The Characterization of the Variational Minimizers for Spatial Restricted 𝑁+1-Body Problems

Abstract

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

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+1-body problems with a zero mass on the vertical axis of the plane for 𝑁 equal masses. Suppose point masses 𝑚₁=⋯=𝑚_𝑁=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,…,𝑚_𝑁. Let 𝜌_𝑗=𝑒^{√−1(2𝜋𝑗/𝑁)} and 𝑞₁(𝑡)=𝑟𝑒^{√−12𝜋𝑡}𝜌₁,…,𝑞_𝑗(𝑡)=𝜌_𝑗𝑞₁(𝑡),…,𝑞_𝑁(𝑡)=𝑟𝑒^{√−12𝜋𝑡} (1) satisfy the 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) The orbit 𝑞(𝑡)=(0,0,𝑧(𝑡))∈𝑅³ for zero mass satisfies the following equation: 𝑞̈=^𝑁_𝑖=1𝑚_𝑖𝑞_𝑖−𝑞𝑞_𝑖−𝑞³. (4)

Obviously, 𝑞(𝑡)=(0,0,0) satisfies (4); it seems that 𝑞(𝑡)≡0 is a variational minimizer, but we will prove it is not this is the goal of this paper.

Define 𝑓𝑞=¹₀12𝑞̇²+^𝑁_𝑖=11𝑞−𝑞_𝑖𝑑𝑡,𝑞∈Λ_𝑖, (5) then 𝑓𝑞=¹₀12𝑧^′2+𝑁√𝑟²+𝑧²𝑑𝑡≜𝑓(𝑧),𝑞∈Λ_𝑖, (6) where Λ₁=𝑞(𝑡)=(0,0,𝑧(𝑡))∣𝑧(𝑡)∈𝑊^1,2𝑅𝑍𝑞()(𝑡),Λ,𝑅−𝑡=−𝑞₂=𝑞(𝑡)=(0,0,𝑧(𝑡))∣𝑧(𝑡)∈𝑊^1,2𝑅𝑍𝑧1,𝑅𝑡+2(𝑡),𝑊=−𝑧_1,2𝑅𝑍=𝑥(𝑡)𝑥(𝑡),𝑥̇(𝑡),𝑅∈𝐿²))𝑥()(𝑡).((0,1,𝑅𝑡+1=𝑥 (7) Notice that the symmetry in Λ₂ 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 Λ_𝑖 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 𝑊^1,2(𝑅/𝑍,𝑅): ⟨𝑢,𝑣⟩=¹₀𝑢𝑣+𝑢^′⋅𝑣^′‖𝑢‖=𝑑𝑡,₁₀|𝑢|²𝑑𝑡^1/2+¹₀𝑢^′2𝑑𝑡^1/2, (8) which is equivalent to ¹₀𝑢^′2𝑑𝑡^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 Λ_𝑖 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,forall𝜎∈𝐺}. Then the critical point of 𝑓 in 𝑆 is also a critical point of 𝑓 in 𝐻.

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.

Lemma 4 (Poincare-Wirtinger Inequality). Let 𝑞∈𝑊^1,2(𝑅/𝑍,𝑅^𝑁) and ∫^𝑇₀𝑞(𝑡)𝑑𝑡=0; then 𝑞(̇^𝑇₀𝑡)²𝑑𝑡≥2𝜋𝑇^2𝑇₀𝑞(𝑡)²𝑑𝑡. (10)

Lemma 5. 𝑓(𝑞) in (6) attains its infimum on Λ₁=Λ₁ or Λ₂=Λ₂.

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

Lemma 6 (Jacobi's Necessary Condition [7]). If the critical point 𝑢=𝑢(𝑡) corresponds to a minimum of the functional ∫^𝑏_𝑎𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))𝑑𝑡 and if 𝐹_{𝑢′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 −𝑑𝑑𝑡𝑃ℎ^′ℎ(𝑎)(𝑐)+𝑄ℎ=0,=0,ℎ=0, (11) has only the trivial solution ℎ(𝑡)≡0, for all 𝑡∈(𝑎,𝑐), where 1𝑃=2𝐹_{𝑢′𝑢′𝑢=}𝑢1𝑄=2𝐹_𝑢𝑢−𝑑𝐹𝑑𝑡^𝑢𝑢′_𝑢=𝑢. (12)

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 Λ_𝑖=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=𝑢(𝑡) is a critical point of the functional ∫^𝑇₀𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))𝑑𝑡 on {𝑢∈𝑊^1,2(𝑅/𝑇𝑍,𝑅),𝑢^′(0)=0} and 𝐹_{𝑢′𝑢′}|_𝑢=𝑢>0. If the open interval (0,𝑇) contains a point 𝑐 conjugate to 0, then 𝑢=𝑢(𝑡) is not a minimum of the functional ∫^𝑇₀𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))𝑑𝑡.

