#  From the Lagrange Triangle to the Figure Eight Choreography: Proof of
Marchal’s Conjecture

Renato Calleja IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal
20-726, C.P. 01000, México D.F., México<EMAIL_ADDRESS>Carlos
García-Azpeitia IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal
20-726, C.P. 01000, México D.F., México<EMAIL_ADDRESS>Olivier
Hénot École Polytechnique, Centre de Mathématiques Appliquées, route de
Saclay, 91120 Palaiseau, France<EMAIL_ADDRESS>Jean-
Philippe Lessard McGill University, Department of Mathematics and Statistics,
805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada.
<EMAIL_ADDRESS>Jason D. Mireles James Florida Atlantic University,
Department of Mathematical Sciences, Science Building, Room 234, 777 Glades
Road, Boca Raton, Florida, 33431, USA<EMAIL_ADDRESS>
###### Abstract

For the three body problem with equal masses, we prove that the most symmetric
continuation class of Lagrange’s equilateral triangle solution, also referred
to as the $P_{12}$ family of Marchal [37], contains the remarkable figure
eight choreography discovered by Moore in 1993 [41], and proven to exist by
Chenciner and Montgomery in 2000 [14]. This settles a conjecture of Marchal
which dates back to the 1999 conference on Celestial Mechanics in Evanston
Illinois, celebrating Donald Saari’s 60th birthday [18].

Key words: three body problem, continuation class of the Lagrange triangle,
figure eight choreography, computer-assisted proof, Marchal’s conjecture

##  1 Introduction

In a paper published in 1993, Moore describes a periodic solution of the
gravitational $N$-body problem where three bodies of equal mass follow one
another around a closed eight-shaped curve in the plane [41]. The orbit was
discovered by numerically minimizing the Newtonian potential over the space of
closed plane curves via gradient descent. This _remarkable solution_ of the
$N$-body problem was discovered independently by Chenciner and Montgomery,
whose variational proof of its existence appeared in the Annals of Mathematics
in the year 2000 [14]. See the right frame of Figure 1 in the present work,
and also [40] for graphical illustrations of the three body eight. These days
such an orbit is called a choreography, as the bodies appear to dance around a
prescribed curve in space. The paper [13] by Chenciner, Gerver, Montgomery and
Simó provides a lovely introduction to the subject of choreographies in
particle systems.

Shortly after the announcement of the eight by Chenciner and Montgomery,
Marchal published a paper investigating the discrete symmetries of $3$-body
choreographies [38]. In particular, recalling that non-circular periodic
solutions of the equal mass three body problem have at most 12 space-time
symmetries, he studied the properties of the most symmetric family of
_relative periodic orbits_ bifurcating from Lagrange’s equilateral triangle by
continuation with respect to the frequency of a rotating frame. By a _relative
periodic orbit_ , we mean a solution which is periodic after changing to an
appropriate co-rotating frame. Marchal referred to this most symmetric
continuation class as the $P_{12}$ family, and in the same reference showed
that:

  * •

The $P_{12}$ family is not empty: there exists a family of relative periodic
orbits bifurcating from the Lagrange triangle and having the maximal allowed
12 symmetries.

  * •

The $P_{12}$ family comprises relative choreographies: in a suitable rotating
coordinate system, each periodic orbit in the $P_{12}$ family can be viewed as
a single closed curve in $\mathbb{R}^{3}$. The three bodies follow one another
around this curve, spaced out by a time shift of $1/3$ the period.

  * •

Variational properties: the $P_{12}$ family minimizes the action between
appropriate terminal conditions.

Next he observed that the three body figure eight evinces all 12 symmetries of
the $P_{12}$ family. Inspired by this, he proposed that $P_{12}$ actually
contains the eight. We refer to this as Marchal’s conjecture.

The local structure of the $P_{12}$ family, and hence the plausibility of
Marchal’s conjecture, is further considered in the papers [10] and also [12],
of Chenciner and Féjoz, and of Chenciner, Féjoz and Montgomery respectively.
In particular, the former paper applies a center manifold analysis in a frame
rotating with the Lagrange triangle and shows that, near the triangle, the
$P_{12}$ family is diffeomorphic to a cylinder, and that the cylinder is
transverse to the plane of the Lagrange triangle (in fact it is a vertical
Lyapunov family attached Lagrange’s relative equilibrium). So, in appropriate
rotating coordinates, $P_{12}$ is described as a smoothly varying one-
parameter family of out-of-plane choreographic orbits, which bifurcate from
the Lagrangian triangle.

In the latter of the two references just cited, the authors make a local
analysis of the eight. They show – via variational techniques – that there are
three families of choreographic orbits bifurcating from the figure eight, and
that one of these has the symmetries of the $P_{12}$ family. They also provide
numerical evidence which suggests that these are the only families of relative
choreographies bifurcating from the figure eight. Together, these two papers
show that there are no local obstructions to Marchal’s conjecture.

It remains to demonstrate that the $P_{12}$ family actually bridges the gap
between the work of [10] and [12]. The contribution of the present work is to
complete this picture, settling Marchal’s conjecture in the affirmative. A
precise statement of our main result is given in Theorem 4.2. We summarize it
informally as follows.

######  Theorem 1.1.

In the three body problem with equal masses, the figure eight choreography of
Moore, Chenciner, and Montgomery is contained in the $P_{12}$ family of
Marchal.

This theorem, combined with Proposition 3 in [8], implies the existence of a
dense subset of $P_{12}$, corresponding to choreographies in the inertial
frame with arbitrarily large period and lying on the surface of a topological
cylinder.

The proof of Theorem 1.1 is constructive, and proceeds by several
interconnected steps.

  1. 1.

Symmetric function space reduction: we reformulate the problem in a rotating
frame and impose (in Fourier space) the symmetries of Marchal’s $P_{12}$
family. This, in particular, removes all three body bifurcations which do not
result in relative choreographies. The result is a system of algebraic
advanced/delay differential equations with discrete symmetries. Our task is to
establish the existence of a branch of solutions of the symmetrized problem,
parameterized by the frequency of the rotating frame, starting from the
Lagrange triangle and extending to the figure eight.

  2. 2.

Desingularization at the endpoints: the Lagrange triangle and the figure eight
are both planar solutions of the three body problem, and the desired $P_{12}$
branch is transverse to each of these planes (see the right frame of Figure
1). Despite the discrete symmetries mentioned above, and because of the planar
symmetry, neither the triangle nor the eight are isolated as choreographies.
Put another way, the $P_{12}$ branch begins and ends at a symmetry breaking
bifurcation, and because of this it is necessary to desingularize the problem
at each endpoint. We introduce an “unfolding parameter” which desingularizes
this symmetry breaking along the entire branch.

  3. 3.

A posteriori argument: we numerically compute a candidate branch of solutions
to the desingularized symmetrized problem (see the left frame of Figure 1),
and apply a uniform contraction argument in a neighborhood of this candidate.
In this way, we prove the existence of a true branch of solutions near (in an
appropriate norm) the candidate solution branch. The procedure is carried out
in a Banach space of rapidly decaying Fourier-Chebyshev series coefficients,
where we are able to express the entire solution branch using a single
Chebyshev expansion in the frequency parameter.

  4. 4.

Analytical bound derivation: deriving the analytical bounds needed for the
uniform contraction argument constitutes a significant portion of the paper.
The assumptions of the uniform contraction theorem are distilled down to a
list of inequalities, involving the Fourier-Chebyshev coefficients of the
candidate branch. Ultimately, verifying these hypotheses demands a very large,
but finite number of calculations.

  5. 5.

Interval arithmetic: given the significant computational complexity of the
inequalities, and the large number of Fourier-Chebyshev coefficients
representing our candidate, manual calculations are not feasible. Instead, we
use a computer for this task, employing an interval arithmetic library [1] as
well as the RadiiPolynomial library [24] implemented in the Julia programming
language [2]. Readers interested in independently verifying the necessary
inequalities at the final step of the proof can use the following Jupyter
Notebook [25]:

https://github.com/OlivierHnt/MarchalConjecture.jl

The necessary checks are completed in less than an hour on a laptop with a M1
chip and 8GB of RAM.

Finally, let us remark that Marchal showed in [38] that if the Lagrange
triangle can be continued to the figure eight through the $P_{12}$ family,
then – by further symmetry considerations – the family continues through the
figure eight, and returns to the Lagrange triangle with opposite orientation.
Following the branch back to the figure eight (with reversed orientation) and
finally back to the Lagrange triangle with the original orientation closes the
loop. So, the branch proved to exist in the present work is precisely one
quarter of the $P_{12}$ family. Hence, as a corollary to Theorem 1.1, we
obtain the global description of $P_{12}$.

(a) Thirty-three orbits sampled from the continuous branch of solutions proven
to exist by Theorem 1.1. Each curve is a relative periodic choreography of the
three body problem with equal masses.

(b) The Lagrange triangle (blue) and the figure eight (red). Each of these
choreographies is planar, and the $P_{12}$ family – comprised of relative
choreographies – is transverse to both planes.

Figure 1: Branch of relative choreographies connecting the Lagrange triangle
to the figure eight. A dense set of the relative choreographies in the family
corresponds to choreographies in inertial frame.

The remainder of the paper is structured as follows. Section 2 covers the
symmetry reduction, the desingularization at the figure eight and the Lagrange
triangle, and the treatment of non-polynomial nonlinearities. These concepts
collectively reformulate the problem as a functional equation. In Section 3,
we define the norms and establish the necessary bounds to prove the existence
of a branch of solutions of the functional equation using the contraction
mapping theorem. Section 4 completes the demonstration of Marchal’s conjecture
by establishing that the branch of zeros identified in Section 3 corresponds
to the $P_{12}$ family of Marchal and ends at the figure eight of Chenciner
and Montgomery. Section 5 concludes the paper by outlining potential future
work.

###  1.1 Some historical remarks

The first choreographic solution of the $N$-body problem appeared in
Lagrange’s 1772 publication of the equilateral triangle solution of the
$3$-body problem [33]. For appropriate initial velocities, this special
solution involves three bodies of equal mass moving in circular motion at
constant speed. The result was generalized more than a century later by Hoppe
[26], who showed that for every $N$ many bodies there is a circular periodic
orbit where the $N$ equal masses are located at the vertices of a rotating
regular $N$-gon. Today we would refer to all of these polygonal solutions as
trivial choreographies. It is a testament to the richness of the gravitational
$N$-body problem that it took more than 200 years for a non-trivial
choreography to be discovered by Moore [41].

The story of Marchal’s conjecture picks up in the Fall of 1999 when Chenciner,
Gerver, Montgomery, and Simó were all congregated at the Paris Observatory’s
Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE). Montgomery
had recently published the paper [39], where he used variational arguments to
prove the existence of many braided periodic solutions of the $N$-body problem
with the strong $1/r^{a}$ potential with $a=1$. He further conjectured that
many of these braids should continue to the Newtonian potential taking $a\to
2$.

Chenciner and Montgomery worked on these and similar problems in the winter of
1999 and, as recalled in the introduction of [13], by December they had proven
the existence of the three body eight for the Newtonian potential. This set
off – in the words of the same introduction – a flurry of work, and soon
Gerver and Simó had computed a whole zoo of new $N$-body choreographies for
$N$ as large as $799$. It was during this period that Simó actually coined the
term _choreography_.

