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A Quantized Hill’s Dynamical System

A Quantized Hill’s Dynamical System Hindawi Advances in Astronomy Volume 2021, Article ID 9963761, 7 pages https://doi.org/10.1155/2021/9963761 Research Article 1 2 3 Elbaz I. Abouelmagd , Vassilis S. Kalantonis , and Angela E. Perdiou Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan–11421, Cairo, Egypt Department of Electrical and Computer Engineering, University of Patras, Patras 26 504, Greece Department of Civil Engineering, University of Patras, Patras 26 504, Greece Correspondence should be addressed to Angela E. Perdiou; aperdiou@upatras.gr Received 10 March 2021; Revised 27 April 2021; Accepted 25 May 2021; Published 7 June 2021 Academic Editor: Jagadish Singh Copyright © 2021 Elbaz I. Abouelmagd et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present a modified version of Hill’s dynamical system that is called the quantized Hill’s three-body problem in the sense that the equations of motion for the classical Hill’s problem are now derived under the effects of quantum corrections. To do so, the position variables and the parameters that correspond to the quantum corrections of the respective quantized three-body problem are scaled appropriately, and then by taking the limit when the parameter of mass ratio tends to zero, we obtain the relevant equations of motion for the spatial quantized Hill’s problem. Furthermore, the Hamiltonian formula and related equations of motion are also derived. Robe problems (see, e.g., [13, 14] and references therein). 1. Introduction However, the simplest of its versions is Hill’s problem, which can be treated as a perturbed two-body problem. In par- In the analysis of dynamical systems that deal with celestial objects, the restricted three-body problem plays a funda- ticular, this problem is considered to be a limiting case of the mental role. (e problem gains its importance from the restricted three-body problem when the parameter mass μ variety of its modifications that approximate different real tends to zero, and it may be used to study the satellite motion systems [1, 2]. In addition, it can be applied and used in both around a planet [15]. A considerable study on the circular stellar and planetary dynamics as well as in space missions Hill’s problem has been established by Henon ´ [16, 17], where [3, 4]. Also, this problem can be effectively used to determine he determined the main families of periodic orbits, revealed the possibility of sub-Jovian and terrestrial planets [5, 6]. the phase space by means of surface of sections of the Due to its extensive astronomical applications, considerable Poincare´ map, and found the stability regions in the pa- rameters’ plane. Recently, for the same problem, Lara et al. in variants have been proposed in order to study a test particle in solar and planetary systems [7–9]. For example, the [18] have employed a normalization approach in complex variables to compute a single perturbation solution with modification in which the more massive body is a source of radiation and the smaller one is either oblate or triaxial body enough accuracy. (e solution captures the main four motivates us to apply this perturbed version in our solar families of periodic orbits for the Hill problem originated system. (is modification may be more realistic than the from the libration points. (ey have also extended the so- classical one since in the solar system, the Sun is radiating lution validity to energy values. In addition, Nishimura et al. and some planets are not spherical but sufficiently oblate (or in [19] used the same model to study spacecraft orbital triaxial) bodies [10–12]. motion. (ey studied 3-dimensional distant retrograde On the other hand, the classical restricted problem has orbits and also found a sufficient condition for a closed orbit some simpler modified versions, instead of the aforemen- to be unstable. To do so, the authors transformed the relative equations of motion into a time-independent form by using tioned complex perturbed models, such as Sitnikov and 2 Advances in Astronomy Fourier series. Also recently, Kalantonis in [20] have studied equations of a test particle within the frame of the quantized the families of spatial periodic orbits bifurcating from the model are given by vertical self-resonant periodic orbits of the basic families of simple planar periodic orbits. € ξ − 2nη _ � Γ ξ , η , ζ 􏼁 , 1 1 1 1 1 1 zξ In the framework of Hill’s problem, Markellos et al. in [21, 22] have proposed the models in which the radiation or oblateness of the primaries are also considered. (e pro- _ η + 2nξ � Γ ξ , η , ζ 􏼁 , (1) 1 1 1 1 1 1 zη posed models were used in order to find estimates for the maximum possible distance of Hill stable direct orbits around the small primary and to estimate the maximum € ζ � Γ ξ ,η ,ζ , 1 1 1 1 1 zζ sizes of accretion disks in binary stars. For Hill’s problem where the larger primary is a source of radiation, Kalantonis where Γ is