# Braids of the $N$-body problem II: carousel solutions
by cabling central configurations

M. Fontaine and C. García-Azpeitia

Abstract. We prove the existence of relative periodic solutions of the planar
$N=\sum_{j=1}^{n}k_{j}$-body problem starting with $n$ bodies moving close to
a non-degenerate central configuration and replacing each of them with
clusters of $k_{j}$ bodies that move close to a small central configuration.
We name these solutions carousel solutions. The proof relies on blow-up
techniques for variational methods used in our previous work [17].

Keywords. $N$-body problem, periodic solutions, perturbation theory.

##  1 Introduction

The existence of braids in the $N$-body problem has been intensively studied
since the pioneering work of Poincaré. In the case of strong forces, the
classical approach exploits the fact that the Euler action functional blows up
at any orbit belonging to the boundary of a braid class, which implies the
existence of minimizers for tied braid classes by using the direct method of
the calculus of variations, see [21], [31] and references therein. Later on,
C. Moore in [32] found braids for gravitational and weak forces by making
numerical continuations of the aforementioned braids for strong forces.

For gravitational forces, the existence of braids in the $3$-body problem
($N=2+1$) was established in the classical setting of the Sun-Earth-Moon
system. These solutions are obtained by replacing one body with two in a
circular motion of the $2$-body problem. There is a large literature related
to this problem [34], [39], [23], [24], [8]. An extension of these solutions
to the $4$-body problem ($N=3+1$), where one body in the Lagrange triangular
configuration is replaced by two, was obtained in [11]. This result was then
extended in [29] to the general case $N=n+1$, where one body in a
nondegenerate central configuration is replaced by two, making use of
symplectic scalings and the Implicit Function Theorem.

In our previous work [17], we established a new approach based on blow-up
techniques to construct solutions of the $N=n+1$-body problem in
$E=\mathbb{R}^{2d}$ for central forces (weak, gravitational and strong), where
one body in a nondegenerate central configuration of $n$ bodies is replaced by
two. Along with such a solution, two of the bodies rotate uniformly around
their center of mass. The other $n-1$ bodies and the center of mass of the
pair remain, at each time, close to a central configuration. When $d=1$, these
solutions are braids obtained by replacing a strand in a braid by a new braid,
a process that was called cabling by C. Moore in [32].

In the present article, we generalize the construction in [17] to replace
several bodies in a non-degenerate central configuration by clusters of bodies
arranged themselves in small central configurations. We name these new
solutions carousel solutions (Figure 1). Due to the higher level of complexity
that involves dealing with multiple clusters, we only treat the case when the
motion takes place in the plane ($d=1$). One may consider symmetry constraints
as in [17] to possibly extend these results to higher even-dimensional spaces.

Figure 1: Carousel solutions of the $12$-body problem. The four bodies in a
polygonal configuration are replaced by clusters of three bodies of equal
masses arranged at the vertices of an equilateral triangle (Lagrange
triangular configuration).

### Model: the $N$-body and $N$-vortex filament problems

We first introduce a multi-index notation to describe the positions of the
bodies in their respective cluster. The positions are given by vectors
$q_{j,k}$ in $E=\mathbb{R}^{2}$ for $k=1,\dots,k_{j}$ and $j=1,\dots,n$. To
each $q_{j,k}$ we attach a positive mass $m_{j,k}>0$. The index $j$ represents
the cluster that contains $k_{j}$ bodies. Setting $N=\sum_{j=1}^{n}k_{j}$, the
equations of motion of the $N$-body problem are

$m_{j,k}\ddot{q}_{j,k}=-\sum_{\begin{subarray}{c}(j^{\prime},k^{\prime})\\\
\left(j^{\prime},k^{\prime}\right)\neq\left(j,k\right)\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\frac{q_{j,k}-q_{j^{\prime},k^{\prime}}}{\|q_{j,k}-q_{j^{\prime},k^{\prime}}\|^{\alpha+1}}\qquad
k=1,\dots,k_{j}\qquad j=1,\dots,n.$ (1.1)

Without loss of generality, we suppose that $k_{j}>1$ for $j=1,\dots,n_{0}$
and $k_{j}=1$ for $j=n_{0}+1,\dots,n$. That is, the $j$-cluster contains only
one body when $j=n_{0}+1,\dots,n$.

The relevant cases from the physical point of view are the gravitational
potential ($\alpha=2$) and the logarithmic potential ($\alpha=1$). In the
latter case, equations (1.1) govern the approximate interaction of $N$ steady
vortex filaments in fluids (Euler equation) [25], [4], [26], [10], Bose-
Einstein condensates (Gross–Pitaevskii equation) and superconductors
(Ginzburg-Landau equation) [13], [9]. Although it is worth mentioning that
there are some subtle differences in the equations for steady vortex filaments
with respect to equations (1.1. Namely, the role of the masses in the models
of filament are played by the circulations, which can be negative, quantized
or weighted by different factors representing the vortex core of filaments.

### Main result: existence of carousel solutions

We construct carousel solutions starting from multiple central configurations.
For $j=1,\dots,n_{0}$ the $j$-cluster is close to a central configuration of
$k_{j}$-bodies $a_{j}=(a_{j,1},\dots,a_{j,k_{j}})$ such that

$m_{j,k}a_{j,k}=\sum_{\begin{subarray}{c}k^{\prime}=1\\\ k^{\prime}\neq
k\end{subarray}}^{k_{j}}m_{j,k}m_{j,k^{\prime}}\frac{a_{j,k}-a_{j,k^{\prime}}}{\left\|a_{j,k}-a_{j,k^{\prime}}\right\|^{\alpha+1}}\qquad
k=1,\dots,k_{j}~{}.$ (1.2)

We assume that each central configuration $a_{j}$ has zero center of mass. For
$j=n_{0}+1,\dots,n$ we assume that the $j$-clusters are made of a single body.

Our main result states that we can replace $n_{0}$ bodies in a central
configuration of $n$-bodies by clusters of $k_{j}$-bodies close to the central
configurations $a_{j}$. For this purpose, we start from a central
configuration $a_{0}=(a_{0,1},\dots,a_{0,n})$ with masses
$M_{j}=\sum_{k=1}^{k_{j}}m_{j,k}$ given by the total masses of the
$j$-clusters for $j=1,...,n_{0}$. This means that

$M_{j}a_{0,j}=\sum_{\begin{subarray}{c}j^{\prime}=1\\\ j^{\prime}\neq
j\end{subarray}}^{n}M_{j}M_{j^{\prime}}\frac{a_{0,j}-a_{0,j^{\prime}}}{\left\|a_{0,j}-a_{0,j^{\prime}}\right\|^{\alpha+1}}\qquad
j=1,\dots,n_{0}.$ (1.3)

We also assume, without loss of generality, that the central configuration
$a_{0}$ has zero center of mass.

In Theorem 3.8, we construct the carousel solutions of (1.1) starting from
$n_{0}$ central configurations $a_{1},\dots,a_{n_{0}}$ satisfying (1.2) and
$a_{0}$ satisfying (1.3). We suppose that $a_{j}$ are $2\pi
p_{j}$-nondegenerate for $p_{1},\dots,p_{n_{0}}\in\mathbb{Z}\backslash\\{0\\}$
fixed integers (Definition 3.3), and also that $a_{0}$ is nondegenerate
(Definition 3.6). We then prove that, for every sufficiently small
$\varepsilon$, there are at least $n_{0}+1$ solutions of the $N$-body problem
(1.1) with components of the form

$\displaystyle q_{j,k}(t)$ $\displaystyle=\exp(tJ)u_{0,j}(\nu
t)+r_{j}\exp(t\omega_{j}J)u_{j,k}(\nu t),\quad j=1,\dots,n_{0},\quad
k=1,\dots,k_{j}$ (1.4) $\displaystyle q_{j,1}(t)$
$\displaystyle=\exp(tJ)u_{0,j}(\nu t),\quad\;j=n_{0}+1,\dots,n.$

The matrix $J$ is a complex structure on $E$. The paths
$(u_{0,1}(s),\dots,u_{0,n}(s))$ and $(u_{j,1}(s),\dots,u_{j,k_{j}}(s))$ remain
$\varepsilon$-close in a space of $2\pi$-periodic paths to the central
configurations $a_{0}$ and $a_{j}$ for $j=1,...,n_{0}$ respectively. The
amplitudes and the frequencies of rotation of the clusters

$r_{j}=\left(1+p_{j}\nu\right)^{-2/(\alpha+1)}\qquad\mbox{and}\qquad\omega_{j}=1+p_{j}\nu$

are controlled uniformly by the parameter $\varepsilon$, where

$\nu=\varepsilon^{-\left(\alpha+1\right)/2}-1$

represents the frequency of the perturbation from the arrangement of the
central configurations in rigid motion. Furthermore, we show that the
solutions are determined by different orientating phases $\vartheta_{j}\in
S^{1}$ such that
$u_{j,k}(s)=\exp(\vartheta_{j}J)a_{j,k}+\mathcal{O}(\varepsilon)$ for
$j=1,...,n_{0}$. Thus the result can be rephrased as

$\displaystyle q_{j,k}(t)$
$\displaystyle=\exp(tJ)a_{0,j}+r_{j}\exp((t\omega_{j}+\vartheta_{j})J)a_{j,k}+\mathcal{O}(\varepsilon)\
\quad j=1,\dots,n_{0},\quad k=1,\dots,k_{j}$ (1.5) $\displaystyle q_{j,1}(t)$
$\displaystyle=\exp(tJ)a_{0,j}+\mathcal{O}(\varepsilon)\quad\;j=n_{0}+1,\dots,n,$

where $\mathcal{O}(\varepsilon)$ are quasi-periodic functions of order
$\varepsilon$. Thus the parameter $\varepsilon$ provides a uniform measure of
the shrink of each cluster because the amplitudes
$r_{j}=\mathcal{O}(\varepsilon)$.

The solutions that we obtain are quasi-periodic if $\nu\notin\mathbb{Q}$. If
the frequency $\nu=p/q$ is rational, the frequencies $\omega_{j}=(q+p_{j}p)/q$
are rational as well. Our theorem implies for this case that, for any fixed
integer $q>0$, there is some integer $p_{0}>0$ such that, for each integer
$p>p_{0}$, the components $q_{j,k}(t)$ are $2\pi q$-periodic. These are braid
solutions, where the centers of mass of the $n$ clusters (close to the central
configuration $a_{0}$) wind around the origin $q$ times, while the
configuration of $k_{j}$ bodies in each cluster winds around its center of
mass $q+p_{j}p$ times (Figure 1). The sign of $p_{j}$ determines whether the
$j$-cluster has a _prograde_ or a retrograde rotation with respect to the
whole system. When the rotation is prograde ($p_{j}>0$), the cluster rotates
in the same direction as the main relative equilibrium. When the rotation is
_retrograde_ ($p_{j}<0$), the cluster rotates in the opposite direction.

Our proof relies on the assumption that the central configurations $a_{j}$ for
$j=1,\dots,n_{0}$ are $2\pi p_{j}$-nondegenerate. In simple terms, the $2\pi
p_{j}$-nondegeneracy condition means that the group orbit of $a_{j}$ is a
nondegenerate critical manifold for the action functional of the $k_{j}$-body
problem defined in the space of $2\pi p_{j}$-periodic paths (Definition 3.3).
In Section 4 we prove that this condition holds for an infinite number of
$k_{j}$-polygonal configurations with equal masses when
$\alpha\in(1,1+\delta)$ for a positive small $\delta$. In the case $\alpha\geq
1$ (but $\alpha\neq 2$) we show that this condition holds for the Lagrange
triangular configuration with different masses provided they satisfy the
relation

$\beta:=27\frac{m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}}{\left(m_{1}+m_{2}+m_{3}\right)^{2}}>9\left(\frac{3-\alpha}{1+\alpha}\right)^{2}.$
(1.6)

We can therefore always replace any body in a nondegenerate central
configuration by regular $k_{j}$-polygons. We conjecture that the $2\pi
p_{j}$-nondegeneracy condition is generic for central configurations in the
case $\alpha\geq 1$ ($\alpha\neq 2$).

When $\alpha=2$, all the central configurations are $2\pi p_{j}$-degenerate
due to the existence of the elliptic homographic solutions which are generated
by a central configuration. This implies that our result cannot be directly
extended to the case $\alpha=2$. In Theorem 3.9 we extend our result to the
gravitational case under two additional assumptions: (i) we can divide only a
central body in a central configuration which is symmetric under
$2\pi/m$-rotations with $m\geq 2$ and (ii) the central configuration $a_{1}$
of the 1-cluster needs to be $2\pi/m$-nondegenerate (Definition 3.5). There
are plenty of central configurations satisfying the first assumption such as
the Maxwell configuration [37], the nested polygonal configurations with a
center appearing in [18] [30] [22]. Regarding the second assumption, we prove
in Section 4 that the $2\pi/m$-nondegeneracy condition of a central
configuration holds for the $k_{1}$-polygonal configurations with equal masses
for $k_{1}=4,...,1000$. We show that this condition also holds true in the
case of the Lagrange triangular configuration when the masses satisfy the
inequality (1.6) with $\alpha=2$. Therefore, we can always replace the central
body in a symmetric central configuration by a regular $k_{1}$-polygon. We
conjecture that the $2\pi/m$-nondegeneracy condition is generic for central
configurations.

### Method: perturbation of nondegenerate critical manifolds

Our method starts by writing down the Euler-Lagrange equations with respect to
the action functional $\mathcal{A}$ of the $N$-body problem. We implement
several changes of coordinates in configuration space that involve Jacobi-like
coordinates, rotating frames and scalings of the amplitudes of the clusters.
We extend the action functional $\mathcal{A}$ to new coordinates

$u=(u_{0},u_{1},...,u_{n})\in E^{n}\times E^{k_{1}}\times...\times E^{k_{n}},$

where the action functional splits into two terms
$\mathcal{A}(u)\mathcal{=A}_{0}(u)+\mathcal{H}(u)$, and the Euler-Lagrange
equations of $\mathcal{A}_{0}(u)$ are uncoupled in the components $u_{j}$ and
are given simply by the $k_{j}$-body problem in $u_{j}$ for $j=1,\dots,n_{0}$
and the $n$-body problem for $u_{0}$. The action of the $N$-body problem
corresponds to the restriction of $\mathcal{A}(u)$ to the subspaces $E_{j}$
defined in (2.4) as the subspace of zero center of mass of the $k_{j}$-body
problem. In particular, we have the convention that $u_{j}=0\in E_{j}$ for the
clusters $j=n_{0}+1,\dots,n$. The coupling term $\mathcal{H=O}(\varepsilon)$
encodes the interactions of the different clusters in the new coordinates,
which depends on $\varepsilon$ through the parameters $r_{j},\omega_{j},\nu$
introduced before and is small of order $\varepsilon$.

The Euler-Lagrange equations of $\mathcal{A}$ are set as a gradient in a
subspace of $2\pi$-periodic paths denoted by $X$. We fix a configuration

$u_{a}=(a_{0},a_{1},\dots,a_{n})$

where $a_{0}$ is a central configuration (1.3) of the $n$-body problem,
$a_{j}$ is a central configuration (1.2) of the $k_{j}$-body problem for
$j=1,\dots,n_{0}$ and, according to our convention, $a_{j}=0$ for
$j=n_{0}+1,...,n$. It turns out that the configuration $u_{a}$ is a critical
point of $\mathcal{A}_{0}$. Furthermore, the functional $\mathcal{A}_{0}(u)$
is invariant under the action of the torus group $G=U(1)^{n+1}$ on $X$ defined
by

$(g_{0},g_{1},\dots,g_{n})\cdot(u_{0},u_{1}\dots,u_{n})=(g_{0}u_{0},g_{1}u_{1},\dots,g_{n}u_{n}).$

where $(g_{0},g_{1},\dots,g_{n})\in U(1)^{n+1}$. The action of $g_{0}$ rotates
the $n$-body problem consisting of the $n$ centers of mass of the clusters
about the origin. The action of each $g_{j}$ rotates the configuration of the
$j$-cluster about its center of mass for $j=1,...,n_{0}$ and acts trivially on
$u_{j}=0$ for $j=n_{0}+1,...,n$.

It follows that the group orbit $G(u_{a})$ is a critical manifold of
$\mathcal{A}_{0}$. The perturbation $\mathcal{H=O}(\varepsilon)$ breaks the
$G$-symmetry, in the sense that it is invariant under the action of the
diagonal subgroup $H=\widetilde{U(1)}$ of $G$. Our theorem is obtained by
proving the persistence of $H$-orbits of critical points of the perturbed
action $\mathcal{A}=\mathcal{A}_{0}+\mathcal{H}$ in a tubular neighbourhood of
the critical manifold $G(u_{a})$ in $X$. The core of the proof (Section $3$)
relies on three Lyapunov-Schmidt reductions: a reduction to finite dimension,
a reduction that regularizes the functional, and a reduction that uses Palais-
slice coordinates near the orbit $U(1)^{n}(a_{1},...,a_{n})$. We conclude that
finding critical points of $\mathcal{A}$ in a neighbourhood of $G(u_{a})$ is
equivalent to doing so for some regular function

$\Psi_{\varepsilon}^{\prime}:U(1)^{n_{0}}\rightarrow\mathbb{R}$

defined on the compact manifold $U(1)^{n_{0}}$ which represents the orbit of
$(a_{1},...,a_{n})$ under the action of the group $U(1)^{n}$. A major
difference regarding our previous work [17] is that to obtain
$\Psi_{\varepsilon}^{\prime}$ we need to solve first the component $u_{0}$.
The delicate part of the procedure is finding uniform estimates in
$\varepsilon$ because the functional $\mathcal{A}_{0}$ explodes when
$\varepsilon\rightarrow 0$ at different scales. The theorem is easily obtained
as a consequence of the fact that the Lyusternik-Schnirelmann category of the
compact manifold $U(1)^{n_{0}}$ is $n_{0}+1$, which gives a lower bound for
the number of critical points of $\Psi_{\varepsilon}^{\prime}$. To each of
these critical points corresponds a different way of orienting the central
configurations $a_{j}$ for $j=1,...,n_{0}$ with respect to the central
configuration $a_{0}$. Generically, from Morse theory, the functional has in
fact $2^{n_{0}}$ critical points.

