# A Hamiltonian approach for point vortices on non-orientable surfaces I: the
Möbius band

Nataliya A. Balabanova, James Montaldi

###### Abstract

This is the first of two companion papers, in which we investigate vortex
motion on non-orientable two dimensional surfaces. We establish the
‘Hamiltonian’ approach to point vortex motion on non-orientable surfaces
through describing the phase space, the Hamiltonian and the local equations of
motion. This paper is primarily focused on the dynamics on the Möbius band. To
this end, we adapt some of the known notions of vortex dynamics to non-
orientable surfaces. We write Hamiltonian-type equations of vortex motion
explicitly and follow that by the description of relative equilibria and a
thorough investigation of motion of one and two vortices, with emphasis on the
periodicity of motion.

###### Contents

  1. 1 Introduction
  2. 2 Hamiltonian approach to point vortex flows on non-orientable manifolds
    1. 2.1 Twisted scalars and vector fields
    2. 2.2 Vorticity and vortex strength
    3. 2.3 Phase space for point vortices
    4. 2.4 The Hamiltonian
    5. 2.5 The momentum map
  3. 3 Vortex motion on a Möbius band
    1. 3.1 Equations of motion
    2. 3.2 Invariants and relative equilibria
    3. 3.3 A general class of equilibria
    4. 3.4 The N-ring relative equilibria
    5. 3.5 Motion of a single vortex
  4. 4 Motion of two vortices
    1. 4.1 Fixed equilibria
    2. 4.2 General motion of two vortices
      1. 4.2.1 Zero momentum
      2. 4.2.2 $\Gamma_{1}=\Gamma_{2}$
  5. 5 Concluding remarks
  6. A Velocity of the N-ring relative equilibria

##  1 Introduction

Neither of the classical approaches to fluid dynamics prohibits considering
fluid motion on non-orientable manifolds: the Euler equations

$\rho\frac{Dv}{Dt}=-\nabla p$

with $\rho$ the density, $p$ the pressure and $v$ the divergence free flow of
the fluid only demand that the manifold in question be Riemannian (see [19]);
secondly, Arnold’s interpretation [6] of the fluid flow as diffeomorphisms
that preserve the volume element is a construction defined in the non-
orientable case as well. On the other hand, the definition of vorticity does
depend on a choice of orientation, so the equation for the vorticity is only
well-defined if the manifold is oriented.

Much has been written about point vortices on a general (oriented) surface
(see [4, 2, 15, 25, 21] and many others), and the setup involves using the
strength $\Gamma\in\mathbb{R}$ of each point vortex, the symplectic form on
the surface and the Hamiltonian which involves the hydrodynamic Green’s
function. The strength of a point vortex is positive if the local direction of
flow agrees with the orientation of the surface, and is negative otherwise.
The symplectic form defines the orientation of the surface.

This is the first of two works (see [7] for the second) describing how to
adapt the Hamiltonian approach to point vortices to non-orientable surfaces.
On such surfaces, the sign of the vortex strength cannot be defined
consistently, and neither is there a symplectic form.

To circumvent this, we define the vortex strength as a _twisted_ or _pseudo-
scalar_ , which means that to each local orientation one defines a scalar, and
opposite orientations give rise to opposite scalar values. The equations of
motion, in turn, will be described locally on oriented charts; however, as we
will see below, their form will allow for globally defined trajectories of
motion.

A system of $N$ point vortices, of strengths $\Gamma_{1},\dots,\Gamma_{N}$, on
an oriented surface $M$ has phase space $M^{N}\setminus\Delta$ where
$M^{N}=M\times M\times\dots\times M$ and $\Delta$ is the big diagonal. The
symplectic form is

$\sigma=\bigoplus_{j=1}^{N}\Gamma_{j}\sigma_{j}$ (1.1)

where $\sigma_{j}$ is the symplectic form on the $j$th copy of $M$, and the
Hamiltonian is equal to the sum

$\mathcal{H}(x_{1},\dots,x_{N})=-\sum_{i\neq
j}\Gamma_{i}\Gamma_{j}\,G(x_{i},x_{j})-\frac{1}{2}\sum_{j}\Gamma_{j}^{2}R(x_{j}),$
(1.2)

where $G(x,y)$ is the hydrodynamic Green’s function and $R(x)$ is the Robin
function (see for example [12, 8] and references therein) which is defined by

$R(x)=\lim_{y\to x}\left(G(x,y)-\frac{1}{2\pi}\log d(x,y)\right).$ (1.3)

where $d(x,y)$ is the geodesic distance between $x$ and $y$.

The simplest example of such a system is one on the plane, consisting of the
standard planar symplectic form and the logarithmic Hamiltonian
$\mathcal{H}=-\frac{1}{2\pi}\sum_{\alpha<\beta}\Gamma_{\alpha}\Gamma_{\beta}\log(|z_{\alpha}-z_{\beta}|)$
(see [3, 23, 16], etc.).

Imposing specific symmetries on a planar system allows for its interpretation
as one on a two dimensional manifold.

For example, dividing the plane into evenly spaced vertical strips of width
$2\pi r$, each with the exact same set of point vortices gives us a system on
a cylinder (see [22, 5] for detailed construction), with the Hamiltonian given
by
$\mathcal{H}=-\frac{1}{2\pi}\sum_{k<l}\Gamma_{k}\Gamma_{l}\log\left|\sin\frac{z_{k}-z_{l}}{2r}\right|$.

Periodic vortex configurations on lattices yield systems on flat tori, the
dynamics of which have been studied, for example, in [24, 29]. In [17, 12] and
[28] systems on curved tori and spheres are investigated; [26] derives an
algorithm determining the motion of a single point vortex on a surface of
constant curvature and of genus greater than one. In a more general setup,
[27] establishes existence and uniqueness of the Green’s function under
hydrodynamic conditions at the boundaries and ends of the surface for an
oriented Riemannian surface of finite topological type. The Green’s function,
together with the Robin function, are then used to describe the motion of
systems of point vortices.

Certain symmetries of systems of point vortices allow us to descend from an
orientable double cover to a non-orientable manifold. The systems with such
symmetries have been considered previously, but the option of writing the
equations intrinsically on non-orientable manifolds has never, to our
knowledge, been addressed.

For instance, [17] studies vortex systems on a sphere with a particular
symmetry: vortices come in pairs of opposite strength that lie on the
antipodal points on the sphere. Though not explicitly mentioned in the
original text, this system is the double cover of one on the projective plane.

In this paper we develop a ‘Hamiltonian’ approach to point vortex motion on
non-orientable surfaces and investigate in detail the case when the non-
orientable manifold is the Möbius band.

### Hamiltonian approach to point vortex flows on non-orientable manifolds

We establish that for flows on non-orientable manifolds vorticity must be
interpreted as a two-form; for point vortex flow it means that its scalar
counterpart, point vortex strength $\Gamma$, must be a so-called twisted
scalar. The same interpretation of vorticity and stream function as twisted
functions was adopted in [31].

Using the approach to construction of the phase space from [20], we
demonstrate that the phase space for the motion of $N$ point vortices on a
non-orientable manifold $M$ is the $N$-fold Cartesian product
$\bigotimes_{i=1}^{N}\widetilde{M}$, where $\widetilde{M}$ is the orientable
double cover of $M$.

We find the explicit form of the Hamiltonian (2.5) and prove that it is a
regular (i.e., not twisted) function on $M$ and $\bigotimes_{i=1}^{N}M$ and is
constructed in a standard way from the Green’s function of the Laplace
operator and its Robin function (Proposition 2.4).

### Explicit formulae for the Möbius band

In this section, we restrict our attention to point vortex motion on the
Möbius band.

The model of the Möbius band that we employ is an infinite strip of width
$\pi$ on the plane with oppositely oriented vertical sides (as in Figure 1),
with a cylinder $\mathbb{C}/2\pi\mathbb{Z}$ as a double cover. We call the
boundary of the strip the imaginary boundary.

Since this model induces local orientation, we assume that point vortices move
inside the strip as usual, but upon reaching the imaginary boundary they
’jump’ to the other side, change the sign of their strength and the sign of
their $y$ coordinate.

Periodizing the formulae on the double cover, we are able to write the
Hamiltonian and the Hamilonian type equations of the system; we introduce a
Möbius flip (formalization of the ’vortex jump’ described above and the
orientation changing isometric involution for the double cover) of the
covering system on a cylinder and prove that the Hamiltonian and the equations
of motion are invariant under the Möbius flip, applied to any number of
vortices. This guarantees that not only the equations of motion are well-
defined, but that it also doesn’t matter where we assume the imaginary
boundary to be located: the motion will be the same regardless of where we
draw it.

### Motion of point vortices

Having derived the equations of motion early in Section 3, we observe that the
symmetry group of the system is $S^{1}$, acting by horizontal translations;
this action has a globally defined momentum map that is a regular (as opposed
to twisted) function. In Section 3.3 we show the existence of a general class
of ‘equatorial’ equilibria, and Section 3.4 is dedicated to particular
instances of relative equilibria that are aligned and staggered rings of
vortices on the Möbius band.

We describe in detail the motion of a single point vortex in Section 3.5. Due
to its simplicity, this case allows for a very clear outlook on how the
trajectories of the global motion are glued together. Indeed, even in this
simple case it becomes clear that the dynamics is not defined on $M$, but on
$\widetilde{M}$.

For comparison with the planar case, we investigate the motion of two point
vortices with opposite strengths, with centres placed with a vertical or
horizontal symmetry. The former configuration will be a relative equilibrium
and an example of a so-called vortex street ([30]); the trajectories of the
system with horizontal symmetry will be closed loops.

The next arrangement we investigate is an asymptotic one: point vortex
strengths are assumed to be arbitrary, but the moduli of $y$-coordinates of
both vortices are taken to be infinitely large. The two will then move in
horizontal lines, in the same or opposite directions, depending on the
strengths of the vortices. However, due to different velocities this motion
will not be a relative equilibrium.

In the final section (Section 4), we describe the motion of a pair of
arbitrary point vortices, starting with a description of the fixed equilibria
of two vortices.

In order to perform the calculations for the system of two point vortices, we
observe that this system on a Möbius band lifts to a system of four point
vortices on a cylinder, and vice versa: the motion of a system of four point
vortices with certain imposed symmetries on a cylinder correspond to that of a
pair of vortices on a Möbius band.

Using the first integral of motion and the dependence of the Hamiltonian
solely on $x_{1}-x_{2}$, we reduce the system and establish the minimum number
of critical points. Numerical calculations point towards existence of two
kinds of arrangements of the level sets of the reduced Hamiltonian; we
describe the motion of the system corresponding to all principal types of the
level sets within each of them.

##  2 Hamiltonian approach to point vortex flows on non-orientable manifolds

###  2.1 Twisted scalars and vector fields

As we have mentioned above, several hydrodynamic notions are directly tied to
local orientation; we need their counterparts that are well-defined in the
non-orientable case in order to describe our system in a Hamiltonian-like
manner.

Let $M$ be a manifold, possibly with boundary. Then at (or in a neighbourhood
of) each point $p\in M$ there are two possible orientations. If we let $(p,o)$
be a point together with an orientation, then the collection of such pairs
forms the orientation bundle over $M$,

$\pi:\widetilde{M}\to M$

which we denote ${\mathop{\mathfrak{missing}}{or}\nolimits}(M)$. The fibre
consists of two points, the two orientations of the manifold at that point.
The total space $\widetilde{M}$ of
${\mathop{\mathfrak{missing}}{or}\nolimits}(M)$ is connected if and only if
$M$ is non-orientable. Furthermore, it has a tautological orientation: the
projection is a local diffeomorphism, so on each sheet one associates the
orientation corresponding to that sheet.

The manifold $\widetilde{M}$ is thereby an oriented double cover of $M$, and
for the three principal 2-dimensional cases this is recognizable globally as
the sphere as the double cover of the projective plane, the cylinder as the
double cover of the Möbius band and the torus as the double cover of the Klein
bottle.

Let $\tau:\widetilde{M}\to\widetilde{M}$ denote the automorphism of
$\mathfrak{or}(M)$ that swaps the two points in each fibre; over each point
$p$ it acts by reversing the orientation at $p$ (for example, for the
projective plane this is the antipodal map on the sphere). Then $M$ is clearly
the quotient of this total space by the action of the group generated by
$\tau$.

Any real-valued function, differential form or vector field on $M$ lifts to
one on $\widetilde{M}$ which is in each case, invariant under $\tau$.
Conversely, any such object (scalar field, vector field etc) on
$\widetilde{M}$ invariant under $\tau$ descends to a similar object on $M$. We
refer to such objects as regular.

