Periodic Solutions for Circular Restricted 𝑁+1-Body Problems

Abstract

For circular restricted 𝑁+1-body problems, we study the motion of a sufficiently small mass point (called the zero mass point) in the plane of 𝑁 equal masses located at the vertices of a regular polygon. By using variational minimizing methods, for some 𝑁, 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 𝑁+1-body problems. Suppose that points of positive masses 𝑚₁=⋯=𝑚_𝑁=1 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 𝑁 equal masses and does not influence the motion of 𝑚₁,…,𝑚_𝑁, but the motion of the zero mass point is affected by the given 𝑁 equal mass points. Let 𝜌_𝑖=𝑒^{√−1(2𝜋𝑖/𝑁)} and we denote the position vectors of the given 𝑁 bodies by 𝑞_𝑖(𝑡), 𝑖=1,…,𝑁, then 𝑞₁(𝑡)=𝑟𝑒^{√−12𝜋𝑡}𝜌₁,…,𝑞_𝑖(𝑡)=𝑟𝑒^{√−12𝜋𝑡}𝜌_𝑖𝑞,…,_𝑁(𝑡)=𝑟𝑒^{√−12𝜋𝑡}, (1) where the radius 𝑟>0. It is known that 𝑞₁(𝑡),…,𝑞_𝑁(𝑡) satisfy the following Newtonian equations: 𝑚_𝑖𝑞̈_𝑖=𝜕𝑈𝜕𝑞_𝑖,𝑖=1,…,𝑁, (2) where 𝑈=_{1≤𝑖<𝑗≤𝑁}𝑚_𝑖𝑚_𝑗𝑞_𝑖−𝑞_𝑗. (3) We also assume that ^𝑁_𝑖=1𝑚_𝑖𝑞_𝑖=^𝑁_𝑖=1𝑞_𝑖=0. (4) The orbit 𝑞(𝑡)∈𝑅² for the zero mass point is governed by the gravitational forces of 𝑚₁,…,𝑚_𝑁 and therefore it satisfies the following equation: 𝑞̈=^𝑁_𝑖=1𝑚_𝑖𝑞_𝑖−𝑞𝑞_𝑖−𝑞³. (5)

For 𝑁-body problems, there are many papers concerned with the periodic solutions by using variational methods; see [1–15] 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 𝑛-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 𝑛-body problems. Zhang and Zhou [10–12] decomposed the Lagrangian action for 𝑛-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 𝑁+1-body problem with some fixed wingding numbers and 𝑁 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 Γ∶𝑥(𝑡), 𝑡∈[𝑎,𝑏] be a given oriented continuous closed curve, and 𝑝 be a point on the plane not on the curve. Then, the mapping 𝜑∶Γ→𝑆¹ given by 𝜑(𝑥(𝑡=𝑥(𝑡)))−𝑝𝑥(𝑡)[]−𝑝,𝑡∈𝑎,𝑏 (6) is defined to be the position mapping of the curve Γ relative to 𝑝, and when the point on Γ goes around the curve once, its image point 𝜑(𝑥(𝑡)) will go around 𝑆¹ a number of times; this number is called the winding number of the curve Γ relative to 𝑝, and we denote it by deg(Γ,𝑝). If 𝑝 is the origin, we write degΓ.

Let 𝑅(𝑛)=cos2𝜋𝑛−sin2𝜋𝑛sin2𝜋𝑛cos2𝜋𝑛∈𝑆𝑂(2) (7) be a counter-clockwise rotation of angle 2𝜋/𝑛 in 𝑅².

