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Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem
In the current article, we study the kite four-body problems with the goal of identifying global regions in the mass parameter space which admits a corresponding central configuration of the four masses. We consider two different types of symmetrical configurations. In each of the two cases, the existence of a continuous family of central configurations for positive masses is shown. We address the dynamical aspect of periodic solutions in the settings considered and show that the minimizers of the classical action functional restricted to the homographic solutions are the Keplerian elliptical solutions. Finally, we provide numerical explorations via Poincaré cross-sections, to show the existence of periodic and quasiperiodic solutions within the broader dynamical context of the four-body problem.
To understand the dynamics presented by a total collision of the masses or the equilibrium state of a rotating system, we are led to the concept of central configurations. A configuration of bodies is central if the acceleration of each body is a scalar multiple of its position [1–4]. Let and denote the position and the mass of the th body, respectively. Also, let represent the distance between the th and th bodies. An -body system forms a planar noncollinear central configuration [5, 6] ifwhere and represent the area of the triangle determined by the sides and .
The four-body problem can be considered from two different perspectives. The perturbative approach where we study the dynamical aspects as a perturbation of the three-body dynamics and assume that one of the masses is vanishingly small, or the global approach where we allow the masses to vary independently and stay positive. In this paper, we take the global approach and will study analytically the problem of central configurations and their dynamical aspects.
The computation of central configurations is a difficult problem for . To overcome this difficulty, symmetries or other restriction methods are used to reduce the number of variables and obtain partial answers; see, for example, Cors and Roberts ; Albouy et al. ; Shoaib et al. ; Érdi and Czirják . In this paper, we consider a four-body problem with one axis of symmetry so that the four different masses make a convex or concave kite.
Since the classification of central configurations as one of the problems for the 21st century by Smale , it has attracted a lot of attention in recent years and has helped in the understanding of the -body problem [12–23]. Ji et al.  and Waldvogel  study a rhomboidal four-body problem with two pairs of masses and use Poincaré sections to find regions of stability for the rhomboidal four-body problem. In addition, Waldvogel  also takes advantage of the simplicity of the equations and study its collisions and escape manifolds. Yan  considers the same problem for four equal masses and studies the linear stability of its periodic orbits. One of our results will consider the same model but with only one pair of masses. We will analytically derive regions of central configurations and will also investigate the existence of periodic orbits. Mello and Fernandez  prove the existence of kite central configurations for four- and five-body problems on a circle. In one special case, our rhomboidal model is similar to their model for which we also identify a number of periodic orbits and discuss its action minimizing orbits. Gordon  has proved that the elliptic Keplerian orbit minimizes the Lagrangian action of the two-body problem with periodic boundary conditions. It is also known that the Eulerian and Lagrangian elliptical solutions for the planar three-body problem are the variational minimizers of the Lagrangian action functional [29, 30]. In the study by Mansur and Offin ; Mansur et al. ; Mansur et al. , the authors have extended these ideas to prove that the homographic solutions to the constrained parallelogram four-body problem are the variational minimizers of the Lagrangian action functional. In this paper, we prove that the minimizers for the action functional restricted to the homographic solutions are the Keplerian elliptical solutions for the four-body problem with three equal and unequal masses. Perez-Chavela and Santoprete  show the existence of kite central configurations for a pair of symmetric masses and show that such a configuration must always possess a symmetry. Similarly, Celli  proves the existence of planar diamond and trapezoidal central configurations for two pairs of equal masses. Corbera and Llibre  give a complete classification of the same problem and show that this setup has exactly 34 different classes of central configurations.
More recently, Deng et al.  and Corbera et al.  prove that any four-body setup with perpendicular diagonals must be a kite [35, 38]. Santoprete  studies a four-body problem with a pair of equal masses and a pair of parallel opposite sides and show that if the opposite masses are equal, then the four-body arrangement must have a line of symmetry and will be a kite.
The paper is organized as follows: Section 2 discusses the equations of motion for the four-body problem. Section 3 discusses the existence of central configurations and the action minimizing orbits for the four-body problem where three masses are equal and arranged at vertices of an isosceles triangle and the fourth mass is on the axis of symmetry. In Section 4, we discuss the variational techniques where the action functional corresponding to these family of solutions is shown to be a minimizer. Section 5 discusses the existence of central configurations and the action minimizing orbits for the four-body problem where two symmetric masses are equal on the horizontal axis and two nonequal masses are on the vertical axis.