It is worth mentioning that many authors have analyzed the persistence of
solutions near a nondegenerate critical manifold under perturbation. In our
case, the critical manifold of $\mathcal{A}_{0}$ is the group orbit $G(u_{a})$
and the perturbation $\mathcal{H}$ is $H$-invariant. The perturbation of
nondegenerate critical manifolds consisting of group orbits has been studied
previously in [12], [40], [27], [7], [2], [16], and references therein. In the
context of the $n$-body problem, the papers [33] and [1] analyze the breaking
of symmetries of the critical manifold of periodic solutions of the Kepler
problem when a non-radial external force is introduced. Our work uses ideas
already appearing in all those works, although a remarkable difference is
that, in our case, the functional $\mathcal{A}_{0}$ explodes as
$\varepsilon\rightarrow 0$ at different scales. We solve this problem by means
of a procedure that is similar to our previous work [17] and was motivated by
the blow-up methods appearing in [5] and [6].

Blow-up techniques have a long history. For instance, they were used by Floer
and Weinstein in [15] to find single-soliton standing waves of the nonlinear
Schrödinger equation in dimension one. The critical manifold in that case is
non-compact due to the action of the group $\mathbb{R}$. Later on, these ideas
were used to obtain multi-bump solitons for Schrödinger equations in higher
dimension - the interested reader can consult the large bibliography in [3].
We think that it is possible to use similar methods to extend our results to
the gravitational case by taking into consideration the non-compact critical
manifold of all the elliptic homographic solutions generated by the central
configurations.

##  2 Problem setting for carousels

Let $E=\mathbb{R}^{2}$ with inner product $\langle\cdot,\cdot\rangle$. We
consider $N=\sum_{j=1}^{n}k_{j}$ bodies moving in $E$ under the influence of a
central force field. We use a multi-index notation which simplifies greatly
the treatment of the problem. The positions of the bodies in $E$ are denoted
by $q_{j,k}$ where $k=1,\dots,k_{j}$ and $j=1,\dots,n.$ To each of the
positions we attach positive masses $m_{j,k}>0$. The index $j$ represents the
cluster of bodies that contains $k_{j}$ bodies.

We define the kinetic energy and the potential function by

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}m_{j,k}\|\dot{q}_{j,k}\|^{2}$
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}(j^{\prime},k^{\prime}),\left(j,k\right)\\\
\left(j^{\prime},k^{\prime}\right)\neq\left(j,k\right)\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\phi_{\alpha}(\|q_{j,k}-q_{j^{\prime},k^{\prime}}\|)$

where $\|\dot{q}_{j,k}\|^{2}=\langle\dot{q}_{j,k},\dot{q}_{j,k}\rangle$ and
$\phi_{\alpha}$ is a function such that
$\phi_{\alpha}^{\prime}(r)=-r^{-\alpha}$. The factor $1/2$ in the potential
$U$ appears due to the double sum of the same term. The Newtonian potential
corresponds to $\phi_{2}(r)=1/r$ and the vortex filament potential corresponds
to $\phi_{1}(r)=-\ln(r)$. Newton’s laws of motion are:

$m_{j,k}\ddot{q}_{j,k}=\nabla_{q_{j,k}}U=-\sum_{\begin{subarray}{c}(j^{\prime},k^{\prime})\\\
\left(j^{\prime},k^{\prime}\right)\neq\left(j,k\right)\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\frac{q_{j,k}-q_{j^{\prime},k^{\prime}}}{\|q_{j,k}-q_{j^{\prime},k^{\prime}}\|^{\alpha+1}},\qquad
k=1,\dots,k_{j},\qquad j=1,\dots,n.$

Let $\mathcal{L}=K+U$ be the Lagrangian of the system. The action functional

$\mathcal{A}(q)=\int_{0}^{T}\mathcal{L}(q(t),\dot{q}(t))dt$

is taken over the Sobolev space $H^{1}([0,T],E^{N})$ of paths
$q:[0,T]\rightarrow E^{N}$ such that $q$ and its first derivative $\dot{q}$
are square integrable in the sense of distributions.

###  2.1 Jacobi-like coordinates

Similarly to [17] we make a change of coordinates so that the action
functional splits into two terms $\mathcal{A}_{0}+\mathcal{H}$. The Euler-
Lagrange equations of $\mathcal{A}_{0}$ give rise to a set of uncoupled
$k_{j}$-body problems and an $n$-body problem. This procedure is a
generalization of [17] which allows us to produce carousel solutions of the
$N$-body problem. For this purpose we define new coordinates

$q_{j,k}=Q_{0,j}+Q_{j,k},\quad k=1,\dots,k_{j},\quad j=1,\dots,n~{}$ (2.1)

which are overdetermined in the coordinates $Q_{0,j}$ and $Q_{j,k}$. To
address this matter we choose components $Q_{j,k}$ satisfying the $n$
constraints

$\sum_{k=1}^{k_{j}}m_{j,k}Q_{j,k}=0,\quad j=1,\dots,n.$ (2.2)

Let $Q=\left(Q_{0},Q_{1},\dots,Q_{n}\right),$ where
$Q_{0}=(Q_{0,1},\dots,Q_{0,n})\in E^{n}$ represent the positions of the
centers of mass of the $n$ clusters, and each
$Q_{j}=(Q_{j,1},\dots,Q_{j,k_{j}})\in E^{k_{j}}$ represents the positions of
the $k_{j}$ bodies in the $j$-cluster. Denoting by
$M_{j}=\sum_{k=1}^{k_{j}}m_{j,k}$ the total mass of the $j$-cluster, we use
(2.1) and (2.2) to write the center of mass of the whole system as

$\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}m_{j,k}q_{j,k}=\sum_{j=1}^{n}M_{j}Q_{0,j}\text{.}$
(2.3)

From now on, we impose the centers of mass of the $n$ clusters and the center
of mass of the $n$-body problem to be zero by considering the coordinates such
that

$Q=\left(Q_{0},Q_{1},\dots,Q_{n}\right)\in E_{\mbox{\tiny
red}}^{n+N}:=E_{0}\times E_{1}\times\dots\times E_{n},$

where the subspaces $E_{j}$ are given by

$E_{0}:=\left\\{Q_{0}\in E^{n}:\sum_{j=1}^{n}M_{j}Q_{0,j}=0\right\\},\quad
E_{j}:=\left\\{Q_{j}\in
E^{k_{j}}:\sum_{k=1}^{k_{j}}m_{j,k}Q_{j,k}=0\right\\},$ (2.4)

for $j=1,\dots,n~{}$. Since there is only one body in a $j$-cluster for
$j=n_{0}+1,\dots,n$, the constraint (2.2) with $k_{j}=1$ implies that
$Q_{j}\in E_{j}=\\{0\\}$. Hence the position of a single body is determined by
$q_{j,1}=Q_{0,j}$ with $M_{j}=m_{j,1}$.

Figure 2: The representation of the Jacobi-like coordinates with
$q_{j,k}=Q_{0,j}+Q_{j,k}$ in the case $j=1,2,3,4$ and $k_{j}=1,2,3$ for each
$j$. The symbol $\odot$ denotes the cabling operation of [33].

The Jacobi-like coordinates allow us to write the action functional as
$\mathcal{A}(Q)=\mathcal{A}_{0}(Q)+\mathcal{H}(Q)$.__ It is important to
remark that we consider this functional extended to $H^{1}([0,T],E^{N+n})$,
and that the solutions to the $N$-body problem are the solutions of the action
$\mathcal{A}(Q)$ with the holonomic constraint that $Q\in E_{\mbox{\tiny
red}}^{n+N}$_._ The first term $\mathcal{A}_{0}$ is called the unperturbed
functional. The remaining part $\mathcal{H}$ is a coupling term invariant by
linear isometries which becomes small when the norms $\|Q_{j,k}\|$
simultaneously become small for every $k,j$.

######  Proposition 2.1.

In Jacobi-like coordinates, the action functional becomes

$\mathcal{A}(Q)=\mathcal{A}_{0}(Q)+\mathcal{H}(Q)=\int_{0}^{T}\sum_{j=0}^{n}\mathcal{L}_{j}(Q_{j}(t),\dot{Q}_{j}(t))dt+\int_{0}^{T}h(Q(t))dt,$

with

$\mathcal{L}_{0}(Q_{0},\dot{Q}_{0})=K_{0}(\dot{Q}_{0})+U_{0}(Q_{0})=\frac{1}{2}\sum_{j=1}^{n}M_{j}\|\dot{Q}_{0,j}\|^{2}+\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\
j<j^{\prime}\end{subarray}}^{n}M_{j}M_{j^{\prime}}\phi_{\alpha}\left(\|Q_{0,j}-Q_{0,j^{\prime}}\|\right),$
(2.5)
$\mathcal{L}_{j}(Q_{j},\dot{Q}_{j})=K_{j}(\dot{Q}_{j})+U_{j}(Q_{j})=\frac{1}{2}\sum_{k=1}^{k_{j}}m_{j,k}\|\dot{Q}_{j,k}\|^{2}+\sum_{\begin{subarray}{c}k,k^{\prime}=1\\\
k<k^{\prime}\end{subarray}}^{k_{j}}m_{j,k}m_{j,k^{\prime}}\phi_{\alpha}\left(\|Q_{j,k}-Q_{j,k^{\prime}}\|\right),$
(2.6)

for $j=1,\dots,n$ and

$h(Q)=\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\
j<j^{\prime}\end{subarray}}^{n}\sum_{\begin{subarray}{c}k\in K_{j}\\\
k^{\prime}\in
K_{j^{\prime}}\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\left(\phi_{\alpha}\left(\|\left(Q_{0,j}-Q_{0,j^{\prime}}\right)+\left(Q_{j,k}-Q_{j^{\prime},k^{\prime}}\right)\|\right)-\phi_{\alpha}\left(\|Q_{0,j}-Q_{0,j^{\prime}}\|\right)\right),$
(2.7)

where $K_{j}=\left\\{1,\dots,k_{j}\right\\}$ is the set of indices for the
$j$-cluster.

Proof. It suffices to show that
$\mathcal{L}(Q,\dot{Q})=\sum_{j=0}^{n}\mathcal{L}_{j}(Q_{j},\dot{Q}_{j})+h(Q)$.
We first compute the kinetic energy $K$ in these new coordinates. We get

$K=\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}m_{j,k}\|\dot{q}_{j,k}\|^{2}=\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}m_{j,k}\left(\|\dot{Q}_{0,j}\|^{2}+\|\dot{Q}_{j,k}\|^{2}+2\langle\dot{Q}_{0,j},\dot{Q}_{j,k}\rangle\right).$

Using the constraint (2.2), it follows that
$\sum_{k=1}^{k_{j}}m_{j,k}\langle\dot{Q}_{0,j},\dot{Q}_{j,k}\rangle=0$. Since
the total mass of the $j$-cluster is $M_{j}$, we get

$K=\frac{1}{2}\sum_{j=1}^{n}M_{j}\|\dot{Q}_{0,j}\|^{2}+\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{k_{j}}m_{j,k}\|\dot{Q}_{j,k}\|^{2}.$

The potential in the new coordinates becomes

$\displaystyle U$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{n}\sum_{\begin{subarray}{c}k,k^{\prime}\in
K_{j}\\\ k\neq
k^{\prime}\end{subarray}}m_{j,k}m_{j,k^{\prime}}\phi_{\alpha}\left(\|Q_{j,k}-Q_{j,k^{\prime}}\|\right)$
$\displaystyle+\frac{1}{2}\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\ j\neq
j^{\prime}\end{subarray}}^{n}\sum_{\begin{subarray}{c}k\in K_{j}\\\
k^{\prime}\in
K_{j^{\prime}}\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\phi_{\alpha}\left(\|\left(Q_{0,j}-Q_{0,j^{\prime}}\right)+\left(Q_{j,k}-Q_{j^{\prime},k^{\prime}}\right)\|\right).$

The Lagrangian is of the form
$\mathcal{L}(Q,\dot{Q})=\sum_{j=0}^{n}\mathcal{L}_{j}(Q_{j},\dot{Q}_{j})+h(Q)$,
where

$\displaystyle h(Q)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\ j\neq
j^{\prime}\end{subarray}}^{n}\sum_{\begin{subarray}{c}k\in K_{j}\\\
k^{\prime}\in
K_{j^{\prime}}\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\phi_{\alpha}\left(\|\left(Q_{0,j}-Q_{0,j^{\prime}}\right)+\left(Q_{j,k}-Q_{j^{\prime},k^{\prime}}\right)\|\right)$
$\displaystyle-\frac{1}{2}\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\ j\neq
j^{\prime}\end{subarray}}^{n}M_{j}M_{j^{\prime}}\phi_{\alpha}\left(\|Q_{0,j}-Q_{0,j^{\prime}}\|\right).$

Replacing $M_{j}$ by $\sum_{k\in K_{j}}m_{j,k}$, we obtain (2.7).

###  2.2 Rotating coordinates

We define rotating coordinates by

$\displaystyle Q_{j}(t)$
$\displaystyle=\exp(t\omega_{j}\mathcal{J}_{k_{j}})v_{j}(t),\quad j=1,\dots,n$
$\displaystyle Q_{0}(t)$ $\displaystyle=\exp(t\mathcal{J}_{n})v_{0}(t),$

where $\omega_{j}$ is a frequency of rotation for each cluster of bodies. The
endomorphism $J$ denotes the standard complex structure on $E$ and
$\mathcal{J}_{n}$ denotes the endomorphism $J\oplus\dots\oplus J$ of $E^{n}$.
The components of $v_{0}=(v_{0,1},\dots,v_{0,n})$ correspond to the $n$
positions of the centers of mass of the clusters in a rotating frame of
frequency one. The components of
$v_{j}=\left(v_{j,1},\dots,v_{j,k_{j}}\right)$ correspond to the positions of
the $k_{j}$ bodies in the $j$-cluster in a rotating frame of frequency
$\omega_{j}$. By (2.2) and (2.3) they satisfy the constraints

$\sum_{j=1}^{n}M_{j}v_{0,j}=0\quad\mbox{and}\quad\sum_{k=1}^{k_{j}}m_{j,k}v_{j,k}=0\text{,}$
(2.8)

which corresponds to $v=(v_{0},v_{1},\dots,v_{n})\in E_{\mbox{\tiny
red}}^{n+N}$.

The unperturbed functional

$\mathcal{A}_{0}(Q)=\int_{0}^{T}\sum_{j=0}^{n}\mathcal{L}_{j}(Q_{j}(t),\dot{Q}_{j}(t))dt,$

is invariant under the transformations that rotate simultaneously each of the
coordinates $Q_{j,k}$. The term $\mathcal{A}_{0}(v)$ is obtained by replacing
the coordinates $Q_{j,k}$ in (2.5) and (2.6) by the coordinates $v_{j,k}$. If
we denote by $I$ the identity matrix on $E$, we define the endomorphisms
$\mathcal{M}_{j}\in\mbox{End}(E^{k_{j}})$ and
$\mathcal{M}_{0}\in\mbox{End}(E^{n})$ by

$\mathcal{M}_{j}=m_{j,1}I\oplus\dots\oplus
m_{j,k_{j}}I\quad\mbox{and}\quad\mathcal{M}_{0}=M_{1}I\oplus\dots\oplus
M_{n}I.$

The variations for $\mathcal{A}_{0}$ with respect to $\delta v_{j}$ are

$\displaystyle\frac{\delta\mathcal{A}_{0}}{\delta v_{j}}$
$\displaystyle=\mathcal{M}_{j}\left(\mathcal{I}_{k_{j}}\partial_{t}+\omega_{j}\mathcal{J}_{k_{j}}\right)^{2}v_{j}+\nabla_{v_{j}}U_{j}(v),\quad
j=1,\dots,n,$ (2.9) $\displaystyle\frac{\delta\mathcal{A}_{0}}{\delta v_{0}}$
$\displaystyle=\mathcal{M}_{0}\left(\mathcal{I}_{n}\partial_{t}+\mathcal{J}_{n}\right)^{2}v_{0}+\nabla_{v_{0}}U_{0}(v).$
(2.10)

The potentials

$U_{0}(v)=\sum_{\begin{subarray}{c}j,j^{\prime}=1\\\
j<j^{\prime}\end{subarray}}^{n}M_{j}M_{j^{\prime}}\phi_{\alpha}\left(\|v_{0,j}-v_{0,j^{\prime}}\|\right)\quad\mbox{and}\quad
U_{j}(v)=\sum_{\begin{subarray}{c}k,k^{\prime}=1\\\
k<k^{\prime}\end{subarray}}^{k_{j}}m_{j,k}m_{j,k^{\prime}}\phi_{\alpha}\left(\|v_{j,k}-v_{j,k^{\prime}}\|\right).$

where obtained in Proposition 2.1. The variations (2.9) correspond to
$k_{j}$-body problems in rotating frame, and (2.10) corresponds to an $n$-body
problem in rotating frame.

######  Definition 2.1.

The amended potential $V_{j}:E^{k_{j}}\rightarrow\mathbb{R}$ for the
$k_{j}$-body problem is defined by

$V_{j}(v_{j})=\frac{1}{2}\left\|\mathcal{M}_{j}^{1/2}v_{j}\right\|^{2}+U_{j}(v_{j}).$
(2.11)

A configuration $a_{j}\in E_{j}$ is a central configurations of the
$k_{j}$-body problem (with frequency one and zero center of mass) if it is is
a critical point of $V_{j}$.

A central configuration $a_{j}\in E_{j}$ of the $k_{j}$-body problem satisfies
the equation $\mathcal{M}_{j}a_{j}=\nabla_{v_{j}}U_{j}(a_{j})$. Similarly, a
central configuration $a_{0}\in E_{0}$ of the $n$-body problem satisfies
$\mathcal{M}_{0}a_{0}=\nabla_{v_{0}}U_{0}(a_{0})$. These equations are
equivalent to (1.2) and (1.3).

###  2.3 Time and space scaling

Given a central configuration $a_{j}=(a_{j,1},\dots,a_{j,k_{j}})$ of the
$k_{j}$-body problem, the scaling $v_{j}=r_{j}a_{j}$ is a central
configuration if the frequency $\omega_{j}$ satisfies
$\omega_{j}^{2}=r_{j}^{-(\alpha+1)}$. Indeed

$\mathcal{M}_{j}\left(\mathcal{I}_{k_{j}}\partial_{t}+\omega_{j}\mathcal{J}_{k_{j}}\right)^{2}v_{j}=-\omega_{j}^{2}r_{j}\mathcal{M}_{j}a_{j}=-\omega_{j}^{2}r_{j}^{\alpha+1}\nabla_{v_{j}}U_{j}(v_{j})\text{.}$

For $\nu\in\mathbb{R}$, we define new coordinates
$u_{0}=(u_{0,1},\dots,u_{0,n})$ and $u_{j}=(u_{j,1},\dots,u_{j,k_{j}})$ by
setting

$\displaystyle v_{j}(t)$ $\displaystyle=$ $\displaystyle r_{j}u_{j}(\nu
t),\quad j=1,\dots,n$ $\displaystyle v_{0}(t)$ $\displaystyle=$ $\displaystyle
u_{0}(\nu t).$

We shall rewrite the action functional
$\mathcal{A}=\mathcal{A}_{0}+\mathcal{H}$ with respect to the $u$-coordinates
and find some restrictions on the set of frequencies $\nu$ and $\omega_{j}$ so
that the functional $\mathcal{A}$ is $2\pi$-periodic with respect to the new
time-parameter $s=\nu t$. We first start looking at the unperturbed functional
$\mathcal{A}_{0}$. The $u$-coordinates are related to the $q$-coordinates as
follows

$q_{j,k}(t)=\exp(tJ)u_{0,j}(\nu t)+r_{j}\exp(\omega_{j}tJ)u_{j,k}(\nu t),$
(2.12)

for $k=1,\dots,k_{j}$ and $j=1,\dots,n$, with the constraint $u\in
E_{\mbox{\tiny red}}^{n+N}$.

######  Proposition 2.2.

The unperturbed functional $\mathcal{A}_{0}(u)$ is given by

$\mathcal{A}_{0}(u)=\int_{0}^{\nu
T}\left(\mathcal{L}_{0}(u_{0},\dot{u}_{0})+\sum_{j=1}^{n}r_{j}^{1-\alpha}\mathcal{L}_{j}(u_{j},\dot{u}_{j})\right)ds$
(2.13)

where

$\displaystyle\mathcal{L}_{j}(u_{j},\dot{u}_{j})=\frac{1}{2}\left\|\mathcal{M}_{j}^{1/2}\left(\frac{\nu}{\omega_{j}}\partial_{s}+\mathcal{J}_{k_{j}}\right)u_{j}(s)\right\|^{2}+U_{j}(u_{j})$
(2.14)
$\displaystyle\mathcal{L}_{0}(u_{0},\dot{u}_{0})=\frac{1}{2}\|\mathcal{M}_{0}^{1/2}\left(\nu\partial_{s}+\mathcal{J}_{n}\right)u_{0}(s)\|^{2}+U_{0}(u_{0}).$

Proof. When $\alpha>1$ the potential $\phi_{\alpha}$ is homogeneous of degree
$1-\alpha$. Then $U_{j}(v_{j})=r_{j}^{1-\alpha}U_{j}(u_{j})$ and the kinetic
energy is

$\displaystyle\left\|\mathcal{M}_{j}^{1/2}\left(\partial_{t}+\omega_{j}\mathcal{J}_{k_{j}}\right)v_{j}(t)\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|r_{j}\mathcal{M}_{j}^{1/2}\left(\nu\partial_{s}+\omega_{j}\mathcal{J}_{k_{j}}\right)u_{j}(s)\right\|^{2}$
$\displaystyle=$ $\displaystyle
r_{j}^{1-\alpha}\left\|\mathcal{M}_{j}^{1/2}\left(\frac{\nu}{\omega_{j}}\partial_{s}+\mathcal{J}_{k_{j}}\right)u_{j}(s)\right\|^{2}.$

For $j=0$ we have $U_{0}(v_{0})=U_{0}(u_{0})$ and the kinetic energy is

$\left\|\mathcal{M}_{0}^{1/2}\left(\partial_{t}+\mathcal{J}_{n}\right)v_{0}(t)\right\|^{2}=\left\|\mathcal{M}_{0}^{1/2}\left(\nu\partial_{s}+\mathcal{J}_{n}\right)u_{0}(s)\right\|^{2}.$

The case $\alpha=1$ is similar, but now for $j=1,\dots,n$ we have
$U_{j}(v_{j})=U_{j}(u_{j})+\ln(r_{j})$ and

$\left\|\mathcal{M}_{j}^{1/2}\left(\partial_{t}+\omega_{j}\mathcal{J}_{k_{j}}\right)v_{j}(t)\right\|^{2}=\left\|\mathcal{M}_{j}^{1/2}\left(\frac{\nu}{\omega_{j}}\partial_{s}+\mathcal{J}_{k_{j}}\right)u_{j}(s)\right\|^{2}.$

The result follows by rescaling $\mathcal{A}$ and adding a constant to it.

######  Proposition 2.3.

The nonlinear term $h(u(s))$ of $\mathcal{H}(u)=\int_{0}^{2\pi}h(u(s))ds$ is
$2\pi$-periodic with respect to the time variable $s$ if and only if

$\omega_{j}=1+p_{j}\nu,\qquad p_{j}\in\mathbb{Z},$ (2.15)

for each $j=1,\dots,n$. In this case,

$\displaystyle h(u(s))$ $\displaystyle=$
$\displaystyle\sum_{j<j^{\prime}}\sum_{\begin{subarray}{c}k\in K_{j}\\\
k^{\prime}\in
K_{j^{\prime}}\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\phi_{\alpha}\left(\|u_{0,j}(s)-u_{0,j^{\prime}}(s)+r_{j}\exp\left(p_{j}s\mathcal{J}\right)u_{j,k}(s)-r_{j^{\prime}}\exp\left(p_{j^{\prime}}s\mathcal{J}\right)u_{j^{\prime},k^{\prime}}(s)\|\right)$
(2.16) $\displaystyle-\sum_{j<j^{\prime}}\sum_{\begin{subarray}{c}k\in
K_{j}\\\ k^{\prime}\in
K_{j^{\prime}}\end{subarray}}m_{j,k}m_{j^{\prime},k^{\prime}}\phi_{\alpha}\left(\|u_{0,j}(s)-u_{0,j^{\prime}}(s)\|\right).$

Proof. In (2.7) the terms
$\phi_{\alpha}\left(\|\left(Q_{0,j}(t)-Q_{0,j^{\prime}}(t)\right)+\left(Q_{j,k}(t)-Q_{j^{\prime},k^{\prime}}(t)\right)\|\right)$
become

$\phi_{\alpha}\left(\|u_{0,j}(s)-u_{0,j^{\prime}}(s)+r_{j}\exp\left(\frac{\omega_{j}-1}{\nu}s\mathcal{J}\right)u_{j,k}(s)-r_{j^{\prime}}\exp\left(\frac{\omega_{j^{\prime}}-1}{\nu}s\mathcal{J}\right)u_{j^{\prime},k^{\prime}}(s)\|\right),$

and the terms $\phi_{\alpha}\left(\|Q_{0,j}(t)-Q_{0,j^{\prime}}(t)\|\right)$
become $\phi_{\alpha}\left(\|u_{0,j}(s)-u_{0,j^{\prime}}(s)\|\right).$ It
follows that the integrand $h(u(s))$ is $2\pi$-periodic with respect to $s$ if
and only if, for all $j=1,\dots,n$, the frequency $(\omega_{j}-1)/\nu$ is an
integer. We can thus fix $n$ integers $p_{j}$ such that
$\omega_{j}=1+p_{j}\nu$. Replacing $(\omega_{j}-1)/\nu$ by $p_{j}$ in the
above expression yields (2.16).

Thus, we obtain the action functional

$\mathcal{A}(u)=\mathcal{A}_{0}(u)+\mathcal{H}(u)=\int_{0}^{2\pi}\left(\mathcal{L}_{0}(u_{0},\dot{u}_{0})+\sum_{j=1}^{n}r_{j}^{1-\alpha}\mathcal{L}_{j}(u_{j},\dot{u}_{j})\right)ds+\int_{0}^{2\pi}h(u(s))ds,$
(2.17)

with $h(u(s))$ as in (2.16) and $\mathcal{L}_{0}(u_{0},\dot{u}_{0})$ and
$\mathcal{L}_{j}(u_{j},\dot{u}_{j})$ as in (2.14) . At this point, the
frequency $\nu\in\mathbb{R}$ is still a free parameter. In analogy with [17],
we choose the frequency $\nu$ as a function of $\varepsilon$ such that the
relation $r_{j}=\varepsilon$ holds in the case that $p_{j}=1$. Thus, we fix
arbitrary integers $p_{1},\dots,p_{n}\in\mathbb{Z}$ and impose the following
conditions

  * (A)

$\omega_{j}=1+p_{j}\nu$ and $r_{j}=\left(1+p_{j}\nu\right)^{-2/(\alpha+1)}$
for each $j=1,\dots,n$.

  * (B)

$\nu=\varepsilon^{-\left(\alpha+1\right)/2}-1$ for some $\varepsilon>0$.

These conditions allow us to express the parameters $\omega_{j},r_{j},\nu$ as
functions of $\varepsilon$ with

$\displaystyle r_{j}$ $\displaystyle=$
$\displaystyle\left(1+p_{j}\nu\right)^{-2/(\alpha+1)}=p_{j}^{-2/(\alpha+1)}\varepsilon+\mathcal{O}\left(\varepsilon^{\left(\alpha+3\right)/2}\right)=p_{j}^{-2/(\alpha+1)}\varepsilon+\mathcal{O}\mathbb{(}\varepsilon^{2}),$
(2.18) $\displaystyle\omega_{j}/\nu$ $\displaystyle=$ $\displaystyle
p_{j}+1/\nu=p_{j}+\mathcal{O(}\varepsilon^{\left(\alpha+1\right)/2})=p_{j}+\mathcal{O}\mathbb{(}\varepsilon).$

We look at solutions of the $N$-body problem as critical points of
$\mathcal{A}(u)$ on some collision-less open set $\Omega$ for small
$\varepsilon$.

###  2.4 Gradient formulation and symmetries

The space $H^{1}(S^{1},E^{n+N})$ is identified with its dual
$H^{1}(S^{1},E^{n+N})^{\ast}$ by the Riesz representation Theorem. This allows
us to define the gradient operator by the relation
$\nabla\mathcal{A}=(-\partial_{s}^{2}+1)^{-1}\delta\mathcal{A}$. We use the
gradient formulation
$\nabla\mathcal{A}=\nabla\mathcal{A}_{0}+\nabla\mathcal{H}$, where
$\nabla\mathcal{H}=\mathcal{O}(\varepsilon)$ is a compact operator and
$\nabla\mathcal{A}_{0}$ is given by

$\displaystyle\nabla_{u_{j}}\mathcal{A}_{0}(u)$
$\displaystyle=(-\partial_{s}^{2}+1)^{-1}r_{j}^{1-\alpha}\left(-\mathcal{M}_{j}\left(\frac{\nu}{\omega_{j}}\mathcal{I}_{k_{j}}\partial_{s}+\mathcal{J}_{k_{j}}\right)^{2}u_{j}+\nabla_{u_{j}}U_{j}(u_{j})\right),$
(2.19) $\displaystyle\nabla_{u_{0}}\mathcal{A}_{0}(u)$
$\displaystyle=(-\partial_{s}^{2}+1)^{-1}\left(-\mathcal{M}_{0}\left(\nu\mathcal{I}_{n}\partial_{s}+\mathcal{J}_{n}\right)^{2}u_{0}+\nabla_{u_{0}}U_{0}(u_{0})\right).$
(2.20)

We know that these equations admit the constant solution
$u_{a}=(a_{0},a_{1},\dots,a_{n})\in X$ where $a_{0}\in E_{0}$ is a central
configuration of the $n$-body problem, each $a_{j}\in E_{j}$ is a central
configuration of the $k_{j}$-body problem for $j=1,...,n_{0}$ and $a_{j}\in
E_{j}=\\{0\\}$ for $j=n_{0}+1,...,n$.

Due to the presence of symmetries, there is in fact a group orbit of solutions
generated by $u_{a}$. The functional $\mathcal{A}_{0}(u)$ is invariant under
the action of the torus $G=U(1)^{n+1}$ defined by

$(g_{0},g_{1},\dots,g_{n})\cdot(u_{0},u_{1}\dots,u_{n})=(g_{0}u_{0},g_{1}u_{1},\dots,g_{n}u_{n}).$

where $(g_{0},g_{1},\dots,g_{n})\in U(1)^{n+1}$. The action of $g_{0}$ rotates
the $n$-body problem consisting of the $n$ centers of mass of the clusters
about the origin. The action of each $g_{j}$ rotates the $k_{j}$-bodies in
each cluster about their center of mass for $j=1,...,n_{0}$ and acts trivially
otherwise. The coupling term $\mathcal{H}$ breaks this symmetry in the sense
that the perturbed functional $\mathcal{A}$ is invariant under the diagonal
subgroup $H=\widetilde{U(1)}$. By $G$-equivariance of the equations (2.19) and
(2.20), the group orbit $G(u_{a})$ is an orbit of solutions.

###  2.5 Euler-Lagrange equations with holonomic constraints

Let $S^{1}=\mathbb{R}/2\pi\mathbb{Z}$ be the standard parametrization of the
circle and denote by

$X=H^{1}(S^{1},E_{\mbox{\tiny red}}^{n+N})\subset H^{1}(S^{1},E^{n+N})$

the real Hilbert space of $2\pi$-periodic paths in $E_{\mbox{\tiny
red}}^{n+N}$. The solutions of the $N$-body problem are the critical points of
the augmented action $\mathcal{A}$ restricted to $X$. That is, the system of
equations of the $N$-body problems is the gradient of $\mathcal{A}$ taking
respect the subspace $X$ and is given by

$P_{X}\nabla\mathcal{A}(u)=0.$ (2.21)

Solving this system is equivalent to finding the critical points of
$\mathcal{A}$ with the holonomic constraints

$g_{0}^{s}(u_{0})=\sum_{j=1}^{n}M_{j}u_{0,j}\cdot e_{s}=0,\qquad
g_{j}^{s}(u_{j})=\sum_{k=1}^{k_{j}}m_{j,k}u_{j,k}\cdot e_{s}=0,\qquad s=1,2.$

Thus, the explicit projection $P_{X}:H^{1}(S^{1},E^{n+N})\rightarrow X$ is
given in components $u_{j}$ by

$P_{X}(u_{j})=u_{j}-\sum_{s=1}^{2}\frac{(u_{j},\nabla_{u_{j}}g_{j}^{s}(u_{j}))}{\|\nabla_{u_{j}}g_{j}^{s}(u_{j})\|^{2}}\nabla_{u_{j}}g_{j}^{s}(u_{j}),\qquad
j=0,\dots,n,$ (2.22)

and the explicit system of equations, equivalent to (2.21), is

$\nabla_{u_{j}}\mathcal{A}(u)=\sum_{s=1}^{2}\frac{(\nabla_{u_{j}}\mathcal{A}(u),\nabla_{u_{j}}g_{j}^{s}(u_{j}))}{\|\nabla_{u_{j}}g_{j}^{s}(u_{j})\|^{2}}\nabla_{u_{j}}g_{j}^{s}(u_{j}),\qquad
j=0,\dots,n.$

with the left hand side given in (2.20) and (2.19). We study these equations
in a collision-less tubular neighbourhood $\Omega\subset X$ of the orbit
$G(u_{a})$.

######  Remark 2.2.

One may consider also the augmented action with the holonomic constraints

$\mathcal{A}^{\ast}(u^{\ast})=\mathcal{A}_{0}(u)+\mathcal{H}(u)+\sum_{j=0}^{n}\sum_{s=1}^{2}\lambda_{j}^{s}\cdot
g_{j}^{s}\text{,}$

where $u^{\ast}=(u,\lambda_{0},...,\lambda_{n})\in
E^{N+n}\times\mathbb{R}^{2(n+1)}$. Solving the action for the augmented system
$\nabla\mathcal{A}^{\ast}(u^{\ast})=0$ is equivalent to solving
$\nabla_{u_{j}}\mathcal{A}^{\ast}(u^{\ast})=P_{X}\nabla_{u_{j}}\mathcal{A}(u)=0$
with $u\in E_{red}^{n+N}$ (because
$\nabla_{\lambda_{j}^{s}}\mathcal{A}^{\ast}(u^{\ast})=g_{j}^{s}(u)=0$). For
each solution $u_{a}$ such that $\nabla_{u}\mathcal{A}_{0}(u_{a})=0$ there is
a unique $u_{a}^{\ast}$ such that
$\nabla_{u^{\ast}}\mathcal{A}_{0}^{\ast}(u_{a}^{\ast})=0$, and a similar
procedure can be implemented for the orbit of $u_{a}^{\ast}$ with the Lagrange
multipliers $\lambda_{j}$ given as variables. The procedures are equivalent
because $(u,\lambda)$ is in the kernel of
$\nabla_{u^{\ast}}^{2}\mathcal{A}^{\ast}(u_{a}^{\ast})$ if and only $u$ is in
the kernel of $P_{X}\nabla_{u}^{2}\mathcal{A}(u_{a})|_{X}$.