Define 𝑊^1,2𝑅𝑍,𝑅²=𝑥(𝑡)∣𝑥(𝑡),𝑥̇(𝑡)∈𝐿²𝑅,𝑅²,𝑥(𝑡+1)=𝑥(𝑡). (8) The norm of 𝑊^1,2(𝑅/𝑍,𝑅²) is ‖𝑥‖=¹₀|𝑥|²𝑑𝑡^1/2+¹₀|𝑥̇|²𝑑𝑡^1/2. (9) We consider the Lagrangian functional of (5) as 𝑓𝑞=¹₀12𝑞̇²+^𝑁_𝑖=11𝑞−𝑞_𝑖𝑑𝑡 (10) on Λ_±, where Λ₋=𝑞∈𝑊^1,2𝑅𝑍,𝑅²𝑞1𝑡+2=𝑅(2)𝑞(𝑡),𝑞(𝑡)≠𝑞_𝑖(𝑡),∀𝑡∈[0,1],𝑖=1,…,𝑁,deg𝑞−𝑞_𝑁=−1,deg𝑞_𝑗−𝑞_𝑁,Λ=1,𝑗=1,…,𝑁−1₊=𝑞∈𝑊^1,2𝑅𝑍,𝑅²𝑞1𝑡+𝑁=𝑅(𝑁)𝑞(𝑡),𝑞(𝑡)≠𝑞_𝑖(𝑡),∀𝑡∈[0,1],𝑖=1,…,𝑁,deg𝑞−𝑞_𝑁=1,deg𝑞_𝑗−𝑞_𝑁.=1,𝑗=1,…,𝑁−1 (11)

 Our main results are the following.

Theorem 2. For 𝑁=2,3, the minimizer of 𝑓(𝑞) on the closure Λ₋ of Λ₋ is a noncollision 1-periodic solution of (5).

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

Theorem 4. For 2≤𝑁≤32402, the minimizer of 𝑓(𝑞) on the closure Λ₊ of Λ₊ 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 𝑟 for the moving orbit of 𝑁 equal mass points is 1𝑟=4𝜋^2/3_{1≤𝑗≤𝑁−1}𝜋𝑐𝑠𝑐𝑁𝑗^1/3. (12)

Lemma 6 (Zhang and Zhou [9]). Suppose that 𝑔∈𝑆𝑂(𝑘) is an element of finite order 𝑠 and it has no fixed point other than the origin (i.e., 1 is not an eigenvalue of 𝑔). Then ∫^𝑇₀𝑋(𝑡)𝑑𝑡=0 for all 𝑋(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑁(𝑡))∈𝐸^𝑁_𝑔, where 𝐸^𝑁_𝑔={(𝑥₁,…,𝑥_𝑁)∣𝑥_𝑖∈𝐻¹(𝑅/𝑇𝑍,𝑅^𝐾),𝑥_𝑖(𝑡+𝑇/𝑠)=𝑔(𝑥_𝑖(𝑡)),∀𝑡∈[0,𝑇],𝑖=1,…,𝑁}.

Lemma 7 (Poincare-Wirtinger inequality [16]). Let 𝑞∈𝑊^1,2(𝑅/𝑇𝑍,𝑅^𝐾) and ∫^𝑇₀𝑞(𝑡)𝑑𝑡=0, and then ^𝑇₀𝑞(𝑡)²𝑇𝑑𝑡≤²4𝜋^2𝑇₀𝑞̇(𝑡)²𝑑𝑡. (13)

Lemma 8 (see [17]). Let 𝑋 be a reflexive Banach space, 𝑆 be a weakly closed subset of 𝑋, and 𝑓∶𝑆→𝑅∪{+∞}. If 𝑓≢+∞ is weakly lower semicontinuous and coercive (𝑓(𝑥)→+∞ as ‖𝑥‖→+∞), then 𝑓 attains its infimum on 𝑆.

Lemma 9 (Palais’s symmetry principle [18]). Let 𝜎 be an orthogonal representation of a finite or compact group 𝐺, let 𝐻 be a real Hilbert space, and let 𝑓∶𝐻→𝑅 satisfy 𝑓(𝜎⋅𝑥)=𝑓(𝑥),∀𝜎∈𝐺,∀𝑥∈𝐻.
Set 𝐹={𝑥∈𝐻∣𝜎⋅𝑥=𝑥,∀𝜎∈𝐺}. Then the critical point of 𝑓 in 𝐹 is also a critical point of 𝑓 in 𝐻.