On the other hand, if application of $\tau$ to say a scalar function $f$ on
$\widetilde{M}$ changes its sign ($f\circ\tau=-f$), then it is called a
twisted or pseudo-function on $M$; the same applies to forms and vector
fields. In other words, the object in question depends on the choice of
orientation, and changing orientation changes the sign of the object. This is
the case for example of the curl of a vector field in $\mathbb{R}^{3}$. (For
in-depth discussion of twisted objects we refer the reader to [1, 10, 9].)

######  Remark 2.1.

Twisted functions are often defined as sections of a twisted line bundle on
$M$. The relationship with the approach above is as follows. Let
$\widetilde{L}$ be the trivial line bundle on $\widetilde{M}$. Then
$\pi_{*}\widetilde{L}$ is a rank-2 vector bundle on $M$ whose fibre over $p\in
M$ is $\widetilde{L}_{p^{\prime}}\oplus\widetilde{L}_{p^{\prime\prime}}$,
where $\pi^{-1}(p)=\\{p^{\prime},p^{\prime\prime}\\}$. Now $\tau$ induces an
action on $\pi_{*}\widetilde{L}$ by swapping the two summands. We can then
decompose $\pi_{*}\widetilde{L}$ as the sum of two line bundles over $M$

$\pi_{*}\widetilde{L}=L_{+}\oplus L_{-}$

where sections of $L_{+}$ are those sections invariant under $\tau$ and
sections of $L_{-}$ are those whose sign is changed by $\tau$. The sections of
$L_{+}$ correspond to genuine (regular) functions on $M$, while sections of
$L_{-}$ correspond to twisted functions on $M$. A similar construction can be
made for vector fields or forms, starting from $T\widetilde{M}$ or
$T^{*}\widetilde{M}$ in place of $\widetilde{L}$. The sections of
$\left(T\widetilde{M}\right)_{-}$ or $\left(T^{\ast}\widetilde{M}\right)_{-}$
are respectively twisted vector fields and twisted one forms.

###  2.2 Vorticity and vortex strength

Consider a divergence-free vector field $v$ on $M$ (tangent to $\partial M$),
and let $\alpha=v^{\flat}$ denote the corresponding 1-form defined using the
Riemannian metric. Then let ${\boldsymbol{\lambda}}=\mathrm{d}\alpha$: this is
the vorticity 2-form. It is defined independently of any orientation.

On the other hand, the vorticity-as-a-scalar does depend on the orientation
and is defined on $\widetilde{M}$ as follows. Since $\widetilde{M}$ is
oriented and Riemannian, it has a natural symplectic form we denote by
$\sigma$. Given the vorticity 2-form ${\boldsymbol{\lambda}}$ on $M$ we can
lift this to $\widetilde{M}$, and find a smooth function $\omega$ on
$\widetilde{M}$ satisfying $\pi^{*}{\boldsymbol{\lambda}}=\omega\sigma$. This
is the scalar vorticity on $\widetilde{M}$.

Now $\tau^{*}\pi^{*}{\boldsymbol{\lambda}}=\pi^{*}{\boldsymbol{\lambda}}$,
while $\tau^{*}\sigma=-\sigma$, and it follows that $\tau^{*}\omega=-\omega$,
which is to say that the scalar vorticity $\omega$ is a _twisted_ scalar on
$M$, as we know from the discussion in the introduction. Indeed, the operation
passing from ${\boldsymbol{\lambda}}$ to $\omega$ is the twisted Hodge star
operator, which on non-oriented manifolds takes regular $p$-forms to twisted
$(n-p)$-forms.

The vorticity equation on $M$ or $\widetilde{M}$ is then

${\boldsymbol{\lambda}}_{t}+L_{v}{\boldsymbol{\lambda}}=0,\quad\text{or}\quad\omega_{t}+\mathrm{d}_{v}\omega=0.$

The first is an equation of 2-forms on $M$ (involving the Lie derivative
$L_{v}$), while the second is of twisted scalars.

In the classical hydrodynamical approach for a small oriented neighbourhood of
the point vortex the _point vortex strength_ $\Gamma$ is the flux of the
vorticity through an open surface $A$ bounded by a curve $C$ that encircles
the vortex. In our terminology, that will be an integral of the one-form
$\mathbf{v}^{\flat}$:

$\Gamma=\oint_{C}\ \mathbf{v}^{\flat}=\int_{\bar{A}}\ {\boldsymbol{\lambda}}$
(2.1)

where $\bar{A}$ is the region $A$ inside $C$ with the given orientation.
Observe how the sign of $\Gamma$ depends on the (local) orientation.

An alternative interpretation of point vortex strength on an oriented manifold
$\widetilde{M}$ is the coefficient $\Gamma$ at the Dirac delta function of the
vorticity-as-a-scalar of a point vortex.

Analogously, for point vortices on the non-orientable $M$, $\Gamma\delta(x)$
is a twisted function (or twisted distribution). As we will see below, the
’twistedness’ of $\Gamma$ will be crucial for the equations of motion to be
well-defined; therefore, we assume that $\delta(x)$ is a regular ‘function’
(distribution), while $\Gamma$ is a twisted scalar (that is, a real number
associated to the point vortex whose sign depends on the local orientation
chosen).

###  2.3 Phase space for point vortices

We follow the approach of Marsden and Weinstein [20], and note that this also
holds for non-orientable surfaces. Given an initial vorticity 2-form
${\boldsymbol{\lambda}}$ on a 2-dimensional Riemannian manifold $M$, the phase
space111Note that if $M$ is oriented then
${\mathop{\mathrm{missing}}{SDiff}\nolimits}(M)$ is not connected, and it
would be more efficient to restrict to
${\mathop{\mathrm{missing}}{SDiff}\nolimits}(M)_{0}$, which consists of
volume-preserving diffeomorphisms isotopic to the identity; this is possibly
what the authors of [20] had in mind. for the dynamics is

$\mathcal{O}_{\boldsymbol{\lambda}}=\left\\{\phi^{*}{\boldsymbol{\lambda}}\mid\phi\in{\mathop{\mathrm{missing}}{SDiff}\nolimits}(M)\right\\},$

where ${\mathop{\mathrm{missing}}{SDiff}\nolimits}(M)$ is the group of volume-
preserving diffeomorphisms of $M$; this can be interpreted as the coadjoint
orbit of ${\mathop{\mathrm{missing}}{SDiff}\nolimits}(M)$ containing
${\boldsymbol{\lambda}}$. The symplectic form is given by the Kostant-
Kirillov-Souriau form,

$\Omega_{\boldsymbol{\lambda}}(L_{u}{\boldsymbol{\lambda}},L_{v}{\boldsymbol{\lambda}})=\int_{M}{\boldsymbol{\lambda}}(u,v)\,\mathrm{d}A$

where $u,v\in\mathcal{X}_{0}(M)$ (divergence-free vector fields), $L_{u}$ and
$L_{v}$ are the Lie derivatives and $\mathrm{d}A$ is the area measure
consistent with the metric. This does not require $M$ itself to be oriented or
symplectic. For a regular vorticity form, the Hamiltonian is given by the
total kinetic energy of the fluid.

The motion of a system of $N$ point vortices on an orientable surface $M$ is
determined by the location and strength of the point vortices; the strength
here being the (scalar) vorticity concentrated at a point, in the form of
delta functions. In this case, it was pointed out in [20] that the phase space
$\mathcal{O}_{\boldsymbol{\lambda}}$ can be identified with $M\times
M\times\dots\times M\setminus\Delta$ (the product of $N$ copies of $M$ with
the collision set removed). The identification is

$(x_{1},\dots,x_{N})\longmapsto\sum_{j}\Gamma_{j}\delta_{x_{j}}\sigma$ (2.2)

where $\sigma$ is the symplectic form on $M$ and $\delta_{p}$ is a delta
function supported at $p$. The symplectic form on
$\mathcal{O}_{\boldsymbol{\lambda}}$ becomes

$\Omega_{\boldsymbol{\lambda}}(\mathbf{u},\mathbf{v})=\sum_{j}\Gamma_{j}\,\sigma_{x_{j}}(u_{j},v_{j})$

where $u_{j},v_{j}\in T_{x_{j}}M$.

The Hamiltonian is given by

$\mathcal{H}(x_{1},\dots,x_{N})=-\sum_{i<j}\Gamma_{i}\Gamma_{j}G(x_{i},x_{j})-\tfrac{1}{2}\sum_{j}\Gamma_{j}^{2}R(x_{j}).$
(2.3)

Now consider the case where $M$ is non-orientable. Two or three problems arise
when adapting the discussion above and we need to employ the twisted
construction described above.

Most obviously, since on $M$ the $\Gamma_{j}$ are twisted scalars, it is not
clear what the first term in the Hamiltonian (2.3) means. Moreover, the phase
space $\mathcal{O}_{\boldsymbol{\lambda}}$ can no longer be identified with a
product of $N$ copies of $M$. This is because, a given $N$-tuple of points
$(x_{1},\dots,x_{N})$ does not determine the interaction between the vortices.
Indeed, when $M$ is non-orientable, there is a volume preserving
diffeomorphism which takes say $x_{1}$ back to $x_{1}$ by a ‘non-orienting’
loop, which has the effect of reversing the contribution to the vorticity
2-form at the point $x_{1}$.

Thus, in $\mathcal{O}_{\boldsymbol{\lambda}}$ there are vorticities supported
at the same points, but with any subset of the local vorticity 2-forms
reversed. That is, if ${\boldsymbol{\lambda}}=\sum_{j}\lambda_{x_{j}}$ where
$\lambda_{x_{j}}$ is a delta-function 2-form (a 2-current), then
$\mathcal{O}_{\boldsymbol{\lambda}}$ also contains
${\boldsymbol{\lambda}}^{\prime}=\sum_{j}(\pm\lambda_{x_{j}})$ for all choices
of signs. This leads to a parametrization of
$\mathcal{O}_{\boldsymbol{\lambda}}$ by the product of $N$ double covers of
$M$.

######  Proposition 2.2.

The phase space $\mathcal{O}_{\boldsymbol{\lambda}}$ for a system of $N$ point
vortices on a non-orientable surface $M$ can be parametrized by
$\widetilde{M}\times\dots\times\widetilde{M}\setminus\widetilde{\Delta}$,
where $\widetilde{\Delta}$ is the ‘very big diagonal’ consisting of points
$(x_{1}^{\prime},\dots,x_{N}^{\prime})\in\widetilde{M}^{N}$ for which $i\neq
j\Rightarrow\pi(x_{i}^{\prime})\neq\pi(x_{j}^{\prime})$.

The parametrization is constructed in the following way. For the vorticity
2-form ${\boldsymbol{\lambda}}$ on $M$, let $\\{x_{1},\dots,x_{N}\\}$ be the
support of ${\boldsymbol{\lambda}}$. For each $x$ in this set let
$\pi^{-1}(x)=\\{x^{\prime},\,\tau(x^{\prime})\\}$ for some
$x^{\prime}\in\widetilde{M}$. The pull-back of ${\boldsymbol{\lambda}}$ to
$\widetilde{M}$ is a 2-form, and the tautological symplectic from $\sigma$
allows us to define the vortex strengths $\Gamma$ at $x^{\prime}$ and
$-\Gamma$ at $\tau(x^{\prime})$ (we do not assume $\Gamma>0$, the vorticity
will be positive at the point where the orientation agrees with the direction
of circulation of the vortex, and negative (of the same magnitude) at the
other point). The system can be parametrized by
$(x_{1}^{\prime},\dots,x_{N}^{\prime})$ and the associated vorticities
$(\Gamma_{1},\dots,\Gamma_{N})$.

###  2.4 The Hamiltonian

In the same notation as above, given a point $x$ on $M$, for the preferred
lift we are making choices between $x^{\prime}$ and $\tau(x^{\prime})$ . The
choice could be made by requiring all $\Gamma_{j}>0$ for example. However,
that turns out not to be convenient for calculations, so we allow an arbitrary
choice for each $x_{j}$.

Thus, given any point
$(x_{1}^{\prime},\dots,x_{N}^{\prime})\in\widetilde{M}^{N}\setminus\widetilde{\Delta}$
the system on $\widetilde{M}$ consists of $2N$ vortices at points
$\\{x_{1}^{\prime},\dots,x_{N}^{\prime},\tau(x_{1}^{\prime}),\dots,\tau(x_{N}^{\prime})\\}$
. The scalar vorticity at $x_{j}^{\prime}$ is $\Gamma_{j}$ and at
$\tau(x_{j}^{\prime})$ is $-\Gamma_{j}$.

