# Determination of stable branches of relative equilibria of the $N$-vortex
problem on the sphere

Constantineau, K.111Department of Mathematics and Statistics, McGill
University, 805 Sherbrooke West, Montreal, QC H3A 0B9, Canada.
<EMAIL_ADDRESS>García-Azpeitia, C.222Departamento de
Matemáticas y Mecánica, IIMAS-UNAM. Apdo. Postal 20-126, Col. San Ángel,
Mexico City, 01000, Mexico<EMAIL_ADDRESS>García-Naranjo, L.
C333Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Via
Trieste 63, 35121 Padova, Italy<EMAIL_ADDRESS>and J.-P.
Lessard444Department of Mathematics and Statistics, McGill University, 805
Sherbrooke West, Montreal, QC H3A 0B9, Canada<EMAIL_ADDRESS>
###### Abstract

We consider the $N$-vortex problem on the sphere assuming that all vorticities
have equal strength. We investigate relative equilibria (RE) consisting of $n$
latitudinal rings which are uniformly rotating about the vertical axis with
angular velocity $\omega$. Each such ring contains $m$ vortices placed at the
vertices of a concentric regular polygon and we allow the presence of
additional vortices at the poles. We develop a framework to prove existence
and orbital stability of branches of RE of this type parametrised by $\omega$.
Such framework is implemented to rigorously determine and prove stability of
segments of branches using computer-assisted proofs. This approach circumvents
the analytical complexities that arise when the number of rings $n\geq 2$ and
allows us to give several new rigorous results. We exemplify our method
providing new contributions consisting in the determination of enclosures and
proofs of stability of several equilibria and RE for $5\leq N\leq 12$.

Keywords: $N$-vortex problem, relative equilibrium, stability, interval
arithmetic, computer-assisted proofs.
2020 MSC: 70K42, 76M60, 65G30, 65G20, 47H10, 37C25.

##  1 Introduction

In the last decades the $N$-vortex problem on the sphere has received
considerable attention. A non-exhaustive list of references contains [11, 10,
31, 43, 13, 60, 12, 44, 63, 16, 23, 53] and other papers that are mentioned
below. The equations of motion go back to Gromeka [26] and Bogomolov [9]. The
importance of the system is usually associated with geophysical fluid dynamics
since it provides a simple model for the dynamics of cyclones and hurricanes
in planetary atmospheres. Moreover, recent numerical work [45] suggests that
the $N$-vortex problem on the sphere plays a crucial role in the long term
behaviour of the incompressible two-dimensional Euler equations on the same
domain. We also mention that the $N$-vortex problem on the sphere equipped
with a general Riemannian metric was recently studied in [64] and, as noted in
[33, 57], this is a convenient approach to treat the $N$-vortex problem on
more general closed surfaces of genus $0$. We refer the reader to the book
[52] and the papers [1, 2] for an overview and extensive bibliography on
vortex dynamics.

The equations of motion of the problem define a Hamiltonian system on the
$2N$-dimensional phase space $M$ which is obtained as the cartesian product of
$N$ copies of the 2-sphere $S^{2}$ minus the collision set. The Hamiltonian
function $H:M\to\mathbb{R}$ accounts for the pairwise interaction between the
vortices, and the symplectic form $\Omega$ on $M$ is a weighted sum by the
vortex strengths of the area form on each copy of $S^{2}$. A fundamental
aspect of the problem is that both $H$ and $\Omega$ are invariant under the
action of $SO(3)$ on $M$ that simultaneously rotates all vortices. As a
consequence, the equations of motion are $SO(3)$-equivariant and, in
accordance with Noether’s Theorem, there exists a momentum map
$\Phi:M\to\mathbb{R}^{3}$, whose components are first integrals. It is well-
known (see e.g. [46]) that the problem is integrable if $N\leq 3$, and on the
zero level set of $\Phi$ if $N=4$, and appears to be non-integrable otherwise.

The simplest solutions to the problem are the equilibrium configurations which
correspond to the critical points of $H$. If all the vortex strengths have the
same sign, then $H$ is bounded from below and it assumes a global minimum on
$M$. The minimising configurations are called  ground states and form a very
important kind of stable equilibria. Unfortunately, the rigorous determination
of the ground states is an extremely complicated problem for which very little
is known. If all vortices have equal strengths, which is the case that we
treat in most of this work, after a suitable normalisation, the Hamiltonian
$H:M\to\mathbb{R}$ becomes

$H(v)=-\sum_{i<j}\ln\left(\left\|v_{i}-v_{j}\right\|^{2}\right),\qquad
v=(v_{1},\dots,v_{N})\in M,$

where, for all $i=1,\dots,N$, $v_{i}$ is a unit vector on $\mathbb{R}^{3}$
which specifies the position of the $i^{th}$ vortex on the unit sphere $S^{2}$
(note that $v_{i}\neq v_{j}$ for all $i$, $j$, since collisions have been
removed). Using the above expression for $H$, it is easy to see that the
ground state configurations are geometrically characterised as those which
maximise the product of the pairwise euclidean distances between the vortices.
These configurations are sometimes called Fekete points, and their
determination corresponds to Smale’s problem $\\#7$ which has only been solved
for a few values of $N$ (2, 3, 4, 5, 6 and 12). We refer the reader to [3],
the nice review [4] and the references therein for more information on Smale’s
problem $\\#7$. In Table 1.1 below we list the known Fekete points together
with a conjecture555In section 6.1 we show that these are at least local non-
degenerate minima of $H$ for the values of $N=8,9,10,11$. We also show in
section 6.2 that the configuration for $N=7$, which is widely conjectured to
be the minimiser [4], is degenerate in the sense of Bott [14]. of their
position for the values $N=7,\dots,11$ (the detailed description of the
configurations in the table is given in section 6.1). The configurations of
the table are illustrated in Figures 1.1 and 1.2. One can analytically show
that these are critical points of $H$ except for $N=11$ whose treatment, as
explained in section 6.1, required the use of a computer-assisted proof.

$N$ | Polyhedron |  $\mathbb{Z}_{m}$ -symmetries
---|---|---
2 | antipodal points | $\mathbb{Z}_{2}$
3 | equilateral triangle | $\mathbb{Z}_{3}$
4 | tetrahedron |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$
5 | triangular bipyramid |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$
6 | octahedron |  $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$ and $\mathbb{Z}_{4}$
7 | pentagonal bipyramid |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{5}$
8 | square antiprism |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$
9 | triaugmented triangular prism |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$
10 | gyroelongated square bipyramid |  $\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$
11 | – (see section 6.1 for details) | $\mathbb{Z}_{2}$
12 | icosahedron |  $\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$ and $\mathbb{Z}_{5}$
Table 1.1: Known ($N=2,3,4,5,6,12$) and conjectured ($N=7,8,9,10,11$) ground
states of the $N$ vortex problem on the sphere assuming identical vortex
strengths. See section 6.1 for details.

Another type of fundamental solutions, whose study is the main topic of this
work, is given by relative equilibria (RE). For our specific problem, these
are periodic solutions in which all vortices rotate uniformly about a fixed
axis at constant angular speed $\omega\in\mathbb{R}$. These solutions
correspond to critical points of $H$ restricted to the level sets of $\Phi$.
Without loss of generality, the rotation axis is chosen as the vertical
$z$-axis in $\mathbb{R}^{3}$ throughout this work.

An important class of RE is that in which the vortices are arranged in $n$
latitudinal rings, each of which consists of a regular polygon of $m$
identical vortices. We allow for the possibility of having $p$ vortices at the
poles ($p=0,1$ or $2$) so the total number of vortices $N=mn+p$. Such RE
possess a discrete $\mathbb{Z}_{m}$-symmetry and will be thus called
$\mathbb{Z}_{m}$-symmetric RE. Several previous works have focused on the
study of this type of RE [56, 13, 40, 38, 15, 7, 39]. In particular, the
seminal work of Lim, Montaldi and Roberts [40] gives several existence results
based on symmetry considerations. However, explicit analytic expressions and
stability results for these RE are mainly known for the case of $n=1$ ring
(see [39] for a comprehensive list of results in this case). When the number
of rings $n\geq 2$, the computational complexity enormously increases and one
has to settle with numerical investigations as in [39]. This is somewhat
unsettling, considering that the $\mathbb{Z}_{m}$-symmetric RE consisting of
only one ring lose stability as the number of vortices $m$ in each ring grows
(see e.g. [39] and our discussion in section 6.3 for precise statements).

In this work we circumvent the difficulty to obtain rigorous existence and
stability results for $\mathbb{Z}_{m}$-symmetric RE having $n\geq 2$ rings by
relying on computer-assisted proofs (CAPs). This approach falls in the
category of CAPs in dynamics, which is by now a well-developed field, with
some famous early pioneering works being the proof of the universality of the
Feigenbaum constant [37] and the proof of existence of the strange attractor
in the Lorenz system [61]. We refer the interested reader to the survey papers
[32, 49, 58, 5, 25, 29], as well as the books [62, 6, 50].

###  1.1 Contributions

Our work focuses on the case of equal vortex strengths and develops a
theoretical framework for existence and stability of branches of
$\mathbb{Z}_{m}$-symmetric666The value of $m$ can be chosen as $1$
corresponding to general asymmetric RE. RE parametrised by the angular speed
$\omega$. This framework is implemented in an INTLab code which establishes
existence and provides rigorous bounds for (segments of) the branch using
CAPs. Moreover, the code also performs a (nonlinear) stability test which may
be validated with a CAP.

The input for the code is a numerical approximation of a non-degenerate RE
(configuration and angular speed) with a prescribed $\mathbb{Z}_{m}$-symmetry.
Provided that the given RE approximation has sufficient precision and is far
from bifurcations, the code returns an enclosure of a segment of the branch
containing the approximated RE and possessing the prescribed symmetry. The
stability procedure involves validation of the positivity of the spectrum of a
certain matrix and limitations arise in the presence of eigenvalues which are
either too close to zero or which cluster.

The fundamental aspects of our theoretical framework and its CAP-
implementation are summarised below. We divide the presentation of our results
on existence, stability and CAPs.

#### Existence.

The first step in our construction is to perform a discrete reduction of the
system by $\mathbb{Z}_{m}$ leading to a reduced Hamiltonian system with an
$SO(2)$-symmetry. This is the content of Theorem 3.5 whose formulation is
inspired by the previous work [24] of the second and third author. This
discrete reduction relies on the permutation symmetry of the vortices and
corresponds to the restriction of the dynamics to the invariant submanifold of
$M$ formed by $\mathbb{Z}_{m}$-symmetric configurations777These are generic
configurations (not necessarily RE) in which the vortices are organised in
regular $m$-gons on $n$ latitudinal rings and possibly in the presence of $p$
vortices at the poles ($p=0,1,2$) so $N=mn+p$. The $\mathbb{Z}_{m}$-symmetries
of each of the equilibria in Table 1.1 is indicated in the last column and is
illustrated in Figures 1.1 and 1.2. In each case, it is easy to determine the
corresponding values of $n$ and $p$ from the figure.. Theorem 3.5 shows that
the dynamics in this invariant manifold is conjugate to the dynamics of a
symplectic Hamiltonian system on a $2n$-dimensional reduced phase space, that
we denote $M_{n}$, and whose reduced Hamiltonian function
$h:M_{n}\to\mathbb{R}$ is given explicitly by (3.4). Moreover, the theorem
also shows that this reduced Hamiltonian system possesses an $SO(2)$-symmetry
corresponding to the simultaneous rotation of the rings.

The discrete reduction allows us to cast the problem of determination of
branches of $\mathbb{Z}_{m}$-symmetric RE parametrised by $\omega$ in terms of
the determination of branches of critical points of the augmented Hamiltonian
of the reduced system, which is the function $h_{\omega}:M_{n}\to\mathbb{R}$
depending parametrically on $\omega$ given by $h_{\omega}=h-\omega\phi$, where
$\phi:M_{n}\to\mathbb{R}$ is the momentum map of the $SO(2)$-action on
$M_{n}$. After introducing Lagrange multipliers to embed $M_{n}$ in
$\mathbb{R}^{3n}$, and introducing an unfolding parameter as in [48, 28] to
deal with an $SO(2)$-degeneracy coming from the symmetries, we reduce the
problem of determining critical points of $h_{\omega}$ to finding zeros of a
suitable map

$F:\mathbb{R}^{d+1}\to\mathbb{R}^{d},\qquad(x,\omega)\mapsto F(x,\omega).$
(1.1)

The map $F$ involves the derivatives of $h_{\omega}$ and its explicit form,
and the dimension $d$, may be read from (3.15). Under the hypothesis that a
non-degenerate zero of $F$ is known (corresponding to a
$\mathbb{Z}_{m}$-symmetric RE of our problem which is non-degenerate in a
sense that we make precise in the text) the existence of a local branch of
$\mathbb{Z}_{m}$-symmetric RE parametrised by $\omega$ is then guaranteed by
the implicit function theorem. The existence results that we have just
outlined are formalised in Theorem 3.10 and Corollary 3.12. In particular, the
corollary implies the existence of local branches of
$\mathbb{Z}_{m}$-symmetric RE emanating from each $\mathbb{Z}_{m}$-symmetry of
the equilibria illustrated in Figures 1.1 and 1.2 having $N\neq 7$.888Due to
the degeneracy of the $N=7$ configuration (mentioned in a previous footnote)
the existence of branches of RE cannot be concluded from the corollary if
$N=7$. Despite this, we analytically determine a $\mathbb{Z}_{5}$-symmetric
branch of RE emanating from the symmetry depicted in Figure 2(b) (containing
only $1$ ring) and determined a region of nonlinear stability for it in
section 6.2. These RE are obtained via a suitable vertical displacement of the
rings, which results in their uniform rotation with small angular frequency
about the vertical axis.

Finally we indicate that this paper also contains an original general
existence result which is not used in our framework but provides a groundwork
for our investigation. It is valid for arbitrary vortex strengths and shows
existence of two local branches of RE parametrised by $\omega$ emanating from
any non-degenerate equilibrium (Theorem 3.2).

(a) $N=4$, $\mathbb{Z}_{2}$

(b) $N=4$, $\mathbb{Z}_{3}$

(c) $N=5$, $\mathbb{Z}_{2}$

(d) $N=5$, $\mathbb{Z}_{3}$

(e) $N=6$, $\mathbb{Z}_{2}$

(f) $N=6$, $\mathbb{Z}_{3}$

(g) $N=6$, $\mathbb{Z}_{4}$

(h) $N=12$, $\mathbb{Z}_{2}$

(i) $N=12$, $\mathbb{Z}_{3}$

(j) $N=12$, $\mathbb{Z}_{4}$

Figure 1.1: Ground states for $N=4,5,6,12$ and their
$\mathbb{Z}_{m}$-symmetries.

(a) $N=7$, $\mathbb{Z}_{2}$

(b) $N=7$, $\mathbb{Z}_{5}$

(c) $N=8$, $\mathbb{Z}_{2}$

(d) $N=8$, $\mathbb{Z}_{4}$

(e) $N=9$, $\mathbb{Z}_{2}$

(f) $N=9$, $\mathbb{Z}_{3}$

(g) $N=10$, $\mathbb{Z}_{2}$

(h) $N=10$, $\mathbb{Z}_{4}$

(i) $N=11$, $\mathbb{Z}_{2}$

Figure 1.2: Conjectured ground states for $N=7,8,9,10,11$ and their
$\mathbb{Z}_{m}$-symmetries.

#### Stability.

Our nonlinear stability analysis relies on the _energy-momentum method_ of
Patrick [55]. Such method concludes Lyapunov stability modulo a subgroup,
which in our case generically translates into orbital stability of a periodic
orbit (see Proposition 2.7). The method examines positivity of a certain
Hessian matrix which we block-diagonalise exploiting the
$\mathbb{Z}_{m}$-symmetries following closely the construction of Laurent-
Polz, Montaldi and Roberts [39]. Our approach differs from [39] in two ways.
On the one hand, instead of using spherical coordinates, we work with the
extrinsic geometry induced by our embedding of $M$ on $\mathbb{R}^{3N}$. This
allows us to interpret tangent vectors to $M$ as vectors in the ambient space
$\mathbb{R}^{3N}$, which is convenient for the implementation of CAPs.
Secondly, we refine the block diagonalisation of [39] by exploiting the
complex structure of some blocks as determined by Theorem 4.8. The great
technicality involved in this block diagonalisation and its refinement is
compensated by obtaining larger ranges of application of CAPs, since the
method prevents clustering of eigenvalues of large matrices.

A summary of the stability test which indicates the matrix blocks which need
to be computed, and their dimension, according to the values of $m$, $n$ and
$p$ is given in subsection 4.5.

#### Computer-assisted proofs (CAPs).

The CAPs in this paper are obtained via a finite dimensional Newton-
Kantorovich like theorem (see [54] for the original version), which is similar
to the well-known interval Newton’s method [27, 47] and Krawczyk’s operator
approach [34, 51]. This method allows us to find zeros of the mapping $F$ in
(1.1) to find enclosures of the branches of $\mathbb{Z}_{m}$-symmetric RE and
also, via a suitable formulation, to validate eigenvalues necessary for the
stability test.

As mentioned above, the CAPs are implemented in an INTLab code whose input is
a numerical approximation of a $\mathbb{Z}_{m}$-symmetric RE (configuration
and angular speed). If the approximation has sufficient precision and is not
too close to a bifurcation, the code proves existence of a branch of RE with
this symmetry, determines an enclosure and validates the stability test.

The range of applications of our framework and our code is exemplified by the
results presented in section 6. In particular, Theorem 6.1 proves stability of
the equilibrium configurations in Table 1.1 for $N=8,9,10,11$ (the case $N=7$
is degenerate and our test is inconclusive, and the stability for other values
of $N$ in the table was known). We then prove existence and find enclosures of
segments of stable branches of RE emanating from the equilibria in Table 1.1.
These results are presented in section 6.2, and allow us to conjecture which
branches of RE minimise $H$ on the level sets $\Phi^{-1}(\mu)\subset M$ for
$\mu$ close to zero for the different values of $N$. Finally, in section 6.3,
we investigated RE near total collision. The results of [39] imply that for
$N\geq 10$ the $\mathbb{Z}_{m}$-symmetric RE consisting of only 1-ring are
unstable. Our results in section 6.3 provide an educated first guess of the
shape of the RE which minimises $H$ on the level sets of $\Phi$ near total
collision for $N=10,11,12$.

###  1.2 Structure of the paper

We start by reviewing some known preliminary material in section 2. This
serves to introduce concepts and notation used throughout the paper. Moreover,
it is useful to recall known details and results on RE, the energy-momentum
method and the CAPs that we use. We then focus on results on existence of
branches of RE in section 3. Our main original results in this section are
Theorem 3.2 on the existence of branches of RE near equilibrium for arbitrary
vortex strengths, Theorem 3.5 on the discrete $\mathbb{Z}_{m}$-reduction in
the case of equal vorticities, and Theorem 3.10 and Corollary 3.12 on the
existence of local branches of $\mathbb{Z}_{m}$-symmetric RE. Section 4
focuses on the application of the energy-momentum method for nonlinear
stability of RE. Most of this section is a development of the block-
diagonalisation of [39] which, as mentioned above, is better suited for the
implementation of CAPs. Our refinement of this construction is given in
Theorem 4.8 which exploits the complex structure of some blocks. A useful
summary of the blocks that need to be constructed and whose eigenvalues need
to be calculated according to the values of $m$, $n$ and $p$ is given in
subsection 4.5. In Section 5 we explain how CAPs are implemented using the
setting of sections 3 and 4 to establish existence and stability of branches
of $\mathbb{Z}_{m}$-symmetric RE. Finally, we give examples of the range of
applications of our framework and our code in section 6. The paper contains a
series of appendices which complement the main body of the text.

##  2 Preliminaries

In this section we recall the equations of motion and basic known properties
of the equations of motion of the $N$-vortex on the sphere. This allows us to
introduce the notation and concepts used ahead. In particular our working
definition of RE is given in Definition 2.1. We then recall some properties of
RE which follow from the general theory of RE for Hamiltonian systems with
symmetry which are used in our construction. We also recall the energy
momentum method for nonlinear stability in subsection 2.4 and the Newton-
Kantorovich like theorem for our CAPs in subsection 2.5.

