An intrinsic approach in the curved n-body problem: the negative curvature case
aa r X i v : . [ m a t h . D S ] S e p AN INTRINSIC APPROACH IN THE CURVED n -BODY PROBLEM: THE NEGATIVE CURVATURECASE Florin DiacuPacific Institute for the Mathematical SciencesandDepartament of Mathematics and StatisticsUniversity of VictoriaP.O. Box 3060 STN CSCVictoria, BC, Canada, VSW 3R4 [email protected]
Ernesto P´erez-ChavelaDepartamento de Matem´aticasUAM-IztapalapaM´exico, D.F. MEXICO [email protected]
J. Guadalupe Reyes VictoriaDepartamento de Matem´aticasUAM-IztapalapaM´exico, D.F. MEXICO [email protected]
November 17, 2018
Abstract.
We consider the motion of n point particles of posi-tive masses that interact gravitationally on the 2-dimensional hy-perbolic sphere, which has negative constant Gaussian curvature.Using the stereographic projection, we derive the equations of mo-tion of this curved n -body problem in the Poincar´e disk, where westudy the elliptic relative equilibria. Then we obtain the equationsof motion in the Poincar´e upper half plane in order to analyze thehyperbolic and parabolic relative equilibria. Using techniques ofRiemannian geometry, we characterize each of the above classesof periodic orbits. For n = 2 and n = 3 we recover some previ-ously known results and find new qualitative results about relativeequilibria that were not apparent in an extrinsic setting. Introduction
We consider the negative curvature case of the curved n -body prob-lem, i.e. the motion of n point masses in spaces of constant Gaussiancurvature under the influence of a gravitational potential that naturallyextends Newton’s law. This article is the completion of an earlier studydone for positive curvature by Ernesto P´erez-Chavela and J. GuadalupeReyes Victoria, [26]. The novelty of the approach used in these twinpapers is the introduction of intrinsic coordinates, which until recentlyseemed to be out of reach due to the elaborated computations theyinvolve. But after Florin Diacu, Ernesto P´erez-Chavela, and ManueleSantoprete obtained the general equations of motion for any number n of bodies in terms of extrinsic coordinates, [8], a glimpse of hope ap-peared for an intrinsic approach, which has been now achieved for the2-dimensional case in [26] and the present paper. An intrinsic study ofthe 3-dimensional case is yet to be done.Whereas [26] had set a basic strategy, which we followed here too, inthe present study we met with more technical difficulties. In [26], theidea to use stereographic projection and analyze the relative equilibriain the spherical plane had been successful, but the similar approachproved not strong enough for negative curvature. Apart from derivingthe equations of motion in the Poincar´e disk to study elliptic relativeequilibria, we also had to use the Poincar´e upper-half-plane model ofhyperbolic geometry in order to overcome the hurdles encountered inthe disk model when analyzing hyperbolic and parabolic relative equi-libria. This outcome seems to confirm an old saying, which claims that“in celestial mechanics, there is no ideal system of coordinates.” Theresults are, of course, model independent, but it appears that each ofthem is easier to prove and understand with the help of a particularmodel.1.1. Motivation and history.
The motivation behind the study ofthe curved n -body problem runs deep. In the early 1820s, Carl Frie-drich Gauss allegedly tried to determine the nature of the physicalspace within the framework of classical mechanics, i.e. under the as-sumption that space and time exist apriori and are independent ofeach other, [24], [13]. He measured the angles of a triangle formed bythree mountain peaks, hoping to learn whether space is hyperbolic orelliptic. But the results of his measurements did not deviate form theEuclidean space beyond the unavoidable measurement errors, so hisexperiments were inconclusive. Since we cannot reach distant stars tomeasure the angles of large triangles, Gauss’s method is of no practicaluse for astronomic distances either. n intrinsic approach in the curved n -body problem of negative curvature 3 But celestial mechanics can help us find a new approach towardsestablishing the geometric nature of the physical space. If we extendNewton’s n -body problem beyond the Euclidian case and also provethe existence of solutions that are specific to each of the negative, zero,and positive constant Gaussian curvature spaces, then the problemof understanding the geometry of the universe reduces to finding, innature, some of the orbits proved mathematically to exist.Therefore obtaining the natural extension of the Newtonian n -bodyproblem to spaces of non-zero constant Gaussian curvature, and study-ing the system of differential equations thus derived, appears to be aworthy endeavour towards comprehending the geometry of the gravita-tional universe. Additionally, an investigation of this system when thecurvature tends to zero may help us better understand the dynamicsof the classical case, viewed as a particular problem within a generalmathematical framework.The attempts to find a suitable extension of Newton’s gravitationallaw to spaces of constant curvature started in the 1830s with the workof J´anos Bolyai, [2], and Nikolai Lobachevsky, [22], who considered a2-body problem in hyperbolic space. Unfortunately, their purely geo-metric approach led to no immediate results. An important step for-ward was made in 1870 by Ernest Schering, who expressed the ideas ofBolyai and Lobachevsky in analytic terms, [27]. This achievement ledto the formulation of the equations of motion of the 2-body problem,both for negative and positive curvature. Since then, many researchersbrought contributions to the curved 2-body problem, including Wil-helm Killing, [15], [16], [17], and Heinrich Liebmann, [19], [20], [21].The latter showed that the orbits of the curved Kepler problem (i.e.the motion of a body around a fixed center) are conics in the hyperbolicplane and proved an analogue of Bertrand’s theorem, [1], which statesthat for the Kepler problem all bounded orbits are closed. But themost convincing argument that the extension of Newton’s law due toBolyai and Lobachevsky is natural rests with the fact that the poten-tial of the Kepler problem is a harmonic function in the 3-dimensionalspace, i.e. a solution of the Laplace-Beltrami equation, in analogy withthe classical potential, which is a solution of Laplace’s equation, i.e. itis a harmonic function in the classical sense, [18].Recently, Florin Diacu, Ernesto P´erez-Chavela, and Manuele Santo-prete proposed a new setting for the problem, which allowed an easyderivation of the equations of motion for any n ≥ F. Diacu, E. P´erez-Chavela, and J.G. Reyes Victoria of the variational approach of constrained Lagrangian dynamics, [12].They also succeeded to solve Saari’s conjecture in the collinear casefor both positive and negative curvature, having settled the Euclideancase earlier, [9], [7]. Remarkable is also their discovery of the fact that,in the curved 3-body problem, all Lagrangian orbits (i.e. rotating equi-lateral triangles) must have equal masses. Since Lagrangian orbits ofnon-equal masses exist in our solar system, such as the rotating equi-lateral triangles formed by the Sun, Jupiter, and the Trojan asteroids,it means that, at distances of the order of 10 AU, space is Euclidean.Since then, several papers analyzed the equations of motion. FlorinDiacu studied the singularities of the equations and of the solutions,[5], Florin Diacu and Ernesto P´erez-Chavela provided a complete clas-sification of the homographic orbits in the curved 3-body problem,[6], Florin Diacu considered the polygonal homographic orbits for anyfinite number of bodies, [4], and Pieter Tibboel solved an open prob-lem stated in [4]. Along the same lines, Regina Mart´ınez and CarlesSim´o studied the stability of the Lagrangian relative equilibria and ho-mographic orbits for the unit sphere, [23]. All these papers treat the2-dimensional case. The only study, so far, of the 3-dimensional curved n -body problem is [3], in which Florin Diacu analyzed relative equilib-ria, a paper in which more details on the history of the problem, aswell as an extensive bibliography, can be found.1.2. Our results.
