Platonic solids and symmetric solutions of the $N$-vortex problem on the sphere

TL;DR

This paper develops a theoretical framework to analyze symmetric solutions of the equal-strength N-vortex problem on the sphere, proving the existence of periodic orbits and oscillations related to platonic solid equilibria.

Contribution

It introduces a novel symmetry-based reduction method for the N-vortex problem on the sphere, extending previous ideas and proving new existence results for periodic orbits.

Findings

Existence of 1-parameter families of periodic orbits.

Families of oscillations near platonic solid equilibria.

Extension of symmetry reduction techniques for vortex dynamics.

Abstract

We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the discrete reduction of the system by twisted subgroups of the full symmetry group that rotates and permutes the vortices. Our approach formalises and extends ideas outlined previously by Tokieda (C. R. Acad. Sci., Paris I 333 (2001)) and Souli\`{e}re and Tokieda (J. Fluid Mech. 460 (2002)) and allows us to prove the existence of several 1-parameter families of periodic orbits. These families either emanate from equilibria or converge to collisions possessing a specific symmetry. Our results are applied to show existence of families of small nonlinear oscillations emanating from the platonic solid equilibria.