2. Equations of Motion
Consider four positive point masses , , , and having position vectors and interbody distances . For a general four-body setup, equation (1) gives the following six central configuration equations when :
Lemma 1. Consider a four-body problem with masses , , , and and position vectors , , , and , where and , then(a)The symmetric masses and are equal.(b)The central configuration equations are
Proof. Consider four positive masses , , , and with position vectors , , , and , where and . Using the definitions of , , and , we obtainwithUsing the symmetry of the problem and the relations (4) and (5), it is trivial to see thatSince and , therefore . This completes the proof of Lemma 1 (a).
From the geometry of the problem, and are both isosceles, and therefore , , and and hence . This also implies that . By a similar argument, it can be shown that . This leaves two independent equations and from the set of equations given in (2). This completes the proof of Lemma 1.
3. Three Equal Masses at the Vertices of a Triangle and a Fourth Mass on the Axis of Symmetry
In this section, we consider a four-body problem where three equal masses are arranged at the vertices of an isosceles triangle and a fourth mass is on the axis of symmetry as shown in Figure 1. We start by showing the existence of central configuration for a concave kite four-body problem and then explicitly find regions where such a configuration exists for positive masses. We also discuss the action minimizing orbits for this particular problem.
3.1. Central Configurations
Theorem 1. Consider four point masses and having position vectors , , , and , where and . Then, there exists a unique mass ratio :such that is a central configuration for in subject to the constraint . The region and the constraint are given below:where and .
Before we attempt to prove Theorem 1, we will need help from the following lemmas.
Lemma 2. The function defined by (8) is negative for all and .
Proof. Let and , then we haveConsider the equation of the straight line segment that lies in the first quadrant:For positive and ,is equivalent toTo prove that , we need to show thatFor , it is trivial to see thatSimilarly, to show that , we needInequality (16) is equivalent toExpanding the left-hand side of (17), we obtain the following:For positive and ,Consequently, for and .
This completes the proof of Lemma 2.
Lemma 3. The partial derivative of defined by (3) satisfies , for all and .
Proof. For positive and , we havewhere and is the derivative of w.r.t. .
Since for all and , thenUsing the fact that and , we getSince and are positive, thereforeThis completes the proof of Lemma 3.
Remark 1. Numerically, it is easy to show that when and . For , and , when , see Figure 2. Therefore, by intermediate value theorem has at least one root when .
Lemma 4. Consider the function defined by (3). Then, for any there exists an interval containing and an interval containing such that there is a unique continuously differentiable function defined on with that satisfies .
Proof. Let be any number in the interval ; then, using Lemma 3, we have . Then, numerically, one can check that for . Thus, by the mean value theorem, there exists at least one , solution of . By Lemma 3, for all , hence the solution is unique. Since has continuous partial derivatives and , with , then by the implicit function theorem, there exists an open interval containing and an interval containing such that there is a unique continuously differentiable function defined on with that satisfies . This completes the proof of Lemma 4.
Proof of Theorem 1. Let and , and then from Lemma 1, we obtain the following central configuration equations:Solving the above equations, we obtainsuch that constraint (3) holds, where and . It is proved in Lemmas 2, 3, and 4 that constraint (8) is satisfied by showing the existence of a smooth curve:To find the region where , we solve the following inequality for and :The functions and are positive in and , respectively, whereTherefore, the configuration shown in Figure 1 forms a central configuration in , wheresuch that (8) holds. Since has an absolute minimum at and is monotonically decreasing for and increasing for . Therefore, for . Hence, the region simplifies toThe region is shown in Figure 3. This completes the proof of Theorem 1.
To be able to comment on the values of in the central configuration region, we use interpolation techniques and write the solution of as , whereThe function accurately approximates the solution of in the central configuration region where is positive. The approximation error is between and . This gives as a function of as follows:The function is a bounded, well-defined continuous function of except when . To identify the values of where , we write it asThe numerical solution of shows that it has three real roots , , and . However, a careful observation of the region of existence of central configuration for the four-body problem in Figure 1 shows that defines a boundary between the region of existence and nonexistence and and are outside the domain of interest. Hence, uniquely defines the positive values of the mass ratio for the four-body problem as described in Theorem 1. The region of existence of central configuration for the four-body problem with three equal masses is given in Figure 3. Taking advantage of the interpolated expression , is given for in Figure 4 which shows to be an increasing function of with the minimum and maximum at the end points of the domain.
