The Rhomboidal 4-Body Problem Revisited

Transcription

1 Qual. Theory Dyn. Syst. (2015) 14: DOI /s Qualitative Theory of Dynamical Systems The Rhomboidal 4-Body Problem Revisited Martha Alvarez-Ramírez 1 Mario Medina 1 Received: 15 November 2014 / Accepted: 12 June 2015 / Published online: 4 July 2015 Springer Basel 2015 Abstract We consider the planar rhomboidal 4-body problem where the masses are equal by pairs. In this paper we briefly review some recent results related to binary and total collision regularization, central configurations and existing periodic orbits of this problem, but some of them are not widely known to the celestial mechanics community. Keywords 4-Body problem Binary collision Quadruple collision Regularization Mathematics Subject Classification Primary 70F07; Secondary 70F15 1 Introduction The motion of systems of n-bodies under their mutual gravitational attraction has always fascinated mathematicians and astronomers. Today the few-body problem is recognized as a standard tool in astronomy and astrophysics, from solar system dynamics to galactic dynamics [20]. Approximately two thirds of the stars in our Galaxy exist as part of multistellar systems. Stellar situations in the solar system can be seen as living in the province of few body problems, where computations of the orbits of these systems must be done in a precise way. The number of quadruple stellar systems in the Galaxy is estimated to be of the order of thousands of millions. It is worth to B Martha Alvarez-Ramírez mar@xanum.uam.mx Mario Medina mvmg@xanum.uam.mx 1 Departamento de Matemáticas, UAM-Iztapalapa, Iztapalapa, Mexico, D.F., Mexico

2 190 M. A. Ramírez, M. Medina note that the 4-body problem is increasingly being used for explaining many complex dynamical phenomena that appear in the solar system and exoplanetary systems. The n-body problem describes the dynamics of n 2 particles with masses m k, k = 1,...,n moving in the plane R 2, and is governed by the Newtonian force law of inverse square distance. Let r i R 2 be the position coordinate of the body with mass m i.the equations which describe the motion of the masses are m i d 2 r i dt 2 = n j=1 j =i Gm i m j (r j r i ) r i r j 3 = U r i, i = 1,...,n (1) where G is the gravitational constant and the potential energy is given by U(r) = 1 i< j n G m im j r i r j. A solution of a system of differential equations, as the given above, is said to experience a singularity at time t < if the solution cannot be analytically extended beyond t. Note that equations (1) are defined everywhere and are real analytic except at points in the physical space that are occupied by at least two bodies. In order to be more precise, by defining the sets ij ={(r 1,...,r n ) R 2n r i = r j } and = i< j ij, we see that potential U is a real analytical function in R 2n. By the existence and uniqueness theorem of solutions for differential equations, given any set of initial position in R 2n and initial velocity in R 2n there is a unique solution for the differential equations (1), and a maximal interval of existence 0 t < t of the solution so that the solution satisfies such initial conditions. It is said that this solution experiences a singularity if t <. In case t is a singularity for a solution r(t) = (r 1 (t),..., r n (t)), the singularity is due to collision or is a collision singularity if there is a point r so that r(t) r as t t. Otherwise we shall say that the singularity is a noncollision singularity. The extension of a collision solution for t > t has been studied in several cases. If the extension is possible in some sense it is said that the collision is regularizable. In general, the n-body problem has two types of singularities: collisions and the infinite expansion in a finite interval of time; a long standing question that many people have dealt with is the following: are these singularities regularizable? The interest of researchers on regularization is not purely theoretical. Indeed, the singularities and their vicinity are a source of large errors and strong instabilities in the numerical integrations; these difficulties are removed when the singularities are eliminated. The goal of this paper is to make an overview for some of the different ideas and techniques that are dispersed in the available literature and have been very useful to study the planar symmetric 4-body problem with equal masses by pairs with a rhomboidal configuration, in which the initial positions and velocities are symmetric with respect to the axes in the plane, thus the system has two degrees of freedom. This problem has been studied among others by Delgado Fernández and Pérez-Chavela [6], Lacomba and Pérez-Chavela [11,12], Ji et al. [9], Waldvogel [36] and it is also mentioned by Shibayama [30] in a study of periodic orbits.

