.15)The simplest non-trivial example for this case is a potential V ( u ) = V ( rL − ). Asuitable choice for s ( u, E, L ) for this case is s ( u, E, L ) = α + α uL = 1 L (cid:101) s ( u ) = 1 r , (3.16)where α and α are constants that are independent of E and L . In order that thefirst term in eq.(3.14) canceled, c ( E ) must be chosen as c ( E ) = α − Em α . (3.17)For an elegant expression of V ( rL − ) all the terms of power a V ( u, E ) ∼ u , then a ( E ) = − α α , (3.18)9hich leads to b ( E ) = 1 λ (cid:115) α + (2 Em − α ) λ α , (3.19)which leads to r = Lλ α ( λ −
1) sgn α + (cid:112) Emλ − ( λ − α cos λϕ , (3.20)Accordingly the potential in eq.(3.14) has the following expression V ( u ) = u α m (cid:0) λ − (cid:1) , (3.21)and therefore V ( r, E, L ) = (cid:0) λ − (cid:1) (cid:18) α m + L mr − α L mr (cid:19) . (3.22)It is obvious that for λ = 1 leads to the free particle case. Otherwise, the first termis just a constant shift in energy, the second term is a centrifugal potential up toa constant λ − − α L/ (2 m ) ∼ L , which is an essential difference from the usual Keplerpotential. For real and positive r (see eq.(2.14)), and using eq.(3.20), the energyvalues for this system is restricted by the following condition α m (cid:0) λ − (cid:1) ≥ E ≥ α mλ (cid:0) λ − (cid:1) , (3.23)moreover, α < λ = 1 /
2, and α > λ >
1. In figure 1, the potential V ( u ) is plotted against u for different values for λ using eq.(3.21). All the curvesare symmetric for any value of λ . In figure 2, the potential V ( u ) is plotted against u for certain values of energy E , and different values of λ . The energies subjectto condition eq.(3.23). In this case u ∈ [ u min , u max ], the value of u min and u max can be calculated for the given λ and E using eq.(3.2). In figure 3, the potential V ( rL − ) is plotted against r . The curves has been plotted for different value of λ , and for L = 1 , , . In the left panel it is obvious that V ( rL − ) → −∞ as r → λ = 1 /
2, while in the middle and right panels V ( rL − ) → ∞ as r → λ > / / ,
2. For λ = 1 /
2, the potential constant α = −
1, which is incompliance with the condition in eq.(2.2), that demands in this case that α < α > λ > / ,
2. Thelast three figures are important for studying the potential for this case. Figure 4 It is must be stressed here that choosing the values of L = 1 , , E , which has been calculated according to our simple choice of m = 1, and α = ± u -10010203040 V(u) λ= / λ= λ = / λ= / α =−1 m =1 Figure 1:
The potential V ( u ) versus u for different values of λ = 1 / , / , , / . Inall the graphs, m = 1 , and α = − . is for the orbit with λ = 1 /
2. For m = 1, α = −
1, the energy E = − / L = 1 , , V ( rL − ) as the rescaling potential,where, for example, the orbit for certain energy and L = 2 is a copy of the orbitwith L = 1, except it is bigger in size. The other orbits in other graphs of figure4 are for other values of energy E = − , − /
4, where it calculated such that thecondition in eq.(3.23) is fulfilled. For these two case, the generalized eccentricity˜ e ( E, L ) = 0 , / , and (cid:112) / e ( E, L ) (cid:54) = 0, the orbit intersects once with itself. In figure 5, thegraphs are for different values of λ , and different energies. For each value of λ and E , three different orbits are plotted for L = 1 , ,
3. We note here that for a halfinteger λ , the orbit intersect with itself, as one can see that from figure 4 whilefor λ an integer, the orbit does not intersect with it self, as one can see in figure5 for the λ = 2. Figure 6 shows that the lowest values of E leads to r → ∞ for λϕ = 0 , π, π, ... . The figures from 3 to 6 show that eq.(2.13) actually works.Another example of a rescaling potential V ( u, E ) = V ( r √− mEL − ) is a poten-tial that leads to a rescaling factor L/ √− mE . It can be generated by using thefollowing substitution (cid:101) s ( u ) = A sinh( σu ) = Lr (3.24)in eq.(3.14), where σ is a constant, which is independent of E . To cancel E in the11 uV(u) u -15-10-50 1 2 3 4 5 6 7-15-10-50 1 2 3 4 5 6 7 u -15-10-50 V(u) u V(u) u u V(u) u V(u) -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81 -0.4 -0.2 0 0.2 0.4 0.6 0.8 u u V(u) -0.4 -0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81 u min u min u min =0u min u min u min u max u min u min E =-7/5 E =-1 E =-3/4 E =8/27 E =1/3 E =1/2 u min u max u max u max u max u max u max u max u max E =3/4 E =1 E =3/7 λ=1/2λ=3/2λ=2 λ=1/2 λ=1/2λ=3/2 λ=3/2λ=2 λ=2 Figure 2:
The potential V ( u ) versus u for different values of λ , and different valuesof the energy E . The values of E are chosen for each value of λ = 1 / , / , suchthat the condition in eq.(3.23) is respected. The variable u has a minimum value of u min , and maximum value u min , which are determined by eq.(3.2), with α = − for λ = 1 / , and α = 1 for λ = 3 / , . In all the graphs m = 1 , and α = − V ( u, E ), we make the following substitution A = √− mE c ( E ) = 12 σ , (3.25)which means that for this case E <
