Applications of Planar Newtonian Four-body Problem to the Central Configurations

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1 Available at Appl. Appl. Math. ISSN: Vol. 1 Issue (December 17) pp Applications and Applied Mathematics: An International Journal (AAM) Applications of Planar Newtonian Four-body Problem to the Central Configurations M. R. Hassan 1 M. S. Ullah Md. Aminul Hassan 3 and Umakant Prasad 4 1 P. G. Department of Mathematics S. M. College T. M. Bhagalpu University Bhagalpur-811 Bihar India hassansmc@gmail.com Department of Mathematics Ambedkar College New Delhi India 3 GTE Bangalore-56 India mahassan1@gmail.com 4 Department of Physics T. N. B. College T. M. Bhagalpur University Bihar India Received: March 16 17; Accepted: November Abstract The present study deals with the applications of the planar Newtonian four-body problem to the different central configurations. The basic concept of central configuration is that the vector force must be in the direction of the position vector so that the origin may be taken at the centre of mass of the four bodies and the force towards the position vector multiplied by corresponding inverse mass is directly proportional to the position vector relative to the centre of mass. For applying the Newtonian four body problem to the central configuration the equations of motion of four bodies have been established in inertial frame. By the methods of previous authors some mathematical tools of the planar problem have been developed and by using them the Newtonian four-body problem have been reduced to central configuration. All the previously generated mathematical models depend upon the directed areas weighted directed areas and position vectors of the centre of the bodies. With the help of these models some conditions have been established which show that the origin is located at the centre of mass of the system. Finally these conditions have been used to some particular cases of concave configuration convex configuration and symmetric configuration to generate some tools which will be helpful for further researches of this field. Keywords: Newtonian four-body problem; Directed area; Weighted directed area; Flattetrahedron; Concave central configuration; Convex central configuration; Symmetric central configuration MSC 1 No.: 37M5 7F7 7F15 188

2 AAM: Intern. J. Vol. 1 Issue (December 17) Introduction The planar Newtonian four-body problem is to determine the motion of four point masses that interacts each other only by Newton s law of gravitation. In other words they behave gravitationally like four particles at a distance apart equal to the distance between their centers. The four masses will be assumed to be sufficiently isolated from each other in the universe so that only force acting; is the inverse square force of their mutual attraction along the line oining their centers. Dziobek (19) for the first time discussed central configuration with relative distances as coordinates. Moulton (191) focused on the straight line solutions in the problem of N bodies. MacMillan and Bartky (193) studied permanent configurations in the problem of four bodies. Brumberg (1957) discussed permanent configurations in the problem of four bodies and their stability also. Cabral (1973) derived integral manifolds of the N body problem. Saari (198) presented the role and properties of N body central configurations. Saari and Hulkower (1981) further studied the manifolds of total collapse orbits and of completely parabolic orbits for the bifurcation of relative equilibria in the four and five-body problem. Schmidt (1988) reviewed the central configurations in. Buck (1991) discussed the collinear central configuration in N equal masses. Albouy (1993) studied integral manifolds of the N-body problem. Casasayas Llibre and Nunes (1994) focused on the central configurations of the planar N 1 body problem. Albouy (1995) studied the symmetric des configuration centrales de quatre corps. McCord (1996) discussed planar central configuration estimated in the N body problem. N body problem. Meyer and Schmidt (1988) discussed the 3 R andr Albouy (1996) studied the symmetric central configurations of four equal masses. McCord Meyer and Wang (1998) studied the integral manifolds of three-body problem. Roberts (1999) presented a continuum of relative equilibria in the five-body problem. Moeckel (1) has worked out genetic finiteness for Dziobek configurations. Kotsireas and Lazard () have studied central configurations of the five-body problem with equal masses in threedimensional space. Albouy and Llibre () have studied spatial central configurations for the 4 1 body problem. In () four-body central configurations with equal masses have been discussed by Lonngi and Sun whereas in () concave central configuration in the four-body problem has been studied by Hampton. Leandro (3) has studied finiteness and bifurcations of some symmetrical classes of central configuration. Cors Llibre and Olle (4) have discussed the central configurations of the planar co-orbital satellite problem whereas Xia (4) has presented convex central configurations for the N body problem. Hampton and Moeckel (6) have studied finiteness of relative equilibria of the four-body problem. Chavela and Santoprete (7) have discussed convex four-body central configurations with some equal masses. Albouy Fu and Sun (8) have focused on the symmetry of planar four-body convex central configurations. Llibre and Mello (8) have discussed new central configurations for the planar five-body problem. Lee and Santoprete (9) have studied central configuration of the five-body problem with equal masses. Bernat Llibre and Perez (9) have discussed the planar central configurations of the fourbody problem with three equal masses. Pina and Lonngi (1) have studied central

