Central Configurations for Newtonian 𝑁+2𝑝+1-Body Problems

Abstract

We show the existence of spatial central configurations for the 𝑁+2𝑝+1-body problems. In the 𝑁+2𝑝+1-body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the 𝑁+2𝑝+1th is located at the the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.

1. Introduction and Main Results

The Newtonian 𝑛-body problems [1–3] concern with the motions of 𝑛 particles with masses 𝑚_𝑗∈𝑅⁺ and positions 𝑞_𝑗∈𝑅³(𝑗=1,2,…,𝑛), and the motion is governed by Newton’s second law and the Universal law: 𝑚_𝑗𝑞̈_𝑗=𝜕𝑈𝑞𝜕𝑞_𝑗, (1) where 𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛) and with Newtonian potential: 𝑈𝑞=_{1⩽𝑗<𝑘⩽𝑛}𝑚_𝑗𝑚_𝑘𝑞_𝑗−𝑞_𝑘. (2) Consider the space 𝑋=𝑞=𝑞₁,𝑞₂,…,𝑞_𝑛∈𝑅^3𝑛∶^𝑁_𝑗=1𝑚_𝑗𝑞_𝑗=0, (3) that is, suppose that the center of mass is fixed at the origin of the space. Because the potential is singular when two particles have the same position, it is natural to assume that the configuration avoids the collision set Δ={𝑞=(𝑞₁,…,𝑞_𝑁)∶𝑞_𝑗=𝑞_𝑘 for some 𝑘≠𝑗}. The set 𝑋⧵Δ is called the configuration space.

Definition 1 (see [2, 3]). A configuration 𝑞 = (𝑞₁,𝑞₂,…,𝑞_𝑛)∈𝑋⧵Δ is called a central configuration if there exists a constant 𝜆 such that ^𝑛_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑚_𝑘𝑞_𝑗−𝑞_𝑘³𝑞_𝑗−𝑞_𝑘=−𝜆𝑚_𝑘𝑞_𝑘,1⩽𝑘⩽𝑛. (4) The value of constant 𝜆 in (4) is uniquely determined by 𝑈𝜆=𝐼, (5) where 𝐼=^𝑛_𝑘=1𝑚_𝑘𝑞_𝑘². (6) Since the general solution of the 𝑛-body problem cannot be given, great importance has been attached to search for particular solutions from the very beginning. A homographic solution is a configuration which is preserved for all time. Central configurations and homographic solutions are linked by the Laplace theorem [3]. Collapse orbits and parabolic orbits have relations with the central configurations [2, 4–6], so finding central configurations becomes very important. The main general open problem for the central configurations is due to Wintner [3] and Smale [7]: is the number of central configurations finite for any choice of positive masses 𝑚₁,…,𝑚_𝑛? Hampton and Moeckel [8] have proved this conjecture for any given four positive masses.

For 5-body problems, Hampton [9] provided a new family of planar central configurations, called stacked central configurations. A stacked central configuration is one that has some proper subset of three or more points forming a central configuration. Ouyang et al. [10] studied pyramidal central configurations for Newtonian 𝑁+1-body problems; Zhang and Zhou [11] considered double pyramidal central configurations for Newtonian 𝑁+2-body problems; Mello and Fernandes [12] analyzed new classes of spatial central configurations for the 𝑁+3-body problem. Llibre and Mello studied triple and quadruple nested central configurations for the planar 𝑛-body problem. There are many papers studying central configuration problems such as [13–22].

Based on the above works, we study stacked central configuration for Newtonian 𝑁+2𝑝+1-body problems. In the 𝑁+2𝑝+1-body problems, 𝑁 bodies are at the vertices of a regular 𝑁-gon 𝑇, and 2𝑝 bodies are symmetrically located on the same straight line which is perpendicular to 𝑇 and passes through the center of 𝑇; the 𝑁+2𝑝+1th body is located at the center of 𝑇. The masses located on the vertices of the regular 𝑁-gon are equal; the masses located on the line and symmetric with respect to the center of 𝑇 are equal. (see Figure 1 for 𝑁=4 and 𝑝=2).

Figure 1 

In this paper we will prove the following result.