0. Accordingly, the expression of V ( u, E ) canbe written as V ( u, E ) = Euσ (cid:0) a ( E ) − uλ (cid:1) cosh( σu ) . (3.26)It is obvious from the above equation, that the expression of a ( E ) cannot be manip-ulated to reduce or further simplify the expression of V ( u, E ). On the other hand,it can be chosen such that the expression of u ( ϕ ) is simpler. Moreover, for a certainexpression of a ( E ), there can be an upper limit on the value of λ . For example, for a ( E ) = a = µσ , µ = n , n = 2 , , , ..., (3.27)which leads to b ( E ) = (cid:112) µ − λ λ σ , (3.28)where µ > λ . Therefore we get u ( ϕ ) = 1 λ (cid:16) µ + (cid:112) µ − λ cos( λϕ ) (cid:17) , (3.29)12 r -6-5-4-3-2-10 V ( r ) r r V ( r ) L = L = L = L = L = L = L = L = L = λ=1/2 λ=3/2 λ=2α =−1 α =1 α =1 Figure 3:
The rescaling potential V ( rL − ) versus r for different values of λ , and L = 1 , , for each value of λ = 1 / , / , . The value α = − for λ = 1 / , and α = 1 for λ = 3 / , . In all the graphs, m = 1 . and r = L √− mE csch (cid:20) λ (cid:16) µ + (cid:112) µ − λ (cid:17) cos( λϕ ) (cid:21) . (3.30)It is obvious from eq.(3.30) that the rescaling factor for the orbit is L/ √− mE .The expression of V ( rL − , E ) for this case is V ( rL − , E ) = L − Emr mr arccsch (cid:18) r √− mEL (cid:19) × (cid:18) λ arccsch (cid:18) r √− mEL (cid:19) − µ (cid:19) . (3.31)In figure 7, the potential V ( rL − , E ) is plotted against r by using eq.(3.31). Thecurves has been plotted for one value of λ = 1 /
2. Roughly speaking, all the graphsshow that the energy is more attractive as | − E | is getting bigger. Again here, itmust be kept in mind the units in all the figures were ignored.Figures 8,9 show again that eq.(2.13) leads to closed orbits. The orbits in figure 8are for λ = 1 /
2. In general they have a common shape with with the orbits in figure4 (the second and the third rows), for example, in both cases the orbit intersect oncewith itself. On the other hand, they are not similar, especially with larger valuesof µ . This is simplify because eq.(3.20) is different than eq.(3.30). Moreover, forthe sinh-potential, the rescaling factor can be taken as L/ √− mE instead of L , as13 L =1 L =2 L =3 L =1 L =2 L =3 L =1 L =2 ~ E =-3/2 E =-3/2 E =-3/2 E =-1 E =-1 E =-13/8 E=- e= √ e =0~ ~ ~~~~ e =1/2 e =1/2 e =1/2 L =3~ 3/8~ E=- e= √ e= √ E=- e =0 e =0 Figure 4:
Closed orbits for a particle moving under the action of the rescaling poten-tial V ( rL − ) for λ = 1 / , and for different values of the energy E = − / , − , − / .For each value of E there is a graph with L = 1 , , . In all the graphs, m = 1 , α = − , and α < . one can readily see from eq.(3.30). For this case, for any E , and any L , we canobtain the same orbit, as long as L/ √− mE is fixed. In physical terms, for a given L/ √− mE , the increasing of L that leads to an increase of the centrifugal force canbe balanced by an increase of | − E | which increase the attraction force (see figure7), such that the orbit is unchanged. This is true as long as L/ √− mE unchanged.In figure 9, the orbits are for different values of λ = 3 / , , /
2. For each valueof λ , three graphs have been plotted for µ = 153 / , /
25 and µ = 3. In allthe graphs L/ √− mE = 1. It is clear that orbits stretched as µ increases for fixedrescaling factor L/ √− mE = 1. V ( r, E, L ) = (cid:80) n =1 A n ( E,L ) r n If the condition in eq.(3.13) is dropped, then the expression of is s ( u, E, L ) is notnecessarily separable. Let us assume that s ( u, E, L ) has the following expression s ( u, E, L ) = uα ( E, L ) + α ( E, L ) u = 1 r (3.32)14 L =1 L =2 L =3 L =1 L =3 L =2 L =1 E =8/27 L =1 L =1 L =1 L =1 L =1 L =3 L =3 L =3 L =3 L =3 L =3 L =2 L =2 L =2 L =2 L =2 L =2 E =1/3 E =1/2 E =3/7 E =3/4 E =1 E =1/2 E =1 E =3/2 λ=3/2 λ=3/2 λ=3/2λ=2 λ=2 λ=2λ=5/2 λ=5/2 λ=5/2 Figure 5:
Closed orbits for a particle moving under the action of the rescaling poten-tial V ( rL − ) for different values of λ , and different values of L = 1 , , correspondto each value of λ = 1 / , / , . In this figure, the energies has been chosen to high-light the stretching of the orbit with the increase of energy. The calculations weredone with, α = − for λ = 1 / , and with α = 1 for λ = 3 / , . For all the graphs m = 1 , and α < . The energy E in the expression of V ( u, E, L ) can be canceled out in eq.(3.4) byusing the following substitution c ( E, L ) = − mEα ( E, L ) L , (3.33)then V ( r, E, L ) can be written as V ( r, E, L ) = (cid:88) n =1 A n ( E, L ) r n , (3.34)where, A ( E, L ) = 4 Eα ( E, L ) − L a ( E, L ) mα ( E, L ) ,A ( E, L ) = L m ( λ − − Eα ( E, L ) + 3 L a ( E, L ) α ( E, L ) mα ( E, L ) ,A ( E, L ) = 4 Eα ( E, L ) − L λ α ( E, L ) m − L α ( E, L ) a ( E, L ) mα ( E, L ) ,A ( E, L ) = − Eα ( E, L ) + L λ α ( E, L ) m + L α ( E, L ) a ( E, L ) mα ( E, L ) . (3.35)15 r -3-2-10123 V ( r ) -4 -3 -2 -1 0 1 2 3 4 5 6 r -20-15-10-505101520 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 r -3-2-10123 V ( r ) L =1 E =-3/8 L =1 E =5/8 E =3/2 L =1 λ=1/2 λ=3/2 λ=2α =−1 α =1 α =1 Figure 6:
