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

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Applications of Planar Newtonian Four-Body Problem to the Central Configurations Available at http://pvamu.edu/aam Applications and Applied Appl. Appl. Math. Mathematics: ISSN: 1932-9466 An International Journal (AAM) Vol. 12, Issue 2 (December 2017), pp. 1088 - 1108 Applications of Planar Newtonian Four-body Problem to the Central Configurations M. R. Hassan1, M. S. Ullah2, Md. Aminul Hassan3 and Umakant Prasad4 1P. G. Department of Mathematics S. M. College, T. M. Bhagalpu University Bhagalpur-812001, Bihar, India [email protected] 2Department of Mathematics, Ambedkar College, New Delhi, India 3GTE, Bangalore-560022, India [email protected] 4Department of Physics, T. N. B. College T. M. Bhagalpur University Bihar, India Received: March 16, 2017; Accepted: November 16, 2017 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; Flat- tetrahedron; Concave central configuration; Convex central configuration; Symmetric central configuration MSC 2010 No.: 37M05, 70F07, 70F15 1088 AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 1089 1. 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 joining their centers. Dziobek (1900) for the first time, discussed central configuration with relative distances as coordinates. Moulton (1910) focused on the straight line solutions in the problem of N bodies. MacMillan and Bartky (1932) 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 (1980) presented the role and properties of body central configurations. Saari and Hulkower (1981) further studied the manifolds of total collapse orbits and of completely parabolic orbits for the body problem. Meyer and Schmidt (1988) discussed the bifurcation of relative equilibria in the four and five-body problem. Schmidt (1988) reviewed the central configurations in RR23and . 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 body problem. 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 (2001) has worked out genetic finiteness for Dziobek configurations. Kotsireas and Lazard (2002) have studied central configurations of the five-body problem with equal masses in three- dimensional space. Albouy and Llibre (2002) have studied spatial central configurations for the 41 body problem. In (2002), four-body central configurations with equal masses have been discussed by Lonngi and Sun whereas in (2002) concave central configuration in the four-body problem has been studied by Hampton. Leandro (2003) has studied finiteness and bifurcations of some symmetrical classes of central configuration. Cors, Llibre and Olle (2004) have discussed the central configurations of the planar co-orbital satellite problem whereas Xia (2004) has presented convex central configurations for the body problem. Hampton and Moeckel (2006) have studied finiteness of relative equilibria of the four-body problem. Chavela and Santoprete (2007) have discussed convex four-body central configurations with some equal masses. Albouy, Fu and Sun (2008) have focused on the symmetry of planar four-body convex central configurations. Llibre and Mello (2008) have discussed new central configurations for the planar five-body problem. Lee and Santoprete (2009) have studied central configuration of the five-body problem with equal masses. Bernat, Llibre and Perez (2009) have discussed the planar central configurations of the four- body problem with three equal masses. Pina and Lonngi (2010) have studied central 1090 M. R. Hassan et al. configurations for the planar Newtonian four-body problem. Pina (2010) 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 (2010). Also, we have applied planar Newtonian four-body problem to the specific configurations so that some mathematical tools can be generated for future research. 2. Equations of Motion Let the system consists of four particles of masses m1, m 2 , m 3 and m 4 situated at PPPP1, 2 , 3 and 4 respectively. Let i i 1,2,3,4 be the position vectors of the four particles 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 (2010) can be written as Gm1221 m Gm 1331 m Gm 1441 m m11 3 3 3 , 12 13 14 Gm2112 m Gm 2332 m Gm 2442 m m22 3 3 3 , 21 23 24 (1) Gm3113 m Gm 3223 m Gm 3443 m m33 3 3 3 , 31 32 34 Gm4 m 1 1 4 Gm 4 m 2 2 4 Gm 4 m 3 3 4 m44 3 3 3 , 41 42 43 th th where ij ji ij 1,2,3,4 are the distances between the ijand particles, mjj 1,2,3,4 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 Gm1 m 2Gm1 m 3 Gm 1 m 4 Gm 2 m 3 Gm 2 m 4 Gm 3 m 4 U . (2) 12 13 14 23 24 34 AAM: Intern. J., Vol. 12, Issue 2 (December 2017) 1091 Pm11 12 Pm22 23 13 Pm33 24 14 1 34 2 3 4 Pm44 O Figure 1. Newtonian Four-body Problem 3. The Planar Problem Following Pina and Lonngi (2010) the directed areas DDDD1,,, 2 3 4 representing twice the area of each of the four faces of a tetrahedron with vertices 1,,, 2 3 4 have been derived in detail as follows: In triangle PPP432, 11 PPPPPPP432 4243 24 34 22 2 PPPD4 3 2 1 4 2 2 3 3 4 Similarly in the triangle PPP1 3 4D2 1 4 4 3 3 1. In triangle PPPD2 1 4 3 1 2 2 4 4 1, (3) and in triangle PPPD1 2 3 4 2 1 3 2 1 3. Adding the above equations of System (3), we get ˆ DDDD1 2 3 4 0 . (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 PPPP1 2 3 4 is important but old because the four vector directed areas are all parallel to unit vector k, i.e., Djj kD , where D j is positive or negative according as Dj 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 1092 M. R. Hassan et al. x1 x 2 x 3 x 4 y y y y . 1 2 3 4 0 0 0 0 The four directed areas are written in terms of these co-ordinates as 111 111 111 111 (5) DxxxDxxxDxxxDxxx1234,,, 21433124 4132 y2 y 3 y 4 y 1 y 4 y 3y1 y 2 y 4 y 1 y 3 y 2 which are the four signed 33 minors forming the matrix 1 1 1 1 x x x x . 1 2 3 4 y1 y 2 y 3 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 DDDD1 2 3 4 0 , (6) D1 x 1 D 2 x 2 D 3 x 3 D 4 x 4 0 , (7) D1 y 1 D 2 y 2 D 3 y 3 D 4 y 4 0 .
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