Abstract and Applied Analysis

Volume 2013 (2013), Article ID 272791, 5 pages

http://dx.doi.org/10.1155/2013/272791

Research Article

Central Configurations for Newtonian -Body Problems

Furong Zhao1,2 and Jian Chen1,3

¹Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

²Department of Mathematics and Computer Science, Mianyang Normal University, Mianyang, Sichuan 621000, China

³Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan 621000, China

Received 29 November 2012; Revised 31 January 2013; Accepted 3 February 2013

Academic Editor: Baodong Zheng

Copyright © 2013 Furong Zhao and Jian Chen. 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 show the existence of spatial central configurations for the -body problems. In the -body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the th is located at the the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.

1. Introduction and Main Results

The Newtonian -body problems [1–3] concern with the motions of particles with masses and positions , and the motion is governed by Newton’s second law and the Universal law:

where and with Newtonian potential:

Consider the space

that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the collision set for some . The set is called the configuration space.

Definition 1 (see [2, 3]). A configuration = is called a central configuration if there exists a constant such that

The value of constant in (4) is uniquely determined by

where

Since the general solution of the -body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning. A homographic solution is a configuration which is preserved for all time. Central configurations and homographic solutions are linked by the Laplace theorem [3]. Collapse orbits and parabolic orbits have relations with the central configurations [2, 4–6], so finding central configurations becomes very important. The main general open problem for the central configurations is due to Wintner [3] and Smale [7]: is the number of central configurations finite for any choice of positive masses ? Hampton and Moeckel [8] have proved this conjecture for any given four positive masses.

For 5-body problems, Hampton [9] provided a new family of planar central configurations, called stacked central configurations. A stacked central configuration is one that has some proper subset of three or more points forming a central configuration. Ouyang et al. [10] studied pyramidal central configurations for Newtonian -body problems; Zhang and Zhou [11] considered double pyramidal central configurations for Newtonian -body problems; Mello and Fernandes [12] analyzed new classes of spatial central configurations for the -body problem. Llibre and Mello studied triple and quadruple nested central configurations for the planar -body problem. There are many papers studying central configuration problems such as [13–22].

Based on the above works, we study stacked central configuration for Newtonian -body problems. In the -body problems, bodies are at the vertices of a regular -gon , and bodies are symmetrically located on the same straight line which is perpendicular to and passes through the center of ; the th body is located at the center of . The masses located on the vertices of the regular -gon are equal; the masses located on the line and symmetric with respect to the center of are equal. (see Figure 1 for and ).

Figure 1

In this paper we will prove the following result.

Theorem 2. For -body problem in where and , there is at least one central configuration such that bodies are at the vertices of a regular -gon , and 2 bodies are symmetric with respect to the center of the regular -gon , and located on a line which is perpendicular to the regular -gon ; the th body is located at the center of . The masses at the vertices of are equal and the masses symmetric with respect to the center of are equal.

2. The Proof of Theorem 2

Our approach to Theorem 2 is inspired by the of arguments of Corbera et al. in [23].

2.1. Equations for the Central Configurations of -Body Problems

To begin, we take a coordinate system which simplifies the analysis. The particles have positions given by , where , ; , , where ; .

The masses are given by , , where .

Notice that is a central configuration if and only if

By the symmetries of the system, (7) is equivalent to the following equations:

that is,

where

In order to simplify the equations, we defined

 , , where ; , when ; , when ;  when ; , , where .

Equations (9)–(11) can be written as a linear system of the form given by

The column vector is given by the variables . Since the is function of , we write the coefficient matrix as .

2.2. For

We need the next lemma.

Lemma 3 (see [12]). Assuming , , there is a nonempty interval , and , such that for each , forms a central configuration of the -body problem.

For , system (12) becomes

If we consider as a function of , then is an analytic function and nonconstant. By Lemma 3, there exists a such that (13) has a unique solution satisfying and .

2.3. For All

The proof for is done by induction. We claim that there exists such that system (12) has a unique solution , for . We have seen that the claim is true for . We assume the claim is true for and we will prove it for . Assume by induction hypothesis that there exists such that system (12) has a unique solution and for .

We need the next lemma.

Lemma 4. There exists such that , for and is a solution of (12).

Proof. Since , we have that the first equation of (12) is satisfied when , for and . Substituting this solution into the last equation of (12), we let

We have that

Therefore there exists at least a value satisfying equation . This completes the proof of Lemma 4.

By using the implicit function theorem, we will prove that the solution of (12) given in Lemma 3 can be continued to a solution with .

Let , we define

It is not difficult to see that the system (12) is equivalent to for .

