IntroductionBy analogy with the constructions and conclusions ofworks [1, 2] and relying on the definition of the center ofmass given in works [3], we postulate its immobility in spacesof constant curvature, in this case in the three-dimensionalLobachevsky space, and consider the problem of two particleswith an internal interaction described by potential, dependingon the separation between particles. The essence of thestatement, which replaces the formulation of the theorem onthe center of mass in the three-dimensional Euclidean space,is that in spaces of constant curvature: Lobachevsky, on the3-sphere and in three-dimensional elliptical space, there isa frame of reference in which the center of mass of the systemof particles is at rest.1. Variables of the center of mass and relativecoordinates for a system of two particlesSince the formalism used below, despite the fact that it allowsone to unify the description of the geometries of a numberof three-dimensional and two-dimensional spaces of constantcurvature (and therefore convenient), is not widely used,we are forced to present some calculations similar to thoseused in [1-3]. The problems associated with the separationof variables, including those in spaces of constant curvature,can also be found in [4]. To formulate and solve the problemin three-dimensional Lobachevsky space, instead of biquaternionsdefined over double numbers, biquaternions overcomplex numbers will be used.The following definition of the center of mass coordinatesof two particles with masses m1 and m2 is usedXC = im1X(1) + m2X(2)p(m1X(1) + m2X(2))(m1 ¯X (1) + m2 ¯X (2)).(1)Here the corresponding biquaternions are given over thecomplex numbers, and not over the double ones, as it wasin the case with the 3-sphere [1,2].Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ru57The three-dimensional independent coordinates of thecenter of mass will be the components of the vectorqC = −iXCX0C= −im1X(1) + m1X(2)m1X(1)0 + m2X(2)0. (2)The coordinates of two material points in the embedding fourdimensionalspace will be the components of the biquaternions:X(1) = iX(1)0 + X(1), X(2) = iX(2)0 + X(2), (3)where i2 = −1. The ends of the vectors (biquaternions) lieon the upper field of the pseudo-Euclidean space hyperboloid,on which the real Lobachevsky space is realized. For convenience,the radius of space curvature is assumed to be unity.ThenX(1) ¯X (1) = −1, X(2) ¯X (2) = −1. (4)As independent coordinates, it is convenient to use the Beltramicoordinates, which are components of vectors on thesphere [5]q(1) = −iX(1)X(1)0, q(2) = −iX(2)X(2)0(5)with the law of addition (subtraction)q” = 〈q,±q’〉 =q ± q’ ± [q, q’]1 ∓ (q, q’)(6)coinciding with the composition law of F.I. Fedorov [6]. Here,parentheses denote the usual scalar, square brackets denotethe vector product of vectors. In variables (5), expression (2)has the formqC =m1q(1)/p1 + (q(1))2 + m2q(2)/p1 + (q(2))2m1/p1 + (q(1))2 + m2/p1 + (q(2))2.(7)As noted earlier, expression (7) for the coordinates of the centerof mass coincides in form with a similar expression forthe coordinates of the center of mass in a three-dimensionalflat space, in which the expressions for constant massesm1 and m2 are replaced by mass expressions with the dependenceof masses on coordinates m1/p1 + (q(1))2 andm2/p1 + (q(2))2.We also note that this definition coincides with the definitiongiven in [6], if we take into account that q2 = −th2r ,where r is the distance between two points. As it follows fromthe formula (7) (and shown in [7]), such a definition can begeneralized to an arbitrary number of particles. The biquaternionanalogue