On a Periodic Solution of the 4-Body Problems

Jian Chen and Bingyu Li 1 Mathematical College, Sichuan University, Chengdu, Sichuan 610064, China 2 School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China Correspondence should be addressed to Jian Chen; jamchen1@163.com Received 25 November 2013; Accepted 6 January 2014; Published 11 February 2014 Academic Editor: Dongho Chae Copyright © 2014 J. Chen and B. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the necessary and sufficient conditions on the masses for the periodic solution of planar 4-body problems, where three particles locate at the vertices of an equilateral triangle and rotate with constant angular velocity about a resting particle. We prove that the above periodic motion is a solution of Newtonian 4-body problems if and only if the resting particle is at the origin and the masses of the other three particles are equal and their angular velocity satisfies a special condition.


Introduction and Main Results
Celestial mechanics is related to the study of bodies (mass points) moving under the influence of their mutual gravitational attractions.The many-body problems, that is, body problems, are the most important problems in celestial mechanics.-body problems are nonlinear systems of ordinary differential equations which describe the dynamical law for the motion of -bodies.The motion equations of the Newtonian -body problems [1][2][3] are where   = (  ,   ,   ) ∈  3 is the position of the th body with mass   , () is the Newtonian potential function and  is the gravitational constant which can be taken as one by choosing suitable units.
Definition 1 (see [1][2][3]).The -bodies form a central configuration at time  if there exists a scalar  ∈  such that where  = / and where It is well-known that for  ≥ 3, the general solution of Newtonian -body problems cannot be given until now, so great importance has been attached to searching for a 2 Advances in Mathematical Physics particular solution from the very beginning.Central configurations have something to do with periodic solutions and collapse orbits and parabolic orbits [1][2][3].So finding central configurations becomes very important.In 1772, Lagrange showed that for three masses at the vertices of an equilateral triangle,the orbit rotating about the center of masses with an appropriate angular velocity is a periodic solution of the three-body problems.In 1985, Perko and Walter [4] showed that for  ≥ 4,  masses at the vertices of a regular polygon rotating about their common center of masses with an appropriate angular velocity describe a periodic solution of the -body problems if and only if the masses are equal.In 1995 and in 2002, Moeckel and Simo [5] and Zhang and Zhou [6] studied the sufficient and necessary conditions for planar nested 2-body problems.In this paper, we study the following problems.
Let three particles locate at the vertices of a unit equilateral triangle; the orbits, describing their rotations with angular velocity  about the 4th particle which is resting at  4 , are given by where   is the th complex roots of the unit; that is.

The Proof of Theorem 2
Differentiating twice in both sides of (7) shows that Multiplying both sides in (10) by   and summing these equations over all  = 1, 2, 3, we have If ( 7) is a solution of (1), then the left side of the above equation must be zero since the total force of the conservative system is zero, which is shown by (7) as hence we have Because of we have Since the 4th particle is resting, which means that the force exerted on  4 by the other particles is zero, then we have Substituting (7) into the above equation and letting then ( 17) is equivalent to the following equation: Because then ( 19) is equivalent to the following linear equations: where  1 ,  2 ,  3 are regarded as unknowns.
It is easy to prove that the rank of the coefficient matrix of the above linear equations ( 21) is two, which means all the solutions of the linear equations satisfy where  is any real number and ( 1 ,  2 ,  3 )  ̸ = 0 is one solution for (21).
Because (1, 1, 1)  is one solution for the linear equations (21), hence we have which means Then we get  4 = 0 and the common center of masses and  1 =  2 =  3 = 1, and (7) becomes so we have By Definition 1,  1 ,  2 ,  3 ,  4 form a central configuration at any time and  2 =  = /.
The proof of Theorem 2 is completed.

The Proof of Theorem 3
By Theorem 2, we have and substituting  1 ,  2 ,  3 into the above equation, we have That is which means that the masses of the particles at the vertices of the equilateral triangle are equal.By Theorem 2 and (6), we have The proof of Theorem 3 is completed.