Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 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 𝑁 + 1-Body Problems
Fengying Li, Shiqing Zhang, and Xiaoxiao Zhao
Yangtze Center of Mathematics and College of Mathematics, Sichuan University, Chengdu 610064, China
Correspondence should be addressed to Fengying Li; lify0308@163.com
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.
We use Jacobi’s necessary condition for the variational minimizer to study the periodic solution for spatial restricted 𝑁 + 1-body
problems with a zero mass on the vertical axis of the plane for 𝑁 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 2 ≤ 𝑁 ≤ 472; 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
where
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 𝑁 + 1body problems with a zero mass on the vertical axis of the
plane for 𝑁 equal masses. Suppose point masses 𝑚1 = ⋅ ⋅ ⋅ =
𝑚𝑁 = 1 move on a circular orbit around the center of masses.
The motion for the zero mass is governed by the gravitational
√
forces of 𝑚1 , . . . , 𝑚𝑁. Let 𝜌𝑗 = 𝑒 −1(2𝜋𝑗/𝑁) and
√−12𝜋𝑡
𝑞1 (𝑡) = 𝑟𝑒
𝜌1 , . . . , 𝑞𝑗 (𝑡)
√−12𝜋𝑡
(1)
𝑈=
𝑁
𝑚 (𝑞 − 𝑞)
𝑞 ̈ = ∑ 󵄨𝑖 𝑖 󵄨3 .
󵄨
󵄨
𝑖=1 󵄨󵄨𝑞𝑖 − 𝑞󵄨󵄨
𝜕𝑈
,
𝜕𝑞𝑖
𝑖 = 1, . . . , 𝑁,
(4)
Obviously, 𝑞̃(𝑡) = (0, 0, 0) satisfies (4); it seems that 𝑞̃(𝑡) ≡
0 is a variational minimizer, but we will prove it is not this is
the goal of this paper.
Define
1
1 󵄨 󵄨2 𝑁 1
𝑓 (𝑞) = ∫ [ 󵄨󵄨󵄨𝑞󵄨󵄨󵄨̇ + ∑ 󵄨󵄨
󵄨󵄨 ] 𝑑𝑡,
2
0
𝑖=1 󵄨󵄨𝑞 − 𝑞𝑖 󵄨󵄨
satisfy the Newtonian equations:
(3)
The orbit 𝑞(𝑡) = (0, 0, 𝑧(𝑡)) ∈ 𝑅3 for zero mass satisfies the
following equation:
= 𝜌𝑗 𝑞1 (𝑡) , . . . , 𝑞𝑁 (𝑡) = 𝑟𝑒
𝑚𝑖 𝑞𝑖̈ =
𝑚𝑖 𝑚𝑗
∑ 󵄨󵄨
󵄨󵄨 .
1≤𝑖<𝑗≤𝑁 󵄨󵄨󵄨𝑞𝑖 − 𝑞𝑗 󵄨󵄨󵄨
𝑞 ∈ Λ 𝑖,
(5)
then
(2)
1
𝑁
1 󵄨 󵄨2
] 𝑑𝑡 ≜ 𝑓 (𝑧) ,
𝑓 (𝑞) = ∫ [ 󵄨󵄨󵄨󵄨𝑧󸀠 󵄨󵄨󵄨󵄨 +
2
√
2
0
𝑟 + 𝑧2
𝑞 ∈ Λ 𝑖 , (6)
2
Abstract and Applied Analysis
Lemma 5. 𝑓(𝑞) in (6) attains its infimum on Λ1 = Λ 1 or Λ2 =
Λ 2.
where
{𝑞 (𝑡) = (0, 0, 𝑧 (𝑡)) | 𝑧 (𝑡) ∈ 𝑊1,2 ( 𝑅 , 𝑅)}
Λ1 = {
𝑍
},
𝑞 (−𝑡) = −𝑞 (𝑡)
}
{
Proof. By Lemmas 3 and 4, it is easy to prove Lemma 5.
1,2 𝑅
{
{𝑞 (𝑡) = (0, 0, 𝑧 (𝑡)) | 𝑧 (𝑡) ∈ 𝑊 ( , 𝑅)}
}
(7)
𝑍
Λ2 = {
,
}
1
{
}
𝑧 (𝑡 + ) = −𝑧 (𝑡)
{
}
2
󵄨
2
󵄨
𝑅
󵄨 𝑥 (𝑡) , 𝑥̇ (𝑡) ∈ 𝐿 ((0, 1) , 𝑅)
}.