###  2.1 Equations of motion

The equations of motion of the $N$-vortex problem on the sphere are a
Hamiltonian system on the phase space $M$ obtained as the cartesian product of
$N$ copies of the unit sphere in $\mathbb{R}^{3}$ minus the collision set
$\Delta$. That is, $M:=(S^{2}\times\dots\times S^{2})\setminus\Delta$ where

$S^{2}:=\\{w\in\mathbb{R}^{3}\,:\,\|w\|=1\\}\qquad\mbox{and}\qquad\Delta:=\\{(v_{1},\dots,v_{N})\in
M\,:\,v_{i}=v_{j}\,\,\mbox{for some $i\neq j$}\\}.$

The Hamiltonian function $H:M\to\mathbb{R}$ and symplectic form $\Omega$ on
$M$ are given by

$H(v):=-\frac{\Gamma_{i}\Gamma_{j}}{4\pi}\sum_{i<j}\ln\left(\left\|v_{i}-v_{j}\right\|^{2}\right),\qquad\Omega:=\sum_{i=1}^{N}\Gamma_{i}\pi^{*}_{i}\omega_{S^{2}}.$

Here $v=(v_{1},\dots,v_{N})\in M$ specifies the position of each of the
vortices whose respective (constant) intensities are denoted by the scalars
$\Gamma_{i}\neq 0$. The map $\pi_{i}:M\to S^{2}$ denotes the projection onto
the $i^{th}$ factor of the cartesian product $S^{2}\times\dots\times S^{2}$
and $\omega_{S^{2}}$ is the standard area form on $S^{2}$. The resulting
equations of motion are

$\dot{v}_{j}=-\frac{1}{\Gamma_{j}}v_{j}\times\nabla_{v_{j}}H=\frac{1}{4\pi}\sum_{i\neq
j}\Gamma_{i}\frac{v_{i}\times v_{j}}{\|v_{i}-v_{j}\|^{2}},\qquad i=1,\dots,N,$
(2.1)

where $\times$ denotes the cross product in $\mathbb{R}^{3}$. Owing to the
Hamiltonian structure, the Hamiltonian function $H$ is a first integral.

###  2.2 Rotational symmetries, center of vorticity, equilibria and relative
equilibria

##### Rotational symmetries.

The position of the vortices in the sphere is specified by
$v=(v_{1},\dots,v_{N})\in M$. This tacitly assumes a choice of an inertial
frame whose origin lies at the center of the sphere which defines cartesian
coordinates for the $v_{j}$’s. There is freedom in the choice of orientation
of this inertial frame and this is reflected by the invariance of the problem
under the $SO(3)$ action that simultaneously rotates all vortices.
Specifically we consider the $SO(3)$ action on $M$ defined by

$(g,v)\mapsto g.v:=(gv_{1},\dots,gv_{N})\in M,\qquad g\in
SO(3),\;v=(v_{1},\dots,v_{N})\in M.$

For $N\geq 3$ this action is free since the collisions do not belong to $M$.
It is also a proper action since $SO(3)$ is compact.

It is easily checked that both the Hamiltonian and the symplectic form are
invariant under this action and as a consequence, the equations of motion
(2.1) are $SO(3)$-equivariant.

##### Center of vorticity.

One may easily show that the components of the map

$\Phi:M\to\mathbb{R}^{3},\qquad\Phi(v)=\Gamma_{1}v_{1}+\dots+\Gamma_{N}v_{N},$
(2.2)

are first integrals of the equations of motion (2.1). It is common to refer to
$\Phi(v)$ as the center of vorticity of the vortex configuration
$v=(v_{1},\dots,v_{N})$. The existence of these first integrals is in fact an
instance of Noether’s theorem. Upon the standard identification of the dual
Lie algebra $\mathfrak{so}(3)^{*}$ with $\mathbb{R}^{3}$, the function $\Phi$
can be geometrically interpreted as the momentum map associated to the
symplectic action of $SO(3)$ on the symplectic manifold $(M,\Omega)$.

For $N\geq 3$, the freeness of the action guarantees that $\Phi$ is a
submersion onto its image. As a consequence, if $\mu\in\mathbb{R}^{3}$ is such
that $\Phi(v)=\mu$ for some $v\in M$, then $\Phi^{-1}(\mu)$ is a smooth
submanifold of $M$ of codimension 3 and $T_{v}\Phi^{-1}(\mu)=\ker d\Phi(v)$.

The momentum map $\Phi$ plays a fundamental role in the study of relative
equilibria. An important property of $\Phi$ is its equivariance with respect
to the coadjoint representation which, in our interpretation, is simply the
standard linear action of $SO(3)$ on $\mathbb{R}^{3}$, namely, one has

$\Phi(g.v)=g\Phi(v),\quad\forall v\in M,\;g\in SO(3).$ (2.3)

##### Equilibrium points and ground states.

The simplest solutions of the equations of motion are the equilibria which are
in one-to-one correspondence with the critical points of $H:M\to\mathbb{R}$.
Due to the $SO(3)$ invariance of the problem, equilibrium points are never
isolated: if $v_{0}\in M$ is an equilibrium of (2.1) so is $g.v_{0}$ for any
$g\in SO(3)$. In other words, the orbit $SO(3).v_{0}\subset M$ is comprised of
equilibrium points.

If all the vortex strengths $\Gamma_{i}$ have the same sign, then $H$ is
bounded from below and it assumes a global minimum on $M$. The minimising
configurations are called ground states and form a very important kind of
equilibria from the physical point of view. Again, the $SO(3)$-invariance of
$H$ implies that if $v_{0}\in M$ is a ground state then the orbit
$SO(3).v_{0}\subset M$ consists of ground states. The ground states of the
system are stable in a sense that will be made precise below. As mentioned in
the introduction, proving that a given equilibrium is the ground state of the
system is a very difficult problem for which very little is known.

##### Relative equilibria.

The next type of fundamental solutions are the so-called relative equilibria
(RE) which are the main subject of this work. This kind of solutions may exist
for any system of differential equations which is equivariant with respect to
the action of a continuous symmetry group. There are several equivalent
definitions of relative equilibria. For our purposes, it suffices to define
them as solutions of the equations that at the same time are orbits of a one-
parameter subgroup of the symmetry group. Thus, in our case, relative
equilibria have the form $v(t)=e^{\xi t}.a$ for a fixed $a\in M$ and a skew-
symmetric matrix $\xi\in\mathfrak{so}(3)$. This is a periodic solution in
which the vortices steadily rotate along the axis passing through the center
of the sphere in the direction of the vector $\xi_{e}\in\mathbb{R}^{3}$
satisfying $\langle\xi_{e}\rangle=\ker\xi$ and whose angular velocity
$\omega_{e}$ satisfies $2\omega_{e}^{2}=\mbox{trace}(\xi\xi^{T})$.

Just like equilibrium points, relative equilibria are never isolated: if
$v(t)=e^{\xi t}.a$ is a relative equilibrium and $g\in SO(3)$ then, by
equivariance of (2.1), $\tilde{v}(t):=g.v(t)$ is also a solution. However, we
may write

$\tilde{v}(t)=g.v(t)=(ge^{\xi t}g^{-1}).(g.a)=e^{(g\xi
g^{-1})t}.(g.a)=e^{\tilde{\xi}t}.\tilde{a},$

where $\tilde{\xi}:=g\xi g^{-1}\in\mathfrak{so}(3)$ and $\tilde{a}:=g.a\in M$,
which shows that $\tilde{v}(t)$ is also a relative equilibrium with axis of
rotation $\tilde{\xi}_{e}=g\xi_{e}$ and the same angular frequency
$\tilde{\omega}_{e}=\omega_{e}$. Physically, this means that the orientation
of the axis of rotation of a relative equilibrium is unessential and may be
arbitrarily chosen via a suitable rotation of the inertial frame. On the other
hand, relative equilibria having distinct angular frequencies are not related
by a rotation. Therefore, for the rest of the paper, we will restrict our
attention to relative equilibria for which the axis of rotation is the
$z$-axis and we will give a prominent role to the angular velocity $\omega$.
In accordance with this, we search for solutions of (2.1) of the form

$v(t)=e^{\omega J_{3}t}.a=(e^{\omega J_{3}t}a_{1},\dots,e^{\omega
J_{3}t}a_{N}),\qquad J_{3}:=\begin{pmatrix}0&-1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},$ (2.4)

for a certain frequency $\omega\in\mathbb{R}$ and a configuration
$a=(a_{1},\dots,a_{N})\in M$. Our discussion leads to the following
definition.

######  Definition 2.1.

A relative equilibrium (RE) of the equations of motion (2.1) is a pair
$(a,\omega)\in M\times\mathbb{R}$ such that $v(t)$ given by (2.4) is a
solution of (2.1). We say that $\omega$ is the angular velocity and $a\in M$
is the configuration of the relative equilibrium.

######  Remark 2.2.

  1. (i)

Equilibrium points may be interpreted as RE with zero angular velocity.

  2. (ii)

If $(a,\omega)$ is a RE with $\omega\neq 0$, then so is
$(e^{J_{3}\theta}.a,\omega)$ for any $\theta\in[0,2\pi)$. This is a
consequence of the autonomous nature of the equations of motion (2.1): if
$v(t)$ is a solution, then so is $v(t+t_{0})$ for any $t_{0}\in\mathbb{R}$. In
other words, $a$ is determined up to a time-shift along the solution with
initial condition $a$. Considering that the solution is periodic, this time-
shift corresponds to a certain $SO(2)$ action. Such $SO(2)$ action is
precisely the one-parameter subgroup of $SO(3)$ of rotations about the $z$
axis. Namely, if we denote $e_{3}:=(0,0,1)\in\mathbb{R}^{3}$, and define

$SO(2):=\\{g\in
SO(3)\,:\,ge_{3}=e_{3}\\}=\left\\{e^{J_{3}\theta}=\begin{pmatrix}\cos\theta&-\sin\theta&0\\\
\sin\theta&\cos\theta&0\\\ 0&0&1\end{pmatrix}\,:\,\theta\in[0,2\pi)\right\\},$
(2.5)

then the orbit $SO(2).a$ coincides with the dynamical orbit of the solution
$v(t)=e^{\omega J_{3}t}.a$, and is comprised of relative equilibrium
configurations with the same angular velocity $\omega$.

  3. (iii)

Some authors instead define the whole orbit $SO(3).a\subset M$ of the relative
equilibrium configuration $a$ as the relative equilibrium since it projects to
a single equilibrium point of the reduced system on the quotient space
$M/SO(3)$. As explained in item (ii), our approach selects instead an
$SO(2)$-orbit of representatives. This is essential for the continuation of
relative equilibria as a function of the angular velocity $\omega$ considered
ahead.

For any $\omega\in\mathbb{R}$ define the augmented Hamiltonian

$H_{\omega}:M\to\mathbb{R},\qquad H_{\omega}(v):=H(v)-\omega\Phi_{3}(v),$
(2.6)

where $\Phi_{3}:M\to\mathbb{R}$ denotes the third component of the momentum
map $\Phi$ given by (2.2). Namely, $\Phi_{3}(v)=e_{3}\cdot\Phi(v)$ where
$e_{3}=(0,0,1)\in\mathbb{R}^{3}$, and $\cdot$ denotes the standard euclidean
product in $\mathbb{R}^{3}$. The following proposition is a particular
instance of general results on relative equilibria on symplectic manifolds
(see e.g. [41]). We present an elementary proof for completeness.

######  Proposition 2.3.

The following statements hold.

  1. (i)

$(a,\omega)\in M\times\mathbb{R}$ is a relative equilibrium if and only if
$a\in M$ is a critical point of $H_{\omega}$.

  2. (ii)

If $(a,\omega)\in M\times\mathbb{R}$ is a relative equilibrium with
$\omega\neq 0$ then $e_{3}\times\Phi(a)=0$.

###### Proof.

(i) Let $\omega\in\mathbb{R}$ and introduce the time dependent change of
variables,

$y_{j}(t)=e^{-\omega J_{3}t}v_{j}(t)\text{,}\qquad j=1,\dots,N.$

According to Definition 2.1, it is clear that $(a,\omega)\in
M\times\mathbb{R}$ is a relative equilibrium if and only if $a\in M$ is an
equilibrium solution for the equations satisfied by
$y(t)=(y_{1}(t),\dots,y_{N}(t))$. A direct calculation shows that

$\dot{y}_{j}=\frac{1}{4\pi}\sum_{i\neq j)}^{N}\Gamma_{i}\frac{y_{i}\times
y_{j}}{\left\|y_{j}-y_{i}\right\|^{2}}-\omega e_{3}\times
y_{j}=-y_{j}\times\left(\nabla_{y_{j}}H_{\omega}(y)\right).$

This shows that the equations for $y(t)$ are Hamiltonian on the symplectic
manifold $(M,\Omega)$ with respect to the Hamiltonian function $H_{\omega}$.
As a consequence, the equilibrium points of $y(t)$ are in one-to-one
correspondence with the critical points of $H_{\omega}$.

(ii) By conservation of $\Phi$ we have $\Phi(a)=\Phi(v(t))$ where $v(t)$ is
given by (2.4). Therefore,

$\begin{split}\Gamma_{1}a_{1}+\dots+\Gamma_{N}a_{N}&=\Gamma_{1}e^{\omega
J_{3}t}a_{1}+\dots+\Gamma_{N}e^{\omega J_{3}t}a_{N}\\\ &=e^{\omega
J_{3}t}(\Gamma_{1}a_{1}+\dots+\Gamma_{N}a_{N})=e^{\omega
J_{3}t}\Phi(a).\end{split}$

Differentiating with respect to $t$ and evaluating at $t=0$ gives $\omega
J_{3}\Phi(a)=0$, which for $\omega\neq 0$ is equivalent to
$e_{3}\times\Phi(a)=0$.

∎

A fundamental observation which follows directly from (2.3) and the definition
of the augmented Hamiltonian $H_{\omega}$ is that, for $\omega\neq 0$,
$H_{\omega}$ is no longer $SO(3)$-invariant but only $SO(2)$-invariant with
$SO(2)$ given by (2.5). As a consequence of this $SO(2)$-symmetry, if $a\in M$
is a critical point of $H_{\omega}$ so is $e^{J_{3}\theta}.a$ for
$\theta\in[0,2\pi)$. This observation (together with Proposition 2.3) is
consistent with item (ii) of Remark 2.2.

Another useful interpretation of RE which follows from the Proposition 2.3 and
the Lagrange multiplier theorem is stated next. We refer the reader to [41,
Chapter 4] for details.

######  Proposition 2.4.

$(a,\omega)\in M\times\mathbb{R}$ is a relative equilibrium if and only if
$a\in M$ is a critical point of the restriction of $H$ to the level set
$\Phi^{-1}(\mu)\subset M$ where $\mu=(0,0,\Phi_{3}(a))\in\mathbb{R}^{3}$.

Recall from our previous discussion on the ground states, that if all the
vortex strengths $\Gamma_{i}$ have equal sign, then the Hamiltonian $H$ is
bounded from below. Therefore, given $\mu\in\mathbb{R}^{3}$ belonging to the
image of $\Phi$, there exists a minimal energy relative equilibrium
$(a,\omega)\in M\times\mathbb{R}$ satisfying that $H(a)$ is the minimum value
of the restriction of $H$ to $\Phi^{-1}(\mu)\subset M$. These minimising RE
are stable (in a sense that we make precise below). The results in section 6
allow us to conjecture which are the positions of these RE in the case of
identical vortex strengths for specific values of $N$, and to prove that they
are at least local minima of the restriction of $H$ to $\Phi^{-1}(\mu)\subset
M$ for certain values of $\mu$ using computer-assisted proofs.

###  2.3 The case of identical vortex strengths.

Most of the results of this paper assume that the vortex strengths
$\Gamma_{i}=\Gamma_{j}$ for all $i,j=1,\dots,N$. With an appropriate scaling
of units, we can assume that the Hamiltonian $H$ and the symplectic form
$\Omega$ are given by

$H(v)=-\sum_{i<j}\ln\left(\left\|v_{i}-v_{j}\right\|^{2}\right),\qquad\Omega=\sum_{i=1}^{N}\
\pi^{*}_{i}\omega_{S^{2}},$ (2.7)

and the equations of motion are

$\dot{v}_{j}=-v_{j}\times\nabla_{v_{j}}H=\sum_{i\neq j}\frac{v_{i}\times
v_{j}}{\|v_{i}-v_{j}\|^{2}},\qquad i=1,\dots,N.$ (2.8)

Finally, the centre of vorticity, or momentum map $\Phi$, whose general form
is given by (2.2) simplifies to

$\Phi:M\to\mathbb{R}^{3},\qquad\Phi(v)=v_{1}+\dots+v_{N}.$ (2.9)

The assumption that the vortices have equal strengths introduces a discrete
symmetry by permutation of the vortices. Specifically, if $S_{N}$ is the
symmetric group of order $N$ and $\hat{G}:=S_{N}\times SO(3)$ then $\hat{G}$
defines a (left) action on $M$ by

$(\sigma,g).(v_{1},\dots,v_{N}):=(gv_{\sigma^{-1}(1)},\dots,gv_{\sigma^{-1}(N)}),\quad(\sigma,g)\in
S_{N}\times SO(3)=\hat{G},\quad(v_{1},\dots,v_{N})\in M.$ (2.10)

The above action is symplectic, preserves the Hamiltonian $H$, and the
equations (2.8) are equivariant. However, in contrast with the $SO(3)$-action
on $M$, the $\hat{G}$-action is not free and this will allow us to extract
valuable information.

##### Equilibrium points and ground states.

We first recall the following known result about equilibria in the case of
identical vorticities (see Theorem 10.1 in [4] and the references therein).

######  Proposition 2.5.

Let $a\in M$ be an equilibrium point of the equations of motion (2.8) for
identical vortices, then $\Phi(a)=0$.

We now recall from the introduction, that determining the ground states in the
case of equal vortex strengths corresponds to Smale’s problem $\\#7$ which has
only been solved for a few values of $N$ (2, 3, 4, 5, 6 and 12). Table 1.1
lists the known ground states together with our conjecture of their position
for the values $N=7,8,9,10,11$. The table also indicates their
$\mathbb{Z}_{m}$-symmetries since they will be essential in our study of
relative equilibria emanating from these configurations at a later stage of
the paper. The contents of the table for $N\geq 4$ are illustrated in Figures
1.1 and 1.2. Note that the center of vorticity of all these configurations
vanishes by virtue of Proposition 2.5.

###  2.4 Stability of relative equilibria

Our discussion of stability of relative equilibria relies on the following
definition.

######  Definition 2.6.

(Patrick [55]). Let $K<SO(3)$ be a subgroup. A relative equilibrium
$(a,\omega)\in M\times\mathbb{R}$ is $K$-stable or  stable modulo $K$ if for
any neighbourhood $V$ of $K.a$ there exists a neighbourhood $U\subset V$ of
$a$ which is invariant under the flow of the equations of motion (2.1) for
$t\geq 0$.

The idea of the definition is that if $(a,\omega)\in M\times\mathbb{R}$ is a
$K$-stable relative equilibrium, then the solutions with initial conditions
near $a$ will stay close to the $K$-orbit through $a$ for all $t\geq 0$.

Let $(a,\omega)$ be a relative equilibrium and suppose that
$\Phi(a)=\mu\in\mathbb{R}^{3}$. As shown by Patrick [55], if $(a,\omega)$
projects to a Lyapunov stable equilibrium point of the symmetry reduced
symplectic system, then the RE is $SO(3)_{\mu}$-stable where $SO(3)_{\mu}$ is
the isotropy subgroup of $\mu$ under the coadjoint action (represented as
$(g,\mu)\mapsto g\mu$). In other words, for the unreduced system (2.1),
initial conditions near $a$ may only drift along the orbits of $SO(3)_{\mu}$.
Therefore, the appropriate notion of stability of the RE $(a,\omega)$ is that
of $SO(3)_{\mu}$-stability.

There are two possibilities for $SO(3)_{\mu}$:

  1. (i)

If $\mu=0$ then $SO(3)_{\mu}=SO(3)_{0}=SO(3)$.

  2. (ii)

If $\mu\neq 0$ then $SO(3)_{\mu}$ is the set of rotation matrices around $\mu$
which is isomorphic to $SO(2)$. (If $\omega\neq 0$, then $SO(3)_{\mu}$ exactly
equals the representation of $SO(2)$ in (2.5) since $\mu=\Phi(a)$ is parallel
to $e_{3}$ by Proposition 2.3(ii).)