3 The Rhomboidal 4-Body Problem Revisited 191 The rhomboidal 4-body problem is well suited for studying many of the features and difficulties that appear in most of the special cases of general n-body problem and even inthen-body problem itself. Namely, regularization of the binary collisions, periodic solutions, homothetic solutions, central configurations, as well as total collision and escape orbits, just to mention some of them. Besides, as mentioned by Waldvogel [36], the rhomboidal 4-body problem can be see as a two degrees of freedom subproblem of the Caledonian 4-body problem (CSFBP) which is the planar symmetric 4-body problem introduced by Roy and Steves [22], where two pairs of equal masses move in a fixed plane, occupying positions of central symmetry with respect to the origin. The CSFBP involves two pairs of masses moving in coplanar, initially circular orbits, starting in a collinear arrangement [23]. The CSFBP as proposed by Steves and Roy is relevant in studying the stability and evolution of symmetric quadruple stellar clusters and exoplanetary systems of two planets orbiting a binary pair of stars, see for example Steves and Roy [29] and Széll et al. [34]. Of course, CSFBP has four degrees of freedom. This paper is organized as follows. In Sect. 2, we set the 4-body problem in a general frame, then we introduce the particular case considered in this work. Section 3 treats about central configurations in the rhomboidal 4-body problem, because they play a particular role on collision orbits. Section 4 is devoted to two different types of orbits for the rhomboidal 4-body problem, collision and collision free orbits, in particular we consider Aarseth and Zare regularization, and also consider the McGehee technique of blowing up, both applied to the rhomboidal 4-body problem in order to regularize the binary and total collisions, respectively. Some numerical results are presented in this section. The symmetric periodic solution with rhomboidal configuration collision-free obtained by Chen [5] is given in Sect Finally, some conclusions are presented. 2 Rhomboidal 4-Body Problem We take a planar inertial coordinate system (x, y) with origin at the center of masses. Let r i = (x i, y i ) be the position of the body with mass m i, i = 1, 2, 3, 4. Then the equations of motion of the 4-body problem are given as below: r 1 = m 2 r12 3 r 12 + m 3 r13 3 r 13 + m 4 r 14, r 14 r 2 = m 1 r12 3 r 21 + m 3 r23 3 r 23 + m 4 r 24, r 24 r 3 = m 1 r 31 + m 2 r 31 r32 3 r 32 + m 4 r 34, r 34 r 4 = m 1 r 41 + m 2 r 42 + m 4 r 43, (2) r 41 r 42 r 43 where r ij = r j r i are the relative positions of the bodies with respect each other, r ij are the distances between the particles with positions r i and r j and units are chosen so that the gravitational constant is G = 1. The rhomboidal 4-body problem configuration means four particles with masses m 1, m 2, m 3, m 4 respectively, which are located in the plane at the vertices of a rhom-

4 192 M. A. Ramírez, M. Medina Fig. 1 The rhomboidal 4-body problem m 2 y x m 1 m 3 m 4 bus, where m 1 = m 3 > 0 are in the horizontal direction and m 2 = m 4 > 0in the vertical direction, see Fig. 1. The particles are given symmetric initial conditions in positions and velocities with respect to the axes in the plane and always keep a symmetric rhomboidal configuration under the law of Newton attraction. Due to the symmetry of the configuration, we have r 4 = r 2, r 3 = r 1, r 34 = r 12, r 14 = r 23. The symmetry conditions applied to initial positions and velocities reduce the dimension of the phase space, making it a useful tool. Therefore, the equations of motion (2) are reduced as follow r 1 = m 2 r12 3 r 12 + m 3 r13 3 r 13 + m 4 r14 3 r, 14 r 2 = m 1 r12 3 r 21 + m 3 r23 3 r 23 + m 4 r24 3 r. (3) 24 Since the resulting forces acting on m 1 and on m 3 have vertical component equal to zero and are of equal magnitude, m 1 and m 3 can move horizontally on the x-axis. In a similar way the motion of m 2 and m 4 is restricted to the y-axis. Then we assume that the coordinates of bodies m 1 = m 3 > 0 lie at (±x, 0), and those of the other two bodies with masses m 2 = m 4 > 0 lie at (0, ±y). In terms of this coordinates, the motion equation (3) can be written as ẍ = m 1 4x 2 2m 2 x (x 2 + y 2 ) 3/2, ÿ = m 2 4y 2 2m 1 y (x 2 + y 2, (4) ) 3/2

