Quantitative kam Theory, with an Application to the Three-Body Problem

Based on quantitative “kam theory”, we state and prove two theorems about the continuation of maximal and whiskered quasi-periodic motions to slightly perturbed systems exhibiting proper degeneracy. Next, we apply such results to prove that, in the three-body problem, there is a small set in phase space where it is possible to detect both such families of tori. We also estimate the density of such motions in proper ambient spaces. Up to our knowledge, this is the first proof of co-existence of stable and whiskered tori in a physical system.

1 Some sets of canonical coordinates for many-body problems 1.1 (1+n)-body problem, Delaunay-Poincaré coordinates and Arnold's theorem In the masterpiece [1], a young a brilliant mathematician, named Vladimir Igorevich Arnold, stated, and partly proved, the following result.
Theorem 1.1 "Theorem of stability of planetary motions", [1, Chapter III, p. 125] For the majority of initial conditions under which the instantaneous orbits of the planets are close to circles lying in a single plane, perturbation of the planets on one another produces, in the course of an infinite interval of time, little change on these orbits provided the masses of the planets are sufficiently small.[...] In particular [...] in the n-body problem there exists a set of initial conditions having a positive Lebesgue measure and such that, if the initial positions and velocities of the bodies belong to this set, the distances of the bodies from each other will remain perpetually bounded.
Let us summarize the main ideas behind the statement above.
After the symplectic reduction of the linear momentum, the (1 + n)-body problem with masses m 0 , m 1 , . .., m n is governed by the 3n-degrees of freedom Hamiltonian (see Appendix A) where x i represent the difference between the position of the i th planet and the mass m 0 , y i are the associated symplectic momenta, x • y = 1≤i≤3 x i y i and |x| := (x • x) 1/2 denote, respectively, the standard inner product in R 3 and the Euclidean norm; The phase space is the "collisionless" domain of R 3n × R 3n (y, x) = (y 1 , . . ., y n ), (x 1 , . . ., x n ) s.t.0 = x i = x j , ∀ i = j , endowed with the standard symplectic form where y ij , x ij denote the j th component of y i , x i .The planetary case is when m 1 , . .., m n are of the same order, and much smaller that m 0 .In such a case, letting m i → µm i , y i → µy i , with 0 < µ 1, one obtains with Consider the two-body Hamiltonians Assume that h i (y i , x i ) < 0 so that the Hamiltonian flow φ t hi evolves on a Keplerian ellipse E i and assume that the eccentricity e i ∈ (0, 1).Let a i , P i denote, respectively, the semimajor axis and the perihelion of E i .Let C i denote the i th angular momentum C i (y j , x j ) := x i × y i .
Define the Delaunay nodes and, for u, v ∈ R 3 lying in the plane orthogonal to a vector w, let α w (u, v) denote the positively oriented angle (mod 2π) between u and v (orientation follows the "right hand rule").
Theorem 1.2 (Birkhoff ) Let H be a Hamiltonian having the form in (17)-(18).Assume that there exists ε > 0 A ⊂ R n and s ∈ N such that H is smooth on an open set M2m+2n Then there exists 0 < ε ≤ ε and a symplectic map ("Birkhoff transformation") which puts the Hamiltonian (17) into the form where the average f av b (Λ, w) := T n f b dl is in BNF of order s: P s being homogeneous polynomial in r of order s, with coefficients depending on Λ.
Theorem 1.3 ("The Fundamental Theorem", V. I. Arnold, [1]) If the Hessian matrix of h and the matrix τ (Λ) do not vanish identically, and if µ is suitably small with respect to ε, the system affords a positive measure set K µ,ε of quasi-periodic motions in phase space such that its density goes to one as ε → 0.
Remark 1.1 (Arnold, Herman) It turns out that such invariants satisfy identically the following two secular resonances Such resonances strongly violate the assumption (19) of Theorem 1.2.
We remark that the former equality in (24) is mentioned in [1], while the latter been pointed out by M. Herman in the 1990s.Note that (24) do not appear in the planar problem, because the matrix Q v , hence the ς i 's, do not exist in that case.Being aware of such difficulty, Arnold completely proved Theorem 1.1 via Theorem 2.2 in the case of the planar three-body problem, checking explicitly the non vanishing of the 2×2 torsion matrix for that case.However, in the case of the spatial problem, the question remained open until 2004, when M. Herman and J. Féjoz [8] proved Theorem 1.1 via a completely different strategy, which does need Birkhoff normal form.We refer to [6] for more details.

