Singular Reduction of the $2$-Body Problem on the $3$-Sphere and the $4$-Dimensional Spinning Top

We consider the dynamics and symplectic reduction of the $2$-body problem on a sphere of arbitrary dimension. It suffices to consider the case when the sphere is $3$-dimensional. As the $3$-sphere is a group it acts on itself by left and right multiplication and these together generate the action of the $SO(4)$ symmetry on the sphere. This gives rise to a notion of left and right momenta for the problem, and allows for a reduction in stages, first by the left and then the right, or vice versa. The intermediate reduced spaces obtained by left or right reduction are shown to be coadjoint orbits of the special Euclidean group $SE(4)$. The full reduced spaces are generically $4$-dimensional and we describe these spaces and their singular strata.
The dynamics of the $2$-body problem descend through a double cover to give a dynamical system on $SO(4)$ which, after reduction and for a particular choice of Hamiltonian, coincides with that of a $4$-dimensional spinning top with symmetry. This connection allows us to “hit two birds with one stone” and derive results about both the spinning top and the $2$-body problem simultaneously. We provide the equations of motion on the reduced spaces and fully classify the relative equilibria and discuss their stability.