A Hamiltonian approach for point vortices on non-orientable surfaces

TL;DR

This paper develops a Hamiltonian framework for point vortices on non-orientable surfaces by leveraging the orientable double cover and introducing twisted vorticity. It defines a regular Hamiltonian on the double cover that descends to the non-orientable base via twisted Green's functions and Robin functions, enabling explicit vortex dynamics on the Möbius band and Klein bottle. The work uncovers symmetry-driven reductions, identifies broad classes of equilibria (including N-ring configurations), and derives detailed one- and two-vortex dynamics, with the Klein bottle analysis relying on Jacobi theta functions to manage periodicities. The resulting framework provides a rigorous, globally consistent description of vortex motion on non-orientable manifolds and offers substantial groundwork for further investigations into multi-vortex configurations and more complex non-orientable geometries.

Abstract

We investigate the motion of point vortices on the Mobius band and Klein bottle. Since these are non-orientable surfaces, the standard Hamiltonian approach does not apply. We therefore begin by establishing a modified Hamiltonian approach which works for arbitrary non-orientable surfaces, through describing the phase space, the Hamiltonian and the local equations of motion. We use a combination of twisted functions and oriented double covers to adapt some of the known notions of vortex dynamics to non-orientable surfaces. For both of the surfaces of interest, we write Hamiltonian-type equations of vortex motion explicitly and follow that by the description of relative equilibria and an investigation of the motion of one and two vortices.

Figures (18)

• Figure 1: Commutative diagram for operators $\delta,$ d, twisted and regular forms on an $n$-dimensional manifold $M$.
• Figure 2: Two copies of the model of the Möbius band, covered by a cylinder, with a vortex and its image under $\tau$. We refer to the vertical lines as the 'imaginary boundary': they are at $x=0,\, \pi r,\, 2\pi r$. The dashed line is $y=0$.
• Figure 3: A typical equilibrium configuration for $N=3$. Note that in the chosen orientation, $\Gamma_1$ and $\Gamma_3$ will be positive numbers, whereas $\Gamma_2$ will be negative.
• Figure 4: The two types of $N$-ring relative equilibrium on a cylinder and the Möbius band. The vortices on the Möbius band are solid, the ones on the "opposite side" of the cylinder are dashed.
• Figure 5: Instantaneous fluid flow induced by a solitary vortex on the Möbius band.

Theorems & Definitions (58)

• Remark 1.1
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Remark 3.1
• Proposition 3.2