Abstract and Applied Analysis

Volume 2013 (2013), Article ID 845795, 5 pages

http://dx.doi.org/10.1155/2013/845795

Research Article

The Characterization of the Variational Minimizers for Spatial Restricted -Body Problems

Fengying Li, Shiqing Zhang, and Xiaoxiao Zhao

Yangtze Center of Mathematics and College of Mathematics, Sichuan University, Chengdu 610064, China

Received 8 February 2013; Accepted 27 April 2013

Academic Editor: Maoan Han

Copyright © 2013 Fengying Li 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.

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted -body problems with a zero mass on the vertical axis of the plane for equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any ; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.

1. Introduction and Main Result

The Newtonian n-body problem [1] is a classical problem. Spatial restricted 3-body model was studied by Sitnikov [2]. Mathlouthi [3] et al. studied the periodic solutions for the spatial circular restricted 3-body problems by mini-max variational methods.

In this paper, we study spatial circular restricted -body problems with a zero mass on the vertical axis of the plane for equal masses. Suppose point masses move on a circular orbit around the center of masses. The motion for the zero mass is governed by the gravitational forces of . Let and

satisfy the Newtonian equations:

where

The orbit for zero mass satisfies the following equation:

Obviously, satisfies (4); it seems that is a variational minimizer, but we will prove it is not this is the goal of this paper.

Define

then

where

Notice that the symmetry in is related to the Italian symmetry [4].

In this paper, our main result is the following.

Theorem 1. The minimizer of on the closure of is a nonconstant periodic solution for ; hence the zero mass must oscillate, so that it can not be always in the same plane with the other bodies.

2. Proof of Theorem 1

We define the inner product and equivalent norm of :

which is equivalent to

Lemma 2 (Palais' Symmetry Principle [5]). By Palais' Symmetry Principle, we know that the critical point of in is a noncollission periodic solution of Newtonian equation (4).

Let be an orthogonal representation of a finite or compact group in the real Hilbert space such that for all , where .

Let . Then the critical point of in is also a critical point of in .

In order to prove Theorem 1, we need the following lemmas:

Lemma 3 (see [6]). Let be a reflexive Banach space, be a weakly closed subset of , , and is weakly lower semicontinuous and coercive ( as ); then attains its infimum on .

Lemma 4 (Poincare-Wirtinger Inequality). Let and ; then

Lemma 5. in (6) attains its infimum on or .

Proof. By Lemmas 3 and 4, it is easy to prove Lemma 5.

Lemma 6 (Jacobi's Necessary Condition [7]). If the critical point corresponds to a minimum of the functional and if along this critical point, then the open interval contains no points conjugate to ; that is, for all , the following boundary value problem

has only the trivial solution , for all , where

Remark 7. It is easy to see that Lemma 6 is suitable for the fixed end problem. In this paper, we consider the periodic solutions of (2) on ; hence we need to establish a similar conclusion as Lemma 6 for the periodic boundary problem.

Lemma 8. Let . Assume that is a critical point of the functional on and . If the open interval contains a point conjugate to , then is not a minimum of the functional .

Proof. Suppose is a minimum of the functional . The second variation of is

where

Set

For all , it is easy to see that . Then by , is a minimum of . The Euler-Lagrange equation which is called the Jacobi equation of (15) is

Since the interval contains a point conjugate to , there exists a nonzero Jacobi field satisfying

Letting

we have and

Notice that we can extend periodically when we take as the period, so . For all , it is easy to check that . Then by (19), one has is a minimum of . Hence we get

Combining with and , by the uniqueness of initial value problems for second-order differential equation, we have on , which contradicts the definition of . Therefore, Lemma 8 holds.

Lemma 9. The radius for the moving orbit of equal masses is

Proof. By (1)–(3), we have

Substituting (1) into (22), we have

Then

Therefore

Proof of Theorem 1. Clearly, is a critical point of on . For the functional (6), let

Then the second variation of (6) in the neighborhood of is given by

where

The Euler equation of (27) is called the Jacobi equation of the original functional (6), which is

that is,

Next, we study the solution of (30) with initial values . It is easy to get

which is not identically zero on , but we will prove for some .

Suppose there exists such that .

Hence, for some

We have

by using (24).

Case 1 (Minimizing on ). Letting ,

It is easy to check that , , , and is a nonzero solution of (30).

If we take ,

It is equivalent to

Let

It is not hard to check that is nonmonotone. But for , (36) holds by writing program to calculate it.

Therefore, for , we have such that

Notice that we can extend periodically when we take 1 as the period, so . Then by Lemma 8, is not a local minimum for on . Hence the minimizers of on are not always at the center of masses; they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; therefore, we get the nonplanar periodic solutions.

Case 2 (Minimizing on ). Let

It is easy to check that , ,, and is a nonzero solution of (30).

We hope ; that is,

It implies that

Calculated by program, for , we have such that .

Notice that we can extend periodically when we take 1 as the period, so . Then by Lemma 8, is not a local minimum for on . Hence the minimizers of on are not always at the center of masses; they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; therefore, we get the nonplanar periodic solutions.

We can use another argument to get much larger . We construct a test function such that for , where is a very large number. Let

and we extend by . We have

Writing program to calculate, we find an such that .

Hence is not a local minimum for on . So the minimizers of on are not always at the center of masses; they must oscillate periodically on the vertical axis; that is, the minimizers are not always coplanar with the other bodies; hence, we get the nonplanar periodic solutions.

Acknowledgments

The authors would like to thank the referee for his/her many valuable comments and suggestions. This paper is Supported by the national Natural Science Foundation of China (11071175) and the Ph.D Programs Foundation of Ministry of Education of China.

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