of the relative variable for two given particlesis the operatorY12 = −X(2) ¯X(1), (8)defined asX(2) = Y12X(1). (9)Independent three-dimensional coordinates of relative motion,defined as components of the relative motion vectorqy =Y12 − ¯ Y12Y12 + ¯ Y12=*−iX(2)X(2)0, iX(1)X(1)0+== 〈q(2),−q(1)〉 =q(2) − q(1) − [q(2), q(1)]1 + (q(1), q(2)). (10)Let us also introduce four-dimensional Y1 and Y2 and threedimensionalq(1)y , q(1)y coordinates of points relative to thecenter of mass, determined similarly to (8) and (9), namelyX(1) = Y1XC, X(2) = Y2XC, (11)moreoverY1 = −X(1) ¯XC, Y2 = −X(2) ¯XC. (12)It is clear thatY12 = Y2 ¯ Y1. (13)Then for the first particleq(1) = −iX(1)X(1)0==*−qy1 + m1m2p1 + q2y,−iXCX0C+== 〈q(1)y , qC〉, (14)and for the second one we get:q(2) = −iX(2)X(1)0==*qy1 + m2m1p1 + q2y,−iXCX0C+== 〈q(2)y , qC〉. (15)where q(1) and q(2) are defined from Y1 and Y2 respectively.It is easy to verify the validity of formulas (14) and (15) by directcalculation. It should be noted that q(1) and q(2) are expressedin terms of relative variables qy and center of massvariables qC.From (13) it follows thatqy = 〈q(2),−q(1)〉 = 〈q(2)y ,−q(1)y〉. (16)The variables introduced satisfy the relationsXC ¯XC = −1, Y12 ¯ Y12 = 1,Y1 ¯ Y1 = 1, Y2 ¯ Y2 = 1. (17)58Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ru2. Two material points on S13 . Non-relativisticclassical problemThe action for the problem of two material points in threedimensionalspace, interacting with forces that depend onlyon the relative variable, we write in the form [1-3]W =Z h12m1 ˙X (1) ˙¯X(1)++m2 ˙X (2) ˙¯X (2)− V (Y12)idt. (18)Here it is immediately taken into account that the operation ofdifferentiation and conjugation are commuting. The dot abovethe letters denotes differentiation with respect to time. Expression(18) will take a standard form if we pass to independentvariables q(1) and q(2). In this caseW =Z h12m1gab(q(1))q˙(1)a q˙(1)b ++m2gab(q(2))q˙(2)a q˙(2)b− ϕ(qy)idt, (19)wheregab =11 + q2"δab − qaqb1 + q2#(20)is the metric tensor of the three-dimensional Lobachevskyspace in variables that are components of vectors on thesphere. In expression (19), according to the accepted assumption,we set qC = 0 and write it in spherical coordinates.ThenW =Z "12m1r˙21 + m1 sh2 r1􀀀θ˙21 + sin2 θ ˙ϕ21++m2r˙22+m2 sh2 r2􀀀θ˙22+sin2 θ ˙ϕ22−U(r12)#dt. (21)Replacing in (21) the coordinates of individual particles withrelative variables r, θ, ϕ in accordance with formulas (14),(15) with qC = 0, we get the following expression for theactionW =Z "12μ∥(r)r˙2++μ⊥(r) sh2 r􀀀θ˙2 + sin2 θ ˙ϕ2− U(r12)#dt, (22)where we have introduced the longitudinal reduced mass oftwo material pointsμ∥ = m1 m22+ m1m2 ch rm21+ m22+ 2m1m2 ch r!2++m2 m21+ m1m2 ch rm21+ m22+ 2m1m2 ch r!2(23)and the transverse reduced massμ⊥ =m1m2(m1 + m2)m21+ m22+ 2m1m2 ch r. (24)The Hamiltonian of the system is therefore equal toH =12hμ∥(r)r˙2++μ⊥(r) sh2 r􀀀θ˙2 + sin2 θ ˙ϕ2i+ U(r12). (25)It is easy to check that the expression for the Hamilton function(25) in the flat limit r −→ 0 transforms into the Hamiltonianfunction of the plane problem for a reduced mass particle.The corresponding coefficients transform into the expressionfor the reduced massμ =m1m2m1 + m2. (26)Thus, in spaces with curvature, reduced particle masses canbe interpreted as dependent on coordinates, as it also seenfrom (7) (see also [8]). The same is true for composite systems:the reduced masses are functions of the coordinates.Taking into account the form of the Hamilton function (25)and the following definitions of generalized momentapr =∂L∂r˙, pθ =∂L∂θ˙, pϕ =∂L∂ ˙ϕ, (27)we consider the Hamilton-Jacobi equation12μ∥(r) ∂W∂r!2+12μ⊥(r) sh2 r××" ∂W∂θ!2+1sin2 θ ∂W∂ϕ!2#+ U(r) +∂W∂t= 0.