According to Proposition 2.5, the first case ($\mu=0$) is always encountered
at equilibria when all vortices have equal strengths. For instance, all
equilibrium configurations in Table 1.1 have $\mu=0$ and are
$SO(3)_{0}=SO(3)$-stable.999this is only a conjecture if $N=7$. For other
values of $N$ see the discussion in Section 6.1 for proofs and earlier
references of this fact. This means that a solution whose initial condition is
close to any of these equilibria will in general not stay close to it, but
will be constrained to evolve in such way that the vortices are at every time
near the vertices of a rotated version of such equilibrium.

On the other hand, the second case ($\mu\neq 0$) is the generic one. For
$\omega\neq 0$, considering that $SO(2).a$ coincides with the dynamical orbit
of the solution $v(t)=e^{\omega J_{3}t}.a$ (item (ii) of Remark 2.2), we come
to the following conclusion that we state as a proposition due to its
relevance.

######  Proposition 2.7.

Let $(a,\omega)\in M\times\mathbb{R}$ be a RE with $\omega\neq 0$. If
$\mu=\Phi(a)\neq 0$ then $SO(3)_{\mu}$-stability of the RE $(a,\omega)$ is
equivalent to orbital stability of the periodic orbit $v(t)=e^{\omega
J_{3}t}.a$.

####  2.4.1 Energy momentum method

The stability analysis of RE that we employ in this work relies on the energy-
momentum method of Patrick [55] and will allow us to determine sufficient
conditions for $SO(3)_{\mu}$-stability.

Let $(a,\omega)\in M\times\mathbb{R}$ be a relative equilibrium and let
$\mu=\Phi(a)\in\mathbb{R}^{3}$. From our discussion above, we know that $a$ is
a critical point of the augmented Hamiltonian $H_{\omega}:M\to\mathbb{R}$. The
energy-momentum method examines the signature of the restriction of the
Hessian $d^{2}H_{\omega}(a):T_{a}M\times T_{a}M\to\mathbb{R}$ to a subspace
$\mathcal{N}_{a}\subset T_{a}M$ satisfying the property

$\mathcal{N}_{a}\oplus(\mathfrak{so}(3)_{\mu}\cdot a)=\ker d\Phi(a).$ (2.11)

Here $\mathfrak{so}(3)_{\mu}$ denotes the Lie algebra of $SO(3)_{\mu}$ and
$\mathfrak{so}(3)_{\mu}\cdot a\subset T_{a}M$ is the tangent space to the
$SO(3)_{\mu}$-orbit through $a$ at $a$. The subspace $\mathcal{N}_{a}$ is
called a symplectic slice and is a symplectic subspace of $T_{a}M$. The above
property states that it is tangent to the level set $\Phi^{-1}(\mu)$ at $a$
and is also a direct complement of the tangent space to the
$SO(3)_{\mu}$-orbit through $a$.

If the restriction of $d^{2}H_{\omega}(a)$ to the symplectic slice
$\mathcal{N}_{a}$ is definite then [41, Theorem 5.1.1] implies that
$(a,\omega)$ is $SO(3)_{\mu}$-stable in $\Phi^{-1}(\mu)$ and $SO(3)$-stable in
$M$. The results of Patrick [55] then imply that $(a,\omega)$ is
$SO(3)_{\mu}$-stable in $M$. In particular we conclude that:

if the restriction of $d^{2}H_{\omega}(a)$ to $\mathcal{N}_{a}$ is positive
definite, then the RE $(a,\omega)$ is $SO(3)_{\mu}$-stable.

###  2.5 computer-assisted proofs

As mentioned in the introduction, the CAPs in this paper are obtained via a
finite dimensional Newton-Kantorovich like theorem (see [54] for the original
version), which is similar to the well-known interval Newton’s method [27, 47]
and Krawczyk’s operator approach [34, 51]. The central idea is to compute an
approximate solution to the problem and to show that a Newton-like operator is
a contraction on a ball centered at the numerical approximation. Additionally,
we combine predictor-corrector continuation methods (e.g. see [30, 20])
together with the uniform contraction principle (e.g. see [17]) to obtain one-
dimensional branches of solutions. Let us give some details.

Consider a smooth map
$F:\mathbb{R}^{d}\times\mathbb{R}\rightarrow\mathbb{R}^{d}$,
$(x,\omega)\mapsto F(x,\omega)$, for some $d\in\mathbb{N}$. Now consider two
numerical approximations $(\bar{x}_{0},\omega_{0}),(\bar{x}_{1},\omega_{1})$
such that $F(\bar{x}_{i},\omega_{i})\approx 0$ (for $i=0,1$). Given
$s\in[0,1]$, let $\bar{x}_{s}:=s\bar{x}_{1}+(1-s)\bar{x}_{0}$ and
$\omega_{s}:=s\omega_{1}+(1-s)\omega_{0}$.

Choose a norm $\|\cdot\|$ on $\mathbb{R}^{d}$, and given a point
$x\in\mathbb{R}^{d}$, define the closed ball of radius $r>0$ by
$\overline{B_{r}(x)}=\\{\xi\in\mathbb{R}^{d}:\|\xi-x\|\leq r\\}$.

######  Theorem 2.8 (Uniform Newton-Kantorovich-like theorem).

Let $F$ be as above and $A\in M_{d}(\mathbb{R})$ be such that $A\approx
D_{x}F(\bar{x}_{0},\omega_{0})^{-1}$. Consider the non-negative bounds $Y$,
$\hat{Y}$, $Z$ and the positive number $r_{*}$ satisfying

$\displaystyle\|AF(\bar{x}_{0},\omega_{0})\|$ $\displaystyle\leq Y,$
$\displaystyle\|A[F(\bar{x}_{s},\omega_{s})-F(\bar{x}_{0},\omega_{0})]\|$
$\displaystyle\leq\hat{Y},\quad\textit{for s}\in[0,1]$
$\displaystyle\|A[I-D_{x}F(c,\omega_{s})]\|$ $\displaystyle\leq
Z,\quad\textit{for all c}\in\overline{B_{r_{*}}(\bar{x}_{s})}\textit{and
s}\in[0,1].$

Define the radii polynomial

$p(r):=(Z-1)r+Y+\hat{Y}.$ (2.12)

If there exists $r_{0}$ such that $0<r_{0}\leq r_{*}$ satisfies $p(r_{0})<0$,
then there exists a function

$\tilde{x}:\\{\omega_{s}:s\in[0,1]\\}\rightarrow\bigcup\limits_{s\in[0,1]}\overline{B_{r_{*}}(\bar{x}_{s})}$

such that $F(\tilde{x}(\omega_{s}),\omega_{s})=0$ for all $s\in[0,1]$.

The above Newton-Kantorovich-like theorem will be used to show existence of
branches of relative equilibria (see Section 5.1) and to prove stability via a
rigorous control of the eigenvalues of a specific matrix (see Section 5.2).

##  3 Existence of branches of relative equilibria

This section is concerned with the existence of branches of RE. We first
present a general non-constructive existence result valid for arbitrary vortex
strengths in subsection 3.1 and then focus on the case of identical vortex
strengths. We develop a constructive framework to prove existence of symmetric
RE in subsection 3.2 which will be implemented using CAPs in subsection 5.1.

###  3.1 Existence of relative equilibria for arbitrary vortex strengths

We start by recalling the following classical definition that is needed in the
formulation of our results.

######  Definition 3.1 (Bott (1954) [14]).

Let $P\subset M$ be a submanifold and suppose each point in $P$ is a critical
point of $f:M\to\mathbb{R}$. We say that $P$ is a nondegenerate critical
submanifold if, in addition, for each $x\in P$ one has101010By
$\ker\left(d^{2}f(x)\right)$ we mean the null-space of the bilinear form
$d^{2}f(x):T_{x}M\times T_{x}M\to\mathbb{R}$, namely, $\ker
d^{2}f(x)=\left\\{{\bf w}\in T_{x}M\,:\,d^{2}f(x)({\bf w},\tilde{{\bf
w}})=0\;\forall\tilde{{\bf w}}\in T_{x}M\,\right\\}$.
$\ker\left(d^{2}f(x)\right)=T_{x}P$.

Note that it is obvious that $T_{x}P\subset\ker\left(d^{2}f(x)\right)$ and the
definition is really saying that $d^{2}f(x)$ is non-degenerate in the
directions transversal to $T_{x}P$ for all $x\in P$.

######  Theorem 3.2.

Let $N\geq 3$ and suppose that $a_{0}\in M$ is an equilibrium of (2.1) with
the property that $SO(3).a_{0}$ is a nondegenerate critical manifold of
$H_{0}:M\rightarrow\mathbb{R}$. Then there exists $\omega_{0}>0$ such that for
any $\omega\in(-\omega_{0},\omega_{0})$ there exist at least two distinct
relative equilibria with angular velocity $\omega$. These relative equilibria
approach the orbit $SO(3).a_{0}$ as $\omega\to 0$.

The proof that we give below is not constructive and will not be used to
determine RE at a later stage of the paper. It relies on topological argument
and techniques for the persistence of relative equilibria under symmetry
breaking (e.g. [22]) which are well suited to our problem since the symmetries
of the augmented Hamiltonian $H_{\omega}$ break from $SO(3)$ to only
$SO(2)<SO(3)$ as $\omega$ passes through zero.

###### Proof.

Let $\mathcal{N}\subset T_{a_{0}}M$ be a vector subspace which is a direct
complement of $T_{a_{0}}(SO(3).a_{0})$. The Palais slice theorem guarantees
the existence of an $SO(3)$ invariant tubular neighbourhood $U$ of
$SO(3).a_{0}$ which is diffeomorphic to $SO(3)\times\mathcal{N}_{0}$, where
$\mathcal{N}_{0}\subset\mathcal{N}$ is a neighbourhood of $0$. We may thus
represent a point $v\in U$ as a pair $v=(g,x)\in SO(3)\times\mathcal{N}_{0}$.
In particular, the point $a_{0}=(e,0)$ (where $e$ denotes the identity element
in $SO(3)$) and elements of the orbit $SO(3).a_{0}$ have the form $(g,0)$. Our
hypothesis that the orbit of $a_{0}$ is a nondegenerate critical manifold of
$H_{0}$ implies that for all $g\in SO(3)$ we have

$dH_{0}(g,0)=0\qquad\mbox{and}\qquad\left.d^{2}H_{0}(g,0)\right|_{\mathcal{N}}\quad\mbox{is
non-degenerate}.$

In particular, for fixed $g_{0}\in SO(3)$ and $\omega\in\mathbb{R}$, the map
$f_{g_{0},\omega}:\mathcal{N}_{0}\to\mathcal{N}^{*}$ given by
$f_{g_{0},\omega}(x)=d_{x}H_{\omega}(g_{0},x)$ satisfies

$f_{g_{0},0}(0)=0\qquad\mbox{and}\qquad D_{x}f_{g_{0},0}(0)\quad\mbox{is
invertible}.$

Therefore, by the implicit function theorem we have

$d_{x}H_{\omega}(g,\tilde{x}_{g_{0}}(g,\omega))=0,$

for a unique function $\tilde{x}_{g_{0}}:U_{g_{0}}\times
I_{g_{0}}\to\mathcal{N}_{0}$, where $U_{g_{0}}$ is a neighbourhood of
$g_{0}\in SO(3)$, $I_{g_{0}}\subset\mathbb{R}$ is a small interval around
zero, and $\tilde{x}_{g_{0}}$ satisfies $\tilde{x}_{g_{0}}(g_{0},0)=0$.

The above argument may be repeated varying the group element $g_{0}\in SO(3)$.
Using compactness of $SO(3)$, one may obtain (see [22] for details) a uniform
version of the implicit function theorem, namely, the existence of
$\omega_{0}>0$ and a unique function
$\tilde{x}:SO(3)\times(-\omega_{0},\omega_{0})\to\mathcal{N}_{0}$ satisfying
$\tilde{x}(g,0)=0$ such that

$d_{x}H_{\omega}(g,\tilde{x}(g,\omega))=0,\qquad\mbox{for all $g\in SO(3)$,
$\omega\in(-\omega_{0},\omega_{0})$}.$

For any $\omega\in(-\omega_{0},\omega_{0})$ define
$\psi_{\omega}:SO(3)\to\mathbb{R}$ by
$\psi_{\omega}(g)=H_{\omega}(g,\tilde{x}(g,\omega))$. By compactness of
$SO(3)$ the function $\psi_{\omega}$ attains its maximum and its minimum at
certain $g_{1},g_{2}\in SO(3)$ and hence $H_{\omega}$ has an extremum at
$a_{1}:=(g_{1},\psi_{\omega}(g_{1}))$ and
$a_{2}:=(g_{2},\psi_{\omega}(g_{2}))$. Therefore, $(a_{1},\omega)$ and
$(a_{2},\omega)$ are the desired relative equilibria. ∎

######  Remark 3.3.

  1. (i)

The proof given above follows the ideas from [22, Theorem 2.2] (see also
references therein), although our case is simpler because the $SO(3)$ action
on $M$ is free when the total number of vortices $N\geq 3$.

  2. (ii)

There are several existence results of RE in Lim, Montaldi and Roberts [40,
Section 3]. Their approach relies on the discrete symmetries arising from
permutations of identical vortices, and the momentum value plays a prominent
role. In particular, section 3.3 in their paper is devoted to bifurcations
from zero momentum. Our Theorem 3.2 is complementary to their results since we
do not assume any equality between the vortex intensities and the emphasis is
given to the angular velocity $\omega$ instead of the momentum. It is
important to notice that equilibrium points (for which $\omega=0$) may have
non-vanishing momentum if the intensities of the vortices are distinct as is
seen by the example of two antipodal vortices with different strengths.

  3. (iii)

It is unclear to us how general is the non-degeneracy assumption in the
theorem. It would be reasonable to expect that it is generically satisfied,
but in the course of our investigations (see Remark 6.3) we discovered that it
does not hold for the pentagonal bipyramid corresponding to $N=7$ in Table 1.1
where all vortex intensities are assumed to coincide.

###  3.2 Existence of symmetric relative equilibria in the case of identical
vortex strengths

From now on we suppose that all vortices have equal strengths, so $H$,
$\Omega$, $\Phi$ and the equations of motion are given by (2.7), (2.8) and
(2.9). We also recall from section 2.3 that the total symmetry group is
$\hat{G}=S_{N}\times SO(3)$ and acts on $M$ symplectically by (2.10), and,
with respect to this action, $H$ is invariant and the equations of motion
(2.8) are equivariant.

####  3.2.1 $\mathbb{Z}_{m}$-symmetric configurations and discrete reduction

Consider a positive integer $m\geq 1$ and denote by $g_{m}\in SO(3)$ the
matrix

$g_{m}:=\exp\left(\frac{2\pi}{m}J_{3}\right)=\begin{pmatrix}\cos\frac{2\pi}{m}&-\sin\frac{2\pi}{m}&0\\\
\sin\frac{2\pi}{m}&\cos\frac{2\pi}{m}&0\\\ 0&0&1\end{pmatrix}.$ (3.1)

Assume that $N=mn+p$ with $p=0,1$ or $2$, and let $\tau_{m}\in S_{N}$ be the
following permutation written as the product of $n$ disjoint cycles

$\tau_{m}:=(1,\dots,m)(m+1,\dots,2m)(2m+1,\dots,3m)\cdots((n-1)m+1,\dots,nm).$
(3.2)

######  Definition 3.4.

Let $v=(v_{1},\dots,v_{N})\in M$. We say that $v$ is
$\mathbb{Z}_{m}$-symmetric if $(\tau_{m},g_{m}).v=v$.

The definition is tailored to identify the configurations $v\in M$ comprised
of $n$ latitudinal rings, each consisting of a regular $m$-gon and in the
presence of $p$ vortices at the poles ($p=0,1$ or $2$). Note that if $m=1$
there is really no ring and the configuration may have no symmetries. In this
case we follow the convention that $n=N$ and $p=0$.

Let $K_{m}$ be the subgroup of $\hat{G}$ generated by
$(\tau_{m},g_{m})\in\hat{G}$. It is clear that $K_{m}$ is isomorphic to the
cyclic group $\mathbb{Z}_{m}$. The set of all $\mathbb{Z}_{m}$-symmetric
configurations is then

$\mbox{Fix}(K_{m}):=\\{v\in M\,:\,(\sigma,g).v=v\;\forall(\sigma,g)\in
K_{m}\\}\subset M.$

Considering that $K_{m}$ is a subgroup of $\hat{G}$ and the equations (2.8)
are $\hat{G}$-equivariant, it follows that $\mbox{Fix}(K_{m})$ is invariant
under the flow of (2.8). Therefore, we may speak of $\mathbb{Z}_{m}$-symmetric
solutions, $\mathbb{Z}_{m}$-symmetric equilibria and
$\mathbb{Z}_{m}$-symmetric relative equilibria.

We wish to understand the structure of the set $\mbox{Fix}(K_{m})$ and the
restriction of (2.8) to it. Suppose first that $m\geq 2$ and consider the set
$M_{n}$ which is the cartesian product of $n$ copies of the unit sphere in
$\mathbb{R}^{3}$ minus the collisions and poles. Namely,

$M_{n}:=\underbrace{S^{2}\times\cdots\times S^{2}}_{n}\setminus(\Delta_{n}\cup
P_{n}),$

where

$\Delta_{n}:=\\{u\in(S^{2})^{n}\,:\,u_{i}=u_{j}\,\mbox{for some $i\neq
j$}\\},\quad\mbox{and}\quad P_{n}:=\\{u\in(S^{2})^{n}\,:\,u_{i}=(0,0,\pm
1)\,\mbox{for some $i$}\\}.$ (3.3)

Elements of $M_{n}$ should be interpreted as generators of the $n$ latitudinal
rings that make up a $\mathbb{Z}_{m}$-symmetric configuration. Note that the
removal of the set $P_{n}$ in the definition of $M_{n}$ above is necessary
since the presence of a ring generator at the pole leads to a collision if
$m\geq 2$. For $m=1$ we will instead assume that $M_{n}=M$ (we do not need to
remove $P_{n}$ since the presence of a “generator” at the pole does not lead
to a collision).

For our purposes it is convenient to equip $M_{n}$ with the symplectic form
$\omega_{M_{n}}:=m\sum_{j=1}^{n}\pi_{j}^{\ast}\omega_{S^{2}}$ and consider the
Hamiltonian function $h:M_{n}\to\mathbb{R}$ given by

$h(u):=\sum_{j=1}^{n}\left(-\frac{m}{4}\sum_{i=1}^{m-1}\ln\left\|g_{m}^{i}u_{j}-u_{j}\right\|^{2}-\frac{m}{2}\sum_{j<j^{\prime}\leq
n}\sum_{i=1}^{m}\ln\left\|g_{m}^{i}u_{j}-u_{j^{\prime}}\right\|^{2}-\frac{m}{2}\sum_{f\in
F_{p}}\ln\left\|u_{j}-f\right\|^{2}\right),$ (3.4)

where $u=(u_{1},\dots,u_{n})\in M_{n}$ and $F_{p}$ is the ordered set defined
according to the value of $p=N-mn\in\\{0,1,2\\}$ as

$F_{0}:=\emptyset,\qquad F_{1}:=\\{(0,0,1)\\},\qquad
F_{2}:=\\{(0,0,1),(0,0,-1)\\}.$ (3.5)

The Hamiltonian vector field on $M_{n}$ corresponding to $h$ and the
symplectic form $\omega_{M_{n}}$ is given by the reduced system111111note that
our use of the terminology “reduced system” does not mean that we are passing
to the orbit space of a group action. As follows from Theorem 3.5, it is more
appropriate to think of a “discrete reduction” in which one restricts the
system to a connected component of the invariant set $\mbox{Fix}(K_{m})$.

$\dot{u}_{j}=-\frac{1}{m}u_{j}\times\nabla_{u_{j}}h(u),\qquad j=1,\dots,n.$
(3.6)

Finally, consider the map $\rho:M_{n}\to M$ defined according to the value of
$p=N-mn\in\\{0,1,2\\}$, by

$\rho(u_{1},\dots,u_{n})=(g_{m}^{1}u_{1},\dots,g_{m}^{m}u_{1},g_{m}^{1}u_{2},\dots,g_{m}^{m}u_{2},\dots,g_{m}^{1}u_{n},\dots,g_{m}^{m}u_{n},F_{p}),$
(3.7)

where $F_{p}$ is given by (3.5). This mapping produces a
$\mathbb{Z}_{m}$-symmetric configuration on $M$ out of the ring generators
$(u_{1},\dots,u_{n})$ in the natural way.

The following theorem is inspired by [24, Theorem 3.5]. Among other things, it
states that if an initial condition is made up of $n$ latitudinal rings, each
of which is made up of a regular $m$-gon (and possibly $p$ poles), then its
evolution by (2.8) will preserve such structure and the reduced system (3.6)
describes the evolution of the ring generators $u_{1},\dots,u_{n}$.