3 19 M. R. Hassan et al. configurations for the planar Newtonian four-body problem. Pina (1) has discussed new coordinates for the four-body problem. In the present work we have proposed to study the concave central configuration convex central configuration and symmetric central configuration in the planar Newtonian four-body problem in the light of Pinna and Lonngi (1). Also we have applied planar Newtonian four-body problem to the specific configurations so that some mathematical tools can be generated for future research.. Equations of Motion m m m and m situated at Let the system consists of four particles of masses P P P and P respectively. Let i be the position vectors of the four particles i respectively with respect to an inertial frame with O as the origin. Considering the masses spherically symmetrical with homogenous layers so that they attract one another like point masses. The only forces acting are the mutual Newtonian gravitational attractions between the bodies. The equations of motion of the four particles given by Pina and Lonngi (1) can be written as Gm m Gm m Gm m m Gm m Gm m Gm m m Gm m Gm m Gm m m Gm m m 4 4 Gm m Gm m (1) th are the distances between the i and where i i i th particles m denotes the masses of four particles and G is the gravitational constant. The right hand sides of the above equations are the gravitational forces which are derived from the potential energy given by U Gm m Gm m Gm m Gm m Gm m Gm m ()

4 AAM: Intern. J. Vol. 1 Issue (December 17) P m P m P m P m The Planar Problem Figure 1. Newtonian Four-body Problem Following Pina and Lonngi (1) the directed areas D1 D D3 D representing twice the 4 area of each of the four faces of a tetrahedron with vertices in detail as follows: In triangle P4P 3P O 1 1 P4 P3 P P4 P P4 P P P P D Similarly in the triangle PP P D have been derived In triangle P PP 1 4D (3) and in triangle PP P D. 3 Adding the above equations of System (3) we get D D D D ˆ. (4) i.e. the vectorial sum of the areas of four faces of the tetrahedron is the zero vector. The concept of particles to be taken at the vertices of flat quadrilateral PP 1 P3 P 4 is important but old because the four vector directed areas are all parallel to unit vector k i.e. D kd where D is positive or negative according as D is parallel towards k or opposite to k. Let us choose the third component of the Cartesian co-ordinates of the four particles as zero and form the matrix of Cartesian co-ordinates as

5 19 M. R. Hassan et al. x1 x x3 x4 y y y y The four directed areas are written in terms of these co-ordinates as D x x x D x x x D x x x D x x x (5) y y y y y y y y y y y y which are the four signed 3 3 minors forming the matrix x1 x x3 x. 4 y 1 y y3 y 4 Addition to the previous matrix of a row equal to any of its three rows; produces a square matrix with zero determinant implying that the necessary and sufficient conditions to have a flat tetrahedron are D1 D D3 D4 (6) D1 x1 D x D3 x3 D4 x4 (7) D1 y1 D y D3 y3 D4 y4. (8) The Equations (7) and (8) may be grouped in the zero-vector condition as D D D D ˆ. (9) From these properties we deduce a necessary condition for flat solutions in terms of distances as