the effective potential, and it is read as et al. in [23] considered homoclinic connections at both the Lyapunov planar periodic orbits and collinear equilibrium 2 2 2 points. Markakis et al. in [24] proposed a respective Hill Γ ξ ,η ,ζ 􏼁 � n 􏼐ξ + η 􏼑 1 1 1 1 1 1 model by combining the radiation pressure and oblateness (2) effects. (ey found approximate expressions for the loca- +(1 − μ)Γ ϱ + μΓ ϱ , 􏼁 􏼁 11 1 21 2 tions of equilibrium points and explored their linear sta- bility. Also, by applying singular perturbations methods, 1 Q Q i1 i2 they determined approximate expressions for the Lyapunov Γ ϱ􏼁 � 􏼠1 + + 􏼡, i � 1, 2. (3) i1 i ϱ ϱ i i orbits emanating from the collinear points in both the co- i planar and spatial cases. For the same problem, Perdiou et al. (e derivatives of this potential with respect to ξ , η , 1 1 [25] studied the network evolution of the basic families, and ζ are determined their stability as well as the stability regions of retrograde satellites in the plane of initial conditions by Γ � (x + μ)φ ϱ 􏼁 + (x + μ − 1)φ ϱ 􏼁 , 1 1 1 2 2 means of appropriate Poincare´ surface of sections. (e el- zξ liptical Hill’s problem is constructed by using the same assumptions of circular Hill’s problem, but assume that the Γ � y φ ϱ + φ ϱ , 􏼂 􏼁 􏼁 􏼃 (4) planet moves on an elliptic orbit around the Sun. (is model 1 1 1 2 2 zη was obtained by Ichtiaroglou in [26] in which few families of periodic orbits were computed. A further study was Γ � z􏽨φ ϱ 􏼁 + φ ϱ 􏼁 − n 􏽩, addressed by Ichtiaroglou and Voyatzis in [27] where they 1 1 1 2 2 zζ explored the stability of periodic orbits. Also, for the elliptic case, Voyatzis et al. in [28] computed a large set of families of where periodic orbits together with their linear stability and classify 2 2 2 2 them according to their resonance condition. ϱ � ξ + μ 􏼁 + η + ζ , 1 1 1 In this paper, we derive a quantum version for Hill’s 2 2 2 ϱ � ξ + μ − 1 + η + ζ , (5) 2 1 1 1 problem that comes from the quantized three-body problem 2 2 2 2 introduced by Alshaery and Abouelmagd [29]. (is special ϱ � ξ + η + ζ , 3 1 1 1 variant is called the quantized Hill’s problem, and its are the distances of the massless body from the two pri- equations of motion are obtained in a similar way as Hill’s maries, while the mean motion n is given by problem is derived from the classical restricted three-body problem; however, the calculations are not direct and further (6) assumptions are considered, which incorporate the per- n � 1 + 2Q + 3Q . 1 2 turbations of quantum corrections. In particular, our work Relations (1)–(6) represent the equations of motion of a presents the equations of motions in both the planar and circular restricted three-body problem in a synodic reference spatial cases as well as the Hamiltonian formula with the frame (see [29] for details). related equations. More precisely, our paper is organized as We would like to refer here that φ (ϱ ) and φ (ϱ ) are follows. First, in Section 2, the quantized three-body 1 1 2 2 explicit functions in the distances ϱ and ϱ , which are read 1 2 problem is discussed. In Section 3, the Hill version of the as latter is systematically derived, while the relevant Hamil- tonian approach is obtained in Section 4. Finally, in Section 1 2Q 3Q 5, we summarize our work and conclude. 11 12 φ ϱ 􏼁 � (1 − μ)􏼢n − − − 􏼣, 1 1 3 4 5 ϱ ϱ ϱ 1 1 1 (7) 2. Quantized Three-Body Problem 1 2Q 3Q 2 21 22 φ ϱ � μ n − − − , 􏼁 􏼢 􏼣 2 2 3 4 5 We will accept the notations and nomenclature of the spatial ϱ ϱ ϱ 2 2 2 quantized restricted three-body problem, which is studied by Alshaery and Abouelmagd in [29]. In this context, the where Advances in Astronomy 3 Q � n 􏼐R + R 􏼑, 1 1 m m 1 2 ξ − 2nη _ � Γ(ξ, η,ζ ), (8) zξ Q � n 􏼐l 􏼑 , 2 2 p η € + 2nξ � Γ(ξ,η,ζ ), (12) zη Q � k 􏼐R + R 􏼑, 11 1 m m 2 z (9) Q � Q � k 􏼐l 􏼑 , ζ � Γ(ξ, η,ζ ), 12 22 2 p zζ Q � k 􏼐R + R 􏼑. 21 3 m m where (e quantities Q , Q , Q , Q , Q , and Q , which are 1 2 11 12 21 22 2 2 2 Γ(ξ, η,ζ ) � n 􏽨(ξ − μ + 1) + η 􏽩 identified by equations 8 and (9), utilize quantum correc- (13) tions. In particular, the first two quantities are due to the mean motion, while the last four are corrections of the + (1 − μ)Γ ρ + μΓ ρ , 􏼁 􏼁 11 1 21 2 potential, which the perturbed test particle motion governs. It is clear that the classical (unperturbed) motion can be while the relevant distances are now obtained when each quantity is assigned to zero. Details for 2 2 2 2 ρ � (1 + ξ) + η + ζ , the estimated quantum corrections from the basic quantum (14) 2 2 2 2 ρ � ξ + η + ζ . principles as well as their clear quantum mechanical meaning were introduced in [30–32]. Also, R , R , and R m m m 1 2 We now follow again Szebehely in [15] to scale the are the gravitational radii of the primaries m and m and the 1 2 variables by introducing massless body m, respectively, while l is the Planck length. 