𝑊1,2 ( , 𝑅) = {𝑥 (𝑡) 󵄨󵄨󵄨
𝑥 (𝑡 + 1) = 𝑥 (𝑡)
󵄨󵄨
𝑍
Notice that the symmetry in Λ 2 is related to the Italian
symmetry [4].
In this paper, our main result is the following.
Theorem 1. The minimizer of 𝑓(𝑞) on the closure Λ𝑖 of Λ 𝑖 (𝑖 =
1, 2) is a nonconstant periodic solution for 2 ≤ 𝑁 ≤ 472; 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
𝑊1,2 (𝑅/𝑍, 𝑅):
1
⟨𝑢, V⟩ = ∫ (𝑢V + 𝑢󸀠 ⋅ V󸀠 ) 𝑑𝑡,
0
1
‖𝑢‖ = [∫ |𝑢|2 𝑑𝑡]
1/2
0
1
󵄨 󵄨2
+ [∫ 󵄨󵄨󵄨󵄨𝑢󸀠 󵄨󵄨󵄨󵄨 𝑑𝑡]
0
(8)
1/2
,
which is equivalent to
1
󵄨 󵄨2
[∫ 󵄨󵄨󵄨󵄨𝑢󸀠 󵄨󵄨󵄨󵄨 𝑑𝑡]
0
1/2
+ |𝑢 (0)| .
(9)
Lemma 2 (Palais’ Symmetry Principle [5]). By Palais’ Symmetry Principle, we know that the critical point of 𝑓(𝑞) in Λ𝑖 is
a noncollission periodic solution of Newtonian equation (4).
Let 𝜎 be an orthogonal representation of a finite or compact
group 𝐺 in the real Hilbert space 𝐻 such that for all 𝜎 ∈ 𝐺, 𝑓(𝜎⋅
𝑥) = 𝑓(𝑥), where 𝑓 : 𝐻 → 𝑅.
Let 𝑆 = {𝑥 ∈ 𝐻 | 𝜎 ⋅ 𝑥 = 𝑥, for all 𝜎 ∈ 𝐺}. Then the critical
point of 𝑓 in 𝑆 is also a critical point of 𝑓 in 𝐻.
Lemma 6 (Jacobi’s Necessary Condition [7]). If the critical
point 𝑢 = 𝑢̃(𝑡) corresponds to a minimum of the functional
𝑏
∫𝑎 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡))𝑑𝑡 and if 𝐹𝑢󸀠 𝑢󸀠 > 0 along this critical point,
then the open interval (𝑎, 𝑏) contains no points conjugate to 𝑎;
that is, for all 𝑐 ∈ (𝑎, 𝑏), the following boundary value problem
−
ℎ (𝑎) = 0,
Lemma 4 (Poincare-Wirtinger Inequality). Let 𝑞 ∈ 𝑊1,2
𝑇
× (𝑅/𝑍, 𝑅𝑁) and ∫0 𝑞(𝑡)𝑑𝑡 = 0; then
𝑇
2
𝑇
2𝜋
󵄨2
󵄨 ̇ 󵄨󵄨2
󵄨
∫ 󵄨󵄨󵄨𝑞(𝑡)
󵄨󵄨 𝑑𝑡 ≥ ( ) ∫ 󵄨󵄨󵄨𝑞(𝑡)󵄨󵄨󵄨 𝑑𝑡.
𝑇
0
0
(10)
(11)
ℎ (𝑐) = 0,
has only the trivial solution ℎ(𝑡) ≡ 0, for all 𝑡 ∈ (𝑎, 𝑐), where
1
󵄨󵄨
𝑃 = 𝐹𝑢󸀠 𝑢󸀠 󵄨󵄨󵄨󵄨
2
󵄨𝑢=̃𝑢
(12)
󵄨󵄨
1
𝑑
󵄨󵄨
𝑄 = (𝐹𝑢𝑢 − 𝐹𝑢𝑢󸀠 )󵄨󵄨 .