Using (2.3), the Hamiltonian is given by

$\begin{split}\mathcal{H}_{\widetilde{M}}(x_{1},\ldots,x_{N})=&-\sum_{i<j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(x_{i}^{\prime},x_{j}^{\prime})-\sum_{i<j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(\tau(x_{i}^{\prime}),\tau(x_{j}^{\prime}))+\sum_{i\neq
j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(x_{i}^{\prime},\tau(x_{j}^{\prime}))\\\
&{}+\sum_{j}\Gamma_{j}^{2}\widetilde{G}(x_{j}^{\prime},\tau(x_{j}^{\prime}))-\frac{1}{2}\sum_{j}\Gamma_{j}^{2}\widetilde{R}(x_{j}^{\prime})-\frac{1}{2}\sum_{j}\Gamma_{j}^{2}\widetilde{R}(\tau(x_{j}^{\prime}))\end{split}$
(2.4)

where $\widetilde{G}(x,y)$ is the hydrodynamic Green’s function and
$\widetilde{R}$ the Robin function on $\widetilde{M}$.

However, an adjustment needs to be made to (2.4): for justification, we return
to the body of fluid on $\widetilde{M}$. Due to the absence of external
forces, the total energy of the fluid is given by the surface integral
$\int_{\widetilde{M}}\frac{1}{2}\rho|v|^{2}\mathrm{d}s$, where $\rho$ is the
constant density of the fluid and $v$ its velocity. Transferring to the flow
on $M$ means that we only consider half of the fluid; therefore, the total
energy must be divided by 2. With this, we conclude that the total energy of
the fluid on $M$ will be given by

$\begin{split}\mathcal{H}(x_{1},\ldots,x_{N})=&-\frac{1}{2}\sum_{i<j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(x_{i}^{\prime},x_{j}^{\prime})-\frac{1}{2}\sum_{i<j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(\tau(x_{i}^{\prime}),\tau(x_{j}^{\prime}))+\frac{1}{2}\sum_{i\neq
j}\Gamma_{i}\Gamma_{j}\,\widetilde{G}(x_{i}^{\prime},\tau(x_{j}^{\prime}))\\\
&{}+\frac{1}{2}\sum_{j}\Gamma_{j}^{2}\widetilde{G}(x_{j}^{\prime},\tau(x_{j}^{\prime}))-\frac{1}{4}\sum_{j}\Gamma_{j}^{2}\widetilde{R}(x_{j}^{\prime})-\frac{1}{4}\sum_{j}\Gamma_{j}^{2}\widetilde{R}(\tau(x_{j}^{\prime}))\end{split}$
(2.5)

Since $\tau$ is a globally defined isometry of $\widetilde{M}$ it follows that
both $\widetilde{G}$ and $\widetilde{R}$ are invariant under $\tau$, which
implies some of these terms coincide.

Suppose that we fix $y^{\prime}$, a preferred lift of $y$ and $x^{\prime}$, a
preferred lift of $x$. Observe how (2.5) is unaffected by which of the two
preimages we choose to be our preferred lift, so we are free to pick either
one. This allows us to express functions on $M$ through ones on
$\widetilde{M}$:

######  Definition 2.3.

We define the twisted Green’s function on $M$ as

$G_{M}(x,y)=\widetilde{G}(x^{\prime},y^{\prime})-\widetilde{G}(x^{\prime},\tau(y^{\prime}))$
(2.6)

and the (regular) Robin function on $M$ as

$R_{M}(x)=\widetilde{R}(x^{\prime})-\widetilde{G}(x^{\prime},\tau(x^{\prime}))$
(2.7)

######  Proposition 2.4.

$G_{M}$ is indeed a twisted Green’s function of the Laplacian on $M$, and
$R_{M}(x)$ is the Robin function of $G_{M}(x,y)$.

###### Proof.

Consider the Laplacian of $G_{M}$ as a function on $\widetilde{M}$:

$\Delta_{x^{\prime}}\left(\widetilde{G}(x^{\prime},y^{\prime})-\widetilde{G}(x^{\prime},\tau(y^{\prime}))\right)=\delta_{y^{\prime}}(x^{\prime})-\delta_{\tau(y^{\prime})}(x^{\prime})$

For a fixed $y$ (and, consequently, $y^{\prime}$)
$\Delta_{x^{\prime}}\left(\widetilde{G}(x^{\prime},y^{\prime})-\widetilde{G}(x^{\prime},\tau(y^{\prime}))\right)$
descends to $M$ as a twisted function $\widetilde{\delta}_{y}(x)$. Analogously
to the regular Dirac delta function, it indicates when one of the preimages of
$x$ on $\widetilde{M}$ coincides with one of the preimages of $y$ on
$\widetilde{M}$.

As one can easily see, $G_{M}$ depends on the choice of the preferred lift,
since it is twisted in both of its variables.

Now we want to check that $R_{M}$ coincides with the desingularization of
$G_{M}$.

Saying that $x\to y$ for two points $x,y\in M$ is equivalent to one of the two
statements for $\widetilde{M}$: $x^{\prime}\to y^{\prime}$ or
$x^{\prime}\to\tau(y^{\prime})$. Observe, however, that $R_{M}$, despite being
a limit, is a function of one variable only and is therefore covered by a
function on $\widetilde{M}$.

Thus, we are restricted to one lift only: either $x^{\prime}\to y^{\prime}$ or
$x^{\prime}\to\tau(y^{\prime})$, depending on $\tau$ and our choice of
preferred lifts. Since $x^{\prime}$ is a ’mute’ variable, it does not matter
which of the two options we choose, as it can be seen from the calculations
below.

We have from (1.3) and (2.7):

$\begin{split}\widetilde{R}(y^{\prime})-\widetilde{G}(y^{\prime},\tau(y^{\prime}))=&\lim_{x^{\prime}\to
y^{\prime}}\left(\widetilde{G}(y^{\prime},x^{\prime})-\frac{1}{2\pi}\log
d(y^{\prime},x^{\prime})-\frac{1}{2}\widetilde{G}(y^{\prime},\tau(x^{\prime}))-\frac{1}{2}\widetilde{G}(x^{\prime},\tau(y^{\prime}))\right)\\\
=&\lim_{x^{\prime}\to
y^{\prime}}\biggl{(}\frac{1}{2}\widetilde{G}(y^{\prime},x^{\prime})+\frac{1}{2}\widetilde{G}(\tau(y^{\prime}),\tau(x^{\prime}))-\frac{1}{2\pi}\log
d(y^{\prime},x^{\prime})-\\\
&\qquad\quad-\frac{1}{2}\widetilde{G}(y^{\prime},\tau(x^{\prime}))-\frac{1}{2}\widetilde{G}(x^{\prime},\tau(y^{\prime}))\biggr{)}\\\
=&\lim_{x^{\prime}\to
y^{\prime}}\left(G_{M}(y^{\prime},x^{\prime})-\frac{1}{2\pi}\log
d(y^{\prime},x^{\prime})\right).\end{split}$ (2.8)

Since we choose one lift out of two, explicit form of
$\widetilde{R}(\tau(y^{\prime}))-\widetilde{G}(y^{\prime},\tau(y^{\prime}))$
is the limit with $\tau(x^{\prime})\to\tau(y^{\prime})$, which can be shown to
coincide with (2.8). Hence, $R_{M}(x)$ is a regular function on $M$,
independent of our choice of the preferred lift. ∎

This enables us to rewrite (2.4) as

$\mathcal{H}(x_{1},\ldots,x_{N})=-\sum_{i<j}\Gamma_{i}\Gamma_{j}\,G_{M}(x_{i},x_{j})-\tfrac{1}{2}\sum_{j}\Gamma_{j}^{2}R_{M}(x_{j})$
(2.9)

Since exchanging $x^{\prime}$ with $\tau(x^{\prime})$ also involves changing
the sign of the corresponding $\Gamma$, it is clear from (2.4) that this
Hamiltonian function $\mathcal{H}$ is well-defined on $M\times\dots\times
M\setminus\Delta$.

Notice that even if the Robin function on $\widetilde{M}$ is constant, the one
on $M$ may not be, because of the term
$\widetilde{G}(x^{\prime},\tau(x^{\prime}))$. For example, on the projective
plane with the usual metric the Robin function is constant, while on the
Möbius band and Klein bottle it is not.

To finish formalizing the Hamiltonian approach, we establish the symplectic
form on the phase space:

######  Lemma 2.5.

The symplectic form on
$\widetilde{M}\times\ldots\times\widetilde{M}\setminus\widetilde{\Delta}$ from
Proposition 2.2 is given by $\bigoplus\limits_{k=1}^{N}\Gamma_{k}\sigma_{k}$,
where $\sigma_{j}$ is the symplectic form on $j$th copy of $\widetilde{M}$.

###### Proof.

The vorticity of the fluid on the $j$th copy of $\widetilde{M}$ is given by
$\Gamma_{j}\delta_{x^{\prime}_{j}}\sigma-\Gamma_{j}\delta_{\tau(x^{\prime}_{j})}\sigma$.
Due to the symmetries of our system, if the velocity of the point
$x^{\prime}_{j}$ is given by a vector $v$, the velocity of
$\tau(x^{\prime}_{j})$ will be $\mathrm{d}\tau(v)$. Hence, the calculation of
the symplectic form as in [20] gives us

$\Omega_{\lambda}(\mathbf{u},\mathbf{v})=\sum_{j}\Gamma_{j}\sigma_{x^{\prime}_{j}}(u_{j},v_{j})-\Gamma_{j}\sigma_{\tau(x^{\prime}_{j})}\left(\mathrm{d}\tau(u_{j}),\mathrm{d}\tau(v_{j})\right)=2\sum_{j}\Gamma_{j}\sigma_{x^{\prime}_{j}}(u_{j},v_{j}),$

since $\tau^{\ast}\sigma=-\sigma$. We have divided the Hamiltonian by 2 and
therefore have to do the same to the symplectic form, obtaining the statement
of the lemma. Note that this symplectic form is independent of our choice of
preferred lift. ∎

######  Remark 2.6.

By the symmetric placement of vortices on $\widetilde{M},$ the stream function
$\psi_{\widetilde{M}}$ on $\widetilde{M}$ is

$\psi_{\widetilde{M}}(x)=\sum_{k}\Gamma_{k}\left(\widetilde{G}(x,x_{k})-\widetilde{G}(x,\tau(x_{k}))\right),$

where $x_{k}$ and $\tau(x_{k})$ are the centres of the point vortices and
their copies respectively.

Observe that $\psi_{\widetilde{M}}$ is a twisted function in $x$:
$\psi_{\widetilde{M}}\left(\tau(x)\right)=-\psi_{\widetilde{M}}(x)$. However,
$\psi_{\widetilde{M}}$ is not sensitive to exchanging $x_{k}$ and
$\tau(x_{k})$, since that changes not only the sign of
$G(x,x_{k})-G(x,\tau(x_{k}))$, but the sign of $\Gamma_{k}$ as well.

This makes sense from the hydrodynamic point of view: the descent of
$\psi_{\widetilde{M}}$ to $M$, the stream function $\psi_{M}$, must be a
twisted function of $x\in M$ ([31]), unaffected by our choice of preferred
lift.

######  Remark 2.7.

For a single vortex on $M$ (corresponding to two on $\widetilde{M}$) the
Laplacian of the stream function will be

$\Delta\psi_{\widetilde{M}}=\Gamma(\delta_{y^{\prime}}(x^{\prime})-\delta_{\tau(y^{\prime})}(x^{\prime})),$
(2.10)

which we write as a function on $\widetilde{M}$ and which, as we already know,
descends to a twisted function on $M$. However, bear in mind (2.10) was
written on $\widetilde{M}$ interpreting $\Gamma$ as a scalar; an assumption
that cannot be made for $M$. Therefore, when writing vorticity-as-a-scalar
intrinsically on $M$, we ’delegate’ the twistedness to $\Gamma$, as we did at
the end of Section 2.2.

This is done in the following way: for a fixed $y\in M$, we lift the regular
delta function $\delta_{y}x$ to $\widetilde{M}$, to obtain
$\delta_{y}^{\prime}x^{\prime}+\delta_{\tau(y^{\prime})}x^{\prime}$. However,
by placing point vortices of strengths $\Gamma$ and $-\Gamma$ at $y^{\prime}$
and $\tau(y^{\prime})$ respectively, we ’twist’ it, making it into (2.10).

###  2.5 The momentum map

Suppose a (connected) Lie group $G$ acts by isometries on $M$ and denote its
Lie algebra by $\mathfrak{g}$. We describe the (possibly local) momentum map
on the phase space associated to this action.

The action lifts to an orientation-preserving action of $G$ by isometries on
$\widetilde{M}$ ([11]); therefore, the action in question is symplectic.

Consider an element $\xi\in\mathfrak{g}$ and let $\xi_{\widetilde{M}}$ be the
associated vector field of the infinitesimal action on $\widetilde{M}$. Then,
since the action of $G$ on $\widetilde{M}$ is the lift of one on $M$,
$\mathrm{d}\tau(\xi(x))=\xi(\tau(x))$, i.e. $\xi_{\widetilde{M}}$ is a regular
vector field.