5 The Rhomboidal 4-Body Problem Revisited 193 where dots denote differentiation with respect to time t. The Hamiltonian associated is H = ẋ2 2 + ẏ2 2 m2 1 4x m2 2 4y 2m 1m 2 x 2 + y 2. 3 On Central Configurations and the Relation of Positions and Masses for the Rhomboidal 4-Body Problem Since the general solution to the n-body problem cannot be obtained, one of the directions to follow consists in finding particular solutions. Some of these particular solutions correspond to homographic solutions and central configurations, where a homographic solution is a configuration which is preserved for all time. A classical problem is to determine the central configurations of n-body motions. Let us recall that a configuration is called a central configuration if the acceleration vector for each body is a common scalar multiple of its position vector. Specifically, in the Newtonian n-body problem with the center of mass at the origin, for each index i, the identity n j=1 j =i Gm i m j (r j r i ) r i r j 3 + λm i r i = 0 (5) holds for some scalar λ. Central configurations are at the core of many and interesting issues in celestial mechanics. A pleiad of authors have made important and deep contributions to their understanding. Collapse, ejection orbits and parabolic orbits are related to them (Saari [25 28]). The works of Euler [7] and Lagrange [13] on relative equilibria, the number of collinear cental configurations (see [19]), the problem of their finiteness (Wintner [37] and Smale [32]) and the fact that central configurations are linked to homographic solutions [37]. These are some of the reasons because central configurations deserve so a special place in the realm of celestial mechanics. The advances obtained in the search for a total comprehension of the central configurations, even for the 4-body problem are not complete, but some partial results have been obtained, although a wide number of researchers have been involved in this quest. For the 4-body problem Albouy [2] gave a complete classification of the possible central configurations for the equal masses case and Hampton and Moeckel [8] proved that the number of central configurations is finite, at least there are 32 central configurations at most 8472 equivalence classes, also they found mass-independent bounds. As we know, central configurations are determined in a unique way up to dilations and rotations, see [37]. Once we set the positions of two of the bodies which are not situated at opposed vertices of the rhombus, the locus of the other two bodies are obtained by symmetry. Long et al. [15] proved that if the semi-distance between the bodies on the vertical axis is given by x and the semi-distance between the bodies on the horizontal direction is equal to y and 1 x < y < 3x then the masses of the 3

6 194 M. A. Ramírez, M. Medina bodies on the horizonal axis are equal (m 1 = m 3 ) and, similarly, the masses of the bodies on the vertical axis are equal too (m 2 = m 4 ). Then the values of the masses m 1 and m 2 are fixed as functions of the semi-distances x and y as given by m 1 = 4y3 (x 2 + y 2 ) 3/2 (8x 3 (x 2 + y 2 ) 3/2 ) 64x 3 y 3 (x 2 + y 2 ) 3 (6) and m 2 = 4x3 (x 2 + y 2 ) 3/2 (8y 3 (x 2 + y 2 ) 3/2 ) 64x 3 y 3 (x 2 + y 2 ) 3. (7) Also, once we give values to the masses m 1 and m 2, we can see that the values of the parameters given by the semi-distances between the vertical and the horizontal vertical binaries, x and y, respectively, are determined in an unique way through Eqs. (6) and (7). One interesting property about central configurations is that, in the plane, if we place n particles of equal masses at the vertices of a regular n-gon or at the vertices of a regular (n 1)-gon and the last mass particle at the center of this (n 1)-gon, we obtain central configurations. Besides, there are some highly symmetrical configurations that have recently been studied, among them there is the set of plane configurations consisting of two regular n-gons (n 2), where an n-gon is a configuration where all the n particles have the same mass and lie at the vertices of a regular n-gon. Two n-gons form a configuration of two nested n-gons if the corresponding vertices are aligned and the particles in the same n-gon have equal masses. A configuration of two twisted n-gons consists of two n-gons so that there is a rotation of the corresponding particles of the outer n-gon by an angle equal to π/n. Observe something that seems to be very naive, in the case of two twisted 2-gons the configuration obtained is a rhomboidal configuration. When studying nested n-gons for n 2 among other symmetrical and spatial configurations and the bifurcations arising when varying ratio of the masses of two nested n-gons, Moeckel and Simó [18] showed that for two nested rings (n 2), and every ratio m 1 /m 2, there are exactly two planar central configurations, one with the ratio of the sizes of the two polygons less than 1, and the other one greater than 1. More recently, when fixing the masses m 1 = m 3 = 1 and by taking to be equal to 1 the size of one of the 2-gons in a configuration of two twisted 2-gons of the 4-body problem, Barrabés et al. [3] have proved that for each value of the mass ratio m, there exist only one central configuration of two twisted 2-gons of the 4-body problem (rhomboidal central configuration). Moreover, the admissible sizes a for the other 2-gon (with masses m 2 = m 4 = m) are contained in the interval (1/ 3, 3). Even more, if a = 1/ 3 then m, while as a = 3 implies m = 0, see Fig. 2. Of course, all the central configurations obtained are convex.