Later the same month, at a conference on Celestial Mechanics in Evanston,
Illinois, honoring the 60th birthday of Donald Saari, Chenciner gave a
presentation announcing the proof of the figure eight. According to the first
paragraph of [38], Marchal was in attendance. Indeed, in his 2010 Mémoire
d’Habilitation [18], Féjoz recalls that upon hearing the talk, Marchal
immediately realized that the figure eight could be related to the Lagrange
triangle through his $P_{12}$ family, and Marchal’s conjecture was born. The
conjecture is referred to as “Partially Numerical Theorem $19$” in [18], and
its status is reviewed in detail in Section 3.4 of the same document. See also
the 2009 paper [11] of Chenciner and Féjoz for a detailed numerical evidence
supporting the conjecture.

After Evanston, interest in choreographic solutions spread rapidly. In 2000
the paper of Chenciner and Montgomery, describing the proof of the figure
eight, appeared in the Annals of Mathematics [14]. Marchal’s paper [38], about
the symmetries of the figure eight, appeared in Celestial Mechanics and
Dynamical Astronomy the same year. This marked the first appearance of the
conjecture in print, and from this point on, the mathematical theory
progressed as discussed in the previous section.

During the next two years, the papers [43, 44] by Simó provided detailed
numerical accounts of many additional symmetric and asymmetric families of
$N$-body choreographies, and reported the surprising numerical result that the
three body figure eight choreography appeared to be stable in the Hamiltonian
sense.

We also mention that, as recalled in both the references of [14] and the
acknowledgements of [13], it was Phil Holmes and Robert MacKay who brought the
paper [41] of Moore to the attention of Chenciner and Montgomery. This
apparently happened during the galley proofing of [14]. Chenciner and
Montgomery in turn seem to have brought the paper of Moore to the attention of
a much larger audience than before. The paper seems to have been cited seven
times between 1993 and 2000, and has more than three hundred citations by the
the date of the present manuscript.

It should also be noted that the first computer-assisted existence theorems
for choreographies appeared fairly soon after the Evanston conference. The
paper [30] of Kapela and Zgliczyński, appearing in Nonlinearity in 2003,
reproved the existence of the figure eight and also established its convexity
(a difficult property to obtain using variational methods). In the same
reference they also prove the existence of some symmetric planar 4, 6, and 8
body choreographies which had appeared earlier in the numerical studies of
Simó [43, 44]. Later in [28], Kapela and Simó proved the linear stability of
the figure eight, and the existence of many additional non-symmetric
choreographies from [43, 44]. In [29], the same authors prove the KAM
stability of the figure eight in an appropriate energy/symmetry submanifold.

Finally, we mention the work [8] by four of the authors of the present
manuscript. This paper provides computer-assisted existence proofs for spatial
torus knot choreographies (including a five body trefoil knot), and laid the
foundation for the current work. More precisely, the computer-assisted proofs
in the reference just cited employ a functional analytic framework based on
advanced/delay equations similar to the one discussed in Section 2. We note
however that the results of [8] are localized in parameter space, while the
results of the present describe a global branch containing the triangle and
the eight.

This is of course by no means a complete review of the literature on
choreographic orbits in celestial mechanics. An excellent overview by
Montgomery, with many additional references, is found at [40]. Similarly,
computer-assisted methods based on a posteriori validation of spectral methods
have a long history going back to the works of Lanford, Eckmann, Koch, and
Wittwer in the mid-1980’s on the Fegenbaum conjectures [34, 17, 16]. The use
of such computer-assisted methods in nonlinear analysis and dynamical systems
theory is by now a thriving subindustry, and to attempt even a terse overview
would be a task beyond the scope of the present paper. We refer the interested
reader to the review articles of [35, 32, 46, 20] and to the books of [45, 47,
42] for thorough discussion of this literature.

##  2 Reformulation of the problem as a functional equation

We consider the motion of three bodies interacting under Newton’s
gravitational law, depending on their mass, initial position and initial
velocity. The positions $q_{1},q_{2},q_{3}$ are solutions of the equations of
motion

$m_{j}\ddot{q}_{j}=-\partial_{q_{j}}U,\qquad
U\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}-\sum_{j<l}\frac{Gm_{j}m_{l}}{\|q_{j}-q_{l}\|},\qquad j=1,2,3,$ (1)

where $(q_{1},q_{2},q_{3})\in\\{(\mathbb{R}^{3})^{3}\,:\,q_{1}\neq
q_{2},\,q_{1}\neq q_{3},\,q_{2}\neq q_{3}\\}$ and $G$ denotes the
gravitational constant.

Changing the units of length, mass and time by the factors $A,B,C$
respectively allows to scale $G=1$ under the condition $A^{3}=BC^{2}$. We
study the case of equal masses, and, by altering the unit of mass, we scale
$m_{1}=m_{2}=m_{3}=1$, leaving out one more degree of freedom. Then, the
motion of the bodies is described by the system

$\ddot{q}_{j}=-\sum_{j\neq l}\frac{q_{j}-q_{l}}{\|q_{j}-q_{l}\|^{3}},\qquad
j=1,2,3.$ (2)

We set the inertial frame by placing the origin at the center of mass of the
Lagrange triangle, whose plane is associated with the horizontal $xy$-plane.
Due to our scaling of mass, the Lagrange triangle constitutes a relative
equilibrium of (2), rotating horizontally with frequency $1$, whose vertices
are located at

$q_{j}(t)=3^{-1/6}e^{(t+2\pi
j/3)\bar{J}}e_{1}=3^{-1/6}\begin{pmatrix}\cos(t+2\pi j/3)\\\ \sin(t+2\pi
j/3)\\\ 0\end{pmatrix},\qquad j=1,2,3,$ (3)

where $e_{1}\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}(1,0,0)$ and
the matrix $\bar{J}$ is defined below in (4). This configuration forms a
circular choreography such that $q_{j}(t)=q_{3}(t+2\pi j/3)$ for $j=1,2$. The
$P_{12}$ family studied by Marchal is a family of solutions of (2) in a
horizontally rotating frame, parameterized by the rotation frequency, see [38]
for details. For this purpose, we introduce the matrices

$\bar{I}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix},\qquad\bar{J}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix}.$ (4)

Then the positions of the bodies in Marchal’s rotating frame coordinates are
given by

$U_{j}(t,\Omega)=e^{\Omega t\bar{J}}q_{j}(t).$ (5)

Under a scaling of time (using our last degree of freedom), we consider
$U_{1},U_{2},U_{3}$ to have period $2\pi$. Note that in the inertial frame,
the solution $q_{1},q_{2},q_{3}$ may be quasiperiodic, depending on $\Omega$.
Proposition 3 in [8] implies that there are infinitely many choreographies, in
the inertial frame, lying on the surface of a topological cylinder.

###  2.1 Symmetry reduction

The $P_{12}$ family consists of a family of a specific class of periodic
solutions with the _choreography symmetry_

$U_{j}(t,\Omega)=U_{3}(t+4\pi j/3,\Omega),\qquad j=1,2.$ (6)

To avoid carrying out an additional subscript, we let $U=U_{3}$. Given that
the masses of the three bodies are equal, in the rotating frame (5) the
equations are equivariant under permutations and time-shifts. Therefore, a
solution satisfying (5) and (6) is represented as a solution to the system of
delay differential equations

$0=G(U)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\ddot{U}-2\Omega\bar{J}\dot{U}-\Omega^{2}\bar{I}U+\sum_{j=1}^{2}\frac{U-S^{j}U}{\|U-S^{j}U\|^{3}},$
(7)

where

$[S^{j}U](t,\Omega)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}U(t+4\pi j/3,\Omega).$ (8)

Notice that in these coordinates, the triangular relative equilibrium (3)
satisfies the symmetries (6) and is represented by

$U(t,1)=3^{-1/6}e^{2t\bar{J}}e_{1}.$

For the sake of simplicity, in this section we study formally the operator
$G:Y\to Y$ in the space

$Y\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{R}^{3}),$
(9)

and conscientiously do not prescribe a norm. In Section 3, we will use a
Banach space of analytic functions contained in $Y$. The operator $G$ is
equivariant under the action of the group
$(\sigma,\tau)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ in $Y$ given by

$\displaystyle\sigma\cdot U(t,\Omega)$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}R_{y}U(-t,\Omega),$ (10) $\displaystyle\tau\cdot U(t,\Omega)$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}R_{z}U(t+\pi,\Omega),$ (11)

where

$R_{y}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&1\end{pmatrix},\qquad
R_{z}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-1\end{pmatrix}.$ (12)

Notice that the fixed-point space of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ is

$\mathcal{U}\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}\Big{\\{}U\in
Y\,:\,U(-t,\Omega)=R_{y}U(t,\Omega)\text{ and
}U(t+\pi,\Omega)=R_{z}U(t,\Omega)\Big{\\}}.$ (13)

This equivariance property implies that we can find solutions by restricting
the operator to $\mathcal{U}$,

$G:\mathcal{U}\to\mathcal{U}.$

######  Remark 2.1.

Many of the necessary symmetries can be imposed directly in coefficient space.
In fact, it can be seen that a function $U\in\mathcal{U}$ with Fourier
coefficients given by

$U(t,\Omega)=\sum_{k\in 2\mathbb{Z}}\begin{pmatrix}a_{1,k}(\Omega)\cos kt\\\
b_{2,k}(\Omega)\sin kt\\\ 0\end{pmatrix}+\sum_{k\in
2\mathbb{Z}+1}\begin{pmatrix}0\\\ 0\\\ a_{3,k}(\Omega)\cos kt\end{pmatrix},$
(14)

has the symmetries of the $P_{12}$ family.

We will prove that Marchal’s $P_{12}$ family begins at the Lagrange triangle
with $\Omega=1$ and ends at the figure eight with $\Omega=0$ by studying
periodic solutions $U\in\mathcal{U}$ of (7) for $\Omega\in[0,1]$. In
particular, we will verify in our proof that we have a figure eight
choreography when $\Omega=0$ as described by the main theorem of [14]. The
characterization of the eight-shape choreography is reported in the next
lemma.

######  Lemma 2.2.

For $\Omega=0$, let $U=(0,U_{y},U_{z})\in\mathcal{U}$ be a solution of
equation (7) with $U_{x}=0$. Then, $q=(q_{1},q_{2},q_{3})$ with $q_{j}=S^{j}U$
is a planar choreography solution of the $3$-body problem with the following
properties:

  1. (i)

The linear momentum of the solution is zero, i.e.

$\sum_{j=1}^{3}q_{j}(t)=0.$ (15)

  2. (ii)

The solution in the $yz$-plane translated by $\pi/2$,
$\hat{q}(t)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}(U_{z}(t+\pi/2),U_{y}(t+\pi/2))$, satisfies the symmetries of the
figure eight choreography proven in [14], i.e.

$\displaystyle\hat{q}(t+\pi)$
$\displaystyle=(-\hat{q}_{1}(t),\hat{q}_{2}(t)),$ (16)
$\displaystyle\hat{q}(-t+\pi)$
$\displaystyle=(\hat{q}_{1}(t),-\hat{q}_{2}(t)).$ (17)

  3. (iii)

The angular momentum is zero, i.e.

$\sum_{j=1}^{3}q_{j}(t)\times\dot{q}_{j}(t)=0.$ (18)

  4. (iv)

If in addition $U\times\dot{U}$ is nowhere vanishing for all $t\in(0,\pi/2)$,
then the orbit of $q(t)$ is an eight-shape figure.

###### Proof.

  1. $(i)$

The conservation of linear momentum implies that

$\sum_{j=1}^{3}q_{j}(t)=ct+b.$

Since our solution is periodic $c=0$. By the symmetries, the $0$-th Fourier
modes of $U_{y}$ and $U_{z}$, for $U=(U_{x},U_{y},U_{z})\in\mathcal{U}$, are
zero. Also, by assumption $U_{x}$ is zero, thus the $0$-th Fourier coefficient
of $q_{j}$ is $0$, which implies that $b=0$.

