Continuous Choreographies as Limiting Solutions of $N$-body Type Problems with Weak Interaction

TL;DR

This paper studies the limit of large N-body problems with weak interactions, deriving an integro-differential equation for the limiting motion and proving the circle as the minimal action solution among certain loops.

Contribution

It introduces a new integro-differential equation for the limiting behavior of N-body systems with weak interactions and proves the circle as the absolute minimizer of the associated action functional.

Findings

The limit N→∞ leads to an integro-differential equation for the system.

Circle solutions minimize the action functional among zero mean loops.

The solutions correspond to traveling waves of the derived equation.

Abstract

We consider the limit $N\to +\infty$ of $N$-body type problems with weak interaction, equal masses and $-\sigma$-homogeneous potential, $0<\sigma<1$. We obtain the integro-differential equation that the motions must satisfy, with limit choreographic solutions corresponding to travelling waves of this equation. Such equation is the Euler-Lagrange equation of a corresponding limiting action functional. Our main result is that the circle is the absolute minimizer of the action functional among zero mean (travelling wave) loops of class $H^1$.