Let $\psi:\widetilde{M}\to\mathfrak{g}^{*}$ be the associated momentum map
(which exists globally provided the cohomological obstruction vanishes—if it
does not vanish then the map exists locally). The map $\psi$ is defined up to
a constant by the equation

$\left<\mathrm{d}\psi_{x}(v),\xi\right>=\sigma_{x}\left(\xi_{\widetilde{M}}(x),v\right),$
(2.11)

where $v$ is any vector field on $\widetilde{M}$.

Now take $v$ such that it descends to $M$ as a regular vector field and
consider the application of $\tau$ to both sides of (2.11):

$\left<\mathrm{d}\psi_{\tau(x)}(\mathrm{d}\tau(v)),\xi\right>=\left((\tau^{-1})^{\ast}\sigma\right)_{\tau(x)}\left(\mathrm{d}\tau\left(\xi_{\widetilde{M}}(x)\right),\mathrm{d}\tau(v)\right)=-\sigma_{x}\left(\xi_{\widetilde{M}}(x),v\right).$

Therefore, $\mathrm{d}\psi(x)=-\mathrm{d}\psi(\tau(x))$, and
$\psi:\widetilde{M}\to\mathfrak{g}^{*}$ is a twisted function.

On the phase space, the momentum map
$\Phi:\widetilde{M}^{N}\to\mathfrak{g}^{*}$ is the usual one given by

$\Phi(z_{1},\dots,z_{N})=\sum_{j}\Gamma_{j}\,\psi(z_{j}).$

Applying $\tau$ to any of the $z_{j}$ reverses both $\psi$ and $\Gamma_{j}$
and thereby leaves this expression unchanged. This momentum map therefore
descends to a map (rather than a twisted map) $M\times\dots\times
M\to\mathfrak{g}^{*}$.

##  3 Vortex motion on a Möbius band

In this section we apply all the apparatus developed above to the setting of
vortex motion on the Möbius band. We adopt the boundary-free model of the
Möbius band as a strip of fixed width $\pi r$ and infinite height in the plane
with opposite sides identified with opposite orientation: this infinite model
of the Möbius band has no boundary, so we do not need to consider vortex-
boundary interactions. If we denote the coordinate on the strip as $z=x+iy$,
its real part has to be considered modulo $\pi r$.

The double cover of this model is a cylinder $\mathbb{C}/2\pi r\mathbb{Z}$ of
circumference $2\pi r$ and coordinates $(x,y)$. We orient this by $dx\wedge
dy$. The imaginary line that we draw on the cylinder to signify the division
we will further refer to as the imaginary boundary. Figure 1 depicts our
model; the blue arrows in it and all following figures indicate the oppositely
oriented sides of the imaginary boundary, and the dashed horizontal line is
the line $y=0$. As a quotient of the cylinder, the Möbius band is obtained by
identifying $(x,y)$ with $\tau(x,y)$, where $\tau$ is the orientation
reversing isometry given by $\tau(x,y)=(x+\pi r,\,-y)$.

Figure 1: Two copies of the model of the Möbius band, covered by a cylinder.
We refer to the vertical lines as the ‘imaginary boundary’. The dashed line is
where $y=0$.

The cylinder has symmetry group $O(2)\times E(1)$, where $E(1)$ is the
Euclidean group for the line. The subgroup consisting of those elements
commuting with $\tau$ descends to symmetries of the Möbius band. This subgroup
is $O(2)\times\mathbb{Z}_{2}$, where $\mathbb{Z}_{2}$ acts by
$(x,y)\mapsto(x,-y)$.

###  3.1 Equations of motion

For brevity, we will refer to the pair $(z,\Gamma)$, i.e., the complex planar
coordinate of the vortex centre and the vortex strength, as vortex parameters,
or just parameters.

The dynamics of a vortex with parameters $(z,\Gamma)$ will be covered by the
dynamics of two vortices on the cylinder: one with parameters $(z,\Gamma)$ and
the other $(\bar{z}+\pi r,-\Gamma$).

The Hamiltonian for the cylinder reads (see [22]):

$\mathcal{H}=-\frac{1}{2\pi}\sum_{k<l}\Gamma_{k}\Gamma_{l}\log\left|\sin\frac{z_{k}-z_{l}}{2r}\right|,$

or, as expressed through $x$-$y$ coordinates,

$\mathcal{H}=-\frac{1}{4\pi}\sum_{k<l}\Gamma_{k}\Gamma_{l}\log\left[\sin^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}-y_{l}}{2r}\right)\right].$

We periodize the Hamiltonian as in Section 2.4 to obtain the explicit formula
for the Möbius band Hamiltonian:

$\begin{split}\mathcal{H}=&-\frac{1}{4\pi}\sum\limits_{k<l}\Gamma_{k}\Gamma_{l}\log\left(\sin^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}-y_{l}}{2r}\right)\right)+\\\
&+\frac{1}{4\pi}\sum\limits_{k<l}\Gamma_{k}\Gamma_{l}\log\left(\cos^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}+y_{l}}{2r}\right)\right)+\\\
&+\frac{1}{8\pi}\sum\limits_{k}\Gamma_{k}^{2}\log\left(1+\sinh^{2}\left(\frac{y_{k}}{r}\right)\right);\end{split}$
(3.1)

The last term is the Robin function $R_{M}$. We can easily see how it is
unaffected by the change of the local orientation or the exchange of the
covering vortices.

Writing the system of equations on one chart, we obtain:

$\begin{cases}\dot{x}_{k}=&-\frac{1}{8\pi r}\sum\limits_{k\neq
l}\Gamma_{l}\frac{\sinh\left(\frac{y_{k}-y_{l}}{r}\right)}{\sin^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}-y_{l}}{2r}\right)}+\frac{1}{8\pi
r}\sum\limits_{k\neq
l}\Gamma_{l}\frac{\sinh\left(\frac{y_{k}+y_{l}}{r}\right)}{\cos^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}+y_{l}}{2r}\right)}\\\
&+\frac{1}{4\pi r}\Gamma_{k}\tanh\left(\frac{y_{k}}{r}\right),\\\\[4.0pt]
\dot{y}_{k}=&\frac{1}{8\pi r}\sum\limits_{k\neq
l}\Gamma_{l}\frac{\sin\left(\frac{x_{k}-x_{l}}{r}\right)}{\sin^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}-y_{l}}{2r}\right)}+\frac{1}{8\pi
r}\sum\limits_{k\neq
l}\Gamma_{l}\frac{\sin\left(\frac{x_{k}-x_{l}}{r}\right)}{\cos^{2}\left(\frac{x_{k}-x_{l}}{2r}\right)+\sinh^{2}\left(\frac{y_{k}+y_{l}}{2r}\right)}.\par\end{cases}$
(3.2)

######  Remark 3.1.

One can see that changing the value of $r$ can be compensated by rescaling
$x,y$ and time, so from now on we assume that $r=1$.

In the framework of this model, a point vortex moves as it would on an
oriented manifold, as long as it does not cross the imaginary boundary. If
that happens, the vortex has to ’jump’ to the other side of the strip, change
the sign of its $y$ coordinate and the sign of its strength.

This setup might seem unnatural, but the intuitive approach to vortex motion
had to be sacrificed in order to have local orientation and be able to treat
$\Gamma$s as scalars. However, this perception does not lead to a
contradiction: crossing the imaginary boundary is the same as changing
orientation of a small area surrounding the vortex, and therefore $\Gamma$ has
to change its sign.

###  3.2 Invariants and relative equilibria

######  Definition 3.2.

The transformation that takes all the vortices $(\Gamma_{i},z_{i})$ on the
cylinder covering the given one on the Möbius band into the system
$(-\Gamma_{i},\bar{z}_{i}-\pi)$ we call the Möbius flip.

This is the formalization of the ’vortex jump’; however, in this case the
vortex need not be on the imaginary boundary.

The Möbius flip on the cylinder is the involution $\tau$ from Sec. 2.1;
therefore, applying it to all the point vortices must leave the Hamiltonian
and the equations unchanged (this is easy to check with the explicit form of
the Hamiltonian (3.1)). Additionally, it is easy to check that the local
Hamiltonian equations are not affected by the change of orientation.

Observe that invariance under the Möbius flip implies that the location of the
imaginary boundary does not affect our motion; this illustrates that the
equations are well-defined.

As described above, the system on the Möbius band has a one-parameter group of
Hamiltonian symmetries inherited from the cyliner: $SO(2)$ acting by
horizontal translations, giving rise to a conserved quantity
$\Phi:=\sum_{i}\Gamma_{i}y_{i}$. As in the discussion in Section 2.5, $\Phi$
is a regular function: indeed, changing the orientation changes the signs of
$y_{j}$ together with $\Gamma_{j}$, and the function stays unchanged.

It is immediate from the form of the symmetry group that relative equilibria
are horizontally moving rigid configurations of point vortices.

###  3.3 A general class of equilibria

It was pointed out in [22] that on the cylinder, given a sequence of $2N$
point vortices placed around a horizontal circle, with vorticities of
alternating signs, then for the given cyclic ordering there is an equilibrium
point (it is not known whether or not this is unique, although numerical
experiment suggests it is for $N=3$).

For such a configuration to arise from one on the Möbius band, the set of
points must be $\tau$-invariant, and moreover $\Gamma_{x}=-\Gamma_{\tau x}$.
For this to be compatible with having alternating signs of vorticity, $N$ must
be odd.

######  Proposition 3.3.

In a preferred chart, consider the set of all configurations $\\{(x_{j},0)\mid
j=1,\dots,N\\}$ with $0\leq x_{1}<x_{2}<\dots<x_{N}\leq\pi$, and with
vorticities of alternating signs: $\Gamma_{j}\Gamma_{j+1}<0$ for
$j=1,\dots,N-1$ and where $N$ is odd. Then there is an equilibrium
configuration in this set (see Figure 2).

###### Proof.

Consider the domain of these configurations. The Hamiltonian is a function of
the $x_{j}$ which tends to $+\infty$ as the configuration tends to the
boundary (i.e. two vortices tend to a collision). It follows that in the
interior of the domain, the Hamiltonian must attain a maximum. This critical
point will be an equilibrium of the system. ∎

A particular instance of this is where, in our preferred chart, the points
have alternating vorticity $\Gamma,\,-\Gamma,\dots$ and placed at the vertices
of a regular $N$-gon (for odd $N$).

$\Gamma_{1}$$\Gamma_{2}$$\Gamma_{3}$ Figure 2: A typical equilibrium
configuration for $N=3$

###  3.4 The N-ring relative equilibria

These configurations are more easily described on the cylinder. Consider a
regular ring of $N\geq 1$ identical vortices with vorticity $\Gamma\neq 0$ on
a horizontal circle with common vertical coordinate $y>0$. Together with this
consider the ‘antipodal ring’, the image of the first ring under the
involution $\tau$, and with vorticities $-\Gamma$. It has common vertical
coordinate $-y$.

Note that if $N$ is even, then the two rings are vertically aligned, while if
$N$ is odd they are vertically staggered. In terms of $O(3)$ and the
Schönflies notation, for even $N$ the configuration has symmetry $D_{Nh}$
while for odd $N$ it has symmetry $D_{Nd}$ (see for example [18, Table 2] for
this notation).

Since these configurations are invariant under $\tau$ they project to single
rings with $N$ vortices on the Möbius band. We call these $N$-rings; they are
illustrated in Figure 3. If $N=1$ this is the single point vortex described in
Section 3.5 below.

$\Gamma$$\Gamma$$\Gamma$$-\Gamma$$-\Gamma$$-\Gamma$ (a) A vertically aligned
$N$-ring RE for even values of $N$ (here $N=6$).

$\Gamma$$\Gamma$$\Gamma$$-\Gamma$$-\Gamma$ (b) A vertically staggered $N$-ring
RE for odd values of $N$ (here $N=5$).

Figure 3: The two types of $N$-ring relative equilibrium on a Möbius band

######  Theorem 3.4.

On the Möbius band, an $N$-ring is a relative equilibrium.

Suppose that the $N$-ring in question is an aligned one: that means $N$ is an
even number. Then if the strength of the vortices in the upper row is $\Gamma$
and their vertical coordinate $y$, the angular velocity is given by

$\xi_{a}=\begin{cases}\displaystyle\frac{\Gamma
N}{4\pi}\coth\left(\frac{Ny}{2}\right)&\text{if $N/2$ is even}\\\\[12.0pt]
\displaystyle\frac{\Gamma N}{4\pi}\coth(Ny)&\text{if $N/2$ is
odd}.\end{cases}$ (3.3)

On the other hand, the angular velocity of staggered equilibria ($N$ odd) is
given by by

$\xi_{s}=\frac{\Gamma}{8\pi}\left(\tanh(y)-\coth(y)\right)+\frac{\Gamma
N}{4\pi}\coth\left(2Ny\right).$ (3.4)

A simple symmetry argument using the symmetries $D_{Nh}$ and $D_{Nd}$
mentioned above, demonstrates that such configurations are, indeed, relative
equilibria. The momentum value at the ring is $N\Gamma y$, so those at
different heights arise for different values of the momentum. For detailed
calculations of the velocities, see Appendix A.