󵄨󵄨𝑢=̃𝑢
2
𝑑𝑡
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 Λ𝑖 = Λ 𝑖 (𝑖 = 1, 2); hence we need to
establish a similar conclusion as Lemma 6 for the periodic
boundary problem.
Lemma 8. Let 𝐹 ∈ 𝐶3 (𝑅 × 𝑅 × 𝑅, 𝑅). Assume that 𝑢 = 𝑢̃(𝑡)
𝑇
is a critical point of the functional ∫0 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡))𝑑𝑡 on
{𝑢 ∈ 𝑊1,2 (𝑅/𝑇𝑍, 𝑅), 𝑢󸀠 (0) = 0} and 𝐹𝑢󸀠 𝑢󸀠 |𝑢=̃𝑢 > 0. If the open
interval (0, 𝑇) contains a point 𝑐 conjugate to 0, then 𝑢 = 𝑢̃(𝑡)
𝑇
is not a minimum of the functional ∫0 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡))𝑑𝑡.
Proof. Suppose 𝑢 = 𝑢̃(𝑡) is a minimum of the
𝑇
functional ∫0 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡))𝑑𝑡. The second variation of
𝑇
∫0 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡))𝑑𝑡 is
𝑇
∫ (𝑃ℎ󸀠2 + 𝑄ℎ2 ) 𝑑𝑡,
0
where
󵄨󵄨
1
𝐹𝑢󸀠 𝑢󸀠 󵄨󵄨󵄨󵄨 ,
2
󵄨𝑢=̃𝑢
𝑃=
󵄨󵄨
𝑑
1
󵄨
𝑄 = (𝐹𝑢𝑢 − 𝐹𝑢𝑢󸀠 )󵄨󵄨󵄨 .
󵄨󵄨𝑢=̃𝑢
2
𝑑𝑡
In order to prove Theorem 1, we need the following
lemmas:
Lemma 3 (see [6]). Let 𝑋 be a reflexive Banach space, 𝑆 be a
weakly closed subset of 𝑋, 𝑓 : 𝑆 → 𝑅 ∪ {+∞}, and 𝑓 ≢ +∞
is weakly lower semicontinuous and coercive (𝑓(𝑥) → +∞ as
‖𝑥‖ → +∞); then 𝑓 attains its infimum on 𝑆.
𝑑
(𝑃ℎ󸀠 ) + 𝑄ℎ = 0,
𝑑𝑡
(13)
(14)
Set
𝑇
𝑄𝑢̃ (ℎ) = ∫ (𝑃ℎ󸀠2 + 𝑄ℎ2 ) 𝑑𝑡.
0
(15)
For all ℎ ∈ {𝑢 ∈ 𝑊1,2 (𝑅/𝑇𝑍, 𝑅), 𝑢󸀠 (0) = 0}, it is easy to
see that 𝑄𝑢̃ (ℎ) ≥ 0. Then by 𝑄𝑢̃ (𝜃) = 0, 𝜃 is a minimum of
𝑄𝑢̃ (ℎ). The Euler-Lagrange equation which is called the Jacobi
equation of (15) is
−
𝑑
(𝑃ℎ󸀠 ) + 𝑄ℎ = 0.
𝑑𝑡
(16)
Abstract and Applied Analysis
3
Since the interval (0, 𝑇) contains a point 𝑐 conjugate to 0, there
exists a nonzero Jacobi field ℎ0 ∈ 𝐶2 ([0, 𝑇], 𝑅) satisfying
𝑑
− (𝑃ℎ󸀠0 ) + 𝑄ℎ0 = 0,
𝑑𝑡
ℎ0 (0) = 0,
ℎ0 (𝑐) = 0,
(17)
ℎ0󸀠
Then
𝑟3 =
(0) = 0.
𝑡 ∈ [0, 𝑐] ,
𝑡 ∈ (𝑐, 𝑇] ,
1 2/3
𝜋
𝑟 = ( ) [ ∑ csc ( 𝑗)]
4𝜋
𝑁
[1≤𝑗≤𝑁−1
]
(18)
𝑇
𝑄𝑢̃ (̂ℎ) = ∫ (𝑃̂ℎ󸀠2 + 𝑄̂ℎ2 ) 𝑑𝑡
= ∫
𝑐
0
(19)
(𝑃ℎ0󸀠2
+
𝑄ℎ02 ) 𝑑𝑡
1 󵄨 󵄨2
𝑁
.
