Lagrangian and Eulerian RE

Abstract

The case $N = 3$ presents particular interest in Euclidean space because the equilateral triangle is a RE for any values of the masses, a property discovered by Joseph Louis Lagrange in 1772, We will further show that this is not the case in ${\text{S}}^{ 2}$ and ${\text{H}}^{ 2}$, where the positive and negative elliptic Lagrangian RE exist only if the masses are equal. This conclusion provides a first step towards understanding with the help of these equations whether space is Euclidean for distances of the order of 10 AU because Lagrangian orbits of unequal masses show up in our solar system, as for example the approximate equilateral triangle formed by the Sun, Jupiter, and the Trojan/Greek asteroids.