  2. $(ii)$

For a $yz$-planar solution $U(t)=(0,U_{y},U_{z})\in\mathcal{U}$ we have that
$\hat{q}$ has the symmetries $\hat{q}(t+\pi)=(-\hat{q}_{1}(t),\hat{q}_{2}(t))$
and

$\hat{q}(-t+\pi)=(U_{z}(-t-\pi/2),U_{y}(-t-\pi/2))=(U_{z}(t+\pi/2),-U_{y}(t+\pi/2))=(\hat{q}_{1}(t),-\hat{q}_{2}(t)).$

These two symmetries agree with the ones found in [14].

  3. $(iii)$

By $(ii)$, the symmetries agree with [14], see also Remark 3.2 in [30]. Using
both symmetries one has that $\hat{q}(-t)=-(\hat{q}_{1}(t),\hat{q}_{2}(t))$,
which implies that at time $\pi/2$ we have the following relations

$\displaystyle q_{1}(\pi/2)$ $\displaystyle=(0,0,0),$ $\displaystyle
q_{2}(\pi/2)$ $\displaystyle=-q_{3}(\pi/2)=(0,*,*),$
$\displaystyle\dot{q}_{2}(\pi/2)$ $\displaystyle=\dot{q}_{3}(\pi/2)=(0,*,*),$

where $(0,*,*)$ denotes a vector whose first component vanishes. From these,
it is straightforward to verify that the angular momentum is zero at $\pi/2$.

  4. $(iv)$

It remains to be prove that the shape of the orbit is a figure eight without
extra small loops or other unpleasant features. This fact is an immediate
consequence of the corollary following Lemma 7 in [14], which requires the
assumption in $(iv)$. ∎

###  2.2 Desingularization

Marchal’s $P_{12}$ family connects to two different families at $\Omega=1$ and
$\Omega=0$ due to the symmetries of the problem. At $\Omega=0$ it connects to
the family of $x$-translations of the figure eight. At $\Omega=1$ it connects
to the family of homothetic Lagrange triangles. In this subsection, we augment
the system to isolate the $P_{12}$ family from those branches.

####  2.2.1 Figure eight

At $\Omega=0$, any translation in the $x$-direction of a solution is also
solution of (7). To isolate the figure eight, we fix the average value of
$U_{x}(t,\Omega)$, with respect to $t$, to be zero along the branch. To
compensate for this restriction, we introduce an unfolding parameter function
$\beta=\beta(\Omega)$ in (7), such that we consider the system

$\beta
e_{1}-\Omega^{2}\bar{I}U-2\Omega\bar{J}\dot{U}+\ddot{U}+\sum_{j=1}^{2}\frac{U-S^{j}U}{\|U-S^{j}U\|^{3}}=0,$
(19)

with the constraint

$\frac{1}{2\pi}\int_{0}^{2\pi}U_{x}(t,\Omega)\,dt=0,\qquad\text{for all
}\Omega\in[0,1].$ (20)

The following lemma shows that periodic solutions of (19) together with (20)
are equivalent to periodic solutions of (7).

######  Lemma 2.3.

If $U=(U_{x},U_{y},U_{z})\in\mathcal{U}$ is a periodic solution of (19)
subjected to the condition (20), then $\beta(\Omega)=0$ for all
$\Omega\in[0,1]$.

###### Proof.

By the invariance of the $3$-body problem under $x$-translations for
$\Omega=0$, given that $U_{j}=S^{j}U$, we have by the conservation of linear
momentum in $x$ that

$0=\int_{0}^{2\pi}\sum_{j=1}^{3}\left\langle\ddot{U}_{j}+\sum_{l\neq
j}\frac{U_{j}-U_{l}}{\|U_{j}-U_{l}\|^{3}},e_{1}\right\rangle\,dt.$

Since this equality still holds for $\Omega\neq 0$, it follows that

$0=\int_{0}^{2\pi}\sum_{j=1}^{3}S^{j}\left\langle\beta
e_{1}-\Omega^{2}\bar{I}U-2\Omega\bar{J}\dot{U},e_{1}\right\rangle\,dt=6\pi\beta,$

since $\int_{0}^{2\pi}\dot{U}_{y}\,dt=0$ by periodicity and
$\int_{0}^{2\pi}U_{x}\,dt=0$ by assumption. This implies $\beta(\Omega)=0$ for
all $\Omega\in[0,1]$. ∎

####  2.2.2 Lagrange triangle

At $\Omega=1$, the (planar) homothetic family of the Lagrange equilateral
triangle meets the (off-plane) $P_{12}$ family. The goal of this section is to
derive an auxiliary system to (19) which only retains the $P_{12}$ family. We
use a _blow-up_ (as in “zoom-in”) method (e.g. see [15, 48]). Let
$u=(u_{1},u_{2},u_{3})$ be defined by the relation

$U=L_{\sqrt{a}}u,$ (21)

where

$L_{\alpha}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&\alpha\end{pmatrix}.$ (22)

Then, the system (19) becomes

$\beta
e_{1}-\Omega^{2}\bar{I}u-2\Omega\bar{J}\dot{u}+\ddot{u}+\sum_{j=1}^{2}\frac{u-S^{j}u}{\|L_{\sqrt{a}}u-S^{j}L_{\sqrt{a}}u\|^{3}}=0,$
(23)

which we supplement with the conditions

$u_{3}(0,\Omega)=1,\qquad\frac{1}{2\pi}\int_{0}^{2\pi}u_{1}(t,\Omega)\,dt=0,\qquad\text{for
all }\Omega\in[0,1].$ (24)

For $a\neq 0$, $u=(u_{1},u_{2},u_{3})$ is a periodic solution of (23) if and
only if $U=L_{\sqrt{a}}u$ is a periodic solution of (19). The following lemma
ensures that the system (23) together with (24) isolates the $P_{12}$ family.

######  Lemma 2.4.

If $\Omega=1$ and $a=0$, in a neighborhood of $u=(u_{1},u_{2},u_{3})$ with

$(u_{1},u_{2})=3^{-1/6}(\cos 2t,-\sin 2t),$

corresponding to the Lagrange triangle, the equation (23) together with the
conditions (24) has an isolated solution $u$.

###### Proof.

Note that the system (23) becomes decoupled in $(u_{1},u_{2})$ and $u_{3}$,
such that the equation for $(u_{1},u_{2})$ is solved by the Lagrange triangle,
and for $u_{3}$ we have the linear differential equation

$\ddot{u}_{3}+C^{-3}\sum_{j=1}^{2}u_{3}-S^{j}u_{3}=0,$

where
$C=\|(u_{1},u_{2},0)-S(u_{1},u_{2},0)\|=\|(u_{1},u_{2},0)-S^{2}(u_{1},u_{2},0)\|=3^{1/3}$.
This linear equation admits a family of periodic solutions, parameterized by
its amplitude, and determined by the Lagrange triangle $(u_{1},u_{2})$. Hence,
the linear scaling $u_{3}(0,\Omega)=1$ characterizes the unique periodic
solution $u_{3}=\cos t$. ∎

###  2.3 Polynomial embedding

Polynomial nonlinearities benefit inherently from the Banach algebra structure
induced by the Fourier-Chebyshev convolutions. We employ automatic
differentiation techniques (e.g. see [5, 9, 21, 23, 27, 31, 36]) to construct
an auxiliary system to (23) only comprised of polynomial nonlinearities.

We introduce

$v=\partial_{t}u,\qquad
w_{j}=\|L_{\sqrt{a}}u-S^{j}L_{\sqrt{a}}u\|^{-1},\quad\text{for }j=1,2.$

Notice that the symmetries of $u\in\mathcal{U}$ extend naturally to $v$ and
$w$

$\displaystyle v(t+\pi,\Omega)$ $\displaystyle=R_{z}v(t,\Omega),\qquad
v(-t,\Omega)=-R_{y}v(t,\Omega),$ $\displaystyle w_{j}(t+\pi,\Omega)$
$\displaystyle=w_{j}(t,\Omega),\qquad w_{j}(-t,\Omega)=w_{3-j}(t,\Omega).$

The only nontrivial identity is the last one, which follows from

$\displaystyle w_{2}(-t,\Omega)$
$\displaystyle=\frac{1}{\left\|L_{\sqrt{a}}u(-t,\Omega)-L_{\sqrt{a}}u(-(t-8\pi/3),\Omega)\right\|}$
$\displaystyle=\frac{1}{\left\|R_{y}L_{\sqrt{a}}u(t,\Omega)-R_{y}L_{\sqrt{a}}u(t-4\pi/3,\Omega)\right\|}=w_{1}(t,\Omega).$

Given a function $h\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{R})$, we define

$[Rh](t,\Omega)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}h(-t,\Omega).$ (25)

With this notation, the above identity reads $w_{2}=Rw_{1}$. To avoid carrying
out an additional subscript, we let $w=w_{1}$, so that $w_{2}=Rw$.

Introducing an unfolding parameter $\alpha=\alpha(\Omega)$, the system (23)
can be re-written as the first-order polynomial system

$\displaystyle\partial_{t}v$ $\displaystyle=-\beta
e_{1}+\Omega^{2}\bar{I}u+2\Omega\bar{J}v-w^{3}(u-Su)-Rw^{3}(u-S^{2}u),$ (26)
$\displaystyle\partial_{t}u$ $\displaystyle=v,$ $\displaystyle\partial_{t}w$
$\displaystyle=-\alpha-w^{3}\big{\langle}u-Su,L_{a}(v-Sv)\big{\rangle},$

which we supplement with the conditions

$u_{3}(0,\Omega)=1,\quad\frac{1}{2\pi}\int_{0}^{2\pi}u_{1}(t,\Omega)\,dt=0,\quad\Big{[}w^{2}\big{\langle}u-Su,L_{a}(u-Su)\big{\rangle}\Big{]}(0,\Omega)=1,\quad\text{for
all }\Omega\in[0,1],$ (27)

and where

$w^{3}(u-Su)=\begin{pmatrix}w^{3}(u_{1}-Su_{1})\\\ w^{3}(u_{2}-Su_{2})\\\
w^{3}(u_{3}-Su_{3})\end{pmatrix}.$

The following lemma shows that periodic solutions of (26) together with (27)
are equivalent to periodic solutions of (23).

######  Lemma 2.5.

If $(u,v,w)$ is a periodic solution of (26) subjected to the conditions (27),
then $\alpha(\Omega)=0$ for all $\Omega\in[0,1]$.

###### Proof.

Let us first prove that $w$ can never vanish. Indeed, suppose that
$w(t_{0},\Omega)=0$ for some $t_{0}$. If $\alpha=0$, then $w=0$ by uniqueness
of the initial value problem, which violates
$\Big{[}w^{2}\big{\langle}u-Su,L_{a}(u-Su)\big{\rangle}\Big{]}(0,\Omega)=1$.
On the other hand, if $\alpha\neq 0$, then $w$ changes sign at $t_{0}$. Since
$\alpha$ is a strictly positive or negative constant, $w$ can never change
sign again which contradicts the fact that $w$ is periodic.