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

Lemma 11. (1) (Gordon’s theorem [19]) Let 𝑥∈𝑊^1,2([𝑡₁,𝑡₂],𝑅^𝐾) and 𝑥(𝑡₁)=𝑥(𝑡₂)=0. Then for any 𝑎>0, one has ^𝑡2_𝑡112|𝑥̇|²+𝑎|𝑥|3𝑑𝑡≥2(2𝜋)^2/3𝑎^2/3𝑡₂−𝑡₁^1/3. (14)
(2) (Long and Zhang [20]) Let 𝑥∈𝑊^1,2(𝑅/𝑇𝑍,𝑅^𝐾), and ∫^𝑇₀𝑥𝑑𝑡=0, then for any 𝑎>0, we have ^𝑇₀12|𝑥̇|²+𝑎|𝑥|3𝑑𝑡≥2(2𝜋)^2/3𝑎^2/3𝑇^1/3. (15)

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 𝑓(𝑞) in (10) on Λ_±.

Lemma 12. 𝑓(𝑞) in (10) attains its infimum on Λ_±.

Proof. It is easy to check that the eigenvalue 𝜆 of 𝑅(𝑛) satisfies 𝜆=cos2𝜋𝑛,sin2𝜋𝑛=0, (16) which implies 𝜆≠1 for 𝑛≥2. Then by Lemma 6, we have ¹₀𝑞(𝑡)=0∀𝑞∈Λ_±. (17) So by using Lemma 7, for all 𝑞∈Λ_±, we see that an equivalent norm of (9) on Λ_± is 𝑞≅^𝑇₀𝑞̇²𝑑𝑡^1/2. (18) Hence, by the definition of 𝑓(𝑞), 𝑓 is coercive on Λ_±.
Similar to the proof of Lemma 2.4 in [13], we can get the conclusion that 𝑓 is weakly lower semi-continuous on Λ_±.
Therefore, by using Lemma 8, it can be concluded that 𝑓(𝑞) in (10) attains its infimum on Λ_±.

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 𝑞, we can estimate the lower bound for the value of Lagrangian action functional.

Lemma 13. For 𝜕Λ₋={𝑞∈𝑊^1,2(𝑅/𝑍,𝑅²)∣𝑞(𝑡+1/2)=𝑅(2)𝑞(𝑡), ∃1≤𝑖₀≤𝑁, 𝑡₀∈[0,1]𝑠.𝑡.𝑞_𝑖0(𝑡₀)=𝑞(𝑡₀)}, we have inf_{𝑞∈𝜕Λ−}𝑓𝑞≥32(4𝜋)^2/3𝑁^−1/3+𝑁−1𝑁32(2𝜋)^2/3𝑁^2/3−12(2𝜋)²𝑟²≜𝑑₀. (19)