Orbits with r → ∞ at certain point or points for a particle moving underthe action of the rescaling potential V ( rL − ) . The three graphs are for values of λ = 1 / , / , . The value α = − for λ = 1 / , and α = 1 for λ = 3 / , . Theenergies are for the upper limit values, which are E = − / , / , / correspond to λ = 1 / , / , respectively. In all the graphs m = 1 , α < , and L = 1 . There are many special cases for the potential in eq.(3.34), an important case is thecase with α ( E, L ) = 0 and λ = 1, then V ( r, E, L ) = − L a ( E, L ) mrα ( E, L ) = − κ ( E, L ) r , (3.36)where κ ( E, L ) = L a ( E, L ) mα ( E, L ) , (3.37)is a totally arbitrary function of E and L . The orbit equation for this case can beobtained by calculating b ( E, L ) from eq.(3.5). The result is r = α ( E, L ) u = L κ ( E, L ) m (1 + e cos( ϕ )) , e = (cid:115) EL mκ ( E, L ) . (3.38)When κ ( E, L ) = κ , then the Kepler potential is retrieved. The different possibilitiesof choosing κ ( E, L ) lead to important applications, which will be discussed later inthis article. 16 r -600-540-480-420-360-300-240-180-120-600 V ( r L - , E ) r -600-540-480-420-360-300-240-180-120-600 0 0.2 0.4 0.6 0.8 1 r -600-540-480-420-360-300-240-180-120-600 V ( r L - , E ) L = L = L = L = L = L = L = L = L = λ=1/2 λ=1/2 λ=1/2 E =-1/4 E =-1/2 E =-1 µ=7/12 µ=7/12 µ=7/12 Figure 7:
The rescaling potential V ( rL − , E ) versus r for λ = 1 / . Each graph is fordifferent energy E = − / , − / , − . The curves in each graph are for L = 1 , , .In all the graphs m = 1 , µ = 7 / , and the value of the generalized eccentricity ˜ e ( E, L ) = √ / . Another special case can arise when α ( E, L ) (cid:54) = 0, and A ( E, L ) = A ( E, L ) = 0,which can be fulfilled when a ( E, L ) = ∓ λα ( E, L ) √− EmL , α ( E, L ) = ± λ L √− Em . (3.39)Accordingly, the potential function for this case is V ( r, E, L ) = ∓ λLr (cid:114) − Em + ( λ − L mr , α ( E, L ) = ± λ L √− Em . (3.40)From eq.(3.5), the value of b ( E, L ) = 0. However, this does not mean that the orbitis a circle. From eq.(3.39) it is clear that a ( E, L ) = 0, which makes the system non-physical because r = 0 for any ϕ , thus, V ( r, E, L ) singular for this case for any valueof ϕ . This is an important reminder that not all the choices of s ( u, E, L ) can leadto a physical solution. Other solutions are dismissed because they lead to negativeor imaginary r for any value of λ , which means that the condition in eq.(2.14) isviolated. These cases are A ( E, L ) = A ( E, L ) = 0, and A ( E, L ) = A ( E, L ) = 0.The are three physical cases, the first physical cases is for A ( E, L ) = A ( E, L ) =0, with α ( E, L ) , α ( E, L ) >
0. Then, in addition to eq.(3.33), the following relations17 L/ √ ( )=1 ( )=2 L/ √ L/ √ ( )=1 L/ √ µ=51/100 µ=51/100 L/ √ ( )=2 ( )=3( )=3 L/ √ ( )=2 L/ √ ( )=1 µ=51/100 ( )=3 L/ √ L/ √µ=14/25 µ=14/25 µ=14/25µ=1µ=1µ=1 -mE -mE -mE-mE -mE -mE-mE -mE -mE Figure 8:
Closed orbits for a particle moving under the action of the rescaling po-tential V ( rL − , E ) for λ = 1 / . The figure has three rows of graphs, for the firstrow σ = 1 / , for the second row σ = 1 / , and for the third σ = 1 . In each row,there are three orbits. The rescaling factor for each orbit are L/ √− mE = 1 , , respectively. For all the graphs m = 1 . are valid a ( E, L ) = ∓ α ( E, L )(1 + 5 λ ) L (cid:115) − Em λ ) , α ( E, L ) = ± L √ λ √− Em . (3.41)The potential for this case is V ( rL − , E ) = ∓ L (1 + λ ) r (cid:115) − E m (1 + 2 λ ) ± L (2 λ − λ − mr (cid:112) − Em (1 + 2 λ ) . (3.42)The trajectory for this case can be obtained from the following relation for λ < r = L (cid:114) λ + 1 − Em (1 − λ ) (1 + sgn α cos( λϕ ))5 λ + 1 + sgn α (1 − λ ) cos( λϕ ) α ( E, L ) = L √ λ + 1 √− Em , (3.43)and the following relation for λ > r = L (cid:114) λ + 1 − Em ( λ −
1) (1 − sgn α cos( λϕ ))5 λ + 1 + ( λ −
1) sgn α cos( λϕ ) α ( E, L ) = − L √ λ + 1 √− Em . (3.44)18 L/ √ ( )=1 ( )=1 L/ √ ( )=1 ( )=1 ( )=1 ( )=1 L/ √µ=153/100 µ=42/25 ( )=1 ( )=1 ( )=1 L/ √ L/ √λ=3/2 λ=3/2 λ=3/2λ=2 λ=2 λ=2λ=5/2 λ=5/2 λ=5/2µ=3 L/ √ L/ √ L/ √ L/ √µ=51/25 µ=66/25 µ=4µ=5µ=70/25µ=255/100 -mE-mE -mE -mE-mE -mE-mE -mE -mE Figure 9:
Closed orbits for a particle moving under the action of the rescaling poten-tial V ( rL − , E ) . The figure has three rows of graphs, for the first row λ = 3 / , forthe second row λ = 2 , and for the third λ = 5 / . In each row there are three orbits,for µ = 153 / , / , respectively. For all the graphs m = 1 , and the rescalingfactor L/ √− mE = 1 . The orbits for this case are closed for any
E <
0, as it shown in figure 10.
