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1 SIAM J. MATH. ANAL. Vol. 37 No. 5 pp c 2006 Society for Industrial and Applied Mathematics N-DIMENSIONAL ELLIPTIC INVARIANT TORI FOR THE PLANAR N + -BODY PROBLEM LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI 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 + -body problem. For small planetary masses such orbits are close to the limiting solutions given by the N planets revolving around the sun on planar circles. The eigenvalues of the linearized secular dynamics are also computed asymptotically. The proof is based on an appropriate averaging and KAM theory which overcomes the difficulties caused by the intrinsic degeneracies of the model. For concreteness we focus on a caricature of the outer solar system. Key words. N-body problem nearly integrable Hamiltonian systems lower-dimensional elliptic tori AMS subject classifications. 70F0 34C27 37J40 70K43 DOI. 0.37/S Introduction and results... 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; see for example the theoretical work of Poincaré [Poi905] Arnold [A63] Herman [H95] and the numerical investigations of Laskar [L96]. Only recently a complete proof based on [H95] of the existence of quasi-periodic motions corresponding to maximal invariant tori of dimension 3N for the N + - body problem for arbitrary N has been produced in [F04]. We recall that the main difficulties that one encounters in the application of general tools such as averaging and KAM theory to particular cases of interest in celestial mechanics are related to the strong degeneracies of the analytical models. The scope of this paper is to show the existence of quasi-periodic orbits lying on N-dimensional invariant elliptic tori for the planar N + -body problem. The main difference from [H95] and [F04] besides the dimension of the constructed tori relies on the explicit evaluations of the eigenvalues of the linearized secular dynamics which allow us to apply more standard KAM methods. 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 + 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 Received by the editors April ; accepted for publication in revised form April ; published electronically January This work was supported by MIUR Variational Methods and Nonlinear Differential Equations. Dipartimento di Matematica Università Roma Tre Largo S. L. Murialdo I-0046 Roma Italy biasco@mat.uniroma3.it luigi@mat.uniroma3.it. Dipartimento di Matematica Università di Roma Tor Vergata Via della Ricerca Scientifica I-0033 Roma Italy valdinoci@mat.uniroma2.it. 560

2 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 56 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 semiaxes 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. below. The above outer model which roughly mimics some traits of physically relevant cases has also the nice feature of providing particularly 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 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 nonplanar three-body problem. More recently [LR95] and [R95] and [BCV03] proved the existence of quasiperiodic orbits for the nonplanar three-body problem lying on 4-dimensional and linearly stable 2-dimensional tori respectively. Two-dimensional invariant tori for the planar three-body problem have been found in [F02]. Periodic orbits of the nonplanar three-body problem winding around invariant tori have been constructed in [BBV04]. Finally the existence of a positive measure set of initial data giving rise to maximal invariant tori for the planetary N + -body problem has been established in [F04]. The paper is organized as follows. In section.2 we give a more precise statement of our main result. In section 2 we write down the N +-body problem Hamiltonian in Delaunay Poincaré 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 [BBV04] in the three-body case and it is based on a general averaging theorem and on KAM theory for lower-dimensional tori see [P96] [BCV03] [BBV04]. For completeness we include a classical but not easy to find description of analytical properties of the Delaunay Poincaré variables see section 2 and Appendix A; in Appendix B we collect some simple linear algebra lemmata that are used in the arguments given in section Statement of results. We denote the N + massive points bodies by P 0...P N and let m 0...m N be their masses interacting through gravity with constant of gravitation. Fix m 0 > 0 and assume that. m i = εμ i i =...N 0 <ε<. Here ε is regarded as a small parameter and μ i is of order in ε. The point P 0 represents the Sun and the points P i i =...N the planets. We assume that all the bodies lie on a fixed plane that will be identified with R 2. The phase space of this dynamical system the planetary planar N + -body system has dimension 4N after reduction by the symmetries of translations.

3 562 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI We will state the result in terms of orbital elements of the osculating ellipses of the two-body problems associated to P 0 P j. Let u 0 and u j denote the coordinates of P 0 and P j 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 P 0 P j with initial data given by u 0 u j u 0 u j. Of course such ellipses describe the motions of the full N +-body problem only approximately; nevertheless they provide a nice set of coordinates allowing for example to describe the true motions in terms of the eccentricities e j and the major semiaxes a j of the osculating ellipses. For further details and pictures of the orbital elements we refer the reader 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 ellipses e i ; here well-separated means that.2 0 <a i <θa i+ i N. for a suitable constant 0 <θ<. For concreteness we shall focus on a caricature of the outer solar system; i.e. we will assume that for some m 0 < μ i < 4m 0.3 μ i = μ i for i = 2 μ i = δ μ i for i =3...N 0 <δ<. In this setting P and P 2 imitate in a very rough way of course the physical features of the giant planets Jupiter and Saturn while P 3 and P 4 represent Uranus and Neptune. 2 A rough description of our main result is given in the following theorem; a more precise and quantitative version is given in Theorem 4.2 below. Theorem.. Consider a planar planetary N +-body system satisfying if N 3. and.3. Let A R N be a compact set of semiaxes where.2 holds for a suitable 0 <θ<. 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 +-body system possesses a family of N-dimensional elliptic invariant Diophantine quasi-periodic tori; such family is parametrized by the osculating major semiaxes varying in a subset of A of density 3 C ε c. These motions correspond to orbits with osculating eccentricities bounded by C 2 ε c2 and the variation in time of the osculating major semiaxes of these orbits is bounded by C 3 ε c3. We have the following 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 depends on N A and δ is related to a KAM smallness condition and rough estimates lead as is well known to ridiculously small quantities for somewhat more serious KAM estimates we refer the reader to [CC03]. Finally the positive constants C i s depend on N A and δ while the c i s depend only on N and could also be easily calculated; see The assumptions.2 and.3 in the theorem are used to check explicitly suitable nondegeneracy conditions. However giving explicit constants and A mathematical motivation for considering two dominant planets is given in Remark 3.2iii. 2 The Jupiter/Saturn mass ratio is approximately 3.34 while the Neptune/Uranus mass ratio is about.8 to have it all the Jupiter/Uranus mass ratio is Here and in what follows the density is intended with respect to Lebesgue measure.