Therefore $w\neq 0$, from which we deduce that

$w=\left(\big{\langle}u-Su,L_{a}(u-Su)\big{\rangle}+C+2\alpha\int_{0}^{t}w(t^{\prime})^{-3}\,dt^{\prime}\right)^{-1/2},$

where $C$ is a constant of integration. Since $w$ is either strictly positive
or negative, the periodicity of $w$ requires $\alpha=0$. ∎

###  2.4 Zero-finding problem

We now write (26) together with (27) as the zero of a mapping $F$.
Specifically, we consider the mapping given by

$F(a,\beta,\alpha,u,v,w)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{pmatrix}\eta(u)\\\ \gamma(a,u,w)\\\ g(\beta,u,v,w)\\\
f(u,v)\\\ h(a,\alpha,u,v,w)\end{pmatrix},$ (28)

where

$\displaystyle\eta(u)$ $\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{(}u_{3}(0,\Omega)-1,\frac{1}{2\pi}\int_{0}^{2\pi}u_{1}(t,\Omega)\,dt\Big{)},$
(29) $\displaystyle\gamma(a,u,w)$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{[}w^{2}\big{\langle}u-Su,L_{a}(u-Su)\big{\rangle}\Big{]}(0,\Omega)-1,$
(30) $\displaystyle g(\beta,u,v,w)$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\partial_{t}v+\beta
e_{1}-\Omega^{2}\bar{I}u-2\Omega\bar{J}v+w^{3}(u-Su)+Rw^{3}(u-S^{2}u),$ (31)
$\displaystyle f(u,v)$ $\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\partial_{t}u-v,$ (32) $\displaystyle h(a,\alpha,u,v,w)$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\partial_{t}w+\alpha+w^{3}\big{\langle}u-Su,L_{a}(v-Sv)\big{\rangle}.$
(33)

The exact domain and image of $F$ is detailed in Section 3. For the time
being, we define the spaces

$\mathcal{V}\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}\Big{\\{}v\in
Y\,:\,v(-t,\Omega)=-R_{y}v(t,\Omega)\text{ and
}v(t+\pi,\Omega)=R_{z}v(t,\Omega)\Big{\\}},$ (34)

and

$\mathcal{W}\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}\Big{\\{}w\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{R})\,:\,w(t+\pi,\Omega)=R_{z}w(t,\Omega)\Big{\\}}.$
(35)

######  Lemma 2.6.

The map
$F:\mathbb{R}^{3}\times\mathcal{U}\times\mathcal{V}\times\mathcal{W}\to\mathbb{R}^{3}\times\mathcal{U}\times\mathcal{V}\times\mathcal{W}$
is well-defined.

###### Proof.

We need to verify that
$F(a,\beta,\alpha,u,v,w)\in\mathbb{R}^{3}\times\mathcal{U}\times\mathcal{V}\times\mathcal{W}$
if
$(a,\beta,\alpha,u,v,w)\in\mathbb{R}^{3}\times\mathcal{U}\times\mathcal{V}\times\mathcal{W}$.
The only term that requires some computation is the nonlinear term

$P(u,w)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}w^{3}(u-Su)+Rw^{3}(u-S^{2}u).$

The fact that $P(u,w)\in\mathcal{U}$ if $(u,w)\in\mathcal{U}\times\mathcal{W}$
follows from

$\displaystyle[P(u,w)](-t,\Omega)$
$\displaystyle=w^{3}(-t,\Omega)\big{(}u(-t,\Omega)-u(-t+4\pi/3,\Omega)\big{)}+w^{3}(t,\Omega)\big{(}u(-t,\Omega)-u(-t-4\pi/3,\Omega)\big{)}$
$\displaystyle=w^{3}(-t,\Omega)\big{(}R_{y}u(t,\Omega)-R_{y}u(t-4\pi/3,\Omega)\big{)}+w^{3}(t,\Omega)\big{(}R_{y}u(t,\Omega)-R_{y}u(t+4\pi/3,\Omega)\big{)}$
$\displaystyle=R_{y}[P(u,w)](t,\Omega).\qed$

##  3 Existence of a branch of solutions of the functional equation

In this section, we prove the existence of a one-parameter family of zeros of
$F$, defined formally in (28), parameterized by $\Omega\in[0,1]$. As a matter
of fact, we consider the complexification of the spaces $\mathcal{U}$,
$\mathcal{V}$ and $\mathcal{W}$, respectively denoted by
$\mathcal{U}^{\mathbb{C}}$, $\mathcal{V}^{\mathbb{C}}$ and
$\mathcal{W}^{\mathbb{C}}$. The mapping
$F:\mathbb{C}^{3}\times\mathcal{U}^{\mathbb{C}}\times\mathcal{V}^{\mathbb{C}}\times\mathcal{W}^{\mathbb{C}}\to\mathbb{C}^{3}\times\mathcal{U}^{\mathbb{C}}\times\mathcal{V}^{\mathbb{C}}\times\mathcal{W}^{\mathbb{C}}$
is still well-defined, as in Lemma 2.6. In Section 4, we conclude the proof of
Marchal’s conjecture by verifying that this zero is indeed in
$\mathbb{R}^{3}\times\mathcal{U}\times\mathcal{V}\times\mathcal{W}$ and that
it represents the branch of the $P_{12}$ family, joining the Lagrange triangle
to the figure eight.

###  3.1 Choice of norms

Let $\nu\geq 1$. Consider the Banach space

$\mathcal{U}_{\nu}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{\\{}u\in\mathcal{U}^{\mathbb{C}}\,:\,\|u\|_{\mathcal{U}_{\nu}}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\sum_{j=1}^{3}\|u_{j}\|_{\nu}<\infty\Big{\\}}$ (36)

where, given $\phi\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})$,

$\|\phi\|_{\nu}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\frac{1}{(2\pi)^{2}}\sum_{k,n\in\mathbb{Z}}\left|\int_{0}^{2\pi}\int_{0}^{1}\frac{\phi(t,\Omega)e^{ikt}\mathcal{T}_{|n|}(2\Omega-1)}{\sqrt{1-(2\Omega-1)^{2}}}\,d\Omega
dt\right|\nu^{|k|}.$ (37)

Here, $\mathcal{T}_{n}:[-1,1]\to[-1,1]$ represents the $n$-th Chebyshev
polynomial of the first kind, given by the recurrence relation

$\mathcal{T}_{0}(s)=1,\qquad\mathcal{T}_{1}(s)=s,\qquad\mathcal{T}_{n}(s)=2s\mathcal{T}_{n-1}(s)-\mathcal{T}_{n-2}(s),\quad
n\geq 2.$ (38)

Importantly, these polynomials satisfy the identity
$\mathcal{T}_{n}(\cos\theta)=\cos(n\theta)$, meaning that infinite series of
Chebyshev polynomials amounts to cosine series. This highlights the
significance of employing them as a basis for series expansions with respect
to the frequency $\Omega$, since the convergence property of analytic Fourier
series holds true for functions defined on the entire range of parameter
values $[0,1]$. This fact contrasts with Taylor expansions, where despite the
function’s analyticity, the presence of poles frequently necessitates
partitioning the function’s domain into numerous subintervals.

Consider also

$\displaystyle\mathcal{V}_{\nu}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{\\{}v\in\mathcal{V}^{\mathbb{C}}\,:\,\|v\|_{\mathcal{V}_{\nu}}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\sum_{j=1}^{3}\|v_{j}\|_{\nu}<\infty\Big{\\}},$ (39)
$\displaystyle\mathcal{W}_{\nu}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{\\{}w\in\mathcal{W}^{\mathbb{C}}\,:\,\|w\|_{\nu}<\infty\Big{\\}},$
(40) $\displaystyle X$ $\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{\\{}\psi:C^{\infty}([0,1],\mathbb{C})\,:\,\|\psi\|_{X}<\infty\Big{\\}},$
(41)

where, given $\psi\in C^{\infty}([0,1],\mathbb{C})$,

$\|\psi\|_{X}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\left|\int_{0}^{1}\frac{\psi(\Omega)\mathcal{T}_{|n|}(2\Omega-1)}{\sqrt{1-(2\Omega-1)^{2}}}\,d\Omega\right|.$
(42)

It will be convenient for the forthcoming norm estimates to observe that the
spaces $\mathcal{U}_{\nu},\mathcal{V}_{\nu},\mathcal{W}_{\nu},X$ form unital
Banach algebras with respect to the product of functions. Let us detail the
demonstration for $X$, namely let us show, for any $\psi_{1},\psi_{2}\in X$,
that $\|\psi_{1}\psi_{2}\|_{X}\leq\|\psi_{1}\|_{X}\|\psi_{2}\|_{X}$:

$\displaystyle\|\psi_{1}\psi_{2}\|_{X}$
$\displaystyle=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\left|\int_{0}^{1}\frac{\psi_{1}(\Omega)\psi_{2}(\Omega)\mathcal{T}_{|n|}(2\Omega-1)}{\sqrt{1-(2\Omega-1)^{2}}}\,d\Omega\right|$
$\displaystyle=\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\left|\int_{0}^{2\pi}\psi_{1}(\cos\theta)\psi_{2}(\cos\theta)\cos(n\theta)\,d\theta\right|$
$\displaystyle=\frac{1}{(2\pi)^{2}}\sum_{n\in\mathbb{Z}}\left|\sum_{n^{\prime}\in\mathbb{Z}}\left(\int_{0}^{2\pi}\psi_{1}(\cos\theta)\cos((n-n^{\prime})\theta)\,d\theta\right)\left(\int_{0}^{2\pi}\psi_{2}(\cos\theta)\cos(n^{\prime}\theta)\,d\theta\right)\right|$
$\displaystyle\leq\left(\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\left|\int_{0}^{2\pi}\psi_{1}(\cos\theta)\cos(n\theta)\,d\theta\right|\right)\left(\frac{1}{2\pi}\sum_{n\in\mathbb{Z}}\left|\int_{0}^{2\pi}\psi_{2}(\cos\theta)\cos(n\theta)\,d\theta\right|\right)$
$\displaystyle=\|\psi_{1}\|_{X}\|\psi_{2}\|_{X}.$

Finally, define

$\mathcal{X}_{\nu}\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}X\times
X\times
X\times\mathcal{U}_{\nu}\times\mathcal{V}_{\nu}\times\mathcal{W}_{\nu},$ (43)

endowed with the norm

$\|x\|_{\mathcal{X}_{\nu}}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\|a\|_{X}+\|\beta\|_{X}+\|\alpha\|_{X}+\|u\|_{\mathcal{U}_{\nu}}+\|v\|_{\mathcal{V}_{\nu}}+\|w\|_{\nu},\qquad\text{for
all }x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu}.$ (44)

Denoting by $\mathcal{D}(F)$ the domain of $F$, we obtain the following.

######  Proposition 3.1.

$F:\mathcal{D}(F)\subset\mathcal{X}_{\nu}\to\mathcal{X}_{\nu}.$

###### Proof.

This follows from Lemma 2.6 (adapted to the complexified space) and

$\mathcal{D}(F)=\Big{\\{}x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu}\,:\,\|\partial_{t}u\|_{\mathcal{U}_{\nu}}<\infty,\|\partial_{t}v\|_{\mathcal{V}_{\nu}}<\infty,\|\partial_{t}w\|_{\nu}<\infty\Big{\\}}.\qed$

###  3.2 Verification of the contraction hypotheses

We now establish the contraction of a Newton-type operator around a numerical
approximation of the branch of solutions. Without deviating from the focus of
this manuscript, it is important to highlight the efficacy of our strategy.
Most existing rigorous continuation techniques employ piecewise-linear – or
even piecewise-constant – approximations of the branch, significantly limiting
the step-size between parameter values and incurring substantial computational
cost. In contrast, our technique, capable of obtaining the entire branch of
solutions as a single fixed-point, is based on the recent works [3, 4]. In
essence, the argument leverages the algebraic structure of $F$, with respect
to the parameter $\Omega$, to verify the contraction mapping theorem around a
high-order approximation of the branch.