𝐹 (𝑧, 𝑧󸀠 ) = 󵄨󵄨󵄨󵄨𝑧󸀠 󵄨󵄨󵄨󵄨 +
2
√𝑟 + 𝑧2
2
(20)
Combining with ̂ℎ(0) = ̂ℎ(𝑐) = 0 and ̂ℎ󸀠 (0) = 0,
by the uniqueness of initial value problems for secondorder differential equation, we have ̂ℎ(𝑡) ≡ 0 on [0, 𝑐],
which contradicts the definition of ̂ℎ. Therefore, Lemma 8
holds.
Lemma 9. The radius 𝑟 for the moving orbit of 𝑁 equal masses
is
𝑟=(
1
)
4𝜋
2/3
[ ∑ csc ( 𝜋 𝑗)]
𝑁
[1≤𝑗≤𝑁−1
]
1
where
󵄨󵄨
1
1
𝐹𝑧󸀠 𝑧󸀠 󵄨󵄨󵄨󵄨 = ,
2
󵄨𝑧=0 2
󵄨󵄨󵄨
𝑑
𝑁
1
𝑄 = (𝐹𝑧𝑧 − 𝐹𝑧𝑧󸀠 )󵄨󵄨󵄨 = − 3 .
󵄨󵄨𝑧=0
2
𝑑𝑡
2𝑟
(28)
𝑃=
The Euler equation of (27) is called the Jacobi equation of the
original functional (6), which is
−
𝑑
(𝑃ℎ󸀠 ) + 𝑄ℎ = 0,
𝑑𝑡
Proof. By (1)–(3), we have
𝑞𝑗 − 𝑞𝑁
̈ = ∑󵄨
𝑞𝑁
󵄨󵄨3 .
󵄨
𝑗 ≠ 𝑁 󵄨󵄨𝑞𝑗 − 𝑞𝑁 󵄨󵄨
󵄨
󵄨
𝑁
(30)
ℎ = 0.
𝑟3
Next, we study the solution of (30) with initial values ℎ(0) =
0, ℎ󸀠 (0) = 1. It is easy to get
ℎ󸀠󸀠 +
√
(31)
𝑁
𝑐 = 𝑘𝜋.
𝑟3
(32)
We have
𝑐=[
−4𝜋2 = ∑
1 − 𝜌𝑗
1
𝜋
4𝜋2 𝑟3 = ∑ 󵄨
=
∑ csc ( 𝑗) .
3
󵄨
󵄨
󵄨
4 1≤𝑗≤𝑁−1
𝑁
𝑗 ≠ 𝑁 󵄨󵄨1 − 𝜌𝑗 󵄨󵄨
󵄨
󵄨
𝑟3
𝑁
⋅ sin √ 3 𝑡,
𝑁
𝑟
which is not identically zero on [0, 1], but we will prove ℎ(𝑐) =
0 for some 𝑐 ∈ (0, 1).
Suppose there exists 𝑐 ∈ (0, 1) such that ℎ(𝑐) = 0.
Hence, for some 𝑘 ∈ 𝑍+
(22)
Substituting (1) into (22), we have
𝜌𝑗 − 𝜌𝑁
󵄨󵄨
󵄨󵄨3 ,
𝑗 ≠ 𝑁 𝑟3 󵄨󵄨𝜌𝑗 − 𝜌𝑁 󵄨󵄨
󵄨
󵄨
(29)
that is,
ℎ (𝑡) = √
(21)
(26)
(27)
1/3
.
(25)
∫ (𝑃ℎ󸀠2 + 𝑄ℎ2 ) 𝑑𝑡,
0
𝑑
(𝑃̂ℎ󸀠 ) + 𝑄̂ℎ = 0.
𝑑𝑡
.
Then the second variation of (6) in the neighborhood of 𝑧 = 0
is given by
= 0.