The rotational degeneracy
In [1], Arnold wrote -without giving the details -that the former resonance in (24) was to be ascribed to the conservation of the total angular momentum of the system: An argument which clearly shows this goes as follows.Using Poincaré coordinates, the planets' angular momenta have the expressions In particular, the two former components of the total angular momentum (25) are given by On the other hand, it is possible to find a canonical transformation (Λ, λ, η, p, ξ, q) → (Λ, λ, η, p, ξ, q) (27) having the form (16) with ρ h = id and ρ v ∈ SO(n) chosen such in a way that the last raw of where fixes the Euclidean norm of (28) to 1.With such choice, we have Λ j p j and, similarly, Therefore, (26) become Now, as the projection of the transformation (27) on λ's is a λ-independent translation, the averaged perturbing function using the new coordinates can be obtained applying such transformation to the function in (15).We denote it as which hold because they are true for f , and C is λ-independent.Using (29), it is immediate to see that (30) imply that the quadratic form is independent of pn , qn .Hence, the n th raw and column of Qv (Λ) vanish identically.This implies that Qv (Λ), hence Q v (Λ), has an identically vanishing eigenvalue, which is ς n (Λ) in (24).

Jacobi reduction of the nodes
In the case n = 2, Arnold in [1] suggested to get rid of the rotation invariance (described in the previous section) by means of the classical so-called Jacobi reduction of the nodes.This is a classical procedure with a remarkable geometric meaning, which goes as follows.Let us consider a reference frame (i, j, k) whose third axis k is along the direction of the total angular momentum C = C 1 + C 2 , while i coincides with the intersection of the planes orthogonal to C 1 , C 2 .Such intersection is well defined provided that C 1 C 2 , namely, when the problem is not planar.With such a choice of the reference frame, one cannot fix Delaunay coordinates completely freely.Indeed, by the choice of i, we have that the ζ j satisfy Moreover, a geometrical analysis of the triangle formed by C 1 , C 2 and C shows that the coordinates Z j satisfy Figure 3: The construction underlying Jacobi reduction of the nodes. where is the Euclidean norm of C. As i moves, the following fact is not obvious at all -in fact proved by R. Radau.
Theorem 1.4 (R. Radau, 1868, [16]) Replacing relations (31)-(32) inside the Hamiltonian (1) with n = 2 written in Delaunay coordinates, one obtains a function, depending on (Λ j , j , G j , g j ) (j = 1, 2) and G, whose Hamilton equations relatively to (Λ j , j , G j , g j ) generate the motions of the coordinates (Λ j , j , G j , g j ) referred to the rotating frame under the action of the Hamiltonian (1) with n = 2.The motion of Z j and ζ j can be recovered via (31)-(32).

Deprit coordinates
Arnold commented on the general problem of rotational degeneracy as follows: [1, Chap.III, §5, n. 5] In the case of more than three bodies [n > 2] there is no such [analogue to Jacobi reduction of the nodes] elegant method of reducing the number of degrees of freedom However, exactly 20 years later, in 1983, A. Deprit [7] discovered a set of canonical coordinates which, after a simple transformation, do the desired job and reduce to Jacobi's when n = 2. Let us describe them.Consider the "partial angular momenta" with C i as in (7).Notice that S n = C is the total angular momentum of the system.Define the "Deprit nodes" with are defined as follows (compare also Figures 4,5 and 6): We have Theorem 1.5 (A.Deprit, 1983, [7]) For later need, we formulate an equivalen statement of Theorem 1.5.We consider the coordinates with where Z i , G i , ζ i , are as in (10), R i , r i are as in (37), and, finally, Figure 5: The frames D i+1 and the coordinates Ψ i , Ψ i−1 , G i+1 and ψ i .
The frames H i and the coordinates g i , G i , i .and The proof of the following fact is left to the reader.
Indeed, we have where i * j is the convex angle formed by k and C j and, finally, verify, as well known, Then, by Lemma 1.1, (42) and as C j * • i = 0, we have We denote as φ where S j+1 , ν j , ν j at the right hand sides are to be written as functions of D e (see (34) and ( 10)): As the right hand sides are defined only in terms of C j , so they are functions of Z, ζ and G, while are independent of R and r.Theorem 1.6 Theorem 1.5 is equivalent to stress that Proof Use Lemma 1.2 and that the coordinates (R, r) are shared by D ep and D e .
Base step We prove the statement 1.5 with n = 2.We first observe that, in such case, (y j , x j ) are expressed, through (R, Ψ, G, r, ψ, ϕ) via the formulae where i is the convex2 angle formed by k and C; i j is the convex angle formed by C and C j and, finally, x j pl , y j pl are as in (41), with φ j replaced by ϕ j .Using Lemma 1.1 twice, one easily finds We have used Cj and we have let Taking the sum of (45) with j = 1, 2 and using (42) and recognizing that we have the proof.
Induction The inductive step is made on the statement (44).The map φ Dep D e in (44) will be named φ n .We assume that (44) holds for a given n ≥ 2 and prove it for n + 1.Consider the map where the tilded arguments have dimension n, we let and then having split We moreover define a map φ * ,n+1 on (Ψ * , G * , ψ * , ϕ * ) acting as on the designed variables, and as the identity on the remaining ones.Note that the arguments at left hand side have dimension 2, that G * = ( Ψ n−1 , G n+1 ), and put ϕ * = (ϕ * ,1 , ϕ * ,2 ).Again by the inductive assumption, we have Let us now look at the composition It acts as and, by ( 47) and (48), verifies It is not difficult to recognize -using (43) -that the map (49) coincides with φ n+1 .For the details, we refer to [11,3].
The Deprit map In this section, we provide the explicit expression of the map The discussion in the previous section shows that each orbital frame H i , i = 1, . .., n, can be reached via a sequence of transformations which overlap the D n+1 := (i, j, k) to H i through the following diagram (named tree by Deprit): In turn, , are described by the sequence of rotations (see figure 5); , are related by the sequence of rotations Then we find that (50) has the expression ) and y * i , x * i as in (41).
are defined as follows.Let S j be as in (33).Define the K-nodes and then the Kcoordinates as follows.
Remark 1.2 Note that the node νn coincides with ν = ν n+1 in (34); the coordinates Z and ζ are the same as in (37) and, finally, the coordinates χ coincide with the coordinates Ψ in (37).In particular, D ep and K share the construction in Figure 4.The geometrical meaning of the other K-coordinates is pointed out in the next section.
A chain of reference frames We consider the following chain of vectors where νj , nj are the K-nodes in (51), given by the skew-product of the two consecutive vectors in the chain.We associate to this chain of vectors the following chain of frames where Ĝn+1 = (i, j, k) is the initial prefixed frame and the frames, while Fj , Ĝj are frames defined via By construction, each frame in the chain has its first axis coinciding with the intersection of horizontal plane with the horizontal plane of the previous frame (hence, in particular, νj ⊥ S j and nj ⊥ x j ).