7 The Rhomboidal 4-Body Problem Revisited 195 Fig. 2 The rhomboidal-body problem as central configuration. This figure was taken from [3] 4 Collision and Collisionless Orbits in the Rhomboidal Problem In this part of the work we shall refer to two different types of orbits that appear in the rhomboidal problem. The first one corresponds to collision orbits, that are related to the very important notions of singularity and regularization. The second kind of orbits refer to orbits that do not suffer of collision at any time of their evolution, as the so called collision free periodic orbits. 4.1 Collision Orbits and the Rhomboidal Problem As known, a singularity at time t 0 is due to collision if as t t 0 each particle approaches a definite position in the inertial coordinate frame. A known fact about singularities without collision states that collisionless singularities are equivalent to the existence of unbounded motion as t t 0. Many important properties on singularities and regularization have been developed along time. First important facts were proved by Painlevè [21], von Zeipel [35] and Levi-Civita [14]. More recently, works by Kustaanheimo and Stiefel [10], Saari [24] and Xue [38] have given us a deeper understanding about singularities and regularization. Of course, we have not intended to be exhaustive on the people who in great measure increased our understanding of the subject. One interesting reference is given by Celletti [4] where the author presents an overview of singularities, regularizations and collisions in gravitational n-body systems. Even that for 4-bodies, non-collision singularities have measure zero, but one does not know whether they exist. In a recent and remarkable advance, Xue [38] has proved that there is a Cantor set of initial conditions in a planar 4-body problem such that

8 196 M. A. Ramírez, M. Medina all the four bodies escape to infinity in finite time avoiding collisions. Also, as we shall see in next subsections, Aarseth and Zare [1] achieved a regularization of every 2-body collision by introducing two KS transformations. Regularization is usually thought of as the process of converting the singularities that appear in a system of differential equations into regular points, and the corresponding solutions are then well behaved. The purpose of the this section is to show some regularization processes that have been applied to the rhomboidal problem. This is primarily accomplished by changing the dependent variables and also the time scale in such a fashion that the new motion equations are free of the singularities suffered by the original equations Regularizing Binary Collisions and Blowing Up of Total Collision Next, we shall concentrate on two independent works, one by Waldvogel [36], and the second, authored by Yan [39], both articles appeared in the same year and different journals. In such papers, the authors direct their attention to the regularization of the simple binary collisions that appear in this planar 4-body problem in order to deal with different aspects involving the rhomboidal 4-body problem. The existing two non total collisions that appear in this planar problem take place at the origin of coordinates. The particles moving on the vertical axis of symmetry collide at the origin as the other particles keep moving on the horizontal axis, symmetrically with respect to the origin. In a similar way, particles moving on the horizontal axis collide at the origin as the other binary move on the vertical axis, symmetrically with respect to the center of mass fixed at the origin. The attention of both authors was focused on a regularization technique for binary collisions implemented by Aarseth and Zare [1] when studying the three-body problem in which they used two KS transformations that permitted one of the bodies to have repeated collisions with the other two bodies, whose relative motion could be described by equations which even being still singular were numerically well behaved. Let us emphasize the importance of the regularization of motions close to collisions from a computational point of view. If a trajectory is close to collision, the velocities of some bodies become large enough, making it very difficult to obtain precise computations in this situation. Observe that the energy on an orbit is constant due to the fact that it is an integral of motion. So, when at least two bodies are close to collision the potential energy becomes very large, so the kinetic energy should be large too. In consequence, the velocities of these close interacting bodies should also be large, which reduces the effectiveness of the calculations for orbits near close approach of a cluster of interacting bodies. As stated before, some regularization techniques allow better ways to compute trajectories for solutions near or through binary collisions; on the other side, there are some other regularizing techniques that are better suited to obtain a good understanding of the geometric and topological properties of solutions passing close to total collision. In the case of the rhomboidal 4-body problem, the use of the McGehhe technique of blowing up total collision together with the use of an adaptation of the Sundman [33] type of regularization which permit to regularize simultaneously the non total