Proof. It follows from (4) that 𝑞̇^𝑁_𝑖=1𝑖=0, (20) which implies 𝑞̇^𝑁_𝑖=1−𝑞̇_𝑖²=^𝑁_𝑖=1𝑞̇²+𝑞̇_𝑖²−2𝑞,̇𝑞̇_𝑖=𝑁𝑞̇²+^𝑁_𝑖=1𝑞̇_𝑖²−2𝑞,̇^𝑁_𝑖=1𝑞̇_𝑖=𝑁𝑞̇²+^𝑁_𝑖=1𝑞̇_𝑖². (21) Therefore, 𝑞̇²=1𝑁^𝑁_𝑖=1𝑞̇−𝑞̇_𝑖²−𝑞̇_𝑖². (22) Hence, 𝑓𝑞=¹₀12𝑞̇²+^𝑁_𝑖=11𝑞−𝑞_𝑖=1𝑑𝑡𝑁¹₀^𝑁_𝑖=112𝑞̇−𝑞̇_𝑖²+𝑁𝑞−𝑞_𝑖−1𝑑𝑡2𝑁¹₀^𝑁_𝑖=1𝑞̇_𝑖²𝑑𝑡. (23) If 𝑞∈Λ₋ is a collision solution, then there exists 𝑡₀∈[0,1] and 1≤𝑖₀≤𝑁 s.t 𝑞(𝑡₀)=𝑞_𝑖0(𝑡₀). Since 𝑞_𝑖(𝑡+1/2)=𝑅(2)𝑞_𝑖(𝑡), one has 𝑞(𝑡₀+𝑘/2)=𝑞_𝑖0(𝑡₀+𝑘/2),∀0≤𝑘≤2. So, by (1) of Lemma 11, we get 1𝑁¹₀12𝑞̇−𝑞̇_𝑖0²+𝑁𝑞−𝑞_𝑖0=2𝑑𝑡𝑁₀^1/212𝑞̇−𝑞̇_𝑖0²+𝑁𝑞−𝑞_𝑖0≥3𝑑𝑡2(4𝜋)^2/3𝑁^−1/3. (24) For noncollision pair 𝑞,𝑞_𝑖(𝑖≠𝑖₀), we claim ∫¹₀(𝑞(𝑡)−𝑞_𝑖(𝑡))𝑑𝑡=0. In fact, it follows from (1) that 𝑞_𝑖+1(𝑡)=𝑞_𝑖(𝑡+1/𝑁), which implies 𝑞_𝑖(𝑡)=𝑞₁𝑡+𝑖−1𝑁. (25) Therefore, by (4), we have ¹₀𝑞₁(𝑡)=𝑑𝑡_01/𝑁𝑞₁(𝑡)𝑑𝑡+⋯+¹_{()𝑁−1/𝑁}𝑞₁(𝑡)=𝑑𝑡_01/𝑁𝑞₁(𝑡)𝑑𝑡+⋯+₀^1/𝑁𝑞₁𝑡+𝑁−1𝑁=𝑑𝑡_{0𝑁1/𝑁𝑖=1}𝑞_𝑖(𝑡)𝑑𝑡=0. (26) Since 𝑞₁(𝑡+1)=𝑞₁(𝑡), by (25) and (26), we obtain ¹₀𝑞_𝑖(𝑡)𝑑𝑡=¹₀𝑞₁𝑡+𝑖−1𝑁𝑑𝑡=¹₀𝑞₁(𝑡)𝑑𝑡=0. (27) Combined with (17) and (27), we have ∫¹₀(𝑞(𝑡)−𝑞_𝑖(𝑡))𝑑𝑡=0. Hence, by (2) of Lemma 11, we can get 1𝑁¹_{0𝑖≠𝑖0}12𝑞̇−𝑞̇_𝑖²+𝑁𝑞−𝑞_𝑖≥𝑑𝑡𝑁−1𝑁32(2𝜋)^2/3𝑁^2/3. (28) For the other term of 𝑓, using the expression for the orbits 𝑞₁,…,𝑞_𝑁 as in (1), one has −12𝑁¹₀^𝑁_𝑖=1𝑞̇_𝑖²1𝑑𝑡=−2(2𝜋)²𝑟². (29) Therefore, it follows from (24), (28), and (29) that inf_{𝑞∈𝜕Λ−}𝑓𝑞≥32(4𝜋)^2/3𝑁^−1/3+𝑁−1𝑁32(2𝜋)^2/3𝑁^2/3−12(2𝜋)²𝑟²≜𝑑₀. (30)