Proof. Suppose 𝑢=𝑢(𝑡) is a minimum of the functional ∫^𝑇₀𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))𝑑𝑡. The second variation of ∫^𝑇₀𝐹(𝑡,𝑢(𝑡),𝑢^′(𝑡))𝑑𝑡 is ^𝑇₀𝑃ℎ^′2+𝑄ℎ²𝑑𝑡, (13) where 1𝑃=2𝐹_{𝑢′𝑢′𝑢=}𝑢,1𝑄=2𝐹_𝑢𝑢−𝑑𝐹𝑑𝑡^𝑢𝑢′_𝑢=𝑢. (14) Set 𝑄𝑢(ℎ)=^𝑇₀𝑃ℎ^′2+𝑄ℎ²𝑑𝑡. (15) For all ℎ∈{𝑢∈𝑊^1,2(𝑅/𝑇𝑍,𝑅),𝑢^′(0)=0}, it is easy to see that 𝑄𝑢(ℎ)≥0. Then by 𝑄𝑢(𝜃)=0, 𝜃 is a minimum of 𝑄𝑢(ℎ). The Euler-Lagrange equation which is called the Jacobi equation of (15) is −𝑑𝑑𝑡𝑃ℎ^′+𝑄ℎ=0. (16) Since the interval (0,𝑇) contains a point 𝑐 conjugate to 0, there exists a nonzero Jacobi field ℎ₀∈𝐶²([0,𝑇],𝑅) satisfying −𝑑𝑑𝑡𝑃ℎ^′₀+𝑄ℎ₀ℎ=0,₀(0)=0,ℎ₀(𝑐)=0,ℎ^′₀(0)=0. (17) Letting ℎ(𝑡)=ℎ₀(𝑡)[],(],𝑡∈0,𝑐0𝑡∈𝑐,𝑇 (18) we have ℎ∈𝐶²([0,𝑇]⧵{𝑐},𝑅),ℎ(0)=ℎ(𝑐)=ℎ(𝑇)=0 and 𝑄𝑢ℎ=^𝑇₀𝑃ℎ^′2+𝑄ℎ²=𝑑𝑡_𝑐0𝑃ℎ₀^′2+𝑄ℎ²₀𝑑𝑡=0. (19) Notice that we can extend ℎ periodically when we take 𝑇 as the period, so ℎ∈𝑊₀^1,2(𝑅/𝑇𝑍,𝑅). For all ℎ∈𝐶¹₀([0,𝑇],𝑅), it is easy to check that 𝑄𝑢(ℎ)≥0. Then by (19), one has ℎ∈𝐶²([0,𝑇]⧵{𝑐},𝑅)∩𝑊₀^1,2(𝑅/𝑇𝑍,𝑅) is a minimum of 𝑄𝑢(ℎ). Hence we get −𝑑𝑃ℎ𝑑𝑡^′+𝑄ℎ=0. (20) Combining with ℎ(0)=ℎ(𝑐)=0 and ℎ^′(0)=0, by the uniqueness of initial value problems for second-order 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 1𝑟=4𝜋^2/3_{1≤𝑗≤𝑁−1}csc𝜋𝑁𝑗^1/3. (21)

Proof. By (1)–(3), we have 𝑞̈_𝑁=_𝑗≠𝑁𝑞_𝑗−𝑞_𝑁𝑞_𝑗−𝑞_𝑁³. (22) Substituting (1) into (22), we have −4𝜋²=_𝑗≠𝑁𝜌_𝑗−𝜌_𝑁𝑟³𝜌_𝑗−𝜌_𝑁³,4𝜋²𝑟³=_𝑗≠𝑁1−𝜌_𝑗1−𝜌_𝑗³=14_{1≤𝑗≤𝑁−1}csc𝜋𝑁𝑗. (23) Then 𝑟³=116𝜋²_{1≤𝑗≤𝑁−1}csc𝜋𝑁𝑗. (24) Therefore 1𝑟=4𝜋^2/3_{1≤𝑗≤𝑁−1}csc𝜋𝑁𝑗^1/3. (25)