9 The Rhomboidal 4-Body Problem Revisited 197 collisions arising in this 4-body problem, we are able to see the geometric behavior of solutions that pass close to total collision Binary Collision Regularization by Aarseth and Zare Method Firstly, we shall concentrate on the work done by Waldvogel [36], who studied the regularization of binary collisions in the problem under consideration. The canonical form of the equations of motion is preserved if we adopt a new Hamiltonian K, where h is the (constant) total energy on the orbit under consideration dt = x 1 x 2 dτ, K = x 1 x 2 (H h). (8) Next, new coordinates ξ j and new momenta π j, j = 1, 2 are introduced as follows x j = ξ 2 j, p j = π j 2ξ j, j = 1, 2. (9) In order to preserve the Hamiltonian form, the transformation of the momenta must be such that the π j are canonical conjugates of the corresponding ξ j. This may be done by using the generating function W (p,ξ),see[31]: π j = W ξ j, j = 1, 2 with W (p,ξ)= p 1 x 1 + p 2 x 2. With the transformation (9) being canonical, it suffices to express the Hamiltonian K of Eq. (8) in terms of ξ j and π j in order to obtain the regularized Hamiltonian K = 1 8 ( π 2 1 ξ 2 2 m 1 + π 2 2 ξ 2 1 m 2 ) 1 4 (m2 1 ξ m2 2 ξ 1 2 ) 2m 1m 2 ξ1 2ξ 2 2 ξ1 4 + ξ 2 4 H 0 ξ 2 1 ξ 2 2 (10) with K (τ) = 0 on the orbit. The regularized equations of motion, for j = 1, 2 with k := 3 j, and = d/dτ, become ξ j = K = π jξk 2, π j 4m j π j = K ξ j = ξ j t = dt dτ = ξ 2 1 ξ 2 2 π 2 k 4m k + m2 k 2 + 4m 1m 2 ( ξ 4 k ξ ξ 4 2 ) 3/2 + 2H 0 ξ 2, by transformation of the variables involved. We have obtained equations of motion which are regular with respect to collisions between any pair of particles. The only cases excepted are those in which collisions between more than one pair occur simultaneously and those in which at least one of the masses vanishes. k

10 198 M. A. Ramírez, M. Medina In an independent research carried out by Yan [39], the author developed a regularization of the existing binary collisions in the rhomboidal 4-body problem when studying the existence and stability of a special type of orbits for the rhomboidal 4- body problem. The method used by Yan yielded the same adaptation for this planar 4-body problem as done by Waldvogel [36] of the technical device used by Aarseth and Zare [1] for the regularization of singularities due to the binary collisions in the 3-body problem that was achieved by generalizing a Levi-Civita type transformation and using an appropriate scaling of time. In this section we shall show that the regularization of simple collisions in the rhomboidal four body problem can be achieved by a generalized Levi-Civita type transformation and an appropriate scaling of time, as adapted from Aarseth and Zare [1]. For a better understanding of the behavior of the motion of the bodies in a neighborhood of binary collision, the standard technique is to make a change of coordinates and a rescaling of time. In the new coordinates, the orbits which approach binary collision can be extended across the collision in a smooth manner with respect to the new time variable. In [39] the author studied the existence and linear stability of rhomboidal periodic orbits for a regularized version of the planar 4-body problem. He also considers 4 equal masses that, at the starting time t = 0, are in the points (±1, 0) and (0, ±1) with initial velocities (±v, 0) and (0, v) respectively. The family of symmetric periodic orbits with regularizable collisions have alternating binary collisions of the symmetric pairs of equal masses. The analytic existence of these periodic orbits in the rhomboidal 4-body 1, m, 1, m problem has been given by Yan [39] form = 1, and by Shibayama [30] for arbitrary m > 0. For m = 1, the linear stability of this periodic orbit in the rhomboidal 4-body problem has been numerically established by Yan [39] using Roberts s symmetry reduction method. In [39] the author studied the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass 4-body problem, for which the two binary collisions are regularized by the Aarseth and Zare [1] method. First we take w 1 = 2ẋ,w 2 = 2ẏ and introduce a new set of canonical variables in two dimensions, denoted by Q 2 1 = x, Q2 2 = y, P 1 = 2Q 1 w 1 and P 2 = 2Q 2 w 2.Anew time variable s is introduced through dt/ds = xy. Then the regularized Hamiltonian H = xy(h h) has the form H = xy(h h) = 1 6 P2 1 Q P2 2 Q Q Q2 2 4Q2 1 Q2 2 hq 2 Q Q2 2, (11) Q4 2 where h is the total energy of the Hamiltonian H and h = h(0) = 2v The regularized Hamiltonian H gives the following differential equations of motion: Q 1 = 1 8 P 1Q 2 2, Q 2 = 1 8 P 2Q 2 1,

