Advances in Mathematical Physics

Volume 2014 (2014), Article ID 478495, 3 pages

http://dx.doi.org/10.1155/2014/478495

Research Article

On a Periodic Solution of the 4-Body Problems

Jian Chen1,2 and Bingyu Li¹

¹Mathematical College, Sichuan University, Chengdu, Sichuan 610064, China

²School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Received 25 November 2013; Accepted 6 January 2014; Published 11 February 2014

Academic Editor: Dongho Chae

Copyright © 2014 Jian Chen and Bingyu 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.

Abstract.

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.

1. 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 [1–3] are

where 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 [1–3]). The -bodies form a central configuration at time if there exists a scalar such that

where and

for all .

A configuration is a central configuration if and only if

where

It is well-known that for , the general solution of Newtonian -body problems cannot be given until now, so great importance has been attached to searching for a particular solution from the very beginning. Central configurations have something to do with periodic solutions and collapse orbits and parabolic orbits [1–3]. 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 , 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 -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 , are given by

where is the th complex roots of the unit; that is.

Theorem 2. If and is a solution of Newtonian -body problems (1), then and the center of masses is at the origin. That is, form a central configuration at any time .

Theorem 3. Let and be a solution of Newtonian -body problems (1) if and only if masses () of the three particles () are equal, and the angular velocity is

2. 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 , 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

that is,

hence we have

Because of

we have

Since the 4th particle is resting, which means that the force exerted on 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 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 is one solution for (21).

Because is one solution for the linear equations (21), hence we have

which means

Then we get and the common center of masses

and , and (7) becomes

so we have

By Definition 1, form a central configuration at any time and .

The proof of Theorem 2 is completed.

3. The Proof of Theorem 3

By Theorem 2, we have

and substituting 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

and we can easily get

The proof of Theorem 3 is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely express their gratitude to Professor Shiqing Zhang for his help and guidance. This work is partially supported by NSF of China.

References

A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, NJ, USA, 1941.

D. G. Saari, Collisions, Rings and Other Newtonian N-Body Problems, American Mathematical Society, Providence, RI, USA, 2005.

R. Moeckel, “On central configurations,” Mathematische Zeitschrift, vol. 205, no. 1, pp. 499–517, 1990.

L. M. Perko and E. L. Walter, “Regular polygon solutions of N-body problem,” Proceedings of the American Mathematical Society, vol. 94, pp. 301–309, 1985.

R. Moeckel and C. Simo, “Bifurcation of spatial central configurations from planar ones,” SIAM Journal on Mathematical Analysis, vol. 26, pp. 978–998, 1995.

S. Zhang and Q. Zhou, “Periodic solutions for planar 2N-body problems,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2161–2170, 2003.