Conjugated equilibrium solutions for the 2 --body problem in the two dimensional sphere M 2 R for equal masses
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug CONJUGATED EQUILIBRIUM SOLUTIONS FOR THE –BODYPROBLEM IN THE TWO DIMENSIONAL SPHERE M R FOREQUAL MASSES
PEDRO PABLO ORTEGA PALENCIAUniversidad de CartagenaDepartamento de Matem´aticasCampus San Pablo, CP 130015Cartagena de Indias, Colombia [email protected]
J. GUADALUPE REYES VICTORIAUniversidad Aut´onoma MetropolitanaDepartamento de Matem´aticasUnidad Iztapalapa, CP 09340, Cd. de M´exico, M´exico [email protected]
August 19, 2019
Abstract.
We study here the behaviour of solutions for conjugated (antipo-dal) points in the 2-body problem on the two-dimensional sphere M R . Weuse a slight modification of the classical potential used commonly in [2], [4]and [10], which avoids the conjugated (antipodal) points as singularities andpermit us obtain solutions through these points, as limit of relative equilibria.Such limit solutions behave as relative equilibria because are invariant underKilling vector fields in the Lie Algebra su(2) and are geodesic curves. MSC: Primary 70F15, Secondary 53Z05 Keywords: Two dimensional conformal sphere M R , The n –body problem, Rel-ative equilibria, Conjugated points.1. Introduction
We consider in this paper the problem of studying the motion of two pointinteracting particles of masses m , · · · , m n on the two-dimensional conformal sphere M R under the action of a suitable (cotangent) potential.We omit along all the document the term curved because the studied space has anon-euclidian metric. Also, we omit the term intrinsic , since when it is chosen thegeometric structure on the conformal sphere M R = b C = C ∪ {∞} as the Riemannsurface with the canonical complex variables ( w, ¯ w ) endowed with the conformalmetric(1) ds = 4 R dwd ¯ w ( R + | w | ) , * * the corresponding differentiable structure in such coordinates is given to this space(see [6, 7] for more details).Following the methods of the geometric Erlangen program as in [8] for the space M R , are defined the conic motions of the n –body problem in terms of the actionof one dimensional subgroups of M¨obius isometric transformations group SU(2),associated to a suitable Killing vector field. By the method of matching the vectorfield in the Lie algebra associated to the corresponding subgroup with the cotan-gent gravitational field we state in each case the functional algebraic conditions(depending on the time t ) that the solutions must hold in order to be a relativeequilibrium. See [5] and [10] where these techniques were also used.In celestial mechanics on surfaces with positive Gaussian curvature, it is verycommon for the cotangent potential to be used as an extension of the Newtonianpotential. This paper presents geometric, dynamic and analytical arguments thatjustify the introduction of a subtle variant of this potential. With the help ofstereographic projection, it is shown that the potential that best approximates theNewtonian potential for the case of constant and positive Gaussian curvature iscot (cid:18) θ (cid:19) instead of the one normally used, cot θ .This paper is organized as follows.In Section 2 we obtain the new potential for the problem which avoids the con-jugated (antipodal) points as singularities and are stated the equations of motionof the problem in complex coordinates in M R as in [5] and [10].In Section 3 we show the algebraic equations for the elliptic relative equilibriafor the general problem with the new potential.In section 4 we obtain the classification of the relative equilibria for the two bodyproblem as Borisov et al in [2].In section 5 we obtain the limit solutions for any conjugated (antipodal) pair ofpoints in M R , which satisfy one regularized system of equations of motion obtainedfrom the original one.2. Equations of motion and relative equilibria
In [10] the authors obtain the equations of motion for this problem using thestereographic projection of the sphere (of radius R ) embedded in R into the com-plex plane C endowed with the metric (1). Moreover, in [1] the classical equationsof the of particles with positives masses m , m , · · · , m n in a Riemannian or semi-Remannian manifold with coordinates ( x , x , · · · , x N ) endowed with a metric ( g ij )and associated connection Γ ijk , and such that the particles move under the influenceof a pairwise acting potential U are given by(2) D ˙ x i dt = ¨ x i + X l,j Γ ilj ˙ x l ˙ x j = X k m k g ik ∂U∂x k , for i = 1 , , · · · , N , where Ddt denotes the covariant derivative and g − = ( g ik ) isthe inverse matrix for the metric g . Remark 1.