On the universal cover (the plane) these $N$-rings lift to von Kármán vortex
streets, see for example [13].

###  3.5 Motion of a single vortex

A well-known fact is that a single vortex on a cylinder, sphere or plane
remains stationary. As we see below, this is not the case in general for the
Möbius band, and the equation of motion of one vortex is derived from (3.2)
and is due solely to the Robin function in (3.1):

$\begin{cases}\dot{x}=\frac{1}{4\pi}\Gamma\tanh{y},\\\ \dot{y}=0;\end{cases}$

showing that a vortex is stationary if and only if it lies on the line $y=0$.

Figure 4: Instantaneous fluid flow induced by a solitary vortex on the Möbius
band.

From the system of equations (3.2) we obtain the streamlines of one vortex on
the band: see Figure 4. As time progresses, the vector field translates
horizontally with the vortex.

For this case, let us demonstrate the construction of a global vector field
$(\dot{x},\dot{y})$ and investigate how the trajectories of motion come
together. Due to non-orientability, the least number of covering charts for
the Möbius strip is 2, which we will denote $U_{1}$ and $U_{2}$.

Figure 5: Two charts cover of a Möbius band. Horizontal arrows denote the
direction of the horizontal vector field that is the solution of the
Hamiltonian equations on each chart.

Let $U_{12}$ be the left intersection of the charts, and $U_{21}$ the right
one as illustrated in Figure 5. The Jacobian of the coordinate change will be
positive on $U_{12}$ and negative on $U_{21}$.

We suppose that the vortex is initially placed on $U_{1}$; moving
horizontally, it reaches $U_{12}$: the sign of its strength in local
coordinates will not change transferring to $U_{2}$. Once it reaches $U_{21}$,
however, its strength will change its sign, and so will its $y$-coordinate.
The resulting vector field is depicted in Figure 5.

Observe that despite each of the vector fields on the charts in Figure 5 being
the solution of the Hamiltonian system of equations (as written on each of
them) the resulting solution is not a vector field on the Möbius band.
However, it is (and that is how we will interpret it) a distribution of the
instantaneous velocities for given initial conditions of the system.

Despite this, the trajectories of motion are globally defined: due to changes
in signs they can be glued together as the vortex moves from one chart to
another.

##  4 Motion of two vortices

The equations of motion for the system with two point vortices are:

$\begin{split}\begin{cases}\dot{x}_{1}=-\frac{1}{8\pi}\Big{(}\Gamma_{2}\frac{\sinh(y_{1}-y_{2})}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}-y_{2}}{2}\right)}-\Gamma_{2}\frac{\sinh(y_{1}+y_{2})}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}+y_{2}}{2}\right)}-2\Gamma_{1}\tanh(y_{1})\Big{)}\\\
\dot{x}_{2}=-\frac{1}{8\pi}\Big{(}\Gamma_{1}\frac{\sinh(y_{2}-y_{1})}{\sin^{2}\left(\frac{x_{2}-x_{1}}{2}\right)+\sinh^{2}\left(\frac{y_{2}-y_{1}}{2}\right)}-\Gamma_{1}\frac{\sinh(y_{2}+y_{1})}{\cos^{2}\left(\frac{x_{2}-x_{1}}{2}\right)+\sinh^{2}\left(\frac{y_{2}+y_{1}}{2}\right)}-2\Gamma_{2}\tanh(y_{2})\Big{)}\\\
\dot{y}_{1}=\frac{\Gamma_{2}\sin(x_{1}-x_{2})}{8\pi}\left(\frac{1}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}-y_{2}}{2}\right)}+\frac{1}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}+y_{2}}{2}\right)}\right)\\\
\dot{y}_{2}=\frac{\Gamma_{1}\sin(x_{2}-x_{1})}{8\pi}\left(\frac{1}{\sin^{2}\left(\frac{x_{2}-x_{1}}{2}\right)+\sinh^{2}\left(\frac{y_{2}-y_{1}}{2}\right)}+\frac{1}{\cos^{2}\left(\frac{x_{2}-x_{1}}{2}\right)+\sinh^{2}\left(\frac{y_{2}+y_{1}}{2}\right)}\right)\end{cases}\end{split}$
(4.1)

To get an intuition about the motion, below we consider some of the simplest
examples.

AB Figure 6: Two vortices with opposite vortex strength and centres arranged
with vertical symmetry (a 2-ring in the terminology of Section 3.4) — see
Example 4.1.

######  Example 4.1.

Consider first, a 2-ring (as in Section 3.4), depicted in Figure 6.

If the initial coordinates of the centres are $(x_{0},\pm y_{0})$, Equation
(4.1), as well as Theorem 3.4, yield $\dot{y}_{i}=0,\
\dot{x}_{i}=\frac{1}{2\pi}\coth\left(2y_{0}\right)$.

In the preferred chart, both vortices will move horizontally towards the
imaginary boundary, reach it, ’jump’ to the other side, exchanging places, and
then continue to move in the same fashion. Since the velocity of both vortex
centres is the same, this configuration is an example of a relative
equilibrium.

######  Example 4.2.

In this case, let the two vortices again have opposite signs:
$\Gamma_{1}=-\Gamma_{2}$, but now be positioned symmetrically with respect to
the line $x=\frac{\pi}{2}$. This configuration is fixed by the corresponding
reflection in $O(2)$.

Denote the initial positions of the centres by $(x_{0},y_{0})$ and $(\pi-
x_{0},y_{0})$ respectively, and suppose that $y_{0}<0$ as illustrated in the
leftmost part of Figure 7. Substituting these values into the equations (4.1)
easily demonstrates that the symmetric arrangement of point vortex centres is
preserved throughout the motion. The restriction of the Hamiltonian to this
fixed point space is

$\mathcal{H}=\frac{1}{4\pi}\log\left(\frac{\cos^{2}x\,\cosh^{2}y}{\sin^{2}x+\sinh^{2}y}\right).$

AB

AB

BA

BA

Figure 7: Schematic motion of two point vortices with horizontally symmetric
placement of the centres; the semi-circles are approximate trajectories of
their motion — see Example 4.2.

Additionally, through drawing the level sets of the Hamiltonian, we can
conclude that the motion goes as follows: two vortices start moving
symmetrically in opposite directions towards the symmetry line; upon crossing
the line $y=0$ they start moving away from each other until they reach the
imaginary boundary after the same finite amount of time, ‘jump over’ to the
other side, go along the trajectory that the other used to occupy, reach the
boundary and exchange places once more. This resembles a relative equilibrium
on the plane, with two vortices going in a circle around their centre of
vorticity, but here the trajectory is neither a circle, nor is such a
configuration a relative equilibrium. Figure 7 illustrates four consecutive
stages of motion.

######  Example 4.3.

Here, we look at the asymptotic case: $y_{1}=\infty$, with
$\Gamma_{1}>\Gamma_{2}>0$. Recalling the existence of the invariant
$\Gamma_{1}y_{1}+\Gamma_{2}y_{2}=C$ and assuming that $C$ is finite, we derive
that $y_{2}=-\infty$ and that
$y_{1}+y_{2}=y_{1}\left(1-\frac{\Gamma_{1}}{\Gamma_{2}}\right)+\frac{C}{\Gamma_{2}}\to-\infty$.
By taking limits of our system of equations, we get

$\begin{cases}\dot{x}_{1}=-\frac{1}{4\pi}(2\Gamma_{2}-\Gamma_{1}),\\\
\dot{x}_{2}=-\frac{1}{4\pi}\Gamma_{2},\\\
\dot{y}_{1}=\dot{y}_{2}=0;\end{cases}$ (4.2)

Due to the difference in velocities the motion will not be a relative
equilibrium. Also, note that whether the two vortices are moving in the
opposite or same directions depends on the relations between $\Gamma_{1}$ and
$\Gamma_{2}$.

###  4.1 Fixed equilibria

We continue by determining whether the general 2-vortex system has any fixed
equilibrium points.

Fixed equilibria, as critical points of the Hamiltonian, correspond to zeros
of (4.1). Observe that fixed equilibria have to have $x_{1}-x_{2}=0$. This and
a few manipulations turn (4.1) into

$\begin{cases}2\Gamma_{2}\cosh y_{1}\cosh
y_{2}=\Gamma_{1}\sinh^{2}y_{1}-\Gamma_{1}\sinh y_{1}\sinh y_{2},\\\
2\Gamma_{1}\cosh y_{1}\cosh y_{2}=\Gamma_{2}\sinh^{2}y_{2}-\Gamma_{2}\sinh
y_{1}\sinh y_{2}.\end{cases}$ (4.3)

Equating the right hand sides and solving as a quadratic equation for
$\frac{\sinh y_{2}}{\sinh y_{1}}$ gives

$\frac{\sinh y_{2}}{\sinh y_{1}}=1\ \mathrm{or}\ \frac{\sinh y_{2}}{\sinh
y_{1}}=-\frac{\Gamma_{1}^{2}}{\Gamma_{2}^{2}}.$

The first case is impossible, since our vortices would then occupy the same
point. The second one enables us to rewrite the first equation in (4.1) as

$2\sqrt{\Gamma_{2}^{4}+\Gamma_{1}^{4}\sinh^{2}y_{1}}=\frac{\Gamma_{1}}{\Gamma_{2}}\frac{\left(\Gamma_{1}^{2}+\Gamma_{2}^{2}\right)\sinh^{2}y_{1}}{\sqrt{1+\sinh^{2}y_{1}}},$
(4.4)

which implies that fixed equilibria can only occur when $\Gamma_{1}$ and
$\Gamma_{2}$ are of the same sign. Solving (4.1) yields

$\begin{split}&y_{1}=\pm\operatorname{arcsinh}\left(\sqrt{\frac{2\Gamma_{2}^{2}\left(\Gamma_{1}^{4}+\Gamma_{2}^{4}+\sqrt{\Gamma_{1}^{8}+\Gamma_{1}^{6}\Gamma_{2}^{2}+\Gamma_{1}^{2}\Gamma_{2}^{6}+\Gamma_{2}^{8}}\right)}{\Gamma_{1}^{2}\left(\Gamma_{1}^{2}-\Gamma_{2}^{2}\right)^{2}}}\right)\\\
&y_{2}=\mp\operatorname{arcsinh}\left(\sqrt{\frac{2\Gamma_{1}^{2}\left(\Gamma_{1}^{4}+\Gamma_{2}^{4}+\sqrt{\Gamma_{1}^{8}+\Gamma_{1}^{6}\Gamma_{2}^{2}+\Gamma_{1}^{2}\Gamma_{2}^{6}+\Gamma_{2}^{8}}\right)}{\Gamma_{2}^{2}\left(\Gamma_{1}^{2}-\Gamma_{2}^{2}\right)^{2}}}\right)\end{split}$
(4.5)

The expression under the outer square roots is well-defined when
$\Gamma_{1}\neq\Gamma_{2}$.Thus, we have shown

######  Proposition 4.4.

When $\Gamma_{1}\Gamma_{2}>0$ and $\Gamma_{1}\neq\Gamma_{2}$, the system (4.1)
is in a state of a fixed equilibrium when $x_{1}=x_{2}$, and $y_{1},\,y_{2}$
are as in (4.5).

Notice from (4.5) that as $\Gamma_{1}-\Gamma_{2}\to 0$ so the equilibrium
points tend to infinity.

For fixed distinct values of $\Gamma_{1}$ and $\Gamma_{2}$ of the same sign,
we will call the two (opposite in sign) values of
$\Phi(y_{1},y_{2})=\Gamma_{1}y_{1}+\Gamma_{2}y_{2}$ at fixed equilibria the
equilibrium values of the momentum map.

###  4.2 General motion of two vortices

In this section, we describe the motion of two point vortices in the more
general case. To analyze our problem, we lift the two vortices of strengths
$\Gamma_{1}$ and $\Gamma_{2}$ on the Möbius band to four on the cylinder, with
strengths $\pm\Gamma_{1},\pm\Gamma_{2}$, positioned with due symmetry.

######  Observation 4.5.

Some key points ease our computations significantly:

  * •

For a fixed orientation on the cylinder, the lift of the system on the Möbius
band is unique;

  * •

The symmetric arrangement of the vortices on the cylinder is preserved by the
motion, therefore on the cylinder the trajectories of one pair of inequivalent
vortices completely determines the motion of the whole system and the motion
on the Möbius band as well. This way, we can observe only that inequivalent
pair;

  * •

Since it does not matter where we draw the imaginary boundary, we can safely
assume the signs of $\Gamma_{i}$: we assume $\Gamma_{1}>\Gamma_{2}>0$.