1/3 Moreover, the numbers n , n , k , k , and k can be esti- 1 2 1 2 3 ξ � μ x, mated from the analysis of Feynman diagrams. Generally, 1/3 (15) η � μ y, the values of these numbers depend on different definitions; therefore, they are different in both sign and magnitude 1/3 ζ � μ z. [30, 31]. In fact, the constants Q , Q , and Q measure the 1 11 21 size of the relativistic effect or post-Newtonian approxi- (e aforementioned scale preserves the magnitude of mation, while Q , Q , and Q measure control quantum 2 12 22 Coriolis and centrifugal terms in the same order in the correction contributions. However, all these effects tend to previous equations. Substituting (15) into equations zero in the case of large distances. (12)–(14), we obtain Utilizing equations (2), (4), and (7), we obtain from (1) − 2/3 the equation of the Jacobi integral in the form x € − 2ny _ � μ Ω(x, y, z), zx 2 2 _ _ 2Γ ξ , η , ζ 􏼁 − 􏼒ξ + η _ + ζ 􏼓 � C, (10) 1 1 1 1 1 1 1 − 2/3 y € + 2nx _ � μ Ω(x, y, z), (16) zy where C is the constant of integration or alternatively the − 2/3 well-known Jacobi constant, which can be utilized for the € z � μ Ω(x, y, z), zz study of the invariant manifolds of the considered system. where 3. Hill Version of Quantized Three- 1 2 2 1/3 2/3 2 Ω(x, y, z) � n 􏼔􏼐μ x − μ + 1􏼑 + μ y 􏼕 Body Problem (17) + (1 − μ)Γ r 􏼁 + μΓ r 􏼁 , In order to find the quantized Hill version, we follow the 11 1 21 2 transformation of Szebehely in [15]. First, we subject the and the scaled distances are equations of motion of the quantized restricted three-body 2 1/3 2/3 2 problem given by system (1) and the related relations to a r � 1 + 2μ x + μ r , translation along the ξ -axis, so we let in this way the co- 2 2/3 2 (18) r � μ r , ordinates center moves to the mass center of the smaller 2 2 2 2 r � x + y + z . primary. (erefore, the relations between the old (ξ , η , ζ ) 1 1 and new (ξ, η,ζ ) coordinates are given by We utilize equations (16) and (17) and let the mass ξ � ξ − μ + 1, parameter μ tends to zero; the existing this limit leads to Hill’s equations. (us, equation (16) can now be rewritten as η � η, (11) x € � L + L + L + L , 10 11 12 13 ζ � ζ . y € � L + L + L + L , (19) 20 21 22 23 Utilizing these relations in equations (1), (2), and (5), we z � L + L + L , get 31 32 33 4 Advances in Astronomy where the limits L , i � 1, 2, 3 and j � 0, 1, 2, 3, are given by they are not defined at μ � 0. (e first two limits have a ij pole of order one due to some particular terms with co- L � lim 􏽨2ny _ + n x􏽩, 1/3 efficients 1/μ , while the last three limits have poles of μ⟶0 order one and two due to some specific terms with co- 1/3 2/3 1 − μ efficients 1/μ and 1/μ , respectively. L � lim κ r , x􏼁 , 11 1 1/3 μ⟶0 On the other hand, the parameters of quantum cor- 1/3 rections are very small and can be scaled by the factors μ (20) 2/3 1 − μ and μ . In this sense, we can scale R , R , R , and l . 1/3 m m m p 1 2 L � −lim α r 􏼁 􏼐1 + μ x􏼑, 12 μ 1 1/3 Specifically, this scale may be considered by taking μ⟶0 1/3 2/3 1/3 1/3 Q � μ α , Q � μ α , Q � μ α , Q � μ α , 1 1 2 2 11 11 21 21 2/3 2/3 Q � μ α , and Q � μ α . After the previous dis- 12 12 22 22 L � −lim c (r)x, 13 μ μ⟶0 cussion and by utilizing these scaled quantum corrections in 1/3 equations (20)–(22), the singularities at the poles 1/μ and with 2/3 1/μ can be removed, so L � −lim 􏽨2nx − n y􏽩, 20 L � 2ny _ + n x, μ⟶0 L � 2x + 2α , 11 1 L � −lim λ r , r , 21 1 (24) μ⟶0 L � −2α , (21) 12 11 L � −lim (1 − μ)α r 􏼁 y, 22 μ 1 μ⟶0 L � −f (r)x, 13 r L � −lim β (r)y, μ⟶0 L � −2x _ + y, L � −h (r), 21 r L � −lim λ r , r z, (25) 31 1 μ⟶0 L � 0, L � −lim (1 − μ)α r 􏼁 z, L � −g (r)y, 32 μ 1 (22) 23 r μ⟶0 L � −lim β (r)z, L � −h (r)z, μ⟶0 31 r L � 0, (26) while we have abbreviated L � −g (r)z, 33 r 1/3 1 + μ x κ r , x􏼁 � n − , where 1 2α 3α 21 22 1 − μ 1 f (r) � 􏼠1 + + 􏼡, 3 2 λ r , r􏼁 � + , 1 r r 3 3 r r 1 3α (27) 2Q 3Q g (r) � 􏼒2α + 􏼓, 11 12 r 21 (23) α r 􏼁 � + , μ 1 4 5 r r 1 1 h (r) � 1 + . 2Q 3Q r 21 22 β (r) � + , 1/3 4 2/3 5 μ r μ r Substituting equations (24)–(26) into system (19), we get x € − 2y _ � 3x + 2 α − α 􏼁 − f (r)x, c (r) � + β (r). 1 11 r μ μ y € + 2x _ � −f (r)y, (28) From equation (18), we observe that r → 1 and r → 0 1 2 € z � −z − f (r)z. 1/3 r when μ→ 0. Furthermore, r is of order O(μ ), thus k k (1/r ) → 1 and (1/r ) is undefined, where k is a positive 1 2 (e last system represents the spatial quantized Hill integer. We also remark here that, if the parameters of problem; it is the limiting case, which is acquired from the quantum corrections are ignored, i.e., Q � Q � 0 and 1 2 spatial quantized restricted three-body problem, which was Q � Q � Q � Q � 0, then the limits (20)–(22) 11 12 21 22 firstly presented and studied by Alshaery and Abouelmagd converge, hence exist, and system (19) is reduced to the [29]. System (28) can be rewritten in a similar manner as that classical planar and spatial Hill three-body systems of the restricted three-body problem, i.e., [15, 33]. In the case where these parameters are not equal x € − 2y _ � U (x, y, z), to zero, we get some divergent limits, such as L , L , L , L , and L . (e limits that diverge corre- y € + 2x _ � U 11 12 13 23 33 (x, y, z), (29) spond