6 AAM: Intern. J. Vol. 1 Issue (December 17) 193 D D D D (a) D D D D D D D D D D D D (b) D D D D D D D D DD DD (c) D D D D D D D D D D D D (d) D D D D D D D D D D D D (e) D D D D D D D D D D D D (1) (f ) D D D D D D D D Adding all equations of System (1) and using Equations (4) and (9) we get Di Di Di D Di i D i1 1 i1 1 i1 1. (11) Here we observed that Equations (7) (8) (9) (1) and (11) are purely geometrical independent of the origin of coordinates and independent of the masses. Besides the condition of the sum of directed areas equal to zero. Equations (7) and (9) were considered by Dziobek and are currently connected to affine geometry and the planar four-body problem. In triangle P P P D then by Heron s formula D D D D D By putting the value of D in the above equation one can find

7 194 M. R. Hassan et al. 4D Similarly in triangle D (1) PP 1 3P4 then D and D D D 13 D 34 D 14 4D D (13) In triangle PP 1 P4 and PP 1 P3 respectively we have 4D (14) 4D (15) The expressions given in Equations (1) (13) (14) and (15) are the required four areas of triangles in the form of square roots of a function of the length of three sides. All the formulas established in Equations (11) (1) (13) (14) and (15) are the tools for further higher research in planar Newtonian four-body problem. 4. Central Configurations Central configurations are assumed under the condition that the force divided by the corresponding mass is proportional to the position vector from the center of mass (as the origin) i.e. i i i Ai where A is the proportionality constant to be found. Thus from Equation (1) we have m m m A (16) m m m A (17) m m m A (18) m m m A. (19) Taking cross-product to the left of both sides of Equation (16) and (17) by 1 and equating the results we get

8 AAM: Intern. J. Vol. 1 Issue (December 17) 195 m m m i.e. m m The directed areas correlated by the equations W m D D D D and weighted directed areas () W W W W are D 134 (1) m where by Laura and Andoyer (8) the weighted directed areas are given by and W W m1 m W W D vertices m3 m4 is the directed area of the triangle with the three particles different from m () at its By using third and fourth equations of System () in () and accordingly for other equations also we get (a) W W4 3 3 (b) W 3 3 W (c) W 3 W (d) W W (e) W 3 3 W1 (f) W W (3) It is obvious that in the non-planar case no pair of these four vectors is parallel for arbitrary masses and position and the unique three-dimensional central configuration satisfying the equations of System (3) requires all the six distances to be same so that the masses lie at the vertices of an equilateral tetrahedron. This configuration has the only solution of straight

9 196 M. R. Hassan et al. paths towards or away from the center of mass. The planar solutions with zero enclosed volume but finite area are obtained by taking into account that the four vectors in Equation (1) are now parallel. They are written in terms of the weighted directed areas with W W k W D m. Suppression of vector k the homogeneous system from Equations (3a) (3b) (3c) (3d) (3e) and (3f) leads to where WX (4) W4 W4 W1 W1 W4 W4 W W W4 W4 W3 W3 W W W3 W3 W W1 W3 W3 W 1 W1 W W W1 X 3 1/ 3 3 1/ / / / 4 3 1/ 43 Since the coefficient matrix of this system is anti-symmetric hence a non-trivial solution exists that is the linear combination of its two eigenvectors with eigenvalue zero namely W W W W WW W W W W W W exist and The first eigenvector is evident to considering the structure of the rows of the matrix and the equilateral solution. The existence of the second eigenvector is secured because the matrix is anti-symmetric of order six. This is the consequence of the fact that a real anti-symmetric matrix becomes Hermitian by multiplying its elements by the imaginary unit. The eigenvalues of the real anti-symmetric matrix are purely imaginary numbers forming complex conugate pairs or zero. If the order of the square matrix is an odd number we have at least one zero eigenvalue. If the order of the anti-symmetric matrix is an even number the zero eigenvalue has even multiplicity. The existence of the second eigenvector for the degenerate zero eigenvalue is a property of Hermitian matrices or anti-symmetric matrices. Also in order to show that there is no other pair of eigenvectors corresponding to the eigenvalue. From Equation (4) the characteristic polynomial of the anti-symmetric matrix can be written as i