The second physical cases is for A ( E, L ) = A ( E, L ) = 0. Then, in additionto eq.(3.33), the following relations are valid a ( E, L ) = ∓ (cid:114) − Em λα ( E, L ) L , α ( E, L ) = ± Lλ √− Em . (3.45) V ( rL − , E ) = − L (1 + λ )2 mr ± L λ mr √− Em . (3.46) r = Lλ √− Em sgn α ( E, L ) − cos( λϕ )2 sgn α ( E, L ) + cos( λϕ ) , α ( E, L ) = − Lλ √− Em . (3.47)The orbits for this case are closed for any
E <
0, as it shown in figure 11. It isimportant to note here that the solution with α ( E, L ) =
Lλ/ √− Em leads to thenon-physical case with r <
0. 19 he third physical case and the physical case is for A ( E, L ) = A ( E, L ) = 0,the solution is V ( rL − , E ) = − L mr (1 + λ − L λ Em r , α ( E, L ) = ± Lλ √− Em . (3.48)For this case b ( E, L ) = 0, then the only closed orbit is a circle with a radius r = Lλ √− Em , (3.49)and, α ( E, L ) = − Lλ √− Em , (3.50)which means that r = contant > E < L . As for α ( E, L ) =
Lλ/ √− Em , then r = contant <
0, therefore this solution is discarded. -0.4 0.2-0.4-0.200.20.4-0.4 -0.2 0 0.2 0.4-0.4-0.200.20.4 -0.2 -0.1 0 0.1 0.2-0.2-0.100.10.2-0.2 -0.1 0 0.1 0.2-0.2-0.100.10.2 -0.1 -0.05 0 0.05 0.1-0.1-0.0500.050.1 -0.1 -0.05 0 0.05 0.1-0.1-0.0500.050.1-0.2 -0.1 0 0.1 0.2 0.3-0.2-0.100.10.20.3 -0.2 -0.1 0 0.1 0.2 0.3-0.2-0.100.10.20.3 0-0.2-0.100.10.2 -0.2 -0.1 0 0.1 0.2-0.2-0.100.10.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6-0.4-0.200.20.40.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6-0.4-0.200.20.40.6-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4-0.200.20.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4-0.200.20.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8-0.6-0.4-0.200.20.40.60.8-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8-0.6-0.4-0.200.20.40.60.8 -0.5 0 0.5-0.500.5 -0.5 0 0.5-0.500.5 A =A =0 A =A =0 A =A =0A =A =0 A =A =0 A =A =0A =A =0 A =A =0 A =A =0 λ=1/3 λ=2/3 λ=4/5λ=1/2 λ=3/2 λ=5/2λ=2 λ=3 λ=4 Figure 10:
Closed orbits for a particle moving under the action of the potential V ( rL − , E ) for the first physical case with A = A = 0 for different values of λ .For all the graphs m = 1 , L/ √− mE = 1 , and α ( E, L ) > . A =A =0 A =A =0 A =A =0A =A =0 A =A =0 A =A =0A =A =0 A =A =0 A =A =0 λ=1/3 λ=2/3 λ=4/5λ=1/2 λ=3/2 λ=5/2λ=1 λ=2 λ=3 Figure 11:
Closed orbits for a particle moving under the action of the potential V ( rL − , E ) for the second physical case with A = A = 0 for different values of λ .For all the graphs m = 1 , L/ √− mE = 1 , and α ( E, L ) > . We start this chapter by explaining some basic properties of the Weierstrasse func-tion. There are several Weierstrasse functions that share one common feature ofbeing doubly periodic functions, or quasi-periodic. In relation to this work, study-ing all Weierstrasse functions would be a very lengthy process. Therefore we willonly study special cases of the Weierstrasse’s elliptic function ℘ which is defined bythe following relation[12–14] ℘ ( z ; g , g ) = 1 z + ∞ (cid:48) (cid:88) m,n = −∞ z − mω − nω ) − mω + 2 nω ) , (4.1)where the prime means that terms in the sum give zero denominators are excluded.The half periods in the above equation are ω and ω . It is clear from eq.(4.1) that ℘ ( z + 2 mω + 2 nω ; g , g ) = ℘ ( z ; g , g ) , m, n = 1 , , , ... (4.2)The Weierstrasse invariants g and g can be evaluated using the following relations[15] g = 60 ∞ (cid:48) (cid:88) m,n = −∞ mω + 2 nω ) g = 140 ∞ (cid:48) (cid:88) m,n = −∞ mω + 2 nω ) . (4.3)21t must be noted here that here that notation used here for different variables areused by most references. The Weierstrasse function ℘ satisfies the following twodifferential equations ℘ (cid:48)(cid:48) ( z ; g , g ) − ℘ ( z ; g , g ) + g , (4.4)and ℘ (cid:48) ( z ; g , g ) − ℘ ( z ; g , g ) + g ℘ ( z ; g , g ) + g = 0 . (4.5)Another feature of ℘ that can be obtained from eq.(4.1) is [12] ℘ ( zt ; g , g ) = 1 t ℘ ( zt ; g t , g t ) . (4.6)From what has been mentioned, it is possible that ℘ to be imaginary and negative,what we are interested in, is a positive real ℘ . Such an example is ℘ ( z ; 3 ,
1) = 32 cot (cid:32)(cid:114) z (cid:33) + 1 . (4.7)Then periodicity orbit function u ( ϕ ) can be written as u ( ϕ ) = ℘ ( ϕ ; g , g ) , (4.8)Therefore, the only ℘ that leads to closed orbits is the one with the periodicity ℘ ( ϕ ; g , g ) = ℘ ( ϕ + 2 nπ ; g , g ) , n = 1 , , , ..., (4.9)which restricts further the values of g and g . For this case, using the above firstdifferential equation eq.(4.4), and eq.(2.6) give γ ( u, E, L ) = 6 u − g ( E, L )2 , (4.10)and therefore eq.(2.8) givesΩ( u, E, L ) = 2 u − g ( E, L ) u − c ( E, L ) . (4.11)Accordingly, eq.(2.13) for this case can be written as V ( u, E, L ) = E + L m (cid:18) s (cid:48) ( u, E, L ) (cid:18) c ( E, L ) + 2 u − g ( E, L ) u (cid:19) − s ( u, E, L ) (cid:19) . (4.12)Furthermore, using eq.(2.12), eq.