Tangential Trapezoid Central Configurations

A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the in-circle or inscribed circle. In this paper we classify all planar four-body central configurations, where the four bodies are at the vertices of a tangential trapezoid.


INTRODUCTION
The classical n-body problem concerns the study of the dynamics of n particles interacting among themselves by their mutual attraction according to Newtonian gravity.
Let x i ∈ R d (i = 1, . . . , n) denote the position vector of the i-body, and let m i ∈ R + (i = 1, . . . , n) denote the mass of the i-body. R d is the Euclidean space (d = 2 or 3). By Newton's law of motion and Newton's gravitational law the equations of motion of the n-body problem are governed bÿ where r ij = |x i − x j | is the mutual Euclidean distance between the i-body and the j-body. Here we take the gravitational constant G = 1.
The vector x = (x 1 , . . . , x n ) ∈ (R d ) n is called the configuration of the system. Define δ(x) as the dimension of a configuration x, i. e., the dimension of the smallest affine space of R d containing all of the points x i . Configurations with δ(x) = 1, 2, 3 are called collinear, planar and spacial, respectively.
When n = 2, the n-body problem has been completely solved. However, for the n-body problem for n 3 the complete solution remains open. Let M = m 1 + · · · + m n , c = m 1 x 1 + · · · + m n x n M be the total mass and the center of masses of the n bodies, respectively.
A configuration x is called a central configuration if the acceleration vectors of the n bodies are proportional to their positions with respect to the center of masses with the same constant of proportionality, i. e., n j=1,j =i where λ is the constant of proportionality.
A central configuration is invariant under isometries and homotheties with respect to the center of masses. So we say that two central configurations x = (x 1 , . . . , x n ),x = (x 1 , . . . ,x n ) are equivalent if we can pass from one to the other through a dilation or a rotation with respect to the center of the mass. This relation is of equivalence. Therefore, when studying central configurations, we only count the classes of central configurations defined by the above equivalence relation.
Central configurations play an important role in celestial mechanics. First, the knowledge of central configurations allows us to obtain homographic solutions of the n-body problem (see [35]). We recall that a homographic solution of the n-body problem is a solution such that the configuration remains constant up to rotation and scaling. Second, there is a relation between central configurations and the bifurcations of the hypersurfaces of constant energy h and angular momentum c (see [37,46]). Third, if the n bodies are going to a simultaneous collision or to a total parabolic escape to infinity, then the configuration of n bodies is asymptotic to a central configuration(see [20,25,43,47]).
In this paper we are interested in the planar 4-body problem. Simó [45] studied numerically the class of central configurations for the 4-body problem with arbitrary masses. In 2006 the finiteness of central configurations for the 4-body problem has been proved by Hampton and Moeckel [26] with the assistance of a computer. Later on Albouy and Kalsoshin [6] proved this result without using a computer.
For m 1 = m 2 = m 3 = m 4 Llibre found all the planar central configurations by studying the intersections of two curves and assuming that the central configurations have a straight line of symmetry, see [30]. Later on Albouy proved the existence of such symmetry and provided a more analytical proof for the central configurations with equal masses.
The central configurations with three equal masses were studied by Bernat et al. They classified the noncollinear kite central configurations. For more details, see [14], also see [29].
In 2010 Piña and Lonngi [42] found new properties of the symmetric and nonsymmetric central configurations for the 4-body problem.
MacMillan and Bartky [34] proved that there is a unique isosceles trapezoid central configuration for the 4-body problem when two pairs of equal masses are located at the adjacent vertices of a trapezoid. Long and Sun [33] studied the convex central configurations with three equal masses and they proved that the central configurations must possess a symmetry. Pérez-Chavela and Santoprete [41] generalized further and proved that central configurations must possess such symmetry when two equal masses are located at the opposite vertices of a quadrilateral and at most only one of the remaining masses is larger than the equal masses. Later on Albouy [5] obtained the symmetry of convex central configuration with two equal masses at the opposite vertices.
For m 1 = m 2 = m 3 = m 4Á lvarez and Llibre [7] characterized the convex and concave central configurations with an axis of symmetry.
Using these previous results on the symmetries, Corbera and Llibre [15] gave a complete description of the families of central configurations with two pairs of equal masses and two equal masses sufficiently small.
Cors and Robert [16] studied the case when 4 masses are located at the vertices of a cyclic quadrilateral, see also [10].
The trapezoid central configurations have been studied in [16]. Here we want to see which of these trapezoid central configurations are tangential.
A tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the in-circle or the inscribed circle. Without loss of generality we take m 1 = 1 and assume that the positions of four bodies at the vertices of a trapezoid are where a > 0.

Lemma 1.
If the configuration of 4 masses is a tangential trapezoid with the vertices x 1 , x 2 , x 3 and x 4 , as shown in Fig. 1, then We characterize all planar 4-body problem central configurations, where the four bodies are at the vertices of a tangential trapezoid. Let We associated with x the matrix: X k denotes the matrix obtained by deleting from the matrix X its kth column and its last row. Then let D k = (−1) k+1 det (X k ) for k = 1, . . . , 4. The equations for the central configurations (1.1) of the 4-body problem were written by Dziobek [17] (see also Eq. (16) of [24]) as the following 12 equations with 12 unknowns: The unknowns of Eq. (2.1) are the mutual distances r ij , the variables D i , and the constants c k (k = 1, 2).
The first six Dziobek's equation (2.1) are Multiplying Eqs. (2.2) in order that each one at the right have the expression c 2 2 D 1 D 2 D 3 D 4 , and since the masses must be positive, we obtain the Dziobek relation: The relation holds for every planar central configuration of the 4-body problem.
We can solve c 1 from the Dziobek relation and we have .

(2.4)
If we define To prove Lemma 1, we shall use Pitot's theorem, which states that in a tangential quadrilateral the two sums of lengths of opposite sides are the same. For a proof see, for instance, [27].
Proof (of Lemma 1). By Pitot's and Pythagoras' theorems and Fig. 1 we find that Isolating c from this equality, we obtain the conclusion of the lemma.
Proof (of Theorem 1). From (1.2) we have (3.1) Substituting these expressions together with the values of D k for k = 1, 2, 3, 4 into the last six equations of (2.1), we find that they are identically zero for a tangential trapezoid configuration.  Dividing conveniently two different equations of (2.2), we obtain  In conclusion, the tangential trapezoid central configurations must satisfy (e 2,3 , e 2,4 , e 3,4 ) = (0, 0, 0). Substituting (3.1) into e 2,3 = 0, e 2,4 = 0, e 3,4 = 0, we find that these three equations are satisfied if and only if the equations have solutions where, respectively, In summary, studying the graph of D = 0 with variation from 1 to √ 3