10 AAM: Intern. J. Vol. 1 Issue (December 17) 197 where W I W W W W W W W W 4 4 W W W W W W W W 3 3 W W W W W W W W W 3 W 5 W W W W W WW W 3 W W W WW W 5 W W W W W WW W W W W 6 WW W W W W W WW W W W W W 5 W W WW W 5 W W WW W W W W 5 WW W I 3 W1 W W3 W4 C C 5W W W W W W W W 5W W W 5W WW W W WW WW WW W W WW W W W 5W 5W W W 5W W W W W W W W W W W W W W 3 W W W WW W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W i.e. C is positive definite because the Schwarz-inequality holds T The eigenvector X of W given by the non-zero solutions of the Equation (4) as corresponding to the eigenvalue are 3 W4 W4 W1 W1 3 3 W4 W4 W W 31 3 W4 W4 W3 W W W3 W3 W 41 3 W1 W3 W3 W W1 W W W 1 43 (5)

11 198 M. R. Hassan et al. By applying the row transformations to the above matrix equation one can find WW 3 W3W 4 WW 3 WW 4 1 W W3 W4 W W3 W W1 W1 3 1 M N 31 W4 W M N 3 41 W1 W3 W1 W W W3 W W (6) where M WW W W WW W W andn W W W W W W The coefficient matrix of the Equation (6) is of rank so these equations have 6 4 linearly independent solutions. Thus there is only two linearly independent eigenvector corresponding to the eigenvalue. These equations can be written as 4 W W W W W W W W W W3 W4 W W3 W4 3 W1 W3W 4 WW 1 3 WW 1 WW 4 W W4 W W3 W 4 W W3 W4 W 4 W W WW WW W W W W3 W4 W W3 W4 W W W W 3 43 W W W 3 3 W (7) Solving the above equations by taking two linearly independent eigenvectors one can find the solutions in terms of two parameters and as (a) WW (b) W W (c) W W (d) W W (e) W W (f) W W (8) One way to take into account the zero volume condition in taking the derivative with respect to of the Cayley-Manager determinant (which is proportional to the square of the volume i of the tetrahedron). These derivatives are proportional to the product DD i if the planar

12 AAM: Intern. J. Vol. 1 Issue (December 17) 199 restrictions are taken into account. An obvious difficulty of that deduction is that the nontrivial entries of the Cayley-Manager determinant are ust the squares square roots in the expression of the directed areas D as function of i. i to obtain the In Equations (1) (13) (14) and (15) one needs to use explicitly the restrictions (7) and (9) connecting these areas. Multiplying both sides of equations in system (8) by mm3 m3m1 m4m1 m4m and m4m3 respectively one can find mm 1 (a) m m m m D D (b) m m m m D D (c) m m m m D D (d) m m m m D D (e) m m m m D D (f ) m m m m D D (9) The system of Equations (9) can be written in compact form as 3 k k k k m m m m D D i 134 (3) Multiplying both sides of Equations (9) by and 43 respectively sum over and k on both sides and using the geometric planar constraint (11) one can find k k mm / k k mm k k. (31) The above expression is positive definite. Either in Equation (8) or (31) it is seen that has dimensions of the reciprocal of a cubed length. The combination of Equations (16) (8a) (8c) and (8d) yields A m m m A A m A m A m A m m m m m m A A m A m A m A A m A m A m (3) Similarly the combination of Equations (17) (8a) (8b) and (8e) gives

