Relative periodic solutions of the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-vortex problem on the sphere

Abstract

This paper gives an analysis of the movement of n vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.

Choreographies in the n-vortex Problem

“…A theorem in [19] establishes the existence of Lyapunov families of periodic orbits that arise from polygonal relative equilibria on the sphere. Specifically, each polygon has a global family of periodic solutions of the form u j (t) = x j (νt), with symmetries (1), for each normal mode of oscillation.…”

“…These results depend only on the symmetries of the equations, and can be extended to the case of the n-vortex problem in radially symmetric domains. The global existence of Lyapunov families that arise from the n-polygon of vortices is established in the plane in [21], and on the sphere in [19]. In our current paper we determine choreographies along Lyapunov families of the n-vortex problem in the plane, in a disk, and on a sphere.…”