Theorem 2. For 𝑁+2𝑝+1-body problem in 𝑅³ where 𝑁≥2 and 𝑝≥1, there is at least one central configuration such that 𝑁 bodies are at the vertices of a regular 𝑁-gon 𝑇, and 2𝑝 bodies are symmetric with respect to the center of the regular 𝑁-gon 𝑇, and located on a line which is perpendicular to the regular 𝑁-gon 𝑇; the 𝑁+2𝑝+1th body is located at the center of 𝑇. The masses at the vertices of 𝑇 are equal and the masses symmetric with respect to the center of 𝑇 are equal.

2. The Proof of Theorem 2

Our approach to Theorem 2 is inspired by the of arguments of Corbera et al. in [23].

2.1. Equations for the Central Configurations of 𝑁+2𝑝-Body Problems

To begin, we take a coordinate system which simplifies the analysis. The particles have positions given by 𝑞_𝑗=(cos𝛼_𝑗,sin𝛼_𝑗,0), where 𝛼_𝑗=((𝑗−1)/𝑁)2𝜋, 𝑗=1,…,𝑁; 𝑞_𝑁+𝑗=(0,0,𝑟_𝑗), 𝑞_{𝑁+𝑗+𝑝}=(0,0,−𝑟_𝑗), where 𝑗=1,…,𝑝; 𝑞_𝑁+2𝑝+1=(0,0,0).

The masses are given by 𝑚₁=𝑚₂=⋯=𝑚_𝑁=1, 𝑚_𝑁+𝑗=𝑚_{𝑁+𝑗+𝑝}=𝑀_𝑗, where 𝑗=1,…,𝑝,𝑚_𝑁+2𝑝+1=𝑀₀.