As mentioned earlier, this paper is a natural contin-uation and completion of [26], which was a study of the curved n -bodyproblem with the help of intrinsic coordinates in 2-dimensional spacesof positive curvature. Here we consider the negative curvature case.In section 2, we start with the extrinsic description of the motion of n point particles of positive masses on the hyperbolic sphere L R , givenby the upper sheet of the hyperboloid of 2 sheets,(1) x + y − z = − R , z > , embedded in the Minkowski space R . Using stereographic projectionthrough the north pole of the hyperbolic sphere L R , we move the prob-lem to the Poincar´e disk D R , where we obtain the equations of motionin complex coordinates and show that the original equations of motionare equivalent to the equations obtained in D R . Then we provide thefirst integrals of the system in D R .In section 3, we consider a suitable Killing vector field in D R andits associated one-dimensional additive subgroup of the Lie algebra su (1 , SU (1 , n intrinsic approach in the curved n -body problem of negative curvature 5 proper isometries which generate the elliptic relative equilibria of thecurved n -body problem in D R . By analyzing the corresponding one-dimensional subgroup of M¨obius transformations, we obtain the equa-tions that characterize all the elliptic relative equilibria. Then we studythese elliptic relative equilibria for n = 2 and n = 3. In the former casewe give a complete description of the elliptic relative equilibria. In thelatter case we study both the Eulerian and the Lagrangian orbits anddescribe their qualitative behavior, thus recovering in D R some resultsproved in [6] for L R . Additionally, we show that other elliptic rela-tive equilibria don’t exists in the curved 3-body problem in the case ofnegative curvature.Although we also obtain in section 3 the one-dimensional subgroupsof proper isometries that generate the hyperbolic and parabolic rela-tive equilibria, their complicated expressions make these orbits hard tounderstand. Therefore, in section 4, we move the problem from thePoincar´e disk D R to the Poincar´e upper half plane H R , in which weobtain the intrinsic equations of motion. Again, we consider the suit-able Killing vector fields in H R and their corresponding one-parameteradditive subgroups in its Lie Algebra sl (2 , R ), which we project via theexponential map onto the Lie group of proper isometries SL (2 , R ). Thesubgroups corresponding to elliptic relative equilibria become compli-cated in this context, but we already studied them in the Poincar´e disk.Fortunately, the subgroups corresponding to hyperbolic and parabolicrelative equilibria are simple enough to allow a complete analysis. Sousing the associated one-parameter subgroups of M¨obius transforma-tions, we obtain the conditions for the existence of hyperbolic andparabolic relative equilibria. Then we proceed as in section 3 with ananalysis of the cases n = 2 and n = 3 for the former orbits, whosequalitative behavior we describe if they are of Eulerian type. Finallywe confirm, within our more general context, a result obtained in [8],which proves that parabolic relative equilibria do not exist.2. Equations of motion
Consider a connected and simply connected 2-dimensional surface ofconstant negative Gaussian curvature κ = − /R . It is well known(see, e.g., [10]) that this surface can be locally represented by (1),the upper sheet of the 2-dimensional hyperbolic sphere of radius iR ,denoted by L R , embedded in the Minkowski space R , which is endowedwith the Lorentz inner product defined by(2) Q ⊙ Q := x x + y y − z z for any points Q = ( x , y , z ) and Q = ( x , y , z ) of R . F. Diacu, E. P´erez-Chavela, and J.G. Reyes Victoria
Let Π be the stereographic projection of L R through the north pole,(0 , , − R ), located at the vertex of the lower sheet of the hyperbolicsphere. The image of Π is the planar disk of radius R , denoted by D R ,having the center at the origin of coordinate system, so we can writethis function as Π : L R → D R , Π( Q ) = q. (3)It is easy to see that Π( L R ) is the entire open disk D R , and that if wetake Q = ( x , y , z ), then q = ( u, v ), where(4) u = R x R + z and v = R y R + z . The inverse of the stereographic projection is given by the equations(5) x = 2 R uR − u − v , y = 2 R vR − u − v , z = R ( R + u + v ) R − u − v . The metric (distance) of the sphere L R is transformed into the metric(6) ds = 4 R ( R − u − v ) ( du + dv ) . In the Poincar´e disk D R , the geodesics are the diameters of the circleof radius R and the arcs of the circles orthogonal to it. In terms of theabove metric, all geodesics have infinite length, so the circle of radius R represents the points at infinity.Since the metric is conformal, with factor of conformity λ ( u, v ) = 4 R ( R − u − v ) , the Christoffel symbols associated to the metric are given by(7) − Γ = Γ = Γ = 12 λ ( u, v ) ∂λ∂u = 2 u ( R − u − v ) , (8) Γ = − Γ = Γ = 12 λ ( u, v ) ∂λ∂v = 2 v ( R − u − v ) , so the geodesics can also be obtained by solving the system of secondorder differential equations (see [11] for more details):¨ u + Γ ˙ u + 2Γ ˙ u ˙ v + Γ ˙ v = 0 , ¨ v + Γ ˙ u + 2Γ ˙ u ˙ v + Γ ˙ v = 0 , which is equivalent to(9) ( ¨ u = − R − u − v ( u ˙ u + 2 v ˙ u ˙ v − u ˙ v )¨ v = − R − u − v ( v ˙ v + 2 u ˙ u ˙ v − v ˙ u ) . n intrinsic approach in the curved n -body problem of negative curvature 7 From now on, we will think of D R as being endowed with the abovemetric. The geodesic distance between two points q k , q j ∈ D R is d kj = d ( q k , q j ) = d ( Q k , Q j ) = R cosh − (cid:18) − Q k ⊙ Q j R (cid:19) . In analogy with [26], where we considered the cotangent potential for κ >
0, we will use here the hyperbolic cotangent potential for κ < Q k ⊙ Q j = 4 R q k · q j − R ( || q k || + R )( || q j || + R )( R − || q k || )( R − || q j || ) , where · is the standard inner product in the plane R . Then(10) coth R (cid:18) d kj R (cid:19) = − R q k · q j − ( || q k || + R )(( || q j || + R ) W , where W = p W − W ,W = (cid:2) R q k · q j − ( || q k || + R )(( || q j || + R ) (cid:3) ,W = ( R − || q k || ) (( R − || q j || ) . We now introduce complex variables in D R , z = u + iv, ¯ z = u − iv, in order to simplify the computations. The inverse of this transforma-tion is given by(11) u = z + ¯ z , v = z − ¯ z i . Then in terms of z and ¯ z , the formulas, (5), for the inverse of thestereographic projection are given by(12) x = R ( z + ¯ z ) R − | z | , y = iR (¯ z − z ) R − | z | , z = R ( | z | + R ) R − | z | , where | . | denotes the absolute value of a complex number.The distance (6) and equation (10) take the form(13) ds = 4 R ( R − | z | ) dzd ¯ z and(14) coth R (cid:18) d kj R (cid:19) = − z k ¯ z j + z j ¯ z k ) R − ( | z k | + R )( | z j | + R ) p Θ , ( k,j ) ( z, ¯ z ) , where(15) Θ , ( k,j ) ( z, ¯ z ) = [2( z k ¯ z j + z j ¯ z k ) R − ( | z k | + R )( | z j | + R )] F. Diacu, E. P´erez-Chavela, and J.G. Reyes Victoria − ( R − | z k | ) ( R − | z j | ) and d kj = d ( z k , z j ) denotes the geodesic distance in the metric (13)between the points z k and z j in D R . From now on we will think of D R as the hyperbolic Poincar´e disk endowed with this new form of themetric.Notice that, in these complex coordinates, system (9), which yieldsthe geodesics, takes the form(16) ¨ z + 2¯ z ˙ z R − | z | = 0 . The intrinsic approach.
Let z = ( z , z , · · · , z n ) ∈ ( D R ) n bethe configuration of n point particles with masses m , m , · · · , m n > D R . We assume that the particles are moving under the action ofthe Lagrangian(17) L R ( z, ˙ z, ¯ z, ˙¯ z ) = T R ( z, ˙ z, ¯ z, ˙¯ z ) + U R ( z, ¯ z ) , where(18) T R ( z, ˙ z, ¯ z, ˙¯ z ) = 12 X ≤ k ≤ n m k λ ( z k , ¯ z k ) | ˙ z k | is the kinetic energy and(19) U R ( z, ¯ z ) = 1 R n X ≤ k
0, obtained in [26], so we omit it here.
Lemma 1.
Let L ( z, ˙ z ) = 12 n X k =1 m k λ ( z k , ¯ z k ) | ˙ z k | + U R ( z, ¯ z ) n intrinsic approach in the curved n -body problem of negative curvature 9 be the Lagrangian defined in (17) for the given problem. Then the solu-tions of the corresponding Euler-Lagrange equations satisfy the systemof second order differential equations (21) m k ¨ z k = − m k ¯ z k ˙ z k R − | z k | + 2 λ ( z k , ¯ z k ) ∂U R ∂ ¯ z k , k = 1 , . . . , n, where (22) ∂U R ∂ ¯ z k = n X j =1 j = k m k m j RP , ( k,j ) ( z, ¯ z )(Θ , ( k,j ) ( z, ¯ z )) / ,P , ( k,j ) ( z, ¯ z ) = ( R − | z k | )( R − | z j | ) ( z j − z k )( R − z k ¯ z j ) , Θ , ( k,j ) ( z, ¯ z ) = [2( z k ¯ z j + z j ¯ z k ) R − ( | z k | + R )( | z j | + R )] − ( R − | z k | ) ( R − | z j | ) , k, j ∈ { , . . . , n } , k = j. Notice that the first term on the right hand side of (21) dependson the kinetic energy alone, whereas the second term depends on thepotential. Therefore, from Lemma 1, we can also draw the followingconclusion.
Corollary 1.
If in D R the potential is constant in the entire space,then the particles move freely along geodesics.Proof. Since the potential is constant, equations (21) take the form(23) m k ¨ z k + 2 m k ¯ z k ˙ z k R − | z k | = 0 , k = 1 , . . . , n, and, if we simplify by m k >
0, we obtain that the coordinates of eachbody satisfy equation (16), so each body moves along a geodesic. (cid:3)
Equivalence of the models.
