A Variational Approach to Planar Choreographies via Ekeland's Principle

TL;DR

This paper introduces a variational method that optimizes the spatial scale in the N-body problem to find periodic solutions like the figure-eight, extending existing solutions and ensuring collision-free curves.

Contribution

It develops a new variational framework that explicitly optimizes spatial scale, proving existence of critical points corresponding to collision-free periodic solutions, including the figure-eight.

Findings

Existence of critical points with a single self-crossing curve.

Critical curves satisfy Newton's equations and are collision-free.

Framework extends to all 0<α<2, including the classical case α=1.

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 $0<\alpha<2$.