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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 173639, 8 pages
http://dx.doi.org/10.1155/2013/173639
Research Article

Periodic Solutions for Circular Restricted -Body Problems

1College of Mathematics, Sichuan University, Chengdu 610064, China
2School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 18 February 2013; Accepted 3 April 2013

Academic Editor: Baodong Zheng

Copyright © 2013 Xiaoxiao Zhao et al. 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

For circular restricted -body problems, we study the motion of a sufficiently small mass point (called the zero mass point) in the plane of equal masses located at the vertices of a regular polygon. By using variational minimizing methods, for some , we prove the existence of the noncollision periodic solution for the zero mass point with some fixed wingding number.

1. Introduction and Main Results

In this paper, we study the planar circular restricted -body problems. Suppose that points of positive masses move on a circular orbit around the center of masses, and that the sufficiently small mass point, called the zero mass point, moves on the moving plane of the given equal masses and does not influence the motion of , but the motion of the zero mass point is affected by the given equal mass points. Let and we denote the position vectors of the given bodies by , , then where the radius . It is known that satisfy the following Newtonian equations: where We also assume that The orbit for the zero mass point is governed by the gravitational forces of and therefore it satisfies the following equation:

For -body problems, there are many papers concerned with the periodic solutions by using variational methods; see [115] and the references therein. In [1], Chenciner and Montgomery proved the existence of the remarkable figure-“8” type periodic solution for planar Newtonian 3-body problems with equal masses. Marchal [4] studied the fixed end problem for Newtonian -body problems and proved that the minimizer for the Lagrangian action has no interior collision. In [6], Simó used computer to discover many new periodic solutions for Newtonian -body problems. Zhang and Zhou [1012] decomposed the Lagrangian action for -body problems into some sum for two body problems and [11, 12] avoided collisions by comparing the lower bound for the Lagrangian action on the symmetry collision orbits and the upper bound for the Lagrangian action on test orbits in some cases.

Motivated by the above works, we use variational methods to study the circular restricted -body problem with some fixed wingding numbers and equal masses.

For the readers’ conveniences, we recall the definition of the winding number, which can be found in many books on the classical differential geometry.

Definition 1. Let ,  be a given oriented continuous closed curve, and be a point on the plane not on the curve. Then, the mapping given by is defined to be the position mapping of the curve relative to , and when the point on goes around the curve once, its image point will go around a number of times; this number is called the winding number of the curve relative to , and we denote it by . If is the origin, we write .

Let be a counter-clockwise rotation of angle in .

Define The norm of is We consider the Lagrangian functional of (5) as on , where

 Our main results are the following.

Theorem 2. For , the minimizer of on the closure of is a noncollision 1-periodic solution of (5).

Remark 3. In proving Theorem 2, we need to use test functions. We find that when , if the test functions are circular orbits, we cannot get the desired results on . Therefore, we select elliptic orbits as test functions.

Theorem 4. For , the minimizer of on the closure of is a noncollision 1-periodic solution of (5).

2. Preliminaries

In this section, we will list some basic lemmas and inequality to prove our theorems.

Lemma 5 (see [8]). The radius for the moving orbit of equal mass points is

Lemma 6 (Zhang and Zhou [9]). Suppose that is an element of finite order and it has no fixed point other than the origin (i.e., 1 is not an eigenvalue of ). Then for all , where .

Lemma 7 (Poincare-Wirtinger inequality [16]). Let and , and then

Lemma 8 (see [17]). Let be a reflexive Banach space, be a weakly closed subset of , and . If is weakly lower semicontinuous and coercive ( as ), then attains its infimum on .

Lemma 9 (Palais’s symmetry principle [18]). Let be an orthogonal representation of a finite or compact group , let be a real Hilbert space, and let satisfy .
Set . Then the critical point of in is also a critical point of in .

Remark 10. By Palais’s symmetry principle and the perturbation invariance for wingding numbers, we know that the critical point of in is a periodic solution of Newtonian equation (5).

Lemma 11. (1) (Gordon’s theorem [19]) Let and . Then for any , one has
(2) (Long and Zhang [20]) Let , and , then for any , we have

3. Proof of Theorems

In order to get our theorems, we need two steps to complete the proof.

Step 1. We will establish the existence of variational minimizers of in (10) on .

Lemma 12. in (10) attains its infimum on .

Proof. It is easy to check that the eigenvalue of satisfies which implies for . Then by Lemma 6, we have So by using Lemma 7, for all , we see that an equivalent norm of (9) on is Hence, by the definition of , is coercive on .
Similar to the proof of Lemma 2.4 in [13], we can get the conclusion that is weakly lower semi-continuous on .
Therefore, by using Lemma 8, it can be concluded that in (10) attains its infimum on .

Step 2. We will prove that the variational minimizer in Lemma 12 is the noncollision 1-period solution of (5).
For any collision generalized solution , we can estimate the lower bound for the value of Lagrangian action functional.

Lemma 13. For , , , we have

Proof. It follows from (4) that which implies Therefore, Hence, If is a collision solution, then there exists and s.t . Since , one has . So, by (1) of Lemma 11, we get For noncollision pair , we claim . In fact, it follows from (1) that , which implies Therefore, by (4), we have Since , by (25) and (26), we obtain Combined with (17) and (27), we have . Hence, by (2) of Lemma 11, we can get For the other term of , using the expression for the orbits as in (1), one has Therefore, it follows from (24), (28), and (29) that

Lemma 14. For , we have

Proof. Similar to the proof of Lemma 13, Lemma 14 holds.

Proof of Theorem 2. In order to get Theorem 2, we are going to find a test loop such that . Then the minimizer of on must be a noncollision solution if .
Let , , and Hence,
It is easy to see that and Therefore by (36)–(42), we get
In order to estimate , we have computed the numerical values of over some selected test loops. The computation of the integral that appears in (43) has been done using the function of Mathematica 7.1 with an error less than . The results of the numerical explorations are given in Table 1.
For the parameters , and given in Table 1, we all have . For , we do not find suitable parameters , , and such that , where is the radius of the first bodies with equal masses. Hence we only consider the case . This completes the proof of Theorem 2.

tab1
Table 1: Parameters of test loops for Theoerm 2.

Proof of Theorem 4. To get Theorem 4, we are going to find a test loop such that . Then the minimizer of on must be a noncollision solution if .
Let ,  , and Hence, It is easy to see that and Therefore by (48)–(53), we get
In order to estimate , we have computed the numerical values of over some selected test loops. The results of the numerical explorations show that by choosing some and , one has for . From the computational data, let , and we find that holds for with some . But for , no matter how we choose the value of , we obtain only if . For , the computational data shows that for all and . Then combined with Figure 1, it is easy to see that for all and . The same holds for from Figure 2 and the computational data. Moreover, we can get the same conclusions for . Hence, we only consider the case . This completes the proof of Theorem 4.

173639.fig.001
Figure 1: .
173639.fig.002
Figure 2: .

Acknowledgments

This study is supported by the national natural Science Foundation of China. The authors would like to thank Professor Shiqing Zhang, Taisong Xiong, Donglun Wu and Yiyang Deng for some helpful discussions, and the referee for his/her many valuable comments and suggestions on this paper.

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