In [8], where e ∇ = ( ∂ x , ∂ y , − ∂ z ), theequations of motion for the n -body problem in the hyperbolic space L R are(24) m j ¨ Q j = e ∇ Q j V κ ( Q ) − m j κ ( ˙ Q j ⊙ ˙ Q j ) Q j , j = 1 , . . . , n, where Q = ( Q , . . . , Q n ) is the configuration of the system, V κ is theforce function,(25) V κ ( Q ) = X ≤ l 0, and the equations we haveobtained in the Poincar´e disk D R . The proof of this result is perfectlysimilar with the proof of Theorem 2.3 in [26], which concerns the case κ > 0, so we omit it here. Theorem 1. The equations of motion of the curved n -body problemon the Poincar´e disk D R , given by system (21) , and the correspond-ing equations on the hyperbolic sphere L R , given by system (24) , areequivalent. Integrals of motion. Since Theorem 1 proves the equivalencebetween the equations of motion of the curved n -body problem in L R and D R , system (21) inherits the properties of system (24). We are in-terested here in how the first integrals of motion translate from system(24) to system (21). Using equations (12), which give the inverse of n intrinsic approach in the curved n -body problem of negative curvature 11 the stereographic projection, we see that the integrals of motion givenby (27), (28), (29), and (30) become, respectively,(31) 12 n X k =1 m k λ ( z k , ¯ z k ) | ˙ z k | − U R ( z, ¯ z ) = h, where h is the same energy constant as in (27), and(32) n X k =1 im k R ( R − | z k | ) [ R ( ˙ z k − ˙¯ z k ) + ˙ z k ¯ z k − ˙¯ z k z k ] = c , (33) n X k =1 m k R ( R − | z k | ) [ R ( ˙ z k + ˙¯ z k ) − ˙ z k ¯ z k − ˙¯ z k z k ] = c , (34) n X k =1 im k R ( R − | z k | ) (¯ z k ˙ z k − z k ˙¯ z k ) = c . A straightforward computation confirms that the left hand sides ofequations (32), (33), and (34) are real functions. Moreover, c , c , c ∈ R are the same constants as in (28), (29), and (30).3. Relative equilibria in D R In this section we start to analyze the dynamics of the particles thatinteract in D R . Our goal is to give some general characterization of therelative equilibria, which we define below with the help of geometrictools. For this purpose, we will need to understand first what are thesingularities of the equations of motion, such that to avoid the singularconfigurations when dealing with regular equilibria, and to introducethe Principal Axis Theorem, which will guide us towards finding aproper definition of the relative equilibria.3.1. The singularities of the equations of motion in D R . Theequations of motion (21) have singularities in D R when at least onedenominator vanishes, i.e. if there exist indices k, j , with 1 ≤ k < j ≤ n , such that Θ , ( k,j ) ( z, ¯ z ) = 0 . A straightforward computation shows that this statement is equivalentto saying that there are k, j , with 1 ≤ k < j ≤ n , such that( R − z κ ¯ z j )( R − z j ¯ z κ )(¯ z κ − ¯ z j )( z j − z κ ) = 0 . Since | z i | , | ¯ z i | < R, i = 1 , . . . , n , it follows that this equation is satisfiedonly for z j = z k , i.e. when a collision takes place. Therefore the set ofsingularities of the equations of motion is∆ = [ ≤ k 2) denote the Lorentzgroup, defined as the Lie group of all the isometric rotations of deter-minant 1 in R that keep the hyperbolic sphere L R invariant. An im-portant result related to this subgroup of orthogonal transformationsis the Principal Axis Theorem, [25], which states that there are 3 1-parameter subgroups of SO (1 , 2) whose elements can be represented,in some suitable basis of R , as(1) matrices of the form(35) A = P cos θ − sin θ θ cos θ 00 0 1 P − , called elliptic rotations in L R around a timelike axis, the z -axisin our case,(2) matrices of the form(36) B = P s sinh s s cosh s P − , called hyperbolic rotations in L R , around a spacelike axis, the x -axis in our case, and(3) matrices of the form(37) C = P − t tt − t / t / t − t / t / P − , called parabolic rotations in L R , around a lightlike axis, whichis given here by x = 0, y = z , where P ∈ SO (1 , 2) in all cases. Remark 1. This result implies that any isometry in L R can be writtenas a composition of an elliptic rotation around the z -axis, a hyperbolicrotation around the x -axis, and a parabolic rotation around the axis y = 0 , z = x . n intrinsic approach in the curved n -body problem of negative curvature 13 Relative equilibria in D R . Let Iso( D R ) be the group of isome-tries of D R , and assume that { G ( t ) } is a 1-parameter subgroup ofIso( D R ) that leaves D nR \ ∆ and ∆ invariant. We can now give ageneral definition for relative equilibria in D R . Definition 1. A relative equilibrium of the negatively curved n -bodyproblem is a solution z of equations (21) that is invariant relative tosome subgroup { G ( t ) } of Iso( D R ) , i.e. the function w , given by w ( t ) = G ( t ) z ( t ) , obtained by the action of some element of G , is also a solutionof system (21) . To understand the implications of this definition and be able to rep-resent the relative equilibria in D R in as precise terms as we did for L R in [8], we need to take first a look at the topological group structureof isometric rotations of D R (for more details, see, e.g., [11]). For thispurpose, consider the matrix˜ I = (cid:18) − (cid:19) and let SU(1 , 1) = { A ∈ GL(2 , C ) | ¯ A T ˜ IA = ˜ I, det A = 1 } be the special orthochronous unitary group . Then some algebraic com-putations show that any matrix A ∈ SU(1 , 1) has the form A = (cid:18) a b ¯ b ¯ a (cid:19) , with a, b ∈ C satisfying | a | − | b | = 1. This last condition implies thatthe group SU(1 , 1) is diffeomorphic with the real 3-dimensional unithyperbolic sphere embedded in C . The term orthochronous meansthat the transformations A do not change the direction of the time t in the standard interpretation of the Minkowski space. In our case,however, this is just another space coordinate.The group of proper orthochronous isometries of D R , i.e. the groupof transformations that also maintain the geometric orientation, is thequotient group SU(1 , / {± I } , where I is the unit 2 × A ∈ SU(1 , / {± I } ,we can associate a M¨obius transformation, f A : D R → D R , f A ( z ) = az + b ¯ bz + ¯ a , for which it is easy to see that f − A ( z ) = f A ( z ). If M (2 , C ) is the set of 2 × Lie algebra of SU(1 , 1) is the 3-dimensional real linear space su (1 , 1) = { X ∈ M(2 , C ) | ˜ I ¯ X T = − X ˜ I, trace X = 0 } spanned by the Killing vector fields in D R associated to the Pauli ma-trices, g = 12 (cid:18) (cid:19) , g = 12 (cid:18) i − i (cid:19) , g = 12 (cid:18) i − i (cid:19) , which form a basis of su (1 , su (1 , → SU(1 , , applied to the one-parameter additive subgroups { tg } , { tg } , and { tg } , which are straight lines form the geometric point of view. Thisoperation leads us to the following one-parameter subgroups of SU(1 , D R :(1) the subgroup G ( t ) = exp( tg ) = (cid:18) cosh( t/ 2) sinh( t/ t/ 2) cosh( t/ (cid:19) , which defines the one-parameter family of M¨obius transforma-tions(38) f G ( z, t ) = cosh( t/ z + sinh( t/ t/ z + cosh( t/ , (2) the subgroup G ( t ) = exp( tg ) = (cid:18) e it/ e − it/ (cid:19) , which defines the one-parameter family of M¨obius transforma-tions(39) f G ( z, t ) = e it z, (3) the subgroup G ( t ) = exp( tg ) = (cid:18) cosh( t/ i sinh( t/ − i sinh( t/ 2) cosh( t/ (cid:19) , which defines the one-parameter family of M¨obius transforma-tions(40) f G ( z, t ) = cosh( t/ z + i sinh( t/ − i sinh( t/ z + cosh( t/ . n intrinsic approach in the curved n -body problem of negative curvature 15 Elliptic relative equilibria. When projected into the Poincar´edisk, a compositions of the elliptic, hyperbolic, and parabolic rotationsof L R corresponds to some composition of the M¨obius transformations(38), (39), and (40). We therefore need to analyze each of these one-parameter subgroups of transformations in order to find their associ-ated relative equilibria. We will first achieve this goal for the secondgroup, G , given by the M¨obius transformations f G ( z ) = e it z , whichcorrespond to the elliptic rotations in L R . The other 2 groups willbe treated in section 4, since their study becomes very tedious in D R because of complicated computations.Notice that the one-parameter rotation subgroup of SO (1 , 2) intro-duced by the matrix(41) A ( t ) = cos t − sin t t cos t 00 0 1 , i.e. the rotation (35) around the z axis of R that leaves invariant anyhyperbolic sphere L R centered at the origin, is the isometric flow forthe basic Killing vector field (42) L Z ( x , y , z ) = ( − y , x , . This observation leads us to the following result. Proposition 1. Let H : SU(1 , / {± I } → SO (1 , be an isomorphismbetween the groups of proper orthochronous isometries of the Poincar´edisk D R and the Lorentz group of the hyperbolic sphere L R . Then H ( G ( t )) = A ( t ) . Proof. The stereographic projection,Π( x , y , z ) = (cid:18) R x R + z , R y R + z (cid:19) , shows that since the rotation tangent vector at ( x , y , z ) in L R is ( − y , x , D Π[( L Z )( x , y , z )] T = RR + z R x ( R + z ) RR + z R y ( R + z ) ! − yx = − R y R + z R x R + z ! , where D Π is the Jacobian matrix and the upper T denotes the trans-posed of the vector. In complex notation this relationship correspondsto − v + iu = i ( u + iv ) = iz, which leads us to the differential equation(44) ˙ z = iz. Its flow is given by f t ( z ) = e it z , associated to the one-parametersubgroup of M¨obius transformations f G . This remark completes theproof. (cid:3) In order to obtain from Proposition 1 some information regardingthe relative equilibria of type G in D R , we consider functions of theform(45) w k ( t ) = e it z k ( t ) , where z = ( z , . . . , z n ) is a solution of equation (21), and look forconditions that the function w = ( w , . . . , w n ) is also a solution ofsystem (21). Straightforward computations show that(46) ˙ w k = ( iz k + ˙ z k ) e it ¨ w k = (¨ z k + 2 i ˙ z k − z k ) e itd ¯ z k d ¯ w k = e it , k = 1 , . . . , n. Using these facts together with the conditions that w is a solution ofequation (21), m k ¨ w k = − m k ¯ w k ˙ w k R − | w k | + ( R − | w k | ) R ∂U R ∂ ¯ w k , k = 1 , . . . , n, we obtain in terms of z that m k (¨ z k + 2 i ˙ z k − z k ) e it = − m k e − it ¯ z k ( iz k + ˙ z k ) e it R − | z k | + ( R − | z k | ) R ∂U R ∂ ¯ z k d ¯ z k d ¯ w k = − m k ¯ z k ( iz k + ˙ z k ) e it R − | z k | + ( R − | z k | ) R ∂U R ∂ ¯ z k e it . Since z is a solution of (21), m k = 0, and e it = 0, the last relationshipbecomes(47) 2 i ˙ z k − z k = − z k (2 iz k ˙ z k − z k ) R − | z k | , which is equivalent to the equation(48) 2 i (cid:20) | z k | R − | z k | (cid:21) ˙ z k = (cid:20) | z k | R − | z k | (cid:21) z k . Equation (48) holds if and only if1 + 2 | z k | R − | z k | = 0 or 2 i ˙ z k = z k , k = 1 , . . . , n. n intrinsic approach in the curved n -body problem of negative curvature 17 The first set of conditions is equivalent to | z k ( t ) | = 0 = R , whichnever holds. The second set of conditions, which provides informationabout how the velocities must behave for this kind of relative equilibria,holds for | z k ( t ) | = r k , where r k ≥ , k = 1 , . . . , n . For this reason, theparticles form a relative equilibrium associated to the Killing vectorfield (42) if they are moving along Euclidean circles centered at theorigin of the coordinate system in D R . In terms of L R , the particlesmove on circles obtained by slicing the hyperbolic sphere with planesorthogonal to the vertical axis, z , of R .We can now prove the following result. Theorem 2. Consider n point particles with masses m , . . . , m n > , n ≥ , moving in D R . A necessary and sufficient condition for thefunction z = ( z , . . . , z n ) to be a solution of system (21) that is a relativeequilibrium associated to the Killing vector field L Z defined by equation (42) is that for all k = 1 , . . . , n, the following equations are satisfied (49) R ( R + r k ) z k R − r k ) = − n X j =1 j = k m j ( r j − R ) ( z j − z k )( R − z k ¯ z j )( ˜Θ , ( k,j ) ( z, ¯ z )) / , where r k = | z k | and ˜Θ , ( k,j ) ( z, ¯ z )= [2( z k ¯ z j + z j ¯ z k ) R − ( r k + R )( r j + R )] − ( R − r k ) ( R − r j ) ,k, j ∈ { , . . . , n } , k = j .Proof. From the equations 2 i ˙ z k = z k , k = 1 , . . . , n , we concluded thata necessary condition for the existence of a relative equilibrium of theaforementioned type is that the particles move along ordinary circlescentered at the origin of the coordinate system in D R . Differentiatingthese conditions and using them again we obtain that(50) − z k = z k , k = 1 , . . . , n. Comparing these equalities with the equations of motion (21), we con-clude that the coordinates of a relative equilibrium must satisfy the n algebraic equations(51) m k z k = − m k | z k | z k R − | z k | − R − | z k | ) R ∂U R ∂ ¯ z k , k = 1 , . . . , n. Substituting r k = | z k | , k = 1 , . . . , n , into the above equations, weobtain the system of n equations (49), which characterize the relativeequilibria given by the group G . This remark completes the proof. (cid:3) Definition 2. We call elliptic relative equilibria the solutions of system (21) that satisfy the conditions (49) . The case n = 2 . We will next prove the existence of ellipticrelative equilibria for 2 particles in D R , both in the case when theymove on the same suitable circle and in the case when they move ondifferent suitable circles. To achieve this goal, notice first that for n = 2and m , m > 0, equations (49) take the form R ( R + r ) z R − r ) = − m ( r − R ) ( z − z )( R − z ¯ z )[ ˜ Q , (1 , ( z, ¯ z )] / R ( R + r ) z R − r ) = − m ( r − R ) ( z − z )( R − z ¯ z )[ ˜ Q , (2 , ( z, ¯ z )] / , where˜ Q , ( k,j ) ( z, ¯ z ) = [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( r − R ) ( r − R ) . Some algebraic manipulations lead us to the equation( R + r )( R − r ) m ( R + r )( R − r ) m = z ( z − z )( R − z ¯ z ) z ( z − z )( R − z ¯ z ) , and if we simplify the right hand side we obtain(52) ( R + r )( R − r ) m ( R + r )( R − r ) m = − R z − z r R z − z r . This equation shows that there are no elliptic relative equilibria for the2-body problem in D R when one particle is fixed at the origin of thecoordinate system. Indeed, if z = 0, then r = 0, and the denominatorof the right hand side vanishes. If z = 0, then r = 0, and the aboveequation becomes ( R + r ) R m ( R − r ) m = 0 , which has, obviously, no solutions.Equation (52) holds for all time, in particular when the particle m reaches the real line. At that time instant, we have z = α ∈ R , andlet us denote z := z and r := | z | . Then, if we solve equation (52) for z , we have z = m ( R + α )( R − r ) R − m r ( R + r )( R − α ) m ( R + α )( R − r ) α − m R ( R + r )( R − α ) α, therefore z is also a real number, so either z = r or z = − r . In otherwords, when m reaches the real line, m reaches it too. Consequentlythe above equation becomes(53) m ( R + α )( R − r ) m ( R + r )( R − α ) = − ( ± r )[ R − α ( ± r )] α [ R − α ( ± r )] . n intrinsic approach in the curved n -body problem of negative curvature 19 Since α, r < R , it follows that the left hand side is positive, so the righthand side must be positive too. Consequently z = − r , and thus theabove equation takes the form(54) m ( R + α )( R − r ) m ( R + r )( R − α ) = rα , which we can use to estimate the value of r that gives the position of m as a function of m , m , R, and α . For this purpose, we considerthe real function f : ( − R, R ) → R ,(55) f ( x ) = m α ( R + α )( R − x ) − m x ( R + x )( R − α ) , whose zeroes give us the desired values of r and, therefore, the ellipticrelative equilibria for the 2-body problem in D R . For this purpose, letus first prove the following result. Lemma 2. The function f defined in (55) has no double roots.Proof. The first derivative of the function f is(56) f ′ ( x ) = − m αx ( R + α )( R − x ) − m ( R + 3 x )( R − α ) . Since x = 0 and − R < x < R , the double zeroes of f must satisfy theequations ( − x f ( x ) = 0( R − x ) f ′ ( x ) = 0 , which are equivalent to the system ( m αx ( R + α )( R − x ) − m ( R x + x )( R − α ) = 0 − m αx ( R + α )( R − x ) − m ( R + 3 x )( R − α ) ( R − x ) = 0 . If we add the equations of the above system, we obtain the quarticequation x + 6 R x + R = 0 , which has only non-real roots, ± p − ± √ R , a fact that completesthe proof. (cid:3) We can now state and prove the following result, which characterizesthe elliptic relative equilibria of the curved 2-body problem, expressedin terms of the Poincar´e disk model, D R , of the hyperbolic plane. Theorem 3. Consider 2 point particles of masses m , m > movingin the Poincar´e disk D R , whose center is the origin, , of the coordinatesystem. Then a function z = ( z , z ) is an elliptic relative equilibriumof system (21) with n = 2 , if and only if for every circle centered at of radius α , with < α < R , along which m moves, there is a unique circle centered at of radius r , which satisfies < r < R and (54) ,along which m moves, such that, at every time instant, m and m are on some diameter of D R , with between them. Moreover, (1) if m > m > and α are given, then r < α ; (2) if m = m > and α are given, then r = α ; (3) if m > m > and α are given, then r > α .Proof. If the solution z of system (21) with n = 2 is an elliptic relativeequilibrium, then equations (49) are satisfied, and they lead to equation(54), from which we can compute r for given m , m > α , with0 < α < R . Then, by Lemma 2, there is a unique r as desired, so theparticles move as described, and the implication follows.To prove the converse, assume that for given m , m > α ,with 0 < α < R , there is a unique r , which satisfies 0 < r < R and(54), such that the bodies move as described. Then the motion mustbe given by the function z = ( z , z ) with z ( t ) = α (cos t + i sin t ) , z ( t ) = − r (cos t + i sin t ) , with the relationship between m , m , α , and r given by (54). A straight-forward computation shows that this function is a solution of system(21) with n = 2 and satisfies equations (49).To see how the relative values of m and m determine the relation-ship between r and α , we evaluate the function f , defined in (55), at x = 0 , α, R , and obtain f (0) = αm ( R + α ) R > ,f ( α ) = α ( m − m )( R + α )( R − α ) ,f ( R ) = − R m ( R − α ) < . Since, by (56), the derivative f ′ is negative in the entire interval (0 , R ),the function f is strictly decreasing. The conclusion follows then fromthe above relationships. This remark completes the proof. (cid:3) The case n = 3 . In the case of 3 particles in the Poincar´e disk D R , equations (49) become(57) R ( R + r ) z R − r ) = − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / , n intrinsic approach in the curved n -body problem of negative curvature 21 (58) R ( R + r ) z R − r ) = − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / , (59) R ( R + r ) z R − r ) = − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / − m ( R − r ) ( z − z )( R − z ¯ z ) { [2( z ¯ z + z ¯ z ) R − ( r + R )( r + R )] − ( R − r ) ( R − r ) } / . Eulerian Solutions. We will start the study of the case n = 3with the Eulerian elliptic relative equilibria in D R for which the