4 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 563 estimates one can show that the thesis of the theorem holds essentially with no hypotheses on the semiaxes a j and the rescaled masses μ j provided a i a j > 0 and μ j > 0; a rigorous argument based on analytic continuation of the eigenvalues could be given along the lines discussed in [F04]. The invariant tori found in Theorem. are lower-dimensional elliptic tori meaning that the dimension of the tori is 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 nondegeneracy 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 H 2 ; quadratic here refers to the symplectic Cartesian variables measuring the eccentricity and the orientation of the osculating ellipses. The diagonalization of H 2 is trivial under the only assumption that a i a j while conditions.2 and.3 will be used to check that the associated eigenvalues are nonzero simple and distinct so that Melnikov conditions are satisfied. The proof is noninductive on N. 2. Poincaré Hamiltonian setting. The results described in this section are classical even if not easy to find and go back to Delaunay and Poincaré; the reader not familiar with Delaunay and Poincaré variables will find a self-contained exposition in Appendix A. Consider N+ bodies P 0...P N in a fixed ecliptic plane of masses m 0...m N interacting through gravity with constant of gravitation. We assume that the mass of P 0 the star is much larger than the mass of the other bodies the planets ; i.e. we assume.. In heliocentric planar suitably rescaled variables the dynamics of the planar N + -body problem is governed as explained in Appendix A by the Hamiltonian 2. H N X x :=H N 0 X x+εh N X x where X := X...X N R 2N and x := x...x N R 2N are conjugated Cartesian symplectic variables and 2.2 H N 0 := H N := N X i 2 m im i 2m i= i x i X i X j μ iμ j m 2 0 i<j N here we have introduced the dimensionless masses 4 ; x i x j M i := + ε μ i μ i m i := = μ i 2.3. m 0 m 0 + εμ i m 0 M i The Hamiltonian H N 0 is simply the sum of N uncoupled planar Kepler problems formed by the star and the ith planet. Being interested in phase region where 4 Beware not to confuse the dimensionless masses m i with the real masses m i introduced at the beginning of section.2.

5 564 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI the uncoupled Kepler problem describes nearly circular orbits we introduce planar Poincaré variables the construction of which is based on the classical 4-dimensional symplectic map 2.8 below. Let 2.4 F t := t 2 F2 t := 4 2 t 4 t < ; 4 let G 0 s t =t+st+ be the function analytic in a neighborhood of 0 0 implicitly defined by 2.5 G 0 0 0=0 G 0 = s sin G 0 + t cos G 0 ; define the following four functions of three variables ˆη ˆξλ real-analytic in a neighborhood of the set {ˆη ˆξ =0 0} T: Gˆη ˆξλ :=G 0 ˆη cos λ ˆξ sin λ F t ˆξ cos λ ˆη sin λ F t E s ˆη ˆξλ := ˆξ cosλ + G+ˆη sinλ + G F t 2.6 Cˆη ˆξλ := cosλ + E s ˆη F t ˆξ E s F t F 2 t Sˆη ˆξλ := sinλ + E s +ˆξ F t ˆη E s F t F 2 t where t is short for t =ˆη 2 + ˆξ 2 G is short for Gˆη ˆξλ and E s is short for E s ˆη ˆξλ. Lemma 2. planar Poincaré variables. Fix ε μ m 0 > 0 and let M:=+ε μ m:= μ m 0 m 0 M m:= μ m 0 M μ Λ2 σ := a = aλ; μ ε := m 0 M m 2. Then for any Λ + > Λ > 0 there exists a ball B around the origin in R 2 such that the 4-dimensional map where 2.8 Ψ P :Λληξ D := Λ Λ + T B X x R 4 η ξ x = x Λληξ; μ ε :=aλ; μ ε C λ Λ Λ η ξ x 2 = x 2 Λληξ; μ ε :=aλ; μ ε S λ Λ Λ X = XΛληξ; μ ε := m4 x m Λ 3 Λληξ; μ ε = λ is real-analytic in D and symplectic: dλ dλ + dη dξ = dx dx + dx 2 dx 2. Furthermore if H 0 denotes the two-body Hamiltonian H 0 X x := 2m X 2 mm x aλ; μ ε 3/2 x λ

6 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 565 then on the phase region of negative energies H 0 σ σ one has 2Λ 2 2Λ 2 + H 0 Ψ P = σ 2Λ 2 ; in the planar coordinates x R 2 the corresponding motion describes an ellipse of major semiaxis a = aλ; μ ε and eccentricity e = η2 + ξ 2 Λ η 2 + ξ 2 η F = 2 + ξ 2 Λ Λ η2 + ξ 2. 4Λ The proof of this lemma can be found in Appendix A. Note that C0 0λ = cos λ S0 0λ = sin λ so that the a λ x transformation is for η = ξ = 0 just polar coordinates. Let now Ψ N P be the 4N-dimensional map parametrized by μ...μ N ε defined by 2. with Ψ N P : 2.2 Then Ψ N P 2.3 Λ λ η ξ...λ N λ N η N ξ N 0 T R 2 N X x X x = X...X N x...x N X i x i =Ψ P Λ i λ i η i ξ i ; μ i ε. is symplectic and H N 0 Ψ N P = 2 N σ i Λ 2 i= i =: H 0 Λ σ i := μi m 0 3 M i. In such Poincaré variables the full planar N + -body Hamiltonian H N becomes 2.4 HΛληξ=H 0 Λ + εh Λληξ H := H N Ψ N P =: H compl + H princ where the so-called complementary part H compl and the principal part H princ of the perturbation are respectively the functions 2.5 i<j N X i X j expressed in Poincaré variables: 5 and i<j N μ i μ j m 2 0 x i x j X i = XΛ i λ i η i ξ i ; μ i ε and x i = xλ i λ i η i ξ i ; μ i ε. 5 X =X X 2 and x =x x 2 denote here the functions defined in 2.8.

7 566 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI Notice that since X i =m 4 i /Λ 3 i λ i x i the λ-average of H compl vanishes. Moreover as it is well known the λ-average of H is an even function of η ξ; see also Appendix A. Hence we may split the perturbation function as 2.6 with 2.7 H Ληξ:= T N H Furthermore H may be written as 2.8 H = H + H dλ 2π N T N H dλ =0. H Ληξ=H 0 Λ + H 2 Ληξ+H Ληξ where H 0 := H Λ 0 0 H 2 is the η ξ-quadratic part of H while H is the remainder of order four : H Ληξ const η ξ The averaged quadratic potential H 2. In this section we analyze the function H 2 i.e. the η ξ-quadratic part of the λ-average of the perturbation defined in 2.8 which may be written as 3. H 2 = 2 ij N ηj Q ij ξ j ηi where Q ij are 2 2 matrices defined as 2 Q ij := ηiη j H 2 η 2 iξ j H 2 ξ 2 iη j H 2 ξ 2. iξ j H 2 Λ00 The aim of this section is to prove that there exists a symplectic linear change of variables p q η ξ putting the quadratic part 3. in the normal form N Ω i p 2 i + qi 2 ; i= see Remark 3.i. A crucial fact in order to apply KAM theory consists in proving that such Ω i s are nondegenerate 6 in the sense that they are nonvanishing and distinct. Such nondegeneracy is proved in Proposition 3.2 in which we manage to compute explicitly the asymptotics of the Ω i s. In view of the definition of the Poincaré variables we look at the rescaled variables ˆη ˆξ rather than η ξ. Therefore we define 3.3 f ij Λ ˆη ˆξ := dλ 2π N x i Λ i λ i Λ iˆη i Λ i ˆξi ; μ i ε x j Λ j λ j Λj ˆη j Λ j ˆξj ; μ j ε. T N ξ i 6 See the Melnikov condition 4.33.

8 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 567 Thus letting 7 a i := aλ i ; μ i ε c ij := /4 Mi M j m 0 a i a j 2ˆηi ˆη fij j 2ˆη fij A ij := i ˆξj 2ˆηj ˆη fij j 2ˆη fij B ij := j ˆξj 3.4 ˆη=ˆξ=0 we find 2ˆξi ˆη j fij 2ˆξi ˆξj fij 2ˆξj ˆη j fij μi μ j c ij A ij if i j Q ij = μk μ j c kj B kj if i = j. k j 2ˆξj ˆξj fij ˆη=ˆξ=0 It is a remarkable fact that for the planar planetary N +-body problem the matrices A ij and B ij are proportional to the 2 2 identity matrix 2 = 0 0 and have simple integral representation. In fact define for a b J a b := 2π Ia b := 2π and denote for a i a j 2π 0 2π 0 7ab cos t +8a 2 + b 2 cos2t+ab cos3t a 2 + b 2 2ab cos t 5/2 dt 7ab +4a 2 + b 2 cos t ab cos2t a 2 + b 2 2ab cos t 5/2 dt 3.5 α ij := a ia j 8 J a ia j β ij := a ia j 4 Ia ia j. Then the following algebraic result holds. Proposition 3.. Assume a i a j for i j. Then A ij = α ij 2 and B ij = β ij 2. Remark 3.. i An immediate corollary of this result is that in the collisionless domain {a i a j } H 2 has the simple form 3.6 H 2 = Mη η + Mξ ξ 2 M being the real symmetric N N matrix with entries μi μ j c ij α ij if i j 3.7 M ij = μk μ j c kj β kj if i = j. k j The Hamiltonian 3.6 can be immediately put in symplectic normal form: if U is the real orthogonal matrix U T = U which diagonalizes M U T MU = diag Ω... Ω N then the map p = U T η q = U T ξ is symplectic and in such variables the new Hamiltonian takes the form 3.2. 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 Ia b =Ib a and satisfy J a b =b 3 J a/b Ia b =b 3 Ia/b a<b. 7 Recall 2.7 and 2.3.

9 568 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI The functions of one real variable s Js and s Is are respectively even and odd in s and satisfy for small s the following asymptotics: 3.8 J s = 5 8 s s4 + Os 6 Is =3s s3 + Os 5. iii Proposition 3. is a suitable version of a well-known result which can be found e.g. in [Poi905]; see also [LR95]. iv The asymptotics of the α ij s and β ij s may be also computed in terms of the Laplace coefficients see e.g. [LR95]; for our purposes it is simpler to derive the needed asymptotics directly from the integral representations given before 3.5. Proof of Proposition 3.. The computations we are going to perform are algebraic in character and it is therefore enough to consider real variables. Fix i j and define 3.9 R ij Λλˆη ˆξ := x i Λ i λ i Λ iˆη i Λ i ˆξi ; μ i ε x j Λ j λ j Λj ˆη j Λ j ˆξj ; μ j ε 2 so that recall By 2.8 we find 3. f ij Λ ˆη ˆξ = 2π N T N dλ Rij. R ij = a 2 i χ 2 i + a 2 jχ 2 j 2a i a j C i C j S i S j where C k S k and χ k are short for respectively C k = Cˆη k ˆξ k λ k S k = Sˆη k ˆξ k λ k and χ k = C 2 k + S2 k. The proof will consist in computing explicitly λ-averages of quantities of the form 3.2 ρ ζi ζ j λ i λ j := ζ 2 iζ j = Rij ˆη=ˆξ=0 3 ζ i R ij ζj R ij 2R ij ζiζ j R ij 4R 5/2 ij ˆη=ˆξ=0 where ζ k denotes either of the variables ˆη k or ˆξ k. Thus what we need to do is to compute suitable orders in the variables ˆη k ˆξ k of the function R ij. For this purpose the following lemma will be useful. Lemma 3.. Define the following elementary functions: C + λ :=+cos 2 λ = 3 + cos2λ 2 C λ :=+sin 2 λ = 3 cos2λ 2 S 0 λ :=cosλsin λ = 2 sin2λ χx y λ := 2y cos λ +2xsin λ Sx y λ :=sinλ + xc + λ+ys 0 λ Cx y λ :=cosλ yc λ xs 0 λ

10 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 569 and denote by O p z...z n a function of the variables z...z n depending possibly on other variables analytic in a neighborhood of and starting with a homogeneous polynomial of degree p in z...z n. Then 3.3 χ 2 = Cˆη k k ˆξ k λ k 2 + Sˆη k ˆξ k λ k 2 = χˆη k ˆξ k λ k +O 2 ˆη k ˆξ k C k = Cˆη k ˆξ k λ k =Cˆξ k ˆη k λ k +O 2 ˆη k ˆξ k S k = Sˆη k ˆξ k λ k =Sˆξ k ˆη k λ k +O 2 ˆη k ˆξ k. The proof of this lemma follows at once from the explicit expressions for C and S given in Lemma 2. and is left to the reader. We consider first the matrices A ij which allow to compute Q ij for i j and then we turn to the matrices B ij which allow to compute Q jj. Computation of the matrices A ij. First observe that the two derivatives involved in the definition of A ij are always mixed in the variables with indexes i and j. Thus we can neglect the terms of third order in ˆη i ˆξ i ˆη j ˆξ j and the terms of second order of the type O 2 ˆη i ˆξ i and O 2 ˆη j ˆξ j. By Lemma 3. the function R ij in 3. has the form 3.4 R ij = a 2 i 2ˆη i cos λ i +2ˆξ i sin λ i +a 2 j 2ˆη j cos λ j +2ˆξ j sin λ j 2a i a j [cos λ i ˆξ i S 0 λ i ˆη i C λ i cos λ j ˆξ j S 0 λ j ˆη j C λ j + sin λ i + ˆξ i C + λ i +ˆη i S 0 λ i sin λ j + ˆξ ] j C + λ j +ˆη j S 0 λ j + O 2 ˆη i ˆξ i +O 2 ˆη j ˆξ j +O 3 ˆη i ˆξ i ˆη j ˆξ j. Therefore letting 0 be short for ˆηi=ˆξ i=ˆη j=ˆξ j=0 one finds 3.5 R ij 0 = a 2 i + a 2 j 2a i a j cosλ i λ j ˆηi R ij 0 = 2a 2 i cos λ i 2a i a j [ C λ i cos λ j + S 0 λ i sin λ j ] ˆηj R ij 0 = 2a 2 j cos λ j 2a i a j [ C λ j cos λ i + S 0 λ j sin λ i ] ˆξi R ij 0 =2a 2 i sin λ i 2a i a j [C + λ i sin λ j S 0 λ i cos λ j ] ˆξj R ij 0 =2a 2 j sin λ j 2a i a j [C + λ j sin λ i S 0 λ j cos λ i ] 2ˆη i ˆη j R ij 0 = 2a i a j [C λ i C λ j +S 0 λ i S 0 λ j ] 2ˆξi ˆξj R ij 0 = 2a i a j [C + λ i C + λ j +S 0 λ i S 0 λ j ] 2ˆη i ˆξj R ij 0 = 2a i a j [C λ i S 0 λ j +S 0 λ i C + λ j ] 2ˆξi ˆη j R ij 0 = 2a i a j [S 0 λ i C λ j +C + λ i S 0 λ j ]. In particular R ij and ˆηi R ij are even in λ i λ j T 2 while R ˆξj ij and 2ˆη R i ˆξj ij are odd. Thus recalling the definition of ρ ζi ζ j λ i λ j in 3.2 we find that λ ρˆηi ˆξ i λ j j is odd in λ i λ j and it has therefore zero average. For the same reasons also λ ρˆξi ˆη i λ j has zero average. Hence the off-diagonal terms of A ij are zero. We j