13 11 M. R. Hassan et al. A m m m m m m A A m A m A m A A m A m A m (33) the combination of Equations (18) (8b) (8c) and (8f) gives A m m m m m m A A m A m A m A A m A m A m (34) and the combination of Equations (19) (8d) (8e) and (8f) gives A m m m m m m A A m A m A m A A m A m A m (35) Adding Equations (3) (33) (34) & (35) and using constraints (7) and (9) one can find 4 A m (36) i1 i which is the required condition that the origin of coordinates is located at the center of mass. A different proof for to be negative was obtained by Albouy et al. (8). The sign of is important because numerical solutions of Equation (4) of the form of Equations (8a) (8b) (8c) (8d) (8e) and (8f) may be found with positive that obey restriction (7) but are not in planar central configurations since they do not satisfy Equation (11). Additional relations between distances are obtained from the quotient of Equations (8a) (8b) (8c) (8d) (8e) and (8f) as W W (a) (b) W W W W (c) (d) W W W W (e) (f ). W W (37) which is independent of the parameter. The last expression on the right hand side of these equations comes from the previous two assuming that they are different. The middle terms imply restrictions that may also be obtained from the original equations namely

14 AAM: Intern. J. Vol. 1 Issue (December 17) 111 WW W W (38) From these equations one also obtains the parameter in terms of the distances as (39) Indeed this result is an identity substituting the reciprocal cubed distances of Equations (8a) (8b) (8c) and (8d) in the right hand side of this equation. Also in order to show that the central configurations of the four-body problem with equal masses classified according to their dimensions. The dimension of the configuration was deduced by using Cayley-Manager determinants. For instance the Cayley-Manager determinant for five-dimensional volume V is given by V 1. (4) Four-dimensional configurations was found by setting the above determinant to zero as V (41) Three-dimensional configurations was found by setting the above determinant to zero as V 1. (4) The right hand side of the above determinant is the Cayley-Manger determinant for the m m m and m. tetrahedron formed by the masses Two-dimensional configurations was found by setting the above determinant to zero as

15 11 M. R. Hassan et al (43) One-dimensional configurations was found by setting the above determinant to zero as (44) P 3 P 3 P 4 P 4 P 1 P P 1 P Figure. Planar central configurations of 4BP with equal masses 5. Applications of Central Configuration to Particular Cases Let us discuss the Equations of the system (8) for negative (Albouy et al. 8) with the help of central configuration. The concave case occurs when one of the particles is in the convex hull of the other three. In the convex case no particle is in the convex hull of the other three. We consider separately concave and convex cases. P 3 P 4 P 3 P 1 P 1 P 4 P Figure 3. Arrangement of four particles with three equal masses in a central concave configuration P Figure 4. Arrangement of four particles with two equal masses in a central convex configuration

16 AAM: Intern. J. Vol. 1 Issue (December 17) Concave Configuration with Three Equal Masses Without loss of generality we may assume that in the concave case W W W W. (45) From the above inequality it is seen that first three Equations (8a) (8b) and (8c) have the positive sign of and the last three Equations (8d) (8e) and (8f) have the negative sign of. By assuming as the largest side of the set the equilateral triangle based on distances 4i i i i overlaps triangle and hence PP 1 P i i. Therefore 3 3 4i i negative i.e. and hence. m is in the convex hull of the other three. Since one of these is positive and the other 5.. Convex Configuration with Two Equal Masses Here the four particles are in the cyclic order PP 1 P3 P4 we have W W W W. (46) From the above inequality it is seen that Equations (8c) and (8e) have the positive sign of and the other four set of Equations (8a) (8b) (8d) and (8f) have the negative sign of. Let us assume that is the longest diagonal 13 or 4 and is the shortest side of the set then i ik. On the contrary we may assume that the four sides would be longer than i quadrangle PP 1 P3 P 4 then i and the two triangles with common side i could be covered by the i would not be longest diagonal. Thus 3 3 must be of opposite signs i.e. and hence. i ik ik but they 3 3 i ik 5.3. Symmetric Configurations P 4 P 3 P 1 P 3 P 1 P 4 P P Figure 5. Kite symmetry Figure 6. Equilateral triangle symmetry

