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

Yuika Kajihara, Eiko Kin, Mitsuru Shibayama

TL;DR

This work connects celestial mechanics and low-dimensional topology by studying braids generated by periodic simple choreographies in the planar Newtonian $N$-body problem with equal masses. Using braid groups, mapping class groups, and the Nielsen–Thurston classification, the authors show that for each $N\,\ge\,3$ and every $m{m{m{m{m{}}}}}\in \Omega_N$, there exists a simple choreography whose primitive braid type is $\alpha_{-\bm{m{\omega}}}=\sigma_1^{-\omega_1}\cdots\sigma_{N-1}^{-\omega_{N-1}}$ and whose full braid type is $(\alpha_{-\bm{\omega}})^N$, with pseudo-Anosov type except when $\bm{\omega}$ is all ones or all minus ones. The paper also characterizes the $m{\omega}$ that maximize or minimize the associated stretch factor, yielding concrete extremal braids such as the figure-eight ($N=3$, $\bm{\omega}=(1,-1)$) and the super-eight ($N=4$, $\bm{\omega}=(1,-1,1)$). The results rely on a detailed treatment of braids associated with compositions of integers and on recursive polynomial formulas for stretch factors, enabling explicit computation of $\lambda$ via Perron–Frobenius theory. Overall, the work provides a topological lens on the complexity of choreographies and highlights new links between dynamical celestial mechanics and mapping-class-group dynamics.

Abstract

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 $N \ge 3$, 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.

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

TL;DR

This work connects celestial mechanics and low-dimensional topology by studying braids generated by periodic simple choreographies in the planar Newtonian -body problem with equal masses. Using braid groups, mapping class groups, and the Nielsen–Thurston classification, the authors show that for each and every , there exists a simple choreography whose primitive braid type is and whose full braid type is , with pseudo-Anosov type except when is all ones or all minus ones. The paper also characterizes the that maximize or minimize the associated stretch factor, yielding concrete extremal braids such as the figure-eight (, ) and the super-eight (, ). The results rely on a detailed treatment of braids associated with compositions of integers and on recursive polynomial formulas for stretch factors, enabling explicit computation of via Perron–Frobenius theory. Overall, the work provides a topological lens on the complexity of choreographies and highlights new links between dynamical celestial mechanics and mapping-class-group dynamics.

Abstract

We study periodic solutions of the planar Newtonian -body problem with equal masses. Each periodic solution traces out a braid with 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 , Guowei Yu established the existence of a family of simple choreographies to the planar Newtonian -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.
Paper Structure (11 sections, 19 theorems, 86 equations, 20 figures, 1 table)

This paper contains 11 sections, 19 theorems, 86 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braid groups and mapping class groups
  4. Nielsen-Thurston classification
  5. Braids as particle dances
  6. Compositions of integers
  7. Braids associated with compositions of integers
  8. Stretch factors of pseudo-Anosov braids
  9. Simple choreographies by Yu
  10. Proofs
  11. Conclusion

Key Result

Theorem 1.2

For each $N \ge 3$ and $\bm{\omega}= (\omega_1, \dots, \omega_{N-1}) \in \Omega_N$, there exists a simple choreography of the planar Newtonian $N$-body problem whose primitive braid type is given by the braid $\sigma_1^{\omega_1} \sigma_2^{\omega_2} \cdots \sigma_{N-1}^{\omega_{N-1}}$. In particular

Figures (20)

  • Figure 1: (1) The figure-eight solution $\bm{z}(t)$ with period $T$. (2)(3) The primitive braid $b:= b(\bm{z}([0, \frac{T}{3}])) = \sigma_1^{-1} \sigma_2$. (4) The braid $b^3= (\sigma_1^{-1} \sigma_2)^3$ represents the braid type of the figure-eight.
  • Figure 2: The closed curve obtained from the simple choreography $\bm{z}_{\bm{\omega}}(t)$ in the case $N=4$ and $\bm{\omega}= (1, -1,-1)$. Thick arrows illustrate the trajectory of the $0$th particle $z_0(t)$ from $t=0$ to $\frac{N}{2}=2$.
  • Figure 3: The super-eight of the planar $4$-body problem.
  • Figure 4: Simple choreographies $\bm{z}_{\bm{\omega}}(t)$ in the case $N=19$. The dots denote the initial condition. The arrows indicate the trajectory of $\bm{z}_{\bm{\omega}}(t)$ from $t=0$ to $\frac{1}{2}$.
  • Figure 5: (1) $\sigma_i \in B_n$. (2) $h_i = \Gamma(\sigma_i) \in \mathrm{MCG}(D_n)$. (3) $\Sigma_{0, n+1}$.
  • ...and 15 more figures

Theorems & Definitions (47)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • ...and 37 more