  * •

The value of the globally defined momentum map on the cylinder is twice the
value of the momentum map on the Möbius band; therefore, the $y$-coordinates
of the two vortices that we choose to observe are related by
$\Gamma_{1}y_{1}+\Gamma_{2}y_{2}=C$

The next step is to reduce the system. As stated above, we assume that
$\Gamma_{1}>0,\Gamma_{2}>0$ and we have an invariant
$\Gamma_{1}y_{1}+\Gamma_{2}y_{2}=C$. Expressing $y_{2}$ through $y_{1}$, we
substitute it into the system:

$\footnotesize\begin{cases}\dot{x}_{1}=-\frac{1}{8\pi}\Bigg{(}\Gamma_{2}\frac{\sinh\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{\Gamma_{2}}\right)}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{2\Gamma_{2}}\right)}-\Gamma_{2}\frac{\sinh\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{\Gamma_{2}}\right)}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{2\Gamma_{2}}\right)}-2\Gamma_{1}\tanh(y_{1})\Bigg{)}\\\
\dot{y}_{1}=\frac{\Gamma_{2}\sin(x_{1}-x_{2})}{8\pi}\left(\frac{1}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{2\Gamma_{2}}\right)}+\frac{1}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{2\Gamma_{2}}\right)}\right)\\\
\dot{x}_{2}=-\frac{1}{8\pi}\Bigg{(}-\Gamma_{1}\frac{\sinh\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{\Gamma_{2}}\right)}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{2\Gamma_{2}}\right)}-\Gamma_{1}\frac{\sinh\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{\Gamma_{2}}\right)}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{2\Gamma_{2}}\right)}-2\Gamma_{2}\tanh\left(\frac{C-\Gamma_{1}y_{1}}{\Gamma_{2}}\right)\Bigg{)}\\\
\dot{y}_{2}=-\frac{\Gamma_{1}}{\Gamma_{2}}\dot{y}_{1}\end{cases}$ (4.6)

We can reduce the system by one more degree of freedom: the right hand sides
of all equations depend only on $x_{1}-x_{2}$ and $y_{1}$. We subtract the
third equation from the first to get a system in two variables:

$\small\begin{cases}\dot{x}_{1}-\dot{x}_{2}&=-\frac{1}{8\pi}\left(\left(\Gamma_{2}+\Gamma_{1}\right)\frac{\sinh\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{\Gamma_{2}}\right)}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{2\Gamma_{2}}\right)}\right.-\\\
&-\left.\left(\Gamma_{2}-\Gamma_{1}\right)\frac{\sinh\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{\Gamma_{2}}\right)}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{2\Gamma_{2}}\right)}-2\Gamma_{1}\tanh(y_{1})+2\Gamma_{2}\tanh\left(\frac{C-\Gamma_{1}y_{1}}{\Gamma_{2}}\right)\right)\\\
\dot{y}_{1}&=\frac{\Gamma_{2}\sin(x_{1}-x_{2})}{8\pi}\Bigg{(}\frac{1}{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}+\Gamma_{1})y_{1}-C}{2\Gamma_{2}}\right)}+\frac{1}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{(\Gamma_{2}-\Gamma_{1})y_{1}+C}{2\Gamma_{2}}\right)}\Bigg{)}\end{cases}$
(4.7)

The same reduction technique can be applied to the Hamiltonian, giving

$\begin{split}\mathcal{H}=&-\frac{\Gamma_{1}\Gamma_{2}}{4\pi}\log\left(\frac{\sin^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}}{2}\left(1+\frac{\Gamma_{1}}{\Gamma_{2}}\right)-\frac{C}{2\Gamma_{2}}\right)}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\sinh^{2}\left(\frac{y_{1}}{2}\left(1-\frac{\Gamma_{1}}{\Gamma_{2}}\right)+\frac{C}{2\Gamma_{2}}\right)}\right)+\\\
&\quad+\frac{\Gamma_{1}^{2}}{4\pi}\log\left(\cosh(y_{1})\right)+\frac{\Gamma_{2}^{2}}{4\pi}\log\left(\cosh\left(\frac{C-\Gamma_{1}y_{1}}{\Gamma_{2}}\right)\right)\end{split}$
(4.8)

Henceforth, our main goal is to reconstruct the motion of the initial system
from the information we are able to obtain about the reduced one.

We start with searching for singular and critical points of the Hamiltonian.
Without loss of generality, we may suppose that $-\pi\leq x_{1}-x_{2}\leq\pi$.
However, in the pictures we draw four periods, to highlight the symmetries
that the trajectories possess.

Noticing that the partial derivative
$\frac{\partial\mathcal{H}}{\partial(x_{1}-x_{2})}$ is proportional to
$\sin(x_{1}-x_{2})$ multiplied by some strictly negative function, we deduce
that all the critical points lie on the lines $x_{1}-x_{2}=0,\pm\pi$.
Additionally, the Hamiltonian is $2\pi$-periodic in $x_{1}-x_{2}$ and
symmetric with respect to the vertical axes listed above.

Assessing asymptotic behaviour of the function with a fixed $x_{1}-x_{2}$ and
$y_{1}\to\pm\infty$ gives
$\mathcal{H}\sim(\Gamma_{1}^{2}-\Gamma_{1}\Gamma_{2})|y|$. Therefore,
$\mathcal{H}\to+\infty$ if $y_{1}\to\pm\infty$.

(a) Level set curves (b) Vector plot

Figure 8: Level sets of the reduced Hamiltonian with the least number of
critical points

  * •

$x_{1}-x_{2}=0$.

$\begin{split}\mathcal{H}=&-\frac{\Gamma_{1}\Gamma_{2}}{2\pi}\log\left(\frac{\sinh^{2}\left(\frac{y_{1}}{2}\left(1+\frac{\Gamma_{1}}{\Gamma_{2}}\right)-\frac{C}{2\Gamma_{2}}\right)}{\cosh^{2}\left(\frac{y_{1}}{2}\left(1-\frac{\Gamma_{1}}{\Gamma_{2}}\right)+\frac{C}{2\Gamma_{2}}\right)}\right)+\\\
&+\frac{\Gamma_{1}^{2}}{\pi}\log(\cosh(y_{1}))+\frac{\Gamma_{2}^{2}}{\pi}\log\left(\cosh\left(\frac{C-\Gamma_{1}y_{1}}{\Gamma_{2}}\right)\right)\end{split}$

Obviously, $y_{1}=\frac{C}{\Gamma_{1}+\Gamma_{2}}$ is a singular point, as the
point of collision of the vortices $\Gamma_{1}$ and $\Gamma_{2}$; here,
$\mathcal{H}\to+\infty$.

As we have stated above, when $y_{1}\to\pm\infty$, $\mathcal{H}\to+\infty$,
which means that on the line $x_{1}-x_{2}=0$ there must exist at least two
more critical points, both saddles: maximum in $x_{1}-x_{2}$, minimum in $y$.

Both of these points are relative equilibria when $C$ is not an equilibrium
value; for each of two equilibrium values of $C$ one of them turns into a
fixed equilibrium.

Through estimates on the first and second derivatives of
$\dot{x}_{1}-\dot{x}_{2}$ as a function of $y_{1}$ it can be shown that no
more critical points exist on this line.

  * •

$x_{1}-x_{2}=\pm\pi$

Here, the simplified Hamiltonian becomes

$\begin{split}\mathcal{H}=&-\frac{\Gamma_{1}\Gamma_{2}}{\pi}\log\left(\frac{\cosh^{2}\left(\frac{y_{1}}{2}\left(1+\frac{\Gamma_{1}}{\Gamma_{2}}\right)-\frac{C}{2\Gamma_{2}}\right)}{\sinh^{2}\left(\frac{y_{1}}{2}\left(1-\frac{\Gamma_{1}}{\Gamma_{2}}\right)+\frac{C}{2\Gamma_{2}}\right)}\right)+\\\
&\quad+\frac{\Gamma_{1}^{2}}{\pi}\log\left(\cosh(y_{1})\right)+\frac{\Gamma_{2}^{2}}{\pi}\log\left(\cosh\left(\frac{C-\Gamma_{1}y_{1}}{\Gamma_{2}}\right)\right).\end{split}$

$y_{1}=\frac{C}{\Gamma_{1}-\Gamma_{2}}$ will always be a singular point, at
which the Hamiltonian will be $-\infty$. Here, the collision happens between
the vortices $\Gamma_{1}$ and $-\Gamma_{2}$.

Configurations as depicted in Figure 8 occur for certain values of the
$\Gamma_{i}$ and $C$; however, another possibility exists, shown in Figure 9.
In that case, the reduced Hamiltonian has an additional saddle point and a
minimum below it. Observe that both are instances of relative equilibria that
have the $\Gamma_{1}$ and $-\Gamma_{2}$ vortices on a vertical line above each
other; however, as in Proposition 4.4 they cannot be fixed equilibria, as
$\Gamma_{1}$ and $\Gamma_{2}$ have different signs on the chart on the Möbius
band.

It seems (though the rigorous proof presents too big a computational
challenge) that these are the two only possibilities. In principle, there
could be any number of the saddle-minimum pairs on the lines
$x_{1}-x_{2}=\pm\pi$, however this appears not to be the case.

For now we assume that $C\neq 0$ (we address the case of $C=0$ separately
below) and that the level sets of the Hamiltonian are similar to the ones in
Figure 8, i.e. the Hamiltonian has no additional critical points on the lines
$x_{1}-x_{2}=0,\pm\pi$.

(a) Level set curves (b) Vector field plot

Figure 9: Level sets of the reduced Hamiltonian with two additional critical
points

The trajectories as given by the level sets of the reduced Hamiltonian lack
the information about the rotation around a cylinder. In what follows, we
attempt to restore the motion of the system from the information provided to
us in Figure 8.

Three types of curves are present in the picture: the closed trajectories
going around the points with $(x_{1}-x_{2},y_{1})$ coordinates
$(2k\pi,\frac{C}{\Gamma_{1}+\Gamma_{2}})$ (those we will refer to as Type I),
closed trajectories around the points of type
$((2k+1)\pi,\frac{C}{\Gamma_{1}-\Gamma_{2}})$ (Type II) and the rest (Type
III). The motion of the system will then be described by the

######  Theorem 4.6.

For two point vortices with $\Gamma_{1},\Gamma_{2},C$ such that the
Hamiltonian has the level sets as in Figure 8, we have the following:

  * •

If the initial coordinates are in Regions I or II, the two vortices in
consideration will rotate around each other, while simultaneously moving
forward on the Möbius band as a pair. Due to continuous dependence of the
integral, this motion will be periodic on a set of trajectories of planar
measure 0.

  * •

On trajectories of Type III, the vertical and the horizontal distances between
the two vortices will change with a certain period, however, they will not
rotate around each other.The two vortices may move in the same or opposite
directions (both cases occur).

###### Proof.

In Regions I and II, the two vortices are rotating around each other; that is
not the case for the Region III. In order to reconstruct the motion fully,
however, we need some additional information, which we will be obtaining from
certain limiting cases.

From the system (4.6) we observe that on every curve
$\mathcal{C}=\\{x_{1}-x_{2}(t),y(t)\\}$ $\dot{x}_{1}$ (as well as
$\dot{x}_{1}-\dot{x}_{2}$ and $\dot{y}_{1}$) is a function of $x_{1}-x_{2}$
and $y_{1}$. In order to restore the motion, we need to compute
$\int\limits_{0}^{T}\dot{x}_{1}\mathrm{d}t$, where $T$ is the period of motion
on the curve $C$. We rewrite it tautologically as
$\int\limits_{\mathcal{C}}\frac{\dot{x}_{1}(x_{1}-x_{2},y_{1})}{\sqrt{(\dot{x}_{1}-\dot{x}_{2})^{2}+(\dot{y}_{1})^{2}}}\mathrm{d}s$,
with $\mathrm{d}s$ the element of length on the curve $\mathcal{C}$. This
quantity tells us how much the first vortex moves on the cylinder with every
period of motion.

The motion of the system will be periodic in two cases: when the integral is
0, or, on a cylinder with circumference $2\pi$, has the form $\frac{p}{q}\pi,\
p,\ q\ \in\mathbb{Z}$.

Firstly, we try to establish whether the integral in question is always equal
to zero. For Region I, we suppose that our trajectory is very close to the
point $\left(0,\frac{C}{\Gamma_{1}+\Gamma_{2}}\right)$ (for Region II, it is
the point $\left(\pm\pi,\frac{C}{\Gamma_{1}-\Gamma_{2}}\right)$). On these
trajectories, the pair of vortices that are very close together will mimic (to
the rest of the system) the behaviour of one vortex with vorticity
$\Gamma_{1}+\Gamma_{2}$ ($\Gamma_{1}-\Gamma_{2})$, see baby vortices section
in [22].