11 570 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI now compute the diagonal terms of A ij. We begin with ρˆηi ˆη j λ i λ j. By 3.2 and the list in 3.5 we find 3.6 with ρˆηi ˆη j λ i λ j := 2ˆη i ˆη j Rij 0 = ρ λ i λ j ρ 2 λ i λ j 3.7 [ ρ λ i λ j :=a i a j 24a 2 i cos2λ i 24a 2 j cos2λ j +8a 2 i + a 2 j cos2λ i λ j 3a i a j cosλ i 3λ j 7a i a j cosλ i λ j + a i a j cos3λ i λ j 3a i a j cos3λ i λ j ] +54a i a j cosλ i + λ j 5/2. ρ 2 λ i λ j :=8 a 2 i + a 2 j 2a i a j cosλ i λ j Thus changing the variable of integration one finds 2π 2 2ˆη i ˆη j dλ i dλ j T 2 Rij 0 = a i a i 7a ia j cos t +8a 2 i + a2 j cos2t+a ia j cos3t 2π T 8a 2 i + a2 j 2a dt ia j cos t 5/2 = a ia j J a i a j =:α ij. 8 The case λ ρˆξi ˆξ i λ j is very similar and will yield the same result. In place of j 3.6 one finds 3.8 with 3.9 ρ 3 λ i λ j :=a i a j ρˆξi ˆξ j λ i λ j := 2ˆξi ˆξj Rij 0 = ρ 3λ i λ j ρ 2 λ i λ j [ 24a 2 i cos2λ i +24a 2 j cos2λ j +8a 2 i + a 2 j cos2λ i λ j +3a i a j cosλ i 3λ j 7a i a j cosλ i λ j + a i a j cos3λ i λ j +3a i a j cos3λ i λ j ] 54a i a j cosλ i + λ j. Integrating one finds again 2π 2 2ˆξi dλ T 2 ˆξj i dλ j = α ij. Rij 0 This proves Proposition 3. in the case of Q ij with i j. Computation of the matrices B ij. Observe that the derivatives involved in the definition of B ij are two derivatives with the same index j. We can therefore neglect the third order terms and set ˆη i = ˆξ i =0.

12 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 57 Recalling 2.0 we see that χ i ˆηi=ˆξ i=0 = and 3.20 R ij ˆηi=ˆξ i=0 Defining ϕ j = ϕ j Λ j λ j ˆη j ˆξ j through the relations we find 3.22 = a 2 i + a 2 jχ 2 j 2a i a j χ j cos λ i C j χ j sin λ i S j χ j cos ϕ j = C j χ j sin ϕ j = S j χ j R ij ˆηi=ˆξ i=0 = a 2 i + a 2 jχ 2 j 2a i a j χ j cosϕ j λ i. Denote by f θτ the average of a function f over the angles θ and τ. Integrating first with respect to λ i and changing variable of integration t = λ i ϕ j one gets 3.23 Rij ˆηi=ˆξi=0 Rij λ iλ j = with 3.24 R ij := a 2 i + a 2 jχ 2 j 2a i a j χ j cos t. At this point the argument is completely analogous to that used above. First we observe that ζ 2 3 ζh Rij ζk Rij 2 h ζ k = Rij 0 λiλj R ij ζ 2 h ζ Rij k R 5/2 ij tλ j 0. tλ j where ζ l denotes here any of the variables ˆη j ˆξ j. From Lemma 3. it follows that R ij can be written as R ij = ft+gth h 2 +a 2 jh 2 + O 3 ˆη j ˆξ j with ft :=a 2 i + a 2 j 2a i a j cos t gt := 2a 2 j +2a i a j cos t h := ˆη j cos λ j ˆξ j sin λ j h 2 := ˆξ j 2 cos 2 λ j +ˆη j 2 sin 2 λ j +ˆη j ˆξj sin2λ j. Thus since h is of order one in ˆη j ˆξ j and h 2 is of order two in ˆη j ˆξ j 0 R ij = ft ηj Rij 0 = gt cos λ j ˆξj Rij 0 = gt sin λ j 2ˆη j ˆη j Rij 0 = 2gt sin 2 λ j +2a 2 j cos 2 λ j 2ˆη j ˆξj Rij 0 = gt+a j 2 sin2λ j 2ˆξj ˆξj Rij 0 = 2gt cos 2 λ j +2a 2 j sin 2 λ j. 8 Physically ϕ j coincides with v j + g j where v j and g j are respectively the true anomaly and the argument of the perihelion of the osculating ellipse associated to the star and the jth planet; compare to Appendix A.