Although the discussion above and the theorem are primarily intended to be
applied when $m\geq 2$, they also hold trivially for $m=1$ (see Remark 3.6).
It is convenient to allow $m=1$ in our discussion to study RE which possess no
symmetries at a later stage of the paper.

######  Theorem 3.5.

Let $N\geq 3$. The following statements hold.

  1. (i)

The set $\mbox{\em Fix}(K_{m})$ is an embedded submanifold of $M$ whose
connected components are diffeomorphic to $M_{n}$ and are invariant under the
flow of (2.8).

  2. (ii)

The mapping $\rho$ defined by (3.7) is a diffeomorphism from $M_{n}$ to a
connected component of $\mbox{\em Fix}(K_{m})$ which satisfies $h=H\circ\rho$
and conjugates the flows of (3.6) and (2.8). That is, $t\rightarrow u(t)$ is a
solution of (3.6) if and only if $t\mapsto\rho(u(t))$ is a solution of (2.8).

  3. (iii)

Let $SO(2)$ be given by (2.5). The diagonal action of $SO(2)$ on $M_{n}$ is
free, symplectic and has momentum map $\phi:M_{n}\to\mathbb{R}$ given by

$\phi(u)=m\left(\sum_{i=1}^{n}u_{i}\right)\cdot e_{3},$ (3.8)

and, up to perhaps a constant, $\phi=\Phi_{3}\circ\rho$. Moreover, the reduced
Hamiltonian $h$ given by (3.4) is $SO(2)$-invariant and hence, as a
consequence, (3.6) is $SO(2)$-equivariant and $\phi$ is a first integral.

  4. (iv)

Assume that $m\geq 2$. The centre of vorticity of elements of $\mbox{\em
Fix}(K_{m})$ is parallel to $e_{3}$ i.e. $\Phi(v)\times e_{3}=0,\,\forall
v\in\mbox{\em Fix}(K_{m})$.

###### Proof.

(i)-(ii) The mapping $\rho$ defined by (3.7) is injective and takes values on
$\mbox{Fix}(K_{m})$. First note that if $p=0$ it is easy to show that

$\mbox{Fix}(K_{m})=\\{(g_{m}^{1}u_{1},\dots,g_{m}^{m}u_{1},g_{m}^{1}u_{2},\dots,g_{m}^{m}u_{2},\dots,g_{m}^{1}u_{n},\dots,g_{m}^{m}u_{n})\,:\,u=(u_{1},\dots,u_{n})\in
M_{n}\\},$

which shows that $\rho$ is onto and hence bijective. It is clear that
$\rho^{-1}$ is smooth so in this case $\rho$ is a diffeomorphism. Now, if
$p=1$ or $p=2$, then $\mbox{Fix}(K_{m})$ is the disjoint union of two
diffeomorphic connected components $\mbox{Fix}(K_{m})^{\pm}$ which are given
by

$\mbox{Fix}(K_{m})^{\pm}:=\\{(g_{m}^{1}u_{1},\dots,g_{m}^{m}u_{1},g_{m}^{1}u_{2},\dots,g_{m}^{m}u_{2},\dots,g_{m}^{1}u_{n},\dots,g_{m}^{m}u_{n},(0,0,\pm
1))\,:\,u=(u_{1},\dots,u_{n})\in M_{n}\\},$

if $p=1$, and instead by

$\begin{split}\mbox{Fix}(K_{m})^{+}&:=\\{(g_{m}^{1}u_{1},\dots,g_{m}^{m}u_{1},\dots,g_{m}^{1}u_{n},\dots,g_{m}^{m}u_{n},(0,0,1),(0,0,-1))\,:\,u=(u_{1},\dots,u_{n})\in
M_{n}\\},\\\
\mbox{Fix}(K_{m})^{-}&=\\{(g_{m}^{1}u_{1},\dots,g_{m}^{m}u_{1},\dots,g_{m}^{1}u_{n},\dots,g_{m}^{m}u_{n},(0,0,-1),(0,0,1))\,:\,u=(u_{1},\dots,u_{n})\in
M_{n}\\},\end{split}$

if $p=2$. In any case, arguing as in the case $p=0$, shows that $\rho$ is a
diffeomorphism from $M_{n}$ onto $\mbox{Fix}(K_{m})^{+}$. The invariance of
(the connected components of) $\mbox{Fix}(K_{m})$ follows from the
$\hat{G}$-equivariance of (2.8) since $K_{m}$ is a subgroup of $\hat{G}$.

Now denote by $\mbox{Fix}(K_{m})^{+}$ the image of $M_{n}$ by $\rho$ for any
$p=0,1,2$ (in other words, $\mbox{Fix}(K_{m})^{+}$ is defined as above for
$p=1,2$ and instead equals $\mbox{Fix}(K_{m})$ if $p=0$).

In order to prove that $\rho$ conjugates the flows of (3.6) and (2.8), we
proceed as in the proof of [24, Theorem 3.5] and argue that $\rho$ is in fact
a symplectomorphism from $M_{n}$ equipped with $\omega_{M_{n}}$ onto
$\mbox{Fix}(K_{m})^{+}$ equipped with the restriction of $\Omega$ (given by
(2.7)). Below we show that $h=H\circ\rho$ (with $H$ given by (2.7)). Therefore
(see e.g. [42, Proposition 5.4.4]), $\rho$ pulls-back the restriction of the
Hamiltonian vector field of $H$ with respect to $\Omega$ to
$\mbox{Fix}(K_{m})^{+}$ onto the Hamiltonian vector field of $h$ with respect
to $\omega_{M_{n}}$. Hence, the solution curves of these vector fields are
mapped onto each other by $\rho$. In other words, $\rho$ maps solutions of
(3.6) into solutions of (2.8) as required.

The proof that $\Omega$ pulls-back to $\omega_{M_{n}}$ by $\rho$ is a simple
generalisation of the proof of [24, Lemma 3.10] that we omit. We now show that
indeed $h=H\circ\rho$ which completes the proof. Starting from (2.7) and
(3.7), we compute for $u=(u_{1},\dots,u_{n})\in M_{n}$:

$H\circ\rho(u)=-\frac{1}{4}\sum_{(i,j)\neq(i^{\prime},j^{\prime})}\ln\left\|g_{m}^{i}u_{j}-g_{m}^{i^{\prime}}u_{j^{\prime}}\right\|^{2}-\frac{1}{2}\sum_{f\in
F_{p}}\sum_{i=1}^{m}\sum_{j=1}^{n}\ln\left\|g_{m}^{i}u_{j}-f\right\|^{2}.$
(3.9)

Using $g_{m}^{i}f=f$ for $f\in F_{p}$, gives

$\frac{1}{2}\sum_{f\in
F_{p}}\sum_{i=1}^{m}\sum_{j=1}^{n}\ln\left\|g_{m}^{i}u_{j}-f\right\|^{2}=\frac{1}{2}\sum_{f\in
F_{p}}\sum_{i=1}^{m}\sum_{j=1}^{n}\ln\left\|g_{m}^{i}(u_{j}-f)\right\|^{2}=\frac{m}{2}\sum_{j=1}^{n}\sum_{f\in
F_{p}}\ln\left\|u_{j}-f\right\|^{2}\text{.}$ (3.10)

On the other hand,

$\begin{split}\frac{1}{4}\sum_{(i,j)\neq(i^{\prime},j^{\prime})}\ln\left\|g_{m}^{i}u_{j}-g_{m}^{i^{\prime}}u_{j^{\prime}}\right\|^{2}&=\frac{1}{4}\sum_{(i,j)\neq(i^{\prime},j^{\prime})}\ln\left\|g_{m}^{i-i^{\prime}}u_{j}-u_{j^{\prime}}\right\|^{2}\\\
&=\frac{1}{4}\sum_{j=1}^{n}\sum_{\begin{subarray}{c}i,i^{\prime}=1\\\ (i\neq
i^{\prime})\end{subarray}}^{m}\ln\left\|g_{m}^{i-i^{\prime}}u_{j}-u_{j}\right\|^{2}+\frac{1}{4}\sum_{j\neq
j^{\prime}}\sum_{i,i^{\prime}=1}^{m}\ln\left\|g_{m}^{i-i^{\prime}}u_{j}-u_{j^{\prime}}\right\|^{2}\\\
&=\sum_{j=1}^{n}\left(\frac{m}{4}\sum_{i=1}^{m-1}\ln\left\|g_{m}^{i}u_{j}-u_{j}\right\|^{2}\right)+\sum_{j\neq
j^{\prime}}\left(\frac{m}{4}\sum_{i=1}^{m}\ln\left\|g_{m}^{i}u_{j}-u_{j^{\prime}}\right\|^{2}\right)\\\
&=\sum_{j=1}^{n}\left(\frac{m}{4}\sum_{i=1}^{m-1}\ln\left\|g_{m}^{i}u_{j}-u_{j}\right\|^{2}+\frac{m}{2}\sum_{j<j^{\prime}\leq
n}\sum_{i=1}^{m}\ln\left\|g_{m}^{i}u_{j}-u_{j^{\prime}}\right\|^{2}\right).\end{split}$
(3.11)

Substitution of (3.10) and (3.11) into (3.9) and comparing with (3.4) shows
that $H\circ\rho=h$ as required.

(iii) The action is clearly symplectic since the area form on the sphere is
invariant under rotations. The action is free since the action of $SO(2)$ on
$S^{2}$ fixes only the North and South poles and these are removed from
$M_{n}$ (see the definition of $P_{n}$ in (3.3)) if $m\geq 2$. If $m=1$, then
$M_{n}=M$ and the freeness follows since $n=N\geq 3$.

The Hamiltonian vector field of $\phi$ defined by (3.8) with respect to the
symplectic form $\omega_{M_{n}}$ defines the equations

$\dot{u}_{j}=-\frac{1}{m}u_{j}\times\nabla_{u_{j}}\phi(u)=e_{3}\times
u_{j}=J_{3}u_{j},\qquad j=1,\dots,n.$

These equations coincide with the corresponding ones for the infinitesimal
generator of the action corresponding to the matrix $J_{3}$ in the Lie algebra
of the group $SO(2)$, which proves that $\phi$ is indeed the momentum map. It
is immediate to check that $h$ is $SO(2)$-invariant directly from (3.4) using
that $g_{m}\in SO(2)$ and $SO(2)$ is abelian. The preservation of $\phi$ along
the flow of (3.6) and the $SO(2)$-equivariance of these equations follow from
general results for Hamiltonian systems with symmetry (see e.g. [42]).

Next, using the expression of $\Phi$ given in (2.9), and the description of
the set $\mbox{Fix}(K_{m})$ in terms of its connected components given in the
proof of (i) above, it is easy to see that if $v\in\mbox{Fix}(K_{m})$ then
there exists $u=(u_{1},\dots,u_{n})\in M_{n}$ such that

$\Phi(v)=\sum_{j=1}^{n}\sum_{i=1}^{m}g_{m}^{i}u_{j}+\sigma_{p}e_{3},$ (3.12)

where $\sigma_{p}=0$ if $p=0,2$ and $\sigma_{1}=\pm 1$ (the sign depending on
whether $v\in\mbox{Fix}(K_{m})^{+}$ or $v\in\mbox{Fix}(K_{m})^{-}$). Hence,
using the identity

$\sum_{i=1}^{m}g_{m}^{i}=\begin{cases}\mbox{Id}_{3}\quad&\mbox{if}\quad
m=1,\\\ me_{3}e_{3}^{T}\quad&\mbox{if}\quad m\geq 2,\end{cases}$ (3.13)

we obtain $\phi=\Phi_{3}\circ\rho$ if $p=0,2$ and $\phi=\Phi_{3}\circ\rho-1$
if $p=1$.

(iv) For $m\geq 2$, using (3.12) and (3.13) we obtain

$\Phi(v)=\left(m\sum_{j=1}^{n}u_{j}\cdot e_{3}+\sigma_{p}\right)e_{3},$

which is parallel to $e_{3}$ as claimed. ∎

######  Remark 3.6.

Looking ahead at the continuation of RE without symmetries, it is useful to
note how the above discussion specialises for $m=1$. In this case $g_{m}$ is
the $3\times 3$ identity matrix and $\tau_{m}$ is the identity permutation so
$K_{m}$ is the trivial subgroup of $\hat{G}$. It follows that
$\mbox{Fix}(K_{m})=M$, $\rho$ is the identity map on $M$, $h=H$, and the
systems (2.8) and (3.6) coincide.

####  3.2.2 Symmetric relative equilibria: definition and main properties

In view of item (iii) of Theorem 3.5, the reduced system (3.6) is equivariant
with respect to the action of $SO(2)$. Therefore, we may consider existence of
relative equilibria of (3.6) with respect to this action. In analogy with
Definition 2.1, we say that the pair $(b,\omega)\in M_{n}\times\mathbb{R}$ is
a RE of (3.6) if

$u(t)=(e^{\omega J_{3}t}b_{1},\dots,e^{\omega J_{3}t}b_{n}),$

is a solution of (3.6) where $b=(b_{1},\dots,b_{n})$. In analogy with item (i)
of Proposition 2.3, we have:

######  Proposition 3.7.

$(b,\omega)\in M_{n}\times\mathbb{R}$ is a relative equilibrium of (3.6) if
and only if $b\in M_{n}$ is a critical point of the augmented Hamiltonian
$h_{\omega}:=h-\omega\phi:M_{n}\to\mathbb{R}$, where $\phi$ is defined in item
(iii) of Theorem 3.5.

It is straightforward to prove this result proceeding in analogy to the proof
that we presented of Proposition 2.3(i). It also follows by general
considerations since the $SO(2)$ action on $M_{n}$ is symplectic and has
momentum map $\phi$ as stated in Theorem 3.5.

In view of Theorem 3.5, we have $h_{\omega}=H_{\omega}\circ\rho$ (up to
perhaps a constant). Considering that $\rho$ is a diffeomorphism from $M_{n}$
onto a connected component of $\mbox{Fix}(K_{m})$ that we denote
$\mbox{Fix}(K_{m})^{+}$, it follows that the critical points of $H_{\omega}$
in $\mbox{Fix}(K_{m})^{+}$ are in one-to-one correspondence with the critical
points of $h_{\omega}$ in $M_{n}$. Therefore, if $(b,\omega)\in
M_{n}\times\mathbb{R}$ is a RE of (3.6) then $(\rho(b),\omega)\in
M\times\mathbb{R}$ is a $\mathbb{Z}_{m}$-symmetric RE of (2.8). Conversely, if
$(a,\omega)\in M\times\mathbb{R}$ is a $\mathbb{Z}_{m}$-symmetric RE of (2.8)
belonging to $\mbox{Fix}(K_{m})^{+}$, there exists a RE $(b,\omega)\in
M_{n}\times\mathbb{R}$ of (3.6) such that $a=\rho(b)$. Therefore, the search
for $\mathbb{Z}_{m}$-symmetric RE of (2.8) reduces to finding critical points
of $h_{\omega}$. If $m\geq 2$, this is a considerable simplification with
respect to finding critical points of $H_{\omega}$ since the $SO(3)$-symmetry
of $H_{0}$ is broken by our construction and, as a consequence, it will be
possible to apply the implicit function theorem to $dh_{\omega}$ close to
$\omega=0$.

Looking ahead at the implementation, we look for critical points of
$h_{\omega}:M_{n}\to\mathbb{R}$ using the method of Lagrange multipliers. We
write $\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{R}^{n}$ and define
the function

$h_{\omega}^{*}:\mathbb{R}^{3n}\times\mathbb{R}^{n}\to\mathbb{R},\qquad
h_{\omega}^{*}(u,\lambda):=h_{\omega}(u)+m\sum_{j=1}^{n}\lambda_{j}R_{j}(u),$
(3.14)

where $R_{j}(u)=\frac{1}{2}\left(\|u_{j}\|^{2}-1\right)$ and with $h_{\omega}$
in (3.14) interpreted as a function of $u\in\mathbb{R}^{3n}$ in the obvious
way. The method of Lagrange multipliers states that $\hat{u}\in M_{n}$ is a
critical point of $h_{\omega}:M_{n}\to\mathbb{R}$ if and only if there exists
$\hat{\lambda}\in\mathbb{R}^{n}$ such that $(\hat{u},\hat{\lambda})$ is a
critical point of $h_{\omega}^{*}$.

######  Remark 3.8.

The domain of $h_{\omega}^{*}$ should actually be restricted to avoid points
$u\in\mathbb{R}^{3n}$ at which the formula for $h_{\omega}$ is not defined.
These include collisions but also elements $u\in\mathbb{R}^{3n}$ for which a
component $u_{j}\in\mathbb{R}^{3}$ is parallel to $e_{3}$. We have decided to
overlook this detail in the definition of $h_{\omega}^{*}$ to facilitate the
presentation.

####  3.2.3 Symmetric relative equilibria: existence as zeros of a map and
continuation

Suppose that $(b_{0},\omega_{0})\in M_{n}\times\mathbb{R}$ is a RE of (2.5).
We do not assume that $\omega_{0}\neq 0$ so $b_{0}$ could be an equilibrium
point, and in fact, this particular case will be very relevant for us. We want
to determine a branch of RE emanating from $(b_{0},\omega_{0})$, namely, we
want to find RE $(b(\omega),\omega)\in M_{n}\times\mathbb{R}$ parametrised by
$\omega$ close to $\omega_{0}$ and satisfying $b(\omega_{0})=b_{0}$. In view
of the discussion above, such RE correspond to critical points of
$h^{*}_{\omega}$ given by (3.14) so the problem can be reformulated as the
continuation of critical points of $h^{*}_{\omega}$ as a function of $\omega$.
The complication that arises is that, for any value $\omega$, the function
$h_{\omega}^{\ast}$ is invariant under the action of $SO(2)$ on
$\mathbb{R}^{3n}\times\mathbb{R}^{n}$ given by
$g.(u,\lambda)=(gu_{1},\dots,gu_{n},\lambda)$ where $g\in SO(2)$ (and $SO(2)$
is given by (2.5) as usual). Therefore, for a fixed value of $\omega$, a
critical point of $h^{*}_{\omega}$ in fact gives rise to an $SO(2)$-orbit of
critical points. Our strategy to deal with this complication is to isolate an
element of the critical orbit $SO(2).b_{0}$ in order to apply the implicit
function theorem. For this we follow the approach of [48, 28] and consider the
zeros of the augmented map:121212a remark similar to Remark 3.8 applies to the
domain of definition of $F$.

$\begin{split}F:\;&\mathbb{R}^{3n+n+1}\times\mathbb{R}\rightarrow\mathbb{R}^{3n+n+1}\\\
&(u,\lambda,\alpha;\omega)\mapsto(\nabla_{u}h_{\omega}^{\ast}(u,\lambda)+\alpha\mathcal{J}_{3}u,R_{1}(u),...,R_{n}(u),\mathcal{J}_{3}b_{0}\cdot(u-b_{0})),\end{split}$
(3.15)

where $\mathcal{J}_{3}:=\mbox{diag}(J_{3},\dots,J_{3})$ is the $3n\times 3n$
diagonal block matrix with $n$ entries equal to $J_{3}$. We employ the
terminology of [48, 28] and refer to $\alpha\in\mathbb{R}$ as the unfolding
parameter. We also recall from these references that the condition
$\mathcal{J}_{3}b_{0}\cdot(u-b_{0})=0$ requires $u\in\mathbb{R}^{3n}$ to
belong to a Poincaré section of $SO(2).b_{0}$ through $b_{0}$. In particular,
such condition cannot be satisfied for $u\in SO(2).b_{0}$ unless $u=b_{0}$ so
$b_{0}$ is the unique representative of the critical orbit $SO(2).b_{0}$ that
produces a zero of $F$. Finally, note that the conditions $R_{j}(u)=0$
guarantee that $u\in M_{n}$.

The following proposition explains how zeros of $F$ give rise to
$\mathbb{Z}_{m}$-symmetric RE of our problem. The proposition is valid for all
$m\geq 1$.

######  Proposition 3.9.

Let $N\geq 3$. The zeros of the augmented map $F$ defined by (3.15) give rise
to relative equilibria of the $N$-vortex problem consisting of $n$-rings of
regular $m$-gons and $p$-poles. More precisely, if
$F(\hat{u},\hat{\lambda},\hat{\alpha};\hat{\omega})=0$, then $\hat{u}\in
M_{n}$ and $(\hat{u},\hat{\omega})$ is a RE of the reduced system (3.6).
Moreover, if we denote $\hat{v}:=\rho(\hat{u})$ then $\hat{v}\in M$ is
$\mathbb{Z}_{m}$-symmetric and $(\hat{v},\hat{\omega})$ is a
$\mathbb{Z}_{m}$-symmetric RE of the full system (2.8).

Before presenting the proof, we note that

$\mathcal{J}_{3}u\neq 0,\qquad\mbox{for all $u\in M_{n}$.}$ (3.16)

The reason is that $\mathcal{J}_{3}u$ is the value of the infinitesimal
generator at $u$ of the $SO(2)$-action corresponding to the Lie algebra
element $J_{3}$. Such infinitesimal generator cannot vanish since the action
is free (see item (iii) in Theorem 3.5). Condition (3.16) will be used in the
proof below and also ahead in the proof of Theorem 3.10.

###### Proof.

Suppose that
$(\hat{u},\hat{\lambda},\hat{\alpha};\hat{\omega})\in\mathbb{R}^{3n+n+1}\times\mathbb{R}$
belongs to the domain of $F$ and satisfies
$F(\hat{u},\hat{\lambda},\hat{\alpha};\hat{\omega})=0$. The conditions
$R_{i}(\hat{u})=0$ imply that $\|\hat{u}_{i}\|^{2}=1$ for all $i=1,\dots,n$.
Moreover, since $F$ is well-defined at
$(\hat{u},\hat{\lambda},\hat{\alpha};\hat{\omega})$ then
$\hat{u}_{i}\neq\hat{u}_{j}$ for all $i,j=1,\dots,n$. For $m\geq 2$ the well-
definiteness of $F$ also implies that $\hat{u}_{i}\neq(0,0,\pm 1)$ for all
$i=1,\dots,n$. These conditions imply that $\hat{u}\in M_{n}$.

Now note that the vanishing of the first component of
$F(\hat{u},\hat{\lambda},\hat{\alpha};\hat{\omega})$ implies

$\nabla_{u}h_{\hat{\omega}}^{\ast}(\hat{u},\hat{\lambda})+\hat{\alpha}\mathcal{J}_{3}\hat{u}=0.$
(3.17)

We claim that this condition can only hold if $\hat{\alpha}=0$. To see this
note that the $SO(2)$-invariance of $h^{*}_{\hat{\omega}}$ implies

$h^{*}_{\hat{\omega}}(\exp(sJ_{3})\hat{u}_{1},\dots,\exp(sJ_{3})\hat{u}_{n},\hat{\lambda})=h^{*}_{\hat{\omega}}(\hat{u}_{1},\dots,\hat{u}_{n},\hat{\lambda})$

for all $s\in\mathbb{R}$. Differentiating with respect to $s$ and evaluating
at $s=0$ gives
$\nabla_{u}h_{\hat{\omega}}^{\ast}(\hat{u},\hat{\lambda})\cdot\mathcal{J}_{3}\hat{u}=0$.
Therefore, taking the scalar product on both sides of (3.17) with
$\mathcal{J}_{3}\hat{u}$ gives $\hat{\alpha}\|\mathcal{J}_{3}\hat{u}\|^{2}=0$.
But this implies that $\hat{\alpha}=0$ because of (3.16) and since $\hat{u}\in
M_{n}$.

Considering that $\hat{\alpha}=0$, the condition (3.17) simplifies to
$\nabla_{u}h_{\hat{\omega}}^{\ast}(\hat{u},\hat{\lambda})=0$ which together
with the relations $R_{i}(\hat{u})=0$ imply that $(\hat{u},\hat{\lambda})$ is
a critical point of $h_{\hat{\omega}}^{\ast}$. By the Lagrange multiplier
theorem we conclude that $\hat{u}\in M_{n}$ is a critical point of
$h_{\hat{\omega}}:M_{n}\to\mathbb{R}$ and hence, by Proposition 3.7,
$(\hat{u},\hat{\omega})$ is a RE of the reduced system (3.6). The conclusions
about $\hat{v}=\rho(\hat{u})$ follow directly from Theorem 3.5 (see the
discussion after Proposition 3.7). ∎

The following theorem establishes the existence of a unique branch
$(b(\omega),\omega)$ of RE emanating from $(b_{0},\omega_{0})$ as discussed
above, and provides a method to determine it as zeros of the augmented map $F$
defined above (3.15). The theorem requires a non-degeneracy assumption on the
critical orbit $SO(2).b_{0}$ of $h_{\omega_{0}}$ (see Definition 3.1), but
$\omega_{0}$ may be zero.

######  Theorem 3.10.

Let $m\geq 1$, $N\geq 3$, and let $(b_{0},\omega_{0})\in
M_{n}\times\mathbb{R}$ be a RE of the reduced system (3.6) and suppose that
$SO(2).b_{0}$ is a non-degenerate critical manifold of
$h_{\omega_{0}}:M_{n}\to\mathbb{R}$. There exists $\varepsilon>0$ such that
the following statements hold for all
$\omega\in(\omega_{0}-\varepsilon,\omega_{0}+\varepsilon)$:

  1. (i)

there exists a unique $b=b(\omega)\in M_{n}$ with $b(\omega_{0})=b_{0}$ and
depending smoothly on $\omega$ such that $(b(\omega),\omega)$ is a a RE of
(3.6). Furthermore, if we denote $a(\omega):=\rho(b(\omega))$, then
$a(\omega)\in M$ is $\mathbb{Z}_{m}$-symmetric and $(a(\omega),\omega)$ is a
$\mathbb{Z}_{m}$-symmetric RE of the full system (2.8).

  2. (ii)

apart from the existence of $b(\omega)$ with the properties stated above,
there exist unique smooth functions $\lambda=\lambda(\omega)\in\mathbb{R}^{n}$
and $\alpha=\alpha(\omega)\in\mathbb{R}$ such that
$F(b(\omega),\lambda(\omega),\alpha(\omega);\omega)=0$.131313in fact one has
$\alpha(\omega)=0$ as can be concluded from the proof of Proposition 3.9.

###### Proof.

Since $(b_{0},\omega_{0})\in M_{n}\times\mathbb{R}$ is a RE then $b_{0}$ is a
critical point of $h_{\omega_{0}}$ and hence there exists
$\lambda_{0}\in\mathbb{R}^{n}$ such that $(b_{0},\lambda_{0})$ is a critical
point of $h_{\omega_{0}}^{*}$ and, therefore,
$F(b_{0},\lambda_{0},0;\omega_{0})=0$. We will apply the implicit function
theorem to prove (ii). The proof of (i) then follows from Proposition 3.9.

The application of the implicit function theorem requires
$D_{(u,\lambda,\alpha)}F(b_{0},\lambda_{0},0;\omega_{0})$ to be invertible. To
prove that this is indeed the case, suppose that the vector $(\delta
u,\delta\lambda,\delta\alpha)\in\mathbb{R}^{3n}\times\mathbb{R}^{n}\times\mathbb{R}$
belongs to the null-space of
$D_{(u,\lambda,\alpha)}F(b_{0},\lambda_{0},0;\omega_{0})$. Below we show that
necessarily $(\delta u,\delta\lambda,\delta\alpha)=0$.

Denote by $b_{j}^{0}$ the components of $b_{0}$ so that
$b_{0}=(b_{1}^{0},\dots,b_{n}^{0})\in M_{n}$. Similarly, suppose that
$\lambda_{0}=(\lambda_{1}^{0},\dots,\lambda_{n}^{0})\in\mathbb{R}^{n}$, that
$\delta u=(\delta u_{1},\dots,\delta u_{n})\in\mathbb{R}^{3n}$ and
$\delta\lambda=(\delta\lambda_{1},\dots,\delta\lambda_{n})\in\mathbb{R}^{n}$.
One computes

$\begin{split}D_{(u,\lambda,\alpha)}&F(b_{0},\lambda_{0},0;\omega_{0})(\delta
u,\delta\lambda,\delta\alpha)=\\\
&\left(\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\delta
u+(b_{1}^{0}\,\delta\lambda_{1},...,b_{n}^{0}\,\delta\lambda_{n})+\mathcal{J}_{3}b_{0}\delta\alpha\,,\,b_{1}^{0}\cdot\delta
u_{1}\,,\,\dots\,,\,b^{0}_{n}\cdot\delta
u_{n}\,,\,\mathcal{J}_{3}b_{0}\cdot\delta u\right).\end{split}$

Hence, the assumption that $(\delta u,\delta\lambda,\delta\alpha)$ is a null-
vector of $D_{(u,\lambda,\alpha)}F(b_{0},\lambda_{0},0;\omega_{0})$ yields

$\begin{split}&\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\delta
u+(b_{1}^{0}\,\delta\lambda_{1},...,b_{n}^{0}\,\delta\lambda_{n})+\mathcal{J}_{3}b_{0}\delta\alpha=0,\\\
&b^{0}_{j}\cdot\delta u_{j}=0,\qquad j=1,\dots,n,\\\
&\mathcal{J}_{3}b_{0}\cdot\delta u=0.\end{split}$ (3.18)

First note that the embedding $M_{n}\hookrightarrow(\mathbb{R}^{3})^{n}$ leads
to the identification

$T_{b_{0}}M_{n}=\\{{\bf
w}=(w_{1},\dots,w_{n})\in(\mathbb{R}^{3})^{n}\,:\,b^{0}_{j}\cdot
w_{j}=0,\;j=1,\dots,n\,\\},$

so the conditions $b^{0}_{j}\cdot\delta u_{j}=0$ for all $j=1,\dots,n$ imply
$\delta u\in T_{b_{0}}M_{n}$.

Next note that the $SO(2)$-invariance of $h_{\omega_{0}}^{\ast}$ implies
$\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\mathcal{J}_{3}b_{0}=0$,
or, equivalently,
$(\mathcal{J}_{3}b_{0})^{T}\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})=0$.
Therefore, multiplying both sides of the first equation of (3.18) on the left
by $(\mathcal{J}_{3}b_{0})^{T}$ gives

$-\sum_{j=1}^{N}(b_{j}^{0})^{T}J_{3}b_{j}^{0}\delta\lambda_{j}+\|\mathcal{J}_{3}b_{0}\|^{2}\delta\alpha=\|\mathcal{J}_{3}b_{0}\|^{2}\delta\alpha=0,$

which implies that $\delta\alpha=0$ in view of (3.16) since $b_{0}\in M_{n}$.
Therefore, the first equation of (3.18) becomes

$\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\delta
u+(b_{1}^{0}\,\delta\lambda_{1},...,b_{n}^{0}\,\delta\lambda_{n})=0.$ (3.19)

Let ${\bf w}=(w_{1},\dots,w_{n})\in T_{b_{0}}M_{n}$ and multiply both sides of
(3.19) on the left by ${\bf w}^{T}$. Given that
${\bf
w}^{T}(b_{1}^{0}\,\delta\lambda_{1},...,b_{n}^{0}\,\delta\lambda_{n})=\sum_{j=1}^{n}(w_{j}\cdot
b_{j}^{0})\delta\lambda_{j}=\sum_{j=1}^{n}0\delta\lambda_{j}=0$, we obtain

${\bf w}^{T}\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\delta u=0.$

Considering that ${\bf
w}^{T}\nabla_{u}^{2}h_{\omega_{0}}^{\ast}(b_{0},\lambda_{0})\delta
u=d^{2}h_{\omega_{0}}(b_{0})({\bf w},\delta u)$, and that the above equation
holds for arbitrary ${\bf w}\in T_{b_{0}}M_{n}$, we conclude that $\delta
u\in\ker d^{2}h_{\omega_{0}}(b_{0})$. The assumption that $SO(2).b_{0}$ is a
non-degenerate critical manifold of $h_{\omega_{0}}$ then implies that $\delta
u\in T_{b_{0}}(SO(2).b_{0})$. But $T_{b_{0}}(SO(2).b_{0})$ is one-dimensional
and is generated by the vector $\mathcal{J}_{3}b_{0}$, so we may write $\delta
u=c\mathcal{J}_{3}b_{0}$ for some $c\in\mathbb{R}$. But then, using the last
equation in (3.18), we obtain $c\|\mathcal{J}_{3}b_{0}\|^{2}=0$ which implies
$c=0$. Therefore, $\delta u=0$ and in view of (3.19) we obtain

$b_{j}^{0}\,\delta\lambda_{j}=0\qquad j=1,\dots,n.$

Considering that $b_{j}^{0}$ is a unit vector for all $j$, the above
conditions imply
$\delta\lambda=(\delta\lambda_{1},\dots,\delta\lambda_{n})=0$. ∎

######  Remark 3.11.

Recall that when $m=1$ one has $M_{n}=M$ and $h_{\omega}=H_{\omega}$ (up to
perhaps a constant). Also, the systems (2.8) and (3.6) coincide and $\rho$ is
the identity map on $M$. In view of these observations, we conclude that, for
$m=1$, Theorem 3.10 specialises as a continuation result for general (possibly
non-symmetric) RE of the problem. It is however important to notice that the
non-degeneracy hypothesis will never be satisfied if $\omega_{0}=0$ since
$H_{0}=H$ is $SO(3)$-invariant. It could however be satisfied for
$\omega_{0}\neq 0$.

The following corollary establishes the existence of
$\mathbb{Z}_{m}$-symmetric RE arising from $\mathbb{Z}_{m}$-symmetric
equilibria. In contrast with Theorem 3.2, this result is constructive and will
be used to find RE emerging from the equilibria in Table 1.1 as zeros of the
map $F$ according to their $\mathbb{Z}_{m}$-symmetries. Similarly to Remark
3.11, we note that the non-degeneracy hypothesis of the $SO(2)$-orbit in its
statement cannot be satisfied if $m=1$ since in such case $M_{n}=M$ and $h=H$
which is $SO(3)$-invariant. On the other hand, such condition may be satisfied
for $m\geq 2$, since in this case $h_{0}=h$ is only $SO(2)$-invariant and this
is the situation met for the equilibria in Table 1.1 (see Remark 3.13 below).

######  Corollary 3.12.

Let $a_{0}\in M$ be a $\mathbb{Z}_{m}$-symmetric equilibrium of (2.8)
satisfying $a_{0}=\rho(b_{0})$ for a certain $b_{0}\in M_{n}$. By Theorem 3.5,
it follows that $b_{0}$ is an equilibrium of (3.6) and hence a critical point
of $h$. Suppose that $SO(2).b_{0}$ is a non-degenerate critical manifold of
$h:M_{n}\to\mathbb{R}$. Then there exists $\omega_{0}>0$ such that for any
$\omega\in(-\omega_{0},\omega_{0})$ there exists a unique
$\mathbb{Z}_{m}$-symmetric RE $(a(\omega),\omega)$ depending smoothly on
$\omega$ and such that $a(\omega)\to a_{0}$ as $\omega\to 0$. Moreover,
$a(\omega)$ is determined by the condition $a(\omega)=\rho(b(\omega))$ where
$b(\omega)$ satisfies $F(b(\omega),\lambda(\omega),0;\omega)=0$ for a unique
$\lambda=\lambda(\omega)\in\mathbb{R}^{n}$ which depends smoothly on $\omega$.

The proof follows by applying Theorem 3.10 to the RE $(b_{0},\omega_{0})$ with
$\omega_{0}=0$. We remark that the hypothesis that $a_{0}=\rho(b_{0})$ for a
certain $b_{0}\in M_{n}$ is always satisfied, perhaps after a reflection about
the equatorial plane that places any vortices at the poles according to the
ordering of the sets $F_{1}$, $F_{2}$ in (3.5).

######  Remark 3.13.

The hypothesis that $SO(2).b_{0}$ is a non-degenerate critical manifold of
$h:M_{n}\to\mathbb{R}$ is automatically satisfied if $SO(3).a_{0}$ is a non-
degenerate critical manifold of $H:M\to\mathbb{R}$. Except for $N=7$, all
configurations in Table 1.1 are non-degenerate minima of $H:M\to\mathbb{R}$
(see Section 6.1). Therefore, Corollary 3.12 implies the existence of a branch
of $\mathbb{Z}_{m}$-symmetric RE emanating from these equilibria for each
symmetry indicated in Table 1.1 and illustrated in Figures 1.1 and 1.2.

##  4 Nonlinear stability analysis of relative equilibria

We now focus on the nonlinear stability of relative equilibria using the
energy-momentum method described in subsection 2.4.1.

Consider a relative equilibrium $(a,\omega)\in M\times\mathbb{R}$ and recall
from Proposition 2.3 that $a$ is a critical point of the augmented Hamiltonian
$H_{\omega}$ (defined by (2.6)). As explained in subsection 2.4.1, the energy
momentum examines the positiveness of the restriction of $d^{2}H_{\omega}(a)$
to a symplectic slice $\mathcal{N}_{a}$ which is an appropriate subspace of
$T_{a}M$. Throughout this section we identify141414Appendix A reviews several
standard constructions that may be useful to follow the geometric calculations
of this section.

$T_{a}M=V_{a}:=\\{{\bf
w}=(w_{1},\dots,w_{N})\in(\mathbb{R}^{3})^{N}\,:\,a_{j}\cdot
w_{j}=0,\;j=1,\dots,N\,\\},$ (4.1)

where $a=(a_{1},\dots,a_{N})\in M$. The space $V_{a}$ is equipped with the
inner product $\langle\cdot,\cdot\rangle$ which coincides with the restriction
of the standard euclidean product on $(\mathbb{R}^{3})^{N}$. Such inner
product is induced by the Riemannian metric on $S^{2}$ that underlies the
vortex dynamics.151515We recall that vortex dynamics on a surface require the
choice of a metric, which in our case is the standard metric on $S^{2}$. We
refer the reader to [8] for a clear explanation of the role of the geometry of
the surface in the equations of vortex motion.

A general symplectic slice $\mathcal{N}_{a}$ valid for any RE $(a,\omega)$
with $\mu=\Phi(a)\neq 0$ is given in Appendix C. Below we suppose that
$(a,\omega)$ is $\mathbb{Z}_{m}$-symmetric with $m\geq 2$ and construct a
specific one that takes this symmetry into account to give a block
diagonalisation of $\left.d^{2}H_{\omega}\right|_{\mathcal{N}_{a}}$. Our
construction is greatly influenced by previous work of Laurent-Polz, Montaldi
and Roberts [39]. We first give a reformulation of their construction of a
symmetry-adapted basis that is more suitable to the implementation of CAPs in
sections 4.1, 4.2 and 4.3, and then present a further simplification based on
the complex structure of some blocks in section 4.4. Finally, we present a
summary of the block matrices and the stability test in section 4.5. Despite
their technicality, the constructions in this section are very useful to
obtain larger ranges of application of CAPs since they prevent the clustering
of eigenvalues of large matrices. The implementation of CAPs of stability of
branches of RE based on this approach is discussed in section 5.2 and
exemplified in section 6.

###  4.1 Isotypic decomposition of $V_{a}=T_{a}M$ and block diagonalisation of
$d^{2}H_{\omega}(a)$

Assume that the RE $(a,\omega)$ is $\mathbb{Z}_{m}$-symmetric with $m\geq 2$
(Definition 3.4). In this section we consider the associated linear action of
the group $K_{m}\cong\mathbb{Z}_{m}$ (defined in section 3.2.1 above) on
$V_{a}=T_{a}M$, give the isotypic decomposition of $V_{a}$ and explain how it
yields a block diagonalisation of $d^{2}H_{\omega}(a)$.

####  4.1.1 A linear action of $K_{m}\cong\mathbb{Z}_{m}$ on $V_{a}=T_{a}M$

Following the notation of subsection 3.2.1, assume that $a$ is comprised of
$n\geq 1$ rings, each with $m\geq 2$ vortices and $p$ poles ($p$ equal to $0$,
$1$ or $2$) so that $N=mn+p$. Because of the permutation symmetry of the
vortices, we may arrange them in a convenient fashion. We assume that if the
North or South pole are present, then they appear in the last entry (or
entries) of $a$. Moreover, in accordance with (3.7), we assume throughout this
section that

$a=(a_{1,1},\dots,a_{1,m},a_{2,1},\dots,a_{2,m},\dots,\dots,\dots,a_{n,1},\dots,a_{n,m};p_{1},p_{2}).$
(4.2)

In our notation, for $j\in\\{1,\dots,n\\}$ the entries
$(a_{j,1},\dots,a_{j,m})$ denote the location of the $m$ vortices of the
$j^{th}$ ring ordered in an easterly direction so that

$a_{j,k}=g_{m}^{k}u_{j},\qquad j=1,\dots,n,\quad k=1,\dots,m,$ (4.3)

where we recall from (3.1) that $g_{m}=\exp\left(\frac{2\pi}{m}J_{3}\right)$
and

$u_{j}:=(x_{j},y_{j},z_{j})=a_{j,m}\in S^{2},$ (4.4)

is a generator of the $j^{th}$ ring. The entries $p_{1}$ and $p_{2}$ in (4.2)
are the poles (if present).