We observe that in equation (2), the left hand side corresponds tothe equation of the geodesic curves, whereas the right hand side corresponds to the onjugated equilibria solutions for the 2–body problem in the two dimensional sphere 3 gradient of the potential in the given metric. This means that if the potential isconstant, then the particles move along geodesics.We state in this section the equations of motion for the n -body problem in theconformal sphere M R .2.1. The new potential.
In the study of the n -body problem on the sphere thepotential has been considered(3) U ( θ ) = cot( θ ) , where θ represents the angle at the center of the sphere by the position vectorsof the particles. This potential is attractive for values of θ ∈ (0 , π/ θ ∈ ( π/ , π ), and vanishes in π/
2, which does not correspond to theproperties of the Newtonian potential, which is always attractive.Consider, two particles with masses m k , m j sited at the points A k = ( x k , y k , z k ), A j = ( x j , y j , z j ) on the sphere S R , d kj their geodesic distance and θ kj ∈ [0 , π ] theangle at the origin between these. Then,cos θ kj = A k · A j R and d kj = θ kj R .From basic trigonometric relationships in the sphere S R we get,(4) cot (cid:18) d kj R (cid:19) = s R + A j · A k R − A j · A k where A k · A j = x k x j + y k y j + z k z j .After of applying the stereographic projection on M R , we obtain A k · A j = R (cid:18) R ( w k ¯ w j + ¯ w k w j ) + ( R − | w k | )( R − | w j | )( R + | w k | )( R + | w j | ) (cid:19) If we put a = 2 R ( w k ¯ w j + ¯ w k w j ) + ( R − | w k | )( R − | w j | ) b = ( R + | w k | )( R + | w j | ) , (5)then the equation (4) becomes,cot (cid:18) d kj R (cid:19) = r b + ab − a = 1 R | R + w k ¯ w j || w k − w j | , since b + a = 2 | R + w k ¯ w j | y b − a = 2 R | w k − w j | .Therefore, if two particles with positive masses m k , m j are located at the points A k , A j on the sphere S R , with respective stereographic projections w k , w j , they willbe under the action of the potential(6) U kjR = m j m k R cot (cid:18) d kj R (cid:19) = m j m k | R + w k ¯ w j | R | w k − w j | Corollary 1.
When R → ∞ , the pairwise acting potential U kjR converges to theNewtonian potential on the complex plane. Ortega-Palencia Pedro Pablo and Reyes-Victoria J. Guadalupe
The authors at [10] have used the pairwise acting potential (see [10]), V kjR = m k m j cot (cid:18) d kj R (cid:19) = m k m j cot( θ kj )Note that when R → ∞ the pairwise acting potential V kjR also converges tothe classic Newtonian potential. However, such potential generates singularitiesdue to collisions and antipodal configurations, while (6) only presents singularitiesin collisions. In addition, the equations of motion that will be obtained usingthe potential (6) are clearer and more precise than those obtained with the V kjR potential, as can be seen in [10].2.2. Equations of motion.