to the involved functions, which depend on the z � U (x, y, z), parameter μ, possessing singularities and in particular, z Advances in Astronomy 5 where as well as for formulations of quantum mechanics; however, the Hamiltonian approach provides deeper insights in 2 2 (30) U(x, y, z) � 􏽨3x + 4 α − α 􏼁 x − z 􏽩 + w(r). several fundamental features of some astronomical dy- 1 11 namical systems, such as planetary orbits in celestial me- chanics [34, 35]. Utilizing equations (29) and (30), the Jacobi integral is read as C � 2U − V , where explicitly it has the form In our system, the time evolution is defined by the following Hamilton’s equations: 2 2 C � 􏽨3x + 4 α − α 􏼁 x − z 􏽩 q 1 11 (31) zH q _ � , + 2w(r) − V , χ zp 2 2 2 2 (33) with V � x _ + y _ + z _ , while we have abbreviated zH p � − , 1 α α 21 22 zq w(r) � 1 + + . (32) 􏼠 􏼡 r r where χ � (x, y, z) and the Hamiltonian H is given by H � P + 􏼐q p − q p 􏼑 + Υ(q) y x x y 4. Hamiltonian Approach (34) In Hamiltonian approach, a dynamical system is described 2 2 2 − 􏽨2q − q − q + 4 α − α 􏼁 q 􏽩, by a set of canonical coordinates (q, p) where they corre- x y z 1 11 x spond to the n-dimensional vectors q � (q , q , q , . . . , q ) 1 2 3 n with and p � (p , p , p , . . . , p ), while each of the aforemen- 1 2 3 n 2 2 2 2 tioned components is indexed to the frame of reference of P � p + p + p , x y z the respective system. (e components q , i � 1, 2, 3, . . . , n, are known as the generalized coordinates and are selected in (35) 1 α α 21 22 order to remove the constraints or to take into consideration Υ(q) � − 􏼠1 + + 􏼡. 3 2 q q the symmetry characteristics of the system, while p are their conjugate momenta. In classical mechanics, the evolution of Utilizing equations (33) and (34), the equations of time is determined by calculating the total force, which is motion of the spatial quantized Hill model with Hamiltonian exerted on each one of the involved bodies, so the time evo- formula can be written as lutions of both positions and velocities are obtained by using Newton’s second law. However, in Hamiltonian mechanics, the q _ � p + q , x x y evolution of time is computed by finding the relevant Ham- q _ � p − q , y y x iltonian function in the generalized coordinates and then using it into the corresponding Hamilton’s equations. q _ � p , z z (36) (e latter method is tantamount to that which is used to p � p + 2􏼂q + α − α 􏼁 􏼃 − f (q)q x y x 1 11 q x the Lagrangian approach. In fact, for the same generalized momenta, both methods result to the same equations, but p � −p − q − f (q)q , y x y q y the main reason to use Hamiltonian approach, instead that p � −q − f (q)q , of the Lagrangian, comes from the symplectic features of a z z q z Hamiltonian system. In fact, Hamilton’s equations com- 2 2 2 2 prised 2n first-order differential equations, while Lagrange’s where q � q + q + q . Furthermore the Hamiltonian can x y z equations are constituted by n second-order differential be rewritten as equations. Although Hamilton’s equations do not generally H � H + σ H + σ H , (37) 0 1 1 2 2 reduce the difficulty of determining analytical solutions, they may offer some crucial theoretical results due to the fact that where the coordinates and momenta are independent variables with nearly symmetric roles. In addition, if the system H � P + 􏼐q p − q p 􏼑 possesses a kind of symmetry for which a coordinate does 0 y x x y not appear in the Hamiltonian, the relevant momentum is conserved and the corresponding coordinate can be 1 1 2 2 − 􏼐3q − q 􏼑 − , neglected by the other equations. (is results in reducing the 2 r number of coordinates of the dynamical system from n to (38) n − 1, while in the framework of the Lagrangian approach, all 12 H � −2β q − , 1 11 x the generalized velocities remain in the Lagrangian; there- fore, we still have to deal with a system of equations in n coordinates, which must be handled. Both approaches are of 22 H � − . fundamental importance in the study of classical mechanics r 6 Advances in Astronomy Here, σ β � α − α , σ β � α , and σ β � α . Data Availability 1 11 1 11 1 12 21 2 22 22 Within the frame of quantum corrections, σ is of order 2 3 All data used to support the findings of this work are in- O(1/c ) and σ is of order O(1/c ), where, as usual, c denotes corporated within the manuscript. the speed of light. (ereby, the second and third terms in 2 3 equation (37) are of order O(1/c ) and O(1/c ), respectively. Conflicts of Interest (us, the perturbed Hamiltonian (37) can be reduced classical to one of the unperturbed Hamiltonian as (e authors declare that there are no conflicts of interest regarding the publication of this paper. H � H + O . (39) 􏼠 􏼡 References (e obtained relation (39) for Hamiltonian can be used to extend the results of the classical Hill’s problem to the [1] N. Pathak, V. O. (omas, and E. I. Abouelmagd, “(e per- turbed photogravitational restricted three-body problem: quantized problem. 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A Quantized Hill’s Dynamical System