In the case of four vortices this means that each pair of two closely placed
vortices behaves like one bigger vortex. Therefore, the motion is very closely
approximated by that of two vortices of opposite strengths on each side of the
cylinder. This construction is stationary if and only if their vertical
coordinates are zero, which is not the case, since $C\neq 0$. Therefore, the
value of the integral is not identically equal to zero and the translational
component is present in the motion of the system.

In order to proceed, we require a few lemmas:

######  Lemma 4.7.

$\int\limits_{C}\frac{\dot{x}_{1}}{\sqrt{(\dot{x}_{1}-\dot{x}_{2})^{2}+(\dot{y}_{1})^{2}}}\mathrm{d}s$
for all curve types I, II and III continuously depends on the curve, i.e. on
its starting point.

###### Proof.

This is quite straightforward, as we can take any bounded subregion $D$ that
contains no critical points of the function but fully contains curves that
intersect it and consider two close curves $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ within it. Then

$\begin{split}&\left|\int\limits_{\mathcal{C}}\frac{\dot{x}_{1}}{\sqrt{(\dot{x}_{1}-\dot{x}_{2})^{2}+(\dot{y})^{2}}}\mathrm{d}s-\int\limits_{\mathcal{C}^{\prime}}\frac{\dot{x}_{1}}{\sqrt{(\dot{x}_{1}-\dot{x}_{2})^{2}+(\dot{y})^{2}}}\mathrm{d}s\right|<\\\
&<\max\left|\frac{\dot{x}_{1}}{\sqrt{(\dot{x}_{1}-\dot{x}_{2})^{2}+\dot{y}_{1}^{2}}}\right|\left(\int\limits_{\mathcal{C}}\mathrm{d}s-\int\limits_{\mathcal{C}^{\prime}}\mathrm{d}s\right)\to
0,\end{split}$

since all the functions $\frac{\partial\mathcal{H}}{\partial(x_{1}-x_{2})}$,
$\frac{\partial\mathcal{H}}{\partial y_{1}}$, $\mathcal{H}$, as well as
tangent vectors to the curves are bounded due to the choice of $D$ ∎

######  Lemma 4.8.

In the general case, the integral depends non-trivially on the path we choose.

###### Proof.

First, we establish this for Region I (for II, the statement can be proven
similarly).

Suppose the configuration is very close to the singular point inside Region I,
i.e. the two vortices are very close to each other. Therefore, when
calculating the speed of rotation around each other, we may disregard the
influence of the opposite pair of vortices, and the time required for the
vortices to go one full circle will be

$T\sim\epsilon^{2}(\sim\epsilon,\ \mathrm{in\ Region\ II}),$

where $\epsilon$ is the distance between vortices. The horizontal velocity of
the vortex pair will be approximately

$\dot{x}_{1}\sim\frac{\Gamma_{2}+\Gamma_{1}}{2\pi}\tanh(y_{1}).$

The rotational speed is dependent on $\epsilon$, unlike $\dot{x}_{1}$.
Therefore, depending on the initial point of the trajectory, a very different
number of full turns will ”fit” into a pair going full circle around the
cylinder. Thus, the integral depends non-trivially on the trajectory.

For Region III two different cases exist: when $C$ is not an equilibrium value
and when it is.

Suppose $C$ is not an equilibrium value; therefore, the two saddle critical
points on the line $x_{1}-x_{2}=0$ are relative equilibria. Since the motion
depends continuously on the initial parameters, $\dot{x}_{1}\neq 0$ on the
trajectories near separatrices (and, consequently, relative equilibrium
points). However, as we approach relative equilibria, $T\to\infty$ and,
therefore, $\int\limits_{0}^{T}\dot{x_{1}}\mathrm{d}t\to\infty$.

When $C$ is an equilibrium value, non-triviality stems from different values
of the integrals at two asymptotic cases. Suppose $C>0$ and the $y$-coordinate
of the relative equilibrium is greater than 0 as well (the opposite case can
be tackled in similar way). As we approach the separatrix containing the
equilibrium point, $\int\limits_{0}^{T}\dot{x}_{1(2)}\mathrm{d}t\to 0$ , since
the motion smoothly depends on the initial conditions. On the other hand, take
$y_{1}\to+\infty,\ y_{2}\to-\infty$ (as in Example 3). Since the velocities
are given by (4.2) and integral of $\dot{x}_{1}-\dot{x}_{2}$ over the period
$T$ must be $2\pi$,

$T=\frac{8\pi^{2}}{\Gamma_{1}-\Gamma_{2}}.$

Thus, when $y_{1}\to+\infty,\ y_{2}\to-\infty$,
$\int\limits_{0}^{T}\dot{x}_{1}\mathrm{d}t\to-\frac{2\pi\left(2\Gamma_{2}-\Gamma_{1}\right)}{\Gamma_{1}-\Gamma_{2}}$
and
$\int\limits_{0}^{T}\dot{x}_{2}\mathrm{d}t\to-\frac{2\pi\Gamma_{2}}{\Gamma_{1}-\Gamma_{2}}$,
at least one of which is not equal to 0. ∎

Thus, for Regions I and II the integral $\int\limits_{0}^{T}\dot{x}_{1}\
\mathrm{d}t$ will almost never be a rational multiplie of $\pi$, and the
motion will consist of rotation around each other and translating around the
cylinder.

In Region III, when $2\Gamma_{2}\geq\Gamma_{1}$, the pair of infinitely
removed vortices will rotate in the same direction (meaning that
$\int\limits_{0}^{T}\dot{x}_{1}\ \mathrm{d}t$ and
$\int\limits_{0}^{T}\dot{x}_{2}\ \mathrm{d}t$ over the period of motion will
have the same signs). When $2\Gamma_{2}\geq\Gamma_{1}$, the directions of
rotation are opposite (with the oppossite signs of the two integrals). From
the non-trivial and smooth dependance on the trajectory we conclude that it is
almost never a rational number multiplied by $\pi$ either.

Therefore, two types of motion can exist: when the two integrals have the same
sign, the particles are moving in the same direction, and a difference between
the horizontal components in their velocities allows the first integral to
differ by $\pm 2\pi$. The picture is very much the same when they are moving
in the opposite directions.

However, there seems to be nothing that would prevent one of the integrals
from turning zero and then switching the sign: in this case the motion will
turn periodic, with one vortex rotating and the other going around the
cylinder. This might allow for the transfers between the two types of motion
in the region. ∎

If additional critical points (Figure 9) appear on lines $x_{1}-x_{2}=\pm\pi$,
the motion around them will be of the same type as for Regions I and II.
Additional non-closed trajectories will result in motion of the same type as
in trajectories in Region III; the required proof of non-trivial and smooth
dependence on the trajectory can be repeated verbatim.

Two cases remain unaddressed: those of $C=0$ and $\Gamma_{1}=\Gamma_{2}$. We
start with the first one.

####  4.2.1 Zero momentum

Putting $C=0$ makes (4.8) a symmetric function of $y_{1}$, as can be easily
observed from Figure 10. Thus, for a curve $\\{(x_{1}-x_{2})(t),y_{1}(t)\\}$
with period $T$ in Regions I and II we have
$(x_{1}-x_{2})(t+\frac{T}{2})=-(x_{1}-x_{2})(t)\dot{y}_{1}(t+\frac{T}{2})=-y_{1}(t)$,
giving

$\begin{split}\int\limits_{0}^{T}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t&=\int\limits_{0}^{\frac{T}{2}}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t+\int\limits_{\frac{T}{2}}^{T}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t=\\\
&=\int\limits_{0}^{\frac{T}{2}}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t+\int\limits^{\frac{T}{2}}_{0}\dot{x}_{1}(-(x_{1}-x_{2})(t),-y_{1}(t))\mathrm{d}t=\\\
&=\int\limits_{0}^{\frac{T}{2}}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t-\int\limits^{\frac{T}{2}}_{0}\dot{x}_{1}((x_{1}-x_{2})(t),y_{1}(t))\mathrm{d}t=0,\end{split}$
(4.9)

owing to the explicit form of $\dot{x}_{1}$. Hence, the vortex pair does not
rotate around the cylinder; the motion consists solely of the two vortices
rotating around each other. However, for Region III the motion does not differ
from the general case.

Figure 10: Level sets of the reduced Hamiltonian with zero value of momentum
map

####  4.2.2 $\Gamma_{1}=\Gamma_{2}$

When $\Gamma_{1}=\Gamma_{2}$, the major difference occurring is that
$\mathcal{H}$ has a finite limit when $y\to\pm\infty$. As can be seen in
Figure 11, the picture is symmetric, but now the symmetry is with respect to
the line $y=\frac{C}{2}$, robbing us of the zero integral as in (4.9).
Additionally, the critical point at $\frac{C}{\Gamma_{1}-\Gamma_{2}}$
disappears, leaving us with trajectories of Types I and III only.

For Region I, we may employ precisely the same reasoning as we have before,
for the non-zero $C$ case. Region III, however, requires more caution.

If we set $\Gamma_{1}=\Gamma_{2}$ in Example 3, the velocities of the two
point vortices will coincide. This contradicts the fact that
$\int\limits_{0}^{T}\dot{x}_{1}-\dot{x}_{2}\ \mathrm{d}t=\pm 2\pi$.

However, this paradox can be explained: as we have mentioned above, with
$\Gamma_{1}>\Gamma_{2}$ and $y_{1}\to\pm\infty,\ \mathcal{H}\to\infty$ for all
values of $x_{1}-x_{2}$. This is not the case when $\Gamma_{1}=\Gamma_{2}$:
here the limit of $\mathcal{H}$ with $y_{1}\to\pm\infty$ will be equal to
$\frac{1}{\cos^{2}\left(\frac{x_{1}-x_{2}}{2}\right)+\cosh(C)}$. Therefore,
the ’level set’ of the Hamiltonian at $\pm\infty$ is not well-defined.

But we have another limiting case to draw the information from: when
$x_{1}-x_{2}=\pm\pi$, we have precisely one relative equilibrium point:
$y_{1}=-y_{2}=\frac{C}{2}$, where two vortices with opposite vorticity are
above each other. As we have demonstrated in Example 2, they will move
parallel to each other in the same direction. Therefore, on the trajectories
very close to the separatrix we will have the two point vortices moving in the
same direction.

######  Remark 4.9.

When $\Gamma_{1}=\Gamma_{2}$ and $C=0$ (see Example 4.2), we do not have
Region III; as can be checked, the Hamiltonian turns infinite in the lines
$\frac{x_{1}-x_{2}}{2}=\pm\pi$.This happens due to the fact that the
separatrix from Region II stretches and goes to infinity as
$\Gamma_{2}\to\Gamma_{1}$ or vice versa. Thus, in those cases, all the motion
is periodic and without additional rotation, as was demonstrated explicitly
above.

Figure 11: Level sets of the reduced Hamiltonian for the case
$\Gamma_{1}=\Gamma_{2}$

##  5 Concluding remarks

A number of questions remain unanswered, the prime example being to prove that
the maximal number of critical points of the Hamiltonian on the Möbius band is
as depicted in Figure 9. Since this function has three independent parameters
and complicated structure, all the standard approaches to a proof present
considerable numerical difficulties.

The other problem is a description of fixed and relative equilibria for three
point vortices. One can demonstrate that fixed equilibria on a straight
vertical line must conform to the condition
$\Gamma_{1}^{2}y_{1}+\Gamma_{2}^{2}y_{2}+\Gamma_{3}^{2}y_{3}=0$, but the
remaining conditions are hard to pin down.

##  Appendix A Velocity of the N-ring relative equilibria

Here we give a proof of the formulae for the angular velocities of the
$N$-ring relative equilibria stated in Theorem 3.4.

We begin with the case of two aligned rings. As we have mentioned, $N=2K$:
therefore, $K$ is the number of point vortices in the upper row on the chart
on the Möbius band.