13 572 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI Therefore using 3.25 one finds 2ˆη 3 2 g2 +2a 2 j gf sin2λ j j ˆξj Rij 0 λ iλ j = 4f 5/2 0 tλ j =0 since the integrand is odd in λ j showing that also B ij is a diagonal matrix. To compute the diagonal elements we calculate 2ˆη j ˆη j = ρ λ j t and 2ˆξj ˆξj ρ 2 λ j t = ρ 3λ j t 3.26 ρ 2 λ j t with Rij 0 Rij ρ = 7 a 2 i a 2 j +9a 2 i a 2 j +8a j 4 cos2 λ j a2 i a 2 j cos2 λ j 2 t 2 a i 3 a j +0a i a j 3 cos2 λ j t+4a i 3 a j + a i a j 3 cost a 2 i a 2 j cos2 t 2 a i 3 a j +0a i a j 3 cos2 λ j + t+ 7 2 a2 i a 2 j cos2 λ j +2t; ρ 2 =4a 2 i + a 2 j 2a i a j cos t 5/2 ; ρ 3 = 7 a 2 i a 2 j 9 a 2 i a 2 j +8a j 4 cos2 λ j 7 2 a2 i a 2 j cos2 λ j 2 t +2a i 3 a j +0a i a j 3 cos2 λ j t+4a i 3 a j + a i a j 3 cost a 2 i a 2 j cos2 t+2a i 3 a j +0a i a j 3 cos2 λ j + t 7 2 a2 i a 2 j cos2 λ j +2t; taking the λ j -average one finds immediately 2ˆη j ˆη j 2ˆξj ˆξj Rij 0 λ iλ j = Rij 0 λ iλ j = 0 a i a j 4 Ia ia j =:β ij. The next result shows that for δ and ε small generically the eigenvalues of M in are nonvanishing simple and distinct. We formulate the result regarding the semiaxis a j as independent variables. Recall the definitions of α ij and β ij in 3.5 and let if N β j := k=2 μk μ j β kj j 3. m 0 4 ak a j Proposition 3.2. Assume that a j and μ j verify 9 α 2 0 and β 2 ±α 2 β i 0 and β i β j for i j. Then there exist 0 <δ < and 0 <ε 0 < such that for all 0 <δ<δ and 0 ε<ε 0 the eigenvalues { Ω... Ω N } of the matrix M are nonvanishing simple 9 Clearly if N = 2 the statements regarding the β j and the eigenvalues Ω j for j 3 have to be omitted.

14 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 573 and distinct. Furthermore the following asymptotics hold: Ω = μ μ 2 m 0 β 2 + α 2 4 a a 2 Ω 2 = μ μ 2 m 0 β 2 α 2 4 a a 2 + O δ ε + O δ ε Ω j = δβ j + δo δ ε 3 j N. As mentioned above see Remark 3.iv the asymptotic of the α ij s and β ij s may be evaluated in terms of the Laplace coefficients see e.g. [L9]. For completeness we give a detailed proof. Proof. First of all from the definition of c ij see 3.4 and 2.3 it follows that 3.30 c ij = + Oε. m 0 ai a j 4 Thus by definition of M by definition of β j and α ij and by the hypothesis on the masses μ i see.3 we find the following asymptotics: M M 2 = M + O μ δ ε where M μ 2 := M 2 M 22 m β2 α a a 2 α 2 β 2 M jj = δβ j + Oδ ε for j 3 M ij = O δ for i = 2 and j 3 or j = 2 and i 3 M ij = Oδ for i j 3 with i j. Therefore where M + O δ ε O δ M = O δ δm + Oδ ε M := diag β 3...β N MatN 2 N 2. The eigenvalues of M are μ μ 2 m β α 2 0 a a 2 and μ μ 2 m 0 4 a a 2 β 2 α 2 which by the first two requirements in 3.28 are nonzero simple and distinct. The matrix M is diagonal and its eigenvalues β j are also nonzero simple and distinct by The claim now follows by elementary linear algebra compare e.g. Lemma B.2 in Appendix B. 0 We use the standard notation a = Oε a constant c > 0 independent of ε and 0 <ε 0 < s.t. a c ε for all ε ε 0 ; Oσ ε =Oσ+Oε. The O δ in the upper right part of M isa2 N 2 matrix while the O δ in the lower left part of M is an N 2 2 matrix.

15 574 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI Remark 3.2. i The hypotheses 3.28 of Proposition 3.2 are easily checked for example if a j verifies.2 for a suitable θ>0. In fact the asymptotics for J s and Is see 3.8 yield immediately α 2 = 5 64 β 2 = 3 a 2 [ 4 a 2 a 2 β j = 3 μj μ 2 4 m 0 a 3/2 j a 3 [ a +O a 2 a 2 2 ] a 2 a 2 ] +O a 2 a2 7/4 [ +O a j β 2 ± α 2 = 3 a 2 [ 5 4 a 2 a 2 6 β j β i = 3 μj μ 2 a2 7/4 4 m 0 a j [ +O a 3/2 j a a 2 7/4 a 7/4 a 2 a a 2 ] + O a 2 a 2 a2 2 ] + O j 3 a j a2 2 ai 3/4] + O + O i>j 3. a j a j Thus if θ is small enough and if.2 holds one sees that α 2 < 0 β 2 ± α 2 > 0 β j > 0 j 3 β j β i > 0 i>j 3 and the hypotheses 3.28 are verified as claimed. ii The O s appearing in 3.29 and in the proof of Proposition 3.2 depend on the a j s and on 2 m 0. Thus the order in fixing the various parameters is important. One way of proceeding is as follows. First determine θ as explained in the previous point i. Then let ā i i N be positive numbers such that.2 holds i.e. ā i /ā i+ <θfor any i N ; the ā i may be physically interpreted as observed mean major semiaxis. Now consider a compact order-one neighborhood A {0 <a < <a N } of ā...ā N for which.2 continues to be valid such neighborhood exists simply by continuity. Finally fix δ and ε 0 so that Proposition 3.2 holds: such numbers will depend only on ā j s and the order-one size of the chosen neighborhood A. iii In the case of only one dominant planet i.e. μ = μ = O μ i = Oδ for i 2 the first two asymptotics in 3.29 do not give any information: in particular we cannot assure that Ω and Ω 2 are different from zero. On the other hand one could also consider the case of three or more dominant planets and the choice of focusing on two dominant planets has been made for simplicity. 4. Existence of N-dimensional elliptic invariant tori. In this section we prove the existence of N-dimensional elliptic invariant tori for the N + -body problem Hamiltonian H in 2.4 for any N 2. Let m 0 < μ j < 4m 0 let θ A δ and ε 0 be as in Remark 3.2ii and fix 0 <δ<δ which henceforth will be kept fixed. In the rest of the paper only ε is 2 Recall that m 0 < μ j < 4m 0 ; compare with the line before.3.