17 114 M. R. Hassan et al. In this section we want to discuss some important symmetric cases from Equations (37a) (37b) (37c) (37d) (37e) and (37f) one by one. (a) Kite Symmetry The necessary and sufficient condition for kite symmetry is the equality of two weighted area W' s i.e. W W 1 3 and and D D m m. (47) The kite symmetry imposes the above restrictions. The values of the weighted areas W 4 are arbitrary but at least one of them must have opposite sign to that of W 1. W and (b) Equilateral Triangle Symmetry The necessary and sufficient condition for equilateral triangle symmetry is the equality of three weighted areas i.e. W' s W W W (48) P P P be at vertices of an equilateral triangle and the particle Since the three particles 1 3 is at the center of this triangle hence by Equations (1) (13) (14) and (15) we get D4 D1 D D3 m1 m m (49) 3 with W 4 as an arbitrary but satisfying the important condition obtained from the last equation of (37) as P 4 W W 3m. (5) m P 4 P 4 P 3 P 3 P 1 P 1 Figure 7. Square symmetry configuration configuration P P Figure 8. Rhombus symmetry

18 AAM: Intern. J. Vol. 1 Issue (December 17) 115 (c) Square Symmetry The necessary and sufficient condition for a square symmetry is the equality of two weighted areas i.e. W' s W W W W and D D D D. (51) Since the four particles P1 P P3 P 4 are at vertices of a square hence the diagonal must be WW W W equal i.e (5) (d) Rhombus Symmetry The necessary and sufficient condition for rhombus symmetry is the equality of two pairs of weighted area W' s i.e. W W W W and D D D D. (53) P 4 P 3 P 4 P 3 P 1 Figure 9. Isosceles trapezium symmetry P P 1 P Figure 1. Parallelogram symmetry (e) Isosceles Trapezium Symmetry The necessary and sufficient condition for isosceles trapezium symmetry is that W W andw W and. (54) WW W W andw W WW hence in such a case of isosceles Since from Equation (54) trapezium symmetry we also find D D andd D m m and m m. (55) (f) Parallelogram Symmetry The necessary and sufficient condition for parallelogram symmetry is the equality of two pairs of weighted areas with opposite signs W' s i.e. W W andw W and. (56) WW W W andw W W W hence in such a case of Since from Equation (56) parallelogram symmetry we also find

19 116 M. R. Hassan et al. D D andd D m m andm m (57) Discussions and Conclusion First we find out the equations of motion of the four point masses. Next we have studied the planar problem and established some formulas by using Heron s formula relating the absolute value of the area of a triangle to be square root of a function of the three sides. The sign of the directed areas D is inherited from the sign of the corresponding constraint W. Restriction (11) determines the values of for a planar solution and from it the value of the six distances and the four masses are found. This is an implicit way to deduce planar central configurations with four masses. Furthermore we have studied central configurations and find six equations in terms of and to discuss the convex and concave case of the configurations. These equations give all the distances as function of the unknown parameters and the four constraint W. Lastly we have studied the symmetry of different configurations. In many non-symmetric cases restriction (7) is sufficient to define a central configuration and hence the restriction (11) is automatically satisfied but in few cases restriction (11) is not automatically satisfied and allows one to discriminate non-physical situations (such as negative distances or evidently non-planar solutions or geometrically impossible figures). We stress the fact that to determine we choose to find the root of (11) as a function of. In any case considered below the solution is numerically unique. This approach to the four-body central configurations is similar to considering the Euler collinear configurations of three particles where instead of giving the ratio of the masses and computing the corresponding ratio of two distances with a quantic equation the ratio of two distances is first given and the ratio of the masses is computed from this approach based on the weighted areas W. The cases of kite symmetry equilateral triangle symmetry square symmetry rhomboidal symmetry and isosceles trapezium symmetry are the applications of this approach based on the weighted areas W. REFERENCES Albouy A. (1993). Integral Manifolds of the N-body Problem Invent. Math Albouy A. (1985). Symmetric des configuration centrales de quatre corps C.R. Acad. Sci. Paris Albouy A. (1996). The Symmetric Central Configurations of four equal masses Hamiltonian Dynamics and Celestial Mechanics Contemp. Math. 198 Albouy A. and Llibre J. (). Spatial Central Configurations for the 41body Problem Contemporary. Math Albouy A. Fu Y. and Sun S. (8). Symmetry of Planar Four-body Convex Central Configurations Proc. R. Soc. A Bernat J. Llibre J. and Perez C.E. (9). On the Planar Central Configurations of the Four-body Problem with three equal masses Math. Anal Brumberg V.A. (1957). Permanent Configurations in the Problem of Four bodies and their Stability Astro. Iv Buck G. (1991). The collinear Central Configuration in n equal masses. Celestial Mechanics and Dynamical Astronomy 51(4)