11 The Rhomboidal 4-Body Problem Revisited 199 with initial conditions P 1 = 1 8 P2 2 Q 1 + Q 1 + P 2 = 1 8 P2 1 Q 2 + Q 2 + 8Q 1 Q 6 2 (Q Q4 2 )3/2 + 2hQ 1Q 2 2, 8Q 2 Q 6 1 (Q Q4 2 )3/2 + 2hQ 2Q 2 1, (12) Q 1 (0) = 1, Q 2 (0) = 1, P 1 (0) = 4v, P 2 (0) = 4v, (13) where derivatives are with respect to s, and h = 2v Yan [39] showed that there exists an initial velocity v 0 such that system (12) has a periodic solution for v = v 0. Theorem 1 [39] At v = v 0, the solution of the regularized Hamiltonian rhomboidal periodic 4-body orbit with binary collisions in R 2. H is a Numerical Simulations Using Zare Aarseth Type Regularization Coordinates Next, we present some numerical simulations in order to show the behavior of trajectories associated to the system of differential equations (12) with initial conditions (13). To obtain the orbits we see in Fig. 3, we considered several values for the parameter v contained in the aforementioned initial conditions and used the numerical integrator in the Matlab R environment. In this way, it is showed that the modification to the Zare Aarseth regularization obtained by Waldvogel and Yan to regularize non total collisions for the rhomboidal 4-body problem is well suited to perform numerical computations Global Flow on the Quadruple Collision Manifold The aim of this section is to describe the geometric nature of the dynamics of trajectories near total collision in the rhomboidal 4-body problem. In order to attain this goal we recall the now classic transformation introduced by McGehee [16], which has the effect of blowing up the singularity associated to total collision, and replaces this singularity with an invariant manifold which is immersed in the full phase space. This invariant set is usually called the total collision manifold, and the flow is smoothly extended over the total collision manifold. Once we understand the flow on this collision manifold completely, then one is able to read off the behavior of orbits which reach, are ejected from or pass close to quadruple collision. By properly rescaling the spatial coordinates and the time, according McGehee s ideas, and then simultaneously regularizing the singularities due to the existing binary collisions with a Sundman type of transformation, we are able to study the behavior of solutions passing close to total collision (or quadruple collision) in the rhomboidal 4-body problem. After using the aforementioned McGehee type of coordinates and then resorting to the simultaneous regularization it is obtained that the dimension of the total collision manifold is two, that the equations of motion for the flow on the

12 200 M. A. Ramírez, M. Medina Fig. 3 Orbits associated to (12) and(13) with values v = 1.1, 1.201, 1.23 and 1.3, in clockwise sense starting at the top left corner total collision manifold present two rest points and the fact that in the new flow, the collision or ejection orbits are asymptotic (for positive or negative time) to one of these rest points. For this section, we follow closely results obtained by Delgado Fernández and Pérez-Chavela [6] where the global flow on the quadruple collision manifold was studied. Furthermore, we have that after scaling the time and all solutions that terminate at quadruple collision in finite time now are made to tend asymptotically to the hyperbolic equilibrium points for the flow that are located on the total collision manifold. Let α be the ratio of the masses α = m 2 /m 1 as a parameter. Then we can take m 1 = m 3 = 1 and m 2 = m 4 = α. We shall concentrate on the nature of the quadruple collision of the rhomboidal 4-body problem in the case of negative energy, so h < 0 throughout this section. When resorting to the McGeehe change of coordinates the potential energy can be written as U(θ) = 1 2 cos θ + α5/2 2sinθ + 4 2α 3/2 α cos 2 θ + sin 2 θ where 0 θ π/2, and the corresponding equations of motion present singularities due to two different kinds of binary collisions which are simultaneously regularized