Without loss of generality we may assume that the leftmost vortex in the top
row is positioned at $(0,y)$: the $i$th vortex in the top row is at
$(\frac{\pi(i-1)}{K},y)$, with similar coordinates for the lower row. Since
the configuration is a relative equilibrium, we only need to determine the
velocity of one of the vortices: we do it for the leftmost one in the upper
row. After substituting the values above into (3.2), we get

$\dot{x}_{1}=\frac{\Gamma}{4\pi}\left(\tanh(y)+\coth(y)\right)+\frac{\Gamma}{8\pi}\sum_{j=1}^{K-1}\frac{\sinh(2y)}{\sin^{2}\left(\frac{\pi
j}{K}\right)+\sinh^{2}(y)}+\frac{\sinh(2y)}{\cos^{2}\left(\frac{\pi
j}{K}\right)+\sinh^{2}(y)}$ (A.1)

######  Lemma A.1.

The two sums above are given by

$\sum_{j=1}^{K}\frac{\sinh(2y)}{\sin^{2}\left(\frac{\pi
j}{K}\right)+\sinh^{2}(y)}=2K\coth(Ky)$

and for the second,

$\sum_{j=1}^{K}\frac{\sinh(2y)}{\cos^{2}\left(\frac{\pi
j}{K}\right)+\sinh^{2}(y)}=\begin{cases}2K\coth(Ky)&\text{if $K$ is
even}\\\\[12.0pt] \displaystyle 2K\tanh(Ky)&\text{if $K$ is odd}.\end{cases}$

###### Proof.

We adapt the method posted on the stackexchange forum [14]: observe that the
function

$\frac{2K}{z(z^{2K}-1)}$

has residues 1 at $e^{\pi ij/K}$ (and $-2K$ at 0). For our first sum, consider
the function

$f(z):=\frac{2K}{z(z^{2K}-1)}\,\frac{\sinh(2y)}{\sinh^{2}(y)+\left(\frac{z-\frac{1}{z}}{2i}\right)^{2}}=\frac{-8K}{(z^{2K}-1)}\,\frac{\sinh(2y)z}{z^{4}-2z^{2}(1+2\sinh^{2}(y))+1}$
(A.2)

This rational function has poles at the $2K$ roots of unity, and at the four
distinct points $\pm e^{y},\pm e^{-y}$, which are the roots of

$z^{4}-2z^{2}(1+2\sinh^{2}(y))+1=(z^{2}-e^{2y})(z^{2}-e^{-2y})$

(all the poles of $f$ are simple).

To evaluate the sum, we note that the residue of $f$ at $z=e^{i\pi j/K}$ is
now

$\frac{\sinh(2y)}{\sin^{2}(\pi j/K)+\sinh^{2}(y)}.$

Since the sum of all the residues of $f$ vanishes, to find the required sum we
only need to calculate the remaining residues, and multiply their sum by
$-1/2$ (since each term in the sum is counted twice). One finds,

$\mathop{\mathrm{missing}}{Res}(f,\pm
e^{y})=\frac{-2K}{e^{2Ky}-1},\quad\text{and}\quad\mathop{\mathrm{missing}}{Res}(f,\pm
e^{-y})=\frac{2K}{e^{-2Ky}-1}.$

Summing these four residues and multiplying by $-\frac{1}{2}$ gives us the
first sum of the lemma.

Analogously, we construct a rational function $g(z)$ for the second sum::

$g(z):=\frac{2K}{z(z^{2N}-1)}\,\frac{\sinh(2y)}{\sinh^{2}(y)+\left(\frac{z+\frac{1}{z}}{2}\right)^{2}}=\frac{8K}{(z^{2N}-1)}\,\frac{z\sinh(2y)}{z^{4}+2z^{2}(1+2\sinh^{2}(y))+1}$

In this case, we need to compute residues of $g(z)$ at poles located at $\pm
ie^{y}$ and $\pm ie^{-y}$, the roots of $z^{4}+2z^{2}(1+2\sinh^{2}(y))+1$. We
have,

$\mathop{\mathrm{missing}}{Res}(g,\pm
ie^{y})=\frac{-2K}{(-1)^{K}e^{2Ky}-1},\quad\text{and}\quad\mathop{\mathrm{missing}}{Res}(g,\pm
ie^{-y})=\frac{2K}{(-1)^{K}e^{-2Ky}-1}.$

Summing these residues and multiplying by $-\frac{1}{2}$ gives us the second
sum of the lemma. ∎

Suppose $K$ is even; using Lemma A.1, (A.1) becomes

$\begin{array}[]{rcl}\dot{x}_{1}&=&\frac{\Gamma}{2\pi}\coth(2y)+\frac{\Gamma}{8\pi}\left(4K\coth(Ky)-\frac{\sinh(2y)}{\sinh^{2}(y)}-\frac{\sinh(2y)}{\cosh^{2}(y)}\right)\\\\[4.0pt]
&=&\frac{\Gamma K}{2\pi}\coth(Ky),\end{array}$ (A.3)

since in Lemma A.1 we have additionally counted $j=K$.

Using the same method for odd $K$, we get

$\dot{x}_{1}=\frac{\Gamma K}{2\pi}\coth(2Ky).$

We act analogously when the two rows are staggered. However, in this case $N$
is an odd number: we suppose that $N=2K-1$, and that the top row of vortices
on the chart on the Möbius band has $K$ vortices in it. Using the same
notation, we observe that the horizontal distance between the leftmost vortex
(which we assume to be above the line $y=0$) and the other vortices in the
upper row is $\frac{2\pi j}{2K-1},\ j=1,\ldots,K-1$. In turn, horizontal
distances between the same vortex and the vortices in the bottom row are given
by $\frac{(2j-1)\pi}{2K-1},\ j=1,\ldots,K-1$. Thus, we have for the velocity:

$\dot{x}_{1}=\frac{\Gamma}{4\pi}\tanh(y)+\frac{\Gamma\sinh(2y)}{8\pi}\sum_{j=1}^{K-1}\frac{1}{\sinh^{2}(y)+\sin^{2}\left(\frac{(2j-1)\pi}{2K-1}\right)}+\frac{1}{\sinh^{2}(y)+\cos^{2}\left(\frac{2j\pi}{2K-1}\right)}$

We consider the expression
$\sum_{j=1}^{K-1}\frac{1}{\sinh^{2}(y)+\sin^{2}\left(\frac{(2j-1)\pi}{2K-1}\right)}+\frac{1}{\sinh^{2}(y)+\cos^{2}\left(\frac{2j\pi}{2K-1}\right)}$
separately. One can observe that due to the fact that $\sin$ and $\cos$ are
squared in the sum,

$\begin{split}\sum_{j=1}^{K-1}&\left(\frac{1}{\sinh^{2}(y)+\sin^{2}\left(\frac{(2j-1)\pi}{2K-1}\right)}+\frac{1}{\sinh^{2}(y)+\cos^{2}\left(\frac{2j\pi}{2K-1}\right)}\right)\\\
&=\frac{1}{2}\sum_{l=1}^{2K-2}\left(\frac{1}{\sinh^{2}(y)+\sin^{2}\left(\frac{l\pi}{2K-1}\right)}+\frac{1}{\sinh^{2}(y)+\cos^{2}\left(\frac{l\pi}{2K-1}\right)}\right),\end{split}$

which reduces this sum to the one from Lemma A.1; in particular, the top limit
of the sum is always an odd number. Therefore,

$\begin{split}\dot{x}_{1}=&\frac{\Gamma\tanh(y)}{4\pi}+\frac{\Gamma}{16\pi}\Bigl{(}(4K-2)\coth\left((2K-1)y\right)+(4K-2)\tanh\left((2K-1)y\right)\\\
&-\frac{\sinh(2y)}{\sinh^{2}(y)}-\frac{\sinh(2y)}{\cosh^{2}(y)}\Bigr{)}\\\
=&\frac{\Gamma}{8\pi}\left(\tanh(y)-\coth(y)\right)+\frac{\Gamma(2K-1)}{4\pi}\coth\left((4K-2)y\right).\end{split}$
(A.4)

and recalling that $2K-1=N$, we get the statement of our theorem.

######  Remark A.2.

It can be explicitly checked that (3.3) turns into
$\frac{\Gamma}{2\pi}\coth(2y)$ when $N=2$: this is Example 4.1. When $y\to 0$,
$\xi_{a}\sim\frac{\Gamma}{4\pi y}$, consistent with the fact that the angular
velocity tends to $+\infty$ when the two rows aligned are infinitely close.

When we substitute $N=1$ in (3.4), we get the motion of one point vortex: and
indeed, (3.4) readily becomes $\frac{\Gamma}{4\pi}\tanh(y)$. When $y=0$,
staggered $N$-rings become fixed equilibria. With $y\to 0$ the angular
velocity $\xi_{s}$ is given by

$\xi_{s}=\frac{\Gamma y\left(1+2N^{2}\right)}{12\pi}+\bar{O}(y^{3}),$

and becomes 0 when $y=0$.

## References

  * [1] R. Abraham, J. E. Marsden, and T. Ratiu.  Manifolds, tensor analysis, and applications, volume 75.  Springer Science & Business Media, 2012.
  * [2] H. Aref.  Motion of three vortices.  The Physics of Fluids, 22(3):393–400, 1979.
  * [3] H. Aref.  Point vortex dynamics: a classical mathematics playground.  Journal of mathematical Physics, 48(6):065401, 2007.
  * [4] H. Aref, J. B. Kadtke, I. Zawadzki, L. J. Campbell, and B. Eckhardt.  Point vortex dynamics: recent results and open problems.  Fluid Dynamics Research, 3(1-4):63, 1988.
  * [5] H. Aref and M. A. Stremler.  On the motion of three point vortices in a periodic strip.  Journal of Fluid Mechanics, 314:1–25, 1996.
  * [6] V. I. Arnold and B. A. Khesin.  Topological methods in hydrodynamics, volume 125.  Springer Science & Business Media, 1999.
  * [7] N. A. Balabanova.  A Hamiltonian approach for point vortices on non-orientable surfaces ii: the Klein bottle.  Preprint, 2022.
  * [8] S. Boatto and J. Koiller.  Vortices on closed surfaces.  In D. Chang, D. Holm, G. Patrick, and T. Ratiu, editors, Geometry, Mechanics and Dynamics, volume 73 of Fields Institute Communications. Springer, New York, NY., 2015.
  * [9] A. Bossavit.  Differential geometry for the student of numerical methods in electromagnetism.  Lecture notes. Available online on http://butler. cc. tut. fi/~ bossavit/Books/DGSNME/DGSNME. pdf, 1990.
  * [10] R. Bott and L. W. Tu.  Differential forms in algebraic topology, volume 82.  Springer Science & Business Media, 2013.
  * [11] G. E. Bredon.  Introduction to compact transformation groups.  Academic press, 1972.
  * [12] D. G. Dritschel and S. Boatto.  The motion of point vortices on closed surfaces.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2176):20140890, 2015.
  * [13] D. Hally.  Stability of streets of vortices on surfaces of revolution with a reflection symmetry.  J. Math. Phys., 21:211–217, 1980.
  * [14] https://math.stackexchange.com/users/13854/robjohn.  Finite sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^{2}\frac{k\pi}{m}}$.  Mathematics Stack Exchange.  URL:https://math.stackexchange.com/q/2661896 (version: 2018-02-22).
  * [15] R. Kidambi and P. K. Newton.  Motion of three point vortices on a sphere.  Physica D: Nonlinear Phenomena, 116(1-2):143–175, 1998.
  * [16] G. Kirchoff.  Vorlesung uber mechanik, 1876.
  * [17] F. Laurent-Polz.  Point vortices on the sphere: a case with opposite vorticities.  Nonlinearity, 15(1):143, 2001.
  * [18] C. Lim, J. Montaldi, and M. Roberts.  Point vortices on the sphere.  Physica D, 148:97–135, 2001.
  * [19] C. Marchioro and M. Pulvirenti.  Mathematical theory of incompressible nonviscous fluids, volume 96.  Springer Science & Business Media, 2012.
  * [20] J. Marsden and A. Weinstein.  Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids.  Phys. D, 7(1-3):305–323, 1983.  Order in chaos (Los Alamos, N.M., 1982).
  * [21] J. Montaldi and C. Nava-Gaxiola.  Point vortices on the hyperbolic plane.  J. Math. Phys, 55:102702, 1–14, 2014.
  * [22] J. Montaldi, A. Soulière, and T. Tokieda.  Vortex dynamics on a cylinder.  SIAM Journal on Applied Dynamical Systems, 2(3):417–430, 2003.
  * [23] P. K. Newton.  The N-vortex problem: analytical techniques, volume 145.  Springer Science & Business Media, 2013.
  * [24] K. A. O’Neil.  On the hamiltonian dynamics of vortex lattices.  Journal of Mathematical Physics, 30(6):1373–1379, 1989.
  * [25] S. Pekarsky and J. Marsden.  Point vortices on a sphere: Stability of relative equilibria.  J. Math. Phys, 39(11):5894–5907, 1998.
  * [26] C. G. Ragazzo.  The motion of a vortex on a closed surface of constant negative curvature.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2206):20170447, 2017.
  * [27] C. G. Ragazzo and H. H. de Barros Viglioni.  Hydrodynamic vortex on surfaces.  Journal of Nonlinear Science, 27(5):1609–1640, 2017.
  * [28] T. Sakajo and Y. Shimizu.  Point vortex interactions on a toroidal surface.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2191):20160271, 2016.
  * [29] M. A. Stremler and H. Aref.  Motion of three point vortices in a periodic parallelogram.  Journal of Fluid Mechanics, 392:101–128, 1999.
  * [30] M. A. Stremler and S. Basu.  On point vortex models of exotic bluff body wakes.  Fluid Dynamics Research, 46(6):061410, 2014.
  * [31] J. Vanneste.  Vortex dynamics on a Möbius strip.  ArXiv preprint ArXiv:2102.07697, 2021.