Action minimizing orbits in the 2-center problems with simple choreography constraint

TL;DR

This paper proves that in a 2+ n-body gravitational system with two fixed masses, the motion minimizing the action under simple choreography constraints is a uniform circular orbit.

Contribution

It demonstrates that the minimal action configuration under simple choreography constraints is a uniform circular motion in the 2-center problem with fixed masses.

Findings

The action functional attains its minimum on a uniform circular orbit.

The study provides a variational approach to the 2-center problem.

The results identify the specific orbit structure minimizing the action.

Abstract

The aim of this paper is to study the motion of $2+n$-body problem where two equal masses are assumed to be fixed. We assume that the value of each fixed mass is equal to $M>0$ and the remaining $n$ moving particles have equal masses $m>0$. According to Newton's second law and the universal gravitation law, the $n$ particles move under the interaction of each other and the affection of the two fixed particles. Also, this motion has a natural variational structure. Under the simple choreography constraint, we show that the Lagrangian action functional attains its absolute minimum on a uniform circular motion.