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Periodic, Quasi-Periodic and Chaotic Motions in Celestial Mechanics:Theory and Applications

Periodic, Quasi-Periodic and Chaotic Motions in CelestialMechanics: Theory and Applications

Selected papers from the Fourth Meeting on Celestial Mechanics, CELMEC IVSan Martino al Cimino (Italy), 11–16 September 2005

Edited byA. CellettiDipartimento di Matematica, Università di Roma “Tor Vergata”, ItalyandS. Ferraz-MelloInstituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Brasil

Reprinted from Celestial Mechanics and Dynamical Astronomy, Volume 95(1–4), 2006

123

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Cover image: Resonances, tori and chaotic motions in a pendulum-like system through FastLyapunov Indicators. Courtesy of: A. Celletti, C. Froeschlé, E. Lega.

CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY

Vol. 95 Nos. 1–4 2006

Preface

Periodic Orbits and Variational Methods

S. TERRACINI / On the variational approach of the periodic n-bodyproblem

A.D. BRUNO and V.P. VARIN / On families of periodic solutions ofthe restricted three-body problem

E. BARRABÉS, J.M. CORS, C. PINYOL and J. SOLER / Hip-hopsolutions of the 2N-body problem

V. BARUTELLO and S. TERRACINI / Double choreographicalsolutions the n-body type problems

J.F. PALACIÁN and P. YANGUAS / From the circular to the spatialelliptic restricted three-body problem

C. BELMONTE, D. BOCCALETTI and G. PUCACCO / Stability ofaxial orbits in galactic potentials

Perturbation Theory and Regularization

A. CELLETTI and L. CHIERCHIA / KAM tori for N-bodyproblems: a brief history

C. FROESCHLÉ, E. LEGA and M. GUZZO / Analysis of the chaot-ic behaviour of orbits diffusing along the Arnold web

E. CANALIAS, A. DELSHAMS, J.J. MASDEMONT and P. ROLDÁN / The scattering map in the planar restrictedthree body problem

M. ALVAREZ, J.M. CORS and J. DELGADO / On final evolutionsin the restricted planar parabolic three-body problem

J. WALDVOGEL / Quaternions and the perturbed Kepler problem

Dynamics of Solar and Extrasolar Systems

A. LEMAITRE, S. D’HOEDT, and N. RAMBAUX / The 3:2 spin-orbit resonant motion of Mercury

J.D. HADJIDEMETRIOU / Symmetric and asymmetric librations inextrasolar planetary systems: a global view

1–2

3–25

27–54

55–66

67–80

81–99

101–116

117–139

141–153

155–171

173–200

201–212

213–224

225–244

F. NAMOUNI and J.L. ZHOU / The influence of mutual perturba-tions on the eccentricity excitation by jet acceleration in extra-solar planetary systems

G. VOYATZIS and J.D. HADJIDEMETRIOU / Symmetric andasymmetric 3:1 resonant periodic orbits with an application tothe 55Cnc extra-solar system

Z. SÁNDOR / Estimations of orbital parameters of exoplanets fromtransit photometry by using dynamical constraints

S. BREITER and A. ELIPE / Critical inclination in the main problem of a massive satellite

M. FOUCHARD, C. FROESCHLÉ, G. VALSECCHI and H. RICKMAN / Long-term effects of the Galactic tide oncometary dynamics

A. CELLETTI and G. PINZARI / Dependence on the observationaltime intervals and domain of convergence of orbital determina-tion methods

Space Dynamics and Applications

A. ROSSI and G.B.VALSECCHI / Collision risk against space debrisin earth orbits

M. DELLNITZ, O. JUNGE, M. POST and B. THIERE / On targetfor Venus – set oriented computation of energy efficient lowthrust trajectories

C. CIRCI and P. TEOFILATTO / Weak stability boundary trajecto-ries for the deployment of lunar spacecraft constellations

R. ARMELLIN, M. LAVAGNA and A. ERCOLI-FINZI / Aero-gravity assist maneuvers: controlled dynamics modelingand optimization

B. DE SAEDELEER / Analytical theory of a lunar artificial satellitewith third body perturbations

A. CACCIANI, R. BRIGUGLIO, F. MASSA and P. RAPEX /Precise measurement of the solar gravitational red shift

245–257

259–271

273–285

287–297

299–326

327–344

345–356

357–370

371–390

391–405

407–423

425–437

Celestial Mechanics and Dynamical Astronomy (2006) 95:1–2DOI 10.1007/s10569-006-9040-y

Preface

Alessandra Celletti · Sylvio Ferraz-Mello

The Fourth International Meeting of Celestial Mechanics—CELMEC IV—took placein the welcoming landscape of San Martino al Cimino, about 100 km north of Roma,during the period 11–16 September 2005. Following the tradition of the previous CEL-MEC meetings (taking place every 4 years), the goal was to gather together scientistsfrom the different communities involved in Celestial Mechanics (such as universities,astronomical observatories, space agencies, research institutes and industries). Thissynergy was aimed to satisfy the demand of comparing complementary ideas and tech-niques on the recent advances in Celestial Mechanics. The meeting was particularlycrowded with respect to the previous editions, thanks to the participation of about130 people coming from all over the world.

The strong interdisciplinary character of modern Celestial Mechanics is witnessedby the different contributions presented in the current publication, ranging fromadvanced mathematical theories to sophisticated numerical investigations of the solarsystem dynamics. Each section is opened by review papers, which introduce to leadingsubjects, like the variational approaches to find periodic orbits, the stability theoryof the N-body problem, the spin-orbit resonances and chaotic dynamics, the spacedebris polluting the circumterrestrial space. The subsequent research papers encom-pass many key topics of Celestial Mechanics, often bridging from theory to applica-tions, from dynamical system theory to planetary science, from natural to artificialsatellite theory. This nice intermingling of subjects was made possible by the enthusi-astic presentations and discussions of the participants.