(4.12) and the second differential equation eq.(4.5)we get c ( E, L ) = g ( E, L )2 . (4.13)22 .1 The Bertrand theorem and Weierstrasse periodicity For this periodicity, to investigate existence of Bertrand potentials such that V ( u, E, L ) = V ( u ) = V ( r ), it has to be proved that the expression of the potential is indepen-dent of the constants of motion E an L . An ansatz like the one of eq.(3.8) can beexamined. Then eq.(4.12) gives V ( u, E, L ) = E + L m (cid:0) − u n + g n u n − + g n u n − − n u n +1 (cid:1) . (4.14)For certain choice of n , it is possible to cancel E in the above expression. On theother hand, there are no choices for n , g or g that cancel the dependency of thepotential on L . Therefore, we can say; for the Weierstrasse periodicity, there is nopossibility of constructing a Bertrand potential using an ansatz s ( u, E, L ) = s ( u ) = u n . It is not clear if other choice of s ( u, E, L ) will lead to a Bertrand potential.Many choices has been examined and failed. Therefore, at this point, we can saythat the second order differential equation periodicity is the only periodicity thatleads to a Bertrand potential. The simplest example for this case is1 r = s ( u, E, L ) = α ( E, L ) u . (4.15)For the E to be canceled in eq.(4.12), then the following equation must be satisfied c ( E, L ) = − Emα ( E, L ) L = g ( E, L )2 , (4.16)or g ( E, L ) = − Emα ( E, L ) L . (4.17)Accordingly, the potential of eq.(4.12) for this case is V ( r, E, L ) = − L mr + g ( E, L ) L mrα ( E, L ) − L α ( E, L ) mr . (4.18)As a special case, consider the case with g ( E, L ) = 0 , α ( E, L ) = mκ L , (4.19)where κ is coupling constant. Then the potential in eq.(4.18) can be written as V ( r, L ) = − L mr − κr . (4.20)23rom eq.(4.15) and eq.(4.19), the expression of r takes the following form r = mκ L ℘ ( ϕ ; 0 , g ) (4.21)For a closed orbits, g ( E, L ) must take certain values, such that the Weierstrassefunction is 2 πn -periodic. This means that g ( E, L ) = C ( i )1 , C ( i )2 , ... , where C ( i )1 isthe value of g such that the Weierstrasse function is 2 π -periodic, C ( i )2 is the valueof g such that the Weierstrasse function is 4 π -periodic, and so on. The values of i = 1 , , .. starts from i = 1 where g has the minimum value. There are infinitenumber of possible values for g correspond to each 2 πn -periodicity. This meansaccording to eq.(4.17), that the orbit is only closed orbit when the energy is relatedto the angular momentum by the following relation E = − L C ( i ) n m κ , i, n = 1 , , , ... (4.22)The above equation, together with the previous argument mean that; there areinfinite number of possible energy values correspond to any value of the angularmomentum which give closed orbits, however, the energy spectrum is discrete, andit is determined by the values of C ( i ) n . The same results can be reached when solvingdirectly the Newton’s orbit equation for the case of a force derived from the potentialin eq.(4.20). This solution serves as a consistency check. Figures 12 shows that theorbits are closed indeed for certain values of g .Another special case when g = 4 λ , g = 8 λ , α ( E, L ) = mκ L , (4.23)where λ any rational number. Accordingly, the potential of eq.(4.18) can be writtenas V ( r, L ) = − L mr + 4 L λ m κr − κr . (4.24)For this, using eq.(4.15),eq.(4.7), and eq.(4.6) give r = mκ L ℘ ( ϕ ; λ , λ ) = 3 mκ L λ (cid:0) cot( λϕ ) (cid:1) . (4.25)The energy spectrum for this case is E = − L λ m κ , (4.26)Again here, there are infinite number of possible energy values correspond to anyvalue of the angular momentum which give closed orbits, however, the energy spec-trum is discrete, and it is determined by the value of λ , which is a rational number.24 g = −2 g = −1 g = g = g = −4 g = −1 g = g = g = −5 g = −3 g = −1 g = g = −6 g = −3 g = −2 g = −1 π− periodicity 2 π− periodicity 2 π− periodicity 2 π− periodicity 4 π− periodicity 4 π− periodicity 4 π− periodicity 4 π− periodicity 6 π− periodicity 6 π− periodicity 6 π− periodicity 6 π− periodicity 8 π− periodicity 8 π− periodicity 8 π− periodicity 8 π− periodicity Figure 12:
Closed orbits for a particle moving under the action of the potential V ( r, L ) of eq.(4.20), when g = 0 . The first upper row is for π -periodicity Weier-strasse function. The first graph from the left is for the lowest possible value of g ,the second graph at the same row is for the second lowest value of g , and so forth.The second, third and fourth rows are for π , π and π -periodicity respectively. Inall the graphs we put α ( E, L ) = mκ/ L = 1 . In this section, the alternative orbit equation will be used in examples in Bohr Som-merfeld quantization, and stellar kinematics. However, the aim here is to explore theusefulness of the present treatment in future investigations in quantum mechanics,and astrophysics.