16 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 575 regarded as a free parameter: at the moment ε is assumed not to exceed ε 0 but later will be required to satisfy stronger smallness conditions. The semimajor axis map 4. a :Λ=Λ...Λ N aλ ; μ ε...aλ N ; μ N ε is a real-analytic diffeomorphism and we define I = a A then the Hamiltonian H is real-analytic and bounded on the domain I T N BR 2N for a suitable R>0 here Br n denotes the n-ball of radius r and center 0 R n. By Proposition 3. the quadratic part H 2 of the averaged Newtonian interaction H has the simple form 3.6 M being the symmetric matrix defined in 3.7. As already pointed out in Remark 3. the matrix M can be diagonalized with eigenvalues which thanks to our assumptions and to Proposition 3.2 have the form in 3.29 and therefore satisfy inf Ω j > c inf Ω i Ω 4.2 j > c I I for any i j =...N and for a suitable positive constant c independent of ε. If U := UΛ is the symmetric matrix which diagonalizes M U T MU = diag Ω... Ω N then the map 4.3 Ξ:I ϕ p q Λληξ is symplectic and real-analytic and 4.4 H 2 Ξ= 2 where N Ω i Ip 2 i + qi 2. i= p = U T η q = U T ξ I =Λ ϕ = λ + hkl Λ U kl U hl η k ξ l Thus the N + -body problem Hamiltonian H in 2.4 in the case we are considering can be written as 4.5 with h := H 0 H ΞI ϕ p q; ε =hi+fi ϕ p q; ε f := f 0 I+ 2 f := εf I p q; ε+εf 2 I ϕ p q; ε N Ω i Ip 2 i + qi 2 + f I p q; ε i= f 0 := H 0 f := H Ξ f 2 := H Ξ. Here h is uniformly strictly concave f const p q 4 and f 2 dϕ =0. T N The construction of elliptic invariant tori for the Hamiltonian 4.5 is based on four steps which we proceed to describe.

17 576 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI 4.. Averaging. Fix τ>n and pick two numbers b b 2 such that 0 <b < <b 2 < 2 b τ +. 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+εf which possesses an elliptic equilibrium at p = q = 0. As we now proceed to describe one may 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. of [BCV03] or Proposition 7. of [BBV04]. Denote by DR n the complex n-ball of center zero and radius R>0 and for any V R N denote by V R the complex neighborhood of radius R>0 of the set V given by V R := x V D R x. Next define the set Î as the following Diophantine subset of I: { Î := I I : I hi k γ } 4.7 k τ k ZN \{0} with γ := const ε b. Notice that as it is standard to prove 4.8 meas I \ Î const γ = const ε b. The Hamiltonian h + f in 4.5 is real-analytic on the complex domain 4.9 with 4.0 D rsρ := Îr T N s D 2N ρ C 4N r := const ε s := const ρ := const ε b2. The definition of Î is motivated by the necessity to have an estimate on small divisors. In fact let I Îr and ε small enough and let I 0 Î be a point at distance less than r from I. Then for any k Z N \{0} such that k K := const ε b2 by the second relation in 4.6 by 4.7 and by Cauchy estimates one finds 4. I hi k I hi 0 k I hi 0 I hi k γ K τ max 2 I h rk γ 2K τ =: α = const εb+τb2 0 < k K := const ε b2. In order to apply averaging theory see e.g. [N77] 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. 0 in [BCV03] f rsρ const α min{rs ρ2 } K where the norm rsρ is defined as the standard sup-fourier norm f rsρ := 4.2 sup f k I p q e k s k Z N Ipq Îr D2N ρ

18 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 577 f k I p q denoting Fourier coefficients of the multiperiodic real-analytic function ϕ fi ϕ p q. Such condition in view of 4.6 can be achieved by taking ε small enough since by 4. and 4.0 one has α min{rs ρ 2 } f rsρ = Oε and = Oε b+τ+b2+/2. K Hence there exists a close-to-identity real-analytic symplectic change of variables I ϕ p q I ϕ p q verifying compare formulae 2.6 and A.7 of [BCV03] 4.3 I I const ε 2 +b2 and p p q q const ε and such that the Hamiltonian expressed in the new symplectic variables becomes 4.4 hi +ĝi p q + ˆfI ϕ p q ĝ := εf I p q +ε ˆf I p q with ˆf and ˆf real-analytic on the complex domain D r/2s/6ρ/2 and satisfying 4.5 ˆf r/2s/6ρ/2 ε αr = const εb2+b3 with b 3 := 2 b τ +b 2 > 0 ˆf r/2s/6ρ/2 const e const K const ε New elliptic equilibrium. Due to the small term ˆf in 4.4 zero is no longer an elliptic equilibrium for the averaged i.e. ϕ-independent Hamiltonian h +ĝ. Using the implicit function theorem we can find a new elliptic equilibrium for h +ĝ which is ε b2+b3 close to zero. Hence we construct a real-analytic symplectic transformation 4.6 J ψ v u I ϕ p q with I = J and ε b2+b3 -close-to-the-identity such that in the new symplectic variables J ψ v u the Hamiltonian takes the form hj + gj v u + fj ψ v u with g having v = u = 0 as elliptic equilibrium; the functions g and f are realanalytic on a slightly smaller complex domain say D r/7s/7ρ/7 where they satisfy bounds similar to those in 4.5. Furthermore for j =...N the eigenvalues Ω j J of the symplectic quadratic part of g are purely imaginary and ε +b2+b3 -close to ε Ω j J Symplectic diagonalization of the quadratic term. Using a wellknown result on the symplectic diagonalization of quadratic Hamiltonians we can find a real-analytic symplectic transformation 4.7 J ψ ṽ ũ J ψ v u with J = J and ε b2+b3 -close-to-the-identity such that the quadratic part of g becomes simply N Ω i= i Jũ 2 j +ṽ2 j. Whence the new Hamiltonian becomes compare formula 2.22 of [BCV03] 4.8 H := h 0 J+ N Ω i Jũ 2 i +ṽi 2 + g 0 Jṽ ũ+ f 0 J ψ ṽ ũ i=

19 578 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI where 4.9 g 0 f 0 Ω j are real-analytic and 4.20 h 0 J :=h J+ε g J0 0 g 0 Jṽ ũ const ε ṽ ũ 3 Ω const ε f r/8s/8ρ/8 const ε 3. Finally because of inf Ω i const ε>0 inf Ω 2 Ω const ε> Applying KAM theory. We rewrite now the Hamiltonian H in 4.8 in a form suitable for applying elliptic KAM theory. Introducing translated variables y := J p and complex variables z z we define 4.22 H = H p + y ψ z + l z z z 2 i 2 here p is regarded as a parameter and the symplectic form is N j= dy j dψ j + i N j= dz j d z j with i :=. The Hamiltonian H is then seen to have the form with 4.23 N = e + ω y + H = N + P N Ω j z j z j e := h 0 p ω := Jh 0 p Ω:= Ωp j= and P a perturbation which can naturally be split into four terms: P = with 4.24 k 4 P k P = h 0 p + y h 0 p Jh 0 p y y 2 n P 2 = Ωj p + y Ω j p z j z j y z z j= P 3 = g 0 p + y z + z z z 2 P 4 = f 0 p + y ψ z + z i ε z + z 3 by z z 2 i = Oε 3. 2 The parameter p runs over the Diophantine set Î defined in 4.7. Notice that the integrable Hamiltonian N affords for any given value of the parameter p the N- dimensional elliptic torus 4.25 {y =0} T N {z = z =0}