######  Remark 2.3.

We may consider also coordinates for $E_{\mbox{\tiny red}}^{n+N}$ to write
directly the action in these coordinates. That is, we may fix coordinates
$w_{j}=(w_{j,1},\dots,w_{j,k_{j}-1})$ on $E_{j}$ such that $u_{j}\in E_{j}$
viewed as an element of $E^{k_{j}}$ can be written of the form
$u_{j}=\Lambda_{j}w_{j}$ for some $k_{j}\times(k_{j}-1)$ matrix $\Lambda_{j}$.
Note that

$\left\langle\nabla_{u_{j}}\mathcal{A}_{0}(u_{j}),\delta
u_{j}\right\rangle=\left\langle\Lambda_{j}^{T}\nabla_{u_{j}}\mathcal{A}_{0}(\Lambda_{j}w_{j}),\delta
w_{j}\right\rangle$

where the first inner product is taken on the ambient space
$H^{1}([0,2\pi],E^{k_{j}})$ and the second on the reduced space
$H^{1}([0,2\pi],E_{j})$. The reduced Euler-Lagrange equations on $X$ are then

$\Lambda_{j}^{T}\nabla_{u_{j}}\mathcal{A}_{0}(\Lambda_{j}w_{j})=0,\qquad
j=0,\dots,n-1.$ (2.23)

This is the method adopted in our previous paper [17] for $j=1$ and $k_{j}=2$.
In that case, the Euler-Lagrange equations of the 2-body problem in rotating
frame are

$-\mathcal{M}_{1}\left(\frac{\nu}{\omega_{1}}\mathcal{I}_{k_{1}}\partial_{s}+\mathcal{J}_{k_{1}}\right)^{2}u_{1}+\nabla_{u_{1}}U_{1}(u_{1})=0$
(2.24)

where $\mathcal{M}_{1}$ is the diagonal matrix whose entries are the masses of
the two bodies $m_{1,1}$ and $m_{1,2}$. The positions are denoted
$u_{1}=(u_{1,1},u_{1,2})$. We parametrize the reduced space $E_{1}$ by the
relative position $w_{1}=u_{1,1}-u_{1,2}$, so that $u_{1}=\Lambda_{1}w_{1}$
with $\Lambda_{1}=\begin{bmatrix}\lambda_{1,1}I&\lambda_{1,2}I\end{bmatrix}$,
with

$\lambda_{1,1}=\frac{m_{1,2}}{m_{1,1}+m_{1,2}},\qquad\lambda_{2,1}=-\frac{m_{1,1}}{m_{1,1}+m_{1,2}}.$

Conjugating equation (2.24) by $\Lambda_{1}^{T}$ on the left and $\Lambda_{1}$
on the right yields the Kepler problem in rotating frame

$-M_{0}\left(\frac{\nu}{\omega_{1}}\partial_{s}+J\right)^{2}w_{1}-m_{1,1}m_{1,2}\frac{w_{1}}{\|w_{1}\|^{\alpha+1}}=0,$

where $M_{0}=\frac{m_{1,1}m_{1,2}}{m_{1,1}+m_{1,2}}$ is the reduced mass.

##  3 Lyapunov-Schmidt reduction

In this section we reduce the problem to finite dimension by writing the paths
in Fourier series and applying a Lyapunov-Schmidt reduction. We have that

$X=\left\\{u\in L^{2}(S^{1},E_{\mbox{\tiny
red}}^{n+N})\mid\sum_{\ell\in\mathbb{Z}}(\ell^{2}+1)\|\hat{u}_{\ell}\|^{2}<\infty\right\\},$

where $(\hat{u}_{\ell})$ is the sequence of Fourier coefficients in
$(E_{\mbox{\tiny red}}^{n+N})^{\mathbb{C}}=E_{\mbox{\tiny red}}^{n+N}\oplus
iE_{\mbox{\tiny red}}^{n+N}$ satisfying
$\hat{u}_{\ell}=\overline{\hat{u}}_{-\ell}$. That is, $u\in X$ has Fourier
series $u=\sum_{\ell\in\mathbb{Z}}\hat{u}_{\ell}e_{\ell}$ where
$e_{\ell}:S^{1}\rightarrow\mathbb{C}$ is given by $e_{\ell}(s)=e^{i\ell s}$.
We can then write $X=X_{0}\oplus W$ and any element $u\in X$ decomposes
uniquely as $u=\xi+\eta$ with

$\xi=\hat{u}_{0},\qquad\eta=\sum_{\ell\neq 0}\hat{u}_{\ell}e_{\ell}.$

The system of equations $P_{X}\nabla\mathcal{A}(\xi+\eta)=0$ splits into

$\displaystyle P_{X_{0}}\nabla\mathcal{A}(\xi+\eta)$ $\displaystyle=0\in
X_{0}~{},$ $\displaystyle P_{W}\nabla\mathcal{A}(\xi+\eta)$
$\displaystyle=0\in W,$ (3.1)

where $P_{X_{0}}:H^{1}(S^{1},E^{n+N})\to X_{0}$ is the canonical projection
from $P:X\to X_{0}$ given by $Pu=\xi$, composed with $P_{X}$ in (2.22). The
projection $P_{W}:H^{1}(S^{1},E^{n+N})\to W$ is defined as the canonical
projection from $(I-P):X\to W$ given by $(I-P)u=\eta$, composed with $P_{X}$.

The Lyapunov-Schmidt reduction requires solving the equation
$P_{W}\nabla\mathcal{A}(\xi+\eta)=0$. For this purpose, we define an operator
$F_{\varepsilon}:\Omega\subset X\rightarrow W$ by

$F_{\varepsilon}(\xi,\eta):=\mathcal{D}_{\varepsilon}P_{W}\nabla\mathcal{A}(\xi+\eta)$

where $\mathcal{D}_{\varepsilon}\in\mbox{End}(E^{n+N})$ is the block diagonal
matrix

$\mathcal{D}_{\varepsilon}=\nu^{-2}\mathcal{I}_{n}\oplus
r_{1}^{\alpha-1}\mathcal{I}_{k_{1}}\oplus\dots\oplus
r_{n}^{\alpha-1}\mathcal{I}_{k_{n}},$ (3.2)

with $r_{j}^{\alpha-1}=\mathcal{O}(\varepsilon^{\alpha-1})$ and
$\nu^{-2}=\mathcal{O}(\varepsilon^{\alpha+1})$. Since
$\mathcal{D}_{\varepsilon}$ is block diagonal, it commutes with $P_{W}$ and we
get

$F_{\varepsilon}(\xi,\eta)=P_{W}\mathcal{D}_{\varepsilon}\nabla\mathcal{A}(\xi+\eta).$

Solving (3.1) is equivalent to solving $F_{\varepsilon}(\xi,\eta)=0$ for
$\varepsilon\neq 0$ because $\mathcal{D}_{\varepsilon}$ is an isomorphism. The
operator $F_{\varepsilon}(\xi,\eta)$ is continuous at $\varepsilon=0$ because
$\lim_{\varepsilon\rightarrow
0}\left(\nu/\omega_{j}\right)^{2}=\left(1/p_{j}\right)^{2}$. The limit

$F_{0}(\xi,\eta)=\lim_{\varepsilon\rightarrow
0}P_{W}\mathcal{D}_{\varepsilon}\nabla\mathcal{A}_{0}(\xi+\eta)$ (3.3)

is well defined since
$\mathcal{D}_{\varepsilon}\nabla\mathcal{H}=\mathcal{O}(\varepsilon)$.
Furthermore, $F_{0}(gu_{a},0)=0$ for all $g\in G$ by equivariance of the
unperturbed gradient. Solving $F_{\varepsilon}(\xi,\eta)=0$ requires the
derivative $\partial_{\eta}F_{0}[(gu_{a},0)]$ to be invertible on $W$.
Although this is true when $\alpha\neq 2$, the operator is not invertible on
the whole space $W$ when $\alpha=2$. We shall then treat these cases
separately.

######  Remark 3.1.

Alternatively, we could have used the matrix

$\mathcal{D}_{\varepsilon}^{\prime}=\varepsilon^{\alpha+1}\mathcal{I}_{n}\oplus\varepsilon^{\alpha-1}\mathcal{I}_{k_{1}}\oplus\dots\oplus\varepsilon^{\alpha-1}\mathcal{I}_{k_{n}}$

according to [17]. Both regularizations allow us to perform the same
reduction, the only difference is that the scaling matrix
$\mathcal{D}_{\varepsilon}$ depends on $p_{j}$’s.

### The case $\alpha\neq 2$

The linearization of (3.3) at $u_{a}$ is given by

$\partial_{\eta}F_{0}[(u_{a},0)]=\lim_{\varepsilon\rightarrow
0}P_{W}\mathcal{D}_{\varepsilon}\nabla_{u}^{2}\left.\mathcal{A}_{0}[u_{a}]\right|_{W}.$

The Hessian operator of $\mathcal{A}_{0}$ at the critical point is block
diagonal of the form

$\nabla_{u}^{2}\mathcal{A}_{0}[u_{a}]=\nabla_{u_{0}}^{2}\mathcal{A}_{0}[a_{0}]\oplus\nabla_{u_{1}}^{2}\mathcal{A}_{0}[a_{1}]\oplus\dots\oplus\nabla_{u_{n}}^{2}\mathcal{A}_{0}[a_{n}].$

The blocks are derived using (2.19), (2.20) and given by

$\displaystyle\nabla_{u_{j}}^{2}\mathcal{A}_{0}[a_{j}]$
$\displaystyle=(-\partial_{s}^{2}+1)^{-1}r_{j}^{\alpha-1}\left(-\left(\nu/\omega_{j}\right)^{2}\mathcal{M}_{j}\partial_{s}^{2}-2\left(\nu/\omega_{j}\right)\mathcal{M}_{j}\mathcal{J}_{k_{j}}\partial_{s}+\nabla_{u_{j}}^{2}V_{j}[a_{j}]\right),$
$\displaystyle\nabla_{u_{0}}^{2}\mathcal{A}_{0}[a_{0}]$
$\displaystyle=(-\partial_{s}^{2}+1)^{-1}\nu^{2}\left(-\mathcal{M}_{0}\partial_{s}^{2}-2\nu^{-1}\mathcal{M}_{0}\mathcal{J}_{n}\partial_{s}+\nu^{-2}\nabla_{u_{0}}^{2}V_{0}[a_{0}]\right),$

where $V_{j}$ are the amended potentials in (2.11).

######  Definition 3.2.

We define the regularized action for the $j$-cluster by
$A_{j}(u_{j})=\int_{0}^{2\pi}L_{j}(u_{j},\dot{u}_{j})ds$ where

$L_{j}(u_{j},\dot{u}_{j})=\lim_{\varepsilon\rightarrow
0}\mathcal{L}_{j}(u_{j},\dot{u}_{j})=\frac{1}{2}\left\|\mathcal{M}_{j}^{1/2}\left(\frac{1}{p_{j}}\partial_{s}+\mathcal{J}_{k_{j}}\right)u_{j}(s)\right\|^{2}+U_{j}(u_{j}).$
(3.4)

Since $\lim_{\varepsilon\rightarrow 0}\left(\omega_{j}/\nu\right)=1/p_{j}$, we
obtain

$\lim_{\varepsilon\rightarrow
0}r_{j}^{\alpha-1}\nabla_{u_{j}}\mathcal{A}_{0}(u_{j})=\nabla_{u_{j}}A_{j}(u_{j}),$

and

$\partial_{\eta}F_{0}[(u_{a},0)]=P_{W}\left(\left.-(-\partial_{s}^{2}+1)^{-1}\mathcal{M}_{0}\partial_{s}^{2}\oplus\nabla_{u_{1}}^{2}A_{1}[a_{1}]\oplus\dots\oplus\nabla_{u_{n}}^{2}A_{n}[a_{n}]\right|_{W}\right),$

where

$\nabla_{u_{j}}^{2}A_{j}[a_{j}]=(-\partial_{s}^{2}+1)^{-1}\left(-(1/p_{j})^{2}\mathcal{M}_{j}\partial_{s}^{2}-2\left(1/p_{j}\right)\mathcal{M}_{j}\mathcal{J}_{k_{j}}\partial_{s}+\nabla_{u_{j}}^{2}V_{j}[a_{j}]\right).$

Since $\eta=\sum_{\ell\neq 0}\hat{u}_{\ell}e_{\ell}\in W$, we can write

$\partial_{\eta}F_{0}[(u_{a},0)]\eta=\sum_{\ell\neq
0}\hat{T}_{\ell}\hat{u}_{\ell}e_{\ell},$ (3.5)

where the matrix $\hat{T}_{\ell}\in\mbox{End}(E_{\mbox{\tiny red}}^{n+N})$ is
block diagonal of the form

$\hat{T}_{\ell}=\hat{T}_{\ell,u_{0}}\oplus\hat{T}_{\ell,u_{1}}\oplus\dots\oplus\hat{T}_{\ell,u_{n}}.$
(3.6)

These blocks are given explicitly by

$\displaystyle\hat{T}_{\ell,u_{0}}$
$\displaystyle=\frac{\ell^{2}}{\ell^{2}+1}P_{E_{0}}\left.\mathcal{M}_{0}\right|_{E_{0}}$
$\displaystyle\hat{T}_{\ell,u_{j}}$
$\displaystyle=\frac{1}{1+\ell^{2}}P_{E_{j}}\left.\left(\left(\frac{\ell}{p_{j}}\right)^{2}\mathcal{M}_{j}-2i\left(\frac{\ell}{p_{j}}\right)\mathcal{M}_{j}\mathcal{J}_{k_{j}}+\nabla_{u_{j}}^{2}V_{j}[a_{j}]\right)\right|_{E_{j}}$
(3.7)

where $\hat{T}_{\ell,u_{0}}\in\mbox{End}(E_{0}^{\mathbb{C}})$ and
$\hat{T}_{\ell,u_{j}}\in\mbox{End}(E_{j}^{\mathbb{C}})$.

######  Definition 3.3.

The central configuration $a_{j}\in E_{j}$ is  $2\pi p_{j}$-nondegenerate if
the group orbit $U(1)(a_{j})$ is a nondegenerate critical manifold of the
functional $A_{j}(u_{j})$ defined by (3.4) in the space $H^{1}(S^{1},E_{j})$.

The orbit $U(1)(a_{j})$ is called a nondegenerate critical manifold of the
functional $A_{j}(u_{j})$ if the kernel of the Hessian at $a_{j}$ in
$H^{1}(S^{1},E_{j})$ is $\mbox{span}(\mathcal{J}_{k_{j}}a_{j})$. Since
$u_{j}\in H^{1}(S^{1},E_{j})$ is orthogonal to
$\mbox{span}(\mathcal{J}_{k_{j}}a_{j})$ if and only if
$u=\sum_{\ell\in\mathbb{Z}}\hat{u}_{\ell}e_{\ell}$ with $\hat{u}_{0}$
orthogonal to $\mbox{span}(\mathcal{J}_{k_{j}}a_{j})$ in $E_{j}$, this
condition is equivalent to the assumption that the blocks
$\hat{T}_{\ell,u_{j}}$ are invertible in $E_{j}^{\mathbb{C}}$ for $\ell\neq 0$
and $\hat{T}_{0,u_{j}}$ is invertible in the complement to
$\mbox{span}(\mathcal{J}_{k_{j}}a_{j})$ in $E_{j}$.

######  Lemma 3.1.

Assume that $\alpha\neq 2$ and $a_{j}$ is $2\pi p_{j}$-nondegenerate for
$j=1,\dots,n_{0}$. Then the operator $\partial_{\eta}F_{0}[(gu_{a},0)]$ is
invertible on $W$ for all $g\in G$, i.e. there is a constant $c>0$ such that

$\|\partial_{\eta}F_{0}[(gu_{a},0)]^{-1}\eta\|\leq c\|\eta\|\quad\mbox{for
each}\quad\eta\in W,~{}g\in G.$

Proof. For $\ell\neq 0$, the block $\hat{T}_{\ell}$ in (3.6) is always
invertible and so are the blocks $\hat{T}_{\ell,u_{j}}$ when $\ell\neq 0$ by
assumption that $a_{j}$ is $2\pi p_{j}$-nondegenerate for $j=1,\dots,n_{0}$.
This implies that the operator $\partial_{\eta}F_{0}[(u_{a},0)]$ is invertible
on $W$ with

$\partial_{\eta}F_{0}[(u_{a},0)]^{-1}\eta=\sum_{\ell\neq
0}\hat{T}_{\ell}^{-1}\hat{u}_{\ell}e_{\ell},\quad\eta\in W.$

Since
$\hat{T}_{\ell,u_{j}}\rightarrow\left(1/p_{j}\right)^{2}P_{E_{j}}\left.\mathcal{M}_{j}\right|_{E_{j}}$
for $j=1,\dots,n$ and $\hat{T}_{\ell,u_{0}}\rightarrow
P_{E_{0}}\left.\mathcal{M}_{0}\right|_{E_{0}}$ when $\ell\rightarrow\infty$,
it follows that

$\|\partial_{\eta}F_{0}[(u_{a},0)]^{-1}\eta\|\leq c\|\eta\|.$ (3.8)

where $c>0$ is a constant such that any eigenvalue $\lambda$ of
$\hat{T}_{\ell}$ in (3.6) satisfies $\left|\lambda\right|\geq c^{-1}$. Note
that the Hessian $\nabla^{2}\mathcal{A}_{0}[gu_{a}]$ is conjugated to
$\nabla^{2}\mathcal{A}_{0}[u_{a}]$ as $\nabla\mathcal{A}_{0}$ is
$G$-equivariant. This also holds for the constraint gradient because the
$G$-action preserves the constraints. Hence $\partial_{\eta}F_{0}[(gu_{a},0)]$
and $\partial_{\eta}F_{0}[(u_{a},0)]$ are conjugated and the estimate (3.8)
holds when replacing $u_{a}$ by $gu_{a}$ because $G$ acts by isometries.

######  Remark 3.4.

To explain further the meaning of the $2\pi p_{j}$-nondegeneracy condition, we
can consider the Hamiltonian system with Hamiltonian

$H_{j}(u_{j},\pi_{j})=K_{j}-U_{j}$

for the $k_{j}$-body problem, where $\pi_{j}=\partial_{\dot{u}_{j}}L$ is
obtained from the Lagrangian $L_{j}=K_{j}+U_{j}$ defined in the space $E_{j}$
by means of the Legendre transformation. The relative equilibrium $a_{j}$ is
linearly stable if the eigenvalues of the linearized Hamiltonian vector field
$\mathfrak{J}^{-1}\nabla^{2}H_{j}[(a_{j},0)]$ are all purely imaginary, except
by the double zero-eigenvalue corresponding to the generator of the group
orbit $\left(\mathcal{J}_{j}a_{j},0\right)$. On the other hands, our $2\pi
p_{j}$-nondegenerate condition for $a_{j}$ can be verified similarly according
to the equivalent condition that the matrix
$\mathfrak{J}^{-1}\nabla^{2}H_{j}[(a_{j},0)]$ has no eigenvalues of the form
$2\pi i\ell$ with $\ell\in\mathbb{Z}$, except by a double zero-eigenvalue
corresponding to the generator of the $U(1)$-orbit of $(a_{j},0)$.

Unfortunately, the $2\pi p_{j}$-nondegenerate condition has not been verified
before in the literature for central configurations. In order to complement
our result, we verify this condition for an infinite number of polygonal
configurations in Section 4. We conjecture that, for $\alpha\neq 2$, the
condition of being $2\pi$-nondegenerate holds for a generic set of central
configurations in a set of parameters of masses.

### Gravitational case $\alpha=2$

When $\alpha=2$ all the central configurations $a_{j}$ are $2\pi
p_{j}$-nondegenerate due to the existence of elliptic homographic solutions,
and the matrices $\hat{T}_{\ell,u_{j}}$ are never invertible for $\ell=\pm
p_{j}$. To study the case $\alpha=2$, we distinguish different type of
symmetric configurations under $2\pi/m$-rotations at the origin. Examples of
symmetric configurations that we can braid are the Maxwell configuration and
nested polygonal configurations. In these cases, we can only divide the
central body. Thus, we require the additional assumptions listed below.

  * (C0)

We consider the $N$-body problem with $N=n+k_{1}-1$, i.e.

$k_{1}>1,\qquad k_{2}=...=k_{n}=1.$

Then $E_{j}=\\{0\\}$ for $j=2,\dots,n$ and $E_{\mbox{\tiny
red}}^{n+N}=E_{0}\times E_{1}$. A path $u\in X$ is then written as
$u=(u_{0},u_{1})$. Denote by $S_{n}$ the permutation group of $n$ letters. We
need a group of symmetries $\Gamma$ that allows dealing with the resonances.
We consider $\Gamma<\mathbb{Z}_{m}\times S_{n}$ to be the discrete subgroup
generated by the element $(\theta,\sigma)$ such that

$\theta=2\pi/m\in\mathbb{Z}_{m},\qquad\sigma^{m}=(1)\in
S_{n},\qquad\sigma(1)=1,$

and that acts on the components of $u=(u_{0},u_{1})$ as follows:

$\displaystyle(\theta,\sigma)u_{1}(s)$ $\displaystyle=u_{1}(s+\theta),$
$\displaystyle(\theta,\sigma)u_{0}(s)$ $\displaystyle=(\exp(-\theta
J)u_{0,\sigma(1)}(s+\theta),\dots,\exp(-\theta J)u_{0,\sigma(n)}(s+\theta)).$

  * (C1)

The next assumption is that the masses for the bodies $q_{j,1}$ for
$j=2,\dots,n$ satisfy

$m_{j,1}=m_{\sigma(j),1}~{},\qquad j=2,\dots,n.$

For the sake of simplicity, we also assume that $p_{1}=1$, which along with
conditions (A)-(B) imply that

$r_{1}=\varepsilon,\qquad\omega_{1}=\varepsilon^{-\left(\alpha+1\right)/2},\qquad\nu=\varepsilon^{-\left(\alpha+1\right)/2}-1.$

######  Lemma 3.2.

For $\alpha=2$, under the assumptions (C0)-(C1), the functional $\mathcal{A}$
is $\Gamma$-invariant.

Proof. Since the variables $u_{j}(s)$ for $j=0,\dots,n$ are uncoupled in
$\mathcal{A}_{0}$, it is an immediate consequence of the assumptions that the
functional $\mathcal{A}_{0}$ is $\Gamma$-invariant. It remains to show that
the coupling term $\mathcal{H}$ is $\Gamma$-invariant. By assumptions
(C0)-(C1), the integrand $h$ is obtained from (2.16) after setting $u_{j,k}=0$
whenever $j\geq 2$, and $p_{1}=1$. We get

$h(u(s))=\sum_{j^{\prime}=2}^{n}\sum_{k\in
K_{1}}m_{1,k}m_{j^{\prime},1}\left(\phi_{\alpha}\left(\|u_{0,1}(s)-u_{0,j^{\prime}}(s)+r_{1}\exp\left(s\mathcal{J}\right)u_{1,k}(s)\|\right)-\phi_{\alpha}\left(\|u_{0,1}(s)-u_{0,j^{\prime}}(s)\|\right)\right).$

Set $s^{\prime}=s+\theta$. Since $\sigma(1)=1$ and the norms are invariant by
rotations, we obtain

$\displaystyle h((\theta,\sigma)u(s))$
$\displaystyle=\sum_{j^{\prime}=2}^{n}\sum_{k\in
K_{1}}m_{1,k}m_{\sigma(j^{\prime}),1}\left(\phi_{\alpha}\left(\|u_{0,1}(s^{\prime})-u_{0,\sigma(j^{\prime})}(s^{\prime})+r_{1}\exp\left(s^{\prime}\mathcal{J}\right)u_{1,k}(s^{\prime})\|\right)\right.$
$\displaystyle\left.-\phi_{\alpha}\left(\|u_{0,1}(s^{\prime})-u_{0,\sigma(j^{\prime})}(s^{\prime})\|\right)\right)=h(u(s^{\prime}))$

by re-indexing the sum at the end. Finally,

$\mathcal{H(}(\theta,\sigma)u)=\int_{0}^{2\pi}h((\theta,\sigma)u(s))ds=\int_{\theta}^{2\pi+\theta}h(u(s^{\prime}))ds^{\prime}=\mathcal{H}(u).$

Thus, by the Palais Principle of Symmetric Criticality [35], we can restrict
the study of critical points to the fixed point space $X^{\Gamma}$. Notice
that a path $u=(u_{0},u_{1})$ belongs to $X^{\Gamma}$ if and only if
$u(s)=(\theta,\sigma)u(s)$. Thus

$X^{\Gamma}=H^{1}(S^{1},E_{0})^{\Gamma}\oplus H^{1}(S^{1},E_{1})^{\Gamma},$

where $H^{1}(S^{1},E_{1})^{\Gamma}$ is the Sobolev space of $2\pi/m$-periodic
functions in $E_{1}$ and $H^{1}(S^{1},E_{0})^{\Gamma}$ is the subspace of
functions $u_{0}$ satisfying the symmetry
$u_{0,\sigma(j)}=\exp(-\theta\mathcal{J})u_{0,\sigma(j)}(s+\theta)$.

  * (C2)

The last assumption (to ensure that $u_{a}\in X^{\Gamma}$) is that the central
configurations $a_{0}\in E_{0}$ satisfies the property

$a_{0,j}=\exp(-\theta\mathcal{J})a_{0,\sigma(j)}.$ (3.9)

Since $\sigma^{m}=1$ and $\theta=2\pi/m$, this condition implies that the
central configuration $a_{0}$ is symmetric by $2\pi/m$-rotations in the plane,
and since $\sigma(1)=1$ that

$a_{0,1}=\exp(-\theta\mathcal{J})a_{0,1}=0.$

This condition holds true in many symmetric configurations: Maxwell
configuration, nested polygons with a center [30] and spiderwebs with a center
[22].

######  Definition 3.5.

The central configuration $a_{1}\in E_{1}$ is  $2\pi/m$-nondegenerate if the
orbit $U(1)(a_{1})$ is a nondegenerate critical manifold of the functional
$A_{1}(u_{1})$ in the fixed point space $H^{1}(S^{1},E_{1})^{\Gamma}$.

Thus $a_{1}$ is a $2\pi/m$-nondegenerate if the kernel of the Hessian of
$A_{1}$ at $a_{1}$ in $H^{1}(S^{1},E_{1})^{\Gamma}$ is
$\mbox{span}(\mathcal{J}_{k_{1}}a_{1})$. This weaker condition is equivalent
to the hypothesis that the matrices $\hat{T}_{\ell,u_{1}}$ are invertible in
$E_{1}^{\mathbb{C}}$ for $\ell\in m\mathbb{Z}/\\{0\\}$ and $\hat{T}_{0,u_{1}}$
is invertible in the orthogonal complement to
$\mbox{span}(\mathcal{J}_{k_{1}}a_{1})$ in $E_{1}$. We conjecture that for
$\alpha=2$, the condition of being $2\pi/m$-nondegenerate holds for a generic
set of central configurations in the set of parameters. This condition is
verified for the $k$-polygonal configuration in Section 4 for $k=4,...,1000$.

######  Lemma 3.3.

Assume $\alpha=2$. Under the conditions (C0)-(C2), if $a_{1}$ is
$2\pi/m$-nondegenerate for $m\geq 2$, the statement of Lemma 3.1 holds true
after replacing $W$ by the fixed point set $W^{\Gamma}$.

Proof. By conditions (C0)-(C1),

$\hat{T}_{\ell}=\hat{T}_{\ell,u_{0}}\oplus\hat{T}_{\ell,u_{1}}$

because $u=(u_{0},u_{1})$. Since $a_{1}$ is $2\pi/m$-nondegenerate, the matrix
$\hat{T}_{\ell,u_{1}}$ is invertible for the Fourier modes $\ell=0,\pm m,\pm
2m,\dots$. A path $u=(u_{0},u_{1})$ belongs to $X^{\Gamma}$ if $u_{1}$ is
$2\pi/m$-periodic. In particular, the Fourier expansion of $u_{1}$ is fixed by
$\Gamma$ only if

$\hat{u}_{1,\ell}=0\text{ for }\ell\neq 0,\pm m,\pm 2m,\dots$

and hence the operator $\partial_{\eta}F_{0}[(u_{a},0)]$ is invertible on
$W^{\Gamma}$. The argument in the proof of Lemma 3.1 applies now in the fixed
point space. Moreover, note that the group action of $G$ commutes with that of
$\Gamma$. Hence by $\mathbf{(C2)}$, the orbit $G(u_{a})$ belongs to
$X_{0}^{\Gamma}\subset X^{\Gamma}$ and the functional $\mathcal{A}_{0}$
restricted to $X^{\Gamma}$ is still $G$-invariant.

### Lyapunov-Schmidt reduction

Because of Lemmas 3.1, 3.2 and 3.3, we can perform a Lyapunov-Schmidt
reduction as in Theorem 3.2, 3.3 and 3.4 in [17].

######  Theorem 3.4 (Lyapunov-Schmidt reduction and uniform estimates ).

Under conditions (A)-(B) and $\alpha\neq 2$, there is $\varepsilon_{0}>0$ such
that for every $\varepsilon\in(0,\varepsilon_{0})$, there is a $G$-invariant
neighbourhood $\mathcal{V}\subset X_{0}$ of $u_{a}$ and an analytic
$H$-equivariant mapping $\varphi_{\varepsilon}:\mathcal{V}\subset
X_{0}\rightarrow W$ such that $P_{W}\nabla\mathcal{A}(\xi+\eta)=0$ for
$\xi\in\mathcal{V}$, $\xi+\eta\in X$ if and only if
$\eta=\varphi_{\varepsilon}(\xi)$. The system reduces to the finite-
dimensional system $P_{X_{0}}\nabla\Psi_{\varepsilon}(\xi)=0$ in $\mathcal{V}$
where $\Psi_{\varepsilon}(\xi)=\mathcal{A}(\xi+\varphi_{\varepsilon}(\xi))$,
and $P_{X_{0}}$ is the projection of $X$ to $X_{0}$ composed with $P_{X}$.
Furthermore, for each $\xi\in\mathcal{V}$ and $g\in G$, the following estimate
holds:

$\|\varphi_{\varepsilon}(\xi)\|\leq N_{1}(\varepsilon+\|\xi-gu_{a}\|^{2}).$
(3.10)

If $\alpha=2$, the same result holds under the additional conditions (C0)-(C2)
after replacing $X_{0}$ and $W$ by their $\Gamma$-fixed point spaces.

We set $\xi=(\xi_{0},\xi_{1},\dots,\xi_{n})$ where $\xi_{j}\in E_{j}$ are
coordinates for the $k_{j}$-body problems, and $\xi_{0}\in E_{0}$ are
coordinates for the $n$-body problem formed by the centers of mass of the $n$
clusters. Once we fix $\varepsilon\in(0,\varepsilon_{0})$, the function
$\Psi_{\varepsilon}=\mathcal{V}\subset X_{0}\rightarrow\mathbb{R}$ is of the
form $\Psi_{\varepsilon}(\xi)=\mathcal{A}_{0}(\xi)+\mathcal{N}(\xi)$ where

$\mathcal{A}_{0}(\xi)=2\pi\left(V_{0}(\xi_{0})+\sum_{j=1}^{n}r_{j}^{1-\alpha}V_{j}(\xi_{j})\right).$

is $G$-invariant, and

$\mathcal{N}(\xi)=\mathcal{A}_{0}(\xi+\varphi_{\varepsilon}(\xi))-\mathcal{A}_{0}(\xi)+\mathcal{H}(\xi+\varphi_{\varepsilon}(\xi))$

is $H$-invariant and satisfies the estimate

$\|\mathcal{C}_{\varepsilon}\nabla\mathcal{N}(\xi)\|\leq N(\varepsilon+\|\xi-
gu_{a}\|^{2}),$ (3.11)

where $\mathcal{C}_{\varepsilon}=\mathcal{I}_{n}\oplus
r_{1}^{\alpha-1}\mathcal{I}_{k_{1}}\oplus\dots\oplus
r_{n}^{\alpha-1}\mathcal{I}_{k_{n}}$ and $g\in G$. The proof of this estimate
follows the same steps as in Theorem 3.5 in our previous work [17].

### Symmetry reduction

Even if the problem is now finite dimensional, it is still not possible to
continue the solutions of $P_{X_{0}}\nabla\Psi_{\varepsilon}(\xi)=0$ from
$\varepsilon=0$ because $\Psi_{\varepsilon}$ still blows up when
$\varepsilon\rightarrow 0$. We obtain a regular function by passing to the
quotient space under the action of $H$ on $\mathcal{V}$. Accordingly we write

$\xi^{\prime}=(\xi_{1},\dots,\xi_{n})\in E^{\prime}=E_{1}\times\dots\times
E_{n}$ (3.12)

and $a^{\prime}=(a_{1},\dots,a_{n})\in E^{\prime}$. The gradient equation
$P_{X_{0}}\nabla\Psi_{\varepsilon}(\xi_{0},\xi^{\prime})=0$ splits into two
parts

$P_{E_{0}}\nabla\Psi_{\varepsilon}(\xi_{0},\xi^{\prime})=0\quad\mbox{and}\quad
P_{E^{\prime}}\nabla\Psi_{\varepsilon}(\xi_{0},\xi^{\prime})=0.$ (3.13)

We perform a second reduction to express the regular part $\xi_{0}$ with
respect to the singular part $\xi^{\prime}$. We solve
$\xi_{0}(\xi^{\prime},\varepsilon)$ from the equation
$P_{E_{0}}\nabla\Psi_{\varepsilon}(\xi_{0},\xi^{\prime})=0$.

The group $G=G_{0}\times G^{\prime}$ with $G_{0}=U(1)$ and
$G^{\prime}=U(1)^{n}$ acts diagonally on $E_{0}\times E^{\prime}$. We define
the one-codimensional subspace of the regular part $E_{0}$,

$E_{0}^{\prime}=\left\\{\zeta_{0}=(\rho_{0,1}e^{i\theta_{0,1}},\dots,\rho_{0,n}e^{i\theta_{0,n}})\mid\sum_{j=1}^{n}\theta_{0,j}=0\right\\}.$
(3.14)

For every $\xi_{0}\in E_{0}$ we can find $h\in H=\widetilde{U(1)}$ such that
$\xi_{0}=h\zeta_{0}$ for some $\zeta_{0}\in E_{0}^{\prime}$. Setting
$\xi^{\prime}=h\zeta^{\prime}$ one uses $H$-invariance to get

$\Psi_{\varepsilon}(\xi_{0},\xi^{\prime})=\Psi_{\varepsilon}(h\zeta_{0},h\zeta^{\prime})=\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime}).$

The function $\Psi_{\varepsilon}$ now only depends on
$(\zeta_{0},\zeta^{\prime})\in E_{0}^{\prime}\times E^{\prime}$ and (3.13)
become

$P_{E_{0}^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})=0\quad\mbox{and}\quad
P_{E^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})=0.$ (3.15)

######  Definition 3.6.

The central configuration $a_{0}\in E_{0}$ is nondegenerate if the orbit
$U(1)(a_{0})$ is a nondegenerate critical manifold of the amended potential
$\left.V_{0}\right|_{E_{0}}:E_{0}\rightarrow\mathbb{R}$ defined in (2.11).

In this case, the kernel of
$P_{E_{0}}\left.\nabla_{u_{0}}^{2}V_{0}[a_{0}]\right|_{E_{0}}$ is generated by
$\mathcal{J}_{n}a_{0}$. This is equivalent to the assumption that the matrix
$\hat{T}_{0,u_{0}}$ in (3.6) is invertible in a complement to
$\mbox{Span}(\mathcal{J}_{n}a_{0})$ in $E_{0}$.