Let us denote by w = ( w , w , · · · , w n ) ∈ ( M R ) n thetotal vector position of n particles with masses m i > w i , i =1 , , · · · , n , on the space M R .The singular set in M R for the n -body problem given by the cotangent relation(6) is the set of zeros of the equation w j − w k = 0 . From here, are obtained the following singular set∆( C ) = ∪ kj ∆( C ) kj , where ∆( C ) kj = { w = ( w , w , · · · , w n ) ∈ ( M R ) n | w k = w j , k = j } corresponds to the pairwise collision of the particles with masses m j and m k .It is clear that the presence of geodesic conjugated points on an arbitrary Rie-mannian manifold does not allows us in general to singularities in the equationsof motion (2) of such mechanical system, by the same reason as in this particularcase.From now on, we will suppose that the n point particles in the space M R , aremoving under the action of the potential(7) U R ( w , ¯w ) = 1 R n X j We start our analysis of the relative equilibria solutions with the so called, fromnow on by short, elliptic solutions , obtained by the action of the canonical one-dimensional parametric subgroup of SU (2) asociated to the diferential equation˙ w k = 2 i w k . The action of the subgroup has been partially studied in [10].We have the following first result. Theorem 1. Let be n point particles with masses m , m , · · · , m n > moving in M R . An equivalent condition for w ( t ) = ( w ( t ) , w ( t ) , · · · , w n ( t )) to be an ellip-tic solution of (9) is that the coordinates satisfy the following rational functionalequations depending on the time. (11) 16 R ( | w k | − R ) w k ( R + | w k | ) = n X j =1 ,j = k m j ( | w j | + R ) ( R + ¯ w j w k )( w j − w k ) | R + ¯ w j w k | | w j − w k | with velocity ˙ w k = 2 i w k at each point.Proof. By straightforward computations, for the first case, we have from equation˙ w k = 2 i w k the equality(12) ¨ w k = − w k , which substituted, into equation (9), gives us the relation (11). (cid:3) The following result give us conditions on the initial positions of the parti-cles to generate an elliptic solution of equation (11). In other words, such solu-tions do depend on such fixed points. Let w (0) = ( w (0) , w (0) , · · · , w n (0)) =( w , , w , , · · · , w n, ) be the vector of initial positions of the particles. Corollary 2. With the hypotesis of Theorem 1, a necessary and sufficient conditionfor the initial positions w , , w , , · · · , w n, to generate an elliptic solution for thesystem (9), invariant under the Killing vector field ˙ w k = 2 i w k , is the followingsystem of algebraic equations, (13)16 R ( R − | w k, | ) w k, ( R + | w k, | ) = n X j = k m j ( w j, − w k, )( R + ¯ w j, w k, )( R + | w j, | ) | w j, − w k, | | R + ¯ w j, w k, | . The velocity of each particle is given by the relation ˙ w k, = 2 iw k, , for k =1 , , · · · , n . Ortega-Palencia Pedro Pablo and Reyes-Victoria J. Guadalupe Proof. Let w k = w k ( t ) = e it w k, be the action of the Killing vector field ˙ w k =2 i w k at the initial condition point w k, , with velocity ˙ w k, = 2 iw k, . If we multiplyequation (13) by e it , and use the equality ¯ w j ( t ) w k ( t ) = ¯ w j, e − it w k, e it = ¯ w j, w k, ,then is obtained the system,(14) 16 R ( R − | w k | ) w k ( R + | w k | ) = n X j =1 ,j = k m j ( | w j | + R ) ( R + ¯ w j w k )( w j − w k ) | R + ¯ w j w k | | w j − w k | , which shows that w k ( t ) is a solution of (11).The converse claim follows directly considering t = 0 in system (11). This provesthe Corollary. (cid:3) In [10] the reader can find examples for the two and three body problems definedon the conformal sphere M R .4. Relative equilibria of the two-body problem in M R In the case of 2-bodies, invariant