Relative Periodic Solutions Of The N-Vortex Problem Via The Variational Method

TL;DR

This paper demonstrates the existence of infinitely many relative periodic orbits in the planar N-vortex problem with positive vorticities, using symplectic capacity theory, expanding understanding of vortex dynamics beyond fixed points and equilibria.

Contribution

It introduces a variational method combined with symplectic capacity theory to prove the existence of numerous relative periodic solutions in the N-vortex problem.

Findings

Existence of infinitely many relative periodic orbits.

These orbits are dense in energy levels.

Orbits are neither fixed points nor relative equilibria.

Abstract

This article studies the N-vortex problem in the plane with positive vorticities. After an investigation of some properties for normalised relative equilibria of the system, we use symplectic capacity theory to show that, there exist infinitely many normalised relative periodic orbits on a dense subset of all energy levels, which are neither fixed points nor relative equilibria.