Notice that we can extend ̂ℎ periodically when we take 𝑇 as
the period, so ̂ℎ ∈ 𝑊01,2 (𝑅/𝑇𝑍, 𝑅). For all ℎ ∈ 𝐶01 ([0, 𝑇], 𝑅),
it is easy to check that 𝑄𝑢̃ (ℎ) ≥ 0. Then by (19), one has ̂ℎ ∈
𝐶2 ([0, 𝑇] \ {𝑐}, 𝑅) ∩ 𝑊01,2 (𝑅/𝑇𝑍, 𝑅) is a minimum of 𝑄𝑢̃ (ℎ).
Hence we get
−
1/3
Proof of Theorem 1. Clearly, 𝑞(𝑡) = (0, 0, 0) is a critical point
of 𝑓(𝑞) on Λ𝑖 = Λ 𝑖 (𝑖 = 1, 2). For the functional (6), let
we have ̂ℎ ∈ 𝐶2 ([0, 𝑇] \ {𝑐}, 𝑅), ̂ℎ(0) = ̂ℎ(𝑐) = ̂ℎ(𝑇) = 0 and
0
(24)
Therefore
Letting
̂ℎ (𝑡) = {ℎ0 (𝑡)
0
1
𝜋
∑ csc ( 𝑗) .
2
16𝜋 1≤𝑗≤𝑁−1
𝑁
=[
[
(23)
by using (24).
𝜋2 𝑘2 𝑟3
]
𝑁
1/2
𝑘2 (∑𝑁−1
𝑗=1 csc ((𝜋/𝑁) 𝑗))
16𝑁
1/2
]
]
(33)
4
Abstract and Applied Analysis
Case 1 (Minimizing 𝑓(𝑞) on Λ1 = Λ 1 ). Letting 0 < 𝑐 < 1/2,
ℎ (𝑡)
{
{
ℎ (𝑡) = {0
{
{−ℎ (1 − 𝑡)
𝑡 ∈ [0, 𝑐] ,
𝑡 ∈ (𝑐, 1 − 𝑐] ,
𝑡 ∈ (1 − 𝑐, 1] .
𝑁−1
It is easy to check that ℎ(𝑡) ∈ 𝐶 ([0, 1] \ {𝑐, 1 − 𝑐}, 𝑅) ∩
𝑊1,2 (𝑅, 𝑅), ℎ(−𝑡) = −ℎ(𝑡), ℎ(0) = ℎ(0) = 0, ℎ(𝑐) = ℎ(𝑐) = 0,
and ℎ is a nonzero solution of (30).
If we take 𝑘 = 1,
𝑐=[
[
16𝑁
1/2
]
]
1
< .
2
(35)
It is equivalent to
𝑁−1
∑ csc (
𝑗=1
𝜋
𝑗) < 4𝑁.
𝑁
∑ csc (
(34)
2
∑𝑁−1
𝑗=1 csc ((𝜋/𝑁) 𝑗)
It implies that
𝑗=1
{
{
{𝑡
𝑧 (𝑡) = { 1
{
{
{4
Let
(37)
It is not hard to check that 𝑓(𝑥) is nonmonotone. But for 2 ≤
𝑁 ≤ 472, (36) holds by writing program to calculate it.
Therefore, for 2 ≤ 𝑁 ≤ 472, we have 𝑐 ∈ (0, 1) such that
sin √
𝑁
𝑐 = sin 𝜋 = 0.
𝑟3
Case 2 (Minimizing 𝑓(𝑞) on Λ2 = Λ 2 ). Let
ℎ (𝑡)
{
{
{
{
{
{
{
0
{
{
{
{
̃ℎ (𝑡) =
1
{
{
−ℎ (𝑡 − )
{
{
{
2
{
{
{
{
{
{0
{
𝑡 ∈ [0, 𝑐] ,
1
𝑡 ∈ (𝑐, ] ,
2
1 1
𝑡 ∈ ( , + 𝑐] ,
2 2
1
𝑡 ∈ ( + 𝑐, 1] .
2
[
∑𝑁−1
𝑗=1 csc ((𝜋/𝑁) 𝑗)
16𝑁
1/2
]
]
1
< .
2
(42)
𝑓 (𝑧) =
1/2
𝑁
1
𝑑𝑡
+ 2∫
2
√
4
0
𝑟 + 𝑧2
=
1/4
1/2
1
𝑁
𝑁
𝑑𝑡
𝑑𝑡 + 2 ∫
+ 2∫
2
2
2
√𝑟 + 𝑡
4
0
1/4 √𝑟 + (1/16)
=
1
1 + √1 + 16𝑟2
2𝑁
.