In non-relativistic as well as relativistic cases, the Bohr Sommerfeld quantizationhas very few applications, mainly because of the limited number of closed orbits,even when we add cases like the motion of a particle on a cone with a Keplerpotential, or an isotropic harmonic potential at the tip of the cone [16]. It hasbeen demonstrated so far that there are so many potentials with nice features, and25 λ=1/3 λ=2/3 λ=4/5λ=1/2 λ=3/2 λ=5/2λ=1 λ=2 λ=3
Figure 13:
Closed orbits for a particle moving under the action of the potential V ( r, L, λ ) of eq.(4.20) for different values of λ when g = 4 λ . The first upper rowis for π -periodicity Weierstrasse function. The first graph from the left is for thelowest possible value of g , the second graph at the same row is for the second lowestvalue of g , and so forth. The second, third and fourth rows are for π , π and π -periodicity respectively. In all the graphs we put α ( E, L ) = mκ/ L = 1 . lead to closed orbits, which has been highlighted by various figures. Therefore, thenumber of cases under which the Bohr Sommerfeld quantization is applicable canbe hugely increase, provided that the integrals in eq.(1.18) are doable. For angularmomentum, the quantization is straightforward, which is (cid:73) dϕp ϕ = L (cid:90) π dϕ = 2 πL = lh, (5.1)which gives L = (cid:126) l (5.2) (cid:73) p r dr = (cid:90) πn p r drdϕ dϕ = (cid:90) πn p r dgdϕ dϕ = (cid:73) p r dg ( u, E, L ) du du. (5.3)As it is well known p r = ± (cid:114) Em − L r − V ( r ) , (5.4)26herefore, from eq.(5.3) one obtains that (cid:73) dup r dg ( u, E, L ) du = ± (cid:73) dug (cid:48) ( u, E, L ) (cid:115) Em − L g ( u, E, L ) − V ( u, E, L ) . (5.5)An important observation can be made from the previous orbit figures for LSODE-periodicity orbits; when λ is an integer, orbits have a certain numbers of sectors. Inaddition, it takes ∆ ϕ = π/λ for a particle to move from the point where r = r min to r = r max . This can be verified from the fact that r = r (cos( λϕ ). The value of r = r min to r = r max can be obtained by applying the following condition drdϕ = dg ( u, E, L ) dϕ = dg ( u, E, L ) du dudϕ = 0 , (5.6)Accordingly, for optimum r , either dg ( u, E, L ) du = 0 , (5.7)or du dudϕ = 0 , (5.8)or both. The previous discussion leads to (cid:73) p r dg ( u, E, L ) du du = ± λ (cid:90) u max u min dug (cid:48) ( u, E, L ) (cid:115) m ( E − V ( u, E, L )) − L g ( u, E, L ) . (5.9)Using eq.(2.7), the above equation can be written as (cid:73) p r dg ( u, E, L ) du du = ± λ (cid:90) u max u min du Lg (cid:48) ( u, E, L ) g ( u, E, L ) (cid:112) ua ( E, L ) − c ( E, L ) − λ u = n r h. (5.10)To check the validity of this treatment, consider the case of the Kepler potential,then g ( u, E, L ) = 1 /u , λ = 1, a ( E, L ) = mκ/L , and c ( E, L ) = − Em/L . Thevalues of u min = a ( E, L ) − b ( E, L ), and u max = a ( E, L ) + b ( E, L ), where b ( E, L ) isgiven by eq.(3.11). Using the table on integrals [17], the integral in eq.(5.10) can beobtained, then solving for E gives E = − mκ (cid:126) ( n r + l ) = − mκ (cid:126) n (5.11)The simplest example of the quantization of an energy angular momentum de-pendent potential system is the potential in eq.(3.36). For this case, the quantizationcan simply be preformed by replacing κ by κ ( E, L ), then eq.(5.11) can be writtenas. E = − mκ ( E, L ) (cid:126) ( n r + l ) = − mκ ( E, L ) (cid:126) n . (5.12)27f κ ( E, L ) assumed to have the following form κ ( E, L ) = κ EL mκ , (5.13)where κ is the force constant, which is simply in this case κ = Ze / π(cid:15) , and EL / ( mκ ) is a unitless quantity. Accordingly, eq.(5.12) and eq.(5.13) give E = − m (cid:126) κ n l . (5.14)What is interesting about the energy spectrum for this system is; it proportionalto n /l , which is a rational number. Moreover, the energy has infinite degeneracy,because there is infinite values of n and l , such that n /l equal to a certain rationalnumber. Assuming that the gravitational potential is modified to an energy dependent poten-tial, like the one in eq.(3.36). There are infinite possibilities for such modifications.However, we are choosing an example without much scrutiny in relation to observa-tion. let us assume that the Newtonian gravity is modified to V ( r, E, L ) = − k exp (cid:16) EL Mκ (cid:17) r , (5.15)where M is the mass of the source, m is the reduced mass, and k = G mM . Tomake the example even more simple, only circular motion is considered. Then, forthis case, the Newton’s second law gives v c exp (cid:18) mr c v c G M (cid:19) − G Mr c = 0 , (5.16)where r c is the radius of the circular orbit around the center of the disk with velocity v c , and A is unitless constant that can fixed from a possible observation. The aboveequation can be solved for v c , the result is v c = (cid:114) G r c (cid:114) M W [ m/ M ] m (5.17)where W is the Lambert-W function which is also named as log product function.When comparing the result velocity in eq.(5.17) with the one that can be obtained28rom Newtonian gravity, which is v c = (cid:112) G M/r c , we find that the difference be-tween the two solution is very small when m (cid:28) M . This can be shown by expandingthe Lambert-W function. Then eq.(5.17) can be written as v c = (cid:114) G Mr c (cid:16) − m M (cid:17) + O (cid:18) m M (cid:19) . (5.18)In the above equation, it is obvious that the second leading term is small, thereforeone would ask what is the importance of such example. The answer is, the secondterm depends on m , which violates the Einstein’s equivalence principal. Whenthe two bodies in the system haves equal masses, then m = M/
