A Variational Approach to Planar Choreographies via Ekeland's Principle

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 $0<\alpha <2$.