bodieslie on a rotating geodesic. Of course, we can assume that this geodesicrotates around the origin of the coordinate system, so then it must bea rotating diameter of D R .Assume that the particle m is located at the center of the disk, i.e. z = 0, and that m reaches at some time instant the positive axis ofthe real line, i.e. z = r =: α > 0. For m , we denote z := z and take | z | = r . Then equations (57), (58), and (59) become, respectively,(60) 0 = − m ( R − α ) α − m z ( R − r ) r , (61) R ( R + α ) α R − α ) = m α [( α + R ) − ( R − α ) ] / − m ( R − r ) ( z − α )( R − α ¯ z ) { [2 α (¯ z + z ) R − ( α + R )( r + R )] − ( R − α ) ( R − r ) } / , (62) R ( R + r ) z R − r ) = m z [( r + R ) − ( R − r ) ] / − m ( R − α ) ( α − z )( R − αz ) { [2 α ( z + ¯ z ) R − ( r + R )( α + R )] − ( R − r ) ( R − α ) } / . The following result will show that, when one particle is fixed at theorigin of the coordinate system at the center of D R , there is just oneclass of Eulerian elliptic relative equilibria, namely orbits for which the distance from the fixed body to the 2 rotating bodies is the same, andconsequently those masses must be equal. In terms of the hyperbolicsphere L R , the configuration is an isosceles triangle that rotates aroundits vertical height. Theorem 4. Consider 3 point particles of masses m , m , m > moving in the Poincar´e disk D R , whose center is the origin, , of thecoordinate system. Take a function z = ( z , z , z ) that describes thepositions of the particles, with z ( t ) = 0 for all t . Then z is an Eulerianelliptic relative equilibrium of system (21) with n = 3 if and only if m and m are at the opposite sides of the same uniformly rotatingdiameter of a circle of radius α in D R , centered at , with < α < R ,and m = m .Proof. From equation (60), we can conclude that z must be a negativereal number, so z = − r =: − r . This implies that (60) becomes(63) m ( R − α ) α = m ( R − r ) r . Let us further consider the previous equation together with (61) and(62) in which we substitute the previous values of z , r , z , r , z , r .Then we obtain the new system(64) 0 = − m ( R − α ) α + m ( R − r ) r , (65) R ( R + α ) α R − α ) = m / R α + m ( R − r ) ( r + α )( R + αr ) { [ − R rα − ( α + R )( r + R )] − ( R − α ) ( R − r ) } / , (66) R ( R + r ) r R − r ) = m / R r + m ( R − α ) ( α + r )( R + αr ) { [ − R rα − ( r + R )( α + R )] − ( R − r ) ( R − α ) } / . If we multiply both sides of (65) by m ( R − α ) , and both sides of(66) by − m ( R − r ) , when we add the resulting equations we get(67) m ( R + α ) α ( R − α ) = m ( R + r ) r ( R − r ) . n intrinsic approach in the curved n -body problem of negative curvature 23 From equations (64) and (67) we obtain the linear system having themasses m and m as unknowns,(68) ( m r ( R − α ) − m α ( R − r ) = 0 m ( R + α )( R − r ) α − m ( R + r )( R − α ) r = 0 , which has nontrivial solutions if and only if the principal determinantvanishes. A straightforward computations shows that this condition isequivalent to( R + r )( R − α ) r − ( R + α )( R − r ) α = 0 , so the principal determinant vanishes when(69) ( R + r ) r ( R − r ) = ( R + α ) α ( R − α ) . To find the values of r that solve equation (69), we consider the function(70) g : [0 , R ) → R , g ( x ) = ( R + x ) x ( R − x ) , which is strictly increasing in its domain. Therefore equation (69) holdsonly for x = r = α . If we substitute these values in the first equationof system (68), it follows that m = m . (cid:3) In Theorem 4, we required that all masses are positive. Let us nowconsider the case when m = 0. We then obtain the following result. Proposition 2. Consider 3 point particles of masses m = 0 and m = m > moving in the Poincar´e disk D R , whose center is theorigin, , of the coordinate system. Take a function z = ( z , z , z ) that describes the positions of the particles, and assume that z (0) =0 and the real parts of z (0) and z (0) are , i.e. m is initially atthe center and m , m are initially on the horizontal diameter of D R .Then a necessary condition for the particles to form an elliptic relativeequilibrium, is that m and m rotate on the same suitable circle, beingat every time instant at the opposite sides of some diameter of thatcircle, and m lies at the the center of the disk for all time.Proof. Without loss of generality, we can take z = r = ¯ z =: c, z = r = ¯ z =: α, and z = ¯ z = − β. Then some straightforward computations show that equations (60),(61), and (62) become, respectively,(71) R ( R + c ) c ( R − c ) = − m ( R − α ) α − c ) ( R − αc ) + m ( R − β ) β + c ) ( R + βc ) , (72) R ( R + α ) α ( R − α ) = − m ( R − β ) β + α ) ( R + βα ) , (73) R ( R + β ) β ( R − β ) = − m ( R − α ) β + α ) ( R + βα ) . Since m = m , equations (72) and (73) lead us to the relationship(74) ( R + α ) α ( R − α ) = ( R + β ) β ( R − β ) . Consider the smooth real function h : [0 , R ) → R , h ( x ) = ( R + x ) x ( R − x ) . It is easy to see that h is strictly increasing, which implies that relation(74) holds if and only if α = β . Therefore m and m must be atopposite sides of the same circle.If we now use the fact that α = β in equation (71), its right handside vanishes, so c = 0, a remark that completes the proof. (cid:3) Lagrangian Solutions. We will next study the Lagrangian ellip-tic relative equilibria in D R . As we proved in [8], such orbits, formedby rotating equilateral triangles, i.e. r := | z | = | z | = | z | , exist onlywhen the 3 positive masses are equal, m := m = m = m > 0. Theconverse is also true, and Florin Diacu gave a proof of this result forpolygons with n ≥ n = 3 follows the idea used for Theorem 5.3 in [26] for positivecurvature. Theorem 5. Assume that 3 point particles of equal masses move alonga circle of radius r centered at the center of the the Poincar´e disk D R .Then a necessary and sufficient condition for the existence of an ellipticrelative equilibrium is that the particles form an equilateral triangle. n intrinsic approach in the curved n -body problem of negative curvature 25 Proof. With the values z =: r , z =: re iθ and z = z =: re iθ , somestraightforward computations bring equations (57), (58), and (59), re-spectively, to(75) R ( R + r )4 m ( R − r ) = − ( e iθ − R − r e − iθ ) { [4 R r cos θ − ( r + R ) ] − ( R − r ) } / − ( e iθ − R − r e − iθ ) { [4 R r cos θ − ( r + R ) ] − ( R − r ) } / , (76) R ( R + r )4 m ( R − r ) = − e − iθ (1 − e iθ )( R − r e iθ ) { [4 R r cos θ − ( r + R ) )] − ( R − r ) } / − e − iθ ( e iθ − e iθ )( R − r e i ( θ − θ ) ) { [4 R r cos( θ − θ ) − ( r + R ) )] − ( R − r ) } / , (77) R ( R + r )4 m ( R − r ) = − e − iθ (1 − e iθ )( R − r e iθ ) { [4 R r cos θ − ( r + R ) ] − ( R − r ) } / − e − iθ ( e iθ − e iθ )( R − r e i ( θ − θ ) )[[4 R r cos( θ − θ ) − ( r + R ) )] − ( R − r ) ] / . Adding equations (75) and (76), subtracting equation (77) from thesum, and separating the real and imaginary parts of the resulting equa-tion, we obtain(78) 1 − cos θ D + 1 − cos( θ − θ ) D = 2(1 − cos θ ) D (79) sin θ D + sin( θ − θ ) D = 0 , where D = 8 / R r [1 − cos θ ] / [ R + r − R r cos θ ] / ,D = 8 / R r [1 − cos( θ − θ )] / [ R + r − R r cos( θ − θ )] / ,D = 8 / R r [1 − cos θ ] / [ R + r − R r cos θ ] / . It is easy to check that θ = π and θ = π , satisfy equations (78) andtherefore the configuration corresponds to an equilateral triangle. Using standard trigonometry, equation (79) becomes " sin ( θ − θ )sin ( θ ) " ( r − R ) + R r sin ( θ − θ )( r − R ) + R r sin ( θ ) = 1 − sin ( θ − θ )1 − sin ( θ ) . Renaming the variables as u = sin ( θ − θ ) and v = sin ( θ ), the aboveequation takes the form u (1 − v )[( r − R ) + R r u ] = v (1 − u )[( r − R ) + R r v ] . This real equation holds only when u = v , that is, sin ( θ − θ ) =sin ( θ ), or equivalently, θ = 2 θ . If we substitute these values inequation of (78), we obtain1 − cos θ − cos 2 θ = ( R + r − R r cos 2 θ ) ( R + r − R r cos θ ) . Taking w = cos θ and s = cos 2 θ , we are led to(1 − w )( R + r − R r s ) = (1 − s )( R + r − R r w ) , which has real solutions only for w = s , i.e. when cos θ = cos 2 θ ,which yields θ = 0 , π , π , π. To avoid singular configurations, we must take θ = π , π , whichcorrespond to an equilateral triangle positioned in the suitable circle ofradius r . This remark completes the proof. (cid:3) The curved n -body problem in H R Our attempts to study the relative equilibria corresponding to thesubgroups G and G in the Poincar´e disk D R led to insurmount-able computations. Therefore we chose to move the problem to thePoincar´e’s upper-half-plane model, H R , to see if the those equilibriawould be easier to approach. In the remaining part of the paper, wewill obtain the equations of motion in the Poincar´e upper half planeand analyze the hyperbolic and parabolic relative equilibria for n = 2and n = 3.4.1. Equations of motion in H R . We will next obtain the equationsof motion of the curved n -body problem in Poincar´e’s upper half planemodel, H R , with the help of a global isometric fractional linear trans-formation defined by(80) z : H R → D R , z = z ( w ) = − Rw + iR w + iR , n intrinsic approach in the curved n -body problem of negative curvature 27 where w is the complex variable in the upper half plane H R = { w ∈ C | Im ( w ) > } . This transformation has the inverse w : D R → H R , w = w ( z ) = iR ( R − z ) R + z . Since dz = − R i ( w + iR ) dw and d ¯ z = 2 R i ( ¯ w − iR ) d ¯ w, the metric (13) of the disk D R is transformed into the metric(81) − ds = 4 R ( w − ¯ w ) dwd ¯ w. Then H R endowed with the metric (81) is called the