solutions will be shown under the field of vectors˙ w k = 2 i w k such that the two masses move along two different circles.Firstly, we note that the system (13) for the two body problem becomes thesimple algebraic system (independent of time t ),16 R ( | w | − R ) w ( R + | w | ) = m ( w − w )( R + ¯ w w )( R + | w | ) | w − w | | R + w ¯ w | , R ( | w | − R ) w ( R + | w | ) = m ( w − w )( R + ¯ w w )( R + | w | ) | w − w | | R + ¯ w w | . (15)From Corollary 2, a necessary and sufficient condition for the existence of invari-ant elliptical solutions under the Killing vector field ˙ w k = 2 i w k is that the initialreal positions w = α and w = β (with 0 ≤ α, β ≤ R ) satisfy the system (15).This is, substituting in such that system it becomes,16 R ( α − R ) α ( R + α ) = m ( β − α )( R + βα )( R + β ) | β − α | | R + αβ | , R ( β − R ) β ( R + β | ) = m ( α − β )( R + αβ )( R + α ) | α − β | | R + αβ | . (16)By dividing the right half hand and the left hand sides in system (16), andsubstituing w = α , w = β insuch system we obtain the relation(17) ( α − R )( R + β ) α ( β − R )( R + α ) β = − m m , or equivalently(18) m ( α − R )( R + β ) α + m ( β − R )( R + α ) β = 0 . Theorem 2. (Borisov et al. [3] , Ortega et al [9] ) There are only two types ofrelative equilibria for the two body problem with equal masses, obtained as solutionsof the system (16), (1) When both particles are sited in opposite sides of the same circle, β = − α ,called isosceles solutions. onjugated equilibria solutions for the 2–body problem in the two dimensional sphere 7 (2) When the two particles are sited in different circles, β = R ( α − R ) α + R , formingright angle, called right-angled solutions.Both solutions for β are located in the real interval ( − R, .Proof. The solutions for system (16) obtained from the equation (18) are given by β = − α, β = R ( α − R ) α + R as can be seen easily.In the first case, β = − α , we obtain the isosceles solutions.For the second one, let ℓ be the geodesic distance from α to β = R ( α − R ) α + R .If we parametrize in coordinates ( x, y ) ∼ = ( w, ¯ w ) of M R , the corresponding arc Γbetween them by, x ( t ) = ty ( t ) = 0(19)in the interval (cid:20) R ( α − R ) α + R , α (cid:21) , then ℓ = Z Γ dℓ = 2 R Z α R ( α − R ) α + R dtR + t = 2 R (cid:20) arctan (cid:16) αR (cid:17) − arctan (cid:18) R ( α − R ) α + R (cid:19)(cid:21) = 2 R arctan αR − α − Rα + R αR α − Rα + R ! = 2 R arctan (1) = Rπ ℓ = Rθ , where θ is the angle between the givenpoints in M R . Therefore, θ = π (cid:3) Conjugated equilibria solutions From Theorem 2 we have two types of relative equilibria and they generate twodifferent types of equilibria when α → R . Corollary 3. (Perez-Chavela et al. [10] ) For the right-angled relative equilibriawe obtain in the limit when α → R one equilibrium for the system when one ofthe particles is fixed at the origin of coordinates and the other is moving along thegeodesic circle | w | = R with velocity ˙ w ( t ) = − iw ( t ) . This become from the fact that R ( α − R ) α + R → , when α → R . Ortega-Palencia Pedro Pablo and Reyes-Victoria J. Guadalupe Now we begin the study of a solutions for conjugate (antipodal) points for thetwo body problem, as one limit case of isosceles solutions when α → R . Wehave the following result which shows that the relative equilibria converge to thegeodesic circle (equator) which behaves as one relative equilibrium because it isinvariant under the canonical Killing vector field. It is not a solution of the generalsystem (9) even when it is a geodesic for M R and the acting potential vanishesalong such solution. Regardless, such that geodesic is a solution of the regularizedsystem obtained from (9) when we avoid the