Positive Elliptic-Elliptic RE

Abstract

In this chapter we will construct examples of positive elliptic-elliptic RE, i.e. orbits with two elliptic rotations on the sphere ${\text{S}}^{ 3}$. The first example is that of a 3-body problem in which 3 bodies of equal masses are at the vertices of an equilateral triangle, which has two rotations of the same frequency. The second example is that of a 4-body problem in which 4 equal masses are at the vertices of a regular tetrahedron, which has two rotations of the same frequency. The third example is that of a 5-body problem in which 5 equal masses lying at the vertices of a pentatope have two rotations of same-size frequencies. This is the only regular polytope that allows RE, because the five other existing regular polytopes of ${\text{R}}^{ 4}$ have antipodal vertices, so they introduce singularities.