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arXiv · 2511.03134

A Variational Approach to Planar Choreographies via Ekeland's Principle

Juan Manuel Sánchez-Cerritos·Mayte Torres-Hernández
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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 $\mathcal F = K^{\alpha/(\alpha+2)} V^{2/(\alpha+2)}$. 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 $\alpha=1$) and extends to the whole range $0<\alpha<2$.

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Juan Manuel Sánchez-Cerritos, Mayte Torres-Hernández. 2025-11-05. A Variational Approach to Planar Choreographies via Ekeland's Principle. https://arxiv.org/abs/2511.03134

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