######  Theorem 3.5.

Suppose that $a_{0}\in E_{0}$ is nondegenerate. Then for
$\varepsilon\in(0,\varepsilon_{0})$, the critical points of
$\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})$ in the (possibly smaller)
neighbourhood $\mathcal{V}$ are in one to one correspondence with the critical
points of the function
$\Psi_{\varepsilon}^{\prime}:\mathcal{V}^{\prime}\subset
E^{\prime}\rightarrow\mathbb{R}$ given by

$\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})=\sum_{j=1}^{n}\left(r_{j}/\varepsilon\right)^{1-\alpha}V_{j}(\zeta_{j})+\mathcal{N}^{\prime}(\zeta^{\prime})\text{,}$

where $\mathcal{V}^{\prime}\subset E^{\prime}$ is a neighbourhood of the orbit
$G^{\prime}(a^{\prime})$, $V_{j}(\zeta_{j})$ is the amended potential and

$\mathcal{N}^{\prime}(\zeta^{\prime})=\varepsilon^{\alpha-1}\left(\frac{1}{2\pi}\mathcal{A}_{0}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})-\sum_{j=1}^{n}r_{j}^{1-\alpha}V_{j}(\zeta_{j})\right)+\frac{\varepsilon^{\alpha-1}}{2\pi}\mathcal{N}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime}),$
(3.16)

where $\zeta_{0}(\cdot,\varepsilon):\mathcal{V}^{\prime}\subset
E^{\prime}\rightarrow\mathbb{R}$ is unique such that
$P_{E_{0}^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})=0.$

Proof. Consider the equations obtained in (3.15)

$P_{E_{0}^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})=0\quad\mbox{and}\quad
P_{E^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})=0.$

The uniform estimate

$\|P_{X_{0}}\mathcal{C}_{\varepsilon}\nabla\mathcal{N}\left(\zeta_{0},\zeta^{\prime}\right)\|\leq\|\mathcal{C}_{\varepsilon}\nabla\mathcal{N}\left(\zeta_{0},\zeta^{\prime}\right)\|\leq
N\left(\|\zeta^{\prime}-g^{\prime}a^{\prime}\|^{2}+\|\zeta_{0}-a_{0}\|^{2}+\varepsilon\right)\quad\mbox{for
each}\quad g^{\prime}\in G^{\prime}$

implies that

$\lim_{\varepsilon\rightarrow
0}P_{E_{0}^{\prime}}\mathcal{C}_{\varepsilon}\nabla\mathcal{N}(a_{0},g^{\prime}a^{\prime})=0$

and $\lim_{\varepsilon\rightarrow
0}P_{E_{0}^{\prime}}\mathcal{C}_{\varepsilon}\nabla^{2}\mathcal{N}[a_{0},g^{\prime}a^{\prime}]=0$
for each $g^{\prime}\in G^{\prime}$. This is because the scaling matrix
$\mathcal{C}_{\varepsilon}$ acts as the identity on the component $\zeta_{0}$.
In particular,
$P_{E_{0}^{\prime}}\nabla\Psi_{0}(a_{0},g^{\prime}a^{\prime})=0$ and the
Hessian

$P_{E_{0}^{\prime}}\left.\nabla^{2}\Psi_{0}[a_{0},g^{\prime}a]\right|_{E_{0}^{\prime}}=P_{E_{0}^{\prime}}\left.\nabla^{2}V_{0}[a_{0}]\right|_{E_{0}^{\prime}}$

is non-singular on $E_{0}^{\prime}$ by the assumption that $a_{0}$ is
nondegenerate. Using the implicit function theorem and the compactness of
$G^{\prime}$, there is, for each $\varepsilon$ sufficiently small, a smooth
function $\zeta_{0}(\zeta^{\prime},\varepsilon)$ defined on a neighbourhood
$\mathcal{V}^{\prime}\subset E^{\prime}$ of the orbit $G^{\prime}(a^{\prime})$
such that

$P_{E_{0}^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})=0$
(3.17)

on this neighbourhood and $\zeta_{0}(a^{\prime},0)=a_{0}$. Hence, when we fix
$\varepsilon\in(0,\varepsilon_{0})$ with $\varepsilon_{0}$ possibly smaller,
and take a smaller neighbourhood $\mathcal{V}\subset E^{\prime}_{0}\times
E^{\prime}$, the critical points of
$\Psi_{\varepsilon}(\zeta_{0},\zeta^{\prime})$ in $\mathcal{V}$ are in one to
one correspondence with the critical points of the function
$\Psi_{\varepsilon}^{\prime}:\mathcal{V}^{\prime}\subset
E^{\prime}\rightarrow\mathbb{R}$ given by

$\Psi_{\varepsilon}^{\prime}(\zeta^{\prime}):=\frac{\varepsilon^{\alpha-1}}{2\pi}\left(\mathcal{A}_{0}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})+\mathcal{N}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})\right)=\sum_{j=0}^{n}\left(r_{j}/\varepsilon\right)^{1-\alpha}V_{j}(\zeta_{j})+\mathcal{N}^{\prime}(\zeta^{\prime}).$

These are solutions of the equation
$P_{E^{\prime}}\nabla\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})=0$. Note that
the regularizing factor of $\varepsilon^{\alpha-1}/2\pi$ leaves the equation
(3.17) unchanged when we fix $\varepsilon>0$.

######  Lemma 3.6.

There are constants $N^{\prime},N^{\prime\prime}>0$ such that, for each
$g^{\prime}\in G^{\prime}$,

$\|\zeta_{0}(\zeta^{\prime},\varepsilon)-a_{0}\|\leq
N^{\prime}(\varepsilon+\|\zeta^{\prime}-g^{\prime}a^{\prime}\|^{2})\quad\mbox{and}\quad\|P_{E^{\prime}}\nabla_{\zeta^{\prime}}\mathcal{N}(\zeta^{\prime})\|\leq
N^{\prime\prime}(\varepsilon+\|\zeta^{\prime}-g^{\prime}a^{\prime}\|^{2}).$

Proof. We first write the Taylor expansion of the operator

$P_{E_{0}^{\prime}}\nabla\mathcal{A}_{0}(\zeta_{0}(\zeta^{\prime},\varepsilon),\zeta^{\prime})=2\pi
P_{E_{0}^{\prime}}\nabla V_{0}(\zeta_{0}(\zeta^{\prime},\varepsilon))$

around $\zeta_{0}=a_{0}$. For simplicity, we omit the dependence of
$\varepsilon$ in the function $\zeta_{0}(\zeta^{\prime},\varepsilon)$. We may
shrink the neighbourhood $\mathcal{V}^{\prime}$ such that for each
$\zeta^{\prime}\in\mathcal{V}^{\prime}$ we get

$\|P_{E_{0}^{\prime}}\nabla
V_{0}(\zeta_{0}(\zeta^{\prime}))+P_{E_{0}^{\prime}}\nabla^{2}V_{0}[a_{0}](\zeta_{0}(\zeta^{\prime})-a_{0})\|\leq
N_{1}^{\prime}\|\zeta_{0}(\zeta^{\prime})-a_{0}\|^{2}$

for some constant $N_{1}^{\prime}$. By the reverse triangle inequality,

$\|P_{E_{0}^{\prime}}\nabla^{2}V_{0}[a_{0}](\zeta_{0}(\zeta^{\prime})-a_{0})\|\leq\|P_{E_{0}^{\prime}}\nabla
V_{0}(\zeta_{0}(\zeta^{\prime}))\|+N_{1}^{\prime}\|\zeta_{0}(\zeta^{\prime})-a_{0}\|^{2}.$
(3.18)

Since $\zeta_{0}(\zeta^{\prime})$ solves uniquely the equation
$P_{E_{0}^{\prime}}\nabla\Psi_{\varepsilon}(\zeta_{0}(\zeta^{\prime}),\zeta^{\prime})=0$,
we have

$2\pi P_{E_{0}^{\prime}}\nabla
V_{0}(\zeta_{0}(\zeta^{\prime}))=-P_{E_{0}^{\prime}}\nabla\mathcal{N}(\zeta_{0}(\zeta^{\prime}),\zeta^{\prime}).$

The estimates for $\mathcal{N}$ in (3.11) yields

$2\pi\|P_{E_{0}^{\prime}}\nabla
V_{0}(\zeta_{0}(\zeta^{\prime})\|^{2}\leq\|\nabla\mathcal{N}(\zeta_{0}(\zeta^{\prime}),\zeta^{\prime})\|^{2}\leq
N\left(\varepsilon+\|\zeta_{0}(\zeta^{\prime})-a_{0}\|^{2}+\|\zeta^{\prime}-a^{\prime}\|^{2}\right)$

and (3.18) becomes

$\|P_{E_{0}^{\prime}}\nabla^{2}V_{0}[a_{0}](\zeta_{0}(\zeta^{\prime})-a_{0})\|\leq
N_{2}^{\prime}(\varepsilon+\|\zeta_{0}(\zeta^{\prime})-a_{0}\|^{2}+\|\zeta^{\prime}-a^{\prime}\|^{2})$

for some $N_{2}^{\prime}$. Since
$P_{E_{0}^{\prime}}\left.\nabla^{2}V_{0}[a_{0}]\right|_{E_{0}^{\prime}}$ is
invertible, there is $c>0$ such that
$\|P_{E_{0}^{\prime}}\left.\nabla^{2}V_{0}[a_{0}]\right|_{E_{0}^{\prime}}\|\geq
2c$. Therefore

$\|\zeta_{0}(\zeta^{\prime})-a_{0}\|(2c-N_{2}^{\prime}\|\zeta_{0}(\zeta^{\prime})-a_{0}\|)\leq
N_{2}^{\prime}(\varepsilon+\|\zeta^{\prime}-a^{\prime}\|^{2}).$

We can take $\mathcal{V}^{\prime}$ smaller such that
$\|\zeta_{0}(\zeta^{\prime})-a_{0}\|\leq c/N_{2}^{\prime}$ and then

$c\|\zeta_{0}(\zeta^{\prime})-a_{0}\|\leq
N_{2}^{\prime}(\varepsilon+\|\zeta^{\prime}-a^{\prime}\|^{2})~{}.$

We now set $N^{\prime}=N_{2}^{\prime}/c$. This allows us to write
$\zeta_{0}(\zeta^{\prime})=a_{0}+R_{\varepsilon}(\zeta^{\prime})$, where
$R_{\varepsilon}(\zeta^{\prime})$ is the remainder which satisfies by the
above

$\|R_{\varepsilon}(\zeta^{\prime})\|\leq
N^{\prime}(\varepsilon+\|\zeta^{\prime}-a^{\prime}\|^{2}).$ (3.19)

To obtain the second estimate, we use the definition (3.16) and we replace
$\zeta_{0}(\zeta^{\prime})$ by $a_{0}+R_{\varepsilon}(\zeta^{\prime})$. By
(3), the first terms of (3.16) become

$\frac{\varepsilon^{\alpha-1}}{2\pi}\mathcal{A}_{0}(a_{0}+R_{\varepsilon}(\zeta^{\prime}),\zeta^{\prime})-\sum_{j=1}^{n}\left(r_{j}/\varepsilon\right)^{1-\alpha}V_{j}(\zeta_{j})=\varepsilon^{\alpha-1}V_{0}(a_{0}+R_{\varepsilon}(\zeta^{\prime})).$

Applying the mean value theorem, there is some $\mu\in[0,1]$ such that

$P_{E^{\prime}}\nabla
V_{0}(a_{0}+R_{\varepsilon}(\zeta^{\prime}))=P_{E^{\prime}}\nabla^{2}V_{0}(a_{0}+\mu
R_{\varepsilon}(\zeta^{\prime}))\left(R_{\varepsilon}(\zeta^{\prime})\right),$

and there is a constant $e$ such that

$\left\|P_{E^{\prime}}\nabla
V_{0}(a_{0}+R_{\varepsilon}(\zeta^{\prime}))\right\|\leq
e\left\|R_{\varepsilon}(\zeta^{\prime})\right\|.$ (3.20)

We finally get

$\|P_{E^{\prime}}\nabla\mathcal{N}(\zeta^{\prime})\|\leq\varepsilon^{\alpha-1}e\left\|R(\zeta^{\prime})\right\|+N(\varepsilon+\|\zeta^{\prime}-a\|^{2}+\|R_{\varepsilon}(\zeta^{\prime})\|^{2})\leq
N^{\prime\prime}(\varepsilon+\|\zeta^{\prime}-a^{\prime}\|^{2})$

for some $N^{\prime\prime}$ after using the uniform estimate (3.11) and
(3.19). The arguments can be repeated replacing $a^{\prime}$ by
$g^{\prime}a^{\prime}$ for each $g^{\prime}\in G^{\prime}$ and the
neighbourhood $\mathcal{V}^{\prime}$ can be taken as a neighbourhood of the
orbit $G^{\prime}(a^{\prime})$ by compactness of $G^{\prime}$.

### Lyusternik-Schnirelmann application

We now show that the function

$\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})=\sum_{j=1}^{n}\left(r_{j}/\varepsilon\right)^{1-\alpha}V_{j}(\zeta_{j})+\mathcal{N}^{\prime}(\zeta^{\prime})\text{,}$
(3.21)

has critical points in the neighbourhood $\mathcal{V}^{\prime}\subset
E^{\prime}$ of the orbit $G^{\prime}(a^{\prime})$.

######  Theorem 3.7.

If $a_{0}$ is a nondegenerate central configuration and $a_{j}$ is a $2\pi
p_{j}$-nondegenerate central configuration for each $j=1,\dots,n$. Then, for
each $\varepsilon\in(0,\varepsilon_{0})$, there is a neighbourhood
$\mathcal{V}^{\prime}\subset E^{\prime}$ of the orbit $G^{\prime}(a^{\prime})$
such that the number of critical points of the function
$\Psi^{\prime}_{\varepsilon}:\mathcal{V}^{\prime}\to\mathbb{R}$ is bounded
below by $\mbox{Cat}(G^{\prime}/K)$ where $K$ is the stabiliser of
$a^{\prime}$.

Proof. Notice that to critical points of the function
$\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})$ restricted to the subspace
$E^{\prime}\subset E^{N}$ are the solutions of
$P_{E^{\prime}}\nabla\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})=0$. Using the
expansion (2.18), we obtain $\lim_{\varepsilon\rightarrow
0}\left(r_{j}/\varepsilon\right)^{1-\alpha}=p_{j}^{2(\alpha-1)/(\alpha+1)}$,
and

$P_{E_{j}}\nabla\Psi_{0}^{\prime}(a^{\prime})=\left(p_{j}^{2(\alpha-1)/(\alpha+1)}P_{E_{j}}\nabla
V_{j}(a_{j})+\lim_{\varepsilon\rightarrow
0}P_{E_{j}}\nabla\mathcal{N}^{\prime}(a^{\prime})\right).$

By the second estimate in Lemma 3.6, $\|\lim_{\varepsilon\rightarrow
0}P_{E^{\prime}}\nabla_{\zeta^{\prime}}\mathcal{N}^{\prime}(\zeta^{\prime})\|\leq
N^{\prime}\|\zeta^{\prime}-a^{\prime}\|^{2}$, we have

$\lim_{\varepsilon\rightarrow
0}P_{E^{\prime}}\nabla\mathcal{N}^{\prime}(a^{\prime})=0,\qquad\lim_{\varepsilon\rightarrow
0}P_{E^{\prime}}\nabla^{2}\mathcal{N}^{\prime}[a^{\prime}]|_{E^{\prime}}=0.$

Since $a_{j}$ is a critical point of $V_{j}$ for each $j=1,\dots,n$, we get
$P_{E_{j}}\nabla V_{j}(a_{j})$ and
$P_{E^{\prime}}\nabla\Psi_{0}^{\prime}(a^{\prime})=0$. Furthermore, the same
estimate implies that the Hessian of $\Psi_{0}^{\prime}$ respect to
$E^{\prime}$ is

$P_{E^{\prime}}\nabla^{2}\Psi_{0}^{\prime}[a^{\prime}]|_{E^{\prime}}=p_{1}^{2(\alpha-1)/(\alpha+1)}\hat{T}_{0,u_{1}}\oplus\dots\oplus
p_{n}^{2(\alpha-1)/(\alpha+1)}\hat{T}_{0,u_{n}},$

where

$\hat{T}_{0,u_{j}}=P_{E_{j}}\nabla^{2}V_{j}[a_{j}]|_{E_{j}}\in\mbox{End}(E_{j}).$

By the $2\pi p_{j}$-nondegeneracy assumption on $a_{j}$, each block
$\hat{T}_{0,u_{j}}$ is non-singular on a subspace $W_{j}$ complementary to
$T_{a_{j}}U(1)(a_{j})$ in $E_{j}$. Consequently, $W=\bigoplus_{j=1}^{n}W_{j}$
is a complement to $T_{a^{\prime}}G^{\prime}(a^{\prime})$ in $E^{\prime}$. The
argument above is valid if we replace $a^{\prime}$ by $g^{\prime}a^{\prime}$
for any $g^{\prime}\in G^{\prime}$. A standard application of the Palais-Slice
coordinates as in [17] allows us to express the normal coordinates in $W$ in
term of the coordinates along the group orbit $G^{\prime}(a^{\prime})$ after
taking $\varepsilon_{0}$ and $\mathcal{V}^{\prime}$ possibly smaller. For
$\varepsilon\in(0,\varepsilon_{0})$, the solutions of
$P_{E^{\prime}}\nabla\Psi_{\varepsilon}^{\prime}(\zeta^{\prime})=0$ in
$\mathcal{V}^{\prime}$ are in one to one correspondence with the critical
points the function
$\Psi_{\varepsilon}^{\prime}:G^{\prime}(a^{\prime})\rightarrow\mathbb{R}$. By
the Lyusternik-Schnirelmann theorem for compact manifolds [14, 28], the number
of critical points of this function is bounded below by
$\mbox{Cat}(G^{\prime}/K)$ where $K$ is the stabiliser of $a^{\prime}$.

### Existence of carousel solutions

We state our main result regarding the existence of carousel solutions of the
$N$-body problem. In summary, we proved that the existence of periodic
solutions near the orbit $G(u_{a})$ with $u_{a}=(a_{0},a_{1},\dots,a_{n})$
reduces to determining whether the function (3.21) admits critical points.
Recall that we assume that $k_{j}>1$ for $j=1,\dots,n_{0}$ and $k_{j}=1$ for
$j=n_{0}+1,\dots,n$. Thus $K=U(1)^{n-n_{0}}$, because $G^{\prime}=U(1)^{n}$
acts trivially on each subspace $E_{j}=\\{0\\}$ for $j=n_{0}+1,\dots,n$. By
Theorem 3.7, there are at least $\mbox{Cat}(G^{\prime}/K)=n_{0}+1$ solutions.
Therefore, we have,

######  Theorem 3.8 (Carousels for non-gravitational potentials).

Set $\alpha\neq 2$ and $k_{j}>1$ for $j=1,\dots,n_{0}$ and $k_{j}=1$ for
$j=n_{0}+1,\dots,n$. Fix integers
$p_{1},\dots,p_{n_{0}}\in\mathbb{Z}\backslash\\{0\\}$ and choose the
frequencies $\omega_{j},\nu$ and the amplitudes $r_{j}$ according to the
conditions (A)-(B). Suppose that $a_{0}$ is a nondegenerate central
configuration of the $n$-body problem and $a_{j}$ is a $2\pi
p_{j}$-nondegenerate central configuration of the $k_{j}$-body problem for
$j=1,\dots,n_{0}$. Then for every sufficiently small $\varepsilon$, there are
at least $\mbox{Cat}(G^{\prime}/K)=n_{0}+1$ solutions of the $N$-body problem
(1.1) with components of the form

$\displaystyle q_{j,k}(t)$ $\displaystyle=$ $\displaystyle\exp(tJ)u_{0,j}(\nu
t)+r_{j}\exp(t\omega_{j}J)u_{j,k}(\nu t),\quad j=1,\dots,n_{0},\quad
k=1,\dots,k_{j}$ $\displaystyle q_{j,1}(t)$ $\displaystyle=$
$\displaystyle\exp(tJ)u_{0,j}(\nu t)\quad\mbox{for}\;j=n_{0}+1,\dots,n.$

where $u_{0,j}(\nu t)=a_{0,j}+\mathcal{O}_{X}(\varepsilon)$ and $u_{j,k}(\nu
t)=e^{\vartheta_{j}J}a_{j,k}+\mathcal{O}_{X}(\varepsilon)$ for
$j=1,\dots,n_{0}$ with some phases $\vartheta_{j}\in S^{1}$.