20 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 579 which is invariant for the Hamiltonian flow generated by N the flow being simply the Diophantine translation x x + ωt with ω as in Since det 2 Jh 0 0 we can use the frequencies ω as parameters rather than the actions p. We therefore set Î { } 4.26 O := Jh 0 = ω = Jh 0 p : p Î. Notice that by 4.7 we have 4.27 meas Jh 0 I \O const ε b. Now if we put p = pω := Jh 0 ω in 4.22 we can rewrite the N + -body Hamiltonian in the form 4.28 where 4.29 N y z z; ω :=eω+ω y + Hy ψ z z; ω :=N y z z; ω+p y ψ z z; ω N Ω j ωz j z j eω :=h 0 pω Ωω := Ωpω j= and the perturbation P y ψ z z; ω is obtained by replacing p with pω in Recalling 4.0 the Hamiltonian H in 4.28 is real-analytic in 4.30 with 4.3 y ψ z z; ω D r 2 srd := D N r 2 TN s D 2N r O N d r := const ε 3/4 s := const d := const ε. We recall now a well-known KAM result concerning the persistence of lowerdimensional elliptic tori for nearly integrable Hamiltonian systems see [M65] [E88] [K88]. The version we present here is essentially a reformulation of Pöschel s theorem in [P89] compare also with Theorem 5. of [BBV04]. Theorem 4.. Let H have the form in and let it be real-analytic on a domain D r 2 srd of the form 4.30 for some r s and d positive. Assume that 4.32 sup ω Ωω ω O d 4 and that the nonresonance or Melnikov condition 4.33 Ωω k γ 0 k 2 k Z N ω O is satisfied for some γ 0 > 0. Then if d γ 0 and P is sufficiently small i.e P rsd := sup ω O d P ; ω r 2 sr const γ 0 r 2 then there exist a normal form N := e ω +ω y +Ω ωz z a Cantor set Oγ 0 Owith 4.35 meas O\Oγ 0 const γ 0

21 580 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI and a transformation F : Dr N 2 /4 TN s/2 D2N r/2 Oγ 0 Dr N 2 TN s Dr 2N O d y ψ z z ; ω y ψ z z; ω real-analytic and symplectic for each ω and Whitney smooth in ω such that 4.36 H F = N + R with j y h z k z R =0 if 2 j + h + k 2. In particular for each ω Oγ 0 the torus {y =0} T N {z = z =0} is an N-dimensional linearly elliptic invariant torus run by the flow ψ ψ + ωt. Finally 4.37 y y r z z r z z const P rsd γ 0. In this section we have shown that the many-body Hamiltonian 2.4 under the hypotheses spelled out at the beginning of the section has indeed the form assumed in the KAM theorem Theorem 4.. Furthermore by 4.2 the elliptic frequencies Ω i verify the Melnikov conditions 4.33 with 4.38 γ 0 = const ε and by 4.24 and 4.3 the perturbation P verifies for small ε the KAM condition 4.34 since 4.39 P rsd = Or 4 + εr 3 + ε 3 =Oε 3 const γ 0 r 2 = Oε 5/2. Thus the existence of the desired quasi-periodic orbits follows at once from Theorem 4.. We may summarize the final result as follows. Theorem 4.2. Let N 2 and let H be the N +-body problem Hamiltonian in Poincaré variables defined in 2.4. Let m 0 < μ j < 4m 0 let θ A δ and ε 0 be as in Remark 3.2ii. Fix 0 <δ<δ and let I = a A where a is the semimajor axis map defined in 4.. Let τ > N and pick b b 2 as in 4.6. Finally let 0 <ε <ε 0 be such that 4.39 holds for any ε ε and such that all conditions on ε required for constructing the symplectic transformations introduced in sections are satisfied for ε<ε. Then for any ε<ε there exist a Cantor set I I with 4.40 meas I \ I const ε b and a Lipschitz continuous family of tori embedding φ :ϑ p T N I Λϑ; p λϑ; p ηϑ; p ξϑ; p I T N B 2N ρ with ρ := const ε b2 such that for any p I φt N ; 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 γτ-diophantine with γ = Oε b. Furthermore the following bounds hold uniformly on T N I : Λϑ; p p const ε 2 +b2 ηϑ; p + ξϑ; p const ε b2.

22 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 58 Theorem. follows now by taking recall the definitions of b k in c := b c 2 := b 2 c 3 := b In particular the statements on the density of the set of the osculating major semiaxes on the bound on the osculating eccentricities and on the variation of the osculating major semiaxes follows from and 4.4. Appendix A. Poincaré variables for the planar N + -body problem. We briefly recall in this appendix the classical derivation of the Poincaré variables for the planar N-body problem 3 showing in particular the validity of Lemma 2. which is proven in subsections A. and A.2; subsections A.3 and A.4 are included for completeness. A.. Canonical variables for the two-body problem. Consider two bodies P 0 P of masses m 0 m and position u 0 u R 2 with respect to an inertial frame. We assume that P 0 and P interact through gravity with gravitational constant. By Newton s laws the equations of motion for such two-body problem are Let ü 0 = m u u 0 u u 0 3 ü = m 0 u 0 u u 0 u 3. M := m 0 + m m := m 0m A. M x := u u 0 X := mẋ. Then the above equations of motion become ẍ = Mx x 3 and the motion of the two bodies is governed by the Hamiltonian A.2 KX x = 2m X 2 mm x with X x R 2 R 2 conjugate variables; i.e. the equations of motion are ẋ = X K Ẋ = x K. As well known such system is integrable and for K < 0 the orbits are ellipses. More precisely one has the following proposition. m Proposition A.. Fix Λ > 0 > K 0 and let Λ + := 3 M 2 2 2K 0 > Λ. Then there exist ˆρ >0 and a real-analytic symplectic transformation } Ψ DP : Ληλ ξ [Λ Λ + ] [ ˆρ ˆρ] T [ ˆρ ˆρ] X x { x ˆρ2 m 2 M casting A.2 into the integrable Hamiltonian m 3 M 2 /2Λ 2. 3 For a review of the Poincaré variables in the nonplanar case see for instance [Ch88] and [BCV03].