2, which givesthe maximum deviation from the Newtonian gravity, however it is still very small.Moreover, such deviation is not supported by observation.One of the most intriguing phenomena in modern physics is stellar kinematicsin galaxies. According to the observation, stars are moving faster than predicted bycalculations based on the mass of the visible material [18]. To explain mass deficit,the hypothesis of dark matter was proposed [19]. An alternative approach called”modified newtonian dynamics”, or shortly as MOND, which is based on modifyingthe Newton’s laws to give an account for the stellar kinematics [20].At this point, it is far from clear if modifying the Newtonian gravitational po-tential would give a successful description of the stellar kinematics, especially if itis modified to an energy and angular momentum dependent potential. The authorwould like to stress here, that the following example is by no means aiming at pre-senting any alternative, or modification to the gravitational potential V ( r ) = − κ /r .It is aiming at demonstrating that a modification of a potential to an energy and an-gular momentum dependent potential, can lead to velocity distant curve that couldbe explained wrongfully as a sign of a missing mass, or a mass that is not accountedfor.Consider a very thin disk z constitute of particles that are not interacting withtheir neighbors, but moving under the action of the collective gravitational fieldof all the particles in the disk. If the disk rotates in a constant angular velocity ω c = v c /r c , then the equation of motion for a test particle with mass m , and circularorbit is κr c = ˜ G M ( r c ) mr c = mv c r c = m, (5.19)where G is the gravitation constant. For ω c = constant at any point on the rotatingdisk, the density of the disk has to obey the following relation ρ ( r ) = ρ rR (5.20)where ρ is a constant, and R is the radius of the disk. At textbook exercise canshow that the outer mass of the disk has no contribution to the equation of motion29f the test particle. The interior mass at r = r c is given by the following relation M ( r c ) = (cid:90) r c πr ρ R z = 2 πz r c R ρ . (5.21)Accordingly, the equation of motion for a particle with a circular orbit of a radius r c is mv c r c = πρ r c z G mr c , (5.22)which leads to v c = r c (cid:114) πzG ρ R = r c ω c . (5.23)Now assuming that the gravitational potential is modified to an energy dependentpotential, like the one in eq.(5.15). For a circular orbits, the constants of motionare given by the following equations E = mv c , L = mv c r c . (5.24)It is clear that the expression EL / ( M κ ) in eq.(5.15) is unitless. For a particlein a thin disk problem, the modified gravitational potential leads to the followingexpression for the velocity v c = (cid:118)(cid:117)(cid:117)(cid:116) π G r c z ρ W (cid:104) mR πr c zρ (cid:105) mR , (5.25)By plotting v c versus r c from eq.(5.25), and v c versus r c for the Newtonian gravity,it can be realized that near the center of the disk, the two curves differ. An ex-perimentalist who is not informed about any modification in the gravity law, wouldread the two curves as a difference in density, where the matter is less dense nearthe center for the modified case, although ”in fact”, the density is equal for the twocases.The values z , R , ρ in the figure 14 are chosen to be roughly close to the measuredvalues for the Milky way’s galactic disk, while m is equal to the mass of the sunin kg.Indeed the result is opposite to the observation. It shows in fact that thismodification does not offer any explanation for the galactic kinematics. In addition,the modification cause a violation to the Einstein equivalence principal. The onlyjustification for this argument is a mere demonstration that a modification to theNewtonian gravity law could lead to the wrong conclusion about an extra mass, ora mass a deficit. 30 r c v c r c v c m =2 10 kg ρ =10 -17 kg/cm G =6.7 10 -11 m kg -1 s -1 R =4.5 10 m m =2 10 kg G =6.7 10 -11 m kg -1 s -1 ρ =10 -17 kg/cm R =4.5 10 m z =1.8 10 m z =1.8 10 m N e w t o n i a n g r a v i t y M o d i f i e d N e w t o n i a n g r a v i t y M o d i f i e d N e w t o n i a n g r a v i t y N e w t o n i a n g r a v i t y Figure 14:
The graphs for v c velocity (m/sec units) versus distant r c (m units) fornon-interacting bodies on a galactic thin disk with a radius R , and thickness z . Thetest mass is equal to the mass of the sun. The upper panel shows the curves forNewtonian gravity and the modified Newtonian gravity near the center of the disk,in a distant range less than 1 pc. The lower panel show the same curves in distantrange of 10 pc. By tracing the concept of potential function from the early beginning to the presenttime, one can conclude that it is a concept that was brilliantly invented to explainobservation, especially in relation to closed orbits. The Bertrand theorem concludedthat; the Kepler potential, and the isotropic harmonic oscillator potential are theonly systems under which all the orbits are closed. It was never stressed enough inthe physical or mathematical literature that this is only true when the potential isindependent of the initial conditions of motion, which, as we know, determine thevalues of the constants of motion E and L . In other words, the Bertrand theoremis correct only when V ≡ V ( r ) (cid:54) = V ( r, E, L ). In fact it has been proved in thiswork that there are infinitely many energy angular momentum potentials V ( r, E, L )that lead to closed orbits. Reaching to this conclusion was done by generalizing thewell known substitution r = 1 /u in Newton’s orbit equation to a general substitu-tion r = g ( u, E, L ) or r = 1 /s ( u, E, L ) in the equation of motion. This led to thederivation of what is equivalence to Newton’s orbit equation, which we called the”alternative orbit equation”. It is not written in terms second order differential equa-31ion, but in terms of an energy angular momentum dependent potential V ( u, E, L )(see eq.