In this paper we consider the planar non-collinear central configurations with n bodies with power-law potentials like <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_442_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum m_im_jr^{-a}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∑</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <msub> <mi>m</mi> <mi>j</mi> </msub> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mi>a</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_442_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$a>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, in which is possible to remove one body and still have a central configuration. This kind of central configurations is called a (n, 1)-stacked central configuration. We prove that the unique planar (n, 1)-stacked central configuration is formed by a regular polygon with equal masses at the vertices and one arbitrary mass at the barycenter, for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_442_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$4\le n <8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>≤</mo> <mi>n</mi> <mo><</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>. However, our results depend on the value of a.

Some Planar (n, 1)-Stacked Central Configurations

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发表日期：

2025年2月6日

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Antonio C. Fernandes,
Anderson M. Silva,
Claudio Vidal

摘要

In this paper we consider the planar non-collinear central configurations with n bodies with power-law potentials like $\sum m_im_jr^{-a}$ $\sum m_{i} m_{j} r^{- a}$ , $a>0$ $a > 0$ , in which is possible to remove one body and still have a central configuration. This kind of central configurations is called a (n, 1)-stacked central configuration. We prove that the unique planar (n, 1)-stacked central configuration is formed by a regular polygon with equal masses at the vertices and one arbitrary mass at the barycenter, for $4\le n <8$ $4 \leq n < 8$ . However, our results depend on the value of a.