23 582 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI This classical proposition is a planar version of the classical one due to Poincaré see [Poi905 Chapter III] and the variables Ληλξ are usually called planar Poincaré variables. The proof of Proposition A. is particularly interesting from the physical point of view and rests upon the introduction of three different famous changes of variables which we now proceed to describe briefly. Let l and g denote respectively the mean anomaly and the argument of the perihelion. Step. The system is set in symplectic polar variables; namely we consider the symplectic map Ψ spc :R Φ r ϕ X x where r>0and ϕ T given by A.3 { x = r cos ϕ Ψ spc : x 2 = r sin ϕ cos ϕ X = sin ϕ sin ϕ r R cos ϕ Φ r and consider the new Hamiltonian K spc := K Ψ spc. Step 2. There is a symplectic map Ψ D :L G l g R Φ r ϕ that integrates the system: Ψ D is obtained via the generating function A.4 SL G r ϕ = m4 M 2 L 2 + 2m2 M r G2 r 2 dr + Gϕ. The variables L G l g are known as planar Delaunay variables. In such variables the new Hamiltonian becomes K D := K spc Ψ D = m3 M 2 2L 2. Also if C is the angular momentum of the planet and a is the major semiaxis by construction one has that G = C and L = m Ma. Step 3. We need now to remove singularities which appear for small eccentricity. To this aim we first introduce planar Poincaré action-angle variables by means of the linear symplectic transformation given by A.5 Ψ Paa : Ψ Paa : ΛH λ h L G l g { Λ=L H = L G λ = l + g h = g. Then we let Ψ P : Λη λ ξ ΛH λ h be the symplectic map defined by 2H cos h = η 2H sin h = ξ. A.6 As Poincaré showed see [Poi905] [Ch88] [BCV03] the symplectic map Ψ DP : Ληλ ξ X x with A.7 Ψ DP := Ψ spc Ψ D Ψ Paa Ψ P

24 ELLIPTIC TORI FOR THE N + -BODY PROBLEM 583 is real-analytic in a complex neighborhood of Λ [Λ Λ + ] η ξ const Λ λ T. Also the two-body Hamiltonian in Poincaré variables is K Ψ= m3 M 2 2Λ 2. Remark A.. i If we denote X x =Φ DP Ληp λ ξ q then X = m4 M 2 A.8 x Λ 3 λ. Indeed from the Hamilton equations one sees that: λ = Λ m3 M 2 2Λ = m3 M 2 2 Λ and 3 Λ = ξ = η =ṗ= q = 0. Thus by the chain rule X = mẋ = m λ x λ = m4 M 2 proving A.8. 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 semiaxis. Then by construction one sees that A.9 Λ=m Ma ξ2 + η 2 = Λ e + Oe 2. Also if C is the angular momentum of the system one infers that A.0 C =Λ e 2 = Λ + Oe 2. iii A proof of the analyticity of Poincaré variables will also follow by directly inspecting the formulae given in Lemma 2. which is proved in the coming section. A.2. Orbital elements. We now sketch a way to explicitly represent some quantities in terms of Poincaré variables. This will also lead to the proof of Lemma 2.. Let u and v denote the eccentric anomaly and the true anomaly respectively. By geometric considerations A. and 4 A.2 where A.3 Also by A.6 u = l + e sin u cos v = cos u e e cos u l = λ + h. H = η2 + ξ 2 A.4. 2 An explicit expression taking into account H the eccentricity and the major semiaxis is given by A.5 A.6 H =Λ e 2 =Λ e2 2 + Oe2 H eh Λ = 2 H. Λ Λ 4 Such relations are classical and we refer the reader to [Ch88] and [BCV03] for a geometric interpretation of these anomalies. Λ 3 x λ

25 584 LUCA BIASCO LUIGI CHIERCHIA AND ENRICO VALDINOCI In light of A.2 A.7 By means of A. we have sin v = e2 sin u e cos u. u l = e sinu l + l =e cos l sinu l+e sin l cosu l. Thus in the notation of Lemma 2. if G 0 is implicitly defined by G 0 x y =x sin G 0 x y+y cos G 0 x y with G = 0 we have that G 0 is real-analytic G 0 x y =y + xy + O 3 x y and A.8 u l = G 0 e cos l e sin l. Therefore we deduce from A.8 and A.3 that A.9 Moreover denoting A.20 u = λ + h + G 0 e cos h cos λ e sin h sin λ e sin h cos λ + e cos h sin λ. ˆη = η/ Λ ˆξ = ξ/ Λ we deduce from A.6 and A.6 that 2H A.2 e sin h = Λ H 2Λ sin h = ˆξF ˆη 2 + ˆξ 2 where F t = t/4 is real-analytic for t < 4 and agrees with the one introduced in Lemma 2.. Analogously A.22 e cos h =ˆηF ˆη 2 + ˆξ 2. Therefore substituting A.2 and A.22 in A.9 we can write G 0 as an analytic expression of ˆη ˆξλ: more formally there exists a real-analytic ˆη ˆξλ Gˆη ˆξλ which agrees with the one introduced in 2.6 by A.2 and A.22 so that G 0 e cos h cos λ e sin h sin λ e sin h cos λ e cos h sin λ =Gˆη ˆξλ. Hence from A.9 A.23 e cos u = e cos h cosλ + G e sin h sinλ + G e sin u = e sin h cosλ + G+ecos h sinλ + G with G = Gˆη ˆξλ. Notice also that from the formulae in A.6 and A.4 e 2 e 2 = F 2 ˆη 2 + ˆξ 2 for a suitable real-analytic function F 2 actually F 2 t = 2 t 4 which agrees with the notation in Lemma 2.. Thus if we set ϕ = λ + v l = v h recalling also

26 A.2 and A.7 we have ELLIPTIC TORI FOR THE N + -BODY PROBLEM 585 sin ϕ = sin v cos h cos v sin h = e cos u [ e 2 sin u cos h cos u sin h + e sin h] = e cos u [sinu h+esin h F 2 e sin u e cos h] A.24 = e cos u [sinλ + e sin u+esin h F 2 e sin u e cos h] for F 2 = F 2 ˆη 2 + ˆξ 2 and analogously A.25 cos ϕ = e cos u [cosλ + e sin u e cos h F 2 e sin u e sin h]. Hence from A.2 A.22 A.23 A.24 and A.25 it follows that sin ϕ and cos ϕ are real-analytic functions in λ ˆη ˆξ for λ T and small ˆξ ˆη. In particular if C S and E s are as defined in Lemma 2. we deduce from A.23 A.2 and A.22 that A.26 and then from A.25 and A.24 that A.27 e sin u = E s e cos u cos ϕ = C and e cos u sin ϕ = S. Finally by geometric considerations we have A.28 r = a e cos u where r is the distance between the planet and the sun. Thus the formulae in Lemma 2. follow at once by A.26 A.27 A.3 and A.8. A.3. Hamiltonian setting for the planar many-body problem. Consider N + bodies P 0...P N of masses m 0... m N all lying in the same plane interacting through gravity with constant of gravitation. Denote by u i the position of P i in a given inertial frame of R 2 with origin in the center of mass of the system. By Newton s laws we have that A.29 ü i = 0 j i N m j u j u i u j u i 3. Thus if U i := m i u i denotes the momentum of P i we see that the equations of motion A.29 come from the Hamiltonian N i=0 2m i U i 2 0 i<j N m i m j u i u j where U = U 0...U N R 2N+ and u = u 0...u N R 2N+ are conjugate symplectic variables. We now consider P 0 as the sun and introduce canonical heliocentric variables via the linear symplectic transformation A.30 u 0 = r 0 u i = r 0 + r i U 0 = R 0 R R N U i = R i for i =...N.