(2.7), and eq.(2.13)). If u is a periodic function ϕ , such that it obeys a secondorder differential equation of the form u (cid:48)(cid:48) = γ ( u, E, L ), where γ ( u, E, L ) depends onthe periodicity of the orbit, then the orbits are closed, because r ( ϕ ) = r ( ϕ + 2 πn ).In the literature, studying different systems using Newton’s orbit equation is doneby using the force or potential as an input. What is different in this work is usingthe periodicity of the orbit given by γ ( u, E, L ), and the choice of r = g ( u, E, L ) or r = 1 /s ( u, E, L ) as inputs, and then find what potentials V ( u, E, L ) = V ( r, E, L )that give closed orbits. Under such consideration, for each periodicity character-ization function given Ω( u, E, L ) = (cid:82) ua γ ( ζ, E, L ) dζ , there are infinite number ofpotentials V ( r, E, L ) for a certain periodicity characterization function Ω( u, E, L )that give closed orbits. Having said that, this does not mean that the choice of r = 1 /s ( u, E, L ) is absolutely random, because r ( ϕ ) must be real and positive forany value of ϕ . This condition is a constant reminder that such problems must behandled with care, because the condition restricts the domain of the energy E for agiven angular momentum L .The most important finding of this work is the expression of the energy an-gular momentum V ( u, E, L ) in eq.(2.7), or an alterative form in eq.(2.13). Oneof them is more suitable for certain application, and the other one is more suit-able for others. However, both of them are called here the alternative orbit equa-tion. In this work, the expression of V ( u, E, L ) in eq.(2.13) is more practical touse in most of the chapters. The first application of the theorem for a certainΩ( u, E, L ) is the for the case of a linear second differential equation orbits, orthe case when u = a ( E, L ) λ − + b ( E, L ) cos λϕ , where λ is a rational number,then Ω( u, E, L ) = a ( E, L ) u − c ( E, L ) − u λ /
2. For such case, the best test forthe theorem is show that it does not contradict with the Bertrand theorem when V ( u, E, L ) = V ( u ). Indeed this is the case when s ( u, E, L ) = u n , where the alter-native orbit equatio can only be satisfied in two cases. The first case when n = 1and λ = 1, which leads to the Kepler potential V ( u ) = V ( r ) = − κ/r . The secondcase when n = 1 / λ = 2, which leads to the isotropic harmonic potential V ( u ) = V ( r ) = mω r /
2. Moreover, for each case, the values of a ( E, L ) and b ( E, L )can be obtained from eq.(2.13), they are exactly the same expressions that can beobtained from solving directly the Newton’s orbit equation.For orbits of such periodicity, a special case solution of the form s ( u, E, L ) =˜ s ( u, E ) /L was considered, then the alternative orbit equation gives the potentialof the form V ( u, E ) = V ( rL − , E ), which is called the rescaling potential. Twoexamples of such potentials were studied carefully. For the first example V ( u ) ∼ u ,an interesting figures for orbits where obtained, like figures 4, 5, moreover, it hasbeen shown that for real positive r , there is a restriction on the value of the energy E (see eq.(3.23)). For the lower possible value of E , the orbits have a certain point,or points for which r → ∞ , this is shown by figure 6. As for the second example,the solution is more complicated as V ( rL − , E ) contains the term arccsch( cr ) (see32q.(3.31)). For such potential, solving the Newton’s orbit equation to get the orbitgraphs is a very difficult task, or maybe not possible. However, r ( ϕ ) is an input inour treatment, therefore plotting the orbits is possible, as one cane see in figures8,9, which is a proof of the usefulness of this approach.For the same Ω( u, E, L ), a more general solution when s ( u, E, L ) is not factored.A simple example in this context is s ( u, E, L ) = u/ ( α ( E, L ) + α ( E, L ) u ) leads to apotential of the form V ( r, E, L ) = (cid:80) n =1 A n ( E, L ) r − n , where A n ( E, L ) are constantsgiven by eq.(3.35). The possibility of reducing this potential to a two term-potentialwas investigated. It has been proved that this is not always possible. The onlyphysical case are three cases, the first case when A = A = 0, then the orbits areshown by figure 10, the second case when A = A = 0, then the orbits are shownby figure 11, the third case when A = A = 0, then the orbits are circles. Forone-term potential, it was proved that this is only possible if α ( E, L ) = 0, then thepotential is for what we call a modified Kepler problem, with κ → κ ( E, L ), whichis also leads to a closed orbit for arbitrary κ ( E, L ).The second application of the theorem is on the Weierstrasse periodicity orbitswith Ω( u, E, L ) = 2 u − g ( E, L ) u − + g − . For this application, one simpleexamples where studied, which is for s ( u, E, L = α ( E, L ) /u , then the potential isgiven by eq.(4.18). Case in this example is when g = 0, then the value of g thatclosed the orbit are given by figure 12. This leads to an interesting result, whichis a discrete energy spectrum E = − L C ( i ) n m − κ − as a condition for the orbit tobe closed. Another important issue here is the potential for this case V ( r, E, L ) = − ( L / mr ) − κ/r , is a system that could be solved directly using the Newton’sorbit equation. The result is exactly the same as the one the we got using thealternative orbit equation. This is an important checking for the validity of ourapproach. The second case for this example is when g = 4 λ /