Under the assumption that $a$ is given as in (4.2), it follows that
$a\in\mbox{Fix}(K_{m})$, where we recall from subsection 3.2.1 that $K_{m}$
denotes the subgroup of $\hat{G}$, isomorphic to $\mathbb{Z}_{m}$, which is
generated by $(\tau_{m},g_{m})\in\hat{G}$ with $\tau_{m}$ given by (3.2). We
simplify the notation and denote

$\kappa:=(\tau_{m},g_{m})\in\hat{G}$

in what follows.

The linearisation of the $K_{m}$ action on $M$ at $a$ defines a linear
$K_{m}$-action on $V_{a}$ which we denote in terms of the representation
$\rho:K_{m}\to GL(V_{a})$ determined by

$\rho(\kappa){\bf
w}=(g_{m}w_{\tau_{m}^{-1}(1)},\dots,g_{m}w_{\tau_{m}^{-1}(N)}),\qquad\forall{\bf
w}=(w_{1},\dots,w_{N})\in V_{a}.$ (4.5)

Such representation is orthogonal with respect to the inner product
$\langle\cdot,\cdot\rangle$ on $V_{a}$, namely

$\langle{\bf w}_{1},{\bf w}_{2}\rangle=\langle\rho(\kappa){\bf
w}_{1},\rho(\kappa){\bf w}_{2}\rangle\quad\forall{\bf w}_{1},{\bf w}_{2}\in
V_{a}.$

Moreover, considering that $H_{\omega}$ is also $K_{m}$-invariant, it follows
that $d^{2}H_{\omega}(a)$ is $K_{m}$-invariant, i.e.,

$d^{2}H_{\omega}(a)(\rho(\kappa){\bf w}_{1},\rho(\kappa){\bf
w}_{2})=d^{2}H_{\omega}(a)({\bf w}_{1},{\bf w}_{2})\quad\forall{\bf
w}_{1},{\bf w}_{2}\in V_{a}.$ (4.6)

####  4.1.2 Isotypic decomposition of $V_{a}$ and block-diagonalisation of
$d^{2}H_{\omega}(a)$

For $j\in\\{1,...,n\\}$ and $k\in\\{1,...,m\\}$ define the vectors
$b_{j,k},\,c_{j,k}\in\mathbb{R}^{3}$ by

$b_{j,k}:=J_{3}a_{j,k},\qquad c_{j,k}:=b_{j,k}\times a_{j,k}.$ (4.7)

Now define the following vectors in $\mathbb{R}^{3N}$ (given below in terms of
block vectors in $\mathbb{R}^{3}$):

$\begin{split}B_{j,k}&=(0,\dots,0,b_{j,k},0,\dots,0;0,0),\quad
j=1,\dots,n,\;k=1,\dots,m,\\\ C_{j,k}&=(0,\dots,0,c_{j,k},0,\dots,0;0,0),\quad
j=1,\dots,n,\;k=1,\dots,m,\\\ \delta x_{1}&=(0,\dots,0;e_{1},0),\\\ \delta
y_{1}&=(0,\dots,0;e_{2},0),\\\ \delta x_{2}&=(0,\dots,0;0,e_{1}),\\\ \delta
y_{2}&=(0,\dots,0;0,e_{2}),\end{split}$ (4.8)

where $e_{1}=(1,0,0)$ and $e_{2}=(0,1,0)$. (The nonzero entry of $B_{j,k}$ and
$C_{j,k}$ occurs at the slot where $a_{j,k}$ appears in the expression (4.2)
for $a$.)

######  Lemma 4.1.

The vectors $B_{j,k}$, $C_{j,k}$, $\delta x_{s}$, $\delta y_{s}$ with
$j\in\\{1,\dots,n\\}$, $k\in\\{1,\dots,m\\}$, $s\in\\{1,\dots,p\\}$ form an
orthogonal basis of $V_{a}$.

###### Proof.

First note that the vectors $B_{j,k},C_{j,k}$ are non-zero and belong to
$V_{a}$ since $b_{j,k}$ and $c_{j,k}$ are non-zero and perpendicular to
$a_{j,k}$. Similarly, the vectors $\delta x_{s},\delta y_{s}\in V_{a}$ because
$e_{1}$ and $e_{2}$ are perpendicular to $p_{s}=\pm e_{3}$. It is also easy to
check that all of these vectors are mutually orthogonal. In particular, they
are linearly independent and since there are $mn+p=\dim M=\dim V_{a}$ of them,
they form a basis. ∎

For the rest of the section we denote

$\zeta:=\frac{2\pi}{m}.$

For $j\in\\{1,\dots,n\\}$, $l\in\\{0,1,\dots,m-1\\}$ we define the complex
vectors $\hat{B}_{j,l},\hat{C}_{j,l}\in V_{a}\oplus iV_{a}$ by

$\hat{B}_{j,l}:=\sum_{k=1}^{m}e^{ilk\zeta}B_{j,k},\qquad\hat{C}_{j,l}:=\sum_{k=1}^{m}e^{ilk\zeta}C_{j,k}.$
(4.9)

It is clear that
$\mbox{Re}(\hat{B}_{j,l}),\mbox{Im}(\hat{B}_{j,l}),\mbox{Re}(\hat{C}_{j,l}),\mbox{Im}(\hat{C}_{j,l})\in
V_{a}$ for any value of the indices $j,l$. It is also useful to notice that
$\hat{B}_{j,l}$ and $\hat{C}_{j,l}$ are real vectors (their imaginary part is
zero) for $l=0$ and also when $l=\frac{m}{2}$ ($m$ even).

For $l=0,1,\dots,\left[\frac{m}{2}\right]$, we define the following _real_
subspaces of $V_{a}$:

$\begin{split}V_{1}&=\bigoplus_{j=1}^{n}\left(\mbox{Re}\,\hat{B}_{j,1}\oplus\mbox{Im}\,\hat{B}_{j,1}\oplus\mbox{Re}\,\hat{C}_{j,1}\oplus\mbox{Im}\,\hat{C}_{j,1}\right)\bigoplus_{s=1}^{p}\left(\delta
x_{s}\oplus\delta y_{s}\right),\\\
V_{l}&=\bigoplus_{j=1}^{n}\left(\mbox{Re}\,\hat{B}_{j,l}\oplus\mbox{Im}\,\hat{B}_{j,l}\oplus\mbox{Re}\,\hat{C}_{j,l}\oplus\mbox{Im}\,\hat{C}_{j,l}\right),\qquad
l=0,2,3\dots,\left[\frac{m}{2}\right].\end{split}$

######  Lemma 4.2.

  1. (i)

The space $V_{a}$ decomposes as a direct sum
$V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$ of mutually orthogonal
subspaces and $d^{2}H_{\omega}(a)$ block diagonalises with respect to this
decomposition. Namely, if ${\bf w}_{1}\in V_{l_{1}}$ and ${\bf w}_{2}\in
V_{l_{2}}$ with $l_{1}\neq l_{2}$ then

$d^{2}H_{\omega}(a)({\bf w}_{1},{\bf w}_{2})=0.$

  2. (ii)

The sets $\tilde{\mathcal{B}}_{l}$ below are orthogonal bases of $V_{l}$.

$\begin{split}&\tilde{\mathcal{B}}_{0}=\left\\{\,\hat{B}_{j,0}\,,\,\hat{C}_{j,0}\,\right\\}_{j=1}^{n},\\\
&\tilde{\mathcal{B}}_{1}=\begin{cases}\left\\{\,\hat{B}_{j,1}\,,\,\hat{C}_{j,1}\,\right\\}_{j=1}^{n}\cup\left\\{\delta
x_{s},\delta y_{s}\right\\}_{s=1}^{p},\qquad&\mbox{if $m=2$},\\\
\left\\{\,\mbox{\em Re }(\hat{B}_{j,1})\,,\,\mbox{\em
Im}(\hat{B}_{j,1}),\mbox{\em Re }(\hat{C}_{j,1}),\mbox{\em
Im}(\hat{C}_{j,1})\,\right\\}_{j=1}^{n}\cup\left\\{\delta x_{s},\delta
y_{s}\right\\}_{s=1}^{p}\qquad&\mbox{if $m>2$},\end{cases}\\\
&\tilde{\mathcal{B}}_{l}=\left\\{\,\mbox{\em Re }(\hat{B}_{j,l})\,,\,\mbox{\em
Im}(\hat{B}_{j,l}),\mbox{\em Re }(\hat{C}_{j,l}),\mbox{\em
Im}(\hat{C}_{j,l})\,\right\\}_{j=1}^{n},\qquad 2\leq
l<\left[\frac{m}{2}\right],\\\
&\tilde{\mathcal{B}}_{\left[\frac{m}{2}\right]}=\begin{cases}\left\\{\hat{B}_{j,\left[\frac{m}{2}\right]},\hat{C}_{j,\left[\frac{m}{2}\right]}\,\right\\}_{j=1}^{n},\qquad&\mbox{if
$m\geq 4$ is even},\\\ \left\\{\,\mbox{\em Re
}(\hat{B}_{j,\left[\frac{m}{2}\right]})\,,\,\mbox{\em
Im}(\hat{B}_{j,\left[\frac{m}{2}\right]}),\mbox{\em Re
}(\hat{C}_{j,\left[\frac{m}{2}\right]}),\mbox{\em
Im}(\hat{C}_{j,\left[\frac{m}{2}\right]})\,\right\\}_{j=1}^{n},\qquad&\mbox{if
$m\geq 5$ is odd.}\end{cases}\end{split}$

In particular, the dimension of the subspaces $V_{l}$ is as indicated in Table
4.1.

| $\dim V_{0}$ | $\dim V_{1}$ | $\dim V_{l},\;2\leq l<\left[\frac{m}{2}\right]$ | $\dim V_{\left[\frac{m}{2}\right]}$
---|---|---|---|---
$m=2$ | $2n$ | $2n+2p$ | - | -
$m=3$ | $2n$ | $4n+2p$ | - | -
$m\geq 4\quad\mbox{ even}$ | $2n$ | $4n+2p$ | $4n$ | $2n$
$m\geq 5\quad\mbox{ odd}$ | $2n$ | $4n+2p$ | $4n$ | $4n$
Table 4.1: Dimension of the subspaces of the isotypic decomposition
$V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$ according to the
values of $m$, $n$ and $p$.

###### Proof.

The mutual orthogonality of the spaces $V_{l}$ is easily established using the
definitions (4.8) of $B_{j,k}$, $C_{j,k}$, $\delta x_{s}$, $\delta y_{s}$, and
(4.9) of $\hat{B}_{j,l}$, $\hat{C}_{j,l}$. Moreover, the vectors $B_{j,k}$,
$C_{j,k}$, $\delta x_{s}$, $\delta y_{s}$ also form an orthogonal basis of the
complex vector space $V_{a}+iV_{a}$ equipped with the restriction of the
standard Hermitian inner product on $\mathbb{C}^{3N}$. By standard results of
the discrete Fourier transform, it follows that the vectors $\hat{B}_{j,l}$,
$\hat{C}_{j,l}$, $j=1,\dots,n$, $l=0,\dots,m-1$, together with $\delta x_{s}$,
$\delta y_{s}$, $s=1,2$, are also an orthogonal basis of $V_{a}+iV_{a}$. Now
let ${\bf w}\in V_{a}$ (a real vector). There exist complex scalars
$\beta_{j,l},\gamma_{j,l},\lambda_{s},\mu_{s}\in\mathbb{C}$ such that

${\bf
w}=\sum_{l=0}^{m-1}\left(\sum_{j=1}^{n}\beta_{j,l}\hat{B}_{j,l}+\gamma_{j,l}\hat{C}_{j,l}\right)+\sum_{s=1}^{p}\lambda_{s}\delta
x_{s}+\mu_{s}\delta y_{s}.$

Denote by $\overline{z}$ the complex conjugation of $z$ (which may be a scalar
or a vector). Using $\overline{{\bf w}}={\bf w}$ together with the relations
$\overline{\hat{B}_{j,l}}=\hat{B}_{j,m-l}$,
$\overline{\hat{C}_{j,l}}=\hat{C}_{j,m-l}$, $\overline{\delta x_{s}}=\delta
x_{s}$, $\overline{\delta y_{s}}=\delta y_{s}$, gives

${\bf
w}=\sum_{l=0}^{m-1}\left(\sum_{j=1}^{n}\overline{\beta_{j,l}}\hat{B}_{j,m-l}+\overline{\gamma_{j,l}}\hat{C}_{j,m-l}\right)+\sum_{s=1}^{p}\overline{\lambda_{s}}\delta
x_{s}+\overline{\mu_{s}}\delta y_{s}.$

Therefore, $\overline{\beta_{j,l}}=\beta_{j,m-l}$,
$\overline{\gamma_{j,l}}=\gamma_{j,m-l}$, and
$\overline{\lambda_{s}}=\lambda_{s}$, $\overline{\mu_{s}}=\mu_{s}$, and we can
write

${\bf
w}=\sum_{l=0}^{\left[\frac{m}{2}\right]}\left(\sum_{j=1}^{n}\beta_{j,l}\hat{B}_{j,l}+\overline{\beta_{j,l}}\overline{\hat{B}_{j,l}}+\gamma_{j,l}\hat{C}_{j,l}+\overline{\gamma_{j,l}}\overline{\hat{C}_{j,l}}\right)+\sum_{s=1}^{p}\lambda_{s}\delta
x_{s}+\mu_{s}\delta y_{s}.$

This shows that the real and imaginary parts of the vectors $\hat{B}_{j,l}$
and $\hat{C}_{j,l}$ for $l=0,1,\dots\left[\frac{m}{2}\right]$ and
$j=1,\dots,n$, together with $\delta x_{s}$, $\delta y_{s}$, $s=1,2$, form a
basis of the real space $V_{a}$. In particular this proves that
$V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$ as required. Moreover,
recalling that $\hat{B}_{j,l}$ and $\hat{C}_{j,l}$ are real vectors for $l=0$
and $l=\frac{m}{2}$, $m$ even, this also proves item (ii).

On the other hand, we claim that

$\begin{split}&\rho(\kappa)\hat{B}_{j,l}=e^{-il\zeta}\hat{B}_{j,l}\qquad\rho(\kappa)\hat{C}_{j,l}=e^{-il\zeta}\hat{C}_{j,l},\qquad
j=1,\dots,n,\;l=0,\dots,\left[\frac{m}{2}\right],\\\ &\rho(\kappa)\left(\delta
x_{s}+i\delta y_{s}\right)=e^{-i\zeta}\left(\delta x_{s}+i\delta
y_{s}\right),\qquad s=1,\dots,p,\end{split}$ (4.10)

where the identities are interpreted as equalities between real and imaginary
parts of both sides of the equations.

To prove (4.10) start by noticing that, in view of (4.3) and (4.7), we have
$e^{J_{3}\zeta}b_{j,k}=b_{j,k+1}$ and $e^{J_{3}\zeta}c_{j,k}=c_{j,k+1}$ (with
the index $k$ taken modulo $m$). These identities together with (4.5) imply

$\rho(\kappa)B_{j,k}=B_{j,k+1},\qquad\rho(\kappa)C_{j,k}=C_{j,k+1},$

where, again, the index $k$ is taken modulo $m$. As a consequence,

$\rho(\kappa)\hat{B}_{j,l}=\sum_{k=1}^{m}e^{ilk\zeta}B_{j,k+1}=e^{-il\zeta}\sum_{k=1}^{m}e^{il(k+1)\zeta}B_{j,k+1}=e^{-il\zeta}\hat{B}_{j,l}~{},$

and similarly $\rho(\kappa)\hat{C}_{j,l}=e^{-il\zeta}\hat{C}_{j,l}$, as
stated. The other identities in (4.10) follow from

$e^{J_{3}\zeta}\,e_{1}=\cos\zeta\,e_{1}+\sin\zeta\,e_{2},\qquad
e^{J_{3}\zeta}\,e_{2}=-\sin\zeta\,e_{1}+\cos\zeta\,e_{2},$

and the fact that the permutation $\tau_{m}$ fixes the last entries where the
poles appear according to our convention (see the definition of $\tau_{m}$ in
(3.2)).

Equations (4.10) show that the subspaces $V_{l}$ are subrepresentations of
$K_{m}$. In fact, $V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$ is
the $K_{m}$-isotypic decomposition of $V_{a}$. The block diagonalization of
$d^{2}H_{\omega}$ with respect to this decomposition is then a consequence of
the well-known Schur’s lemma. ∎

###  4.2 Construction of a symmetry-adapted basis of a symplectic slice
$\mathcal{N}_{a}$

We will now construct a symplectic slice $\mathcal{N}_{a}$ (i.e. a vector
subspace of $V_{a}$ satisfying (2.11)) by specifying its components with
respect to the isotypic decomposition of $V_{a}$ given in Lemma 4.2. For this
matter we first investigate the position of the subspaces $\ker d\Phi(a)$ and
$\mathfrak{so}(3)_{\mu}\cdot a$ with respect to the isotypic decomposition of
$V_{a}$. This is respectively done in subsections 4.2.1 and 4.2.2. Using this
information, we proceed to define the sought symplectic slice
$\mathcal{N}_{a}$ in subsection 4.2.3.

####  4.2.1 Position of $\ker(d\Phi(a))$ relative to the isotypic
decomposition of $V_{a}$ of Lemma 4.2

Considering that $\Phi:M\to\mathbb{R}^{3}$ is given by
$\Phi(v)=v_{1}+\dots+v_{N}$, its derivative at $a$ is given by

$d\Phi(a):V_{a}\to\mathbb{R}^{3},\qquad d\Phi(a)({\bf w})=w_{1}+\dots+w_{N},$
(4.11)

where ${\bf w}=(w_{1},\dots,w_{N})\in V_{a}$. Therefore,

$\ker d\Phi(a)=\left\\{{\bf w}=(w_{1},\dots,w_{N})\in
V_{a}\,:\,w_{1}+\dots+w_{N}=0\right\\}.$ (4.12)

If $N\geq 3$ then $d\Phi(a)$ is onto and hence $\ker d\Phi(a)$ is a subspace
of $V_{a}$ of codimension 3. The next proposition indicates the position of
$\ker d\Phi(a)$ relative to the isotypic decomposition of $V_{a}$. In its
statement, and in what follows, we find it convenient to denote

$\eta_{j}:=x_{j}-iy_{j}\in\mathbb{C},\qquad j=1,\dots,n,$ (4.13)

where we recall that $u_{j}=(x_{j},y_{j},z_{j})$ is the generator of the
$j^{th}$ ring (Eq (4.4)).

######  Proposition 4.3.

Consider the decomposition
$V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$ established in Lemma
4.2. The following statements hold.

  1. (i)

$V_{l}\subset\ker d\Phi(a)$ for all $l\geq 2$.