Lemma 14. For 𝜕Λ₊={𝑞∈𝑊^1,2(𝑅/𝑍,𝑅²)∣𝑞(𝑡+1/𝑁)=𝑅(𝑁)𝑞(𝑡),∃1≤^∼𝑖₀≤𝑁,^∼𝑡₀∈[0,1]𝑠.𝑡.𝑞^∼_𝑖0(^∼𝑡₀)=𝑞(^∼𝑡₀)}, we have inf_{𝑞∈𝜕Λ+}𝑓𝑞≥32(2𝜋)^2/3𝑁^1/3+𝑁−1𝑁32(2𝜋)^2/3𝑁^2/3−12(2𝜋)²𝑟²≜𝑑₁. (31)

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 𝑞^∼∈Λ₋ such that 𝑓(𝑞^∼)=𝑑₂. Then the minimizer of 𝑓 on Λ₋ must be a noncollision solution if 𝑑₂<𝑑₀.
Let 𝑎>0, 𝑏>0, 𝜃∈[0,2𝜋) and 𝑞^∼−𝑞_𝑁=(𝑎cos(−2𝜋𝑡+𝜃),𝑏sin(−2𝜋𝑡+𝜃))^𝑇. (32) Hence, 𝑞^∼−𝑞₁=𝑟cos2𝜋𝑡−𝑟cos2𝜋𝑡+2𝜋𝑁+𝑎cos(−2𝜋𝑡+𝜃),𝑟sin2𝜋𝑡−𝑟sin2𝜋𝑡+2𝜋𝑁+𝑏sin(−2𝜋𝑡+𝜃)^𝑇,_∼⋮𝑞−𝑞_𝑖=𝑟cos2𝜋𝑡−𝑟cos2𝜋𝑡+𝑖𝑁()𝑖2𝜋+𝑎cos−2𝜋𝑡+𝜃,𝑟sin2𝜋𝑡−𝑟sin2𝜋𝑡+𝑁2𝜋+𝑏sin(−2𝜋𝑡+𝜃)^𝑇,_∼⋮𝑞−𝑞_𝑁−1=𝑟cos2𝜋𝑡−𝑟cos2𝜋𝑡+𝑁−1𝑁()2𝜋+𝑎cos−2𝜋𝑡+𝜃,𝑟sin2𝜋𝑡−𝑟sin2𝜋𝑡+𝑁−1𝑁2𝜋+𝑏sin(−2𝜋𝑡+𝜃)^𝑇. (33)(34)(35)
It is easy to see that 𝑞^∼∈Λ₋ and 𝑞̇^∼−𝑞̇₁²=(2𝜋)²𝑎²+𝑏²2−𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²2−2cos2𝜋𝑁−()𝑟×()𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+2𝜋𝑁+()𝑟−𝜃𝑎−𝑏cos𝜃−cos2𝜋𝑁+𝜃_∼,𝑞−𝑞₁=𝑎²+𝑏²2+𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²2−2cos2𝜋𝑁+()𝑟×()𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+2𝜋𝑁+()𝑟−𝜃𝑎−𝑏cos𝜃−cos2𝜋𝑁+𝜃^1/2,⋮𝑞̇^∼−𝑞̇_𝑖²=(2𝜋)²𝑎²+𝑏²2−𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²𝑖2−2cos𝑁−()𝑟×()𝑖2𝜋𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+𝑁+()𝑟𝑖2𝜋−𝜃𝑎−𝑏cos𝜃−cos𝑁2𝜋+𝜃_∼,𝑞−𝑞_𝑖=𝑎²+𝑏²2+𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²𝑖2−2cos𝑁+()𝑟×()𝑖2𝜋𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+𝑁