Poincar´e upperhalf plane model of hyperbolic geometry, for which the conformal factoris given by µ ( w, ¯ w ) = − R ( w − ¯ w ) . In terms of the metric (81), the geodesics are either half circles orthog-onal to the real axis ( y = 0) or half lines perpendicular to it. All thesecurves have infinite length.Applying transformation (80) to equation (19), we obtain the newpotential in the coordinates ( w, ¯ w ), given by(82) V R ( w, ¯ w ) = 1 R n X ≤ k Let L ( w, ˙ w ) = 12 n X k =1 m k µ ( w k , ¯ w k ) | ˙ w k | + V R ( w, ¯ w ) be the Lagrangian for the given problem in H R . Then the solutioncurves of the corresponding Euler-Lagrange equations satisfy the systemof n second order differential equations, k = 1 , . . . , n , (85) m k ¨ w k − m k ˙ w k w k − ¯ w k = 2 µ ( w k , ¯ w k ) ∂V R ∂ ¯ w k = − ( w k − ¯ w k ) R ∂V R ∂ ¯ w k , where (86) ∂V R ∂ ¯ w k = n X j =1 j = k m k m j R ( ¯ w k − w k )( ¯ w j − w j ) ( w k − w j )( ¯ w j − w k )(Θ , ( k,j ) ( w, ¯ w )) / and Θ , ( k,j ) ( w, ¯ w ) is defined by (83) .Proof. From the definition (80) of the linear fractional transformation z , we obtain in terms of the coordinates that˙ z k = − R i ( w k + iR ) ˙ w k , ¨ z k = 4 R i ( w k + iR ) ˙ w k − R i ( w k + iR ) ¨ w k ,∂ ¯ w k ∂ ¯ z k = ( ¯ w k − iR ) R i . Therefore the equations of motion (23) of the curved n -body problemin the Poincar´e disk get transformed into system (85) in the Poincar´eupper half plane. This remark completes the proof. (cid:3) Integrals of motion in H R . We will further compute the 4 in-tegrals of system (85) by transforming the 4 integrals of system (21)via the transformation (80).The energy integral is given by(87) 12 n X k =1 m k µ ( w k , ¯ w k ) | ˙ w k | − V R ( w, ¯ w ) = h, where V R is given by (82) and h is the energy constant, the same thatoccurs in (31), which was used to obtain (87). n intrinsic approach in the curved n -body problem of negative curvature 29 With the transformation (80), the 3 integrals of the total angularmomentum, (32), (33), and (34), become(88) n X k =1 m k R [ | w k | + R − iR ( w k − ¯ w k )] w k − ¯ w k ) ( w k + iR ) ( ¯ w k − iR ) [ ˙¯ w k w k − ˙ w k ¯ w k − R ( ˙¯ w k − ˙ w k )] = c , (89) n X k =1 m k R [ | w k | + R − iR ( w k − ¯ w k )] ( w k − ¯ w k ) ( w k + iR ) ( ¯ w k − iR ) ( ˙ w k ¯ w k + ˙¯ w k w k ) = c , (90) n X k =1 m k R [ | w k | + R − iR ( w k − ¯ w k )] w k − ¯ w k ) ( w k + iR ) ( ¯ w k − iR ) [ ˙¯ w k w k + ˙ w k ¯ w k + R ( ˙¯ w k + ˙ w k )] = c , which are the integrals of the total angular momentum for system (85).A straightforward computation confirms that the left hand sides ofequations (88), (89), and (90) are real functions. Moreover, the con-stants c , c , c ∈ R are the same that occur in equations (32), (33),and (34).4.3. Relative equilibria in H R . In this subsection we give conditionsfor the existence of hyperbolic and parabolic relative equilibria in thenegative curved problem by using the Poincar´e upper half plane model, H R . The definitions of these concepts are given in the same geometricterms we used in Definitions 1 and 3.Let SL(2 , R ) = { A ∈ GL(2 , R ) | det A = 1 } , be the special linear real 2-dimensional group , which is a 3-dimensionalsimply connected, smooth real manifold. It is well known (see, e.g.,[11]) that the group of proper isometries of H R is the projective quotientgroup SL(2 , R ) / {± I } . Every class A = (cid:18) a bc d (cid:19) ∈ SL(2 , R ) / {± I } has also associated a unique M¨obius transformation f A : H R → H R ,where f A ( z ) = az + bcz + d , for which it is easy to see that f − A ( z ) = f A ( z ). The Lie algebra ofSL(2 , R ) is the 3-dimensional real linear space s l (2 , R ) = { X ∈ M(2 , R ) | trace X = 0 } , spanned by the following suitable set of Killing vector fields, (cid:26) X = 12 (cid:18) − (cid:19) , X = (cid:18) (cid:19) , X = (cid:18) − (cid:19)(cid:27) As in section 3.3, we consider the exponential map of matrices,exp : s l (2 , R ) → SL(2 , R ) , applied to the one-parameter additive subgroups (straight lines) { tX } , { tX } , and { tX } , to obtain the following one-parameter subgroups ofthe Lie group SL(2 , R ):(1) The isometric dilatation subgroup φ ( t ) = exp( tX ) = (cid:18) e t/ e − t/ (cid:19) , which defines the one-parameter family of acting M¨obius trans-formations(91) f ( w, t ) = e t w ( t );(2) The isometric shift subgroup φ ( t ) = exp( tX ) = (cid:18) t (cid:19) , which defines the one-parameter family of acting M¨obius trans-formations(92) f ( w, t ) = w ( t ) + t ;(3) The isometric rotation subgroup φ ( t ) = exp( tX ) = (cid:18) cos t sin t − sin t cos t (cid:19) , which defines the one-parameter family of acting M¨obius trans-formations(93) f ( w, t ) = (cos t ) w ( t ) + sin t ( − sin t ) w ( t ) + cos t . We proved in subsection 3.4 the existence of elliptic relative equilibriafor the initial problem by using the proprieties of the Poincar´e model D R , i.e. we showed that for elliptic relative equilibria, each particlemoves along a circle centered at the origin of the coordinate system.From the Theorem of the invariance of the domain , [14], the isom-etry (80) carries the interior of D R into the interior of H R . Thereforesimple closed curves contained in D R are taken to simple closed curves n intrinsic approach in the curved n -body problem of negative curvature 31 contained in H R (see [14] for more details). For the circle z ( t ) = z e it in the Poincar´e disk, D R , the corresponding curve, w ( t ) = iR ( R − z e it ) R + z e it , in the Poincar´e upper half plane, H R , satisfies w (0) = w (2 π ) and musttherefore belong to the class of the M¨obius transformations (93), whichthus corresponds to elliptic relative equilibria. Since we already studiedthose orbits in the Poincar´e disk, D R , we don’t need to further analyzethem here, the more so since the M¨obius transformations (93) lead tocomplicated computations in H R . However, the M¨obius transforma-tions (91) and (92), corresponding to the Killing vector fields X and X , respectively, are simpler in H R than the analogue transformationsin D R , so they will be the object of our further analysis. The formerwill lead us to hyperbolic relative equilibria and the latter to parabolicrelative equilibria.4.4. Hyperbolic relative equilibria. We will further study the rel-ative equilibria associated to the subgroup (91), which defines the one-parameter family of acting M¨obius transformations f ( w, t ) = e t w ( t )in the Poincar´e upper half plane, H R . Let ξ = ( ξ , . . . , ξ n ), with ξ k ( t ) = e t w k ( t ) , k = 1 , . . . , n , be the action orbit for a solution w =( w , . . . , w n ) of system (85). Then,˙ ξ k = ( w + ˙ w ) e t , ¨ ξ k = ( ¨ w + 2 ˙ w + w ) e t , k = 1 , . . . , n, and therefore the curve ξ is also a solution of the equations of motion(85) if and only if m k ( ¨ w k + 2 ˙ w k + w k ) e t = 2 m k e t ( w k + ˙ w k ) e t w k − e t ¯ w k − ( e t w k − e t ¯ w k ) R ∂V R ∂ ¯ w k d ¯ w k d ¯ ξ k = 2 m k ( w k + 2 w k ˙ w k + ˙ w k ) e t w k − ¯ w k − ( w k − ¯ w k ) R ∂V R ∂ ¯ w k e t , k = 1 , . . . , n. Since w is a solution of (85), d ¯ w k d ¯ ξ k = e − t = 0, and m k = 0 , k = 1 , . . . , n ,the last relationships become(94) 2 ˙ w k − w k ˙ w k w k − ¯ w k = − w k w k − ¯ w k − w k , k = 1 , . . . , n. So if we fix a body m k , the above condition holds if and only if1 − w k w k − ¯ w k = 0 or(95) 2 ˙ w k = − w k . The first condition in equation is equivalent to w k + ¯ w k = 0 andcorresponds to the geodesic given by the vertical half line that formsthe imaginary axis. Via the linear fractional transformation (80), thisgeodesic corresponds to the horizontal geodesic, Im( z ) = 0, of thePoincar´e disk D R .The second condition, (95), holds for the body m k when w k is afunction of the form w k ( t ) = w k (0) e − t/ , where w k (0) is some initialcondition with Im( w k (0)) > 0. Therefore the particle m k moves along ahalf line through the origin of the coordinate system, which is a pointat infinity that is reached when t → ∞ . As t → −∞ , m k goes toinfinity. It is instructive to notice that such integral curves of equation(95) correspond in the Poincar´e disk D R to the parametric curves z k ( t ) = − Rw k (0) e − t/ + iR w k (0) e − t/ + iR = iR e t/ − Rw k (0) iRe t/ + w k (0) , which are equidistant from the horizontal geodesic diameter, all of themstarting at the point ( − R, t → −∞ , and ending in the point( R, t → ∞ . When Re[ w k (0)] = 0, this curve becomes the hor-izontal diameter, which is a geodesic. The above non-geodesic curvestogether with the geodesic horizontal diameter foliate the Poincar´e disk.On the hyperbolic sphere, L R , the above non-geodesic curves aregiven by the equations(96) x = α, y = r sinh t, z = r cosh t, where α = 0 is a constant and r = √ R + α . When α = 0, we recoverthe geodesic. When these curves are taken to the Poincar´e disk via thestereographic projection (4), we obtain the complex curves(97) z ( t ) = u ( t ) + iv ( t ) = 2 Rr ( e t − Re t + 2 r ( e t + 1) + Rαe t iRe t + 2 r ( e t + 1) , which also start at the point ( − R, t → −∞ , and end at thepoint ( R, t → ∞ . In other words, we have obtained two familiesof equidistant curves with the same initial and final directions as thegeodesic diameter.So for the particles m , . . . , m n to form a relative equilibrium in H R associated to the Killing vector field X , they have to move along theupper half lines converging to the origin of the coordinate system in H R as t → ∞ . These half lines are not geodesics, except in the case ofthe vertical half line. Moreover, each