These solutions are quasi-periodic if $\nu\notin\mathbb{Q}$, and periodic if
$\nu\in\mathbb{Q}$. In the case that $\nu=(p-q)/q$ is rational, then
$\varepsilon^{-\left(\alpha+1\right)/2}=p/q$ and $\omega_{j}=(q+p_{j}p)/q$ are
rational. Thus, for any fixed integer $q>0$, there is some $p_{0}>0$ such
that, for each $p>p_{0}$ , the components $q_{j}(t)$ are $2\pi q$-periodic. In
these solutions the centers of mass of $n$ clusters (close to the central
configuration $a_{0}$) wind around the origin $q$ times, while each
$j$-cluster winds around its center of masses $q+p_{j}p$ times. The sign of
the frequency $\omega_{j}$ is determined by $p_{j}$ and represents whether the
$j$-cluster has prograde or retrograde rotation with respect to the whole
system. In the case of rational $\nu=\left(p-q\right)/q$, the _prograde_
($p_{j}>0$) refers to the case that the cluster rotates in the same direction
as the main relative equilibrium, while _retrograde_ ($p_{j}<0$) refers to the
case that the cluster rotates in the opposite direction.

The strategy still applies for the gravitational potential under extra
assumptions. In this case, we have from condition (C1) that

$r_{1}=\varepsilon,\qquad\omega_{1}=\varepsilon^{-\left(\alpha+1\right)/2},\qquad\nu=\varepsilon^{-\left(\alpha+1\right)/2}-1.$

######  Theorem 3.9 (Carousels for gravitational potentials).

Set $\alpha=2$, assume conditions (C0)-(C2) and choose the frequencies
$\omega_{1},\nu$ and the amplitude $r_{1}$ according to the conditions (C1).
Suppose that $a_{0}$ is a nondegenerate central configuration of the $n$-body
problem, and $a_{1}$ is a $2\pi/m$-nondegenerate central configuration. Then
for every sufficiently small $\varepsilon$, there are at least
$\mbox{Cat}(G^{\prime}/K)=2$ solutions of the $N$-body problem with components
of the form

$\displaystyle q_{1,k}(t)$ $\displaystyle=$ $\displaystyle\exp(tJ)u_{0,1}(\nu
t)+r_{1}\exp(t\omega_{1}J)u_{1,k}(\nu t),$ $\displaystyle q_{j,1}(t)$
$\displaystyle=$ $\displaystyle\exp(tJ)u_{0,j}(\nu
t)\quad\mbox{for}\;j=2,\dots,n.$

where $u_{0,j}(\nu t)=a_{0,j}+\mathcal{O}_{X}(\varepsilon)$ and $u_{1,k}(\nu
t)=e^{\vartheta_{1}J}a_{1,k}+\mathcal{O}_{X}(\varepsilon)$ for some phase
$\vartheta_{1}\in S^{1}$. Furthermore, each $u_{1,k}(s)$ is $2\pi/m$-periodic
with $m\geq 2$ and

$u_{0,j}(s)=\exp(-\theta J)u_{0,\sigma(j)}(s+\theta)$ (3.22)

where $(\theta,\sigma)$ is the generator of the discrete symmetry group
$\Gamma$ introduced in (C0).

######  Example 3.7.

The particular case $k_{1}=2$ and $k_{2},\dots,k_{n}=1$ is studied in [17] in
the general setting $E=\mathbb{R}^{2d}$. This follows from the fact that the
Euler-Lagrange equations (2.9) for $j=1$ are equivalent to a $2$-body problem
in a rotating frame of frequency $\omega$. That is, the system for $u_{1}\in
E_{1}$ can be parametrized by $w_{1}\in E$ as

$E_{1}=\left\\{u_{1}=(u_{1,1},u_{1,2})\in
E^{2}:u_{1,1}=\frac{m_{1,2}}{m_{1,1}+m_{1,2}}w_{1},u_{1,2}=-\frac{m_{1,1}}{m_{1,1}+m_{1,2}}w_{1}\right\\}.$

In these coordinates $\left(u_{0},w_{1}\right)\in E_{0}\times E$ the action
$\mathcal{A}_{0}(u_{0},u_{1})$ becomes

$\mathcal{A}_{0}(u_{0},w_{1})=\int_{0}^{2\pi}\left(\mathcal{L}_{0}(u_{0},\dot{u}_{0})+r_{1}^{1-\alpha}\mathcal{L}_{1}(w_{1},\dot{w}_{1})\right),$

where

$\mathcal{L}_{1}(w_{1},\dot{w}_{1})=\frac{m_{1,1}m_{1,2}}{m_{1,1}+m_{1,2}}\left\|(\partial_{t}+\omega
J)w_{1}\right\|^{2}+m_{1,1}m_{1,2}\phi_{\alpha}(\left\|w_{1}\right\|),$

which corresponds to the Lagrangian of the Kepler problem in a rotating frame
of frequency $\omega$. We are then left with a Kepler problem for
$\mathcal{L}_{1}(w_{1},\dot{w}_{1})$ and an $n$-body problem for
$\mathcal{L}_{0}(u_{0},\dot{u}_{0})$. We can normalize the total mass of the
cluster by setting $m_{1,1}+m_{1,2}=1$. The solutions (2.12) are of the form

$\displaystyle q_{1,1}(t)$ $\displaystyle=\exp(tJ)u_{0,1}(\nu t)+\varepsilon
m_{1,2}\exp(\omega_{1}tJ)w_{1}(\nu t),$ $\displaystyle q_{1,2}(t)$
$\displaystyle=\exp(tJ)u_{0,1}(\nu t)-\varepsilon
m_{1,1}\exp(\omega_{1}tJ)w_{1}(\nu t),$ $\displaystyle q_{j,1}(t)$
$\displaystyle=\exp(tJ)u_{0,j}(\nu t),\quad j=2,\dots,n,$

which are exactly the solutions obtained in [17] for the particular case
$E=\mathbb{R}^{2}$.

##  4 The $2\pi p$-nondegeneracy property of the $k$-polygon

In this section we verify the $2\pi p$-nondegeneracy property of a polygonal
central configuration (with frequency one) for $k$ bodies with masses equal to
one. We shall denote by $E_{red}^{k}$ the subspace of $E^{k}$ of central
configurations with center of mass fixed at the origin. With our previous
notations, we have $E_{j}=E_{red}^{k_{j}}$. By (3), a central configuration
$a\in E_{red}^{k}$ is $2\pi p$-nondegeneracy if the block
$\hat{T}_{\ell,u}\in\mbox{End}((E_{red}^{k})^{\mathbb{C}})$ given by

$\hat{T}_{\ell,u}=(\ell^{2}+1)^{-1}P_{E_{red}^{k}}M_{a}(\ell/p)|_{E_{red}^{k}}\in\mbox{End}((E_{red}^{k})^{\mathbb{C}})$

is a non-singular matrix. Here

$M_{a}(\lambda)=\lambda^{2}\mathcal{I}-2i\lambda\mathcal{J}+\nabla_{u}^{2}V[a]\in\mbox{End}((E^{k})^{\mathbb{C}})$
(4.1)

and $V$ is the amended potential

$V(u)=\frac{1}{2}\sum_{j=1}^{k}\left\|u_{j}\right\|^{2}+\sum_{1\leq i<j\leq
k}\phi_{\alpha}\left(\left\|u_{j}-u_{i}\right\|\right)\text{.}$ (4.2)

Specifically, the $2\pi p$-nondegeneracy property of $a$ is equivalent to the
conditions:

  1. (a)

$\hat{T}_{\ell,u}$ is invertible for all $\ell\neq 0$,

  2. (b)

$\hat{T}_{0,u}$ has a one dimensional kernel generated by $\mathcal{J}_{k}a$.

###  4.1 Spectrum of the polygonal configuration

First we find the ratio of the polygonal relative equilibrium with frequency
one.

######  Proposition 4.1.

The polygonal configuration

$a=(s_{1})^{\frac{1}{\alpha+1}}(\exp(J\zeta)e_{1},\dots,\exp(kJ\zeta)e_{1}),\quad
e_{1}=\begin{bmatrix}1\\\ 0\end{bmatrix}$ (4.3)

is a central configuration of frequency one, where $\zeta=2\pi/k$ and

$s_{1}=\frac{1}{2^{\alpha}}\sum_{j=1}^{k-1}\frac{1}{\sin^{\alpha-1}(j\zeta/2)}.$

Proof. We shall show that $a$ is a critical point of the amended potential
$V$. Consider the function

$\widetilde{V}(u;\omega)=\frac{\omega}{2}\sum_{j=1}^{k}\left\|u_{j}\right\|^{2}+\sum_{1\leq
i<j\leq k}\phi_{\alpha}\left(\left\|u_{j}-u_{i}\right\|\right).$ (4.4)

It satisfies $\widetilde{V}(u;1)=V(u)$ and has the scaling property

$\nabla\widetilde{V}(ru;r^{-\left(\alpha+1\right)}\omega)=r^{-\alpha}\nabla\widetilde{V}(u;\omega)\quad\mbox{for
any}\quad r>0.$ (4.5)

Thus $(\widetilde{a};\omega)$ is critical point of $\widetilde{V}$ if and only
if $(r\widetilde{a};r^{-\left(\alpha+1\right)}\omega)$ is a critical points of
$\widetilde{V}$ for any $r>0$. By [18], the unitary polygon

$\widetilde{a}=(\exp(J\zeta)e_{1},\dots,\exp(kJ\zeta)e_{1}),\quad
e_{1}=\begin{bmatrix}1\\\ 0\end{bmatrix}$

with $\omega=s_{1}$ is a critical point of $\widetilde{V}$. It follows that
$(a;1)$ where $a=(s_{1})^{\frac{1}{\alpha+1}}\widetilde{a}$ is a critical
point of $\widetilde{V}(u;1)=V(u)$. In other words, $a$ is a central
configuration with frequency one.

#### Block diagonalisation

Let $S_{k}$ be the permutation group of $k$ letters. The group $G=S_{k}\times
SO(2)$ acts on $E^{k}$ by

$(\sigma,\theta)\cdot(u_{1},\dots,u_{k})=\left(\exp(-J\theta)u_{\sigma(1)},\dots,\exp(-J\theta)u_{\sigma(k)}\right).$

The amended potential (4.2) is $G$-invariant. Let $a\in E^{k}$ be the
polygonal configuration (4.3). Its stabiliser is the subgroup $G_{a}=C_{k}$
generated by the element $(\sigma,\zeta)\in G$ where $\sigma=(1\,2\,\dots\,k)$
and $\zeta=\frac{2\pi}{k}$. The $G_{a}$-equivariant property of the Hessian
$\nabla_{u}^{2}V[a]$ is used in Proposition 7 of [18] to find the irreducible
representations of $(E^{\mathbb{C}})^{k}$. By Schur’s lemma the Hessian of $V$
is equivalent to a block diagonal matrix with $k$ blocks corresponding to the
isotypic components.

######  Definition 4.1.

For $j=1,\dots,k$, we define isomorphisms $T_{j}:E^{\mathbb{C}}\rightarrow
W_{j}$ by

$T_{j}(w)=\frac{1}{\sqrt{k}}(\exp((ijI+J)\zeta)w,\dots,\exp(k(ijI+J)\zeta)w),$
(4.6)

where

$W_{j}=\left\\{(\exp((ijI+J)\zeta)w,\dots,\exp(k(ijI+J)\zeta)w)\mid w\in
E^{\mathbb{C}}\right\\}\subset(E^{\mathbb{C}})^{k}.$

Specifically, in [18] is proved that the subspaces $W_{j}$ are the isotypic
components under the action of $G_{a}$. The group $G_{a}$ acts on each
subspace $W_{j}$ by rotating each component by $\exp(ij\zeta J)$. Since the
subspaces $W_{j}$ are mutually orthogonal, the endomorphism
$P\in\mbox{End}((E^{\mathbb{C}})^{k})$ defined by

$P(w_{1},\dots,w_{k})=\sum_{j=1}^{k}T_{j}(w_{j})$

is orthogonal. Since $P$ rearranges the coordinates of the isotypic
decomposition, it follows by Schur’s Lemma that

$P^{-1}\nabla_{u}^{2}V[a]P=B_{1}\oplus\dots\oplus B_{k}$ (4.7)

where each $B_{j}\in\mbox{End}(E^{\mathbb{C}})$ satisfies

$\nabla_{u}^{2}V[a]T_{j}(w)=T_{j}(B_{j}w).$ (4.8)

Define

$s_{j}=\frac{1}{2^{\alpha}}\sum_{l=1}^{k-1}\frac{\sin^{2}(jl\zeta/2)}{\sin^{\alpha+1}(l\zeta/2)},\qquad\zeta=\frac{2\pi}{k}\text{.}$
(4.9)

Following [18], the numbers $s_{j}$ have the following properties: They are
$k$-periodic, that is $s_{k+j}=s_{j}$. They are symmetric with respect to
$[k/2]$, that is $s_{[k/2]-j}=s_{[k/2]-j}$. They increase as $j$ increases,
that is

$s_{j+1}>s_{j},\qquad 0\leq j\leq n/2\text{.}$

In particular, $s_{j}>s_{0}=0$ for $j=1,...,[n/2]$ with its maximum attained
at $j=[n/2].$

######  Proposition 4.2 (Normal form of the amended potential).

Each endomorphism $B_{j}$ is a matrix of the form

$B_{j}=(1+\alpha_{j})I-\beta_{j}R-\gamma_{j}iJ$ (4.10)

where $I$ is the identity matrix,

$R=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\qquad J=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}$

and the coefficients are given by

$\alpha_{j}=\frac{\alpha-1}{4s_{1}}(s_{j+1}+s_{j-1}),\qquad\beta_{j}=\frac{\alpha+1}{2s_{1}}(s_{j}-s_{1}),\qquad\gamma_{j}=\frac{\alpha-1}{4s_{1}}(s_{j+1}-s_{j-1}).$
(4.11)

Proof. By the scaling property (4.5) of the amended potential $\widetilde{V}$
given in (4.4), we have

$\nabla_{u}^{2}V[a]=\nabla_{u}^{2}\widetilde{V}[(a\mathbf{;}1)]=\frac{1}{s_{1}}\nabla_{u}^{2}\widetilde{V}[(\widetilde{a}\mathbf{;}s_{1})].$
(4.12)

By (4.8) we get

$\nabla_{u}^{2}V[a]T_{j}(w)=\frac{1}{s_{1}}\nabla_{u}^{2}\widetilde{V}[(\widetilde{a}\mathbf{;}s_{1})]T_{j}(w)=T_{j}\left(\frac{1}{s_{1}}\widetilde{B}_{j}w\right).$

The matrices $\widetilde{B}_{j}$ are computed in [18] and are given by

$\widetilde{B}_{j}=(s_{1}+\widetilde{\alpha}_{j})I-\widetilde{\beta}_{j}R-\widetilde{\gamma}_{j}iJ$

where

$\widetilde{\alpha}_{j}=\frac{\alpha-1}{4}(s_{j+1}+s_{j-1}),\qquad\widetilde{\beta}_{j}=\frac{\alpha+1}{2}(s_{j}-s_{1}),\qquad\widetilde{\gamma}_{j}=\frac{\alpha-1}{4}(s_{j+1}-s_{j-1}).$

The result follows from $\nabla_{u}^{2}V[a]T_{j}(w)=T_{j}(B_{j}w)$ with
$B_{j}=\frac{1}{s_{1}}\widetilde{B}_{j}$.

We can now find a block diagonalisation of the matrix

$M_{a}(\lambda)=\lambda^{2}\mathcal{I}-2i\lambda\mathcal{J}+\nabla_{u}^{2}V[a]$

that appears in (4.1). With respect to the isotypic decompositions, we get by
(4.7)

$P^{-1}M_{a}(\lambda)P=m_{1}(\lambda)\oplus\dots\oplus m_{k}(\lambda),$

where each block $m_{j}(\lambda)\in\mbox{End}(E^{\mathbb{C}})$ is given by

$m_{j}(\lambda)=\lambda^{2}I-2\lambda iJ+B_{j}.$ (4.13)

#### Zero eigenvalues arising from symmetries

The analysis of the spectrum of the blocks $m_{j}(\lambda)$ for $j=1,k-1,k$ is
special because the block $m_{k}(\lambda)$ contains the generator of the
$SO(2)$-orbit and the blocks $m_{1}(\lambda)$ and $m_{k-1}(\lambda)$ contain
vectors in the orthogonal complement to $(E_{red}^{k})^{\mathbb{C}}$. These
cases are treated separately in Lemma 4.3 and Lemma 4.4.

######  Lemma 4.3.

Given the block diagonal matrix
$P^{-1}M_{a}(\lambda)P=m_{1}(\lambda)\oplus\dots\oplus m_{k}(\lambda)$,

  1. 1.

The block $m_{k}(\ell/p)$ is invertible for all $\ell\in\mathbb{Z}/\\{0\\}$
when $p\left(3-\alpha\right)^{1/2}\notin\mathbb{N}$.

  2. 2.

The block $m_{k}(0)\in\mbox{End}(E^{\mathbb{C}})$ is invertible on the
orthogonal complement of the line spanned by $e_{2}=Je_{1}$ which is a
generator of the space tangent to the $SO(2)$-orbit at $a$.

Proof.

  1. 1.

The coefficients in (4.11) have the property $s_{j}=s_{-j}=s_{k-j}$ and
$s_{k}=0$. The coefficients (4.11) are then given by

$\alpha_{k}=\frac{\alpha-1}{2},\qquad\beta_{k}=-\frac{\alpha+1}{2},\qquad\gamma_{k}=0.$

From (4.10) we get Thus we have that

$m_{k}(\lambda)=\lambda^{2}I-2\lambda
iJ+\frac{\alpha+1}{2}(I+R)=\left(\begin{array}[c]{cc}\lambda^{2}+\alpha+1&2\lambda
i\\\ -2\lambda i&\lambda^{2}\end{array}\right)$

which is the linearisation of the Kepler problem as in [17] and corresponds to
the invariant manifold of homographic solutions. This matrix has eigenvalues

$\mu_{k}^{\pm}(\lambda)=\frac{\alpha+1}{2}+\lambda^{2}\pm\frac{1}{2}\sqrt{(\alpha+1)^{2}+16\lambda^{2}}.$

The eigenvalue $\mu_{k}^{+}(\lambda)\neq 0$ for all $\lambda$ and
$\mu_{k}^{-}(\lambda)\neq 0$ if $\lambda\notin\\{0,\sqrt{3-\alpha}\\}$. In our
analysis, $\lambda=\ell/p$ where $\ell,p$ are integers and $\ell\neq 0$. In
particular, the operator is invertible if we suppose that
$p\left(3-\alpha\right)^{1/2}\notin\mathbb{N}$. When $\alpha=2$, note that the
eigenvalues $\mu_{k}^{-}(\ell/p)$ for $\ell=\pm p$ are equal to zero due to
the existence of the homographic elliptic orbits of the gravitational $k$-body
problem.

  2. 2.

Since
$a=s_{1}^{\frac{1}{\alpha+1}}(\exp(J\zeta)e_{1},\dots,\exp(kJ\zeta)e_{1})$,
the generator of the tangent space of its $SO(2)$-orbit is

$\mathcal{J}_{k}a=s_{1}^{\frac{1}{\alpha+1}}(J\exp(J\zeta)e_{1},\dots,J\exp(kJ\zeta)e_{1})=s_{1}^{\frac{1}{\alpha+1}}T_{k}(e_{2})$

since $Je_{1}=e_{2}$. Therefore, the matrix $m_{k}(0)=\mbox{diag}(\alpha+1,0)$
is singular only on the space tangent to the $U(1)$-orbit of $a$ which is
generated by $e_{2}$ in $E^{\mathbb{C}}$.

We now analyze the spectrum of $m_{1}(\lambda)$ and $m_{k-1}(\lambda)$. In our
analysis, the matrix $M_{a}(\lambda)$ is an endomorphism of the subspace
reduced by the symmetry of translations $(E_{red}^{k})^{\mathbb{C}}$.
Therefore, the spectrum of the blocks $m_{1}(\lambda)$ and $m_{k-1}(\lambda)$
must be analyzed on the subspaces $T_{j}^{-1}((W_{j})_{red})$ for $j=1$ and
$j=k-1$ respectively. We use the notation

$(W_{j})_{red}=W_{j}\cap(E_{red}^{k})^{\mathbb{C}}.$

######  Lemma 4.4.

If $\alpha>1$, the matrix $m_{j}(\lambda)$ restricted to the subspace
${T_{j}^{-1}((W_{j})_{red})}$ for $j=1,k-1$ is invertible for any $\lambda$.

Proof.

##### Case $j=1$.

By (4.13) and Proposition 4.2, the coefficients in (4.10) are

$\alpha_{1}=\gamma_{1}=\frac{\alpha-1}{4s_{1}}s_{2}\qquad\beta_{1}=0$

and then

$m_{1}(\lambda)=(\lambda^{2}+1+\alpha_{1})I-\left(2\lambda+\alpha_{1}\right)iJ=\left(\begin{array}[c]{cc}\lambda^{2}+1+\alpha_{1}&\left(2\lambda+\alpha_{1}\right)i\\\
-\left(2\lambda+\alpha_{1}\right)i&\lambda^{2}+1+\alpha_{1}\end{array}\right)\text{.}$

The eigenvalues of $m_{1}(\lambda)$ are
$\mu_{1}(\lambda)=\left(\lambda-1\right)^{2}\ $with eigenvector $w_{1}=(1,i)$
and $\mu_{2}(\lambda)=\left(\lambda+1\right)^{2}+2\alpha_{1}$ with eigenvector
$w_{2}=(1,-i)$. The first eigenvalue $\mu_{1}(\lambda)$ vanishes when
$\lambda=1$, but we will show that the corresponding eigenvector $w_{1}$ does
not belong to $T_{1}^{-1}((W_{1})_{red})$. By (4.6)

$T_{1}(w_{1})=\frac{1}{\sqrt{k}}(\exp((iI+J)\zeta)w_{1},\dots,\exp(k(iI+J)\zeta)w_{1}).$

Observe that

$\exp(j\zeta J)w_{1}=\left(\begin{array}[c]{cc}\cos j\zeta&-\sin j\zeta\\\
\sin j\zeta&\cos j\zeta\end{array}\right)\left(\begin{array}[c]{c}1\\\
i\end{array}\right)=\left(\begin{array}[c]{c}\cos\left(j\zeta\right)-i\sin\left(j\zeta\right)\\\
i\cos\left(j\zeta\right)+\sin\left(j\zeta\right)\end{array}\right)=e^{-ij\zeta}Iw_{1}$

for $j=1,\dots,k$ from which it follows that

$T_{1}(w_{1})=\frac{1}{\sqrt{k}}\left(w_{1},\dots,w_{1}\right).$

In particular, $T_{1}(w_{1})$ does not belong to $(W_{1})_{red}$. The matrix
$m_{1}(\lambda)$ restricted to $T_{1}^{-1}((W_{1})_{red})$ is given by
$\left(\lambda+1\right)^{2}+2\alpha_{1}$, which is invertible for $\alpha>1$.

##### Case $j=k-1$.

By (4.13) and Proposition 4.2, the coefficients in (4.10) are

$\alpha_{k-1}=-\gamma_{k-1}=\frac{\alpha-1}{4s_{1}}s_{2}=\alpha_{1}\qquad\beta_{1}=0$

and then

$m_{k-1}(\lambda)=\left(\begin{array}[c]{cc}\lambda^{2}+1+\alpha_{1}&\left(2\lambda-\alpha_{1}\right)i\\\
-\left(2\lambda-\alpha_{1}\right)i&\lambda^{2}+1+\alpha_{1}\end{array}\right)$

are $\mu_{1}(\lambda)=\left(\lambda+1\right)^{2}$ with eigenvector
$w_{1}=(1,-i)$ and $\mu_{2}(\lambda)=\left(\lambda-1\right)^{2}+2\alpha_{1}$
with eigenvector $w_{2}=(1,i)$. Since $\exp(j\zeta J)w_{1}=e^{ij\zeta}Iw_{1}$
for $j=1,\dots,k$, we get

$T_{k-1}(w_{1})=\frac{1}{\sqrt{k}}(w_{1},\dots,w_{1}).$

In particular, $T_{k-1}(w_{1})$ does not belong to $(W_{k-1})_{red}$. The
matrix $m_{k-1}(\lambda)$ restricted to $T_{k-1}^{-1}((W_{k-1})_{red})$ is
given by $\left(\lambda-1\right)^{2}+2\alpha_{1}$, which is invertible for
$\alpha>1$.

###  4.2 The $2\pi p$-nondegeneracy property of the $k$-polygon for weak
forces

We now study the special case of week forces.

######  Proposition 4.5.

Assume that

$\mbox{(i)}\quad
p\left(3-\alpha\right)^{1/2}\notin\mathbb{N}\qquad\mbox{(ii)}\quad
p^{2}\frac{j(k-j)}{k-1}\notin\mathbb{N}\quad j=2,...,k-2.$

Then there exists $\delta>0$ such that the $k$-polygon is $2\pi
p$-nondegenerate for any $\alpha\in(1,1+\delta)$.

Proof. By Proposition 4.3 and assumption (i), the block $m_{k}(\ell/p)$ is
invertible for $\ell\neq 0$ and $m_{k}(0)$ is invertible in an orthogonal
complement to the generator of the orbit. By Proposition 4.4 and the
assumption that $\alpha>1$, the blocks $m_{j}(\ell/p)$ restricted to
$T_{j}^{-1}((W_{j})_{red})$ are invertible for $j=1,k-1$. It remains to show
that the blocks $m_{j}(\ell/p)$ are invertible for all $j=2,\dots,k-2$ and
$\ell\in\mathbb{Z}$.

In the logarithmic case $\alpha=1$ the numbers $s_{j}$ can be computed
explicitly [19] and are given by

$s_{j}=\frac{j(k-j)}{2}\text{.}$

In this case, the coefficients in (4.11) are

$\alpha_{j}=\gamma_{j}=0,\qquad\beta_{j}=\frac{s_{j}}{s_{1}}-1.$

We can compute the matrix (4.13) explicitly and the determinant is

$\det\left(m_{j}(\lambda)\right)=(\lambda^{2}-1)^{2}-\beta_{j}^{2}.$

Thus $\det\left(m_{j}(\lambda)\right)\neq 0$ if and only if

$\lambda^{2}\neq\frac{s_{j}}{s_{1}}=\frac{j(k-j)}{k-1}.$

Therefore, the blocks $m_{j}(\ell/p)$ are invertible if and only if assumption
(ii) holds. Notice that there is $\ell_{0}>0$ such that the blocks
$m_{j}(\ell/p)$ are always invertible for $\left|\ell\right|>\ell_{0}$, which
reflects the compactness nature of the operators. By continuity of
$m_{j}(\ell/p)$ with respect to $\alpha$, there is a $\delta>0$ such that for
any $\alpha\in(1,1+\delta)$, the remaining blocks $m_{j}(\ell/p)$ for
$\left|\ell\right|<\ell_{0}$ are invertible for all $j=2,\dots,k-2$. The
result follows.

For $\alpha\in(1,1+\delta)$ and $k\in\mathbb{N}$ set

$\mathcal{C}_{k,\alpha}=\left\\{p\in\mathbb{N}:p\notin(k-1)\mathbb{N}\cup(3-\alpha)^{-1/2}\mathbb{N}\right\\}.$

For each prime $k-1$ and $p\in\mathcal{C}_{k,\alpha}$ we have that conditions
(i) and (ii) hold. This follows from the fact that $k-1$ does not divide
$p^{2}$ if $p\in\mathcal{C}_{k,\alpha}$, neither $j(k-j)$ for $j=2,...,k-1$
because $k-1$ is prime.

Now, in the case that $\left(3-\alpha\right)^{-1/2}$ is irrational, the set
$\mathcal{C}_{k,\alpha}$ consists of the integers $p$ that are not divided by
$k-1$. While if $\left(3-\alpha\right)^{-1/2}=\mathfrak{p}/\mathfrak{q}$ is
rational, then $\left(3-\alpha\right)^{-1/2}>1/2$ for $\alpha\in(1,1+\delta)$,
$\mathfrak{p}>1$ and
$\mathcal{C}_{k,\alpha}=\left\\{p\in\mathbb{N}:p\notin(k-1)\mathbb{N}\cup\mathfrak{p}\mathbb{N}\right\\}$
is the infinite set of integers $p$ that are not a multiple of $k-1$ or
$\mathfrak{p}$. In both cases $\mathcal{C}_{k,\alpha}$ is an infinite set. We
have the following theorem,

######  Theorem 4.6.

There is a small $\delta>0$ such that for any $\alpha\in(1,1+\delta)$, the
$k$-polygon is $2\pi p$-nondegenerate if $k-1$ is prime and $p$ is chosen from
the infinite set $\mathcal{C}_{k,\alpha}$.

###  4.3 The $2\pi/m$-nondegeneracy property of the $n$-polygon for
gravitational forces

In this section we verify the $2\pi/m$-nondegeneracy property in the
gravitational case. Before proving this property, we use computer-assisted
proofs to validate that there are no integers $\ell\in\mathbb{N}$ such that
$\det\left(m_{j}(\ell)\right)=0$ for $j=2,\dots,k-2$.

######  Proposition 4.7.

For each $k$ from $4$ to $1000$, the polynomial

$P_{j}(\lambda)=\det\left(m_{j}(\lambda)\right)$

has no integer roots for $j=2,\dots,k-2$.

Proof. When $\alpha=2$, we get by (4.11)

$\alpha_{j}-\gamma_{j}=\frac{s_{j-1}}{2s_{1}}\geq
0\qquad\alpha_{j}+\gamma_{j}=\frac{s_{j+1}}{2s_{1}}\geq 0.$

Since

$\displaystyle P_{j}(\lambda)$
$\displaystyle=(\left(\lambda-1\right)^{2}+\alpha_{j}-\gamma_{j})(\left(\lambda+1\right)^{2}+\alpha_{j}+\gamma_{j})-\beta_{j}^{2}$
$\displaystyle\geq\left(\lambda-1\right)^{2}\left(\lambda+1\right)^{2}-\beta_{j}^{2}\geq\left(\lambda^{2}-1\right)^{2}-\beta_{j}^{2}$

The polynomial has not roots for $\lambda^{2}-1\geq\beta_{j}^{2}$. The result
is obtained by validating rigorously, using interval arithmetics in the
package INTLAB in MATLAB, that $P_{j}(\ell)\neq 0$ for
$\ell=0,\dots,\sqrt{\beta_{j}^{2}+1}$ and $j=2,\dots,k-2$.

######  Theorem 4.8.

For $\alpha=2$, the $k$-polygon is $2\pi/m$-nondegenerate with $m>1$ for any
$k=4,\dots,1000$. In addition, there is a small $\delta>0$ such that for all
$\alpha\in(2-\delta,2+\delta)/\\{2\\}$, the $k$-polygon is
$2\pi$-nondegenerate for any $k=4,\dots,1000$.

Proof. By Proposition 4.3 and the fact that $\sqrt{3-\alpha}$ is integer only
if $\alpha=2$ with $\sqrt{3-\alpha}=1$, we have that the block $m_{k}(\ell)$
is fine for $\ell\in m\mathbb{Z}$ if $\alpha=2$ and for $\ell\in\mathbb{Z}$ if
$\alpha\in(2-\delta,2+\delta)\setminus\\{2\\}$. The blocks $m_{j}(\ell)$
restricted to ${T_{j}^{-1}((W_{j})_{red})}$ are always invertible for
$j=1,k-1$ by Proposition 4.4. Proposition 4.7 and the continuity respect
$\alpha$ imply that the blocks $m_{j}(\ell)$ are invertible for all
$\alpha\in(2-\delta,2+\delta)$, $j=2,\dots,n-2$ and $\ell\in\mathbb{Z}$. The
result follows.

###  4.4 The $2\pi p$-nondegeneracy property of the Lagrange triangle for
different masses

Since the conditions of Remark 3.4 for the Hamiltonian is equivalent to the
$2\pi p$–nondegeneracy property, we can use previous computations made for the
analysis of the stability of the Lagrange triangular configuration. In the
case $\alpha=2$, according to the Gascheau result regarding the $3$-body
problem [38], the linear hamiltonian system at the Lagrange triangular
configuration with masses $m_{j}$ has four pairs of zero-eigenvalues and four
complex roots off the imaginary axis (leading to instability) when

$\beta=27\frac{m_{1}m_{2}+m_{1}m_{3}+m_{2}m_{3}}{\left(m_{1}+m_{2}+m_{3}\right)^{2}}>1\text{.}$

The four pairs of zero-eigenvalues correspond to the center of mass and the
homographic elliptic orbits. The linearization of the homographic elliptic
orbits leads to our block $m_{k}(\lambda)$. Therefore, the Lagrange triangular
configuration is $2\pi/m$-nondegenerate for any $m>1$ when $\beta>1$.

In the case $\alpha\neq 2$, after fixing the center of mass equal to zero,
there are four pairs of non-zero eigenvalues. Two pairs correspond to the
linearization in the submanifold of homographic solutions having the block of
the Kepler problem in our analysis $m_{k}(\lambda)$. This block is $2\pi
p$-nonresonant when $p\notin(3-\alpha)^{-1/2}\mathbb{N}$. The other two pairs
of eigenvalues can be found explicitly in [36], and correspond in our setting
to

$\displaystyle\lambda_{1}^{\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{6}i\sqrt{18(1-\alpha)+6\sqrt{9(\alpha-1)^{2}-\beta(\alpha+3)^{2}}}$
$\displaystyle\lambda_{2}^{\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{6}i\sqrt{18(1-\alpha)-6\sqrt{9(\alpha-1)^{2}-\beta(\alpha+3)^{2}}}.$

These eigenvalues are off the imaginary axis when

$\beta>9\left(\frac{3-\alpha}{1+\alpha}\right)^{2}.$ (4.14)

Therefore, we conclude that for $\alpha\neq 2$, the Lagrange triangular
configuration is $2\pi p$-nondegenerate when the inequalities
$p\notin(3-\alpha)^{-1/2}\mathbb{N}$ and (4.14) hold. Furthermore, since the
eigenvalues are analytic in $\alpha$ and $\beta$, the Lagrange triangular
configuration is generically $2\pi p$-nondegenerate for the homogeneous
exponent $\alpha$ and the set of masses $m_{j}$. Observe that for equal masses
$m_{j}=1$ the inequality (4.14) does not hold precisely for $\alpha=1$. This
is the degeneracy found in the previous analysis for the blocks $m_{1}$ and
$m_{n-1}$ in the case of $\alpha=1$.

##### Acknowledgements.

We acknowledge the assistance of Ramiro Chavez Tovar with the preparation of
the figures. M. Fontaine is supported by the FWO-EoS project G0H4518N. C.
García-Azpeitia is supported by PAPIIT-UNAM grant IA100121.

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MF<EMAIL_ADDRESS>
Departement Wiskunde
Universiteit Antwerpen
2020 Antwerpen, BE.

CGA<EMAIL_ADDRESS>
Depto. Matemáticas y Mecánica IIMAS
Universidad Autónoma de México
Apdo. Postal 20-726, Ciudad de México, MX.