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Choreographies with Dihedral Symmetry in the Planar $n$-Body Problem

Juan Manuel Sánchez Cerritos

TL;DR

The paper addresses the planar $n$-body problem with a homogeneous potential of degree $-\\alpha$, seeking $D_n$-equivariant choreographies characterized by a winding number $W$ coprime to $n$. It reduces the problem to the rotating-frame equation $L u = N(u)$ for a single generating curve $u$, and applies Mawhin’s coincidence degree under a nonresonance condition to prove existence. The main contribution is establishing the existence of a $T$-periodic, collision-free choreography for any coprime $(W,n)$ by showing $L$ is a Fredholm isomorphism and $N$ is $L$-compact, together with uniform energy bounds and uniform separation that enforce collision exclusion. This provides a robust topological framework for dihedral-symmetric choreographies, extending beyond variational and numerical methods and offering structural insights into symmetry and compactness in celestial mechanics.

Abstract

We prove the existence of planar $D_n$--equivariant choreographies in the $n$--body problem with homogeneous potential of degree $-α$, $0<α<2$. Each body follows the same closed path, rotated and time-shifted, forming a choreography whenever the winding number $W$ is coprime with $n$. Using Mawhin's coincidence degree, we establish collision-free periodic solutions under a simple nonresonance condition. The proof relies on the spectral structure of the linearized operator, symmetry-induced separation of the bodies, and uniform energy bounds ensuring compactness of the nonlinear term. This provides a topological route to choreographies beyond variational and numerical frameworks.

Choreographies with Dihedral Symmetry in the Planar $n$-Body Problem

TL;DR

The paper addresses the planar -body problem with a homogeneous potential of degree , seeking -equivariant choreographies characterized by a winding number coprime to . It reduces the problem to the rotating-frame equation for a single generating curve , and applies Mawhin’s coincidence degree under a nonresonance condition to prove existence. The main contribution is establishing the existence of a -periodic, collision-free choreography for any coprime by showing is a Fredholm isomorphism and is -compact, together with uniform energy bounds and uniform separation that enforce collision exclusion. This provides a robust topological framework for dihedral-symmetric choreographies, extending beyond variational and numerical methods and offering structural insights into symmetry and compactness in celestial mechanics.

Abstract

We prove the existence of planar --equivariant choreographies in the --body problem with homogeneous potential of degree , . Each body follows the same closed path, rotated and time-shifted, forming a choreography whenever the winding number is coprime with . Using Mawhin's coincidence degree, we establish collision-free periodic solutions under a simple nonresonance condition. The proof relies on the spectral structure of the linearized operator, symmetry-induced separation of the bodies, and uniform energy bounds ensuring compactness of the nonlinear term. This provides a topological route to choreographies beyond variational and numerical frameworks.

Paper Structure

This paper contains 38 sections, 12 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dihedral symmetry $D_n$ and problem formulation
  3. $D_n$–equivariant choreographies and the winding number $W$
  4. Equations of motion in the rotating frame
  5. Uniform separation and collision exclusion
  6. Functional framework and operators
  7. Spaces $X$ and $Z$
  8. Definition of the operators $L$ and $N$
  9. Covariant derivative and compact form of $L$
  10. Geometric and energetic interpretation
  11. Mawhin’s framework and coincidence degree
  12. Fredholm operators and canonical decomposition
  13. Operational decomposition of $Lu=Nu$.
  14. $L$--compactness and admissible homotopies
  15. Definition of the coincidence degree
  16. ...and 23 more sections

Key Result

Theorem 1

Let $n\ge3$ and $W\ge1$ be coprime integers ($\gcd(W,n)=1$), and consider the planar $n$--body problem with homogeneous potential $r^{-\alpha}$, $0<\alpha<2$. Assume the nonresonance condition $\Omega \neq \pm k\,\tfrac{2\pi}{T}$ for all $k\in\mathbb N$. Then there exists at least one $T$--periodic where $L:H^2_{\mathrm{per}}\to L^2_{\mathrm{per}}$ is the linear Fredholm operator associated with

Figures (1)

  • Figure 1: Generating curves $u(t)$ with dihedral symmetry $u(t+\tfrac{T}{n})=R_{2\pi/n}u(t)$ and winding number $W>1$. Each curve is composed of admissible Fourier modes $k=1+\ell n$, ensuring that $\gcd(W,n)=1$. These examples correspond to $(n,W)=(3,4)$, $(4,5)$, $(5,6)$, $(5,11)$, $(6,13)$, and $(7,22)$, respectively.

Theorems & Definitions (22)

  • Theorem : Existence of $D_n$--equivariant choreographies
  • Proposition 2.1: Coprimality condition for choreographies
  • proof
  • Remark 2.2: Spectral structure of the $D_n$–symmetry
  • Proposition 2.3: Uniform separation under $D_n$ symmetry for all $W\ge1$
  • proof
  • Corollary 2.4: Collision exclusion
  • Theorem 4.1: Mawhin Mawhin1972Mawhin1977Mawhin1979
  • Remark 5.1
  • Lemma 5.2: Periodic Poincaré inequality
  • ...and 12 more