(28)The last equation allows separation of variablesW = −Et +Wr(r) +Wθ(θ) +Wϕ(ϕ), (29)and decomposes into the following equations∂W∂ϕ= Mϕ, (30)∂W∂θ2+M2ϕsin2 θ= M2, (31)∂W∂r2+μ∥(r)μ⊥(r)M2sh2 r= 2μ∥(r)(E − U(r)). (32)These equations are easily integrated. WhereinWϕ = Mϕϕ, (33)Wθ =Z sM2 −M2ϕsin2 θdθ, (34)Wr =Z s2μ∥[E − U(r)] − μ∥μ⊥M2sh2 rdr. (35)Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ru59Substituting the last expressions into (29) and differentiatingwith respect to constants, we obtain equations for the particletrajectory∂W∂Mϕ= ϕ1 − ϕ2−−Z θ2θ1Mϕsin2 θqM2 − M2ϕsin2 θdθ = 0, (36)∂W∂M=Z θ2θ1Mdθ qM2 − M2ϕsin2 θ−−Z r2r1μ∥(r)μ⊥(r)Mdrsh2 rq2μ∥[E − U(r)] − μ∥μ⊥M2sh2 r. (37)The law of motion is given by the expression∂W∂E= t2 − t1−−Z r2r1μ∥(r)dr q2μ∥[E − U(r)] − μ∥μ⊥M2sh2 r. (38)3. Schrödinger equation for two material pointsin Lobachevsky spaceThe general formula for the classical kinetic energy of anysystem isTcl =12Xi,jgij(q)q˙iq˙j , (39)where q˙i – generalized speeds, gij(q) – generalized masses.The corresponding operator in quantum mechanics isTq =−ℏ22ΔBL, (40)where ΔBL – the Laplace-Beltrami operator, which can beobtained from the general expressionΔBL =√1g∂∂qi√ggij ∂∂qj. (41)Since in our case the kinetic energy expression can be seenfrom (22), the Laplace-Beltrami operator has the formΔBL =1μ⊥√μ∥ sh2 r∂∂r"μ⊥ sh2 r√μ∥∂∂r#++1μ⊥ sh2 rΔθϕ, (42)whereΔθϕ =1sin θ∂∂θsin θ∂∂θ+1sin2 θ∂2∂ϕ2 . (43)Accordingly, the Schrödinger equation becomesi∂ψ∂t= Hψ, (44)where the Hamiltonian isH = −12ΔLB + U(r). (45)It is clear that the equation we have allows the separation ofvariablesψ = R(r)Y ml (θ, ϕ), (46)where Y ml (θ, ϕ) are the spherical functions satisfying theequationΔθϕY ml (θ, ϕ) =ΔθϕY ml (θ, ϕ) = −l(l + 1)Y ml (θ, ϕ) (47)for l = 0, 1, 2, . . . and the radial part of the wave functionis the solution for the equationd2Rdr2 +√μ∥μ⊥ sh2 rddr"μ⊥ sh2 r√μ∥#dRdr++ 2μ∥(E − U) − μ∥l(l + 1)μ⊥ sh2 r!R = 0. (48)4. A particular problem of a rigid rotator inspaces with constant curvatureAs we know, in the case of a constant relative distancebetween two points, a mechanical system is obtained, whichis called a rigid rotator. Despite the apparent simplicity, thismodel for a flat space, both classical and quantum mechanical[9], find interesting applications, including in the theoryof molecules and nuclear physics. From the approach developedabove, as well as in accordance with works [1-3, 9], itfollows that in spaces of constant curvature, a rigid rotatorhas features associated with the dependence of the reducedmass on the distance between points. These features are exploredbelow. By formulas (21), (25) and the correspondingformulas in [2,3], the Lagrange function of a rigid rotator inthree spaces: in the Lobachevsky space, on the 3-sphere andin the Euclidean space has the formL = Aθ˙2 + sin2 θ ˙ϕ2, (49)where the quantity A has the following expressions in thethree spaces under consideration, respectivelyAlob =12R2μ⊥lob sh2 r0R,Asph =12R2μ⊥sph sin2 r0R,Aflat =12μflatr20, (50)whereμ⊥lob =m1m2(m1 + m2)m21+ m22+ 2m1m2 ch(r0/R), (51)μ⊥sph =m1m2(m1 + m2)m21+ m22+ 2m1m2 cos(r0/R), (52)μflat =m1m2m1 + m2. (53)60Известия Коми научного центра Уральского отделения Российской академии наук № 4 (62), 2023Серия «Физико-математические науки»www.izvestia.komisc.ruSolutions for the corresponding Hamilton-Jacobi equationin the case of a rigid rotator have the formθ = arccos r1 − M24AEcos rEAt!!, (54)ϕ = arctg √4AEMtg rEAt!!. (55)It follows from the last formulas that the period of oscillationsT ∼√A. Let us analyze how the period of oscillations dependson the ratio of the masses of particles (with a constanttotal mass). Denote the mass ratio β = m1/m2 . ThenAlob =mR22sh2r0Rββ2 + 1 + 2β ch(r0/R), (56)Asph =mR22sin2r0Rββ2 + 1 + 2β cos(r0/R),(57)A♭ =mr202β(1 + β)2 . (58)If we fix the values of the constants E = 1,M = 1 thenfrom the condition 1 − M24AE > 0 it follows that A > 1/4and therefore, for the solutions to make sense, the mass ratiocannot be arbitrary. Let’s take the values R = 2.5, r0 =2,m = 1. Then the condition A > 1/4 for three spacesleads to the following restrictions on the mass ratio of theparticles of a rigid rotatorAlob > 1/4 : 0.142 < β < 7.04; (59)Asph > 1/4 : 0.207 < β < 4.83; (60)Aflat > 1/4 : 0.172 < β < 5.83; (61)We construct graphs of dependence of periods of oscillationson the ratio of masses in the range 0.21 < β < 4.8. Figure1 shows that the period of rotation of a rigid rotator, andhence the magnitude of the angular momentum, depends onthe radius of space curvature R.Figure 1. Graphs of rotation period on the ratio of the masses of the constituentparticles at a fixed total mass of the rotator. Top graph – Lobachevskyspace, average – flat space, bottom – 3-sphere.Рисунок 1. Графики периода вращения в зависимости от отношения масссоставляющих частиц при фиксированной общей массе ротатора. Верх-ний график – пространство Лобачевского, средний – плоское простран-ство, нижний – трехмерная сфера.For the equal distances between the material points of therotator and equal radius of the curvature for the Lobachevskyand 3-sphere spaces, the rotation periods are maximum forthe Lobachevsky space, minimum for the 3-sphere. The correspondingcurve for flat space lies between the two mentionedcurves, with each of them tending to the flat spacecurve at R −→ ∞. All three curves are similar.In the quantum case, the Hamiltonian operator of sucha system has the formH =ℏ22IΔθ,ϕ, (62)where the moment of inertia of the system isIlob = 2mR2 sh2 r02R, Iflat =mr202,Isph = 2mR2 sin2 r02R. (63)The Schrödinger equationHψ = Eψ gives the energy levelsof the rigid rotatorEl =ℏ22Il(l + 1), (64)and the eigenfunctions of the Hamilton operator are equal tothe spherical functions for all three spaces ψ = Y ml (θ, ϕ).The levels are degenerate, since each value of the orbitalquantum number corresponds to 2l + 1 magnetic numbervalues. Figure 2 shows energy levels for quantum rotator inthe spaces under consideration (we set here R = 2.5, r0 =2,m = 1).Figure 2. Energy levels for rigid rotator in spaces of constant curvature. Toppoint – 3-sphere, average – flat space, bottom – Lobachevsky space.Рисунок 2. Уровни энергии для жесткого ротатора в пространствах посто-янной кривизны. Верхняя точка – трехмерная сфера, средняя – плоскоепространство, нижняя – пространство Лобачевского.ConclusionThe paper solves the classical and quantum problems ofmotion of two particles in three-dimensional Lobachevskyspace, relative to the center of mass. The Hamilton-Jacobiequation of the problem is formulated and its solutions arefound. The corresponding Schrödinger equation allows theseparation of radial and angular variables. It is shown that thereduced masses of the system depend on the relative distancebetween the particles. The classical and quantum problemsof a rigid rotator in three-dimensional Lobachevsky space areformulated and solved. The dependences of the rotator oscillationperiod on the ratio of the masses of the forming particlesat a fixed distance between them and fixed total massare obtained for three cases: Lobachevsky space, 3-sphereand three-dimensional Euclidean space.