13 The Rhomboidal 4-Body Problem Revisited 201 through a Sundman type of regularization together a time change of scale. For full details see [6]. After the McGehee change of coordinates and the simultaneous regularization, the equations of motion (4) become dr dτ = sin(2θ)rv 2W (θ), dv dτ = ( W (θ) 1 sin(2θ) ) 4W (θ) (v2 4rh), dθ dτ = w, ( dw dτ = cos(2θ) 1 + sin(2θ) ) vw sin(2θ) 2W (θ) 4 + Ẇ (θ) W (θ) 2W (θ) (sin(2θ) w2 ) (14) where Ẇ denote the derivative of W with respect to θ,w (θ) = 2 1 U(θ) sin(2θ) is the regularized potential, and the time has been rescaled by dt dτ = r 3/2 sin(2θ) 2. This last W (θ) system is an analytic vector field where the singularities due to binary collisions have now been removed. Since the topological picture of the dynamics is the same for all the energy manifolds as they keep the same sign for the energy value, a specific negative value for h is chosen so that the total energy yields w 2 sin 2θ) 1 = sin(2θ) 2W (θ) (rh v2 We are now in the position to introduce the so called total collision manifold, which is defined by the set 2 ). = { (r,v,θ,w) r = 0, w 2 + v2 sin 2 (2θ) 4W (θ) } = sin(2θ). (15) Notice that is an invariant set under the flow which is topologically equivalent to a 2-sphere S 2 minus the north and south poles. Using the energy integral, we reduce the system to obtain the set of differential equations that governs the flow on : dv dτ = ( W (θ) 1 sin(2θ) ) 4W (θ) v2 = w 2 W (θ) sin(2θ), dθ dτ = w, ( dw dτ = cos(2θ) 1 v 2 sin(2θ) ) vw sin(2θ) 2W (θ) 4 + Ẇ (θ) W (θ) 2W (θ) (sin(2θ) w2 ). (16)

14 202 M. A. Ramírez, M. Medina Point out that trajectories in the rhomboidal 4-body problem which previously began or ended at quadruple collision in finite time are now slowed down so that they tend to the total collision manifold as time goes to infinity or minus infinity. And trajectories which pass close to quadruple collision now behave very much like the trajectories on the total collision manifold itself. We summarize here the main features of the flow on the total collision manifold (see [6]). (i) The flow on is gradient-like with respect to the coordinate v. (ii) If (v 0,θ 0,w 0 ) is an equilibrium point for the flow on, then w 0 = 0, U(θ 0 ) = 0 and v0 2 = 2U(θ 0). (iii) For each value of α the potential U(θ) has exactly one critical point. (iv) The flow on has exactly two hyperbolic equilibrium points which are denoted by A and B, and correspond to v = 2U(θ 0 ) and v = 2U(θ 0 ), respectively. The sign being associated to orbits directing to total collision and the sign + is associated to orbits going out from to total ejection. (vi) For W u,+( ) A the branch of the unstable submanifold in A which corresponds to w>0(w<0), and W s,+( ) A the branch of the unstable submanifold in A corresponding to w>0(w<0) we have that W u,+ A = W s,+ B if only if W u, A = W s, B. (vii) For α = α 1 = we have W u, A = W s,+ B. (viii) For α small enough and positive, there is no intersection between the invariant submanifolds associated to the rest points in. An intersection occurs in the limiting case α = 0. Theorem 2 [6] In the rhomboidal 4-body problem there are only two values of the mass ratio α, one of them is α1 1 > 1 for which there are connections between the invariant submanifolds of the equilibrium points. By using numerical methods they proved that there are two equilibrium points for the flow on total manifold, and that there are only two values of α for which there is a connection between the invariant submanifolds of the equilibrium points. For these values of α the problem is not regularizable. 4.2 A Collision-Free Symmetric Periodic Solution of the Rhomboidal 4-Body Problem by Variational Approach Next, we shall describe the work carried out by Chen [5], who discovers and proves the existence of a new orbit for the Newtonian planar 4-body problem where all four masses are equal. The configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time. He makes use of the variational structure of the 4-body problem; firstly, he uses Jacobi-type coordinates in order to parametrize the configuration space, then uses the Hopf fibration to reduce the dimension by one, and obtains the three-dimensional reduced configuration space. Each point on the unit sphere, as in the planar 3-body problem, represents a similarity of parallelograms. In order to avoid the occurrence of collisions, he studied the behavior