  2. (ii)

$V_{0}\cap\ker d\Phi(a)$ is a codimension 1 subspace of $V_{0}$ (i.e.
$\dim(V_{0}\cap\ker d\Phi(a))=\dim V_{0}-1$) and a basis for $V_{0}\cap\ker
d\Phi(a)$ is given by

$\tilde{\tilde{\mathcal{B}_{0}}}:=\left\\{\hat{B}_{j,0}\right\\}_{j=1}^{n}\,\bigcup\,\left\\{|\eta_{1}|^{2}\hat{C}_{j,0}-|\eta_{j}|^{2}\hat{C}_{1,0}\right\\}_{j=2}^{n}.$
(4.14)

  3. (iii)

$V_{1}\cap\ker d\Phi(a)$ is a codimension 2 subspace of $V_{1}$ (i.e.
$\dim(V_{1}\cap\ker d\Phi(a))=\dim V_{1}-2$) and a basis of $V_{1}\cap\ker
d\Phi(a)$ is given by
$\mathcal{B}_{1}:=\mathcal{B}^{(1)}_{1}\cup\mathcal{B}^{(2)}_{1}\cup\mathcal{B}^{(3)}_{1}\cup\mathcal{B}^{(4)}_{1}$
where

    1. $\bullet$

If $m\geq 3$ we define

$\begin{split}&\mathcal{B}^{(1)}_{1}:=\left\\{\,\mbox{\em
Re}(z_{1}\hat{B}_{1,1}+i\,\hat{C}_{1,1}),\mbox{\em
Im}(z_{1}\hat{B}_{1,1}+i\,\hat{C}_{1,1})\,\right\\},\\\
&\mathcal{B}^{(2)}_{1}:=\left\\{\,\mbox{\em
Re}(\eta_{j}\hat{B}_{1,1}-\eta_{1}\hat{B}_{j,1})\,,\,\mbox{\em
Im}(\eta_{j}\hat{B}_{1,1}-\eta_{1}\hat{B}_{j,1})\,\right\\}_{j=2}^{n},\\\
&\mathcal{B}^{(3)}_{1}:=\left\\{\,\mbox{\em
Re}(z_{j}\eta_{j}\hat{B}_{1,1}+i\eta_{1}\hat{C}_{j,1})\,,\,\mbox{\em
Im}(z_{j}\eta_{j}\hat{B}_{1,1}+i\eta_{1}\hat{C}_{j,1})\,\right\\}_{j=2}^{n},\\\
&\mathcal{B}^{(4)}_{1}:=\left\\{\mbox{\em
Re}\left(2\hat{B}_{1,1}+im\eta_{1}(\delta x_{s}+i\delta
y_{s})\right)\,,\,\mbox{\em Im}\left(2\hat{B}_{1,1}+im\eta_{1}(\delta
x_{s}+i\delta y_{s})\right)\,\right\\}_{s=1}^{p}.\end{split}$ (4.15)

    2. $\bullet$

If $m=2$ we take instead

$\begin{split}&\mathcal{B}^{(1)}_{1}:=\left\\{z_{1}\mbox{\em
Re}(\eta_{1}\bar{\eta_{j}})\hat{B}_{1,1}-z_{1}|\eta_{1}|^{2}\hat{B}_{j,1}-\mbox{\em
Im}(\eta_{1}\bar{\eta_{j}})\hat{C}_{1,1}\right\\}_{j=2}^{n},\\\
&\mathcal{B}^{(2)}_{1}:=\left\\{z_{1}z_{j}\mbox{\em
Im}(\eta_{1}\bar{\eta_{j}})\hat{B}_{1,1}-z_{1}|\eta_{1}|^{2}\hat{C}_{j,1}+z_{j}\mbox{\em
Re}(\eta_{1}\bar{\eta_{j}})\hat{C}_{1,1}\right\\}_{j=2}^{n},\\\
&\mathcal{B}^{(3)}_{1}:=\left\\{z_{1}\mbox{\em
Im}(\eta_{1})\hat{B}_{1,1}+\mbox{\em
Re}(\eta_{1})\hat{C}_{1,1}-2z_{1}|\eta_{1}|^{2}\delta
x_{s}\,\right\\}_{s=1}^{p},\\\
&\mathcal{B}^{(4)}_{1}:=\left\\{\,z_{1}\mbox{\em
Re}(\eta_{1})\hat{B}_{1,1}-\mbox{\em
Im}(\eta_{1})\hat{C}_{1,1}-2z_{1}|\eta_{1}|^{2}\delta
y_{s}\,\right\\}_{s=1}^{p}.\end{split}$ (4.16)

The proof of Proposition 4.3 requires knowledge of the value of $d\Phi(a)$
acting on the basis vectors of the subspaces $V_{l}$ of Lemma 4.2. This
information is contained in the following proposition.

######  Proposition 4.4.

The following identities hold

$\begin{split}&d\Phi(a)\hat{B}_{j,l}=0,\qquad
l=0,2,\dots\left[\frac{m}{2}\right],\quad j=1,\dots,n,\\\
&d\Phi(a)\hat{B}_{j,1}=\begin{cases}2(-y_{j},x_{j},0)\qquad&m=2,\quad
j=1,\dots,n,\\\ -i\frac{m}{2}\eta_{j}\left(1,i,0\right),\qquad&m\geq 3,\quad
j=1,\dots,n,\end{cases}\\\ &d\Phi(a)\hat{C}_{j,0}=-m|\eta_{j}|^{2}e_{3},\qquad
j=1,\dots,n,\\\
&d\Phi(a)\hat{C}_{j,1}=\begin{cases}2z_{j}(x_{j},y_{j},0),\qquad&m=2,\quad
j=1,\dots,n,\\\ \frac{m}{2}z_{j}\eta_{j}(1,i,0),\qquad&m\geq 3,\quad
j=1,\dots,n,\end{cases}\\\ &d\Phi(a)\hat{C}_{j,l}=0,\qquad j=1,\dots,n,\quad
l=2,\dots,\left[\frac{m}{2}\right],\\\ &d\Phi(a)(\delta x_{s}+i\delta
y_{s})=(1,i,0).\end{split}$ (4.17)

###### Proof.

We will make use of the following formula which can be verified using standard
trigonometric identities

$e^{ikl\zeta}e^{kJ_{3}\zeta}=\frac{1}{2}\left(Ae^{k(l-1)J_{3}\zeta}+\overline{A}e^{k(l+1)J_{3}\zeta}\right)+e^{ikl\zeta}e_{3}e_{3}^{T},\qquad
A=\left(\begin{array}[]{ccc}1&-i&0\\\ i&1&0\\\ 0&0&0\end{array}\right).$
(4.18)

We will also use the following identities which follow from geometric series
calculations

$\small\sum_{k=1}^{m}e^{k(l-1)J_{3}\zeta}=\begin{cases}m\mathbb{I}_{3},\quad&l=1,\\\
me_{3}e_{3}^{T},\quad&l=0,2,\dots,\left[\frac{m}{2}\right]\end{cases},\quad\sum_{k=1}^{m}e^{k(l+1)J_{3}\zeta}=\begin{cases}2\mathbb{I}_{3},\quad&m=2,\,l=1,\\\
me_{3}e_{3}^{T},\quad&\begin{array}[]{l}\mbox{in any other case},\\\ 0\leq
l\leq\left[\frac{m}{2}\right],\end{array}\end{cases}$ (4.19)

where $\mathbb{I}_{3}$ denotes the $3\times 3$ identity matrix.

We begin by noticing that

$d\Phi(a)B_{j,k}=b_{j,k}=J_{3}a_{j,k}=J_{3}e^{kJ_{3}\zeta}u_{j}.$

Therefore, using (4.18) we find

$\begin{split}d\Phi(a)\hat{B}_{j,l}&=\sum_{k=1}^{m}e^{ikl}J_{3}e^{kJ_{3}\zeta}u_{j}\\\
&=\frac{1}{2}J_{3}A\left(\sum_{k=1}^{m}e^{k(l-1)J_{3}\zeta}\right)u_{j}+\frac{1}{2}J_{3}\overline{A}\left(\sum_{k=1}^{m}e^{k(l+1)J_{3}\zeta}\right)u_{j}+\left(\sum_{k=1}^{m}e^{ikl\zeta}\right)J_{3}e_{3}e_{3}^{T}u_{j}.\end{split}$

Using $J_{3}e_{3}=0$ it is seen that the last term on the right vanishes. On
the other hand, for $l\neq 1$ the other two terms also vanish in view of
(4.19) and since $Ae_{3}=\overline{A}e_{3}=0$. Finally, for $l=1$ we must
distinguish the cases $m=2$ and $m\geq 3$. In the former case we have in view
of (4.19),

$\begin{split}d\Phi(a)\hat{B}_{j,1}=J_{3}(A+\overline{A})u_{j}=2(-y_{j},x_{j},0).\end{split}$

For $m\geq 3$ using again (4.19) and $\overline{A}e_{3}=0$, we have instead,

$\begin{split}d\Phi(a)\hat{B}_{j,1}=\frac{m}{2}J_{3}Au_{j}+\frac{m}{2}J_{3}\overline{A}e_{3}e_{3}^{T}u_{j}=\frac{m}{2}(-i\eta_{j},\eta_{j},0)=-i\frac{m}{2}\eta_{j}(1,i,0).\end{split}$

The above calculations show that all given formulas for
$d\Phi(a)\hat{B}_{j,l}$ in (4.17) indeed hold. In order to prove those for
$d\Phi(a)\hat{C}_{j,l}$ we proceed analogously. We first notice that

$\begin{split}d\Phi(a)C_{j,k}&=c_{j,k}=\left(J_{3}a_{j,k}\right)\times
a_{j,k}=\left(J_{3}e^{kJ_{3}\zeta}u_{j}\right)\times\left(e^{kJ_{3}\zeta}u_{j}\right)=e^{kJ_{3}\zeta}\left(J_{3}u_{j}\right)\times
u_{j}=e^{kJ_{3}\zeta}(z_{j}u_{j}-e_{3})\\\
&=z_{j}e^{kJ_{3}\zeta}u_{j}-e_{3}.\end{split}$

Therefore,

$\begin{split}d\Phi(a)\hat{C}_{j,l}&=z_{j}\sum_{k=1}^{m}e^{ikl\zeta}e^{kJ_{3}\zeta}u_{j}-\sum_{k=1}^{m}e^{ikl\zeta}e_{3}.\end{split}$

Using (4.18) we obtain

$d\Phi(a)\hat{C}_{j,l}=\frac{z_{j}}{2}A\left(\sum_{k=1}^{m}e^{k(l-1)J_{3}\zeta}\right)u_{j}+\frac{z_{j}}{2}\overline{A}\left(\sum_{k=1}^{m}e^{k(l+1)J_{3}\zeta}\right)u_{j}+(z_{j}^{2}-1)\left(\sum_{k=1}^{m}e^{ikl\zeta}\right)e_{3}.$

The last term on the right hand side equals
$m(z_{j}^{2}-1)e_{3}=-m|\eta_{j}|^{2}e_{3}$ if $l=0$, and it instead vanishes
if $1\leq l\leq\left[\frac{m}{2}\right]$. Indeed, this follows from

$\sum_{k=1}^{m}e^{ikl\zeta}=\begin{cases}m&\mbox{if $l=0$},\\\ 0&\mbox{if
$1\leq l\leq\left[\frac{m}{2}\right]$}\end{cases}.$

Thus, in view of (4.19), and since $Ae_{3}=\overline{A}e_{3}=0$, we conclude
that the given formulas for $d\Phi(a)\hat{C}_{j,l}$ indeed hold for
$l=0,2,\dots,\left[\frac{m}{2}\right]$. We now treat the case $l=1$. Using
again (4.19) and $\overline{A}e_{3}=0$, we find that for $m\geq 3$ we have

$d\Phi(a)\hat{C}_{j,1}=\frac{m}{2}z_{j}Au_{j}=\frac{m}{2}z_{j}\eta_{j}\left(1,i,0\right).$

On the other hand, if $m=2$ we obtain, again in view of (4.19),

$d\Phi(a)\hat{C}_{j,1}=z_{j}(A+\overline{A})u_{j}=2z_{j}\left(x_{j},y_{j},0\right).$

Finally, it is obvious from their definition, that $d\Phi(a)\delta
x_{s}=e_{1}$ and $d\Phi(a)\delta y_{s}=e_{2}$ which immediately yields the
last identity in (4.17). ∎

Having established the validity of (4.17), we now give a proof of Proposition
4.3.

###### Proof of Proposition 4.3.

(i) Using the bases of $V_{l}$ given in item (ii) of Lemma 4.2, it follows
immediately from (4.17) that $V_{l}\subset\ker d\Phi(a)$ for $l\geq 2$.

(ii) It is clear that $V_{0}$ is not contained in $\ker d\Phi(a)$ since
$d\Phi(a)\hat{C}_{1,0}\neq 0$ and therefore $\dim V_{0}\cap\ker
d\Phi(a)\leq\dim V_{0}-1$. On the other hand, using (4.17), it is easily
verified that the set $\tilde{\tilde{\mathcal{B}_{0}}}$ is contained in
$V_{0}\cap\ker d\Phi(a)$. But, given that $\tilde{\mathcal{B}_{0}}$ as given
in Lemma 4.2 is a basis of $V_{0}$, it is easily seen that
$\tilde{\tilde{\mathcal{B}_{0}}}$ is linearly independent. Considering that it
has $2n-1=\dim V_{0}-1$ elements, it must be a basis of $V_{0}\cap\ker
d\Phi(a)$ and hence the dimension of this space is $\dim V_{0}-1$ as asserted.

(iii) Using that $\ker d\Phi(a)$ is a codimension 3 subspace of $V_{a}$ and
$V_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}V_{l}$, a dimension count,
which takes into account items (i) and (ii), implies that $V_{1}\cap\ker
d\Phi(a)$ is a codimension 2 subspace of $V_{1}$. In view of item (ii) of
Lemma 4.2, the elements of
$\mathcal{B}_{1}=\mathcal{B}_{1}^{(1)}\cup\mathcal{B}_{1}^{(2)}\cup\mathcal{B}_{1}^{(3)}\cup\mathcal{B}_{1}^{(4)}$
(given by either (4.15) or (4.16)) are linearly independent. Moreover, using
again (4.17) one checks that they are contained in $V_{1}\cap\ker d\Phi(a)$.
To finish the proof that $\mathcal{B}_{1}$ is a basis it suffices to do a
count of its elements and compare with Table 4.1. Regardless of the value of
$m$, one sees that the cardinality of $\mathcal{B}_{1}$ equals $\dim V_{1}-2$.
∎

####  4.2.2 Position of $\mathfrak{so}(3)_{\mu}\cdot a$ relative to the
isotypic decomposition of $V_{a}$

As above, let $(a,\omega)\in M\times\mathbb{R}$ be a
$\mathbb{Z}_{m}$-symmetric ($m\geq 2$) RE and denote by
$\mu=\Phi(a)\in\mathbb{R}^{3}$. Recall that $SO(3)_{\mu}$ is the isotropy
group of $\mu\in\mathbb{R}^{3}$ with respect to the standard action of $SO(3)$
on $\mathbb{R}^{3}$, that $\mathfrak{so}(3)_{\mu}$ denotes its Lie algebra,
and $\mathfrak{so}(3)_{\mu}\cdot a\subset T_{a}M=V_{a}$ is the tangent space
to the $SO(3)_{\mu}$-orbit through $a$. The following proposition specifies
the position of $\mathfrak{so}(3)_{\mu}\cdot a$ with respect to the isotypic
decomposition of $V_{a}$ given in Lemma 4.2.

######  Proposition 4.5.

If $\mu\neq 0$ then $\mathfrak{so}(3)_{\mu}\cdot a$ is a 1-dimensional
subspace of $V_{a}$ which is contained in $V_{0}\cap\ker d\Phi(a)$. Moreover,
it is generated by

$s_{a}:=\sum_{j=1}^{n}\hat{B}_{j,0}.$ (4.20)

On the other hand, if $\mu=0$ then $\mathfrak{so}(3)_{0}\cdot a$ coincides
with $\mathfrak{so}(3)\cdot a$ which is a 3-dimensional subspace of $V_{a}$
which satisfies

$\dim\left((\mathfrak{so}(3)\cdot a)\cap V_{0}\cap\ker
d\Phi(a)\right)=1,\qquad\dim\left((\mathfrak{so}(3)\cdot a)\cap V_{1}\cap\ker
d\Phi(a)\right)=2.$

Furthermore, also in this case, the intersection $(\mathfrak{so}(3)\cdot
a)\cap V_{0}\cap\ker d\Phi(a)$ is generated by $s_{a}$ given by (4.20).

###### Proof.

It is easy to see that
$\mathfrak{so}(3)_{\mu}=\\{\xi\in\mathfrak{so}(3)\,:\,\xi\mu=0\\}\subseteq\mathfrak{so}(3)$.
Denoting $a=(a_{1},\dots,a_{N})$, we have

$\mathfrak{so}(3)_{\mu}\cdot a=\\{(\xi a_{1},\dots,\xi
a_{N})\,:\,\xi\in\mathfrak{so}(3)_{\mu}\\}\subset V_{a}.$

Now, using (4.11), for $\xi\in\mathfrak{so}(3)_{\mu}$ we have

$d\Phi(a)(\xi a_{1},\dots,\xi a_{N})=\xi a_{1}+\dots+\xi
a_{N}=\xi(a_{1}+\dots+a_{N})=\xi\mu=0,$

which shows that $\mathfrak{so}(3)_{\mu}\cdot a\subset\ker d\Phi(a)$
regardless of the value of $\mu$.

Suppose that $\mu\neq 0$. Then it is clear that $\mathfrak{so}(3)_{\mu}$ is
1-dimensional and, in virtue of Theorem 3.5(iv), it is generated by $J_{3}$.
In view of the discussion above it follows that $\mathfrak{so}(3)_{\mu}\cdot
a$ is generated by $(J_{3}a_{1},\dots,J_{3}a_{N})\in V_{a}$. Tracing back the
definitions of the vectors $\hat{B}_{j,0}$ it is seen that the vector
$(J_{3}a_{1},\dots,J_{3}a_{N})$ is precisely $s_{a}$. The assertion that
$s_{a}\in V_{0}$ is immediate since it is a linear combination of vectors in
its basis (see Lemma 4.2).

Suppose now that $\mu=0$. It is clear that
$\mathfrak{so}(3)_{0}=\mathfrak{so}(3)$ and $\mathfrak{so}(3)\cdot a$ is
3-dimensional since the $SO(3)$ action on $M$ is free. It is also clear that
in this case $s_{a}\in\mathfrak{so}(3)\cdot a\cap V_{0}$. Thus, to finish the
proof, we only need to show that the intersection $(\mathfrak{so}(3)\cdot
a)\cap V_{1}$ is 2-dimensional. For this matter, we use that
$\mathfrak{so}(3)\cdot a=\\{(\xi a_{1},\dots,\xi
a_{N})\,:\,\xi\in\mathfrak{so}(3)\\}\subset V_{a}$, and take $\xi$ given by
$J_{1},J_{2}\in\mathfrak{so}(3)$ where

$J_{1}:=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
0&1&0\end{pmatrix}\qquad\mbox{and}\qquad J_{2}:=\begin{pmatrix}0&0&1\\\
0&0&0\\\ -1&0&0\end{pmatrix}.$

This leads to the conclusion that the following two independent vectors

$t_{a}^{(1)}:=(e_{1}\times a_{1},\dots,e_{1}\times a_{N}),\qquad
t_{a}^{(2)}:=(e_{2}\times a_{1},\dots,e_{2}\times a_{N}),$

belong to $\mathfrak{so}(3)\cdot a\subset V_{a}$. We will now show that both
$t_{a}^{(1)},t_{a}^{(2)}\in V_{1}$. For this recall the definition of the
linear map $\rho(\kappa):V_{a}\to V_{a}$ given by (4.5). Using the refinement
of the notation of $a$ of (4.2) and the convention (4.3), a simple calculation
(which uses $g_{m}\left((e_{1}+ie_{2})\times
a_{j,k}\right)=\left(g_{m}(e_{1}+ie_{2})\right)\times\left(g_{m}a_{j,k}\right)=e^{-i\zeta}(e_{1}+ie_{2})\times
a_{j,k+1}$, where $k$ is taken modulo $m$) yields

$\rho(\kappa)\left(t_{a}^{(1)}+it_{a}^{(2)}\right)=e^{-i\zeta}\left(t_{a}^{(1)}+it_{a}^{(2)}\right).$

Comparing this with (4.10), it is seen that the vector
$t_{a}^{(1)}+it_{a}^{(2)}\in V_{a}+iV_{a}$ lies in the
$e^{-i\zeta}$-eigenspace of $\rho(\kappa)$. Therefore, $t_{a}^{(1)}$ and
$t_{a}^{(2)}$ belong to the $V_{1}$ component in the isotypic decomposition of
$V_{a}$. ∎

####  4.2.3 Definition of the symplectic slice

Assume that $\mu\neq 0$ and recall from subsection 2.4.1 that a symplectic
slice is any subspace of $\mathcal{N}_{a}\subset V_{a}$ which satisfies $\ker
d\Phi(a)=\mathcal{N}_{a}\oplus(\mathfrak{so}(3)_{\mu}\cdot a)$. Proposition
4.5 implies that, under our assumption that $\mu\neq 0$, we have
$\mathfrak{so}(3)_{\mu}\cdot a=\langle s_{a}\rangle$ and hence
$\mathcal{N}_{a}$ should satisfy

$\ker d\Phi(a)=\mathcal{N}_{a}\oplus\langle s_{a}\rangle.$ (4.21)

In particular, $\mathcal{N}_{a}$ is a $2N-4$-dimensional subspace of $V_{a}$.

