A Variational Approach to Planar Choreographies via Ekeland's Principle

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 spatial scale within the Lagrangian action, leading to the functional F = Kα/(α+2) V2/(α+2). We prove the existence of critical points of 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 $0<α<2$.