The $k$-th Fourier coefficient of a function $\phi\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})$ is denoted by a
subscript $k$ as follows

$\phi_{k}(\Omega)=\frac{1}{2\pi}\int_{0}^{2\pi}\phi(t,\Omega)e^{ikt}\,dt,\qquad\Omega\in[0,1],\quad
k\in\mathbb{Z},$

so that $\phi_{k}\in X$ for all $k\in\mathbb{Z}$. The Fourier-Chebyshev
coefficients are denoted by the double subscript $n,k$ as follows

$\phi_{n,k}=\frac{1}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{1}\frac{\phi(t,\Omega)e^{ikt}\mathcal{T}_{n}(2\Omega-1)}{\sqrt{1-(2\Omega-1)^{2}}}\,d\Omega
dt,\qquad n\in\mathbb{N},\quad k\in\mathbb{Z}.$

Lastly, using a single subscript $n$, we denote the $n$-th Chebyshev
coefficient of a function $\psi\in C^{\infty}([0,1],\mathbb{C})$, namely

$\psi_{n}=\frac{1}{2\pi}\int_{0}^{1}\frac{\psi(\Omega)\mathcal{T}_{n}(2\Omega-1)}{\sqrt{1-(2\Omega-1)^{2}}}\,d\Omega,\qquad
n\in\mathbb{N}.$

Our goal is to prove that a fixed-point operator $T$ (yet to be constructed)
is contracting around a finite Fourier-Chebyshev series approximating the
$P_{12}$ family. This approximation lives in a finite dimensional subspace of
$\mathcal{X}_{\nu}$, which we now detail. Define the _truncation operators_
$\Pi_{K},\Pi_{N,K}:C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})\to
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})$ by

$(\Pi_{K}u)_{k}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}u_{k},&|k|\leq K,\\\
0,&|k|>K,\end{cases}\qquad(\Pi_{N,K}u)_{n,k}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}(\hat{\Pi}_{N}u_{k})_{n},&|k|\leq K,\\\
0,&|k|>K,\end{cases}$ (45)

with $\hat{\Pi}_{N}:X\to X$ given by

$(\hat{\Pi}_{N}u)_{n}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}u_{n},&n\leq N,\\\ 0,&n>N.\end{cases}$ (46)

Consider also the _tail operator_
$\Pi_{\infty(K)}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}I-\Pi_{K}$. Both truncation operators $\Pi_{K}$ and $\Pi_{N,K}$
naturally extend to all $x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu}$ by
acting component-wise

$\Pi_{K}x=(a,\beta,\alpha,\Pi_{K}u,\Pi_{K}v,\Pi_{K}w),\qquad\Pi_{N,K}x=(\hat{\Pi}_{N}a,\hat{\Pi}_{N}\beta,\hat{\Pi}_{N}\alpha,\Pi_{N,K}u,\Pi_{N,K}v,\Pi_{N,K}w).$

Fixing $N,K>0$, one may construct a (non-rigorous) interpolation of the zeros
of $F$ in $\Pi_{N,K}\mathcal{X}_{\nu}$, denoted by $\bar{x}$. For the sake of
completeness, we detail the procedure we followed for this proof. For each
$j=0,\dots,N$, we find approximate unfolding parameters
$a,\beta,\alpha\in\mathbb{C}$ and $2\pi$-periodic solution $u,v\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z},\mathbb{C}^{3}),w\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z},\mathbb{C})$ to (26) satisfying (27) at
the Chebyshev nodes

$\Omega_{j}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\frac{1}{2}(1+\cos(j\pi/N)).$

To be concrete, the periodic solution
$u(t,\Omega_{j}),v(t,\Omega_{j}),w(t,\Omega_{j})$ are represented by finite
Fourier series, with the appropriate symmetries of the $P_{12}$ family as
described in (14):

$\displaystyle
u_{1}(t,\Omega_{j})=\sum_{k\in\mathbb{Z}}(u_{1})_{k}(\Omega_{j})e^{ikt},\qquad$
$\displaystyle(u_{1})_{k}=$ $\displaystyle(u_{1})_{-k},\quad$
$\displaystyle(u_{1})_{2k+1}$ $\displaystyle=0,$ $\displaystyle
u_{2}(t,\Omega_{j})=\sum_{k\in\mathbb{Z}}(u_{2})_{k}(\Omega_{j})e^{ikt},\qquad$
$\displaystyle(u_{2})_{k}=-$ $\displaystyle(u_{2})_{-k},\quad$
$\displaystyle(u_{2})_{2k}$ $\displaystyle=0,$ $\displaystyle
u_{3}(t,\Omega_{j})=\sum_{k\in\mathbb{Z}}(u_{3})_{k}(\Omega_{j})e^{ikt},\qquad$
$\displaystyle(u_{3})_{k}=$ $\displaystyle(u_{3})_{-k},\quad$
$\displaystyle(u_{3})_{2k+1}$ $\displaystyle=0,$

and similarly for $v$ and $w$. Hence, we solve numerically $N+1$ finite
dimensional problems by using Newton’s method on
$\Pi_{K,0}F|_{\Omega_{j}}\circ\Pi_{K,0}$, and we denote by
$\bar{x}_{j}\in\Pi_{K,0}\mathcal{X}_{\nu}$ such an approximate zero, i.e.
$\Pi_{K,0}F|_{\Omega_{j}}(\bar{x}_{j})\approx 0$. Then, we retrieve
numerically the Chebyshev polynomial associated to each component of
$\bar{x}_{j}$ via the _inverse discrete Fourier transform_ , thereby yielding
$\bar{x}\in\Pi_{K,N}\mathcal{X}_{\nu}$. The resulting Fourier-Chebyshev series
coefficients can be found in [25]. Incidentally, it is practical to opt for an
appropriate number of Chebyshev nodes to take full advantage of the _fast
Fourier transform_ algorithm.

Given two Banach spaces $\mathfrak{X},\mathfrak{Y}$, we denote by
$\mathscr{B}(\mathfrak{X},\mathfrak{Y})$ the set of bounded linear operators
from $\mathfrak{X}$ to $\mathfrak{Y}$. To prove the existence of a zero of
$F$, let us construct $A\in\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})$
so that

$T(x)\stackrel{{\scriptstyle\textnormal{\tiny def}}}{{=}}x-AF(x)$ (47)

satisfies the Banach Fixed-Point Theorem. The injectivity of $A$, required to
obtain a zero of $F$, will be a by-product of the contraction.

We define the bounded linear operator
$A:\mathcal{X}_{\nu}\to\mathcal{X}_{\nu}$ given by

$A=A_{\textnormal{finite}}\Pi_{K}+A_{\textnormal{tail}}\Pi_{\infty(K)},$

where
$A_{\textnormal{finite}}=A_{\textnormal{finite}}(\bar{x})\in\mathscr{B}(\Pi_{K}X,\Pi_{K}X)$
is an approximation of $(\Pi_{N,K}DF(\bar{x})\Pi_{N,K})^{-1}$ and
$A_{\textnormal{tail}}:\Pi_{\infty(K)}\mathcal{X}_{\nu}\to\Pi_{\infty(K)}\mathcal{X}_{\nu}$
is defined, for all $x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu}$, by

$\displaystyle A_{\textnormal{tail}}x\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}(0,0,0,u^{\prime},v^{\prime},w^{\prime}),\qquad(u_{j}^{\prime})_{k}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}0,&|k|\leq K,\\\
(ik)^{-1}(u_{j})_{k},&|k|>K,\end{cases}\quad j=1,2,3,$
$\displaystyle(v_{j}^{\prime})_{k}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}0,&|k|\leq K,\\\
(ik)^{-1}(v_{j})_{k},&|k|>K,\end{cases}\quad j=1,2,3,$
$\displaystyle(w^{\prime})_{k}$
$\displaystyle\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\begin{cases}0,&|k|\leq K,\\\ (ik)^{-1}w_{k},&|k|>K.\end{cases}$

We also emphasize that computing $A_{\textnormal{finite}}$ can be done without
numerically inverting the entire square matrix representing
$\Pi_{N,K}DF(\bar{x})\Pi_{N,K}$. Since $DF(\bar{x})$ amounts to a
multiplication operator with respect to $\Omega$, we compute numerically
$A_{j}\in\mathscr{B}(\Pi_{K,0}\mathcal{X}_{\nu},\Pi_{K,0}\mathcal{X}_{\nu})$
such that
$A_{j}(\Pi_{K,0}DF|_{\Omega_{j}}(\bar{x}_{j})\Pi_{K,0})\approx\Pi_{K,0}$ and,
as done for $\bar{x}$, we use the inverse discrete Fourier transform to obtain
the Chebyshev polynomial associated to each component of $A_{j}$.

By construction of $A$, we have that $T\in C^{2}(\mathcal{X}_{\nu})$. For the
remainder of this paper, we fix

$K=70,\qquad N=20,\qquad\nu=\frac{11}{10},\qquad r=10^{-6}.$ (48)

We now give three lemmata leading to the proof that $T$ is a contraction in
$\textnormal{cl}(B_{r}(\bar{x}))$.

######  Lemma 3.2.

$\|AF(\bar{x})\|_{\mathcal{X}_{\nu}}\leq 10^{-8}.$

###### Proof.

We have that

$\displaystyle\|AF(\bar{x})\|_{\mathcal{X}_{\nu}}$
$\displaystyle=\|A_{\textnormal{finite}}\Pi_{K}F(\bar{x})+A_{\textnormal{tail}}\Pi_{\infty(K)}F(\bar{x})\|_{\mathcal{X}_{\nu}}$
$\displaystyle\leq\|A_{\textnormal{finite}}\Pi_{K}F(\bar{x})\|_{\mathcal{X}_{\nu}}+\frac{1}{K+1}\left(\|\Pi_{\infty(K)}g(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathcal{V}_{\nu}}+\|\Pi_{\infty(K)}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\nu}\right).$

Since $g$ and $h$ are polynomials, we can find an upper bound for
$\|AF(\bar{x})\|_{\mathcal{X}_{\nu}}$ by a finite number of computations,
which are carried out by a computer with interval arithmetic. ∎

######  Lemma 3.3.

$\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}\leq\frac{9}{10}.$

###### Proof.

We have that

$\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}=\max\Big{(}\|\Pi_{2K}-ADF(\bar{x})\Pi_{2K}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})},\|\Pi_{\infty(2K)}-ADF(\bar{x})\Pi_{\infty(2K)}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}\Big{)}.$

On the one hand,

$\displaystyle\|\Pi_{2K}-ADF(\bar{x})\Pi_{2K}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\leq\|\Pi_{K}-A_{\textnormal{finite}}\Pi_{K}DF(\bar{x})\Pi_{2K}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\hskip
14.22636pt+\frac{1}{K+1}\max\Big{(}\|\Pi_{\infty(K)}\partial_{a}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(X,\mathcal{C}_{\nu})},$
$\displaystyle\hskip
42.67912pt\max_{i=1,2,3}\Big{(}\sum_{j=1}^{3}\|\partial_{u_{i}}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{u_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)},$
$\displaystyle\hskip
42.67912pt\max_{i=1,2,3}\Big{(}\sum_{j=1}^{3}\|\partial_{v_{i}}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})-\delta_{ji}\partial_{t}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{v_{i}}f_{j}(\bar{u},\bar{v})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{v_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)},$
$\displaystyle\hskip
75.39963pt\sum_{j=1}^{3}\|\partial_{w}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{w}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})-\partial_{t}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)}$
$\displaystyle\leq\|\Pi_{K}-A_{\textnormal{finite}}\Pi_{K}DF(\bar{x})\Pi_{2K}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\hskip
14.22636pt+\frac{1}{K+1}\max\Big{(}\|\Pi_{\infty(K)}\bar{w}^{3}\big{\langle}\bar{u}_{3}-S\bar{u}_{3},\bar{v}_{3}-S\bar{v}_{3}\big{\rangle}\|_{\nu},$
$\displaystyle\hskip
85.35826pt\max_{i=1,2,3}\Big{(}\|\Omega^{2}\|_{X}(\delta_{1i}+\delta_{2i})+\sqrt{3}\big{(}2\|\bar{w}^{3}\|_{\nu}+\|\bar{w}^{3}(L_{\bar{a}}(\bar{v}-S\bar{v}))_{i}\|_{\nu}\big{)}\Big{)},$
$\displaystyle\hskip
85.35826pt1+\max_{i=1,2,3}\Big{(}\|2\Omega\|_{X}(\delta_{1i}+\delta_{2i})+\sqrt{3}\|\bar{w}^{3}(L_{\bar{a}}(\bar{u}-S\bar{u}))_{i}\|_{\nu}\Big{)},$
$\displaystyle\hskip
85.35826pt\|3\bar{w}^{2}\big{\langle}\bar{u}-S\bar{u},L_{\bar{a}}(\bar{v}-S\bar{v})\big{\rangle}\|_{\nu}+\sum_{i=1}^{3}\|3\bar{w}^{2}(\bar{u}_{i}-S\bar{u}_{i})\|_{\nu}+\|3\bar{w}^{2}(\bar{u}_{i}-S^{2}\bar{u}_{i})\|_{\nu}\Big{)},$

where

$\mathcal{C}_{\nu}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\Big{\\{}\phi\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})\,:\,\|\phi\|_{\nu}<\infty\Big{\\}}.$
(49)

We also slightly abuse notation and write $\|\psi(\Omega)\|_{X}$ to denote
$\|\psi\|_{X}$ for $\psi\in X$, e.g. $\|\Omega^{2}\|_{X}=\|\psi\|_{X}$ with
$\psi(\Omega)=\Omega^{2}$. Moreover, the symbol $\delta_{ji}$ denotes the
Kronecker delta, and we use the fact that
$\|R\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}=1$,
$\|I-S\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}=\|I-S^{2}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}=\sqrt{3}$.

On the other hand, the multiplication operator associated with $\phi\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})$, and given by
$\mathcal{M}_{\phi}\psi\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\phi\psi$ for all $\psi\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z}\times[0,1],\mathbb{C})$, satisfies

$\big{(}\Pi_{K}\mathcal{M}_{\phi}\Pi_{\infty(2K)}\big{)}_{k,k^{\prime}}=\big{(}\Pi_{\infty(K)}\phi\big{)}_{k^{\prime}-k},\qquad
k,k^{\prime}\in\mathbb{Z}.$

Hence,

$\displaystyle\|\Pi_{\infty(2K)}-ADF(\bar{x})\Pi_{\infty(2K)}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\leq\|A_{\textnormal{finite}}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\times\max\Big{(}\max_{i=1,2,3}\Big{(}\|\partial_{u_{i}}\eta_{1}(\bar{u})\Pi_{\infty(2K)}\|_{\mathscr{B}(\mathcal{C}_{\nu},X)}+\|\partial_{u_{i}}\gamma(\bar{a},\bar{u},\bar{w})\Pi_{\infty(2K)}\|_{\mathscr{B}(\mathcal{C}_{\nu},X)}+\sum_{j=1}^{3}\|\Pi_{\infty(K)}\partial_{u_{i}}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}$
$\displaystyle\hskip
293.917pt+\|\Pi_{\infty(K)}\partial_{u_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)},$
$\displaystyle\hskip
32.72049pt\max_{i=1,2,3}\|\Pi_{\infty(K)}\partial_{v_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})},$
$\displaystyle\hskip
37.84221pt\|\partial_{w}\gamma(\bar{a},\bar{u},\bar{w})\Pi_{\infty
32K)}\|_{\mathscr{B}(\mathcal{C}_{\nu},X)}+\sum_{j=1}^{3}\|\Pi_{\infty(K)}\partial_{w}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\Pi_{\infty(K)}\partial_{w}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)}$
$\displaystyle+\frac{1}{K+1}\max\Big{(}\max_{i=1,2,3}\Big{(}\sum_{j=1}^{3}\|\partial_{u_{i}}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{u_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)},$
$\displaystyle\hskip
58.8974pt\max_{i=1,2,3}\Big{(}\sum_{j=1}^{3}\|\partial_{v_{i}}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})-\delta_{ji}\partial_{t}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{v_{i}}f_{j}(\bar{u},\bar{v})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{v_{i}}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)},$
$\displaystyle\hskip
91.04872pt\sum_{j=1}^{3}\|\partial_{w}g_{j}(\bar{\beta},\bar{u},\bar{v},\bar{w})\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}+\|\partial_{w}h(\bar{a},\bar{\alpha},\bar{u},\bar{v},\bar{w})-\partial_{t}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu})}\Big{)}$
$\displaystyle\leq\|A_{\textnormal{finite}}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$
$\displaystyle\hskip
14.22636pt\times\max\Big{(}\sqrt{3}\max_{i=1,2,3}\Big{(}\big{(}\delta_{3i}+\|[2\bar{w}^{2}(L_{\bar{a}}(\bar{u}-S\bar{u}))_{i}](0,\Omega)\|_{X}\big{)}\frac{1}{\nu^{2K+1}}+2\|\Pi_{\infty(K)}\bar{w}^{3}\|_{\nu}+\|\Pi_{\infty(K)}\bar{w}^{3}(L_{\bar{a}}(\bar{v}-S\bar{v}))_{i}\|_{\nu}\Big{)},$
$\displaystyle\hskip
46.94687pt\sqrt{3}\max_{i=1,2,3}\|\Pi_{\infty(K)}\bar{w}^{3}(L_{\bar{a}}(\bar{u}-S\bar{u}))_{i}\|_{\nu},$
$\displaystyle\hskip
46.94687pt\|[2\bar{w}\big{\langle}\bar{u}-S\bar{u},L_{\bar{a}}(\bar{u}-S\bar{u})\big{\rangle}](0,\Omega)\|_{X}\frac{1}{\nu^{2K+1}}$
$\displaystyle\hskip
71.13188pt+\|\Pi_{\infty(K)}3w^{2}\big{\langle}u-Su,L_{\bar{a}}(v-Sv)\big{\rangle}\|_{\nu}+\sum_{i=1}^{3}\|\Pi_{\infty(K)}(3\bar{w}^{2}(\bar{u}_{i}-S\bar{u}_{i}))\|_{\nu}+\|\Pi_{\infty(K)}3\bar{w}^{2}(\bar{u}_{i}-S^{2}\bar{u}_{i})\|_{\nu}\Big{)}$
$\displaystyle\hskip
14.22636pt+\frac{1}{K+1}\max\Big{(}\max_{i=1,2,3}\Big{(}\|\Omega^{2}\|_{X}(\delta_{1i}+\delta_{2i})+\sqrt{3}(2\|\bar{w}^{3}\|_{\nu}+\|\bar{w}^{3}(L_{\bar{a}}(\bar{v}-S\bar{v}))_{i}\|_{\nu})\Big{)},$
$\displaystyle\hskip
72.83897pt1+\max_{i=1,2,3}\Big{(}\|2\Omega\|_{X}(\delta_{1i}+\delta_{2i})+\sqrt{3}\|\bar{w}^{3}(L_{\bar{a}}(\bar{u}-S\bar{u}))_{i}\|_{\nu}\Big{)},$
$\displaystyle\hskip
72.83897pt\|3\bar{w}^{2}\big{\langle}\bar{u}-S\bar{u},L_{\bar{a}}(\bar{v}-S\bar{v})\big{\rangle}\|_{\nu}+\sum_{i=1}^{3}\|3\bar{w}^{2}(\bar{u}_{i}-S\bar{u}_{i})\|_{\nu}+\|3\bar{w}^{2}(\bar{u}_{i}-S^{2}\bar{u}_{i})\|_{\nu}\Big{)}.$

Since the above expressions only involve polynomials, as well as the product
of finite matrices, we can find an upper bound for
$\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}$ by a
finite number of computations, which are carried out by a computer with
interval arithmetic. ∎

######  Lemma 3.4.

$\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\|AD^{2}F(x)\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}\leq
10^{4}.$

###### Proof.

Note that

$\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\|AD^{2}F(x)\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}\leq\|A\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\|D^{2}F(x)\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))},$

with
$\|A\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}\leq\max\left(\|A_{\textnormal{finite}}\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})},\frac{1}{K+1}\right)$.
Moreover, observe that, for an operator $\mathfrak{T}\in C^{2}(\ell^{1})$,

$\displaystyle\|D^{2}\mathfrak{T}(x)\|_{\mathscr{B}(\ell^{1},\mathscr{B}(\ell^{1},\ell^{1}))}$
$\displaystyle=\sup_{h\in\ell^{1}}\|D^{2}\mathfrak{T}(x)h\|_{\mathscr{B}(\ell^{1},\ell^{1})}$
$\displaystyle=\sup_{h\in\ell^{1}}\max_{j\in\mathbb{N}}\sum_{i\in\mathbb{N}}|\partial_{j}\sum_{l\in\mathbb{N}}\partial_{l}\mathfrak{T}_{i}(x)h_{l}|$
$\displaystyle\leq\max_{j\in\mathbb{N}}\sum_{i\in\mathbb{N}}\sup_{h\in\ell^{1}}|\partial_{j}\sum_{l\in\mathbb{N}}\partial_{l}\mathfrak{T}_{i}(x)h_{l}|$
$\displaystyle=\max_{j\in\mathbb{N}}\sum_{i\in\mathbb{N}}\|\partial_{j}\nabla\mathfrak{T}_{i}(x)\|_{\ell^{\infty}}$
$\displaystyle=\max_{j\in\mathbb{N}}\sum_{i\in\mathbb{N}}\max_{l\in\mathbb{N}}|\partial_{j}\partial_{l}\mathfrak{T}_{i}(x)|.$

Since the norm $\|\,\cdot\,\|_{\mathcal{X}_{\nu}}$ amounts to a weighted
$\ell^{1}$ space, it is straightforward to obtain an analogous formula for
$\|D^{2}F(x)\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}$
of the form

$\displaystyle\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\|D^{2}F(x)\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}$
$\displaystyle\leq\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\max\Bigg{(}$
$\displaystyle\max\Big{(}\max_{i=1,2,3}\|\partial_{u_{i}}\partial_{a}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(X,X))},\|\partial_{w}\partial_{a}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(X,X))}\Big{)}$
$\displaystyle\quad+\max\Big{(}\max_{i=1,2,3}\|\partial_{u_{i}}\partial_{a}h\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\max_{i=1,2,3}\|\partial_{v_{i}}\partial_{a}h\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\|\partial_{w}\partial_{a}h\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}\Big{)},$
$\displaystyle\max_{i=1,2,3}\Big{\\{}\max\Big{(}\|\partial_{a}\partial_{u_{i}}\gamma\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},X))},\max_{j=1,2,3}\|\partial_{u_{j}}\partial_{u_{i}}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},X))},\|\partial_{w}\partial_{u_{i}}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},X))}\Big{)}$
$\displaystyle\quad+\|\partial_{w}\partial_{u_{i}}g_{i}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}$
$\displaystyle\quad+\max\Big{(}\|\partial_{a}\partial_{u_{i}}h\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\max_{j=1,2,3}\|\partial_{v_{j}}\partial_{u_{i}}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\|\partial_{w}\partial_{u_{i}}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}\Big{)}\Big{\\}},$
$\displaystyle\max_{i=1,2,3}\Big{\\{}\|\partial_{a}\partial_{v_{i}}h\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\max_{j=1,2,3}\|\partial_{u_{j}}\partial_{v_{i}}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\|\partial_{w}\partial_{v_{i}}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}\Big{\\}},$
$\displaystyle\max\Big{(}\|\partial_{a}\partial_{w}\gamma\|_{\mathscr{B}(X,\mathscr{B}(\mathcal{C}_{\nu},X))},\max_{i=1,2,3}\|\partial_{u_{i}}\partial_{w}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},X))},\|\partial_{w}^{2}\gamma\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},X))}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{3}\max\Big{(}\max_{j=1,2,3}\|\partial_{u_{j}}\partial_{w}g_{i}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\|\partial_{w}^{2}g_{i}\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}\Big{)}$
$\displaystyle\quad+\max\Big{(}\|\partial_{a}\partial_{w}h\|_{X,\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\max_{i=1,2,3}\|\partial_{u_{i}}\partial_{w}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},\max_{i=1,2,3}\|\partial_{v_{i}}\partial_{w}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))},$
$\displaystyle\quad\qquad\qquad\|\partial_{w}^{2}h\|_{\mathscr{B}(\mathcal{C}_{\nu},\mathscr{B}(\mathcal{C}_{\nu},\mathcal{C}_{\nu}))}\Big{)}\Bigg{)}$
$\displaystyle\leq\max\Big{(}6\max\big{(}\hat{w}^{2}\hat{u}_{3},\hat{w}\hat{u}_{3}^{2}\big{)}+3\max\big{(}\hat{w}^{3}\hat{v}_{3},\hat{w}^{3}\hat{u}_{3},3\hat{w}^{2}\hat{u}_{3}\hat{v}_{3}\big{)},$
$\displaystyle\hskip
42.67912pt\max_{i=1,2,3}\Big{\\{}6\max\big{(}\delta_{3i}\hat{w}^{2}\hat{u}_{i},(1+\delta_{3i}(\hat{a}-1))\hat{w}^{2},2(1+\delta_{3i}(\hat{a}-1))\hat{w}\hat{u}_{i}\big{)}$
$\displaystyle\hskip 78.24507pt+6\sqrt{3}\hat{w}^{2}$ $\displaystyle\hskip
78.24507pt+3\max\big{(}\delta_{3i}\hat{w}^{3}\hat{v}_{i},(1+\delta_{3i}(\hat{a}-1))\hat{w}^{3},3(1+\delta_{3i}(\hat{a}-1))\hat{w}^{2}\hat{v}_{i}\big{)}\Big{\\}},$
$\displaystyle\hskip
42.67912pt\max_{i=1,2,3}\Big{\\{}3\max\big{(}\delta_{3i}\hat{w}^{3}\hat{u}_{i},(1+\delta_{3i}(\hat{a}-1))\hat{w}^{3},3(1+\delta_{3i}(\hat{a}-1))\hat{w}^{2}\hat{u}_{i}\big{)}\Big{\\}},$
$\displaystyle\hskip
42.67912pt6\max\big{(}\hat{w}\hat{u}_{3}^{2},2\hat{w}\hat{u}_{1},2\hat{w}\hat{u}_{2},2\hat{w}\hat{a}\hat{u}_{3},\hat{u}_{1}^{2}+\hat{u}_{2}^{2}+\hat{a}\hat{u}_{3}^{2}\big{)}$
$\displaystyle\hskip
78.24507pt+6\sqrt{3}\sum_{i=1}^{3}\max(\hat{w}^{2},2\hat{w}\hat{u}_{i})$
$\displaystyle\hskip
78.24507pt+9\max\big{(}\hat{w}^{2}\hat{u}_{3}\hat{v}_{3},\hat{w}^{2}\hat{v}_{1},\hat{w}^{2}\hat{v}_{2},\hat{w}^{2}\hat{a}\hat{v}_{3},\hat{w}^{2}\hat{u}_{1},\hat{w}^{2}\hat{u}_{2},\hat{w}^{2}\hat{a}\hat{u}_{3},2\hat{w}(\hat{u}_{1}\hat{v}_{1}+\hat{u}_{2}\hat{v}_{2}+\hat{a}\hat{u}_{3}\hat{v}_{3})\big{)}\Bigg{)},$

where we used the Banach algebra property, and
$\hat{a}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\|\bar{a}\|_{X}+r$,
$\hat{u}_{i}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\|\bar{u}_{i}\|_{\nu}+r$,
$\hat{v}_{i}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\|\bar{v}_{i}\|_{\nu}+r$,
$\hat{w}\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}\|\bar{w}\|_{\nu}+r$.

Therefore, we can find an upper bound for
$\sup_{x\in\textnormal{cl}(B_{r}(\bar{x}))}\|AD^{2}F(x)\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}$
by a finite number of computations, which are carried out by a computer with
interval arithmetic. ∎

Lemma 3.2, 3.3 and 3.4 yield the desired contraction of $T$.

######  Theorem 3.5.

$T:\textnormal{cl}(B_{r}(\bar{x}))\to\textnormal{cl}(B_{r}(\bar{x})),$

and, for all $x,x^{\prime}\in\textnormal{cl}(B_{r}(\bar{x}))$,

$\|T(x)-T(x^{\prime})\|_{\mathcal{X}_{\nu}}\leq\kappa\|x-x^{\prime}\|_{\mathcal{X}_{\nu}},$

where $\kappa=\frac{91}{100}<1$.

###### Proof.

Let $x,x^{\prime}\in\textnormal{cl}(B_{r}(\bar{x}))$. Taylor’s theorem yields

$\displaystyle\|T(x)-\bar{x}\|_{\mathcal{X}_{\nu}}$
$\displaystyle=\|T(\bar{x})-\bar{x}+[DT(\bar{x})](x-\bar{x})+\int_{0}^{1}[DT(\bar{x}+t(x-\bar{x}))-DT(\bar{x})](x-\bar{x})\,dt\|_{\mathcal{X}_{\nu}}$
$\displaystyle\leq\|AF(\bar{x})\|_{\mathcal{X}_{\nu}}+\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}r+\frac{1}{2}\sup_{x^{\prime\prime}\in\textnormal{cl}(B_{r}(\bar{x}))}\|AD^{2}F(x^{\prime\prime})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}r^{2}$
$\displaystyle\leq 10^{-8}+\frac{9}{10}r+\frac{10^{4}}{2}r^{2}$
$\displaystyle\leq r,$

and, by the mean value theorem, we have

$\displaystyle\|T(x)-T(x^{\prime})\|_{\mathcal{X}_{\nu}}$
$\displaystyle\leq\sup_{x^{\prime\prime}\in\textnormal{cl}(B_{r}(\bar{x}))}\|DT(\bar{x})-DT(\bar{x})+DT(x^{\prime\prime})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}\|x-x^{\prime}\|_{\mathcal{X}_{\nu}}$
$\displaystyle\leq\left(\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}+\sup_{x^{\prime\prime}\in\textnormal{cl}(B_{r}(\bar{x}))}\|AD^{2}F(x^{\prime\prime})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu}))}r\right)\|x-x^{\prime}\|_{\mathcal{X}_{\nu}}$
$\displaystyle\leq\left(\frac{9}{10}+10^{4}\times
10^{-6}\right)\|x-x^{\prime}\|_{\mathcal{X}_{\nu}}$
$\displaystyle=\kappa\|x-x^{\prime}\|_{\mathcal{X}_{\nu}}.$

Therefore, $T$ satisfies the Banach Fixed-Point Theorem in
$\textnormal{cl}(B_{r}(\bar{x}))$. ∎

At last, we have the existence of a family, parameterized by $\Omega\in[0,1]$,
of relative choreographies.

######  Corollary 3.6.

There exists a unique zero of $F$ in $\textnormal{cl}(B_{r}(\bar{x}))$.

###### Proof.

By Theorem 3.5,
$\|DT(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}=\|I-ADF(\bar{x})\|_{\mathscr{B}(\mathcal{X}_{\nu},\mathcal{X}_{\nu})}<1$
which implies that $ADF(\bar{x})$ is invertible. Hence, $A$ is surjective. To
show injectivity, note that
$A_{\textnormal{tail}}:\Pi_{\infty(K)}\mathcal{X}_{\nu}\to\Pi_{\infty(K)}\mathcal{X}_{\nu}$
is injective. Moreover,
$A_{\textnormal{finite}}:\Pi_{K}\mathcal{X}_{\nu}\to\Pi_{K}\mathcal{X}_{\nu}$
is by construction equivalent to a finite dimensional square matrix which
implies that surjectivity is equivalent to injectivity. Thus, by injectivity
of $A$, we have that the fixed-point of $T$ is a zero of $F$. ∎

##  4 Proof of Marchal’s Conjecture

In this section, we prove that the branch of solutions of the functional
equation (28) proven to exist in Section 3 corresponds to the $P_{12}$
Marchal’s family of relative periodic choreographies and that this family ends
in the figure eight of Chenciner and Montgomery. This concludes the proof of
Marchal’s conjecture. Before doing so, let us consider the following auxiliary
result giving sufficient conditions for an isolated zero of a map to satisfy
an invariance property.

######  Lemma 4.1.

Let $\mathfrak{F}:\mathcal{D}(\mathfrak{F})\subset\mathfrak{X}\to\mathfrak{X}$
and $\mathfrak{S}\in\mathscr{B}(\mathfrak{X},\mathfrak{X})$ such that
$\mathfrak{F}\circ\mathfrak{S}=\mathfrak{S}\circ\mathfrak{F}$ and
$\|\mathfrak{S}\|_{\mathscr{B}(\mathfrak{X},\mathfrak{X})}\leq 1$. Consider
$\bar{x},\tilde{x}\in\mathcal{D}(\mathfrak{F})$ and $r>0$ such that
$\tilde{x}$ is the unique element in
$\textnormal{cl}(B_{r}(\bar{x}))\cap\mathcal{D}(\mathfrak{F})$ satisfying
$\mathfrak{F}(\tilde{x})=0$. If $\mathfrak{S}\bar{x}=\bar{x}$, then
$\mathfrak{S}\tilde{x}=\tilde{x}$.

###### Proof.

The property $\mathfrak{F}\circ\mathfrak{S}=\mathfrak{S}\circ\mathfrak{F}$
implies that $\mathfrak{S}\tilde{x}$ is a zero of $\mathfrak{F}$. From
$\|\mathfrak{S}\|_{\mathscr{B}(\mathfrak{X},\mathfrak{X})}\leq 1$, it follows
that $\mathfrak{S}\tilde{x}\in\textnormal{cl}(B_{r}(\bar{x}))$. The fact that
$\tilde{x}$ is the unique zero of $\mathfrak{F}$ in
$\textnormal{cl}(B_{r}(\bar{x}))$ concludes the proof. ∎

We now finish the demonstration of our main result. For the sake of clarity,
we give a precise statement of Theorem 1.1 as follows.

######  Theorem 4.2.

The $3$-body problem with the equation (1) has the family of relative periodic
solutions $q=(q_{1},q_{2},q_{3})$ of the form

$q_{j}(t)=e^{-\Omega\bar{J}t}U(t+4\pi j/3,\Omega),\qquad j=1,2,3,$

where $U(t,\Omega):\mathbb{R}/2\pi\mathbb{Z}\times[0,1]\to\mathbb{R}^{3}$ is
an analytic function obtained as the unique fixed-point of $T$ in Section 3.
At $\Omega=1$, the solution is the Lagrange triangle. At $\Omega=0$, the
solution is of the form $q=(0,q_{y},q_{z})$, and its orbit is an eight-shaped
figure with exactly the same properties as the main theorem of [14], and
reported in Lemma 2.2.

###### Proof.

From Theorem 3.5, we have proven existence (and local uniqueness) of a zero
$\tilde{x}=(\tilde{a},\tilde{\beta},\tilde{\alpha},\tilde{u},\tilde{v},\tilde{w})\in\mathcal{X}_{\nu}$
of $F$, defined in (28). The functions $\tilde{u},\tilde{v},\tilde{w}$ are
analytic with respect to $t$ due to our choice of norm (37). As a matter of
fact, since the fixed-point operator $T$, defined in (47), is analytic with
respect to the parameter $\Omega$, it follows from the uniform contraction
that all the functions
$\tilde{a},\tilde{\beta},\tilde{\alpha},\tilde{u},\tilde{v},\tilde{w}$ are
analytic.

Furthermore, according to Lemma 2.3 and 2.5, it follows that
$\tilde{\alpha}(\Omega)=\tilde{\beta}(\Omega)=0$ for all $\Omega\in[0,1]$.
Therefore,

$U=L_{\sqrt{\tilde{a}}}\tilde{u}$

is an analytic periodic solution of (7).

To show that $U(t,1)$ is the Lagrange triangle, we note that
$\tilde{x}|_{\Omega=1}$ is the unique zero of $F|_{\Omega=1}$ in
$\textnormal{cl}(B_{r}(\bar{x}|_{\Omega=1}))$. By imposing that
$\bar{x}|_{\Omega=1}$ is the Lagrange triangle, it follows that
$\tilde{x}|_{\Omega=1}=\bar{x}|_{\Omega=1}$. Specifically, recalling Lemma
2.4, we set

$\bar{x}|_{\Omega=1}=(0,0,0,\bar{u}(t,1),\partial_{t}\bar{u}(t,1),3^{-1/3}),$

where

$\bar{u}(t,1)=\begin{pmatrix}3^{-1/6}\cos 2t\\\ -3^{-1/6}\sin 2t\\\ \cos
t\end{pmatrix}.$

Now, the Banach space $\mathcal{X}_{\nu}$ contains complex-valued functions,
hence we cannot a priori conclude that $U$ is the $P_{12}$ family. To solve
this, we define $\Sigma:\mathcal{X}_{\nu}\to\mathcal{X}_{\nu}$ by

$\Sigma x\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}(a^{*},\beta^{*},\alpha^{*},u^{*},v^{*},w^{*}),\qquad\text{for all
}x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu},$

where, given a function $h:\mathbb{R}\to\mathbb{C}$ (resp.
$h:\mathbb{R}^{2}\to\mathbb{C}$), we use the notation
$[h^{*}](\Omega)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}h(\Omega)^{*}$ (resp.
$[h^{*}](t,\Omega)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}h(t,\Omega)^{*}$), with the superscript ∗ representing the complex
conjugate. The functions $a,\beta,\alpha,u,v,w$ are real-valued if and only if
$\Sigma x=x$. Again, we constraint the Fourier-Chebyshev coefficients (which
are finite in number) of the approximation of the $P_{12}$ family such that
$\Sigma\bar{x}=\bar{x}$. It is straightforward to verify that Lemma 4.1 is
satisfied with $\mathfrak{F}=F$, $\mathfrak{S}=\Sigma$. Therefore,

$\Sigma\tilde{x}=\tilde{x},$

which completes the proof that $U$ is real-valued and represents the $P_{12}$
family.

Lastly, we show that $U(t,0)$ is the figure eight described in Lemma 2.2.
First, let us prove that $U(t,0)$ lies in the $yz$-plane for all $t$. Let
$\Sigma_{0}:\mathcal{X}_{\nu}\to\mathcal{X}_{\nu}$ be given by

$\Sigma_{0}x\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}(a,\beta,\alpha,-u_{1},u_{2},u_{3},-v_{1},v_{2},v_{3},w),$

for all $x=(a,\beta,\alpha,u,v,w)\in\mathcal{X}_{\nu}$, with
$u=(u_{1},u_{2},u_{3})$ and $v=(v_{1},v_{2},v_{3})$. The functions
$u_{1}(\,\cdot\,,0),v_{1}(\,\cdot\,,0)$ are zero if and only if
$\Sigma_{0}x|_{\Omega=0}=x|_{\Omega=0}$. As done for the Lagrange triangle, we
constraint the finite number of Fourier-Chebyshev coefficients defining the
approximation of the $P_{12}$ family such that
$\Sigma_{0}\bar{x}|_{\Omega=0}=\bar{x}|_{\Omega=0}$. Note that
$\tilde{x}|_{\Omega=0}$ is the unique zero of $F|_{\Omega=0}$ in
$\textnormal{cl}(B_{r}(\bar{x}|_{\Omega=0}))$. Then, Lemma 4.1 holds with
$\mathfrak{F}=F|_{\Omega=0}$, $\mathfrak{S}=\Sigma_{0}$. Whence,

$\Sigma_{0}\tilde{x}|_{\Omega=0}=\tilde{x}|_{\Omega=0},$

meaning that $U(t,0)$ lies in the $yz$-plane for all $t$ as desired.

To establish that the orbit of $U(t,0)$ is the same as the one proven by
Chenciner and Montgomery [14], we satisfy Lemma 2.2. Points
$(i)$-$(ii)$-$(iii)$ follow immediately from the symmetries considered in
$\mathcal{X}_{\nu}$. It remains to verify $(iv)$. To do so, first observe that

$\mu(t)\stackrel{{\scriptstyle\textnormal{\tiny
def}}}{{=}}U(t,0)\times\dot{U}(t,0)=\begin{pmatrix}U_{y}(t,0)\dot{U}_{z}(t,0)-U_{z}(t,0)\dot{U}_{y}(t,0)\\\
0\\\ 0\end{pmatrix}.$

From our symmetries, detailed in (14), it is straightforward to prove that
$\mu(\pi/2)=0$. Since the function $\mu$ is known with a margin of error
related to $r=10^{-6}$ (see Theorem 3.5), we cannot directly determine its
sign close to $\pi/2$. In fact, it holds that
$\dot{\mu}(\pi/2)=\ddot{\mu}(\pi/2)=0$. Thus, using the computer and interval
arithmetic, we perform the following steps:

  1. 1.

we show that $\mu(t)<0$ on $[0,3/2]$. It remains to study the sign of $\mu$ on
$[3/2,\pi/2]$.

  2. 2.

we show that $\dddot{\mu}(t)>0$ on $[3/2,\pi/2]$. Note that we are able to
control derivatives since $U\in\mathcal{U}_{\nu}$ with $\nu>1$, indeed the
Fourier coefficients $a_{k}$ of a function in this space enjoy an exponential
decay rate $|a_{k}|\leq(\sum_{k\in\mathbb{Z}}|a_{k}|)\nu^{-|k|}$. Hence,
$\ddot{\mu}$ is strictly monotone on $[3/2,\pi/2]$, therefore
$\ddot{\mu}(t)=0$ if and only if $t=\pi/2$ which means that $\ddot{\mu}$ has a
sign on $[3/2,\pi/2)$. Repeating this argument, it follows that $\dot{\mu}$
has a sign on $[3/2,\pi/2)$, and in turns $\mu(t)<0$ for all $t\in[3/2,\pi/2)$
as desired.

This concludes the proof that $U$ is the $P_{12}$ family, starting at the
Lagrange triangle and ending at the figure eight ∎

##  5 Future work

By way of conclusions, we consider several additional directions which we view
as natural continuations of the present work.

First, the figure eight choreography is a planar curve having three obvious
symmetry axes: two of them in the plane of the eight, and a third one in the
direction orthogonal to the plane. In our setup, $U_{z}$ corresponds to the
direction along which the eight is the longest, $U_{y}$ the orthogonal
direction to $U_{z}$ in the plane that contains the eight, and $U_{x}$
orthogonal to this plane. In [12], Chenciner, Féjoz and Montgomery find three
families of relative choreographies that bifurcate out of the eight
corresponding to each of the symmetry axes, named $\Gamma_{1}$, $\Gamma_{2}$,
and $\Gamma_{3}$ respectively.

As stated in [12], $\Gamma_{1}$ corresponds to the $P_{12}$ family studied
here. The family $\Gamma_{2}$, which is related to rotations about the $U_{y}$
direction, was studied numerically in [7] where numerical evidence suggests
that there is a bifurcation from $P_{12}$ to a branch of orbits having axial
symmetry in the rotating frame. It would be interesting to extend the proof
done here, to have a different mechanism to arrive from Lagrange’s triangle to
the figure eight using this connection of $P_{12}$ to $\Gamma_{2}$.

The family $\Gamma_{3}$ was already known numerically to Michel Hénon using
computations similar to [22] and has been considered in other works, see for
example [6]. Numerical evidence suggests that there is no off-plane
bifurcation from $\Gamma_{3}$ and making this statement rigorous is also an
interesting future direction.

A numerical study of linear stability is reported in [12]. Chenciner, Féjoz
and Montgomery mention that the $P_{12}$ family becomes unstable as soon as
the family leaves the eight: so, we do not expect to have an interval of
stability. However [12] and [7] suggest an interval of stability in
$\Gamma_{2}$ and $\Gamma_{3}$ from the eight.

Finally, we remark that in the paper [7], four of the authors of the present
work stated the following generalization of Marchal’s conjecture:

######  Conjecture 5.1.

For any odd number of bodies, the $n$-gon choreography and the $n$-body figure
eight are in the same continuation class.

After [7] was published, Chenciner stressed, in private communication, that
the conjecture of Marchal was not just about having a connection between the
triangle and the eight. It was also about showing that both choreographies
actually belong to the $P_{12}$ family. In this manner, Conjecture 5.1 should
not be called a generalization, but a conjecture in its own right, supported
by the numerical evidence in [7].

Specifically, Theorem 38 in [19] states that for any $n\geq 3$ and for each
integer $k$ such that $1\leq k\leq n/2$, the $n$-polygonal relative
equilibrium has a least one family of spatial periodic solutions with
prescribed symmetries in terms of $k$. In [7] the numerical computations
support the conjecture that the $n$-gon choreography and the figure eight
choreography are in the same continuation class through the family with
symmetries $k=2$. On the other hand, numerical computations in [11] support
the conjecture that the $n$-gon choreography and the figure eight are in the
same continuation class through the generalized $P_{12}$ family, which
corresponds in [19] to the family with symmetries $k=(n-1)/2$.

Therefore, the generalized Marchal’s conjecture for odd number of bodies
should read as follows.

######  Conjecture 5.2.

(Generalized Marchal’s Conjecture - Updated) For any odd number of bodies, the
$n$-gon choreography and the figure eight are in the same continuation class
through two different families of relative choreographies.

This conjecture, which is supported by the available numerical evidence,
requires settling many open problems. For example the existence and the
uniqueness of the figure eight for any odd number of bodies. This uniqueness
is still open for $n=3$. Actually, only for $n=3$ do the two different
families start with the same symmetries. Furthermore, the numerical evidence
in [7] shows that the second family that connects the triangle with the eight
arises precisely as a symmetry-breaking from the $P_{12}$ family. This fact
does not contradict the local uniqueness obtained in this paper because the
second family is precisely a symmetry-breaking outside the space of discrete
symmetries $\mathcal{U}$ considered in Section 2.

The verification of the generalized Marchal’s conjecture for $n=3$, and also
for other odd number of bodies, should be amenable to the methods developed in
the present paper.

##  6 Acknowledgments

We would like to thank A. Chenciner, J. Féjoz, R. Montgomery and C. Simó for
fruitful discussions. R. Calleja was partially supported by UNAM-PAPIIT under
grant No. IN 103423. O. Hénot was supported by the Fondation Mathématique
Jacques Hadamard. J.-P. Lessard was supported by NSERC. J. D. Mireles James
was partially supported by NSF under grant No. DMS 2307987.

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