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Hindawi Advances in Astronomy Volume 2021, Article ID 9963761, 7 pages https://doi.org/10.1155/2021/9963761 Research Article 1 2 3 Elbaz I. Abouelmagd , Vassilis S. Kalantonis , and Angela E. Perdiou Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan–11421, Cairo, Egypt Department of Electrical and Computer Engineering, University of Patras, Patras 26 504, Greece Department of Civil Engineering, University of Patras, Patras 26 504, Greece Correspondence should be addressed to Angela E. Perdiou; aperdiou@upatras.gr Received 10 March 2021; Revised 27 April 2021; Accepted 25 May 2021; Published 7 June 2021 Academic Editor: Jagadish Singh Copyright © 2021 Elbaz I. Abouelmagd et al. (is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we present a modified version of Hill’s dynamical system that is called the quantized Hill’s three-body problem in the sense that the equations of motion for the classical Hill’s problem are now derived under the effects of quantum corrections. To do so, the position variables and the parameters that correspond to the quantum corrections of the respective quantized three-body problem are scaled appropriately, and then by taking the limit when the parameter of mass ratio tends to zero, we obtain the relevant equations of motion for the spatial quantized Hill’s problem. Furthermore, the Hamiltonian formula and related equations of motion are also derived. Robe problems (see, e.g., [13, 14] and references therein). 1. Introduction However, the simplest of its versions is Hill’s problem, which can be treated as a perturbed two-body problem. In par- In the analysis of dynamical systems that deal with celestial objects, the restricted three-body problem plays a funda- ticular, this problem is considered to be a limiting case of the mental role. (e problem gains its importance from the restricted three-body problem when the parameter mass μ variety of its modifications that approximate different real tends to zero, and it may be used to study the satellite motion systems [1, 2]. In addition, it can be applied and used in both around a planet [15]. A considerable study on the circular stellar and planetary dynamics as well as in space missions Hill’s problem has been established by Henon ´ [16, 17], where [3, 4]. Also, this problem can be effectively used to determine he determined the main families of periodic orbits, revealed the possibility of sub-Jovian and terrestrial planets [5, 6]. the phase space by means of surface of sections of the Due to its extensive astronomical applications, considerable Poincare´ map, and found the stability regions in the pa- rameters’ plane. Recently, for the same problem, Lara et al. in variants have been proposed in order to study a test particle in solar and planetary systems [7–9]. For example, the [18] have employed a normalization approach in complex variables to compute a single perturbation solution with modification in which the more massive body is a source of radiation and the smaller one is either oblate or triaxial body enough accuracy. (e solution captures the main four motivates us to apply this perturbed version in our solar families of periodic orbits for the Hill problem originated system. (is modification may be more realistic than the from the libration points. (ey have also extended the so- classical one since in the solar system, the Sun is radiating lution validity to energy values. In addition, Nishimura et al. and some planets are not spherical but sufficiently oblate (or in [19] used the same model to study spacecraft orbital triaxial) bodies [10–12]. motion. (ey studied 3-dimensional distant retrograde On the other hand, the classical restricted problem has orbits and also found a sufficient condition for a closed orbit some simpler modified versions, instead of the aforemen- to be unstable. To do so, the authors transformed the relative equations of motion into a time-independent form by using tioned complex perturbed models, such as Sitnikov and 2 Advances in Astronomy Fourier series. Also recently, Kalantonis in [20] have studied equations of a test particle within the frame of the quantized the families of spatial periodic orbits bifurcating from the model are given by vertical self-resonant periodic orbits of the basic families of simple planar periodic orbits. € ξ − 2nη _ � Γ ξ , η , ζ 􏼁 , 1 1 1 1 1 1 zξ In the framework of Hill’s problem, Markellos et al. in [21, 22] have proposed the models in which the radiation or oblateness of the primaries are also considered. (e pro- _ η + 2nξ � Γ ξ , η , ζ 􏼁 , (1) 1 1 1 1 1 1 zη posed models were used in order to find estimates for the maximum possible distance of Hill stable direct orbits around the small primary and to estimate the maximum € ζ � Γ ξ ,η ,ζ , 1 1 1 1 1 zζ sizes of accretion disks in binary stars. For Hill’s problem where the larger primary is a source of radiation, Kalantonis where Γ is the effective potential, and it is read as et al. in [23] considered homoclinic connections at both the Lyapunov planar periodic orbits and collinear equilibrium 2 2 2 points. Markakis et al. in [24] proposed a respective Hill Γ ξ ,η ,ζ 􏼁 � n 􏼐ξ + η 􏼑 1 1 1 1 1 1 model by combining the radiation pressure and oblateness (2) effects. (ey found approximate expressions for the loca- +(1 − μ)Γ ϱ + μΓ ϱ , 􏼁 􏼁 11 1 21 2 tions of equilibrium points and explored their linear sta- bility. Also, by applying singular perturbations methods, 1 Q Q i1 i2 they determined approximate expressions for the Lyapunov Γ ϱ􏼁 � 􏼠1 + + 􏼡, i � 1, 2. (3) i1 i ϱ ϱ i i orbits emanating from the collinear points in both the co- i planar and spatial cases. For the same problem, Perdiou et al. (e derivatives of this potential with respect to ξ , η , 1 1 [25] studied the network evolution of the basic families, and ζ are determined their stability as well as the stability regions of retrograde satellites in the plane of initial conditions by Γ � (x + μ)φ ϱ 􏼁 + (x + μ − 1)φ ϱ 􏼁 , 1 1 1 2 2 means of appropriate Poincare´ surface of sections. (e el- zξ liptical Hill’s problem is constructed by using the same assumptions of circular Hill’s problem, but assume that the Γ � y φ ϱ + φ ϱ , 􏼂 􏼁 􏼁 􏼃 (4) planet moves on an elliptic orbit around the Sun. (is model 1 1 1 2 2 zη was obtained by Ichtiaroglou in [26] in which few families of periodic orbits were computed. A further study was Γ � z􏽨φ ϱ 􏼁 + φ ϱ 􏼁 − n 􏽩, addressed by Ichtiaroglou and Voyatzis in [27] where they 1 1 1 2 2 zζ explored the stability of periodic orbits. Also, for the elliptic case, Voyatzis et al. in [28] computed a large set of families of where periodic orbits together with their linear stability and classify 2 2 2 2 them according to their resonance condition. ϱ � ξ + μ 􏼁 + η + ζ , 1 1 1 In this paper, we derive a quantum version for Hill’s 2 2 2 ϱ � ξ + μ − 1 + η + ζ , (5) 2 1 1 1 problem that comes from the quantized three-body problem 2 2 2 2 introduced by Alshaery and Abouelmagd [29]. (is special ϱ � ξ + η + ζ , 3 1 1 1 variant is called the quantized Hill’s problem, and its are the distances of the massless body from the two pri- equations of motion are obtained in a similar way as Hill’s maries, while the mean motion n is given by problem is derived from the classical restricted three-body problem; however, the calculations are not direct and further (6) assumptions are considered, which incorporate the per- n � 1 + 2Q + 3Q . 1 2 turbations of quantum corrections. In particular, our work Relations (1)–(6) represent the equations of motion of a presents the equations of motions in both the planar and circular restricted three-body problem in a synodic reference spatial cases as well as the Hamiltonian formula with the frame (see [29] for details). related equations. More precisely, our paper is organized as We would like to refer here that φ (ϱ ) and φ (ϱ ) are follows. First, in Section 2, the quantized three-body 1 1 2 2 explicit functions in the distances ϱ and ϱ , which are read 1 2 problem is discussed. In Section 3, the Hill version of the as latter is systematically derived, while the relevant Hamil- tonian approach is obtained in Section 4. Finally, in Section 1 2Q 3Q 5, we summarize our work and conclude. 11 12 φ ϱ 􏼁 � (1 − μ)􏼢n − − − 􏼣, 1 1 3 4 5 ϱ ϱ ϱ 1 1 1 (7) 2. Quantized Three-Body Problem 1 2Q 3Q 2 21 22 φ ϱ � μ n − − − , 􏼁 􏼢 􏼣 2 2 3 4 5 We will accept the notations and nomenclature of the spatial ϱ ϱ ϱ 2 2 2 quantized restricted three-body problem, which is studied by Alshaery and Abouelmagd in [29]. In this context, the where Advances in Astronomy 3 Q � n 􏼐R + R 􏼑, 1 1 m m 1 2 ξ − 2nη _ � Γ(ξ, η,ζ ), (8) zξ Q � n 􏼐l 􏼑 , 2 2 p η € + 2nξ � Γ(ξ,η,ζ ), (12) zη Q � k 􏼐R + R 􏼑, 11 1 m m 2 z (9) Q � Q � k 􏼐l 􏼑 , ζ � Γ(ξ, η,ζ ), 12 22 2 p zζ Q � k 􏼐R + R 􏼑. 21 3 m m where (e quantities Q , Q , Q , Q , Q , and Q , which are 1 2 11 12 21 22 2 2 2 Γ(ξ, η,ζ ) � n 􏽨(ξ − μ + 1) + η 􏽩 identified by equations 8 and (9), utilize quantum correc- (13) tions. In particular, the first two quantities are due to the mean motion, while the last four are corrections of the + (1 − μ)Γ ρ + μΓ ρ , 􏼁 􏼁 11 1 21 2 potential, which the perturbed test particle motion governs. It is clear that the classical (unperturbed) motion can be while the relevant distances are now obtained when each quantity is assigned to zero. Details for 2 2 2 2 ρ � (1 + ξ) + η + ζ , the estimated quantum corrections from the basic quantum (14) 2 2 2 2 ρ � ξ + η + ζ . principles as well as their clear quantum mechanical meaning were introduced in [30–32]. Also, R , R , and R m m m 1 2 We now follow again Szebehely in [15] to scale the are the gravitational radii of the primaries m and m and the 1 2 variables by introducing massless body m, respectively, while l is the Planck length. 