singularities in the force system dueto conjugated antipodal points. Theorem 3. For the isosceles relative equilibria we obtain in the limit when α → R one equilibrium for the system when the particles with equal masses are geodesicconjugated points. Such that particles are moving along the geodesic circle | w | = R with velocity ˙ w ( t ) = − iw ( t ) .Proof. We recall that for the two body problem with equal masses, the potential is(21) U R ( w , ¯w ) = m R (cid:18) | R + w ¯ w || w − w | (cid:19) , Let w ( t ) = αe it and w ( t ) = − αe it be the components of the relative equilib-rium(22) w ( t ) = ( w ( t ) , w ( t ))Then equation (21) becomes in U R ( w ( t ) , ¯w ( t )) = m R (cid:18) | R + ( − αe it )( αe − it ) || αe it | (cid:19) = m R | R − α || α | = m ( R − α )2 αR (23)If we denote by z ( t ) = Re it and w ( t ) = − Re it be the components of thefunction(24) z ( t ) = ( z ( t ) , z ( t )) , whose each of its coordinates parametrizes the geodesic circle of radius R withvelocity ˙ z k ( t ) = − iz k ( t ), then, lim α → R w ( t ) = z ( t ) . From continuity of the potential, and since conjugated points are no longersingularities for the potential, we have U R ( z ( t ) , ¯z ( t )) = lim α → R U R ( w ( t ) , ¯w ( t ))= lim α → R m ( R − α )2 αR = 0(25)which shows that the potential vanishes along (24). onjugated equilibria solutions for the 2–body problem in the two dimensional sphere 9 A simple substitution of α = − β = R in (18) shows that it also a solution for thatreal equation, and for the regularized system, obtained from the relative equilibriacondition system (16), (cid:18) R ( α − R ) α ( R + α ) (cid:19) | R + αβ | = m ( β − α )( R + βα )( R + β ) | β − α | , (cid:18) R ( β − R ) β ( R + β ) (cid:19) | R + αβ | = m ( α − β )( R + αβ )( R + α ) | α − β | . (26)It is easy to see that the function (24) is not a solution of (9) but another simplesubstitution shows that it satisfy the regularized system obtained of (9) when weavoid singularities due also to conjugated antipodal points, (cid:18) ¨ w − w ˙ w R + | w | (cid:19) | R + ¯ w w | ( R + | w | ) = m ( w − w )( R + ¯ w w )( R + | w | )4 R | w − w | (cid:18) ¨ w − w ˙ w R + | w | (cid:19) | R + ¯ w w | ( R + | w | ) = m ( w − w )( R + ¯ w w )( R + | w | )4 R | w − w | (27)This ends the proof. (cid:3) Definition 1. We call to the solution (24) of the regularized system (27), invariantunder the Killing vector field ˙ z = 2 iz a conjugated equilibrium solution for the twobody problem in M R . We remark the the possibility of having a relative equilibrium in antipodal con-jugated points was claimed in (Perez-Chavela et al. [10]), but in that case thesepoints were singularities for the potential there used. Funding Sources. This work was supported by the Universidad de Cartagena,Cartagena, Colombia [Grant Number 066-2013] and The Universidad Aut´onomaMetropolitana Iztapalapa, Cd de M´exico, M´exico. References [1] Abraham, R., Marsden, J., Foundations of Mechanics, Second Edition. Addison-Wesley Pub.Comp. Inc. , Redwood City, CA., USA, 1987[2] Borisov, A.V., Mamaev, I.S, Kilin, A.A. Two-Body Problem on a Sphere. Reduction, stochas-ticity, Periodic Orbits . Regular and Chaotic Dynamics., 2004, vol.9, no.3, pp.265-279.[3] Borisov, A.V., Garca-Naranjo, L., Mamaev, I.S, Montaldi, J. Reduction and relative equilibriafor the two-Body on spaces of constant curvature . Celest Mech Dyn Astr, 130:43, 2018.[4] Diacu, F., Relative equilibria of the curved n -body problem, Atlantis Studies in DynamicalSystem, Atlantic Press, USA, (2012).[5] Diacu, F., P´erez-Chavela, E., Reyes, J.G., An intrinsic approach in the curved n –body prob-lem. The negative case. 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