3.2. Action Minimizing Orbits in the Triangular Four-Body Problem
In this section, we introduce the analytical description of a family of periodic solutions in the four-body problem using variational techniques.
Theorem 2. For the four-body problem considered in Theorem 1, the minimizers for restricted to the homographic solutions are precisely the Keplerian elliptical solutions and the minimum of the action is equal to .
Define the Lagrangian action aswhere the Lagrangian is defined byLet us call the -coordinate of the center of mass in the configuration described earlier in Section 2, thenand the center of mass isIn this case, we have the following Cartesian coordinates for the points :Observe thatWe focus on solutions of the form . These solutions are called homographic solutions. We will restrict the action functional to solutions of this type.
Proof of Theorem 2. The kinetic energy term is equal towhere we have used . The potential energy is given byand using , we getMultiplying and dividing by , we obtainAs defined previously, we use , and by letting , we haveNow, we are ready to compute the action restricted to this class of homographic solutions. We haveLetThen,The infimum of isBy Gordon’s result, we haveThen,Let , then attains its infimum at if and only if attains its infimum at . It is challenging to see that for positive values of and , the function is convex and coercive. However, we have proved that can be written as and used interpolation to find as given in equation (31). Hence, we can now write , and as functions of one variable. For convexity, we use the second derivative test and show that when Hence, is convex when . For coercivity, we see that is continuous for all positive values of , as , and when , , which implies is coercive. Hence, attains at unique and satisfies .
4. Three Unequal Masses
In this section, we discuss a four-body problem which has two symmetric equal masses on the horizontal axis and two nonequal masses and on the vertical axis. The position vectors of the four masses , , , and are , , , and , respectively. For , the four masses make an isosceles triangle with inside the triangle (Figure 5), and for , the convex kite configuration is obtained (Figure 6).
4.1. Central Configurations
Theorem 3. Consider four point masses , having position vectors , , , and , where and such that is a central configuration.(a)Then, there exist unique mass ratios:(b)The region of existence of central configuration where the four positive masses are arranged in a concave kite configuration is given by(c)The region of existence of central configuration where the four positive masses are arranged in a convex kite configuration is given by
Proof of Theorem 3. Consider four point masses with position vectors and masses , , and , where . The solution of and giveswhere , , and . The mass ratios and are well-defined functions of and except at and .
To find central configuration regions where , it is sufficient to show that and have the same sign. It is trivial to see that when . Similarly, when . Hence, is positive inThe complement of , where , is given byIt is to be noted that the sign of will be determined according to whether the four-body configuration is concave or convex. After some simplifications, the central configuration region for is given bySimilarly, the central configuration region for is given byConsider the mass ratio . Let . Since , therefore for , must have the same sign. It is trivial to see that when and for all and . Therefore, inSimilarly, when (rhomboidal configuration), the central configuration region where is given byHence, the region of existence of central configuration for the concave kite four-body problem where all the masses , , and are positive is given by and the corresponding convex kite central configuration region is given by , whereThe regions and are shown in Figures 7 and 8, respectively.
4.2. Action Minimizing Orbits in the Convex Kite Four-Body Problem
In this section, we discuss the minimization property of a four-body problem which has two equal masses on the horizontal axis and two positive masses and on the vertical axis, which is also the axis of symmetry.
Theorem 4. For the four-body problem considered in Theorem 1, the minimizers for restricted to the homographic solutions are precisely the Keplerian elliptical solutions, and the minimum of the action is equal to .
Let us call the -coordinate of the center of mass in the configuration described earlier in Section 3.1, thenand the center of mass is . Observe that
Proof of Theorem 4. The kinetic energy term is equal towhere we have used the fact that .
On the other hand, the potential energy is given byand using , we getMultiplying and dividing by , we obtainDefining , , and ,Now, we are ready to compute the action restricted to this class of homographic solutions: