<p>For Newtonian 8-body problems, we investigate a class of spatial configurations consisting of two parallel rhombuses separated by a distance <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(h&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In these configurations, four masses <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m_{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m_{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m_{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m_{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> are positioned counterclockwise at the vertices of one rhombus, while the other four masses <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m^{\prime }_{1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mn>1</mn> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m^{\prime }_{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mn>2</mn> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m^{\prime }_{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mn>3</mn> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(m^{\prime }_{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>m</mi> <mn>4</mn> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> are positioned counterclockwise at the vertices of the second rhombus. We obtain the necessary conditions and sufficient conditions for the existence of these spatial central configurations. Specifically, in the central configurations, we find that if either <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq10.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\(m_{j}=m^{\prime }_{j}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mi>j</mi> </msub> <mo>=</mo> <msubsup> <mi>m</mi> <mi>j</mi> <mo>′</mo> </msubsup> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1375_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="106" /> </InlineMediaObject> <EquationSource Format="TEX">\(j\in \{1,2,3,4\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>j</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, or the two rhombuses have equal sizes, then both rhombuses must necessarily be squares.</p>

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A class of spatial central configurations in Newtonian 8-body problems

  • Liang Ding,
  • Zaili Yang,
  • Pengfei Yuan

摘要

For Newtonian 8-body problems, we investigate a class of spatial configurations consisting of two parallel rhombuses separated by a distance \(h>0\) h > 0 . In these configurations, four masses \(m_{1}\) m 1 , \(m_{2}\) m 2 , \(m_{3}\) m 3 , and \(m_{4}\) m 4 are positioned counterclockwise at the vertices of one rhombus, while the other four masses \(m^{\prime }_{1}\) m 1 , \(m^{\prime }_{2}\) m 2 , \(m^{\prime }_{3}\) m 3 , and \(m^{\prime }_{4}\) m 4 are positioned counterclockwise at the vertices of the second rhombus. We obtain the necessary conditions and sufficient conditions for the existence of these spatial central configurations. Specifically, in the central configurations, we find that if either \(m_{j}=m^{\prime }_{j}\) m j = m j for all \(j\in \{1,2,3,4\}\) j { 1 , 2 , 3 , 4 } , or the two rhombuses have equal sizes, then both rhombuses must necessarily be squares.