+()𝑟𝑖2𝜋−𝜃𝑎−𝑏cos𝜃−cos𝑁2𝜋+𝜃^1/2,⋮𝑞̇^∼−𝑞̇_𝑁²=(2𝜋)²𝑎²+𝑏²2−𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃_∼),𝑞−𝑞_𝑁=𝑎²+𝑏²2+𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃),𝑞̇₁²=⋯=𝑞̇_𝑁²=(2𝜋)²𝑟². (36)(37)(38)(39)(40)(41)(42) Therefore by (36)–(42), we get 𝑓𝑞^∼=1𝑁¹₀^𝑁_𝑖=112𝑞̇^∼−𝑞̇_𝑖²+_∼𝑁𝑞−𝑞_𝑖1𝑑𝑡−2𝑁¹₀^𝑁_𝑖=1𝑞̇_𝑖²=𝑎𝑑𝑡₂+𝑏²𝜋²+(2𝜋)²𝑟²+2𝜋²()𝑟1𝑎−𝑏cos𝜃−𝑁^𝑁_𝑖=1𝑖cos𝑁+2𝜋+𝜃_𝑁𝑖=110𝑎²+𝑏²2+𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²𝑖2−2cos𝑁+()𝑟×()𝑖2𝜋𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+𝑁+()𝑟𝑖2𝜋−𝜃𝑎−𝑏cos𝜃−cos𝑁2𝜋+𝜃^−1/2−1𝑑𝑡2(2𝜋)²𝑟²=2𝜋²𝑎²+𝑏²2+𝑟²+2𝜋²()𝑟1𝑎−𝑏cos𝜃−𝑁^𝑁_𝑖=1𝑖cos𝑁+2𝜋+𝜃_𝑁𝑖=110𝑎²+𝑏²2+𝑎²−𝑏²2cos(4𝜋𝑡−2𝜃)+𝑟²𝑖2−2cos𝑁+()𝑟×()𝑖2𝜋𝑎+𝑏cos4𝜋𝑡−𝜃−cos4𝜋𝑡+𝑁+()𝑟𝑖2𝜋−𝜃𝑎−𝑏cos𝜃−cos𝑁2𝜋+𝜃^−1/2𝑑𝑡=𝑑₂(𝑎,𝑏,𝜃). (43)
In order to estimate 𝑑₂, we have computed the numerical values of 𝑑₂=𝑓(𝑞) over some selected test loops. The computation of the integral that appears in (43) has been done using the function {𝑞𝑢𝑎𝑑} of Mathematica 7.1 with an error less than 10⁻⁶. The results of the numerical explorations are given in Table 1.
For the parameters 𝑎,𝑏, and 𝜃 given in Table 1, we all have 𝑑₂<𝑑₀. For 4≤𝑁≤1000, we do not find suitable parameters 𝑎∈[0.01,2𝑟), 𝑏∈[0.01,2𝑟), and 𝜃=𝜋,𝜋/2,𝜋/3,…,𝜋/29,𝜋/30 such that 𝑑₂<𝑑₀, where 𝑟 is the radius of the first 𝑁 bodies with equal masses. Hence we only consider the case 𝑁=2,3. This completes the proof of Theorem 2.