1/3 Moreover, the numbers n , n , k , k , and k can be esti- 1 2 1 2 3 ξ � μ x, mated from the analysis of Feynman diagrams. Generally, 1/3 (15) η � μ y, the values of these numbers depend on different definitions; therefore, they are different in both sign and magnitude 1/3 ζ � μ z. [30, 31]. In fact, the constants Q , Q , and Q measure the 1 11 21 size of the relativistic effect or post-Newtonian approxi- (e aforementioned scale preserves the magnitude of mation, while Q , Q , and Q measure control quantum 2 12 22 Coriolis and centrifugal terms in the same order in the correction contributions. However, all these effects tend to previous equations. Substituting (15) into equations zero in the case of large distances. (12)–(14), we obtain Utilizing equations (2), (4), and (7), we obtain from (1) − 2/3 the equation of the Jacobi integral in the form x € − 2ny _ � μ Ω(x, y, z), zx 2 2 _ _ 2Γ ξ , η , ζ 􏼁 − 􏼒ξ + η _ + ζ 􏼓 � C, (10) 1 1 1 1 1 1 1 − 2/3 y € + 2nx _ � μ Ω(x, y, z), (16) zy where C is the constant of integration or alternatively the − 2/3 well-known Jacobi constant, which can be utilized for the € z � μ Ω(x, y, z), zz study of the invariant manifolds of the considered system. where 3. Hill Version of Quantized Three- 1 2 2 1/3 2/3 2 Ω(x, y, z) � n 􏼔􏼐μ x − μ + 1􏼑 + μ y 􏼕 Body Problem (17) + (1 − μ)Γ r 􏼁 + μΓ r 􏼁 , In order to find the quantized Hill version, we follow the 11 1 21 2 transformation of Szebehely in [15]. First, we subject the and the scaled distances are equations of motion of the quantized restricted three-body 2 1/3 2/3 2 problem given by system (1) and the related relations to a r � 1 + 2μ x + μ r , translation along the ξ -axis, so we let in this way the co- 2 2/3 2 (18) r � μ r , ordinates center moves to the mass center of the smaller 2 2 2 2 r � x + y + z . primary. (erefore, the relations between the old (ξ , η , ζ ) 1 1 and new (ξ, η,ζ ) coordinates are given by We utilize equations (16) and (17) and let the mass ξ � ξ − μ + 1, parameter μ tends to zero; the existing this limit leads to Hill’s equations. (us, equation (16) can now be rewritten as η � η, (11) x € � L + L + L + L , 10 11 12 13 ζ � ζ . y € � L + L + L + L , (19) 20 21 22 23 Utilizing these relations in equations (1), (2), and (5), we z � L + L + L , get 31 32 33 4 Advances in Astronomy where the limits L , i � 1, 2, 3 and j � 0, 1, 2, 3, are given by they are not defined at μ � 0. (e first two limits have a ij pole of order one due to some particular terms with co- L � lim 􏽨2ny _ + n x􏽩, 1/3 efficients 1/μ , while the last three limits have poles of μ⟶0 order one and two due to some specific terms with co- 1/3 2/3 1 − μ efficients 1/μ and 1/μ , respectively. L � lim κ r , x􏼁 , 11 1 1/3 μ⟶0 On the other hand, the parameters of quantum cor- 1/3 rections are very small and can be scaled by the factors μ (20) 2/3 1 − μ and μ . In this sense, we can scale R , R , R , and l . 1/3 m m m p 1 2 L � −lim α r 􏼁 􏼐1 + μ x􏼑, 12 μ 1 1/3 Specifically, this scale may be considered by taking μ⟶0 1/3 2/3 1/3 1/3 Q � μ α , Q � μ α , Q � μ α , Q � μ α , 1 1 2 2 11 11 21 21 2/3 2/3 Q � μ α , and Q � μ α . After the previous dis- 12 12 22 22 L � −lim c (r)x, 13 μ μ⟶0 cussion and by utilizing these scaled quantum corrections in 1/3 equations (20)–(22), the singularities at the poles 1/μ and with 2/3 1/μ can be removed, so L � −lim 􏽨2nx − n y􏽩, 20 L � 2ny _ + n x, μ⟶0 L � 2x + 2α , 11 1 L � −lim λ r , r , 21 1 (24) μ⟶0 L � −2α , (21) 12 11 L � −lim (1 − μ)α r 􏼁 y, 22 μ 1 μ⟶0 L � −f (r)x, 13 r L � −lim β (r)y, μ⟶0 L � −2x _ + y, L � −h (r), 21 r L � −lim λ r , r z, (25) 31 1 μ⟶0 L � 0, L � −lim (1 − μ)α r 􏼁 z, L � −g (r)y, 32 μ 1 (22) 23 r μ⟶0 L � −lim β (r)z, L � −h (r)z, μ⟶0 31 r L � 0, (26) while we have abbreviated L � −g (r)z, 33 r 1/3 1 + μ x κ r , x􏼁 � n − , where 1 2α 3α 21 22 1 − μ 1 f (r) � 􏼠1 + + 􏼡, 3 2 λ r , r􏼁 � + , 1 r r 3 3 r r 1 3α (27) 2Q 3Q g (r) � 􏼒2α + 􏼓, 11 12 r 21 (23) α r 􏼁 � + , μ 1 4 5 r r 1 1 h (r) � 1 + . 2Q 3Q r 21 22 β (r) � + , 1/3 4 2/3 5 μ r μ r Substituting equations (24)–(26) into system (19), we get x € − 2y _ � 3x + 2 α − α 􏼁 − f (r)x, c (r) � + β (r). 1 11 r μ μ y € + 2x _ � −f (r)y, (28) From equation (18), we observe that r → 1 and r → 0 1 2 € z � −z − f (r)z. 1/3 r when μ→ 0. Furthermore, r is of order O(μ ), thus k k (1/r ) → 1 and (1/r ) is undefined, where k is a positive 1 2 (e last system represents the spatial quantized Hill integer. We also remark here that, if the parameters of problem; it is the limiting case, which is acquired from the quantum corrections are ignored, i.e., Q � Q � 0 and 1 2 spatial quantized restricted three-body problem, which was Q � Q � Q � Q � 0, then the limits (20)–(22) 11 12 21 22 firstly presented and studied by Alshaery and Abouelmagd converge, hence exist, and system (19) is reduced to the [29]. System (28) can be rewritten in a similar manner as that classical planar and spatial Hill three-body systems of the restricted three-body problem, i.e., [15, 33]. In the case where these parameters are not equal x € − 2y _ � U (x, y, z), to zero, we get some divergent limits, such as L , L , L , L , and L . (e limits that diverge corre- y € + 2x _ � U 11 12 13 23 33 (x, y, z), (29) spond to the involved functions, which depend on the