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<p>We study periodic solutions of the planar Newtonian <i>N</i>-body problem with equal masses. Each periodic solution traces out a braid with <i>N</i> strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch factor greater than 1, which reflects the complexity of the corresponding periodic solution. For each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N \ge 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian <i>N</i>-body problem. We prove that braids arising from Yu’s periodic solutions are of pseudo-Anosov types, except in the special case where all particles move along a circle. We also identify the simple choreographies whose braid types have the largest and smallest stretch factors, respectively.</p>

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A study of braids arising from simple choreographies of the planar Newtonian N-body problem

  • 期刊论文

  • 发表日期：

    2025年9月25日

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  • Yuika Kajihara,
  • Eiko Kin,
  • Mitsuru Shibayama

摘要

We study periodic solutions of the planar Newtonian N-body problem with equal masses. Each periodic solution traces out a braid with N strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch factor greater than 1, which reflects the complexity of the corresponding periodic solution. For each \(N \ge 3\) N ≥ 3 , Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian N-body problem. We prove that braids arising from Yu’s periodic solutions are of pseudo-Anosov types, except in the special case where all particles move along a circle. We also identify the simple choreographies whose braid types have the largest and smallest stretch factors, respectively.