non-geodesic vertical half line is n intrinsic approach in the curved n -body problem of negative curvature 33 equidistant from the geodesic vertical half line, the distance being largerwhen the angle between the non-geodesic half line and the geodesicvertical half line is larger. The sizes of these angles range between 0and π/ 2. Notice also that in the case of the geodesic vertical half line,equation (95) is satisfied as well, a fact that we will use in the proof ofthe following result. Theorem 6. Consider n point particles with masses m , . . . , m n > , n ≥ , moving in H R . A necessary and sufficient condition for thefunction w = ( w , . . . , w n ) to be a solution of system (85) and, atthe same time, a relative equilibrium associated to the Killing vectorfield X defined by equation (91) is that, for every k = 1 , . . . , n, thecoordinates satisfy the conditions (98) R ( w k + ¯ w k ) w k w k − ¯ w k ) = n X j =1 j = k m j ( w j − ¯ w j ) ( w k − w j )( ¯ w j − w k )[Θ , ( k,j ) ( w, ¯ w )] / , where (99) Θ , ( k,j ) ( w, ¯ w )= [( ¯ w k + w k )( ¯ w j + w j ) − | w k | + | w j | )] − ( ¯ w k − w k ) ( ¯ w j − w j ) ,k, j ∈ { , . . . , n } , k = j .Proof. We showed previously that for relative equilibria of the afore-mentioned type the bodies move along straight half lines converging tothe origin of the coordinate system and must therefore satisfy equation(95), which implies that(100) 4 ¨ w k = w k , k = 1 , . . . , n. Using this equation together with equation (95), we can conclude fromthe equations of motion (85) that(101) R m k ( w k + ¯ w k ) w k w k − ¯ w k ) = ∂V R ∂ ¯ w k , k = 1 , . . . , n. Using (86), we obtain the relationships given in (98). This remarkcompletes the proof. (cid:3) Definition 3. We will call hyperbolic relative equilibria the solutionsof system (85) in H R that satisfy equations (98) . We remark that equation (95) also gives the condition the velocity ofthe particle m k must satisfy in order to produce a hyperbolic relativeequilibrium. The case n = 2 . We will further provide a description of thehyperbolic relative equilibria for 2 interacting particles in H R . For this,we observe that for the particles of masses m and m , the equations(98), which characterize hyperbolic relative equilibria become(102) R ( w + ¯ w ) w w − ¯ w ) = m ( ¯ w − w ) ( w − w )( ¯ w − w )[Θ , (1 , ( w, ¯ w )] / , (103) R ( w + ¯ w ) w w − ¯ w ) = m ( ¯ w − w ) ( w − w )( ¯ w − w )[Θ , (2 , ( w, ¯ w )] / , whereΘ , (1 , ( w, ¯ w ) = Θ , (2 , ( w, ¯ w ) = 4( w − w )( ¯ w − ¯ w )( ¯ w − w )( ¯ w − w ) . Straightforward computations lead to the equation(104) ( w + ¯ w ) w ( w + ¯ w ) w = − m ( ¯ w − w ) ( ¯ w − w ) m ( ¯ w − w ) ( ¯ w − w ) , provided that w + ¯ w = 0, which is equivalent with the equation(105) m ( w − ¯ w ) ( ¯ w − w )( w + ¯ w ) w + m ( w − ¯ w ) ( ¯ w − w )( ¯ w + w ) w = 0 , subject to the restriction w + ¯ w = 0.Let us first prove a negative result about hyperbolic relative equilib-ria in the curved 2-body problem in H R . In general terms, unrelatedto any model of hyperbolic geometry, this result states that, on onehand, 2 particles cannot follow each other along a geodesic and main-tain a constant distance between each other and, on the other hand,one particle cannot move along the geodesic, while the other particlemoves along a non-geodesic curve equidistant from that geodesic, suchthat the bodies maintain all the time the same distance between eachother. Proposition 3. Consider 2 point particles of masses m , m > mov-ing in H R . Then there are no hyperbolic relative equilibria as solutionsof system (85) with n = 2 for which both particles move along the geo-desic vertical half line on the imaginary axis or for which one particlemoves along the geodesic vertical half line and the other particle movesalong a non-geodesic half line converging to the origin of the coordinatesystem.Proof. As previously seen, if the particle of mass m moves along ge-odesic vertical half line, then w + ¯ w = 0. Therefore equation (105)becomes m ( ¯ w − w ) ( ¯ w − w )( ¯ w + w ) w = 0 , n intrinsic approach in the curved n -body problem of negative curvature 35 which is impossible because none of the above factors vanishes in eitherof the two scenarios given in the statement. This remark completes theproof. (cid:3) The previous result showed that 2 bodies that form a hyperbolicrelative equilibrium cannot move along the same geodesic. It is thennatural to ask whether they could move along the same non-geodesiccurve that is equidistant from a given geodesic and maintain all thetime the same distance from each other. As we show in the followingresult, expressed in terms of H R , the answer is also negative. Proposition 4. Consider 2 point particles of masses m , m > mov-ing in H R . Then there are no hyperbolic relative equilibria as solutionsof system (85) with n = 2 for which both particles move along the samehalf line converging to the origin of the coordinate system.Proof. Recall that for hyperbolic relative equilibria the expression ofthe orbit of the particle m k is w k ( t ) = w k (0) e − t/ , where Im( w k (0)) > w ( t ) = w (0) e − t/ and w ( t ) = αw ( t ) = αw (0) e − t/ , for some α > 0. If we substitute relations (106) into equation (105), astraightforward computation gives us the condition( αm + m ) w (0) = ( m + αm ) ¯ w (0) . If Re( w (0)) = 0, i.e. when the particles are on the same non-geodesichalf line, then there is no α that can satisfy the equation. This remarkcompletes the proof. If Re( w (0)) = 0, then the bodies are on thegeodesic vertical half line, and only α = − α must be positive, the equation is not satisfied either, so wefound another proof for the first statement of Proposition 3. (cid:3) From Propositions 3 and 4 we learned that, in general terms, inde-pendent of the hyperbolic model used, 2 particles cannot form a hy-perbolic relative equilibrium if they move along the same non-geodesiccurve equidistantly places from a geodesic, along the same geodesic, orif one particle moves on a geodesic and the other particle on a non-geodesic curve equidistant to the geodesic. It is then natural to checkthe last possibility, whether there exist hyperbolic relative equilibriawith one body moving along a non-geodesic curve and the other alonganother non-geodesic curve, both equidistant from a given geodesic.The answer is positive and the motion takes place when the distances from the geodesic to the 2 non-geodesic curves satisfy a certain rela-tionship that depends on the values of the masses.To write these conditions in the language of H R , recall that the slope β of any straight line in the complex plane is defined by the formula iβ = w − ¯ ww + ¯ w . Without loss of generality, we choose the initial conditions such thatthe heights satisfy w (0) − ¯ w (0) = 2 iy and w (0) − ¯ w (0) = 2 iy .Then, in terms of the slopes β and β of the straight lines, equation(105) becomes(107) m β y ( | w | − w w ) = − m β y ( | w | − w w ) . We can now state and prove the following result. Theorem 7. Consider 2 point particles of masses m , m > movingin H R . Then some necessary and sufficient conditions for the existenceof a hyperbolic relative equilibrium as a solution of system (85) with n = 2 are that one particle moves along a non-geodesic half line, whilethe other particle moves along another non-geodesic half line, both halflines converging to the origin of the coordinates system, such that thesupporting lines have slopes of opposite signs that satisfy the relation-ship (108) m m = − β β y y , and that at every time instant there is a geodesic half circle centered atthe origin of the coordinate system on which both particles are located.Proof. If we take the real and imaginary parts of equation (107), weobtain the equations(109) m β y | w | − m β y Re( w w ) = − m β y | w | + m β y Re( w w ) , (110) − m β y Im( w w ) = m β y Im( w w ) . From equation (110), it follows that − m β y = m β y , so condition (108) must be satisfied. Equation (109) implies that m β y | w | = − m β y | w | , which, by condition (108), is equivalent to | w | = | w | . This factproves that, at every time instant, there must exist a geodesic halfcircle centered at the origin of the coordinate system on which theparticles are located. This remark completes the proof. (cid:3) n intrinsic approach in the curved n -body problem of negative curvature 37 Remark 2. Notice that, in the above result, if the masses are equal, m = m > 0, then both half lines have slopes equal in absolute value,but of opposite sign. Indeed, since, at every time instant, the particlesmust be located on the same geodesic half circle centered at the originof the coordinate system, the slopes β and β of the half lines alongwhich they move satisfy β = − β if and only if y = y . Therefore,from equation (108), we have that m = m if and only if β = − β .4.4.2. The case n = 3 . We will next study the case of 3 bodies in thePoincar´e upper half plane, H R , with masses m , m , m > 0. In thiscontext, the system of algebraic equations (98) becomes(111) R ( w + ¯ w ) w w − ¯ w ) = m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / + m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / , (112) R ( w + ¯ w ) w w − ¯ w ) = m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / + m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / , (113) R ( w + ¯ w ) w w − ¯ w ) = m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / + m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − | w | + | w | )] − ( w − ¯ w ) ( w − ¯ w ) } / . We assume that the particles