Notice that (𝑞₁,…,𝑞_𝑁,𝑞_𝑁+1,…,𝑞_𝑁+2𝑝,𝑞_𝑁+2𝑝+1) is a central configuration if and only if ^𝑁+2𝑝+1_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑚_𝑘𝑞_𝑗−𝑞_𝑘³𝑞_𝑗−𝑞_𝑘=−𝜆𝑚_𝑘𝑞_𝑘,1⩽𝑘⩽𝑁+2𝑝. (7) By the symmetries of the system, (7) is equivalent to the following equations: ^𝑁+2𝑝+1_{𝑗=1,𝑗≠𝑘}𝑚_𝑗𝑚_𝑘𝑞_𝑗−𝑞_𝑘³𝑞_𝑗−𝑞_𝑘=−𝜆𝑚_𝑘𝑞_𝑘,𝑘=1,𝑁+1,…,𝑁+𝑝, (8) that is, −𝜆(1,0,0)=−𝛽(1,0,0)−(1,0,0)𝑀₀−^𝑝_𝑖=121+𝑟²_𝑖^3/2𝑀_𝑖(1,0,0), (9) where 𝛽=(1/4)∑^𝑁−1_𝑗=1csc(𝜋𝑗/𝑁)−𝜆0,0,𝑟₁𝑁=−1+𝑟²₁^3/20,0,𝑟₁−10,0,𝑟²₁𝑀₀−12𝑟₁²𝑀₁()+0,0,1_𝑝𝑖=21𝑟_𝑖−𝑟₁²−1𝑟_𝑖+𝑟₁²𝑀₁(0,0,1),−𝜆0,0,𝑟_𝑗𝑁=−1+𝑟²_𝑗^3/20,0,𝑟_𝑗−10,0,𝑟²_𝑗𝑀₀−_{1≤𝑖<𝑗}1𝑟_𝑖−𝑟_𝑗²+1𝑟_𝑖+𝑟_𝑗²𝑀_𝑖()−10,0,12𝑟_𝑗²𝑀_𝑗(0,0,1)+_{𝑗<𝑖≤𝑝}1𝑟_𝑖−𝑟_𝑗²−1𝑟_𝑖+𝑟_𝑗²𝑀_𝑖(0,0,1),𝑗=2,…,𝑝−1−𝜆0,0,𝑟_𝑝𝑁=−1+𝑟²_𝑝^3/20,0,𝑟_𝑝−10,0,𝑟²_𝑝𝑀₀−_{1≤𝑖<𝑝}1𝑟_𝑖−𝑟_𝑝²+1𝑟_𝑖+𝑟_𝑝²𝑀_𝑖()−10,0,12𝑟_𝑝²𝑀_𝑝(0,0,1). (10)(11) In order to simplify the equations, we defined 𝑎_0,𝑗=−2/|1+𝑟²_𝑗|^3/2, 𝑎_𝑗,𝑗=−1/4𝑟³_𝑗, where 𝑗=1,…,𝑝; 𝑎_𝑖,𝑗=(1/|𝑟_𝑖−𝑟_𝑗|²𝑟_𝑖−1/|𝑟_𝑖+𝑟_𝑗|²𝑟_𝑖), when 𝑖<𝑗; 𝑎_𝑖,𝑗=−(1/|𝑟_𝑖−𝑟_𝑗|²𝑟_𝑖+1/|𝑟_𝑖+𝑟_𝑗|²𝑟_𝑖), when 𝑖>𝑗; 𝑎_0,0=𝑎_𝑖,0=1 when 𝑖=1,…,𝑝; 𝑏₀=𝛽+𝑀₀, 𝑏_𝑖=𝑁/|1+𝑟²_𝑖|^3/2+𝑀₀/𝑟³_𝑖, where 𝑖=1,…,𝑝.Equations (9)–(11) can be written as a linear system of the form 𝐴𝑋=𝑏 given by 1𝑎_0,1𝑎_0,2𝑎_0,3⋯𝑎_0,𝑝1𝑎_1,1𝑎_1,2𝑎_1,3⋯𝑎_1,𝑝1𝑎_2,1𝑎_2,2𝑎_2,3⋯𝑎_2,𝑝⋮⋮⋮⋱⋮⋮1𝑎_𝑝,1𝑎_𝑝,2𝑎_𝑝,3⋯𝑎_𝑝,𝑝𝜆𝑀₁𝑀₂𝑀₃⋮𝑀_𝑃=𝑏₀𝑏₁𝑏₂𝑏₃⋮𝑏_𝑝. (12) The column vector is given by the variables 𝑋=(𝜆,𝑀₁,𝑀₂,…,𝑀_𝑝)^𝑇. Since the 𝑎_𝑖𝑗 is function of 𝑟₁,𝑟₂,…,𝑟_𝑝, we write the coefficient matrix as 𝐴_𝑝(𝑟₁,𝑟₂,…,𝑟_𝑝).

2.2. For 𝑝=1

We need the next lemma.

Lemma 3 (see [12]). Assuming 𝑚₁=⋯=𝑚_𝑁=1, 𝑚_𝑁+1=𝑚_𝑁+2=𝑀₁, there is a nonempty interval 𝐼⊂𝑅, 𝑀₀(𝑟₁) and 𝑀₁(𝑟₁), such that for each 𝑟₁∈𝐼, (𝑞₁,…,𝑞_𝑁,𝑞_𝑁+1,𝑞_𝑁+2,𝑞_𝑁+3) forms a central configuration of the 𝑁+2+1-body problem.

For 𝑝=1, system (12) becomes 1𝑎_0,11𝑎_1,1𝜆𝑀₁=𝑏₀𝑏₁,𝐴₁𝑟₁=𝑎_1,1−𝑎_0,11=−4𝑟³₁+21+𝑟²₁^3/2. (13)(14) If we consider |𝐴₁(𝑟₁)| as a function of 𝑟₁, then |𝐴₁(𝑟₁)| is an analytic function and nonconstant. By Lemma 3, there exists a 𝑟^̌‌₁∈𝐼 such that (13) has a unique solution (𝜆,𝑀₁) satisfying 𝜆>0 and 𝑀₁>0.