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 𝑞∈Λ₊ such that 𝑓(𝑞)=𝑑₃. Then the minimizer of 𝑓 on Λ₊ must be a noncollision solution if 𝑑₃<𝑑₁.
Let 𝑎>0,  𝜃∈[0,2𝜋), and 𝑞−𝑞_𝑁=𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃. (44) Hence, 𝑞−𝑞₁=𝑞_𝑁+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃−𝑞₁=𝑟𝑒^{√−12𝜋𝑡}1−𝑒^{√−1(2𝜋/𝑁)}+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃,⋮𝑞−𝑞_𝑖=𝑞_𝑁+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃−𝑞_𝑖=𝑟𝑒^{√−12𝜋𝑡}1−𝑒^{√−1(𝑖/𝑁)2𝜋}+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃,⋮𝑞−𝑞_𝑁−1=𝑞_𝑁+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃−𝑞_𝑁−1=𝑟𝑒^{√−12𝜋𝑡}1−𝑒^{√−1((𝑁−1)/𝑁)2𝜋}+𝑎𝑒^{√−12𝜋𝑡}𝑒^√−1𝜃. (45)(46)(47) It is easy to see that 𝑞∈Λ₊ and 𝑞̇−𝑞̇₁²=(2𝜋)²𝑟²2−2cos2𝜋𝑁+𝑎²+2𝑎𝑟cos𝜃−cos𝜃−2𝜋𝑁,𝑞−𝑞₁=𝑟²2−2cos2𝜋𝑁+𝑎²+2𝑎𝑟cos𝜃−cos𝜃−2𝜋𝑁^1/2,⋮𝑞̇−𝑞̇_𝑖²=(2𝜋)²𝑟²2−2cos𝑖𝑁2𝜋+𝑎²𝑖+2𝑎𝑟cos𝜃−cos𝜃−𝑁2𝜋,𝑞−𝑞_𝑖=𝑟²𝑖2−2cos𝑁2𝜋+𝑎²𝑖+2𝑎𝑟cos𝜃−cos𝜃−𝑁2𝜋^1/2,⋮𝑞̇−𝑞̇_𝑁²=(2𝜋)²𝑎²,𝑞−𝑞_𝑁=𝑎,𝑞̇₁²=⋯=𝑞̇_𝑁²=(2𝜋)²𝑟². (48)(49)(50)(51)(52)(53) Therefore by (48)–(53), we get 𝑓𝑞=1𝑁¹₀^𝑁_𝑖=112𝑞̇−𝑞̇_𝑖²+𝑁𝑞−𝑞_𝑖−1𝑑𝑡2𝑁¹₀^𝑁_𝑖=1𝑞̇_𝑖²𝑑𝑡=2𝜋²𝑎²+(2𝜋)²𝑟²+(2𝜋)²arcos𝜃−𝑎𝑟𝑁^𝑁_𝑖=1𝑖cos𝜃−𝑁2𝜋+𝑎⁻¹+^𝑁−1_𝑖=1𝑟²𝑖2−2cos𝑁2𝜋+𝑎²𝑖+2𝑎𝑟cos𝜃−cos𝜃−𝑁2𝜋^−1/2−12(2𝜋)²𝑟²=2𝜋²𝑎²+𝑟²+(2𝜋)²arcos𝜃−𝑎𝑟𝑁^𝑁_𝑖=1𝑖cos𝜃−𝑁2𝜋+𝑎⁻¹+^𝑁−1_𝑖=1𝑟²𝑖2−2cos𝑁2𝜋+𝑎²𝑖+2𝑎𝑟cos𝜃−cos𝜃−𝑁2𝜋^−1/2=𝑑₃(𝑎,𝜃). (54)
In order to estimate 𝑑₃, we have computed the numerical values of 𝑑₃=𝑓(𝑞) over some selected test loops. The results of the numerical explorations show that by choosing some 𝑎∈[0.11,12.50] and 𝜃=𝜋, one has 𝑑₃<𝑑₁ for 2≤𝑁≤32402. From the computational data, let 𝜃=𝜋,𝜋/2,𝜋/3,…,𝜋/19,𝜋/20, and we find that 𝑑₃<𝑑₁ holds for 2≤𝑁≤19 with some 𝑎∈[0.11,0.90]. But for 20≤𝑁≤32402, no matter how we choose the value of 𝑎∈[0.11,12.50], we obtain 𝑑₃<𝑑₁ only if 𝜃=𝜋. For 𝑁=32403, the computational data shows that 𝑑₃≫𝑑₁ for all 𝑎∈[0.11,10.00] and 𝜃=𝜋,𝜋/2,𝜋/3,…,𝜋/99,𝜋/100. Then combined with Figure 1, it is easy to see that 𝑎.𝑒.𝑑₃>𝑑₁ for all 𝑎∈[0.11,12.50] and 𝜃=𝜋,𝜋/2,𝜋/3,…,𝜋/99,𝜋/100. The same holds for 𝑁=32404 from Figure 2 and the computational data. Moreover, we can get the same conclusions for 32405≤𝑁≤100000. Hence, we only consider the case 2≤𝑁≤32402. This completes the proof of Theorem 4.

Figure 1 

𝑁=32403.

Figure 2 

𝑁=32404.

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.