Let be the solution of system (12) given in Lemma 4. The differential of (17) with respect to the variables is

We have assumed that exists such that system (12) with instead of has a unique solution, so ; therefore . Applying the Implicit Function Theorem, there exists a neighborhood of , and unique analytic functions , for and , such that is the solution of the system (12) for all . The determinant is calculated as

where is the algebraic cofactor of .

We see that , and for do not contain the factor . If we consider as a function of , then is analytic and nonconstant. We can find sufficiently close to such that and therefore, a solution of system (12) is satisfying , for .

The proof of Theorem 2 is completed.

Acknowledgments

The authors express their gratitude to Professor Zhang Shiqing for his discussions and helpful suggestions. This work is supported by NSF of China and Youth found of Mianyang Normal University.

References

R. Abraham and J. E. Marsden, Foundation of Mechanics, Benjamin, New York, NY, USA, 2nd edition, 1978.

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

A. Wintner, The Analytical Foundations of Celestial Mechanics, vol. 5 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1941.

D. G. Saari, “Singularities and collisions of Newtonian gravitational systems,” Archive for Rational Mechanics and Analysis, vol. 49, pp. 311–320, 1973.

D. G. Saari, “On the role and the properties of n-body central configurations,” Celestial Mechanics and Dynamical Astronomy, vol. 21, no. 1, pp. 9–20, 1980.

D. G. Saari and N. D. Hulkower, “On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem,” Journal of Differential Equations, vol. 41, no. 1, pp. 27–43, 1981.

S. Smale, “Mathematical problems for the next century,” Mathematical Intelligencer, vol. 20, pp. 141–145, 1998.

M. Hampton and R. Moeckel, “Finiteness of relative equilibria of the four-body problem,” Inventiones Mathematicae, vol. 163, no. 2, pp. 289–312, 2006.

M. Hampton, “Stacked central configurations: new examples in the planar five-body problem,” Nonlinearity, vol. 18, no. 5, pp. 2299–2304, 2005.

T. Ouyang, Z. Xie, and S. Zhang, “Pyramidal central configurations and perverse solutions,” Electronic Journal of Differential Equations, vol. 2004, no. 106, pp. 1–9, 2004.

S. Zhang and Q. Zhou, “Double pyramidal central configurations,” Physics Letters A, vol. 281, no. 4, pp. 240–248, 2001.

L. F. Mello and A. C. Fernandes, “New classes of spatial central configurations for the n-body problem,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 723–730, 2011.

A. Albouy, “The symmetric central configurations of four equal masses,” in Hamiltonian Dynamics and Celestial Mechanics, vol. 198 of Contemporary Mathematics, pp. 131–135, American Mathematical Society, Providence, RI, USA, 1996.

A. Albouy, Y. Fu, and S. Sun, “Symmetry of planar four-body convex central configurations,” Proceedings of The Royal Society of London. Series A, vol. 464, no. 2093, pp. 1355–1365, 2008.

F. N. Diacu, “The masses in a symmetric centered solution of the n-body problem,” Proceedings of the American Mathematical Society, vol. 109, no. 4, pp. 1079–1085, 1990.

J. Lei and M. Santoprete, “Rosette central configurations, degenerate central configurations and bifurcations,” Celestial Mechanics & Dynamical Astronomy, vol. 94, no. 3, pp. 271–287, 2006.

R. Moeckel, “On central configurations,” Mathematische Zeitschrift, vol. 205, no. 4, pp. 499–517, 1990.

R. Moeckel and C. Simó, “Bifurcation of spatial central configurations from planar ones,” SIAM Journal on Mathematical Analysis, vol. 26, no. 4, pp. 978–998, 1995.

F.R. Moulton, “The straight line solutions of the n-body problem,” Annals of Mathematics, Second Series, vol. 12, pp. 1–17, 1910.

L. M. Perko and E. L. Walter, “Regular polygon solutions of the N-body problem,” Proceedings of the American Mathematical Society, vol. 94, no. 2, pp. 301–309, 1985.

J. Shi and Z. Xie, “Classification of four-body central configurations with three equal masses,” Journal of Mathematical Analysis and Applications, vol. 363, no. 2, pp. 512–524, 2010.

Z. Xia, “Central configurations with many small masses,” Journal of Differential Equations, vol. 91, no. 1, pp. 168–179, 1991.

M. Corbera, J. Delgado, and J. Llibre, “On the existence of central configurations of p nested n-gons,” Qualitative Theory of Dynamical Systems, vol. 8, no. 2, pp. 255–265, 2009.