A. Celletti (B)Dipartimento Di Matematica, Universita’ Di Roma Tor Vergata,Via Della Ricerca Scientifica, Roma 133, Italye-mail: [email protected]

S. Ferraz-MelloIAG-Universidade de Sao Paulo, Rua do Matao,1226, Cidade Universitária, CEP 05508-900Sao Paulo, Brazile-mail: [email protected]

2 Preface

The CELMEC IV organizing committee was composed by Alessandra Celletti(Università di Roma “Tor Vergata”), Andrea Milani (Università di Pisa), EttorePerozzi (Telespazio, Roma) and Giovanni B. Valsecchi (Istituto Nazionaledi Astrofisica—IASF, Roma). The realization of the meeting was made possiblethanks to the financial supports provided by the following institutions: Universityof Roma “Tor Vergata” and its Department of Mathematics, Gruppo Nazionale perla Fisica Matematica (GNFM), Telespazio, European Space Agency (ESA), Univer-sity of Pisa, National Institute for Astrophysics (INAF), Balletti Park Hotel; a specialthank to Antonio Giorgilli for his financial contribution through the University ofMilano Bicocca. These sponsors allowed many young people and researchers fromdeveloping countries to attend the meeting. CELMEC IV was promoted by the Ital-ian Society of Celestial Mechanics and Astrodynamics (SIMCA) and it was hostedby the Balletti Park Hotel (San Martino al Cimino, Viterbo), which provided a veryhandsome atmosphere.

All authors were encouraged to write papers of a length that they consideredsuitable for the presentation of their results. The editors and the editorial board of“Celestial Mechanics and Dynamical Astronomy” arranged for competent and fastrefereeing so that all papers could be reviewed and, when necessary, revised beforepublication.

Alessandra Celletti and Sylvio Ferraz-Mello

Celestial Mechanics and Dynamical Astronomy (2006) 95:3–25DOI 10.1007/s10569-006-9025-x

R E V I E W A RT I C L E

On the variational approach to the periodic n-bodyproblem

S. Terracini

Received: 17 November 2005 /Revised: 6 March 2006 /Accepted: 12 April 2006 / Published online: 7 July 2006© Springer Science+Business Media B.V. 2006

Abstract This expository paper gathers some of the results obtained by the authorin recent works in collaboration with Davide Ferrario and Vivina Barutello, focusingon the periodic n-body problem from the perspective of the calculus of variationsand minimax theory. These researches were aimed at developing a systematic vari-ational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems andachievements of this approach. The material here was exposed in the talk that givenat the Meeting CELMEC IV promoted by SIMCA (Società italiana di MeccanicaCeleste).

Keywords Symmetric periodic orbits · 3-body problem · Collisions · Minimizers ofthe Lagrangian action

1 Introduction

Among all periodic solutions of the planar 3-body problem, the relative equilibriummotions—the equilateral Lagrange and the collinear Euler-Moulton solutions—aredefinitely the simplest and most known. They both show an evident symmetry (SO(2)and O(2) respectively), that is, they are equivariant with respect to the symmetrygroup of dimension 1 acting as SO(2) (resp. O(2)) on the time circle and on theplane, and trivially on the set of indexes {1, 2, 3}. Furthermore, they are minimizers ofthe Lagrangian action functional in the space of all loops having their same symme-try group. Hence, for given a symmetry group G, G-equivariant minimizers for theaction functional can be thought as the natural generalization of relative equilibriummotions.

S. Terracini (B)Dipartimento di Matematica e Applicazioni,Università di Milano Bicocca, Via Cozzi 53,20125 Milano, Italye-mail: [email protected]

4 S. Terracini

This perspective has known a wide popularity in the recent literature and hasproduced a new boost in the study of periodic trajectories to the n-body problem; therecent discovery of the Chenciner and Montgomery eight-shaped orbit is emblematicof this renewed interest (see, for instance Chenciner and Montgomery 2000; Chenciner2002; Ferrario and Terracini 2004 and the major part of the bibliographical referenceshere). Indeed, by Palais principle of symmetric criticality, periodic and quasi-periodicsolutions of the n-body problem can be found as critical points of the Lagrangianaction functional restricted to suitable spaces of symmetric paths.

Let us consider n point particles with masses m1, m2, . . . , mn and positions x1, x2, . . . ,xn ∈ R

d, in dimension d ≥ 2. We denote by X the space of configurations with centerof mass in 0, and by X = X � � the set of collision-free configurations (collisionmeans xi = xj for some i �= j). On the configuration space we define the homogeneous(Newton) potential of degree −α < 0:

U(x) =∑i<j

Ui,j(|xi − xj|) Ui,j(|xi − xj|) = mimj

|xi − xj|α .

In many cases we shall simply require the Ui,j’s be asymptotically homogeneus onlyat the singularity. It is worthwhile noticing that the major part of our analysis can beextended to logarithmic potentials.

The potential U has an actractive singularity on collisions. We are interested in(relative) periodic solutions to the system of differential equations:

mixi = ∂U∂xi

.

We associate with the equation the Lagrangian integrand

L(x, x) = L = K + U =∑

i

12

mi|xi|2 +∑i<j

Ui,j(|xi − xj|)

and the action functional:

A(x) =∫ T

0L(x(t), x(t))dt.

Sometimes it will be preferable to consider the problem in a frame rotating uni-formly about the vertical axis, with an angular speed ω; the corresponding action Aω

then contains a gyroscopic term, associated with Coriolis force. Relative equilibria arestationary solutions in this rotating system; they correspond to those configurations(termed central) which are critical for the restriction of the potential U to the ellipsoidI = λ where I denotes the momentum of inertia:

I(x) =∑

i

mi|xi|2.

We shall seek periodic solutions as critical points of the action functional on theSobolev space of T-periodic trajectories: � = H1(T, X ) or, to be more precise, of theaction constrained on suitable linear subspaces �0 ⊂ �.

Two are the major difficulties in following the variational approach; the first is dueto the lack of coercivity (or of Palais–Smale) due to the vanishing at infinity of the forcefields: indeed sequences of almost–critical points (such as minimizing sequences) mayvery well diverge. Furthermore, as the potential U is singular on collisions, miminizers

On the variational approach to the periodic n-body problem 5

or other critical points can a priori be collision trajectories. Many strategies were pro-posed in the literature in order to face these problems. The development of a suitableCritical Point theory taking into account of the contribution of fake periodic solutions(the critical points at infinity) was proposed by some authors (Bahri and Rabinowitz1991; Majer and Terracini 1993, 1995a,b; Riahi 1999) and returned a good estimateof the number of periodic trajectories satisfying an appropriate bound on the length,with the main disadvantage of requiring a stronger order of infinity at the collisions(the strong force condition). Another suitable strategy in order to recover coercivityof action functional consists in imposing a symmetry constraint on the loop space. Sur-prisingly enough, once coercivity is recovered, also the problem of collisions loosespart of its dramatic character. This fact was remarked for the first time in Degiovanniet al. (1987) and Degiovanni and Giannoni, (1988) for two–body problems, in Serraand Terracini (1994) for three–body problems and, since then, widly exploited in theliterature. This is indeed a general fact: in many cases, minimizing trajectories are freeof collisions; a recent breakthrough in this direction is due of the neat idea, due to C.Marchal, of averaging over all possible variations (generalized and exposed in section3; Marchal’s idea was first exposed in Chenciner 2002). This argument can be used inmost of the known cases to prove the absence of collisions for minimizing trajecories.

2 Symmetry groups and equivariant orbits

Let us start by introducing some basics concepts and definitions from Ferrario andTerracini (2004). Let G be a finite group endowed with:

– an orthogonal representation of dimension 2, say, τ : G → O(2) (on cyclic timeT ∼= S1);

– an orthogonal representation (on the euclidean space Rd), say, ρ : G → O(d);

– and an homomorphism on the symmetric group on n elements n = {1, 2, . . . , n},say, σ : G → n.

Then G acts on time (translation and reversal) T via τ and it acts on the configurationspace X via ρ and σ in the following way

∀i = 1 . . . n : (gx)i = ρ(g)xσ(g)−1(i).

As a consequence we have an action on the space of trajectories:

Definition 2.1 A continuous function x(t) = (x1(t), . . . , xn(t)) is G–equivariant if

∀g ∈ G : x(gt) = (gx)(t).

The linear subspace �0 = �G ⊂ � denotes the set of periodic trajectories in � whichare equivariant with respect to the G-action:

The Palais Principle of symmetric criticality (Palais 1979) guarantees that the criti-cal points of an invariant functional restricted to the space of equivariant trajectoriesare free critical points. We stress that by the special form of the interaction potentials,in our setting invariance is simply implied by equality of those masses which are inter-changed by the action of G on the set of the indices. We will always implicitly makethis assumption.

6 S. Terracini

2.1 Cyclic and dihedral actions

Consider the normal subgroup kerτ � G and the quotient G = G/ ker τ . Since G actseffectively on T, it is either a cyclic group or a dihedral group.

– If the group G acts trivially on the orientation of T, then G is cyclic and we saythat the action of G on � is of cyclic type

– If the group G consists of a single reflection on T, then we say that action of G on� is of brake type.

– Otherwise, we say that the action of G on � is of dihedral type.

maximalisotropyisotropy

principal

fundamentaldomain

Proposition 2.2 Let I be the fundamental domain (for a dihedral type), or anyinterval having as length the minimal angle of time–rotation in G (for a cyclic type).Then the G-equivariant minimization problem is equivalent to problem of minimizingthe action over all paths x : I → X ker τ subject to the boundary conditions x(0) ∈ X H0

and x(1) ∈ X H1 , where H0 and H1 are the maximal isotropy subgroups of the boundaryof I.

2.2 Coercivity

Let us consider the A restricted to the space of symmetric loops �G .


Proposition 2.3 The action functional A is coercive in �G if and only if X G = 0.Consequently, if X G = 0 then a minimizer of AG in �G exists.

Given ρ and σ , we can compute dim X G as:

dim X G = 1|G|

∑g∈G

Tr(ρ(g))#Fix(σ (g))− d.

In the frame rotating with constant angular speed ω the action Aω is generallycoercive, except for a (possible) discrete set values of ω.

2.3 The symmetries of the Chenciner and Montgomery eight-shaped trajectory

1

2

3

In the celebrated paper (Chenciner and Montgomery 2000), a variational argumentwas performed to show the existence of a new periodic trajectory for the three bodyproblem, where the three particles move one a single eight-shaped curve, interchang-ing their positions after a fixed time. One of the the simplest symmetry giving rise tothe Chenciner and Montgomery solution is described below. Denote

x(t) = (x1(t), x2(t), x3(t)) ∈ R6

and impose that

x1(−t) = −x1(t), x2(−t) = −x3(t), x3(−t) = −x2(t)x1(1 − t) = −x2(t), x2(1 − t) = −x1(t), x3(1 − t) = −x3(t)

Here G = D6 be the dihedral group generated by two following reflections: g1 =(τ1, ρ1, σ1) is τ1(t) = −t, ρ1(x) = −x and σ1(1, 2, 3) = (1, 3, 2), and g2 = (τ2, ρ2, σ2) isτ2(t) = 1 − t, ρ2(x) = −x and σ2(1, 2, 3) = (2, 1, 3).

Chenciner and Montgomery minimized the action on the space of symmetric tra-jectories and they showed, by a level estimate that minimals are free of collisions.Actually, for technical reasons, they used the supergroup D12 of D6. There are indeedthree possible groups yielding an eight–shaped trajectory. First, we consider the groupof cyclic action type C6 (the cyclic eight having order 6, which acts cyclically on T (i.e.by a rotation of angleπ/3), by a reflection in the plane E, and by the cyclic permutation(1, 2, 3) in the index set.

The second group, which we denote by D12, is the group of order 12 obtained byextending C6 with the element h defined as follows: τ(h) is a reflection in T, ρ(h) isthe antipodal map in E (thus, the rotation of angle π), and σ(h) is the permutation(1, 2). This is the symmetry group used by Chenciner and Montgomery (2000).

The third group is the dihedral group of order 6D6 we described in this section.This can be seen as the subgroup of D12 generated by h and the subgroup C3 of order3 of C6 ⊂ D12.

The symmetry groups D12 and D6 are of dihedral type. The choreography groupC3 is a subgroup of all the three groups, thus the action is coercive on G-equivariantloops.

8 S. Terracini

3 Generalized orbits and singularities

The existence of a G-equivariant minimizer of the action is a simple consequence ofthe direct method of the Calculus of Variations, whenever the action is coercive on thespace of equivariant loops. These trajectories, however, solve the associated differen-tial equations only where they are free of collisions. Generally speaking, G-equivariantminimizers may present collision, even though the collision set must clearly be of van-ishing Lebesgue measure. A first natural question concerns the number of possiblecollision instants. Here below we follow Barutello et al. (2005).

Definition 3.1 A path x: (a, b) → X is called a generalized solution of the n–bodyproblem if, for every t0 ∈ (a, b), there exists δ > 0 such that the restriction of xto [t0 − δ, t0 + δ] is a local minimizer for the action with respect to G–equivariantcompactly supported variations.

We remark that:

– Equivariant minimizers are generalized solutions.– If the potential is of class C2 outside collisions then every non collision solution is

a generalized solution.– The generalized solutions possess an index: the minimal number of intervals Ij

needed to cover (a, b) such that the restriction of x to Ij is a local minimizer forthe action.

– There is a natural notion of maximal existence interval for generalized solutions,even without the unique extension property.

Let us consider the following assumptions on the interaction potentials:

(A1) ∥∥∥∥∂Ui,j

∂t

∥∥∥∥ ≤ CUi,j(t, x);

(A2) there exist α ∈ (0, 2),γ , C > 0 such that

∇Ui,j(t, x) · x + αUi,j(t, x) ≥ −C|x|γ Ui,j(t, x);

(A3) there exists a continuous positive function on the sphere Ui,j such that

limr→0

rαUi,j(t, rs) = Ui,j(t, s);

uniformly in t and s ∈ Sd−1.

The following theorem is proved in Barutello et al. (2005) (see also Barutello(2004) Ferrario and Terracini (2004), and is based on a suitable variant of Sundman’sinequality and the asymptotic analysis of possible collisions outlined in the sequel.

Theorem 3.2 Let x: (a, b) → X be a generalized solution of the n-body problem. Thencollision instants are isolated in (a, b). Furthermore, if (a, b) is the finite maximal exten-sion interval of x and no escape in finite time occurs then the number of collision instantsis finite.


The theorem extends to the logarithmic potentials, under the following hypotheses:

∇Ui,j(t, x) · x ≥ −mi,j(t)− C|x|γ Ui,j(t, x)

with mi,j(t) > 0 and continuous, C > 0, and

limr→0

Ui,j(t, rs)+ mi,j(t) log(r) = Ui,j(t, s)

uniformly in t and s ∈ Sd−1, and Ui,j is a continuous positive function on the sphere.

3.1 Generalized Sundman–Sperling estimates revisited

Theorem 3.2 can be interpreted as a regularity result. The proof relies on the exten-sion, given in Barutello et al. (2005), to the framework of generalized solutions of theclassical asymptotic estimates on collisions by Sundman and Sperling (1970) (see alsoFerrario and Terracini (2004)).

To start with, let assume t = 0 be a collision instant (possibly involving morethan one colliding cluster). We denote by k ⊂ n the colliding cluster, by (x0 =∑

i∈k mixi/m0, m0 = ∑i∈k mi):

Ik =∑i∈k

mi(xi − x0)2

the momentum of inertia with respect to the center of mass (all the bodies in k collidein x0 if and only if Ik = 0). Moreover, we introduce a system of polar coordinates

r = √Ik s = x/r.

Finally we denote the partial kinetic energy Kk:

Kk =∑i∈k

mi

2|xi|2,

the partial potential function

Uk =∑

i,j∈k,i<j

Ui,j(t, x),

together with the partial energy

Ek = Kk − Uk.

and the partial Lagrangian: Lk = Kk + Uk.

Theorem 3.3

– For every maximal (in the number of bodies) colliding cluster k ⊂ n the partialenergy Ek is bounded.

– There is κ > 0 such that the following asymptotic estimates hold:

r ∼ (κt)2

2+α(

r ∼ κ|t|√− log(|t|))

Kk ∼ Uk ∼ 14 − 2α

Ik ∼ 2(2+α)2 κ

2(κt)−2α2+α (∼ − log |t|) .

10 S. Terracini

– Let s be the normalized configuration of a colliding cluster s = (x − x0)/r. Then

limt→0

r2+α|s|2 = 0

limt→0

U(s(t)) = b < +∞

– Moreover there exists the angular blow–up, that is angular scaled family (s(λt))λ isprecompact for the topology of uniform convergence on compact sets of R\{0}.

As a consequence we have the vanishing of the total angular momentum and theabsence of partial collisions in a neighbourhood of the maximal collision.

3.2 Dissipation (Mc Gehee revisited)

While binary collisions admit the Levi–Civita regularization, simultaneous collisionsinvolving three bodies or more can not be fully regularized, for the possible occur-rence of accumulations of partial collisions. In order to study the motion close to acollision, (McGehee 1974) attached to the phase space a manifold (named the colli-sion manifold) and performed a suitable change of coordinates, showing how someof the motions could be extended also through the singularity. The system writtenin McGehee coordinates in no longer conservative: on the contrary, it possesses aLyapounov function.

It is interesting to give a new look at this dissipation phenomenon, from the lagrang-ian point of view. Here we follow (Barutello 2004); Barutello et al. 2005). To fix ourminds, let us think to a homogeneous potential Uα and perform the following changeof variables

ρ = r2−α

4 , ρ′ = 2 − α

4r−

2+α4 r′.

In this way, we obtain a new action functional depending on (ρ, s)

A(ρ, s) =∫ τ∗

0

12

(4

2 − α

)2

(ρ′)2 + ρ2(

12|s′|2 + U(s)

),

where β := 2(2+α)/(2−α) > 2. Here, similarly to Mc Gehee, we have reparametrizedthe time as

dt = r2+α

2 dτ ,

and hence we have an extra constraint:∫ τ∗

0ρβdτ = T,

while τ ∗ is a free parameter. On the other hand, one can prove that at a collision thereholds

τ ∗ =∫ T

0r−

2+α2 dt = +∞.

In these coordinates the action integral corresponds to the action associated withthe homoclinic problem of a Duffing equation, coupled with the n-body angularsystem. In particular, since the function ρ acts as viscosity time–varying (decreasing)


parameter for the angular lagrangian, we find that the energy of the angular systemincreases when ρ decreases (and vice-versa). This monotonicity property is indeedequivalent to the Sundman inequality.

3.3 Blow-ups and parabolic collision trajectories

For simplicity, from now on we shall consider only the case of α–homogeneus poten-tials: the logarithmic case requires some different arguments and can be found inBarutello (2004) and Barutello et al. (2005). Having understood the behaviour of theradial variable close to a collision, we turn to the asymptotic analysis of the angles(t) as t goes to the collision instant. The convergence of the angular variable is notat all obvious and is the object of a classical problem in Celestial Mechanics (theinfinite-spin problem). In general, there is no hope of proving the full convergence ofthe variable s. Even more, it has to be noticed that our assumptions on the interac-tion potentials are so weak that there does not even exists a limiting problem for theangular variable. However, thanks to Theorem 3.3, there is enough compactness forcompleting the asymptotic analysis through a blow–up argument. For every λ > 0 let

xλ(t) = λ−2/(2+α)x(λt)

If {λn}n is a sequence of positive real numbers such that s(λn) converges to a nor-malized configuration s, then ∀t ∈ (0, 1) : limn→∞ s(λnt) = limn→∞ s(λn) = s. Hencethe rescaled sequence will converge uniformly to the blow-up of x(t) relative tothe colliding cluster k ⊂ n (in t = 0). Moreover, the blow-up x is parabolic: here aparabolic collision trajectory for the cluster k is the path

xi(t) = |t|2/(2+α)ξi, i ∈ k, t ∈ R.

Proposition 3.4 The sequences xλn and dxλn/dt converge to the blow-up x and its deriv-ative ˙x respectively, in the H1-topology. Moreover x is a minimizing trajectory in thesense of Definition 3.1. ∫ T

0

[L(x + ϕ)− L(q) ≥ 0

]dt.

for any compactly supported variation ϕ, G0-equivariant, where G0 is the isotropygroup at t = 0.

Let us assume, in addition to our assumptions, that the total force is asymptoticallyhomogeneous:

limr→0

rα+1∇U(t, rs) = ∇U(t, s),

uniformly in t and in s on compact subset of Snd−1 \�.In this case, the parabolic motions has a configuration ξ = (ξi)i∈k which is central

for the limiting force field, namely it annihilates the tangential part of ∇U(0, s).Hence we are left with the problem of collision parabolic solutions in the gen-

eralized sense we gave in Definition 3.1. Next step consist in proving by a suitablevariation that parabolic solutions can not be local minimizer.

12 S. Terracini

4 Avoiding collisions

4.1 The standard variation and the displacement differential potential

Let G0 be the isotropy group at the collision time, then the blow–up procedureimplies the existence of q, a G0-equivariant minimizing parabolic collision trajectory.Following Ferrario and Terracini (2004), we first define a class of suitable variationsas follows:

Definition 4.1 The standard variation associated to δ and T is defined as

vδ(t) =⎧⎨⎩δ if 0 ≤ |t| ≤ T − |δ|(T − t) δ

|δ| if T − |δ| ≤ |t| ≤ T0 if |t| ≥ T,

By chosing δ to be fixed by G0, we can always manage to have vδG0-equivariant.Our next goal is to find a suitable δ such that its associated standard variation vδ

decreases the action, or, in other words, such that

�A :=∫ +∞

−∞[Lk(q + vδ)− Lk(q)]dt < 0.

Definition 4.2 Let us define the displacement potential differential associated tothe standard variation vδ :

S(ξ , δ) =∫ +∞

0

(1∣∣ξ t2/(2+α) − δ

∣∣α − 1∣∣ξ t2/(2+α)∣∣α)

dt

where ξ , δ ∈ R2.

It can proven that the function S measures the limiting (as T → ∞) differential ofthe potential energy. Thus we obtain the fundamental estimate:

Theorem 4.3 Let q = {q}i = {t2/(2+α)ξi}, i = 1, . . . , k be a parabolic collision trajectoryand vδ any standard variation. Then, as δ → 0

�A = 2|δ|1−α/2∑

i<ji,j∈k

mimjS(ξi − ξj,

δi − δj

|δ|)

+ O(|δ|).

We observe that

S(λξ ,µδ) = |λ|−1−α/2 |µ|1−α/2S(ξ , δ)

and hence the sign of S depends on the angle between ξ and δ. Let

�(ϑ) =∫ +∞

0

1(t

4α+2 − 2 cosϑ t

2α+2 + 1

)α/2 − 1

t2αα+2

dt, α ∈ (0, 2)

�(θ) represents the potential differential needed for displacing the colliding particleoriginarily traveling on the x–axis to the point eiθ .


4.2 Averaging properties involving � and Marchal’s Principle

The value of �(θ) ranges from decrease to +∞, in a way also depending on theexponent α. Hence it can take positive and negative values. However, thanks to someharmonic analysis one can prove that suitable averages are always negative: the firstinequality is particularity useful when dealing with reflected triple collisions from theLagrange central configuration:

�

(2π3

+ γ

)+�

(2π3

− γ

)< 0, ∀γ ∈ [0,π/2].

A crucial estimate was proved in Ferrario and Terracini (2002) about the averages of� on circles:

Theorem 4.4 For every α > 0, ξ ∈ R3

� {0} and for every circle S ⊂ Rd with center

in 0,

S(ξ , S) = 1|S|∫

S

S(ξ , δ)dδ = |ξ |−1−α/2 |δ|1−α/2 12π

∫ 2π

0�(θ)dθ < 0.

Let q be a parabolic collision solution associated to the configuration ξ . Considerξ = xi − xj and δ ranging in a circle. Then the above inequality allows us to statethe existence of at least one δ for which the associated standard variation lowers theaction.

The first who conjectured that the method of averaged variations could be usedto avoid collisions on minimizers was Marchal (2002), who remarked that, being theNewton potential harmonic on R

3, averaging it on a sphere results in a truncation inthe interior. In fact, is not so much a matter of harmonicity rather than a subtle bal-ance between the averaged potential and the speed of collisions, as already Marchalobserved by considering the case of the planar Keplerian potential generated by aplate, which expression is known since the 19th century. Our estimates on the averagesof S(ξ , δ) show that this is a general fact and Marchal’s principle can be extended inthe following way (see Ferrario and Terracini (2004) for homogeneous potentials andBarutello (2004); Barutello et al. (2005) for logarithmic potentials):

Theorem 4.5 (The rotating circle principle) For homogeneus and logarithmic poten-tials it is always more convenient (from the point of view of the integral of the potentialon the time line) to replace one of the colliding particles with a homogeneous circle ofsame mass and fixed radius which is moving keeping its center in the position of theoriginal particle

Of course by a circle we mean the intersection of the sphere with any two-dimen-sional plane.

14 S. Terracini

5 The rotating circle property

As a consequence of the analysis of collisions we find that if the action of G on T and Xfulfills some conditions (computable) then (local) minimizers of the action functionalA in �G ⊂ � do not have collisions. This is the rotating circle condition, introducedin Ferrario and Terracini (1977).

For a group H acting orthogonally on Rd, a circle S ⊂ R

d (with center in 0) istermed rotating under H if S is invariant under H (that is, for every g ∈ HgS = S)and for every g ∈ H the restriction g|S : S → S is a rotation (the identity is meant asa rotation of angle 0).

Let i ∈ n be an index and H ⊂ G a subgroup. A circle S ⊂ Rd = V (with center in

0) is termed rotating for i under H if S is rotating under H and

S ⊂ VHi ⊂ V = Rd,

where Hi ⊂ H denotes the isotropy subgroup of the index i in H relative to theaction of H on the index set n induced by restriction (that is, the isotropyHi = {g ∈ H | gi = i}).

Definition 5.1 A group G acts with the rotating circle property if for every T-isotropysubgroup Gt ⊂ G and for at least n − 1 indexes i ∈ n there exists in R

d a rotatingcircle S under Gt for i.

If the action has the rotating circle property, then for every g ∈ G the linear map1 − g sends the rotating circle into another circle (thus we can use the rotating circleprinciple). In most of the known examples the property is fulfilled. In Ferrario andTerracini (2004) we proved the following results.

Theorem 5.2 Consider a finite group K acting on � with the rotating circle property.Then a minimizer of the K-equivariant fixed–ends (Bolza) problem is free of collisions.

Corollary 5.3 For every α > 0, minimizers of the fixed-ends (Bolza) problem are freeof interior collisions.

Corollary 5.4 If the action of G on � is of cyclic type and ker τ has the rotating circleproperty then any local minimizer of AG in �G is collisionless.

Corollary 5.5 If the action of G on � is of cyclic type and ker τ = 1 is trivial then anylocal minimizer of AG in �G is collisionless.

Theorem 5.6 Consider a finite group G acting on � so that every maximal T-isotropysubgroup of G either has the rotating circle property or acts trivially on the index set n.Then any local minimizer of AG yields a collision-free periodic solution of the Newtonequations for the n-body problem in R

d.

6 The 3-body problem

The major achievement of Barutello et al. (2004) is to give the complete descriptionof the outcome of the equivariant minimization procedure for the planar three-bodyproblem. First we can ensure that minimizers are always collisionless.


Theorem 6.1 Let G a symmetry group of the Lagrangian in the 3-body problem (in arotating frame or not). If G is not bound to collision (i.e. every equivariant loop hascollisions), then any (possible) local minimizer is collisionless.

A symmetry group G of the Lagrangian functional A is termed

– bound to collisions if all G-equivariant loops actually have collisions,– fully uncoercive if for every possible rotation vector ω the action functional AG

ω inthe frame rotating around ω with angular speed |ω| is not coercive in the space ofG-equivariant loops (that is, its global minimum escapes to infinity);

– homographic if all G-equivariant loops are constant up to orthogonal motions andrescaling.

– The core of the group G is the subgroup of all the elements which do not move thetime t ∈ T.

If, for every angular velocity, G is a symmetry group for the Lagrangian functionalin the rotating frame, then we will say that G is of type R. This is a fundamentalproperty for symmetry groups. In fact, if G is not of type R, it turns out that theangular momentum of all G-equivariant trajectories vanishes.

6.1 The classification of planar symmetry groups for 3-body

Theorem 6.2 Let G be a symmetry group of the Lagrangian action functional in theplanar 3-body problem. Then, up to a change of rotating frame, G is either bound tocollisions, fully uncoercive, homographic, or conjugated to one of the symmetry groupslisted in Table 1 (RCS stands for Rotating Circle Property and HGM for HomographicGlobal Minimizer) (Fig.1).

6.2 Planar symmetry groups

– The trivial symmetry. Let G be the trivial subgroup of order 1. It is clear that itis of type R, it has the rotating circle property. It yields a coercive functional on�G = � only when ω is not an integer. If ω = 1

2 mod 1 then the minimizers areminimizers for the anti-symmetric symmetry group (also known as Italian symme-try) x(at) = ax(t), where a is the antipodal map on T and E. The masses can bedifferent.

Table 1 Planar symmetry groups with trivial core

Name |G| Type R Action type Trans. Dec. RCP HGM

Trivial 1 yes 1 + 1 + 1 yes yesLine 2 yes brake 1 + 1 + 1 (no) no2-1-choreography 2 yes cyclic 2 + 1 yes noIsosceles 2 yes brake 2 + 1 no yesHill 4 yes dihedral 2 + 1 no no3-choreography 3 yes cyclic 3 yes yesLagrange 6 yes dihedral 3 no yesC6 6 no cyclic 3 yes noD6 6 no dihedral 3 yes noD12 12 no dihedral 3 no no

16 S. Terracini

Fig. 1 The poset of symmetry groups for the planar 3-body problem

Proposition 6.3 For every ω /∈ Z and every choice of masses the minimum for thetrivial symmetry occurs in the relative equilibrium motion associated to the Lagrangecentral configuration.

– The line symmetry. Another case of symmetry group that can be extended to rotat-ing frames with arbitrary masses is the line symmetry: the group is a group of order2 acting by a reflection on the time circle T, by a reflection on the plane E, andtrivially on the set of indexes. That means, at time 0 and π the masses are collinear,on a fixed line l ⊂ E. It is coercive only when ω �∈ Z. In this case the Lagrangiansolution cannot be a minimum, while the relative equilibrium associated with theEuler configuration can be (Fig. 2).

– The 2-1-choreography symmetry. Consider the group of order 2 acting as follows:ρ(g) = 1, τ(g) = −1 (that is, the translation of half-period) and σ(g) = (1, 2) (thatis, σ(g)(1) = 2, σ(g)(2) = 1, and σ(g)(3) = 3. That is, it is a half-period choreogra-phy for the bodies 1 and 2. It can be extended to rotating frames and coercive fora suitable choice of ω �= 0, 1 mod 2.

3

4

5

6

7

8

9

10

0.2 0.3 0.4 0.5 0.6 0.7 0.8

actio

n

omega

Euler1Hill1Hill2

Euler2

Fig. 2 Action levels for the line symmetry


The Euler’s orbit with k = 1 and the Hill’s orbits with k = ±1 are equivariant forthe 2-1-choreography symmetry, while the Euler’s orbit with k = 0 is not equivari-ant for this symmetry. In Fig. 3 Euler 1 represents the action levels on the Euler’sorbit with k = 1, Hill 1–2 the ones on the Hill’s with k = ±1.

– The isosceles symmetry. The isosceles symmetry can be obtained as follows: thegroup is of order 2, generated by h; τ(h) is a reflection in the time circle T, ρ(h)is a reflection along a line l in E, and σ(h) = (1, 2) as above. The constraint istherefore that at time 0 and π the 3-body configuration is an isosceles triangle withone vertex on l (the third).

Proposition 6.4 For every ω /∈ Z and every choice of masses the minimum forthe isosceles symmetry occurs in the relative equilibrium motion associated to theLagrange configuration.

– The Euler–Hill symmetry. Now consider the symmetry group with a cyclic gener-ator r of order 2 (i.e. τ(r) = −1) and a time reflection h (i.e. τ(h) is a reflectionof T) given by ρ(r) = 1, σ(r) = (1, 2), ρ(h) is a reflection and σ is the identity in{1, 2, 3}. It contains the 2-1-choreography (as the subgroup ker det(τ )), the isoscelessymmetry (as the isotropy of π/2 ∈ T) and the line symmetry (as the isotropy of0 ∈ T) as subgroups.

Proposition 6.5 The minimum of the Euler–Hill symmetry is not homographic,provided that the angular velocity ω is close to 0.5 and the values of the masses areclose to 1.

– The choreography symmetry. The choreography symmetry is given by the groupC3 of order 3 acting trivially on the plane E, by a rotation of order 3 in the timecircle T and by the cyclic permutation (1, 2, 3) of the indices.

Proposition 6.6 For every ω the minimal choreography of the 3-body problem is arotating Lagrange configuration.

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

actio

n

omega

Hill1Euler1

Hill2

Fig. 3 Action levels for the 2-1-choreography symmetry

18 S. Terracini

– The Lagrange symmetry. The Lagrange symmetry group is the extension of thechoreography symmetry group by the isosceles symmetry group. Thus, it is a dihe-dral group of order 6, the action is of type R. Hence, the relative equilibriummotions associated to the Lagrange configuration are admissible motions for thissymmetry and, again, the minimizer occurs in the relative equilibrium motionassociated to the Lagrange configuration.

– The Chenciner–Montgomery symmetry group and the eights. There are three sym-metry groups (up to change of coordinates) that yield the Chenciner–Montgomeryfigure eight orbit: they are the only symmetry groups which do not extend to therotating frame and we have already described them in Sec. 2.3. One can provethat all G-equivariant trajectories have vanishing angular momentum, wheneverthe group is not of type R. Moreover, we were able to partially answer to the openquestion (posed by Chenciner) whether their minimizers coincide or not: for twoof them (D6 and D12) the minimizer is necessarily the same.

6.3 Space three-body problem

Based on the classification of planar groups, by introducing a natural notion of spaceextension of a planar group, Ferrario (2004) gave a complete answer to the classi-fication problem for the three-body problem in the space and at the same time todetermine the resulting minimizers and describe its more relevant properties.

Theorem 6.7 Symmetry groups not bound to collisions, not fully uncoercive and nothomographic are, up to a change of rotating frame, either the three-dimensional exten-sions of planar groups (if trivial core) listed in Table 2 or the vertical isosceles triangle(if non-trivial core).

The next theorem is the answer to the natural questions about collisions anddescription of some main features of minimizers.

Theorem 6.8 Let G be a symmetry group not bound to collisions and not fully unco-ercive. Then

(i) Local minima of Aω do not have collisions.(ii) In the following cases minimizers are planar trajectories:

(a) If G is not of type R: D+,−6 , D−,+

6 and D−,+12 (and then G-equivariant

minimizers are Chenciner–Montgomery eights).

Table 2 Space extensions of planar symmetry groups with trivial core

Name Extensions

Trivial C−1

Line L+,−2 ,L−,+

2

Isosceles H+,−2 ,H−,+

2

Hill H+,−4 ,H−,+

4

3-choreography C+3 , C−

3

Lagrange L+,+6 ,L+,−

6 ,L−,+6

D6 D+,−6 ,D−,+

6

D12 D−,+12


(b) If there is a G-equivariant minimal Lagrange rotating solution: C−1 ,

H+,−2 , C+

3 , L+,+6 , L+,−

6 (and then the Lagrange solution is of coursethe minimizer).

(c) If the core is non-trivial and it is not the vertical isosceles (and thenminimizers are homographic).

(iii) In the following cases minimizers are always non-planar (Fig. 4):(a) The groups L−,+

6 and C−3 for all ω ∈ (−1, 1)+ 6Z, ω �= 0 (the minimiz-

ers for L−,+6 are the elements of Marchal family P12, and minimizers of

C−3 are a less-symmetric family P′

121).

(b) The extensions of line and Hill-Euler type groups, for on open subset ofmass distributions and angular speeds ω: L+,−

2 , L−,+2 , H+,−

4 and H−,+4

(for L−,+2 this happens also with equal masses).

(c) The vertical isosceles for suitable choices of masses and ω.

7 Is the choreographical minimizer a homografic motion?

The results of this sections are contained in Barutello and Terracini (2004) and Arioliet al. (2005).

Definition 7.1 A simple choreography is a trajectory where the bodies lie on the samecurve and exchange their mutual positions after a fixed time, namely,

qi(t) = x(t + (i − 1)τ ), i = 1, . . . , n, t ∈ R,

where τ = 2π/n.

In the space of symmetric (choregraphical) loops, the action takes the form

A(x) = 12

n−1∑h=0

∫ τ

0|x(t + hτ)|2dt + 1

2

n−1∑h, l = 0h �= l

∫ τ

0

dt|x(t + lτ)− x(t + hτ)|α

1

2

3

1

2

3

Fig. 4 Non planar minimizers for the groups L−,+6 and L+,−

2 , plotted in the fixed frame

1 Highly likely they are not distinct families: this is the recurring phenomenon of “more symmetriesthan expected” in n-body problems.

20 S. Terracini

Unfortunately, the bare minimization among choreographical loops returns onlytrivial motions:

Theorem 7.2 For every α ∈ R+ and d ≥ 2, the absolute minimum of A on � is attainedon a relative equilibrium motion associated to the regular n-gon.

This theorem extends some related result for the italian symmetry by Chencinerand Desolneux (1998), and the results in section 6. The proof is based on a (quiteinvolved) convexity argument together with the analysis of some spectral propertiesrelated to the choreographical constraint. Now, in order to find nontrivial minimizers,we look at the same problem in a rotating frame. In order to take into account of theCoriolis force, the new action functional has to contain a gyroscopic term:

A(y) = 12

∫ 2π

0|y(t)+ Jωy(t)|2dt + 1

2

n−1∑h=1

∫ 2π

0

dt|y(t)− y(t + hτ)|α .

Consider the function h: R∗+ → N, h(ω) = minn∈N∗ (ω−n)2

n2 and let ω∗ = 43 . The

same technique used for the inertial system extends to rotating systems having smallangular velocity; this gives the following result.

Theorem 7.3 If ω ∈ (0,ω∗)\{1}, then the action attains its minimum on a circle withminimal period 2π and radius depending on n, α and ω.

7.1 When ω is close to an integer

The situation changes dramatically when ω is close to some integer. To understandthis phenomenum, let us first check the result of the minimization procedure when ω

is an integer:

Proposition 7.4

(1) If ω = n, then the action has a continuum of minimizers.(2) If ω = k, coprime with n, then the action does not achieve its infimum (escape of

minimizing sequences).(3) If ω = k and k divides n, then the action does not achieve its infimum (clustering of

minimizing sequences).

As a consequence, we have the following result:

Theorem 7.5 Suppose that n and k are coprime. Then there exist ε = ε(α, n, k) suchthat if ω ∈ (k − ε, k + ε) the minimum of the action is attained on a circle with minimalperiod 2π/k that lies in the rotating plane with radius depending on n, α and ω.

An interesting situation appears when the integer closest to the angular velocity isnot coprime with the number of bodies. In this case we prove that the minimal orbitit is not circle anymore, as the following theorem states.

Theorem 7.6 Take k ∈ N and g.c.d.(k, n) = k > 1, k �= n. Then there exists ε =ε(α, n, k) > 0 such that if ω ∈ (k−ε, k+ε)\{k} the minimum of the action is attained ona planar 2π-periodic orbit with winding number k which is not a relative equilibriummotion (Fig. 5).

Also, it has be noticed that, for large number of bodies and angular velocities closeto the half on an integer, the minimizer apparently is not anymore planar (Fig. 6).


–4 –3 –2 –1 0 1 2 3 4

–3

–2

–1

0

1

2

3

–6 –4 –2 0 2 4 6–6

–4

–2

0

2

4

6

Fig. 5 Examples for Theorem 7.6 close to an integer that divides n.

–2 –1

01

2

–2 –1

01

20.40.2

00.20.4

–3 –2 –1 0 1 2 3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

Fig. 6 Non planar minimizers of the action with angular velocities close to the half on an integer

7.2 Mountain pass solutions for the choreographical 3-body problem

The discussion carried in the previous section shows that, as the angular velocity var-ies, the minimizer’s shape must undergoes some transitions (for example it has to passfrom relative equilibrium having different winding numbers). This scenario suggeststhe presence of other critical points, such as local minimizers or mountain pass. Thiswas indeed discovered numerically in Barutello and Terracini (2004) and then provedby a computer assisted proof in Arioli et al. (2005).

To begin with, let us look at Fig. 7, where the values of the action functional Aω onthe branches of circular orbits Lω

k are plotted:The analysis of this picture suggests the presence of critical points different from

the Lagrange motions. Indeed, let us take the angular velocity ω = 1.5: in this casethere are two distinct global minimizers, the uniform circular motions with minimalperiod 2π and π , lying in the plane orthogonal to the rotation direction. This is a wellknown structure in Critical Point Theory, referred as the Mountain Pass geometryand gives the existence of a third critical point, provided the Palais-Smale condition

22 S. Terracini

Fig. 7 Action levels for theLagrange solutions

–2 –1 0 1 2 3 4 50

5

10

15

20

25

is fulfilled, with an additional information on the Morse index. Next theorem followsfrom the application of the Mountain Pass Theorem to the action functional A3/2:

Theorem 7.7 There exists a (possibly collision) critical point for the action functionalA3/2 with Morse index smaller then 1 and distinct from any Lagrange motion.

Once the existence of a Mountain Pass critical point was theoretically established,we studied its main properties in order to understand whether it belonged to someknown families of periodic trajectories. To this aim we applied the bisection algorithmproposed in Barutello and Terracini (2004) to approximate the maximal of a locallyoptimal path joining the two strict global minimizers, finding in this a good numericalcandidate. Of course, there could be a gap between the mountain pass solution whoseexistence is ensured by Theorem 7.7 and the numerical candidate found by applyingthe bisection algorithm. In order to fill this gap we proved the existence of an actualsolution very close to the numerical output of the Mountain Pass algorithm (Fig. 8).The argument was based upon a fixed point principle and involved a rigorous com-puter assisted proof. As a consequence, we obtained the existence of a new branch ofsolution for the spatial 3-body problem (see Fig. 9).

–1.5–1

–0.50

0.51

–1–0.5

00.5

11.5

–0.5

0

0.5

–1.5 –1 –0.5 0 0.5 1 1.5

–1

–0.5

0

0.5

1

1.5

Fig. 8 Mountain pass solution with angular velocities close to the half on an integer


0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12mountain passL1L2

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 1.017.45

7.5

7.55

7.6

7.65

7.7

7.75

7.8

mountan passP12L2

Fig. 9 Action levels for the Lagrange and the mountain pass solution in the 3-body problem. Onthe x-axes the angular velocity varies in the interval [0, 3). The left picture focuses on the biburcationfrom the P12 family

Here are some relevant features of the new solution: the orbit is not planar, itswinding number with respect, for instance, to the line x = −0.2, y = 0 is 2 and it doesnot intersect itself. A natural question is whether this solution can be continued as afunction of the parameterω. We were able to extend the numerical-rigorous argumentto cover a full interval of values of the angular velocity, providing the existence of afull branch of solution.

Theorem 7.8 There exists a smooth map B(ω) giving the (locally unique, up to sym-metries) branch of solution of the choreographical 3-body problem for all ω ∈ [1, 2],starting at the Mountain Pass solutions for ω = 1.5.

A natural question is whether the mountain pass branch meets one of the knownbranches of choreographical periodic orbits: either one of the Lagrange or Marchal’sP12 (described by Marchal, (2000)) families. Lead from a wrong intuition, lookingat Fig. 9 (left), we first conjectured that the branch should bifurcate from L1 at theangular velocity value ω = 1. It was A. Chenciner who brough this mistake to ourattention, pointing out how this fact would contradict the numerical computation onthe local bifurcation structure at ω = 1 that he and J. Féjoz were carrying on Chencin-er and Féjoz (2005), see also the results in Chenciner et al. (2005). He also suggestedus that very likely, this new branch was bifurcating, by symmetry breaking, from theP12–family. Indeed, this was shown by our computations, both numerical and rigorous.The details are depicted in Fig. 9 (right).

References

Albouy, A.: The symmetric central configurations of four equal masses. Contemp. Math. 198, 131–135(1996)

Albouy, A., Chenciner, A.: Le problème des n corps et les distances mutuelles. Invent. Math. 131,151–184 (1998)

Ambrosetti, A., Coti Zelati, V.: Periodic solutions of singular Lagrangian systems. In: of Progress inNonlinear Differential Equations and their Applications, vol. 10, Birkhäuser Boston Inc., Boston,MA (1993)

Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications.J. Funct. Anal. 14, 349–381 (1973)

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