2.3. For All 𝑝>1

The proof for 𝑝≥1 is done by induction. We claim that there exists 0<𝑟₁<𝑟₂<⋯<𝑟_𝑝 such that system (12) has a unique solution 𝜆=𝜆(𝑟₁,…,𝑟_𝑝)>0, 𝑀_𝑖=𝑀_𝑖(𝑟₁,…,𝑟_𝑝)>0 for 𝑖=1,…,𝑝. We have seen that the claim is true for 𝑝=1. We assume the claim is true for 𝑝−1 and we will prove it for 𝑝. Assume by induction hypothesis that there exists 0<𝑟^∼₁<𝑟^∼₂<⋯<𝑟^∼_𝑝−1 such that system (12) has a unique solution 𝜆^∼=𝜆(𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1)>0 and 𝑀^∼_𝑖=𝑀_𝑖(𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1)>0 for 𝑖=1,2,…,𝑝−1.

We need the next lemma.

Lemma 4. There exists 𝑟^∼_𝑝>𝑟^∼_𝑝−1 such that 𝜆^∼=𝜆(𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1), 𝑀^∼_𝑖=𝑀_𝑖(𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1) for 𝑖=1,2,…,𝑝−1 and 𝑀^∼_𝑝=0 is a solution of (12).

Proof. Since 𝑀_𝑝=𝑀^∼_𝑝=0, we have that the first 𝑝−1 equation of (12) is satisfied when 𝜆=𝜆^∼, 𝑀_𝑖=𝑀^∼_𝑖 for 𝑖=1,2,…,𝑝−1 and 𝑀_𝑝=𝑀^∼_𝑝=0. Substituting this solution into the last equation of (12), we let 𝑓𝑟_𝑝=𝜆^∼+𝑎^∼_𝑝,1𝑀₁+⋯+𝑎^∼_{𝑝,𝑝−1}𝑀_𝑝−1−𝑁1+𝑟²_𝑝^3/2−𝑀₀𝑟³_𝑝. (15) We have that lim_{𝑟𝑝→+∞}𝑓𝑟_𝑝=𝜆^∼>0,lim_{𝑟𝑝→∼𝑟+𝑝−1}𝑓𝑟_𝑝=−∞. (16) Therefore there exists at least a value 𝑟_𝑝=𝑟^∼_𝑝>𝑟^∼_𝑝−1 satisfying equation 𝑓(𝑟_𝑝)=0. This completes the proof of Lemma 4.

By using the implicit function theorem, we will prove that the solution of (12) given in Lemma 3 can be continued to a solution with 𝑀_𝑝>0.

Let 𝑠=(𝜆,𝑟₁,…,𝑟_𝑝,𝑀₁,…,𝑀_𝑝), we define 𝑔₀(𝑠)=𝜆+𝑎_0,1𝑀₁+⋯+𝑎_0,𝑝𝑀_𝑝−𝑏₀,𝑔₁(𝑠)=𝜆+𝑎_1,1𝑀₁+⋯+𝑎_1,𝑝𝑀_𝑝−𝑏₁,𝑔₂(𝑠)=𝜆+𝑎_2,1𝑀₁+⋯+𝑎_2,𝑝𝑀_𝑝−𝑏₂,⋮𝑔_𝑝(𝑠)=𝜆+𝑎_𝑝,1𝑀₁+⋯+𝑎_𝑝,𝑝𝑀_𝑝−𝑏_𝑝. (17) It is not difficult to see that the system (12) is equivalent to 𝑔_𝑖(𝑠)=0 for 𝑖=0,1,…,𝑝.