+ 2𝑁 ln
+
√1 + 16𝑟2
4
4𝑟
(43)
Writing program to calculate, we find an 𝑁0 = 9 ∗ 1010
such that 𝑓(𝑧) < 𝑓(0).
Hence 𝑞(𝑡) = (0, 0, 0) is not a local minimum for 𝑓(𝑞) on
Λ𝑖 = Λ 𝑖 (𝑖 = 1, 2). So the minimizers of 𝑓(𝑞) on Λ 𝑖 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
(39)
It is easy to check that ̃ℎ(𝑡) ∈ 𝐶2 ([0, 1]\{𝑐, 1/2, (1/2)+𝑐}, 𝑅)∩
𝑊1,2 (𝑅, 𝑅), ̃ℎ(𝑡 + (1/2)) = −̃ℎ(𝑡), ̃ℎ(0) = ℎ(0) = 0, ̃ℎ(𝑐) =
ℎ(𝑐) = 0, and ̃ℎ is a nonzero solution of (30).
We hope 𝑐 ∈ (0, 1/2); that is,
𝑐=[
1
𝑡 ∈ [0, ] ,
4
1 1
𝑡 ∈ [ , ],
4 2
and we extend 𝑧(𝑡) by 𝑧(𝑡 + (1/2)) = −𝑧(𝑡). We have
(38)
Notice that we can extend ℎ periodically when we take 1 as
the period, so ℎ ∈ Λ 1 . Then by Lemma 8, 𝑞(𝑡) = (0, 0, 0) is
not a local minimum for 𝑓(𝑞) on Λ 1 . Hence the minimizers
of 𝑓(𝑞) on Λ 1 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.
(41)
Calculated by program, for 2 ≤ 𝑁 ≤ 𝑁0 = 472, we have
𝑐 ∈ (0, 1/2) such that ℎ(𝑐) = 0.
Notice that we can extend ̃ℎ periodically when we take 1
as the period, so ̃ℎ ∈ Λ 2 . Then by Lemma 8, 𝑞(𝑡) = (0, 0, 0) is
not a local minimum for 𝑓(𝑞) on Λ 2 . Hence the minimizers
of 𝑓(𝑞) on Λ 2 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 𝑁0 . We
construct a test function 𝑧(𝑡) such that 𝑓(𝑧) < 𝑓(0) = 𝑁/𝑟
for 𝑁 ≤ 𝑁0 , where 𝑁0 is a very large number. Let
(36)
𝜋
𝑓 (𝑥) = ∑ csc ( 𝑗) .
𝑥
𝑗 ≠ 𝑥
𝜋
𝑗) < 4𝑁.
𝑁
(40)
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.
References
[1] A. Wintner, The Analytical Foundations of Celestial Mechanics,
vol. 5 of Princeton Mathematical Series, Princeton University
Press, Princeton, NJ, USA, 1941.
[2] K. Sitnikov, “Existence of oscillating motions for the three-body
problem,” Doklady Akademii Nauk USSR, vol. 133, pp. 303–306,
1960.
Abstract and Applied Analysis
[3] S. Mathlouthi, “Periodic orbits of the restricted three-body
problem,” Transactions of the American Mathematical Society,
vol. 350, no. 6, pp. 2265–2276, 1998.
[4] U. Bessi and V. Coti Zelati, “Symmetries and noncollision closed
orbits for planar 𝑁-body-type problems,” Nonlinear Analysis.
Theory, Methods & Applications, vol. 16, no. 6, pp. 587–598, 1991.
[5] R. S. Palais, “The principle of symmetric criticality,” Communications in Mathematical Physics, vol. 69, no. 1, pp. 19–30, 1979.
[6] M. Struwe, Variational Methods, vol. 34 of Results in Mathematics and Related Areas, Springer, Berlin, Germany, 3rd edition,
2000.
[7] I. Gelfand and S. Formin, Calculus of Variations, Nauka,
Moscow, Russia; Prentice-Hall, Englewood Cliffs, NJ, USA,
English edition, 1965.
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