z � U (x, y, z), parameter μ, possessing singularities and in particular, z Advances in Astronomy 5 where as well as for formulations of quantum mechanics; however, the Hamiltonian approach provides deeper insights in 2 2 (30) U(x, y, z) � 􏽨3x + 4 α − α 􏼁 x − z 􏽩 + w(r). several fundamental features of some astronomical dy- 1 11 namical systems, such as planetary orbits in celestial me- chanics [34, 35]. Utilizing equations (29) and (30), the Jacobi integral is read as C � 2U − V , where explicitly it has the form In our system, the time evolution is defined by the following Hamilton’s equations: 2 2 C � 􏽨3x + 4 α − α 􏼁 x − z 􏽩 q 1 11 (31) zH q _ � , + 2w(r) − V , χ zp 2 2 2 2 (33) with V � x _ + y _ + z _ , while we have abbreviated zH p � − , 1 α α 21 22 zq w(r) � 1 + + . (32) 􏼠 􏼡 r r where χ � (x, y, z) and the Hamiltonian H is given by H � P + 􏼐q p − q p 􏼑 + Υ(q) y x x y 4. Hamiltonian Approach (34) In Hamiltonian approach, a dynamical system is described 2 2 2 − 􏽨2q − q − q + 4 α − α 􏼁 q 􏽩, by a set of canonical coordinates (q, p) where they corre- x y z 1 11 x spond to the n-dimensional vectors q � (q , q , q , . . . , q ) 1 2 3 n with and p � (p , p , p , . . . , p ), while each of the aforemen- 1 2 3 n 2 2 2 2 tioned components is indexed to the frame of reference of P � p + p + p , x y z the respective system. (e components q , i � 1, 2, 3, . . . , n, are known as the generalized coordinates and are selected in (35) 1 α α 21 22 order to remove the constraints or to take into consideration Υ(q) � − 􏼠1 + + 􏼡. 3 2 q q the symmetry characteristics of the system, while p are their conjugate momenta. In classical mechanics, the evolution of Utilizing equations (33) and (34), the equations of time is determined by calculating the total force, which is motion of the spatial quantized Hill model with Hamiltonian exerted on each one of the involved bodies, so the time evo- formula can be written as lutions of both positions and velocities are obtained by using Newton’s second law. However, in Hamiltonian mechanics, the q _ � p + q , x x y evolution of time is computed by finding the relevant Ham- q _ � p − q , y y x iltonian function in the generalized coordinates and then using it into the corresponding Hamilton’s equations. q _ � p , z z (36) (e latter method is tantamount to that which is used to p � p + 2􏼂q + α − α 􏼁 􏼃 − f (q)q x y x 1 11 q x the Lagrangian approach. In fact, for the same generalized momenta, both methods result to the same equations, but p � −p − q − f (q)q , y x y q y the main reason to use Hamiltonian approach, instead that p � −q − f (q)q , of the Lagrangian, comes from the symplectic features of a z z q z Hamiltonian system. In fact, Hamilton’s equations com- 2 2 2 2 prised 2n first-order differential equations, while Lagrange’s where q � q + q + q . Furthermore the Hamiltonian can x y z equations are constituted by n second-order differential be rewritten as equations. Although Hamilton’s equations do not generally H � H + σ H + σ H , (37) 0 1 1 2 2 reduce the difficulty of determining analytical solutions, they may offer some crucial theoretical results due to the fact that where the coordinates and momenta are independent variables with nearly symmetric roles. In addition, if the system H � P + 􏼐q p − q p 􏼑 possesses a kind of symmetry for which a coordinate does 0 y x x y not appear in the Hamiltonian, the relevant momentum is conserved and the corresponding coordinate can be 1 1 2 2 − 􏼐3q − q 􏼑 − , neglected by the other equations. (is results in reducing the 2 r number of coordinates of the dynamical system from n to (38) n − 1, while in the framework of the Lagrangian approach, all 12 H � −2β q − , 1 11 x the generalized velocities remain in the Lagrangian; there- fore, we still have to deal with a system of equations in n coordinates, which must be handled. Both approaches are of 22 H � − . fundamental importance in the study of classical mechanics r 6 Advances in Astronomy Here, σ β � α − α , σ β � α , and σ β � α . Data Availability 1 11 1 11 1 12 21 2 22 22 Within the frame of quantum corrections, σ is of order 2 3 All data used to support the findings of this work are in- O(1/c ) and σ is of order O(1/c ), where, as usual, c denotes corporated within the manuscript. the speed of light. (ereby, the second and third terms in 2 3 equation (37) are of order O(1/c ) and O(1/c ), respectively. Conflicts of Interest (us, the perturbed Hamiltonian (37) can be reduced classical to one of the unperturbed Hamiltonian as (e authors declare that there are no conflicts of interest regarding the publication of this paper. H � H + O . (39) 􏼠 􏼡 References (e obtained relation (39) for Hamiltonian can be used to extend the results of the classical Hill’s problem to the [1] N. Pathak, V. O. (omas, and E. I. Abouelmagd, “(e per- turbed photogravitational restricted three-body problem: quantized problem. 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Published: Jun 7, 2021

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