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A Variational Approach to Planar Choreographies via Ekeland's Principle

Juan Manuel Sánchez-Cerritos, Mayte Torres-Hernández

Abstract

We present a variational approach to obtain periodic solutions of the $N$-body problem, in particular the 'figure-eight' solution for three equal masses. The central idea is to explicitly optimize the \emph{spatial scale} within the Lagrangian action, leading to the functional $\mathcal F = K^{α/(α+2)} V^{2/(α+2)}$. We prove the existence of critical points of $\mathcal F$ that enforce a curve with a single self-crossing, and show that every reparametrized critical curve satisfies Newton's equations and is free of collisions. This framework recovers the Chenciner-Montgomery 'eight' (for $α=1$) and extends to the whole range $0<α<2$.

A Variational Approach to Planar Choreographies via Ekeland's Principle

Abstract

We present a variational approach to obtain periodic solutions of the -body problem, in particular the 'figure-eight' solution for three equal masses. The central idea is to explicitly optimize the \emph{spatial scale} within the Lagrangian action, leading to the functional . We prove the existence of critical points of that enforce a curve with a single self-crossing, and show that every reparametrized critical curve satisfies Newton's equations and is free of collisions. This framework recovers the Chenciner-Montgomery 'eight' (for ) and extends to the whole range .

Paper Structure

This paper contains 16 sections, 16 theorems, 95 equations, 1 figure.

Table of Contents

  1. Introduction and background
  2. Variational setting
  3. Symmetric class $(A_x)$–$(Y^\star)$ for the “eight’’ (case $N=3$)
  4. Effect of the symmetries.
  5. A scale–invariant functional and its scale envelope
  6. From the scale–invariant functional to Newton’s equation
  7. Ekeland’s principle: statement and application to $\mathcal{F}$
  8. Collision avoidance via Marchal’s criterion
  9. Regularity of the critical point
  10. Step 1: Weak formulation.
  11. Step 2: $L^\infty$ regularity and $C^1$ regularity.
  12. Step 3: Start of the bootstrap.
  13. Step 4: Inductive step.
  14. Step 5: Conclusion of the bootstrap.
  15. Step 6: Analyticity.
  16. ...and 1 more sections

Key Result

Lemma 2.1

One has $\int_0^{2\pi} x(t)\,dt=0$. In particular, the constant Fourier coefficient of $x$ vanishes.

Figures (1)

  • Figure 1: Idealized figure–eight choreography for three equal masses in the symmetric class $\mathcal{H}_{\mathrm{eight}}$, satisfying $(A_x)$ and $(Y^\star)$, $\alpha=3/2$. The configuration shown corresponds to a collinear instant of the periodic motion.

Theorems & Definitions (37)

  • Lemma 2.1: Zero mean of $x$ from $Y^\star$
  • proof
  • Remark 2.2: Pointwise consequences at $t=\frac{\pi}{2}$
  • Proposition 2.3: Allowed Fourier spectrum
  • proof
  • Remark 2.4: Option (NC1) on $y$
  • Proposition 3.1: Equivalences
  • proof
  • Lemma 3.2: Scale invariance
  • proof
  • ...and 27 more