Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

TL;DR

This paper establishes the existence of a continuous family of periodic solutions for singular first-order Hamiltonian systems of N-vortex type, extending known solutions from the plane to more general domains with complex boundary conditions.

Contribution

It introduces a novel degree theory for $S^1$-equivariant gradient maps to handle the singular Hamiltonian and proves global continua of periodic solutions emanating from relative equilibria.

Findings

Existence of a global continuum of periodic solutions

Solutions originate from relative equilibria like Thomson configurations

Applicable to domains with complex topology and boundary singularities

Abstract

The paper deals with singular first order Hamiltonian systems of the form \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in \Omega \subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi} \sum_{j\neq k=1}^N \Gamma_j \Gamma_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in \Omega^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $\Omega\subset\mathbb{R}^2$. The function $F:\Omega^N\to\mathbb{R}$ may have additional singularities near the boundary of $\Omega^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots,z_N(t))\in\Omega^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $\Omega$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators.