N-dimensional invariant tori for the planar (N+1)-body problem

N -dimensional invariant tori for the planar (N + 1)-body problem ∗ Luca Biasco

Luigi Chierchia

Enrico Valdinoci

May 6, 2004

Abstract For any N ≥ 2 we prove the existence of quasi–periodic orbits lying on N –dimensional invariant elliptic tori for the planetary planar (N +1)–body problem. For concreteness, we focus on a caricature of the outer solar system.

Keywords: N –body problem, nearly–integrable Hamiltonian systems, lower dimensional elliptic tori. MSC2000 numbers: 70F10, 34C27, 37J40, 70K43.

Contents 1 Introduction and results

2

2 Poincar´ e Hamiltonian setting

4

3 The averaged quadratic potential H1,2

7

4 Existence of N –dimensional elliptic invariant tori

14

A Poincar´ e variables for the planar (N + 1)–body problem

20

B Simple eigenvalue perturbations

26

∗ Acknowledgments. Supported by MIUR Variational Methods and Nonlinear Differential Equations. We thank Carlangelo Liverani for fruitful suggestions.

1

1

Introduction and results

1.1 Quasi–periodic motions in the many–body problem The existence of stable trajectories of the many body problem viewed as a model for the solar system has been the subject of researches of many distinguished scientists both in the past and in recent years. Despite eminent contributions given, among others, by Poincar´e [Poi1905], Arnold [A63] and Herman [H95] and some numerical evidence of “stability” of the giant planets of the solar system (compare [L96]), the literature is still lacking in rigorous general proofs1 . The main difficulty consists in the application of general tools (such as averaging and KAM theory) to particular cases of interest in celestial mechanics, where strong degeneracies appear. The scope of this paper is to make some progress in this direction, showing the existence of quasi–periodic orbits lying on N –dimensional invariant elliptic tori for the planar (N + 1)–body problem. Though the method exposed here is quite general, for concreteness we will focus our attention on a caricature of the outer solar system. More precisely, our model will be given by a Sun and N planets with relatively small masses (say, of order ε). All these (N + 1) bodies are considered as point masses in mutual gravitational interaction. Two planets (such as Jupiter and Saturn in the real world) will be assumed to have mass considerably bigger than the other planets. The bodies lie in a given plane and we assume that the initial configuration is far from collisions. We also assume, mimicking the case of the outer solar system, that the two big planets have an orbit which is internal with respect to the orbits of the small planets. We will establish, for a large set of semi–axes, the existence of quasi–periodic orbits with small eccentricities filling up N –dimensional invariant elliptic tori. Such orbits can be seen as continuations of “limiting” circular trajectories of the system obtained by neglecting the mutual interactions among the planets. A more precise statement is given in Theorem 1.1 below. The above “outer model”, which roughly mimics some traits of physically relevant cases, has also the nice feature of providing particulary simple expressions in the related perturbing functions, as we will see in Section 3 below. We stress, however, that many other situations (such as one large planet plus N − 1 small planets; “inner” or “mixed” models, etc.) may be easily dealt with using the techniques and results presented in this paper. The proof of our result is based on techniques developed in [BCV03] and on the explicit computation of the eigenvalues of the quadratic part of the so–called principal part of the perturbation for the planar many–body problem. The first result on quasi–periodic orbits of interest in celestial mechanics goes back to [A63], where quasi– periodic orbits lying on 4–dimensional tori are shown to exist for the planar three–body problem (the general case was discussed there, but no complete proof was given). Related results were given in [JM66], which found linearly unstable quasi–periodic orbits lying on 2–dimensional tori for the non planar three– body problem. More recently, [LR95]–[R95] and [BCV03] proved the existence of quasi–periodic orbits for the non planar three–body problem, lying on 4–dimensional and linearly stable 2–dimensional tori, respectively. Periodic orbits of the non planar three–body problem winding around invariant tori have been constructed in [BBV03]. The result presented below, in the case N = 2, has been obtained, with different methods, in [F02]. The paper is organized as follows. In Section 1.2, we give a more precise statement of our main result. In Section 2 we write down the (N +1)–body problem Hamiltonian in Delaunay–Poincar´e variables. In Section 3 (which, in a sense, is the crucial part of the paper) we discuss degeneracies. In Section 4 we give the proof of the main result. The scheme of proof is similar to the one presented in [BCV03] (see also [BBV03]) in the three–body case and it is based on a “general” averaging theorem and on KAM theory for lower–dimensional tori ([P96], [BCV03], [BBV03]). For completeness, we include a classical (but not–easy–to–find) description of analytical properties of the Delaunay–Poincar´e variables (see § 2 1 See,

however, [F03].

2

and Appendix A); in Appendix B we collect some simple linear algebra lemmata that are used in the arguments given in Section 3.

1.2 Statement of results We denote the N + 1 massive points (“bodies”) by P0 , . . . , PN and let m0 , . . . , mN be their masses interacting through gravity (with constant of gravitation 1). Fix m0 > 0 and assume that mi = εµi ,

i = 1, . . . , N ,

0<ε<1.

(1.1)

Here, ε is regarded as a small parameter and µi is of order 1 in ε. The point P0 represents the “Sun” and the points Pi , i = 1, . . . , N , the “planets”. We assume that all the bodies lie on a fixed plane, that will be identified with R2 . The phase space of this dynamical system (“the planetary, planar (N + 1)–body system”) has dimension 4N . We will state the result in terms of orbital elements of the “osculating ellipses” of the two–body problems associated to (P0 , Pj ). Let u(0) and u(j) denote the coordinates of P0 and Pj (at a given time) and let u˙ (0) and u˙ (j) denote the corresponding velocities. By definition, the “osculating ellipse” is the ellipse described by the solution of the two–body problem (P0 , Pj ) with initial data given by (u(0) , u(j) , u˙ (0) , u˙ (j) ). Of course, such ellipses describe the motions of the full (N + 1)–body problem only approximately; nevertheless, it provides a nice set of coordinates allowing, for example, to describe the true motions in terms of the eccentricities ej and the major semi–axes aj of the osculating ellipses. For further details and pictures of the orbital elements, we refer to [Ch88] and [BCV03]. In this paper we consider a planetary (planar) model with planets evolving from phase points corresponding to well separated nearly–circular ellipse (ei  1); here “well separated” means that 1≤i≤N −1 .

0 < ai < θ ai+1 ,

(1.2)

for a suitable constant 0 < θ < 1. For concreteness, we shall focus on a caricature of the outer solar system, i.e., we will assume that, for some m0 < µ ¯i < 4m0 , µi = µ ¯i µi = δ µ ¯i

for for

i = 1, 2 , i = 3, . . . , N ,

0 < δ < 1.

(1.3)

In this setting, P1 and P2 imitate (in a very rough way, of course) the physical features of the giant planets Jupiter and Saturn, while P3 and P4 represent Uranus and Neptune2 . Theorem 1.1 Consider a planar, planetary (N + 1)–body system satisfying (if N ≥ 3) (1.3). Let A ⊂ RN be a compact set of semi–axes where (1.2) hold. Then, there exists δ ? > 0 and for any 0 < δ < δ ? there exists ε? > 0 so that the following holds. For any 0 < ε < ε? , the planetary, planar (N + 1)–body system possesses a family of N –dimensional elliptic invariant tori, travelled with Diophantine frequencies, provided that the osculating major semi–axes belong to a subset of A of density3 1 − C1 εc1 . These motions correspond to orbits with osculating eccentricities bounded by C2 εc2 and the variation in time of the osculating major semi–axes of these orbits is bounded by C3 εc3 . A few comments: • The numbers δ ? and θ can be easily computed in the course of the proof and are not “very small”; in fact θ is a “universal” constant while δ ? depends only on N and A. On the other hand ε? , which depend on N , A and δ, is related to a KAM smallness condition and rough estimates lead, as well known, to ridiculously small quantities (for somewhat more serious KAM estimates, we refer to [CC03]). Finally, the positive constants Ci ’s depend on N , A and δ, while the ci ’s depend ony on N (and could, also, be easily calculated: see (4.93)). 2 The 3 Here

Jupiter/Saturn mass ratio is approximately 3.34, while the Neptun/Uranus mass ratio is about 1.18. and in the sequel, the “density” is intended with respect to Lebesgue measure.

3

• The assumptions (1.2) and (1.3) in the theorem are used to check explicitely suitable “non degeneracy” conditions. Howerver, giving up explicit constants and estimates, one can show that the thesis of the theorem holds, essentially, with no hypotheses on the semi–axes aj and the rescaled masses µj (provided ai 6= aj > 0 and µj > 0). • The invariant tori found in Theorem 1.1 are lower dimensional elliptic tori meaning that the dimension of the tori are strictly smaller than (in fact, half of) the dimension of the Lagrangian (maximal) tori, which have dimension 2N . “Elliptic” means that the tori are linearly stable. It is not difficult to show that such elliptic tori are surrounded by a set of positive measure of maximal tori. • The proof given below is based on a well known elliptic KAM theorem, which works under “non degeneracy” (or Melnikov) conditions. To check these conditions one has to study the eigenvalues of the “secular” (or averaged) quadratic part of the Newtonian many–body interaction, which will be denoted H1,2 ; “quadratic”, here, referes to the symplectic Cartesian variables measuring the eccentricity and the orientation of the osculating ellipses. The diagonalization of H1,2 is trivial (under the only assumption that ai 6= aj ), while conditions (1.2) and (1.3) will be used to check that the associated eigenvalues are non zero, simple and distinct so that Melnikov conditions are satisfied. The proof is non inductive on N . • A more detailed and quantitative version of Theorem 1.1 is Theorem 4.2 below.

2

Poincar´ e Hamiltonian setting

The results described in this section are classical and go back to Delaunay and Poincar´e; however since they are not so easy to find, for convenience of the reader, we reproduce them in Appendix A. Consider N +1 bodies P0 , . . . , PN , in a fixed (ecliptic) plane, of masses m0 , . . . , mN interacting through gravity (with constant of gravitation 1). We assume that the mass of P0 (the “star”) is much larger that the mass of the other bodies (the “planets”), i.e., we assume (1.1). In heliocentric planar (suitably rescaled) variables, the dynamics of the planar (N + 1)–body problem is governed (as explained in Appendix A) by the Hamiltonian (N )

(N )

H(N ) (X, x) := H0 (X, x) + εH1 (X, x) , where X := (X variables and

(1)

, ..., X

(N )

)∈R

2N

(N )

H0

(N )

H1

(1)

and x := (x

(N )

, ..., x

)∈R

2N

(2.4)

are conjugated Cartesian symplectic

N  X 1 mi Mi  |X (i) |2 − (i) , 2mi |x | i=1   X 1 µi µj := ; X (i) · X (j) − m20 |x(i) − x(j) | 1≤i<j≤N

:=

(2.5)

here we have introduced the dimensionless masses Mi := 1 + ε

µi , m0

mi :=

(N )

µi µi 1 = . m0 + εµi m0 Mi

(2.6)

The Hamiltonian H0 is simply the sum of N un–coupled planar Kepler problems (formed by the star and the ith planet). Being interested in phase region where the un–coupled Kepler problem describe nearly–circular orbits, we introduce planar Poincar´e variables, the construction of which are based on the following classical 4–dimensional symplectic map.

4

Lemma 2.1 (Planar Poincar´ e variables) Let  1  −1 t 2 1 t F1 (t) := 1 − , F2 (t) := 1− , 4 2 4



 |t| < 1/4 ;

(2.7)

let G0 (s, t) = t + st + · · · be the function analytic in a neighborhood of (0, 0) implicitely defined by G0 (0, 0) = 0 ,

G0 = s sin G0 + t cos G0 ;

(2.8)

ˆ λ) real–analytic in a neighborhood of the set define the following four functions of three variables (ˆ η , ξ, ˆ {(ˆ η , ξ) = (0, 0)} × T:   ˆ λ) := G0 (ˆ G(ˆ η , ξ, η cos λ − ξˆ sin λ) F1 (t) , (ξˆ cos λ − ηˆ sin λ) F1 (t) ,   ˆ λ) := Es (ˆ η , ξ, ξˆ cos(λ + G) + ηˆ sin(λ + G) F1 (t) , ˆ λ) := cos(λ + Es ) − ηˆ F1 (t) − ξˆ Es F1 (t) F2 (t) , C(ˆ η , ξ, ˆ λ) := sin(λ + Es ) + ξˆ F1 (t) − ηˆ Es F1 (t) F2 (t) , S(ˆ η , ξ, ˆ λ) and Es where t is short for t = ηˆ2 + ξˆ2 , G is short for G(ˆ η , ξ, and let µ µ 1 , M := 1 + ε , m := m0 m0 M  µ 3 1 σ := , a = a(Λ; µ, ε) := m0 M

(2.9)

ˆ λ). Fix ε, µ, m0 > 0 is short for Es (ˆ η , ξ, m :=

1 µ √ , m0 M

Λ2 . m2

(2.10)

Then, for any Λ+ > Λ− > 0, there exists a ball B around the origin in R2 such that the 4–dimensional map ΨP : (Λ, λ, η, ξ) ∈ D := (Λ− , Λ+ ) × T × B → (X, x) ∈ R4 where 

 η ξ √ , √ ,λ , Λ Λ   η ξ x2 = x2 (Λ, λ, η, ξ; µ, ε) := a(Λ; µ, ε) S √ , √ , λ , Λ Λ 4 m ∂x m ∂x X = X(Λ, λ, η, ξ; µ, ε) := 3 (Λ, λ, η, ξ; µ, ε) = , Λ ∂λ a(Λ; µ, ε)3/2 ∂λ x1 = x1 (Λ, λ, η, ξ; µ, ε) := a(Λ; µ, ε) C

(2.11)

is real-analytic in D and symplectic: dΛ ∧ dλ + dη ∧ dξ = dX1 ∧ dx1 + dX2 ∧ dx2 . (1)

Furthermore, if H0

denotes the two-body Hamiltonian

1 mM |X|2 − , 2m |x|  −1   (1) then, on the phase region of negative energies H0 − 2Λσ2 , − 2Λσ2 , one has (1)

H0 (X, x) :=

−

(1) H0

+

σ ◦ ΨP = − 2 ; 2Λ

in the planar coordinates x ∈ R2 the corresponding motion describes an ellipse of major semiaxis a = a(Λ; µ, ε) and eccentricity r r  η2 + ξ 2  η2 + ξ 2 η2 + ξ 2  η2 + ξ 2  e= F1 = 1− . (2.12) Λ Λ Λ 4Λ

5

The proof of this lemma may be found in Appendix A. Note that C(0, 0, λ) = cos λ ,

S(0, 0, λ) = sin λ ,

(2.13) (N )

so that the (a, λ) → x transformation is, for η = ξ = 0, just polar coordinates. Let, now, ΨP 4N –dimensional map, parameterized by (µ1 , ..., µN , ε), defined by (N )

ΨP

N    → (X, x) : (Λ1 , λ1 , η1 , ξ1 ), ..., (ΛN , λN , ηN , ξN ) ∈ (0, ∞) × T × R2

be the

(2.14)

with   (X, x) = (X (1) , ..., X (N ) ), (x(1) , ..., x(N ) ) , (X (i) , x(i) ) = ΨP (Λi , λi , ηi , ξi ; µi , ε) . (N )

Then, ΨP

(2.15)

is symplectic and N

(N )

H0

(N )

◦ ΨP

=−

1 X σi =: H0 (Λ) , 2 i=1 Λ2i

σi :=

 µ 3 1 i . m0 Mi

(2.16)

In such Poincar´e variables the full planar (N + 1)–body Hamiltonian H(N ) becomes (N )

H(Λ, λ, η, ξ) = H0 (Λ) + εH1 (Λ, λ, η, ξ) ,

H1 := H1

(N )

◦ ΨP

=: H1compl + H1princ ,

(2.17)

where the so-called “complementary part” H1compl and the “principal part” H1princ of the perturbation are, respectively, the functions X

X (i) · X (j)

1 µi µj m20 |x(i) − x(j) |

X

and

1≤i<j≤N

1≤i<j≤N

(2.18)

expressed in Poincar´e variables4 : X (i) = X(Λi , λi , ηi , ξi ; µi , ε)

and

x(i) = x(Λi , λi , ηi , ξi ; µi , ε) .

Notice that, since X (i) = (m4i /Λ3i )∂λi x(i) the λ–average of H1compl vanishes. Moreover, as it is well known, the λ–average of H1 is an even function of (η, ξ); see, also, Appendix A. Hence, we may split the perturbation function as e1 , H1 = H1 + H with

Z H1 (Λ, η, ξ) :=

H1 TN

dλ , (2π)N

(2.19) Z e1 dλ = 0 . H

(2.20)

TN

Furthermore, H1 may be written as H1 (Λ, η, ξ) = H1,0 (Λ) + H1,2 (Λ, η, ξ) + H1,∗ (Λ, η, ξ) ,

(2.21)

where H1,0 := H1 (Λ, 0, 0), H1,2 is the (η, ξ)–quadratic part of H1 while H1,∗ is the “remainder of order four”: |H1,∗ (Λ, η, ξ)| ≤ const |(η, ξ)|4 . 4X

= (X1 , X2 ) and x = (x1 , x2 ) denote here the functions definded in (2.11).

6

3

The averaged quadratic potential H1,2

In this section we analyze the function H1,2 (i.e., the (η, ξ)–quadratic part of the λ–average of the perturbation) defined in (2.21), which may be written as     X 1 ηj ηi H1,2 = Qij · (3.22) ξj ξi 2 1≤i,j≤N

where Qij are (2 × 2)–matrices defined as  2 ∂ηi ,ηj H1,2 Qij := ∂ξ2i ,ηj H1,2

 ∂η2i ,ξj H1,2 . ∂ξ2i ,ξj H1,2 (Λ,0,0)

ˆ In view of the definition of the Poincar´e variables, it is natural to look at the rescaled variables (ˆ η , ξ) rather then (η, ξ). Therefore, we define ˆ := f¯ij (Λ, ηˆ, ξ) Z dλ 1 . p p √ √ N (i) (2π) ˆi , Λi ξˆi ; µi , ε) − x(j) (Λj , λj Λj ηˆj , Λj ξˆj ; µj , ε)| TN |x (Λi , λi , Λi η

(3.23)

Thus, letting5 ai := a(Λi ; µi , ε) , Aij :=

∂η2ˆi ηˆj f¯ij ∂ 2 f¯ij ξˆi η ˆj

we find

cij := ! ∂η2ˆ ξˆ f¯ij i j ∂ 2 f¯ij ξˆi ξˆj

1 m0

, Bij :=



Mi Mj ai aj 2 ∂ηˆj ηˆj f¯ij ∂ 2 f¯ij ξˆj ηˆj

ˆ ηˆ=ξ=0

 √    µi µj cij Aij , X√ Qij =  µk µj ckj Bkj ,  

1/4 , ∂η2ˆ

ˆ

j ξj

∂ξ2ˆ ξˆ

j j

! f¯ij f¯ij

,

(3.24)

ˆ ηˆ=ξ=0

if i 6= j , if i = j .

k6=j

It is a remarkable fact that, for the planar planetary (N + 1)–body problem the matrices Aij and Bij  1 0 are proportional to the (2 × 2)–identity matrix 12 = and have simple integral representation. 0 1 In fact, define, for a 6= b, Z 2π −17ab cos t + 8(a2 + b2 ) cos(2t) + ab cos(3t) 1 J (a, b) := dt , 2π 0 (a2 + b2 − 2ab cos t)5/2 Z 2π 1 −7ab + 4(a2 + b2 ) cos t − ab cos(2t) I(a, b) := dt , 2π 0 (a2 + b2 − 2ab cos t)5/2 and denote, for ai 6= aj , αij :=

ai aj J (ai , aj ) , 8

βij :=

ai aj I(ai , aj ) . 4

Then, the following “algebraic” result holds. Proposition 3.1 Assume ai 6= aj for i 6= j. Then Aij = αij 12 and Bij = βij 12 . 5 Recall

(2.10) and (2.6).

7

(3.25)

Remark 3.1 (i) An immediate corollary of this result is that, in the collisionless domain {ai = 6 aj }, H1,2 has the simple form  1 H1,2 = Mη · η + Mξ · ξ , (3.26) 2 M being the real, symmetric (N × N )–matrix with entries  √ if i 6= j ,   µi µj cij αij , X √ Mij = (3.27) µk µj ckj βkj , if i = j .   k6=j

The Hamiltonian (3.26) can be immediately put in symplectic normal form: if U is the real orthogonal ¯ 1 , ..., Ω ¯ N )), then the map p = U T η, matrix (U T = U −1 ) which diagonalize M (U T M U = diag (Ω T q = U ξ is symplectic and, in such variables, the new Hamiltonian takes the form N

1X¯ 2 Ωi (pi + qi2 ) . 2 i=1 (ii) The functions J and I (which admit simple representations in terms of Gauss hypergeometric functions) are symmetric (J (a, b) = J (b, a) and I(a, b) = I(b, a)) and satisfy J (a, b) = b−3 J (a/b, 1) ,

I(a, b) = b−3 I(a/b, 1) ,

a
The function of one real variables s ∈ (−1, 1) → J (s, 1) and s ∈ (−1, 1) → I(s, 1) are, respectively, even and odd in s, and satisfy, for small s, the following asymptotics J (s, 1) = −

15 2 105 4 s − s + O(s6 ) , 8 8

I(s, 1) = 3s +

45 3 s + O(s5 ) . 8

(3.28)

Proof (of Proposition 3.1) The computations we are going to perform are algebraic in character and it is therefore enough to consider real variables. Fix i 6= j and define 2 p p p p ˆ := x(i) (Λi , λi , Λi ηˆi , Λi ξˆi ; µi , ε) − x(j) (Λj , λj Λj ηˆj , Λj ξˆj ; µj , ε) , Rij (Λ, λ, ηˆ, ξ) (3.29) so that (recall (3.23)) ˆ = f¯ij (Λ, ηˆ, ξ)

1 (2π)N

Z TN

dλ p . Rij

(3.30)

By (2.11) we find   Rij = a2i χ2i + a2j χ2j − 2ai aj Ci Cj − Si Sj ,

(3.31)

where Ck , Sk and χk are short for, respectively, Ck = C(ˆ ηk , ξˆk , λk ) ,

Sk = S(ˆ ηk , ξˆk , λk )

and χk =

q

Ck2 + Sk2 .

The proof will consist in computing explicitely λ–averages of quantities of the form 3(∂ζi Rij )(∂ζj Rij ) − 2Rij (∂ζi ζj Rij ) 1 2 ρζi ,ζj (λi , λj ) := ∂ζi ζj p = 5/2 Rij ˆ 4R ij

η ˆ=ξ=0

,

(3.32)

ˆ ηˆ=ξ=0

where ζk denotes either of the variables ηˆk or ξˆk . Thus, what we need to do is to compute suitable orders in the variables (ˆ ηk , ξˆk ) of the function Rij . For this purpose it will be useful the following

8

Lemma 3.1 Define the following elementary functions 3 + cos(2λ) , 2 3 − cos(2λ) C− (λ) := 1 + sin2 λ = , 2 1 S0 (λ) := cos λ sin λ = sin(2λ) , 2 χ(x, ¯ y, λ) := 1 − 2y cos λ + 2x sin λ S(x, y, λ) := sin λ + xC+ (λ) + yS0 (λ) , C(x, y, λ) := cos λ − yC− (λ) − xS0 (λ) , C+ (λ) := 1 + cos2 λ =

and denote by Op (z1 , ..., zn ) a function of the variables (z1 , ..., zn ) (depending possibly on other variables) analytic in a neighborhood of (0, ..., 0) and starting with a homogeneous polynomial of degree p in (z1 , ..., zn ). Then, χ2k = C(ˆ ηk , ξˆk , λk )2 + S(ˆ ηk , ξˆk , λk )2 = χ(ˆ ¯ ηk , ξˆk , λk ) + O2 (ˆ ηk , ξˆk ) , Ck = C(ˆ ηk , ξˆk , λk ) = C(ξˆk , ηˆk , λk ) + O2 (ˆ ηk , ξˆk ) , Sk = S(ˆ ηk , ξˆk , λk ) = S(ξˆk , ηˆk , λk ) + O2 (ˆ ηk , ξˆk ) .

(3.33)

The proof of this lemma follows at once from the explicit expressions for C and S given in Lemma 2.1 and is left to the reader. We consider first the matrices Aij (which allow to compute Qij for i 6= j) and then we turn to the matrices Bij (which allow to compute Qjj ).

3.1 Computation of the matrices Aij First, observe that the two derivatives involved in the definition of Aij are always mixed in the variables with indexes i and j. Thus, we can neglect the terms of third order in (ˆ ηi , ξˆi , ηˆj , ξˆj ) and the terms of ˆ ˆ second order of the type O2 (ˆ ηi , ξi ) and O2 (ˆ ηj , ξj ). By Lemma 3.1, the function Rij in (3.31) has the form Rij

= a2i (1 − 2ˆ ηi cos λi + 2ξˆi sin λi ) + a2j (1 − 2ˆ ηj cos λj + 2ξˆj sin λj ) h −2ai aj (cos λi − ξˆi S0 (λi ) − ηˆi C− (λi ))(cos λj − ξˆj S0 (λj ) − ηˆj C− (λj )) i +(sin λi + ξˆi C+ (λi ) + ηˆi S0 (λi ))(sin λj + ξˆj C+ (λj ) + ηˆj S0 (λj )) +O2 (ˆ ηi , ξˆi ) + O2 (ˆ ηj , ξˆj ) + O3 (ˆ ηi , ξˆi , ηˆj , ξˆj ) .

Therefore, letting (·)|0 be short for (·)|ηˆi =ξˆi =ˆηj =ξˆj =0 , one finds Rij 0 ∂ηˆi Rij 0 ∂ηˆj Rij 0 ∂ξˆi Rij 0 ∂ξˆj Rij 0 ∂η2ˆi ηˆj Rij 0

= a2i + a2j − 2ai aj cos(λi − λj ) , = −2a2i cos λi − 2ai aj [−C− (λi ) cos λj + S0 (λi ) sin λj ] , = −2a2j cos λj − 2ai aj [−C− (λj ) cos λi + S0 (λj ) sin λi ] , =

2a2i sin λi − 2ai aj [C+ (λi ) sin λj − S0 (λi ) cos λj ] ,

=

2a2j sin λj − 2ai aj [C+ (λj ) sin λi − S0 (λj ) cos λi ] ,

= −2ai aj [C− (λi )C− (λj ) + S0 (λi )S0 (λj )] ,

9

(3.34)

∂ξ2ˆ ξˆ Rij 0 i j ∂η2ˆ ξˆ Rij 0 i j ∂ξ2ˆ ηˆ Rij 0 i j

= −2ai aj [C+ (λi )C+ (λj ) + S0 (λi )S0 (λj )] , = −2ai aj [C− (λi )S0 (λj ) + S0 (λi )C+ (λj )] , = −2ai aj [S0 (λi )C− (λj ) + C+ (λi )S0 (λj )] .

(3.35)

In particular, Rij and ∂ηˆi Rij are even in (λi , λj ) ∈ T2 , while ∂ξˆj Rij and ∂η2ˆ ξˆ Rij are odd. Thus, i j

recalling the definition of ρζi ,ζj (λi , λj ) in (3.32), we find that ρηˆ ,ξˆ (λi , λj ) is odd in (λi , λj ) and it has i j therefore zero average. For the same reasons, also ρξˆ ,ηˆ (λi , λj ) has zero average. Hence, the off–diagonal i j terms of Aij are zero. We now compute the diagonal terms of Aij . We begin with ρηˆi ,ηˆj (λi , λj ). By (3.32) and the list in (3.35), we find ρ1 (λi , λj ) 1 (3.36) ρηˆi ,ηˆj (λi , λj ) := ∂η2ˆi ηˆj p = ρ2 (λi , λj ) Rij 0

with ρ1 (λi , λj )

h := ai aj · − 24a2i cos(2λi ) − 24a2j cos(2λj ) + 8(a2i + a2j ) cos(2(λi − λj ))

ρ2 (λi , λj )

−3ai aj cos(λi − 3λj ) − 17ai aj cos(λi − λj ) +ai aj cos(3(λi − λj )) − 3ai aj cos(3λi − λj ) i +54ai aj cos(λi + λj ) ,  5/2 := 8 a2i + a2j − 2ai aj cos(λi − λj ) .

(3.37)

Thus, changing variable of integration, one finds Z 1 1 2 p ∂ dλi dλj η ˆ η ˆ (2π)2 T2 i j Rij 0

−17ai aj cos t + 8(a2i + a2j ) cos(2t) + ai aj cos(3t) 1 ai ai · dt 2π T 8(a2i + a2j − 2ai aj cos t)5/2 ai aj J (ai , aj ) =: αij . = 8 Z

=

The case ρξˆ ,ξˆ (λi , λj ) is very similar (and will yield the same result). In place of (3.36) one finds i j 1 ρ3 (λi , λj ) ρξˆ ,ξˆ (λi , λj ) := ∂ξ2ˆ ξˆ p (3.38) = i j i j ρ2 (λi , λj ) Rij 0

with ρ3 (λi , λj )

h := ai aj · 24a2i cos(2λi ) + 24a2j cos(2λj ) + 8(a2i + a2j ) cos(2(λi − λj )) +3ai aj cos(λi − 3λj ) − 17ai aj cos(λi − λj ) +ai aj cos(3(λi − λj )) + 3ai aj cos(3λi − λj ) i −54ai aj cos(λi + λj ) .

Integrating one finds again 1 (2π)2

Z T2

1 ∂ξ2ˆ ξˆ p dλi dλj = αij . i j Rij 0

10

(3.39)

This proves Proposition 3.1 in the case of Qij , with i 6= j.

3.2 Computation of the matrices Bij Observe that the derivatives involved in the definition of Bij are two derivatives with the same index j. We can, therefore, neglect the third order terms and set ηˆi = ξˆi = 0. Recalling (2.13) we see that χi |ηˆi =ξˆi =0 = 1 and Rij

ηˆi =ξˆi =0

  Cj Sj = a2i + a2j χ2j − 2ai aj χj (cos λi ) − (sin λi ) . χj χj

(3.40)

Defining ϕj = ϕj (Λj , λj , ηˆj , ξˆj ) through the relation6 cos ϕj = we find

Rij

ηˆi =ξˆi =0

Cj , χj

sin ϕj =

Sj , χj

(3.41)

= a2i + a2j χ2j − 2ai aj χj cos(ϕj − λi ) .

(3.42)

Denote by h f iθ,τ the average of a function f over the angles θ and τ . Integrating first with respect to λi and changing variable of integration (t = λi − ϕj ) one gets * + * + 1 1 p = q (3.43) Rij ˆ ˜ ij R η ˆi =ξi =0 λi ,λj t,λj

with ˜ ij := a2i + a2j χ2j − 2ai aj χj cos t . R

(3.44)

At this point, the argument is completely analogous to that used above. First, we observe that + + * * ˜ ij )(∂ζ R ˜ ij ) − 2R ˜ ij (∂ 2 R ˜ 3(∂ζh R 1 k ζh ζk ij ) 2 = , (3.45) ∂ζh ζk p ˜ 5/2 Rij 4R 0

ij

λi ,λj

0

t,λj

˜ ij can be written as where ζ` denotes here any of the variables ηˆj , ξˆj . From Lemma 3.1 it follows that R   ˜ ij = f (t) + g(t) h1 − h2 + a2j h21 + O3 (ˆ R ηj , ξˆj ) , with f (t) := a2i + a2j − 2ai aj cos t , g(t) := −2a2j + 2ai aj cos t , h1 := ηˆj cos λj − ξˆj sin λj h2 := ξˆj2 cos2 λj + ηˆj2 sin2 λj + ηˆj ξˆj sin(2λj ) . Thus, since h1 is of order one in (ˆ ηj , ξˆj ) and h2 is of order two in (ˆ ηj , ξˆj ), ˜ ij = f (t) R 0 ˜ ∂ηj Rij = g(t) cos λj , 0

6 Physically, ϕ coincide with v + g where v and g are, respectively, the true anomaly and the argument of the j j j j j perihelion of the osculating ellipse associated to the star and the j th planet; compare Appendix A.

11

˜ ij ∂ξˆj R 0 2 ˜ ij ∂ηˆj ηˆj R 0 2 ˜ ∂ηˆ ξˆ Rij j j 0 2 ˜ ∂ξˆ ξˆ Rij j j

0

Therefore, using (3.45), one finds + * 1 ∂η2ˆ ξˆ q j j ˜ ij R 0

= −g(t) sin λj , = −2g(t) sin2 λj + 2a2j cos2 λj , = −(g(t) + aj )2 sin(2λj ) , = −2g(t) cos2 λj + 2a2j sin2 λj .

* =

−

3 2 2g

+
+ 2(a2j − g)f sin(2λj ) 4f 5/2 0

λi ,λj

= 0

t,λj

(since the integrand is odd in λj ), showing that also Bij is a diagonal matrix. To compute the diagonal elements we calculate ρ ˜ (λ , t) 1 1 1 j 2 2 = ρ˜3 (λj , t) ∂ηˆj ηˆj q and ∂ξˆ ξˆ q (3.46) = ρ˜2 (λj , t) j j ρ˜2 (λj , t) ˜ ij ˜ ij R R 0

0

with ρ˜1

ρ˜2 ρ˜3

7 2 2 a a cos(2 λj − 2 t) 2 i j −(2 ai 3 aj + 10 ai aj 3 ) cos(2 λj − t) + 4(ai 3 aj + ai aj 3 ) cos(t) 7 −a2i a2j cos(2 t) − (2 ai 3 aj + 10 ai aj 3 ) cos(2 λj + t) + a2i a2j cos(2 λj + 2 t) ; 2 2 2 5/2 = 4(ai + aj − 2ai aj cos t) ; 7 = −7 a2i a2j − (9 a2i a2j + 8 aj 4 ) cos(2 λj ) − a2i a2j cos(2 λj − 2 t) 2 +(2 ai 3 aj + 10 ai aj 3 ) cos(2 λj − t) + 4(ai 3 aj + ai aj 3 ) cos(t) 7 −a2i a2j cos(2 t) + (2 ai 3 aj + 10 ai aj 3 ) cos(2 λj + t) − a2i a2j cos(2 λj + 2 t) ; 2 = −7 a2i a2j + (9 a2i a2j + 8 aj 4 ) cos(2 λj ) +

taking the λj –average, one finds immediately + + * * 1 1 2 2 q q = ∂ξˆ ξˆ ∂ηˆj ηˆj j j ˜ ij ˜ ij R R 0

0

λi ,λj

=

ai aj I(ai , aj ) =: βij . 4

λi ,λj

Next result shows that, for δ and ε small, generically the eigenvalues of M in (3.26)–(3.27) are non vanishing, simple and distinct. We formulate the result regarding the semi–axis aj as independent variables. Recall the definitions of αij and βij in (3.25) and let (if N ≥ 3) X √µ ¯k µ ¯j 1 βj := βkj , j≥3. (3.47) √ 4 m0 ak aj k=1,2

Proposition 3.2 Assume that aj and µ ¯j verify7 α12 6= 0 7 Clearly,

and

β12 6= ±α12 ,

βi 6= 0

and

βi 6= βj for i 6= j .

¯ j for j ≥ 3 have to be omitted. if N = 2, the statements regarding the βj and the eigenvalues Ω

12

(3.48)

Then, there exist 0 < δ ? < 1 and 0 < ε0 < 1 such that for all 0 < δ < δ ? and 0 ≤ ε < ε0 the eigen¯ 1 , ..., Ω ¯ N } of the matrix M are non vanishing, simple and distinct. Furthermore the following values {Ω asymptotics holds8 : √

µ ¯1 µ ¯2 m0 √ ¯1 µ ¯2 ¯2 = µ Ω m0 ¯1 = Ω

√ β12 + α12 + O( δ, ε) , √ 4 a a 1 2

√ β12 − α12 + O( δ, ε) , √ 4 a a 1 2 √ √ X √µ √ ¯k µ ¯j βij ¯j = δ + δ O( δ, ε) , Ω √ 4 a a m0 k j

3≤j≤N.

(3.49)

k=1,2

Proof First of all, from the definition of cij (see (3.24) and (2.6)) it follows that cij =

1 1 + O(ε) . √ 4 m0 ai aj

(3.50)

Thus, by definition of M , by definition of βj and αij and by the hypothesis on the masses µi (see (1.3)) we find the following asymptotics:     √ √ µ ¯1 µ ¯2 M11 M12 β12 α12 where M ? := = M ? + O( δ, ε) , , √ M21 M22 α12 β12 m0 4 a1 a2 √ for j ≥ 3, Mjj = δβj + O(δ, ε) , √ Mij = O( δ) , for i = 1, 2 and j ≥ 3, or j = 1, 2 and i ≥ 3, Mij = O(δ) , for i, j ≥ 3, with i 6= j . Therefore9

 M =

√ O( δ)

√ M ? + O( δ, ε) √ O( δ)

√

 ,

δM? + O(δ, ε)

where M? := diag (β3 , ..., βN ) ∈ Mat((N − 2) × (N − 2)) . The eigenvalues of M ? are √ µ ¯1 µ ¯2 (β12 + α12 ) √ 4 m0 a1 a2

and

√ µ ¯1 µ ¯2 (β12 − α12 ) , √ 4 m0 a1 a2

which by the first two requirements in (3.48), are non zero, simple and distinct. The matrix M? is diagonal and its eigenvalues βj are, also, non zero, simple and distinct by (3.48). The claim, now, follows by elementary linear algebra (compare,e.g., Lemma B.2 in Appendix B).

8 We use the standard notation: a = O(ε) ⇐⇒ ∃ a constant c > 0 (independent of ε) and 0 < ε < 1 s.t. |a| ≤ c|ε| for 0 all |ε| ≤ ε0 ; √ O(σ, ε) = O(σ) + O(ε). √ 9 The O( δ) in the upper–right part of M is a (2 × (N − 2)) matrix while the O( δ) in the lower–left part of M is a ((N − 2) × 2) matrix.

13

Remark 3.2 (i) The hypotheses (3.48) of Proposition 3.2 are easily checked, for example, if aj verifies (1.2) for a suitable θ > 0. In fact the asymptotics for J (s, 1) and I(s, 1) (see (3.28)) yields immediately  a 2 i 15 1  a1 3 h 1 1+O , 64 a2 a2 a2  a 2 i 3 1  a1 2 h 1 1+O , β12 = 4 a2 a2 a2 √   h  a 7/4   a 2 i ¯j µ ¯2 3 µ 1 a2 7/4 1 2 βj = 1 + O + O , j≥3, 3/2 4 m0 aj a2 aj aj  a i 3 1  a1 2 h 1 1+O , β12 ∓ α12 = 4 a2 a2 a2 √ ¯j µ ¯2 3 µ 1  a2 7/4 βj − βi = × 3/2 4 m0 aj aj  a 2   a 13/4 i h  a 7/4  2 i 1 +O +O , i>j≥3. × 1+O a2 aj aj α12 = −

Thus, if θ is small enough and if (1.2) holds, one sees that α12 < 0 , β12 ∓ α12 > 0 , βj > 0 , ∀j≥3, βj − βi > 0 , ∀i>j≥3, and the hypotheses (3.48) are verified as claimed. (ii) The O(·)’s appearing in (3.49) (and in the proof of Proposition 3.2) depend on the aj ’s (and on10 m0 ). Thus, it is important the order in fixing the various parameters. One way of proceeding is as follows. First determine θ as explained in the previous point (i). Then, let a ¯i , 1 ≤ i ≤ N , be positive numbers such that (1.2) holds, i.e., a ¯i /¯ ai+1 < θ for any 1 ≤ i ≤ N − 1; (the a ¯i may be physically interpreted as observed mean major semi–axis). Now, consider a compact order–one neighborhood A ⊂ {0 < a1 < · · · < aN } of (¯ a1 , ..., a ¯N ) for which (1.2) continues to be valid (such neighborhood exists simply by continuity). Finally, fix δ ? and ε0 so that Proposition 3.2 holds: such numbers will depend only on a ¯j ’s and the (order–one) size of the choosen neighborhood A.

4

Existence of N –dimensional elliptic invariant tori

In this Section we prove the existence of N –dimensional elliptic invariant tori for the (N + 1)-body problem Hamiltonian H in (2.17) for any N ≥ 2. Let m0 < µ ¯j < 4m0 , let θ, A, δ ? and ε0 be as in point (ii) of Remark 3.2 and fix 0 < δ < δ ? , which henceforth will be kept fixed. In the rest of the paper only ε is regarded a free parameter: at the moment, ε is assumed not to exceed ε0 but later will be required to satisfy stronger smallness conditions. The semi–major axis map
~a : Λ = (Λ1 , ..., ΛN ) 7→ a(Λ1 ; µ1 , ε), ..., a(ΛN ; µN , ε) (4.51) is a real–analytic diffeomorphysm and we define I = ~a −1 (A) , 10 Recall

that m0 < µ ¯j < 4m0 ; compare line before (1.3).

14

2N then the Hamiltonian H is real–analytic (and bounded) on the domain I × TN × BR for a suitable n n R > 0 (here Br denotes the n–ball of radius r and center 0 ∈ R ).

By Proposition 3.1, the quadratic part H1,2 of the averaged Newtonian interaction H1 has the simple form (3.26), M being the symmetric matrix defined in (3.27). As already pointed out in Remark 3.1, the matrix M can be diagonalized with eigenvalues, which, thanks to our assumptions and to Proposition 3.2, have the form in (3.49) and, therefore, satisfy ¯ ¯ j | > c¯ , ¯ ¯, (4.52) inf |Ω inf Ω i − Ωj > c I

I

for any i 6= j = 1, . . . , N and for a suitable positive constant c¯ independent of ε. If U := U (Λ) is the ¯ 1 , ..., Ω ¯ N ), then the map symmetric matrix which diagonalize M , U T M U = diag (Ω  p = UT η , q = UT ξ ,    I = Λ, X Ξ : (I, ϕ, p, q) 7→ (Λ, λ, η, ξ) , where (4.53)  (∂Λ Uk` )Uh` ηk ξ` ,   ϕ=λ+ h,k,`

is symplectic (and real–analytic) and N

H1,2 ◦ Ξ =

1X¯ Ωi (I)(p2i + qi2 ) . 2 i=1

(4.54)

Thus, the (N + 1)-body problem Hamiltonian H in (2.17), in the case we are considering, can be written as H ◦ Ξ(I, ϕ, p, q; ε) = h(I) + f (I, ϕ, p, q; ε) (4.55) with h := H0 ,

f := εf1 (I, p, q; ε) + εf2 (I, ϕ, p, q; ε) , N

f1 := f1,0 (I) + f1,0 := H1,0 ,

1X¯ Ωi (I)(p2i + qi2 ) + f˜1 (I, p, q; ε) , 2 i=1 f˜1 := H1,∗ ◦ Ξ ,

e1 ◦ Ξ . f2 := H

Here h is uniformly strictly concave, Z

|f˜1 | ≤ const |(p, q)|4 ,

and

f2 dϕ = 0 . TN

The construction of elliptic invariant tori for the Hamiltonian (4.55) is based on four steps, which we proceed to describe.

4.1 Averaging Fix τ > N − 1 ≥ 1 and pick two numbers b1 , b2 such that 0 < b1 <

1 , 2

0 < b2 <

1 2

− b1



1 . τ +1

(4.56)

Since the integrable Hamitlonian h depends only on the action I, the conjugated variable ϕ is a “fast” angle and, in “first approximation”, the (h + f )-motions are governed by the averaged Hamiltonian h + εf1 , which posses an elliptic equilibrium at p = q = 0. As we, now, proceed to describe, one may

15

remove the ϕ-dependence of the perturbation function f up to high order in ε by using averaging theory; for detailed information on averaging theory in similar situations, see Proposition A.1 of [BCV03] or Proposition 7.1 of [BBV03]. n Denote by DR the complex n–ball of center zero and radius R > 0 and, for any V ⊂ RN , denote by VR the complex neighborhood of radius R > 0 of the set V given by VR := ∪x∈V DR (x). Next, define the set ˆI as the following “Diophantine subset” of I:   ˆI := I ∈ I : |∂I h(I) · k| ≥ γ¯ , ∀k ∈ ZN \ {0} (4.57) with γ¯ := const εb1 . |k|τ

Notice that (as it is standard to prove)   meas I \ ˆI ≤ const γ¯ = const εb1 .

(4.58)

The Hamiltonian h + f in (4.55) is real–analytic on the complex domain 2N 4N Dr,s,ρ := ˆIr × TN , s × Dρ ⊂ C

with

√ r := const ε ,

s := const ,

ρ := const εb2 .

(4.59) (4.60)

The definition of ˆI is motivated by the necessity to have an estimate on small divisors. In fact, let I ∈ ˆIr (and ε small enough) and let I0 ∈ ˆI be a point at distance less than r from I. Then, for any k ∈ ZN \{0} such that |k| ≤ K := const ε−b2 , by the second relation in (4.56), by (4.57) and by Cauchy estimates, one finds |∂I h(I) · k|

≥ |∂I h(I0 ) · k| − |∂I h(I0 ) − ∂I h(I)| |k| γ¯ ≥ − max |∂I2 h| r K Kτ   γ¯ b1 +τ b2 −b2 ≥ =: α = const ε , 0 < |k| ≤ K := const ε . 2K τ

(4.61)

In order to apply averaging theory so as to remove the ε–dependence up to order exp(− const K), one has to verify the following “smallness condition” (compare condition (A.2), p. 110 in [BCV03]) kf kr,s,ρ ≤ const

α min{rs, ρ2 } , K

where the norm k · kr,s,ρ is defined as the standard “sup–Fourier norm”  X  kf kr,s,ρ := sup |fk (I, p, q)| e|k|s , k∈ZN

(4.62)

(I,p,q)∈ˆ Ir ×Dρ2N

(fk (I, p, q) denotig Fourier coefficients of the multi–periodic, real–analytic function ϕ 7→ f (I, ϕ, p, q)). Such condition, in view of (4.56), can be achieved by taking ε small enough since, by (4.61) and (4.60), one has that α min{rs, ρ2 } = O(εb1 +(τ +1)b2 +1/2 ) . kf kr,s,ρ = O(ε) and K Hence, there exists a close–to–identity (real–analytic) symplectic change of variables (I 0 , ϕ0 , p0 , q 0 ) 7→ (I, ϕ, p, q) verifying (compare formulae (2.16) and (A.7) of [BCV03]) √ 1 and |p0 − p| , |q 0 − q| ≤ const ε , |I 0 − I| ≤ const ε 2 +b2 , (4.63) and such that the Hamiltonian expressed in the new symplectic variables becomes h(I 0 ) + gˆ(I 0 , p0 , q 0 ) + fˆ(I 0 , ϕ0 , p0 , q 0 ) ,

gˆ := εf1 (I 0 , p0 , q 0 ) + εfˆ1 (I 0 , p0 , q 0 )

16

(4.64)

with fˆ1 and fˆ real–analytic on the complex domain Dr/2,s/6,ρ/2 and satisfying ε 1 = const εb2 +b3 , with b3 := − b1 − (τ + 1)b2 > 0 , αr 2 ≤ const e− const K  const ε3 .

kfˆ1 kr/2,s/6,ρ/2 ≤ kfˆkr/2,s/6,ρ/2

(4.65)

4.2 New elliptic equilibrium Due to the (small) term fˆ1 in (4.64), zero is no longer an elliptic equilibrium for the “averaged” (i.e. ϕ-independent) Hamiltonian h + gˆ. Using the Implicit Function Theorem, we can find a new elliptic equilibrium for h + gˆ, which is εb2 +b3 close to zero. Hence we construct a real–analytic symplectic transformation (J 0 , ψ 0 , v 0 , u0 ) 7→ (I 0 , ϕ0 , p0 , q 0 ) ,

with I 0 = J 0

and εb2 +b3 -close-to-the-identity,

(4.66)

such that in the new symplectic variables (J 0 , ψ 0 , v 0 , u0 ) the Hamiltonian takes the form h(J 0 ) + g˜(J 0 , v 0 , u0 ) + f˜(J 0 , ψ 0 , v 0 , u0 ) , with g˜ having v 0 = u0 = 0 as elliptic equilibrium; the functions g˜ and f˜ are real–analytic on a slighty smaller complex domain, say Dr/7,s/7,ρ/7 , where they satisfy bounds similar to those in (4.65). Further˜ j (J 0 ) of the symplectic quadratic part of g˜ are purely imaginary more, for j = 1, . . . , N , the eigenvalues Ω 1+b2 +b3 0 ¯ and ε –close to εΩj (J ).

4.3 Symplectic diagonalization of the quadratic term Using a well known result on the symplectic diagonalization of quadratic Hamiltonians, we can find a real–analytic, symplectic transformation ˜ v˜, u ˜ ψ, (J, ˜) 7→ (J 0 , ψ 0 , v 0 , u0 ) ,

with J 0 = J˜ and εb2 +b3 -close-to-the-identity, (4.67) PN ˜ ˜ 2 such that the quadratic part of g˜ becomes, simply, i=1 Ω uj + v˜j2 ). Whence, the new Hamiltonian i (J) (˜ becomes (compare formula (2.22) of [BCV03]) e := h0 (J) ˜ + H

N X

˜ v˜, u ˜ i (J) ˜ (˜ ˜ v˜, u ˜ ψ, Ω u2i + v˜i2 ) + g˜0 (J, ˜) + f˜0 (J, ˜) ,

(4.68)

i=1

where ˜ := h(J) ˜ + ε˜ ˜ 0, 0) , h0 (J) g (J,

(4.69)

˜ j are real-analytic and g˜0 , f˜0 , Ω ˜ v˜, u |˜ g0 (J, ˜)| ≤ const ε |(˜ v, u ˜)|3 ,

˜ ≤ const ε , |Ω|

kf˜kr/8,s/8,ρ/8 ≤ const ε3 .

(4.70)

Finally, because of (4.52), ˜ i | ≥ const ε > 0 , inf |Ω

˜2 − Ω ˜ 1 | ≥ const ε > 0 . inf |Ω

17

(4.71)

4.4 Applying KAM Theory e in (4.68) in a form suitable for applying (elliptic) KAM theory. We rewrite, now, the Hamiltonian H Introducing translated variables y := J˜ − p and complex variables z, z¯, we define  + z¯ z − z¯  e p + y, ψ, z√ , √ , (4.72) H=H 2 i 2 PN PN here p is regarded as a parameter and the symplectic form is j=1 dyj ∧ dψj + i j=1 dzj ∧ d¯ zj with √ i := −1. The Hamiltonian H is then seen to have the form H =N +P with N =e+ω·y+

N X

Ωj zj z¯j ,

e := h0 (p) ,

ω := ∂J˜h0 (p) ,

˜ Ω := Ω(p)

(4.73)

j=1

and P a perturbation, which can be naturally be splitted in four terms: X P = Pk , 1≤k≤4

with P1 P2

= h0 (p + y) − h0 (p) − ∂J˜h0 (p) · y ∼ y 2 , n   X ˜ j (p + y) − Ω ˜ j (p) zj z¯j ∼ y|z||¯ = Ω z| , j=1

P3 P4

 z + z¯ z − z¯  = g˜0 p + y, √ , √ ∼ ε(|z| + |¯ z |)3 , 2 i 2  z + z¯ z − z¯  = f˜0 p + y, ψ, √ , √ = O(ε3 ) . 2 i 2



 by (4.70) , (4.74)

The parameter p runs over the Diophantine set ˆI defined in (4.57). Notice that the integrable Hamiltonian N affords, for any given value of the parameter p, the N −dimensional elliptic torus {y = 0} × TN × {z = z¯ = 0} ,

(4.75)

which is invariant for the Hamiltonian flow generated by N , the flow being, simply, the Diophantine translation x 7→ x + ωt, with ω as in (4.73). Since det ∂J2˜h0 6= 0, we can use the frequencies ω as parameters rather than the actions p. We, therefore, set o
n O := ∂J˜h0 ˆI = ω = ∂J˜h0 (p) : p ∈ ˆI . (4.76) Notice that, by(4.57), we have   meas ∂J˜h0 (I) \ O ≤ const εb1 .

(4.77)

Now, if we put p = p(ω) := (∂J˜h0 )−1 (ω) in (4.72), we can rewrite the (N + 1)–body Hamiltonian in the form H(y, ψ, z, z; ω) := N (y, z, z¯; ω) + P (y, ψ, z, z¯; ω) (4.78) where N (y, z, z; ω) := e(ω) + ω · y +

N X

Ωj (ω)zj z¯j ,

e(ω) := h0 (p(ω)) ,

j=1

18

e Ω(ω) := Ω(p(ω))

(4.79)

and the perturbation P (y, ψ, z, z¯; ω) is obtained by replacing p with p(ω) in (4.74). Recalling (4.60) the Hamiltonian H in (4.78) is real–analytic in 2N N (y, ψ, z, z¯; ω) ∈ Dr2 ,s,r,d := DrN2 × TN s × Dr × Od

with r := const ε3/4 ,

√ d := const ε .

s := const ,

(4.80) (4.81)

We recall, now, a well known KAM result concerning the persistence of lower dimensional elliptic tori for nearly–integrable Hamiltonian systems ([M65], [E88], [K88]). The version we present here is, essentially, a reformulation of P¨ oschel’s Theorem in [P89] (compare, also, with Theorem 5.1 of [BBV03]). Theorem 4.1 Let H have the form in (4.78), (4.79) and let it be real–analytic on a domain Dr2 ,s,r,d of the form (4.80) for some r, s and d positive. Assume that sup |∂ω Ω(ω)| ≤ ω∈Od

1 4

(4.82)

and that the non-resonance (or Melnikov) condition ∀ 1 ≤ |k| ≤ 2, k ∈ ZN ,

|Ω(ω) · k| ≥ γ0 ,

∀ ω ∈ O,

(4.83)

is satisfied for some γ0 > 0. Then, if d ≥ γ0 and P is sufficiently small, i.e., |||P |||r,s,d := sup kP ( · ; ω)kr2 ,s,r ≤ const γ0 r2 ,

(4.84)

ω∈Od

then there exist a normal form N∗ := e∗ (ω) + ω · y∗ + Ω∗ (ω)z∗ z¯∗ , a Cantor set O(γ0 ) ⊂ O with   meas O \ O(γ0 ) ≤ const γ0 , (4.85) and a transformation 2N N N 2N F : DrN2 /4 × TN s/2 × Dr/2 × O(γ0 ) −→ Dr2 × Ts × Dr × Od (y∗ , ψ∗ , z∗ , z¯∗ ; ω) 7−→ (y, ψ, z, z¯; ω)

real–analytic and symplectic for each ω and Whitney smooth in ω, such that H ◦ F = N∗ + R∗

with

∂yj∗ ∂zh∗ ∂z¯k∗ R∗ = 0 ,

if

2|j| + |h + k| ≤ 2 .

(4.86)

In particular, for each ω ∈ O(γ0 ), the torus {y∗ = 0} × TN × {z∗ = z¯∗ = 0} is an N –dimensional, linearly elliptic, invariant torus run by the flow ψ∗ → ψ∗ + ωt. Finally |y∗ − y| , r|z∗ − z| , r|¯ z∗ − z¯| ≤ const

|||P |||r,s,d γ0

.

(4.87)

In this section we have shown that the many–body Hamiltonian (2.17) (under the hypotheses spelled out at the beginning the section) has indeed the form assumed in the KAM Theorem 4.1. Furthermore, by (4.71), the elliptic frequencies Ωi verify the Melnikov conditions (4.83) with γ0 = const ε ,

(4.88)

and, by (4.74) and (4.81), the perturbation P verifies, for small ε, the KAM condition (4.84), since |||P |||r,s,d = O(r4 + εr3 + ε3 ) = O(ε3 ) ≤ const γ0 r2 = O(ε5/2 ) .

(4.89)

Thus, the existence of the desired quasi-periodic orbits follows at once from Theorem 4.1. We may summarize the final result as follows.

19

Theorem 4.2 Let N ≥ 2 and let H be the (N + 1)–body problem Hamiltonian in Poincar´e variables defined in (2.17). Let m0 < µ ¯j < 4m0 , let θ, A, δ ? and ε0 be as in point (ii) of Remark 3.2. Fix ? −1 0 < δ < δ and let I = ~a (A) where ~a is the semi–major axis map defined in (4.51). Let τ > N − 1 and pick b1 , b2 as in (4.56). Finally, let 0 < ε? < ε0 be such that (4.89) holds for any ε ≤ ε? and such that all conditions on ε required for constructing the symplectic transformations introduced in § 4.1÷§ 4.3 are satisfied for ε < ε? . Then, for any ε < ε? , there exist a Cantor set I∗ ⊂ I, with meas (I \ I∗ ) ≤ const εb1 ,

(4.90)

and a Lipschitz continuous family of tori embedding   φ : (ϑ, p) ∈ TN × I∗ 7→ Λ(ϑ; p), λ(ϑ; p), η(ϑ; p), ξ(ϑ; p) ∈ I × TN × Bρ2N , ∗ with ρ∗ := const εb2 such that, for any p ∈ I∗ , φ(TN ; p) is a real-analytic elliptic H-invariant torus, on which the H−flow is analytically conjugated to the linear flow ϑ → ϑ + ω∗ t, ω∗ being (γ, τ )–Diophantine with γ = O(εb1 ). Furthermore, the following bounds hold uniformly on TN × I∗ : 1

|Λ(ϑ; p) − p| ≤ const ε 2 +b2 , |η(ϑ; p)| + |ξ(ϑ; p)| ≤ const εb2 .

(4.91) (4.92)

Theorem 1.1 follows, now, by taking (recall the definitions of bk in (4.56)) c1 := b1 ,

c2 := b2 ,

c3 := b2 +

1 . 2

(4.93)

In particular the statements on the density of the set of the osculating major semi–axes, on the bound on the osculating eccentricities and on the variation of the osculating major semi–axes, follows from (4.90), (2.12), (4.92), (2.10) and (4.91).

A

Poincar´ e variables for the planar (N + 1)–body problem

We briefly recall in this appendix the classical derivation of the Poincar´e variables for the planar N –body problem, showing also the validity of Lemma 2.1. Though the methods exposed here essentially go back to [Poi1905], we believe that this sketchy presentation can be of use for pointing out some less known features of this set of variables that have been employed in this paper11 . The question of the Poincar´e variables for the N –body problem boils down to the two–body problem case. The set of coordinates thus obtained will then be extended to the one claimed in Lemma 2.1, just by taking the Poincar´e variables given by the interaction between the sun and any of the planets.

A.1 Canonical variables for the two–body problem Consider two bodies P0 , P1 of masses m0 , m1 and position u(0) , u(1) ∈ R2 (with respect to an inertial frame). We assume that P0 and P1 interact through gravity, with gravitational constant 1. By Newton’s laws, the equations of motion for such two-body problem are

11 For

u ¨(0)

= m1

(u(1) − u(0) ) , |u(1) − u(0) |3

u ¨(1)

= m0

(u(0) − u(1) ) . |u(0) − u(1) |3

a review of the Poincar´ e variables in the non-planar case, see, for instance, [Ch88] and [BCV03].

20

Let

m0 m1 , M Then, the above equations of motion become M := m0 + m1 ,

m :=

x ¨=

x := u(1) − u(0) ,

X := mx˙ .

(A.94)

Mx , |x|3

and the motion of the two bodies is governed by the Hamiltonian K(X, x) =

1 mM |X|2 − , 2m |x|

(A.95)

with (X, x) ∈ R2 × R2 conjugate variables, i.e., the equations of motion are x˙ = ∂X K, X˙ = −∂x K. As well known, such system is integrable and for K < 0 the orbits are ellipses. More precisely, one has Proposition A.1 Fix Λ− > 0 > K0 and let Λ+ := real–analytic symplectic transformation



m 3 M2 −2K0

 12

> Λ− . Then, there exist ρˆ > 0 and a

      n ΨDP : (Λ, η), (λ, ξ) ∈ [Λ− , Λ+ ] × [−ˆ ρ, ρˆ] × T × [−ˆ ρ, ρˆ] 7→ (X, x) ∈ |x| ≥

ρˆ2 o , m2 M

casting (A.95) into the integrable Hamiltonian (−m3 M2 )/(2Λ2 ). This classical proposition is a planar version of the classical one, due to Poincar´e (see [Poi1905], chapter III) and the variables (Λ, η, λ, ξ) are, usually, called (planar) Poincar´e variables. The proof of Proposition A.1 is particularly interesting from the physical point of view and rests upon the introduction of three different (famous) change of variables, which we, now, proceed to describe briefly. Let ` and g denote, respectively, the mean anomaly and the argument of the perihelion. Step 1. The systemis set in “symplectic” polar variables: namely, we consider the symplectic map Ψspc : (R, Φ), (r, ϕ) 7→ (X, x) (where r > 0 and ϕ ∈ T) given by

Ψspc :

  x1 = r cos ϕ 

x2 = r sin ϕ

 ,

 X=

cos ϕ sin ϕ

sin ϕ r cos ϕ r

−

  

R

 

(A.96)

Φ

and consider the new Hamiltonian Kspc := K ◦ Ψspc .     Step 2. There is a symplectic map ΨD : (L, G), (`, g) 7→ (R, Φ), (r, ϕ) that integrates the system: ΨD is obtained via the generating function Z r m4 M2 2m2 M G2 S(L, G, r, ϕ) = − + − 2 dr + G ϕ . (A.97) L2 r r   The variables (L, G), (`, g) are known as (planar) Delaunay variables. In such variables, the new Hamiltonian becomes m3 M2 KD := Kspc ◦ ΨD = − . 2L2 Also, if C is the angular momentum of the planet and a is the major semi–axis, by construction, one has that √ G = |C| , and L = m Ma .

21

Step 3. We need now to remove singularities, which appear fro samll eccentricity. To this aim, we first introduce (planar) Poincar´e action–angle variables by means of the linear symplectic transformation     ΨPaa : (Λ, H), (λ, h) 7→ (L, G), (`, g) given by 

Λ = L, H = L − G, λ = ` + g , h = −g .     Then, we let ΨP : (Λ, η), (λ, ξ) 7→ (Λ, H), (λ, h) be the symplectic map defined by ΨPaa :

√

(A.98)

√ 2H cos h = η ,

2H sin h = ξ .

(A.99)

As Poincar´e showed (see [Poi1905], [Ch88], [BCV03]), the symplectic map   ΨDP : (Λ, η), (λ, ξ) 7→ (X, x) with ΨDP := Ψspc ◦ ΨD ◦ ΨPaa ◦ ΨP

(A.100)

is real–analytic in a (complex) neighborhood of h p Λ ∈ Λ− , Λ+ ] , |η|, |ξ| ≤ const Λ− , ,

λ ∈ T. 3

2

Also, the two–body Hamiltonian, in Poincar´e variables, is K ◦ Ψ = − m2ΛM 2 .   Remark A.1 (i) If we denote (X, x) = ΦDP (Λ, η, p), (λ, ξ, q) then, X=

m4 M2 ∂x . Λ3 ∂λ

 Indeed, from the Hamilton equations one sees that : λ˙ = ∂Λ − q˙ = 0. Thus, by the chain rule, X = mx˙ = m(∂λ x) λ˙ =

(A.101) 

3 2 m 3 M2 = mΛM , and 3 2Λ2 m4 M2 ∂x Λ3 ∂λ , proving (A.101).

Λ˙ = ξ˙ = η˙ = p˙ =

(ii) We collect some useful relations among the above introduced quantities. Let, as usual, e denote the eccentricity of the Keplerian ellipse and let a denote the major semi–axis. Then, by construction, one sees that p √ √ Λ = m Ma , ξ 2 + η 2 = Λ e (1 + O(e2 )) . (A.102) Also, if C is the angular momentum of the system, one infers that p |C| = Λ 1 − e2 = Λ(1 + O(e2 )) .

(A.103)

The angular variables ` and g have the astronomical meaning of the mean anomaly and the longitude of the perihelion, respectively (see, for instance, [Ch88] and [BCV03] for pictures of the orbital elements and for some geometric insight). (iii) A proof of the analyticity of Poincar´e variables will also follow by directly inspecting the formulae given in Lemma 2.1, which is proved in the coming section.

22

A.2 Orbital elements We now sketch a way to explicitly represent some quantities in terms of Poincar´e variables. This will also lead to the proof of Lemma 2.1. Let u and v denote the eccentric anomaly and the true anomaly, respectively. By geometric considerations, u = ` + e sin u (A.104) and12 cos v =

cos u − e . 1 − e cos u

(A.105)

where ` = λ + h.

(A.106)

Also, by (A.99), η2 + ξ 2 . (A.107) 2 An explicit expression taking into account H, the eccentricity and the major semi–axis is given by H=

p e2 H = Λ (1 − 1 − e2 ) = Λ (1 + O(e2 )) , 2 r  H H 2− . e(H, Λ) = Λ Λ In light of (A.105),

(A.108) (A.109)

√

1 − e2 sin u . 1 − e cos u

sin v =

(A.110)

By means of (A.104), we have u − ` = e sin(u − ` + `) = e cos ` sin(u − `) + e sin ` cos(u − `) . Thus, in the notation of Lemma 2.1, if G0 is implicitely defined by G0 (x, y) = x sin G0 (x, y) + y cos G0 (x, y) , with G0 (0, 0) = 0, we have that G0 is real–analytic, G0 (x, y) = y + xy + O3 (x, y) and u − ` = G0 (e cos `, e sin `) .

(A.111)

Therefore, we deduce from (A.111) and (A.106) that u = λ + h + G0 (e cos h cos λ − e sin h sin λ, e sin h cos λ + e cos h sin λ) . Moreover, denoting

√ ηˆ = η/ Λ ,

we deduce from (A.99) and (A.109) that r e sin h

=

2H · Λ

√ ξˆ = ξ/ Λ ,

r 1−

H sin h = ξˆ F1 (ˆ η 2 + ξˆ2 ) , 2Λ

(A.112) (A.113)

(A.114)

p where F1 (t) = 1 − (t/4) is real–analytic for |t| < 4 (and agrees with the one introduced in Lemma 2.1). Analogously, e cos h = ηˆ F1 (ˆ η 2 + ξˆ2 ) . (A.115) 12 Such

relations are classical and we refer to [Ch88] and [BCV03] for a geometric interpretation of these anomalies.

23

Therefore, substituting (A.114) and (A.115) in (A.112), we can write G0 as an analytic expression of ˆ λ): more formally, there exists a real–analytic (ˆ ˆ λ) 7→ G(ˆ ˆ λ) (which agrees with the one (ˆ η , ξ, η , ξ, η , ξ, introduced in Lemma by (A.114) and (A.115)), so that ˆ λ) . G0 (e cos h cos λ − e sin h sin λ, e sin h cos λ − e cos h sin λ) = G(ˆ η , ξ, Hence, from of (A.112), e cos u e sin u

= e cos h cos(λ + G) − e sin h sin(λ + G) , = e sin h cos(λ + G) + e cos h sin(λ + G) ,

(A.116)

ˆ λ). Notice also that, from the formulae in (A.109) and (A.107), with G = G(ˆ η , ξ, √ 1 − 1 − e2 = F2 (ˆ η 2 + ξˆ2 ) , e2  −1 for a suitable real–analytic function F2 (actually, F2 (t) = 21 1 − 4t , which agrees with the notation in Lemma 2.1). Thus, if we set ϕ = λ + v − ` = v − h, recalling also (A.105) and (A.110), we have sin ϕ

sin v cos h − cos v sin h p 1 = [ 1 − e2 sin u cos h − cos u sin h + e sin h] 1 − e cos u 1 [sin(u − h) + e sin h − F2 · (e sin u) · (e cos h)] = 1 − e cos u 1 = [sin(λ + e sin u) + e sin h − F2 · (e sin u) · (e cos h)] , 1 − e cos u =

(A.117)

for F2 = F2 (ˆ η 2 + ξˆ2 ) and analogously cos ϕ =

1 [cos(λ + e sin u) − e cos h − F2 · (e sin u) · (e sin h)] . 1 − e cos u

(A.118)

Hence, from (A.114), (A.115), (A.116), (A.117) and (A.118), it follows that sin ϕ and cos ϕ are real– ˆ for λ ∈ T and small ξ, ˆ ηˆ. In particular, if C, S and Es are as defined in analytic functions in λ, ηˆ, ξ, Lemma 2.1, we deduce from (A.116), (A.114) and (A.115) that e sin u = Es ,

(A.119)

and then from (A.118) and (A.117) that (1 − e cos u) cos ϕ = C

and

(1 − e cos u) sin ϕ = S .