We now claim that a symplectic slice can be defined as the direct sum

$\mathcal{N}_{a}:=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}U_{l},$ (4.22)

with the following choice of subspaces $U_{l}$ which are contained in $V_{l}$:

  * •

$U_{l}:=V_{l}$ for all $l\geq 2$,

  * •

$U_{1}:=V_{1}\cap\ker d\Phi(a)$,

  * •

$U_{0}$ is any codimension $1$ subspace of $V_{0}\cap\ker d\Phi(a)$ not
containing $s_{a}$.

Indeed, it is immediate to check, using Propositions 4.3 and 4.5, that (4.21)
indeed holds under the above conditions. For future reference we collect this
information in the following lemma that makes a specific choice of $U_{0}$ by
spelling out a basis. Such choice of $U_{0}$ is arbitrary and not the most
natural from the geometric point of view, but has the benefit of being simple
to implement for our CAPs.

######  Lemma 4.6.

Suppose that $\mu=\Phi(a)\neq 0$. Let $U_{0}\subset V_{a}$ be the subspace
with basis

$\mathcal{B}_{0}:=\left\\{\hat{B}_{j,0}\,,\,|\eta_{1}|^{2}\hat{C}_{j,0}-|\eta_{j}|^{2}\hat{C}_{1,0}\right\\}_{j=2}^{n},$
(4.23)

$U_{1}=V_{1}\cap\ker d\Phi(a)$ and $U_{l}:=V_{l}$ for all $l\geq 2$. Then
$\mathcal{N}_{a}:=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}U_{l}$ is a
symplectic slice satisfying $U_{l}\subset V_{l}$. In particular,
$\dim\mathcal{N}_{a}=2N-4$.

###### Proof.

In view of the discussion before the statement of the lemma, we only need to
show that the subspace $U_{0}$ defined above is a codimension $1$ subspace of
$V_{0}\cap\ker d\Phi(a)$ not containing $s_{a}$. To see this note that
$\mathcal{B}_{0}$ is obtained by removing the vector $\hat{B}_{1,0}$ from the
basis $\tilde{\tilde{\mathcal{B}_{0}}}$ of $V_{0}\cap\ker d\Phi(a)$ given in
(4.14) and, therefore, the vector $s_{a}$ in (4.20) cannot be written as a
linear combination of the vectors in $\mathcal{B}_{0}$. ∎

The dimensions of the spaces $U_{l}$ in terms of the parameters $m$, $n$, $p$
are given in Table 4.2. Recalling that $N=mn+p$, one can verify that in all
cases we indeed have $\dim\mathcal{N}_{a}=2N-4$.

| $\dim U_{0}$ | $\dim U_{1}$ | $\dim U_{l},\;2\leq l<\left[\frac{m}{2}\right]$ | $\dim U_{\left[\frac{m}{2}\right]}$
---|---|---|---|---
$m=2$ | $2n-2$ | $2n+2p-2$ | - | -
$m=3$ | $2n-2$ | $4n+2p-2$ | - | -
$m\geq 4\quad\mbox{even}$ | $2n-2$ | $4n+2p-2$ | $4n$ | $2n$
$m\geq 5\quad\mbox{odd}$ | $2n-2$ | $4n+2p-2$ | $4n$ | $4n$
Table 4.2: Dimension of the subspaces in the decomposition of the symplectic
slice $\mathcal{N}_{a}=\bigoplus_{l=0}^{\left[\frac{m}{2}\right]}U_{l}$
according to the values of $m$, $n$ and $p$ (valid in the case
$\mu=\Phi(a)\neq 0$).

The following lemma provides the necessary modifications to the symplectic
slice that are needed when $\mu=0$.

######  Lemma 4.7.

Suppose that $\mu=\Phi(a)=0$. Let $U_{0},\dots,U_{\left[\frac{m}{2}\right]}$
be defined as in Lemma 4.6 and let $\tilde{U}_{1}\subset U_{1}$ be any
subspace satisfying $U_{1}=\tilde{U}_{1}\oplus((\mathfrak{so}(3)\cdot a)\cap
U_{1})$. Then

$\tilde{\mathcal{N}_{a}}:=U_{0}\oplus\tilde{U}_{1}\bigoplus_{l=2}^{\left[\frac{m}{2}\right]}U_{l},$

is a symplectic slice satisfying $\tilde{U}_{1}\subset V_{1}$ and
$U_{l}\subset V_{l}$ for all other $l\neq 1$. In particular,
$\dim(\tilde{\mathcal{N}_{a}})=2N-6$.

The proof of the lemma is again an immediate consequence of Propositions 4.3
and 4.5 ($\tilde{U}_{1}$ is a codimension 2 subspace chosen such that
$\tilde{\mathcal{N}_{a}}\oplus(\mathfrak{so}(3)\cdot a)=\ker d\Phi(a)$). For
further reference we note that the relation between the symplectic slices
$\tilde{\mathcal{N}_{a}}$ and $\mathcal{N}_{a}$ defined above is

$\mathcal{N}_{a}=\tilde{\mathcal{N}_{a}}\oplus((\mathfrak{so}(3)\cdot a)\cap
U_{1}).$ (4.24)

###  4.3 Block diagonalisation of
$\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$ and construction of the
block matrices

As a consequence of item (i) of Lemma 4.2 and the definition of the spaces
$U_{l}\subset V_{l}$, the symmetric bilinear form
$\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$ block diagonalises with
respect to the symmetric decomposition (4.22).

In order to obtain a matrix representation of each of the blocks it is
necessary to use Lagrange multipliers as explained in Appendix A. Namely,
given that $a=(a_{1},\dots,a_{N})\in M$ is a critical point of the augmented
Hamiltonian $H_{\omega}:M\to\mathbb{R}$, there exist Lagrange multipliers
$(c_{1},\dots,c_{N})\in\mathbb{R}^{N}$ such that $(a,c)$, interpreted as a
point in $(\mathbb{R}^{3})^{N}\times\mathbb{R}^{N}$, is a critical point of
the function

$H^{*}_{\omega}:(\mathbb{R}^{3})^{N}\times\mathbb{R}^{N}\to\mathbb{R},\qquad
H^{*}_{\omega}(x_{1},\dots,x_{N},\lambda_{1},\dots,\lambda_{N})=H_{\omega}(x_{1},\dots,x_{N})+\sum\lambda_{j}R_{j}(x).$

Here $R_{j}(x)=\frac{1}{2}(1-\|x_{j}\|^{2})$ and on the right hand side we
think of $H_{\omega}$ as a function whose domain is (an open subset of )
$(\mathbb{R}^{3})^{N}$ in the natural way. We have,

$d^{2}H_{\omega}(a)({\bf w}_{1},{\bf w}_{2})={\bf
w}_{1}^{T}\nabla_{x}^{2}H_{\omega}^{\ast}(a,c){\bf w}_{2},\qquad\forall{\bf
w}_{1},{\bf w}_{2}\in V_{a},$ (4.25)

where $\nabla_{x}^{2}H^{*}_{\omega}(a,c)$ denotes the standard Hessian matrix
of the function $(\mathbb{R}^{3})^{N}\ni x\mapsto
H_{\omega}^{*}(x,c)\in\mathbb{R}$ evaluated at $a$, and ${\bf w}_{1},{\bf
w}_{2}\in V_{a}\subset(\mathbb{R}^{3})^{N}$ are column vectors (see Appendix
A).

Now, in order to give a matrix form for the restriction of
$d^{2}H_{\omega}(a)$ to the block $U_{l}$ we need a basis for $U_{l}$. A basis
$\mathcal{B}_{0}$ for $U_{0}$ is given by (4.23). For $U_{1}$ it is given by
(4.15) or (4.16) (according to whether $m$ differs or equals $2$). Finally,
considering that for $2\leq l\leq\left[\frac{m}{2}\right]$ one has
$U_{l}=V_{l}$, we can simply set $\mathcal{B}_{l}:=\tilde{\mathcal{B}_{l}}$,
where $\tilde{\mathcal{B}_{l}}$ is the basis of $V_{l}$ given in the statement
of item (ii) of Lemma 4.2.

In view of the above considerations, the block matrix $\mathcal{P}_{l}$
representing $\left.d^{2}H_{\omega}(a)\right|_{U_{l}}$ in the
$\mathcal{B}_{l}$ basis is given by

$\mathcal{P}_{l}=P_{l}^{T}\nabla^{2}_{v}H^{\ast}_{\omega}(a)P_{l},\qquad 0\leq
l\leq\left[\frac{m}{2}\right],$ (4.26)

where $P_{l}$ denotes the real matrix of size $3N\times\dim U_{l}$ whose
columns are the vectors of $\mathcal{B}_{l}$, $0\leq
l\leq\left[\frac{m}{2}\right]$. Note that $\mathcal{P}_{l}$ is a square matrix
of size $\dim U_{l}$, which is specified in Table 4.2 according to the values
of $m$, $n$, $p$.

From the block diagonalisation of
$\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$, it follows that

$\mbox{signature}\left(\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}\right)=\sum_{l=0}^{\left[\frac{m}{2}\right]}\mbox{signature}\left(\mathcal{P}_{l}\right).$
(4.27)

In particular, $\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$ is positive
definite if and only if each block $\mathcal{P}_{l}$ is positive definite for
all $l=0,\dots,\left[\frac{m}{2}\right]$.

###  4.4 Complex structure of some blocks of the diagonalisation of
$\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$.

A further simplification in the calculation of the signature of
$\left.d^{2}H_{\omega}(a)\right|_{\mathcal{N}_{a}}$ is gained by noticing that
some of the block matrices $\mathcal{P}_{l}$ in its diagonalisation have a
complex structure. As a consequence, the signature of $\mathcal{P}_{l}$ is
determined by the signature of a complex Hermitian matrix $\mathcal{Q}_{l}$
which has half the size of $\mathcal{P}_{l}$. This simplification is only
possible when $m\geq 3$, and for the blocks $\mathcal{P}_{l}$ with

$l=\begin{cases}1,\dots,\left[\frac{m}{2}\right],\quad&\mbox{if $m$ is
odd},\\\ 1,\dots,\left[\frac{m}{2}\right]-1,\quad&\mbox{if $m$ is
even}.\end{cases}$ (4.28)

For the rest of this subsection it is assumed that $m\geq 3$.

####  4.4.1 Definition of the Hermitian blocks $\mathcal{Q}_{l}$

Let $Q_{1}$ be the $3N\times(2n+p-1)$ complex matrix whose columns are the
vectors

$\begin{split}&z_{1}\hat{B}_{1,1}+i\,\hat{C}_{1,1},\\\
&\eta_{j}\hat{B}_{1,1}-\eta_{1}\hat{B}_{j,1},\qquad j=2,\dots,n,\\\
&z_{j}\eta_{j}\hat{B}_{1,1}+i\eta_{1}\hat{C}_{j,1},\qquad j=2,\dots,n,\\\
&2\hat{B}_{1}^{1}+im\eta_{1}\left(\delta x_{s}+i\delta y_{s})\right),\qquad
s=1,\dots,p.\end{split}$ (4.29)

On the other hand, for $l>1$ satisfying (4.28) we define $Q_{l}$ as the
$3N\times 2n$ complex matrix whose columns are the vectors

$\begin{split}\hat{B}_{j,l},\;\hat{C}_{j,l}\qquad j=1,\dots,n.\end{split}$
(4.30)

For all $l$ satisfying (4.28) we define the Hermitian matrix

$\mathcal{Q}_{l}:=\bar{Q_{l}}^{T}\nabla^{2}_{v}H^{\ast}_{\omega}(a)Q_{l},$
(4.31)

Note that $\mathcal{Q}_{1}$ is $(2n+p-1)\times(2n+p-1)$ whereas
$\mathcal{Q}_{l}$ is $2n\times 2n$ for all other values of $l$. Comparing with
Table 4.2, it is seen that $\mathcal{Q}_{l}$ has indeed half the dimension of
$\mathcal{P}_{l}$.

####  4.4.2 The signature of $\mathcal{P}_{l}$ is twice the signature of
$\mathcal{Q}_{l}$

The relation between the signature of the complex Hermitian matrices
$\mathcal{Q}_{l}$ and the real symmetric matrices $\mathcal{P}_{l}$ is given
by the following.

######  Theorem 4.8.

Let $l$ as in (4.28) and consider the block matrices $\mathcal{P}_{l}$ and
$\mathcal{Q}_{l}$ defined above. Then

$i_{+}(\mathcal{P}_{l})=2i_{+}(\mathcal{Q}_{l}),\qquad
i_{-}(\mathcal{P}_{l})=2i_{-}(\mathcal{Q}_{l}),\qquad\dim\ker\mathcal{P}_{l}=2\dim\ker\mathcal{Q}_{l},$

where $i_{+}$ and $i_{-}$ denote the positive and negative indices of inertia
of the corresponding matrices. In particular,
$\mbox{signature}(\mathcal{P}_{l})=2\mbox{signature}(\mathcal{Q}_{l})$, and
$\mathcal{P}_{l}$ is positive definite if and only if $\mathcal{Q}_{l}$ is
positive definite.

The proof of Theorem 4.8 relies on a general linear algebra result, which we
state as Lemma 4.9 below and whose proof is given in Appendix D.

######  Lemma 4.9.

Let $V$ be a real vector space of dimension $2d$ and $f:V\times
V\to\mathbb{R}$ a symmetric bilinear form. Suppose that there exists a basis
$\\{\alpha_{1},\beta_{1},\dots,\alpha_{d},\beta_{d}\\}$ of $V$ satisfying

$f(\alpha_{j},\alpha_{k})=f(\beta_{j},\beta_{k}),\qquad
f(\alpha_{j},\beta_{k})=-f(\alpha_{k},\beta_{j}),\qquad\forall j,k=1,\dots,d.$
(4.32)

Let $\mathcal{Q}$ be the $d\times d$ complex matrix with entries

$\mathcal{Q}_{jk}=f(\alpha_{j},\alpha_{k})+f(\beta_{j},\beta_{k})+i\left(f(\alpha_{j},\beta_{k})-f(\beta_{j},\alpha_{k})\right),\qquad\forall
j,k=1,\dots,d.$

Then $\mathcal{Q}$ is Hermitian and

$i_{+}(f)=2i_{+}(\mathcal{Q}),\qquad
i_{-}(f)=2i_{-}(\mathcal{Q}),\qquad\dim\ker f=2\dim\ker\mathcal{Q},$

where $i_{+}$ and $i_{-}$ denote the positive and negative indices of inertia
of the corresponding form/matrix. In particular,
$\mbox{signature}(f)=2\mbox{signature}(\mathcal{Q})$, and $f$ is positive
definite if and only if $\mathcal{Q}$ is positive definite.

######  Remark 4.10.

For our purposes it is convenient to notice that the expression for
$\mathcal{Q}_{jk}$ in the statement of the lemma coincides with the formal
expansion of $f(\alpha_{j}+i\beta_{j},\alpha_{k}+i\beta_{k})$ assuming that
$f$ is a sesquilinear form with the convention that is conjugate-linear in the
first component and linear in the second.

The applicability of Lemma 4.9 to our analysis relies on the following
observation.

######  Proposition 4.11.

Fix $l$ satisfying (4.28) and denote

$\begin{split}&\alpha_{j}:=\mbox{\em
Re}(\hat{B}_{j,l}),\qquad\beta_{j}:=\mbox{\em Im}(\hat{B}_{j,l}),\qquad
j=1,\dots,n,\\\ &\alpha_{n+j}:=\mbox{\em
Re}(\hat{C}_{j,l}),\qquad\beta_{n+j}:=\mbox{\em Im}(\hat{C}_{j,l}),\qquad
j=1,\dots,n.\end{split}$

Then we have

$\begin{split}&d^{2}H_{\omega}(a)\left(\alpha_{j},\alpha_{k}\right)=d^{2}H_{\omega}(a)\left(\beta_{j},\beta_{k}\right),\\\
&d^{2}H_{\omega}(a)\left(\alpha_{j},\beta_{k}\right)=-d^{2}H_{\omega}(a)\left(\beta_{j},\alpha_{k}\right),\qquad\forall
j,k=1,\dots,2n.\end{split}$ (4.33)

Furthermore, for $l=1$ denote

$\alpha_{2n+s}:=\delta x_{s},\qquad\beta_{2n+s}:=\delta y_{s},\qquad
s=1,\dots,p,$

then (4.33) holds on the extended index range $j,k=1,\dots,2n+p$.

###### Proof.

The proof follows by combining the $K_{m}$-invariance of $d^{2}H_{\omega}(a)$
(see (4.6)) with the relations (4.10) which imply

$\rho(\kappa)\alpha_{j}=\cos l\zeta\,\alpha_{j}+\sin
l\zeta\,\beta_{j},\qquad\rho(\kappa)\beta_{j}=-\sin l\zeta\,\alpha_{j}+\cos
l\zeta\,\beta_{j},$

for all $j=1,\dots,2n$ if $l\neq 1$ and all $j=1,\dots,2n+p$ if $l=1$.
Therefore, abbreviating $f:=d^{2}H_{\omega}(a)$, and using its
$K_{m}$-invariance, we have

$f(\alpha_{j},\alpha_{k})=f(\rho(\kappa)\alpha_{j},\rho(\kappa)\alpha_{k})=f(\cos
l\zeta\,\alpha_{j}+\sin l\zeta\,\beta_{j}\,,\,\cos l\zeta\,\alpha_{k}+\sin
l\zeta\,\beta_{k}),$

which by bilinearity of $f$ implies

$\sin^{2}l\zeta f(\alpha_{j},\alpha_{k})=\sin^{2}l\zeta
f(\beta_{j},\beta_{k})+\sin l\zeta\cos
l\zeta\left(f(\beta_{j},\alpha_{k})+f(\alpha_{j},\beta_{k})\right).$ (4.34)

On the other hand, again by $K_{m}$-invariance of $d^{2}H_{\omega}(a)$, we
have

$f(\alpha_{j},\beta_{k})=f(\rho(\kappa)\alpha_{j},\rho(\kappa)\beta_{k})=f(\cos
l\zeta\,\alpha_{j}+\sin l\zeta\,\beta_{j},-\sin l\zeta\,\alpha_{k}+\cos
l\zeta\,\beta_{k}),$

which by bilinearity of $f$ implies

$\sin^{2}l\zeta f(\alpha_{j},\beta_{k})=-\sin^{2}l\zeta
f(\beta_{j},\alpha_{k})+\sin l\zeta\cos
l\zeta\left(f(\beta_{j},\beta_{k})-f(\alpha_{j},\alpha_{k})\right).$ (4.35)

Now recalling that we are working under the assumption that $m\geq 3$, $l$
satisfies (4.28) and that $\zeta=\frac{2\pi}{m}$. These conditions imply that
that $\sin l\zeta\neq 0$ which, via a simple manipulation of (4.34) and (4.35)
proves that

$f(\alpha_{j},\alpha_{k})=f(\beta_{j},\beta_{k}),\qquad
f(\alpha_{j},\beta_{k})=-f(\alpha_{k},\beta_{j}),$

as required. ∎

###### Proof of Theorem 4.8.

For $l>1$ the proof follows at once from Lemma 4.9 and Proposition 4.11.
Indeed, the hypothesis of the lemma are verified in view of (4.33) and it is
easy to check that the Hermitian matrix $\mathcal{Q}$ in the lemma is
precisely the matrix $\mathcal{Q}_{l}$ defined in subsection 4.4.1 above (this
is a consequence of (4.25) and Remark 4.10) .

The conclusion for $l=1$ follows by complementing Lemma 4.9 and Proposition
4.11 with the following observation. Suppose that
$\\{\alpha_{1},\beta_{1},\dots,\alpha_{d},\beta_{d}\\}$ are vectors in
$\mathbb{R}^{D}$ satisfying (4.32) for a certain real bilinear form
$f:\mathbb{R}^{D}\times\mathbb{R}^{D}\to\mathbb{R}$. Let $z_{jk}\in\mathbb{C}$
with $j,k\in\\{1,\dots,d\\}$. Then the vectors $\tilde{\alpha}_{j}$,
$\tilde{\beta}_{j}$ defined by

$\tilde{\alpha}_{j}:=\mbox{Re}\left(\sum_{k}z_{jk}(\alpha_{k}+i\beta_{k})\right),\qquad\tilde{\beta}_{j}:=\mbox{Im}\left(\sum_{k}z_{jk}(\alpha_{k}+i\beta_{k})\right),$

also satisfy (4.32). As a consequence of this observation and Proposition
4.11, it follows that the real and imaginary parts of the vectors (4.29),
which form a basis of $U_{1}$, also satisfy the hypothesis (4.32) of Lemma