Let ^∼𝑠=(𝜆^∼,𝑀^∼₁,𝑀^∼₂,…,𝑀^∼_𝑝,𝑟^∼₁,…,𝑟^∼_𝑝) be the solution of system (12) given in Lemma 4. The differential of (17) with respect to the variables (𝜆,𝑀₁,𝑀₂,…,𝑀_𝑝−1,𝑟_𝑝) is 𝐷_𝑝^∼𝑠=1𝑎_0,1𝑎_0,2⋯𝑎_0,𝑝−101𝑎_1,1𝑎_1,2⋯𝑎_1,𝑝−10⋮⋮⋮⋱⋮⋮1𝑎_𝑝−1,1𝑎_𝑝−1,2⋯𝑎_{𝑝−1,𝑝−1}01𝑎_𝑝,1𝑎_𝑝,2⋯𝑎_{𝑝,𝑝−1}𝜕𝑔_𝑝^∼𝑠𝜕𝑟_𝑝=𝐴𝑟_𝑝−1^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1𝜕𝑔_𝑝^∼𝑠𝜕𝑟_𝑝,𝜕𝑔_𝑝^∼𝑠𝜕𝑟_𝑝=^𝑝−1_𝑖=12𝑟_𝑝−𝑟_𝑖³𝑟_𝑝+1𝑟_𝑝−𝑟_𝑖²𝑟²_𝑝+2𝑟_𝑝+𝑟_𝑖³𝑟_𝑝+1𝑟_𝑝+𝑟_𝑖³𝑟²_𝑝^∼𝑀_𝑖+3𝑁𝑟_𝑝1+𝑟²_𝑝^5/2+3𝑀₀𝑟⁴_𝑝>0. (18) We have assumed that 0<𝑟^∼₁<𝑟^∼₂<⋯<𝑟^∼_𝑝−1exists such that system (12) with 𝑝−1 instead of 𝑝 has a unique solution, so |𝐴_𝑝−1(𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝−1)|≠0; therefore 𝐷(^∼𝑠)≠0. Applying the Implicit Function Theorem, there exists a neighborhood 𝑈 of (𝑀^∼_𝑝,𝑟^∼₁,𝑟^∼₁,…,𝑟^∼_𝑝−1), and unique analytic functions 𝜆=𝜆(𝑀^∼_𝑝,𝑟^∼₁,𝑟^∼₁,…,𝑟^∼_𝑝−1), 𝑀_𝑖=𝑀_𝑖(𝑀^∼_𝑝,𝑟^∼₁,𝑟^∼₁,…,𝑟^∼_𝑝−1) for 𝑖=1,…,𝑝−1 and 𝑟_𝑝=𝑟_𝑝(𝑀_𝑝,𝑟₁,𝑟₂,…,𝑟_𝑝−1), such that (𝜆,𝑀₁,𝑀₂,…,𝑀_𝑝) is the solution of the system (12) for all (𝑀_𝑝,𝑟₁,𝑟₂,…,𝑟_𝑝−1)∈𝑈. The determinant is calculated as 𝐴_𝑝𝑟₁,𝑟₂,…,𝑟_𝑝=𝑎_𝑝,𝑝𝐴_𝑝−1𝑟₁,𝑟₂,…,𝑟_𝑝−1+^𝑝−1_𝑖=0𝑎_𝑝,𝑖𝐵_𝑝,𝑖=−12𝑟_𝑝²𝐴_𝑝−1𝑟₁,𝑟₂,…,𝑟_𝑝−1+^𝑝−1_𝑖=0𝑎_𝑝,𝑖𝐵_𝑝,𝑖, (19) where 𝐵_𝑝,𝑖 is the algebraic cofactor of 𝑎_𝑝,𝑖.

We see that |𝐴_𝑝−1(𝑟₁,𝑟₂,…,𝑟_𝑝−1)|, 𝑎_𝑝,𝑖 and 𝐵_𝑝,𝑖 for 𝑖=0,1,…,𝑝−1 do not contain the factor 1/|2𝑟_𝑝|². If we consider |𝐴_𝑝(𝑟₁,𝑟₂,…,𝑟_𝑝)| as a function of 𝑟_𝑝, then |𝐴_𝑝(𝑟₁,𝑟₂,…,𝑟_𝑝)| is analytic and nonconstant. We can find (𝑟^∼∼₁,𝑟^∼∼₂,…,𝑟^∼∼_𝑝) sufficiently close to (𝑟^∼₁,𝑟^∼₂,…,𝑟^∼_𝑝) such that |𝐴_𝑝(𝑟^∼∼₁,𝑟^∼∼₂,…,𝑟^∼∼_𝑝)|≠0 and therefore, a solution (𝜆,𝑀₁,…,𝑀_𝑝) of system (12) is satisfying 𝜆>0, 𝑀_𝑖>0 for 𝑖=1,…,𝑝.

The proof of Theorem 2 is completed.

Acknowledgments

The authors express their gratitude to Professor Zhang Shiqing for his discussions and helpful suggestions. This work is supported by NSF of China and Youth found of Mianyang Normal University.