15 The Rhomboidal 4-Body Problem Revisited 203 of minimizers in the reduced configuration space, obtaining that their action are greater that on the collision-free rhomboid motions. Let x i R 2 and m i be the position and the mass of the ith body, respectively (i = 1, 2, 3, 4). In position coordinates the motion equations are given by where ẍ k = x k U(x), for k = 1, 2, 3, 4, (17) U(x) = U(x 1, x 2, x 3, x 4 ) = i< j 1 x i x j, is the potential energy. If we assumed that the position vector is given by x = (x 1, x 2, x 3, x 4 ) and we associate with the equation the Lagrangian function L(x(t), ẋ(t)), the action functional becomes A(x) = T 0 L(x(t), ẋ(t)) dt, defined on the Sobolev space X = H 1 ([0, T ], V ), where V is the four-dimensional subspace (called the configuration space) ofr 8 defined by the set V ={x = (x 1, x 2, x 3, x 4 ) (R 2 ) 4 : x 1 = x 3, x 2 = x 4 }. We shall seek symmetric periodic solutions as critical points of the action functional on the Sobolev space of T -periodic trajectories. Let us consider the action functional A restricted to the space of symmetric loops X which start at a square configuration and have collinear configuration at t = T, and let X D be the collection of paths in X which have rhomboidal configuration for all time. So, in this paper we are interested in the minimizers of the action functional A restricted on X D. For simplicity, we consider four particles with equal masses, that is, m 1 = m 2 = m 3 = m 4 = 1, and impose the following symmetrical initial conditions x 1 (0) = x 3 (0), ẋ 1 (0) = ẋ 3 (0), x 2 (0) = x 4 (0), ẋ 2 (0) = ẋ 4 (0). (18) Thus, we can construct a periodic solution x = (x 1 (t), x 2 (t), x 3 (t), x 4 (t)) of (17)for time t [0, T ]. Since the system (17) with (18) has the same number of degrees of freedom as the planar 3-body problems, we may visualize the motion by choosing an appropriate coordinate system as in [17]. To do so, consider Jacobi s coordinates given by (z 1, z 2 ) := (x 2 x 1, x 2 x 1 ) C 2

16 204 M. A. Ramírez, M. Medina Fig. 4 The unit shape sphere and level curves of U(θ, φ). This figure was taken from [5] and the composition of them with the Hopf fibration, getting (z 1, z 2 ) ( z 1 2 z 2 2, 2 z 1 z 2 ) =: (u 1, u 2 + iu 3 ) R C. This quotients out the SO(2) symmetry from the coordinates. A level set I 1 (c), c > 0, of the moment of inertia I(x) := x x is a 3-sphere which is mapped onto the 2-sphere u 2 1 +u2 2 +u2 3 = c2 via this mapping. By using spherical coordinates, (u 1, u 2, u 3 ) = (r 2 cos φ cos θ,r 2 cos φ sin θ,r 2 sin φ), the motion can now be more easily visualized by projecting it to the unit sphere r = 1 (see Fig. 4). Spheres of the form r = c > 0 are called shape spheres as in the planar 3-body problem. This shape space is the space of oriented-congruence classes of parallelograms. Now, we have already formed the quotient of the configuration space by the rotation group SO(2), we have also reduced by direct isometries (translations and rotations) the 4-body problem in the plane. The three dimensional space V/SO(2) is called the reduced configuration space, and it is homeomorphic to R 3. Points on this sphere, represent parallelogram oriented configurations whose moment of inertia I are equal to 1. In this section we denote the semi-distance between the masses of the horizontal binary by ξ, and η corresponds to the semi-distance between the bodies of the vertical binaries in the rhomboidal configuration. In terms of ξ and η the action functional is given by ( ) T A(ξ, η) = ξ 2 + η ξ + 1 2η + 4 dt ξ 2 + η 2 0 The aim of Chen s analysis is to show that any minimizer x([0, T ]) must lie in the interior of an octant formed by three coordinate planes, those associated to the

17 The Rhomboidal 4-Body Problem Revisited 205 Fig. 5 The action minimizing rhomboidal motion. This figure was taken from [5] collinear, rhomboidal, and rectangular configurations, see Fig. 4. It follows that the minimizer has no collision. Proposition 1 [5] There are unique positive constants a 0 and v 0 such that (ξ 0,η 0 ) = (ξ, η) solves (4) on [0, T ], and satisfies initial conditions of the form with a = a 0,v = v 0, and ξ(0) = η(0) = a > 0, ξ(0) = η(0) = v>0 η 0 (T ) = ξ 0 (T ) = 0. This solution minimizes A over the set rhomboidal configuration X D. From this result, it follows the existence of a periodic orbit with period T = 2T, which has a rhomboidal configuration for all time and this is a collision-free orbit, as illustrated in Fig. 5. Acknowledgments The authors would like to thank the anonymous referees for their deep review and constructive comments that helped us to bring the overall quality of this paper to a higher level of rigor and presentation. References 1. Aarseth, S.J., Zare, K.: A regularization of the three-body problem. Celest. Mech. 10, (1974) 2. Albouy, A.: Symetrie des configurations centrales de quatre corps. C. R. Acad. Sci. Paris Soc. I Math. 320(2), (1995) 3. Barrabés, E., Cors, J.M., Roberts, G.: On central configurations of twisted rings (2014). (preprint) 4. Celletti, A.: Singularities, collisions and regularization theory. In: Benest, D., Froeschle, C. (eds.) Singularities in Gravitational Systems. Lecture Notes in Physics, vol. 590, pp (2002)

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