Proof of Theorem 1. Clearly, 𝑞(𝑡)=(0,0,0) is a critical point of 𝑓(𝑞) on Λ_𝑖=Λ_𝑖(𝑖=1,2). For the functional (6), let 𝐹𝑧,𝑧^′=12𝑧^′2+𝑁√𝑟²+𝑧². (26) Then the second variation of (6) in the neighborhood of 𝑧=0 is given by ¹₀𝑃ℎ^′2+𝑄ℎ²𝑑𝑡, (27) where 1𝑃=2𝐹_{𝑧′𝑧′𝑧=0}=12,1𝑄=2𝐹_𝑧𝑧−𝑑𝐹𝑑𝑡^𝑧𝑧′_𝑧=0𝑁=−2𝑟³. (28) The Euler equation of (27) is called the Jacobi equation of the original functional (6), which is −𝑑𝑑𝑡𝑃ℎ^′+𝑄ℎ=0, (29) that is, ℎ^′′+𝑁𝑟³ℎ=0. (30) Next, we study the solution of (30) with initial values ℎ(0)=0,ℎ^′(0)=1. It is easy to get ℎ(𝑡)=𝑟³𝑁⋅sin𝑁𝑟³𝑡, (31) 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⁺𝑁𝑟³𝑐=𝑘𝜋. (32) We have 𝜋𝑐=²𝑘²𝑟³𝑁^1/2=𝑘²∑^𝑁−1_𝑗=1csc(𝜋/𝑁)𝑗16𝑁^1/2 (33) by using (24).

Case 1 (Minimizing 𝑓(𝑞) on Λ₁=Λ₁). Letting 0<𝑐<1/2, ℎ(𝑡)=ℎ(𝑡)[],(],()(].𝑡∈0,𝑐0𝑡∈𝑐,1−𝑐−ℎ1−𝑡𝑡∈1−𝑐,1 (34) It is easy to check that ℎ(𝑡)∈𝐶²([0,1]⧵{𝑐,1−𝑐},𝑅)∩𝑊^1,2(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=^{𝑁−1𝑗=1}csc(𝜋/𝑁)𝑗16𝑁^1/2<12. (35) It is equivalent to ^𝑁−1_𝑗=1csc𝜋𝑁𝑗<4𝑁. (36)
Let 𝑓(𝑥)=_𝑗≠𝑥csc𝜋𝑥𝑗; (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 sin𝑁𝑟³𝑐=sin𝜋=0. (38) Notice that we can extend ℎ periodically when we take 1 as the period, so ℎ∈Λ₁. Then by Lemma 8, 𝑞(𝑡)=(0,0,0) is not a local minimum for 𝑓(𝑞) on Λ₁. Hence the minimizers of 𝑓(𝑞) on Λ₁ 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 𝑓(𝑞) on Λ₂=Λ₂). Let ℎ(𝑡)=ℎ(𝑡)[],1𝑡∈0,𝑐0𝑡∈𝑐,2,1−ℎ𝑡−21𝑡∈2,12,1+𝑐0𝑡∈2+𝑐,1. (39) It is easy to check that ℎ(𝑡)∈𝐶²([0,1]⧵{𝑐,1/2,(1/2)+𝑐},𝑅)∩𝑊^1,2(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=^{𝑁−1𝑗=1}csc(𝜋/𝑁)𝑗16𝑁^1/2<12. (40) It implies that ^𝑁−1_𝑗=1csc𝜋𝑁𝑗<4𝑁. (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∈Λ₂. Then by Lemma 8, 𝑞(𝑡)=(0,0,0) is not a local minimum for 𝑓(𝑞) on Λ₂. Hence the minimizers of 𝑓(𝑞) on Λ₂ 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 𝑧(𝑡)=1𝑡𝑡∈0,4,141𝑡∈4,12, (42) and we extend 𝑧(𝑡) by 𝑧(𝑡+(1/2))=−𝑧(𝑡). We have 𝑓(𝑧)=14+2₀^1/2𝑁√𝑟²+𝑧²=1𝑑𝑡4+2₀^1/4𝑁√𝑟²+𝑡²𝑑𝑡+2^1/2_1/4𝑁𝑟²+()=11/16𝑑𝑡4√+2𝑁ln1+1+16𝑟²+4𝑟2𝑁√1+16𝑟². (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 Λ_𝑖=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

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.