(A.120)

Finally, by geometric considerations, we have r = a(1 − e cos u)

(A.121)

where r is the distance between the planet and the sun. Thus, the formulae in Lemma 2.1 follow at once by (A.119), (A.120), (A.96) and (A.101).

A.3 Hamiltonian setting for the planar many body problem

24

Consider (N + 1) bodies P0 , . . . , PN of mass m0 , . . . , mN , all lying in the same plane, interacting through gravity (with constant of gravitation 1). Denote by u(i) the position of Pi in a given inertial frame of R2 , with origin in the center of mass of the system. By Newton’s laws, we have that u ¨(i) =

X 0≤j6=i≤N

mj (u(j) − u(i)) . |u(j) − u(i) |3

(A.122)

Thus, if U (i) := mi u(i) denotes the momentum of Pi , we see that the equations of motion (A.122) come from the Hamiltonian N X X mi mj 1 , |U (i) |2 − (i) − u(j) | 2m |u i i=0 0≤i<j≤N where U = (U (0) , . . . , U (N ) ) ∈ R2(N +1) and u = (u(0) , . . . , u(N ) ) ∈ R2(N +1) are conjugate symplectic variables. We now consider P0 as the “sun” and introduce canonical heliocentric variables via the linear symplectic transformation u(0) = r(0) , u(i) = r(0) + r(i) , U (0) = R(0) − R(1) − . . . − R(N ) ,

U (i) = R(i) ,

for i = 1, . . . , N .

(A.123)

Notice that, by the conservation of the center of mass and our choice of coordinates, R(0) = 0. Thus, the planar many body problem is governed by the (2N )-degree-of-freedom Hamiltonian N  X m0 + mi i=1

2m0 mi

|R(i) |2 −

N  R(i) · R(j) X m0 mi  mi mj  + − (i) . (i) m0 |r | |r − r(j) | 1≤i<j≤N

If mi = εµi for i = 1, . . . , N , i.e., if the “planets” are way smaller than the “sun”, the momenta R(i) are of order ε. Therefore, it is convenient to introduce the following rescaled symplectic variables: R(i)

X (i) =

5/3

,

x(i) =

εm0

r(i) 2/3

,

i = 1, . . . , N.

(A.124)

m0 7/3

In such variables, after a time scale of factor εm0 we obtain the Hamiltonian in (2.4). Notice that, in (N ) that setting, the Hamiltonian H0 corresponds to the sum of N integrable Hamiltonians of the form (A.95), with m and M replaced by mi and Mi , respectively.

A.4 A parity property We recall here the well known fact that the λ–average of H1 (as in (2.17)) is even in (η, ξ). The proof of this will be accomplished by a 180 degree rotation of the perihelia: Proposition A.2 Let f1 (Λ, η, ξ) :=

1 ε

Z H1 (Λ, η, λ, ξ) dλ . T2

Then, f1 (Λ, −η, −ξ) = f1 (Λ, η, ξ). The rescaling by function.

1 ε

is made so that f1 is a (real–analytic) uniformly bounded (by an order–one constant)

Proof The eccentricity ei , the semiaxis ai and the mean anomaly `i of the osculating ellipse of Pi are invariant under the map (Λi , λi , Hi , hi ) 7→ (Λi , λi − π, Hi , hi + π), while the argument of the perihelion

25

gi changes of π. Let us denote ~π := (π, . . . , π) ∈ RN . In light of the consideration above we have that the map (Λ, λ, H, h) 7→ (Λ, λ − ~π , H, h + ~π ) (A.125) leaves |r(i) − r(j) | invariant, for any i, j = 1, . . . , N . The map in (A.125) corresponds to (Λ, λ, ξ, η) 7→ (Λ, λ − π, −ξ, −η) , usually referred to as “space inversion”.

B

Simple eigenvalue perturbations

Lemma B.1 Let M ? ∈ Mat(m × m), M? ∈ Mat(k × k), M ] ∈ Mat(m × k). Then,   M? M] det = det(M ? ) det(M? ) . 0k×m M? Proof Induction over m. Lemma B.2 Let M ? ∈ Mat(m × m), M? ∈ Mat(k × k), M ] ∈ Mat(m × k), M] ∈ Mat(k × m). Let   M ? + O() M ] + O(2 ) . M :=  2 2 M] + O( ) M? + O( ) Then: ¯ 6= 0 is a simple eigenvalue of M ? , then there exists λ ¯ = λ ¯ + O() which is an eigenvalue of • if λ M , provided || is suitably small; ˜ is a simple eigenvalue of M? and det(M ? ) 6= 0, then there exists λ ˜ = λ ˜ + O() so that λ ˜  is • if λ an eigenvalue of M , provided || is suitably small. Proof For the first claim, apply the Implicit Function Theorem to F1 (t, ) := det(M − t1(m+k) ) , noticing that F1 (t, 0) = (−1)k tk det(M ? − t1m ). For the second claim, apply the Implicit Function Theorem to  ?  M − t1m M] F2 (t, ) := det . , M] M? − t1k noticing that k F2 (t, ) = det(M −t1(m+k) ) and that, by Lemma B.1, F2 (t, 0) = det(M ? ) det(M? −t1k ).

26

References [A63] Arnold, V. I.: Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 91–192. [BBV03] Berti, M.; Biasco, L.; Valdinoci, E.: Periodic orbits in Hamiltonian systems with elliptic invariant tori and applications to the planetary three–body problem, to appear in Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 2004 13 . [BCV03] Biasco, L.; Chierchia, L.; Valdinoci, E.: Elliptic two–dimensional invariant tori for the planetary three–body problem, Arch. Rational Mech. Anal. 170 (2003), no. 2, 91–135. [CC03] Celletti, A.; Chierchia, L.: KAM Stability and Celestial Mechanics, Preprint (December 2003); http://www.mat.uniroma3.it/users/chierchia/PREPRINTS/SJV 03.pdf [Ch88] Chenciner, A.: Int´ egration du probl` eme de Kepler par la m´ ethode de Hamilton–Jacobi: coordonn´ ees “action-angles” de Delaunay and coordonn´ ees de Poincar´ e, Notes scientifiques et techniques du B.D.L., S026 (1988). [E88] Eliasson, L.: Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 15 (1988), 115–147. [F02] F´ ejoz, J.: Quasiperiodic motions in the planar three–body problem, J. Differential Equations 183 (2002), no. 2, 303–341. [F03] F´ ejoz, J.: D´ emonstration du “th´ eor` eme d’Arnold” sur la stabilit´ e du syst` eme plan´ etaire (d’apr` es Michael Herman); preprint available on http://www.math.jussieu.fr/∼fejoz/ [H95] Herman M.: Proof of Arnold’s theorem on the existence of maximal tori for planetary systems, Private communication (1995). [JM66] Jefferys, W. H.; Moser, J.: Quasi–periodic solutions for the three–body problem, Astronom. J. 71 (1966) 568–578. [K88] Kuksin, S. B.: Perturbation theory of conditionally periodic solutions of infinite–dimensional Hamiltonian systems and its applications to the Korteweg–de Vries equation. (Russian) Mat. Sb. (N.S.) 136 (178) (1988), no. 3, 396–412, 431; translation in Math. USSR–Sb. 64 (1989), no. 2, 397–413. [L96] Laskar, J.: Large scale chaos and marginal stability in the solar system, Celestial Mechanics and Dynamical Astronomy 64 (1996), 115–162. [LR95] Laskar, J.; Robutel, P.: Stability of the planetary three–body problem. I. Expansion of the planetary Hamiltonian, Celestial Mech. Dynam. Astronom. 62 (1995), no. 3, 193–217. [M65] Melnikov, V. K.: On certain cases of conservation of almost periodic motions with a small change of the Hamiltonian function. (Russian) Dokl. Akad. Nauk SSSR 165 (1965), 1245–1248. [N77] Nekhoroshev N. N.: An exponential estimate of the time of stability of nearly–integrable Hamiltonian systems, I, Usp. Mat. Nauk. 32 (1977), 5–66; Russ. Math. Surv. 32 (1977), 1–65. [Poi1905] Poincar´ e, H.: Le¸cons de M´ ecanique C´ eleste, Tome 1, Gauthier–Villars, Paris (1905)14 . [P89] P¨ oschel, J.: On elliptic lower dimensional tori in Hamiltonian system, Math. Z. 202 (1989), 559–608. [P96] P¨ oschel, J.: A KAM–theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 23 (1996), 119–148. [R95] Robutel, P.: Stability of the planetary three–body problem. II. KAM theory and existence of quasi–periodic motions, Celestial Mech. Dynam. Astronom. 62 (1995), no. 3, 219–261. Luca Biasco & Luigi Chierchia Dipartimento di Matematica, Universit` a “Roma Tre”, Largo S. L. Murialdo 1, I-00146 Roma (Italy) e-mail: biasco, [email protected]

Enrico Valdinoci Dipartimento di Matematica, Universit` a di Roma Tor Vergata, Via della Ricerca Scientifica, 1 I-00133 Roma (Italy) e-mail: [email protected]

13 A

preliminary version is available on–line at http://www.math.utexas.edu/mparc on–line at http://gallica.bnf.fr/scripts/ConsultationTout.exe?E=0&O=N095010

14 Available

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