20 AAM: Intern. J. Vol. 1 Issue (December 17) 117 Cabral H. (1973) On then Integral Manifolds of the N-body Problem. Invent. Math 59-7 Casasayas J. Llibre J. and Nunes A. (1994). Central Configurations of the Planar 1 N body Problem Celestial Mechanics and Dynamical Astronomy 6() Cors J.M. Llibre J. and Olle M. (4). Central Configurations of the Planar Co-orbital Satellite Problem Celestial Mechanics and Dynamical Astronomy Dziobek O. (19). Uber einer werkwurdinger fall des Vielkorper Problems Astron. Nach Hampton M. (). Concave Central Configurations in the Four-body Problem. Thesis University of Washington Hampton M. and Moeckel R. (6). Finiteness of Relative Equilibria of the Four-body Problem Invent. Math Kotsireas I. and Lazard D. (). Central Configurations of the Five-body Problem with Equal Masses in three-dimensional Space J. Math Sci Leandro E.S.G. (3). Finiteness and Bifurcations of some Symmetrical Classes of Central Configuration Arch. Rational Mech Lee T.L. and Santoprete M. (9). Central Configuration of the Five-body Problem with Equal Masses. Celestial Mechanics and Dynamical Astronomy 14(4) Llibre J. Mello L.F. (8). New Central Configurations for the Planar Five-body Problem. Celestial Mechanics and Dynamical Astronomy 1() Long Y. and Sung S. (). Four-body Central Configurations with Equal Masses Arch. Rational. Mech. Anal MacMillan W.D. and Bartky W. (193). Permanent Configurations in the Problem of Four bodies Trans. Am. Math. Soc McCord C. (1996). Planar Central Configuration estimates in the N-body Problem Ergod. Th. and Dynam. Sys McCord C.K. Meyer K.R. and Wang Q. (1998). The Integral Manifolds of Three-body Problem Providence RI: Am. Math. Soc. Mello L.F. and Fernandes A. C. (11). New Spatial Central Configurations in the Fivebody Problem Anais da Academia Brasileira de Ciencias Meyer K.R. and Schmidt D. (1988). Bifurcation of Relative Equilibria in the Four-body and Five-body problem Ergod. Th. and Dynam. Sys Moeckel R. (1). Genetic finiteness for Dziobek Configurations Transactions of the American Mathematical society 353 Moeckel Moulton F.R. (191). The Straight line Solutions the Problem of N bodies Ann. Math Perez-Chavela E. and Santoprete M. (7). Convex Four-body Central Configurations with some Equal Masses Archive for Rational Mechanics and Analysis 185(3) Pina E. and Lonngi P. (1). Central Configurations for the Planar Newtonian Four-body Problem Celestial Mechanics and Dynamical Astronomy 18(1) Pina E. (1). New Co-ordinates for the Four-body Problem Revista mexicana de fisica Roberts G. (1999). A Continuum of Relative Equilibria in the Five-body Problem Physics D: Nonlinear Phenomena 17(3-4)

21 118 M. R. Hassan et al. Saari D.G. (198). On the Role of Properties of N-body Central Configurations Celestial Mechanics and Dynamical Astronomy 1(1) 9- Saari D.G. and Hulkower D. (1981). On the Manifolds of Total Collapse Orbits and of completely Parabolic Orbits for the N-body Problem Journal of Differential Equations 41(1) Xia Z. (4). Convex Central Configurations for the N-body Problem Journal of Differential Equations ()