# Regular Polygon Central Configuration of the Restricted 1 + N-Body Problem

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## Abstract

**:**

## 1. Introduction

**TO SOLVE**and there

**ARE**no general solutions; so, people try to search for particular solutions. Central configurations of the N-body problem is one of the most classical topics in celestial mechanics. Central configurations [2] allow us to construct exact solutions for the N-body problem. Collapse orbits and parabolic orbits have relations with the central configurations, and central configurations also have other interesting properties; so, finding central configurations is very important. Central configurations are configurations such that the total Newtonian acceleration of every body is equal to a constant multiplied by the position vector of this body with respect to the center of mass of the configurations.

**Definition 1**.

- (i)
- When N is odd, the mass parameters of all the infinitesimal bodies must be equal, i.e., ${\mu}_{1}={\mu}_{2}=\cdots ={\mu}_{N}$;
- (ii)
- When N is even, the mass parameters of the alternate infinitesimal bodies are equal, i.e., ${\mu}_{1}={\mu}_{3}=\cdots ={\mu}_{N-1}$ and ${\mu}_{2}={\mu}_{4}=\cdots ={\mu}_{N}$.

## 2. Propositions and Corollaries

**Definition 2**.

**Proposition 1**.

**Proof.**

**Proposition 2**.

**Proposition 3**.

**Corollary 1**.

**Proof.**

**Corollary 2**.

**Proof.**

## 3. Preliminary

**Lemma 1**.

**Proof.**

**Corollary 3**.

**Corollary 4**.

**Proof.**

**Corollary 5**.

**Proof.**

**Remark 1**.

**Corollary 6**.

**Proof.**

**Lemma 2**.

**Lemma 3**.

## 4. Theorems and Proofs

**Theorem 1**.

**Proof.**

**Theorem 2**.

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Configuration of a regular polygon. The blue dots represent the infinitesimal bodies and red dotted line used in the image represents the co-orbital circle.

**Figure 2.**Central configuration for odd N. The blue dots represent the infinitesimal bodies and red dotted line used in the image represents the co-orbital circle.

**Figure 3.**Central configuration for even N. The blue dots and red dotted line used in the image are as above.

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**MDPI and ACS Style**

Chen, J.; Bi, P.; Yang, M. Regular Polygon Central Configuration of the Restricted 1 + *N*-Body Problem. *Universe* **2023**, *9*, 254.
https://doi.org/10.3390/universe9060254

**AMA Style**

Chen J, Bi P, Yang M. Regular Polygon Central Configuration of the Restricted 1 + *N*-Body Problem. *Universe*. 2023; 9(6):254.
https://doi.org/10.3390/universe9060254

**Chicago/Turabian Style**

Chen, Jian, Peng Bi, and Mingfang Yang. 2023. "Regular Polygon Central Configuration of the Restricted 1 + *N*-Body Problem" *Universe* 9, no. 6: 254.
https://doi.org/10.3390/universe9060254