m and m move along non-geodesichalf lines and that m moves along the geodesic vertical half line, suchthat, at every time instant, there is a geodesic half circle centered atthe orgin of the coordinate system on which all particles are located.In other words, if w = ( w , w , w ) represents the configuration of thesystem, we have w + ¯ w = 0 and | w | = | w | = | w | . We can also write that w ( t ) = w (0) e − t/ , w ( t ) = w (0) e − t/ , w ( t ) = w (0) e − t/ , with | w (0) | = | w (0) | = 1 and w (0) = i . The latter conditions are notrestrictive, since the only requirement for the initial conditions is to lieon the half lines on which the particles are assumed to move. Substi-tuting the above forms of w , w , w into equations (111), (112), (113),the factors e − t/ get cancelled, and after redenoting w (0) , w (0) , w (0)by w , w , w , respectively, we obtain the following equations:(114) R ( w + ¯ w ) w w − ¯ w ) = m ( w + 1)2[4 + ( w − ¯ w ) ] / + m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − − ( w − ¯ w ) ( w − ¯ w ) } / , (115) 0 = m ( w − ¯ w ) ( w + ¯ w )[4 + ( w − ¯ w ) ] / + m ( w − ¯ w ) ( w + ¯ w )[4 + ( w − ¯ w ) ] / , (116) R ( w + ¯ w ) w w − ¯ w ) = m ( w + 1)2[4 + ( w − ¯ w ) ] / + m ( w − ¯ w ) ( w − w )( ¯ w − w ) { [( w + ¯ w )( w + ¯ w ) − − ( w − ¯ w ) ( w − ¯ w ) } / . Equation (115) can take place only if w + ¯ w and w + ¯ w haveopposite signs, so we can rewrite this equation as(117) m ( w − ¯ w ) | w + ¯ w | [2( w + ¯ w )] = m ( w − ¯ w ) | w + ¯ w | [2( w + ¯ w )] . To express w and w in terms of the angles the half lines make withthe horizontal axis, we put w = e iθ = cos θ + i sin θ and w = e iθ = cos θ + i sin θ , with θ , θ ∈ ( − π/ , ∪ (0 , π/ m tan θ = m tan θ , which shows what relationship exists between the masses and the anglesof the non-geodesic half lines along which the corresponding particlesmove.We can now state and prove the main result of this section, whichshows that Eulerian relative equilibria for which one body moves alonga geodesic exist only if the masses moving on non-geodesic curvesequidistant from the geodesic are equal and those curves are on op-posite parts of the geodesic and at the same distance from it. n intrinsic approach in the curved n -body problem of negative curvature 39 Theorem 8. Consider 3 point particles of masses m , m , m > moving in H R . Assume that m and m move along non-geodesic halflines emerging from the origin of the coordinate system at angles θ and θ , respectively, and that m moves along the geodesic vertical half line.Moreover, at every time instant, there is a geodesic half circle on whichall 3 bodies are located, and the motion of the particles is given by thefunction w = ( w , w , w ) . Then w is a hyperbolic relative equilibriumthat is a solution of system (85) with n = 3 if and only if θ = − θ and m = m , with θ , θ ∈ ( − π/ , ∪ (0 , π/ .Proof. We already showed that for w = e iθ , w = i, and w = e iθ ,equation (115) takes the form (118). With the same substitutions,equations (114) and (116) become, respectively,(119) R cos θ sin θ = 8 m cos θ + 8 m sin θ (cos θ − cos θ ) , (120) R cos θ sin θ = 8 m sin θ (cos θ − cos θ ) + 8 m cos θ . If we divide equation (119) by cos θ and equation (120) by cos θ ,using relation (118) we obtain that(121) cos θ sin θ = cos θ sin θ . Consider now the function f : ( − π/ , ∪ (0 , π/ , f ( x ) = cos x sin x , which is obviously even. It is easy to see that f is increasing in theinterval ( − π/ , 0) and decreasing in the interval (0 , π/ θ = ± θ . Since θ = θ induces a collision configuration, which is a singularity, the only pos-sible solution is θ = − θ . The fact that m = m follows now fromequation (118). This remark completes the proof. (cid:3) Parabolic relative equilibria. In this section we will study therelative equilibria associated to the subgroup φ ( t ) = exp( tX ) = (cid:18) t (cid:19) , generated by the Killing vector field X and which defines the one-parametric family of acting M¨obius transformations f ( w, t ) = w ( t ) + t, in the upper half plane H R . These orbits correspond to parabolic rel-ative equilibria, and we will show that they do not exist in H R .Let ζ = ( ζ , . . . , ζ n ), with ζ k ( t ) = w k ( t ) + t , be the action orbit for w = ( w , . . . , w n ), which is a solution of the equations of motion (85).Then ˙ ζ k = ˙ w k + 1 and ¨ ζ k = ¨ w k , k = 1 , . . . , n, therefore ζ is also a solution of system (85) if and only if m k ¨ ζ k = 2 m k ˙ ζ k ζ k − ¯ ζ k − ( ζ k − ¯ ζ k ) R ∂V R ∂ ¯ ζ k , k = 1 , . . . , n, which can be written as m k ¨ w k = 2 m k ( ˙ w k + 1) w k − ¯ w k − ( w k − ¯ w k ) R ∂V R ∂ ¯ w k d ¯ w k d ¯ ζ k , k = 1 , . . . , n, which, since d ¯ w k d ¯ ζ k = 1, is the same as(122) m k ¨ w k = 2 m k ( ˙ w k + 2 ˙ w k + 1) w k − ¯ w k − ( w k − ¯ w k ) R ∂V R ∂ ¯ w k , k = 1 , . . . , n. But since w is a solution of system (85), we also have that(123) m k ¨ w k = 2 m k ˙ w k w k − ¯ w k − ( w k − ¯ w k ) R ∂V R ∂ ¯ w k , k = 1 , . . . , n, Comparing now equations (122) and (123), we obtain that(124) 2 ˙ w k = − , k = 1 , . . . , n, which holds if and only if(125) w k ( t ) = − t w k (0) , k = 1 , . . . , n, where w k (0) , k = 1 , . . . , n, are initial conditions. Consequently, a nec-essary condition for the particles m , . . . , m n to form a relative equilib-rium associated to the Killing vector field X is that they move alonghorizontal straight lines in H R passing through w k (0) , k = 1 , . . . , n . Interms of the Poincar´e disk D R , these orbits correspond to the paramet-ric curves z k ( t ) = − R [ − t + w k (0)] + iR − t + w k (0) + iR , k = 1 , . . . , n, which start at the point ( − R, t → −∞ , and end at the samepoint ( − R, t → ∞ . These curves have the same topology as theboundary circle of D R . In terms of the hyperbolic sphere L R , these linesare the parabolas obtained by intersecting L R with a plane orthogonalto the rotation axis, the line y = 0 , z = x . n intrinsic approach in the curved n -body problem of negative curvature 41 We can now state and prove the following result. Theorem 9. Consider n ≥ point particles of masses m , . . . , m n > moving in H R . Then a necessary and sufficient condition for thefunction w = ( w , . . . , w n ) to be a solution of system (85) that is arelative equilibrium associated to the Killing vector field X is that thecoordinate functions satisfy the equations (126) − R w k − ¯ w k ) = n X j =1 j = k m j ( ¯ w j − w j ) ( w k − w j )( ¯ w j − w k )[ ˜Θ , ( k,j ) ( w, ¯ w )] / ,k = 1 , . . . , n , where (127) ˜Θ , ( k,j ) ( w, ¯ w ) = [( ¯ w k + w k )( ¯ w j + w j ) − | w k | + | w j | )] − ( ¯ w k − w k ) ( ¯ w j − w j ) , k, j ∈ { , . . . , n } , k = j. Proof. We saw that relative equilibria associated with the Killing vectorfield X must satify equations (125), which imply that(128) ¨ w k = 0 , k = 1 , . . . , n. Therefore, from equations (123), we can conclude that the coordinatesof a relative equilibrium w satisfy the equations(129) m k R ( w k − ¯ w k ) = ∂V R ∂ ¯ w k , k = 1 , . . . , n. If we now compare the above equations to the expressions (86) of ∂V R ∂ ¯ w k , k = 1 , . . . , n , we obtain the desired relationships (127). Thisremark completes the proof. (cid:3) Definition 4. We will call parabolic relative equilibria the solutions ofsystem (85) in H R that satisfy equations (126) . Notice that equations (124) provide the velocities of the particles incase they form a parabolic relative equilibrium. However, as we willfurther prove, parabolic relative equilibria do not exist in the curved n -body problem. The following statement generalizes a result obtainedin [8] for curvature κ = − 1. Using the same idea as in [8], this resultwas generalized in [3] to the 3-dimensional case. Theorem 10. In the curved n -body problem with negative curvaturethere are no parabolic relative equilibria.Proof. Using the notation w k = a k + ib k and w j = a j + ib j , the real and imaginary part of equations (126) become(130) − R b k = n X j =1 j = k m j b j [( a k − a j ) + ( b j − b k )][ ˜Θ , ( k,j ) ( w , ¯ w )] / , k = 1 , . . . , n, (131) 0 = n X j =1 j = k m j b j b k ( a k − a j )[ ˜Θ , ( k,j ) ( w , ¯ w )] / , k = 1 , . . . , n. Since b k > , k = 1 , . . . , n , equations (131) hold for any k, j ∈{ , . . . , n } , with k = j , if and only if a k = a j . This fact implies thatall the particles are located on the same vertical line. Without lossof generality, we can assume that they are on the vertical half line x = 0 , y > 0. Therefore w k = b k i and w j = b j i . When we substitutethese values into equation (126) we obtain(132) − R b k = n X j =1 j = k m j b j ( b j − b k ) | b k − b j | , k = 1 , . . . , n. Since the particles do not collide, we can assume, without loss of gen-erality, that 0 < b < · · · < b n . Then, for k = 1, we can conclude from (132) that − R b = n X j =2 m j b j ( b j − b ) | b − b j | . But the left hand side of this equation is negative, whereas the righthand side is positive. This contradiction completes the proof. (cid:3) Acknowledgments. Florin Diacu acknowledges the partial supportof an NSERC Discovery Grant, whereas Ernesto P´erez-Chavela andJ. Guadalupe Reyes Victoria acknowledge the partial support receivedfrom Grant 128790 provided by CONACYT of M´exico. References [1] J. Bertrand, Th´eor`eme relatif au mouvement d’un point attir´e vers un centrefixe, C. R. Acad. Sci. (1873), 849-853.[2] W. Bolyai and J. Bolyai, Geometrische Untersuchungen , Hrsg. P. St¨ackel,Teubner, Leipzig-Berlin, 1913.[3] F. 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