Explicit expression of the K-map
We now derive the explicit formulae of the map which relates the coordinates (52) to the coordinates (y 1 , . . ., y n , x 1 , . . ., x n ).We shall prove that such map has the expression where where Tj , Ŝj have the expressions Indeed, Tj is the rotation matrix which describes the change of coordinates from Ĝj+1 to Fj , while Ŝj describes the change of coordinates from Fj to Ĝj , as it follows from the definitions of ( Θ, χ, θ, κ) in (52) (see also Figures 7,8 and 9).The formulae (56)-( 59) are obtained considering the following sequence of transformations connecting Ĝn to any other frame in the chain.From this, and the definitions of the frames (55), one finds and finally Collecting such formulae, one finds (56)-(59).
Canonical character of K Lemma 1.3 K preserves the standard Liouville 1-form: The proof of Lemma 1.3 again relies in Lemma 1.1.
Proof We use the expression in (56).We also define Applying Lemma 1.1 twice, we get Continuing in this way, after n − j + 1 iterates we arrive at We take the sum of (61) with j = 1, . .., n.Exchanging the sums and recognizing that with χ0 := Θ1 and that, by (57), the last term in (61) is In the following section, we shall use the following byproduct of Lemma 1.3.Recall the coordinates D e in (38) and denote Consider the family of projections which, as it is immediate to see, is independent of r and R.
Lemma 1. 4 The projections (62) verify The reduction of perihelia P The P-coordinates have been described in [14].Here, as in the case of K, we change4 notations a little bit and denote them as Figure 10: The references F j and the P-coordinates κ j−1 , j = 2, . .., n.
where Λ, are as in (10), while are defined as follows.Consider a phase space where the Kepler Hamiltonians (6) take negative values.Let S j be as in (33) and P j the perihelia of the instantaneous ellipses generated by ( 6), assuming they are not circles.The coordinates Λ, are the same as in Delaunay, while, roughly, (Θ, χ, ϑ, κ) in (63) are defined as the ( Θ, χ, θ, κ) of K, "replacing x j with P j " (see Figures 10,11,12).Exact definitions are below.Define the P-nodes Then the P-coordinates are To prove that (63) are canonical, we consider the map which is independent of Λ, (even though this will not be used).
Lemma 1.5 φP D e ,aa coincides with the map φK D e in (62).Combining Lemmas 1.4 and 1.5, we have Lemma 1.6 The map Explicit expression of the P-map We now provide the explicit formulae of the map which relates the coordinates (65) to the coordinates (y 1 , . . ., y n , x 1 , . . ., x n ).We shall prove that such map has the expression where where T j , S j have the expressions and Qj = Cj a j as in (10), n j = Mj a 3 j the mean motion, and ξ j the eccentric anomaly, solving ξ j − e j sin ξ j = j .
These formulae are easily obtained using the well-known relations x j = a j (cos ξ j − e j )P j + 1 − e 2 j sin ξ j Q j y j := µ j n j a j 1 − e j sin ξ j − sin ξ j P j + 1 − e 2 j cos ξ j Q j with P j the j th perihelion and Q j = Cj Cj × P j , and the relations which relate C j , P j , Q j to P, which, similarly to how done for K, are: 1.7 The behavior of K and P under reflections The maps K and P have a nice behavior under reflections, which turns to be useful if they are applied to Hamiltonians which are reflection-invariant.We denote as the vector obtained from x = (x 1 , x 2 , x 3 ) by reflecting its second coordinate, and as the simultaneous reflection of the second coordinate of all the y j and all the x j in the system of Cartesian coordinates (y, x) = (y 1 , . . ., y n ), (x 1 , . . ., x n ) .We aim to show that Similarly, using P, it is obtained by changing Proof We prove for K.We write (70) as Now use the formulae in (56)-(59) and that acts on the functions in (59) as The proof for P is similar.
Lemma 1.7 reflects on the Hamiltonian (1) as well as in all Hamiltonians which are R − 2 -invariant as follows.

Arnold's Theorem
Here we retrace the main ideas of the proof of Theorem 1.1 given in [5].Such proof uses on the coordinates (35).The first step is to switch from the coordinates (35) to a new set of coordinates which are well fitted with the close-to-be-integrable form of the Hamiltonian (4).Then we modify the coordinates (35) to the following form which we call action-angle Deprit coordinates, where which integrate Kepler Hamiltonian ( 6).This step is necessary to carry the integrable part in (4) to the form Recall that the new angles γ i provide the direction of the perihelion of the instantaneous ellipse generated by ( 6), however they have a different meaning compared to the analogous angles g i appearing in the set of Delaunay coordinates (10), as, by construction, the γ i 's are measured relatively to the nodes ν i in (34) (because the ϕ i were), while the angles g i in the Delaunay set are measured relatively to ni in (8).The 3n − 2 degrees of freedom Hamiltonian which is obtained is still singular.Singularities appear when the coordinates are not defined and in correspondence of collisions among the planets.The latter case will be later excluded through a careful choice of the reference frame.The singularities of the coordinates appear when the some of the convex angles (Deprit inclinations) i * j := (S j , S j+1 ) j = 1 , . . ., n , S n+1 := k (72) take the values 0 or π, because in such situations the angle ψ j is not defined (see Figures 4, 5, 6) and when the instantaneous orbits of some of the Kepler Hamiltonians ( 6) is a circle, because in that case, the corresponding γ i is not defined.Such singularities are important from the physical point of view, because the eccentricities and the inclinations of the planets of the solar system are very small, hence the system is in a configuration pretty close to the singularity.To deal with this situation, a regularization similar to the Poincaré regularization (11) of Delaunay coordinates has been introduced in [5].Note that, in principle, there are 2 n singular configurations (corresponding to any choice of i * j ∈ {0 , π}, besides e j = 0 for some j).Here we discuss the case i j = 0 for some j.Another regularization will be discussed in Section 2.3.

rps coordinates and Birkhoff normal form
The rps variables are given by (Λ, λ, z) := (Λ, λ, η, ξ, p, q) with (again) the Λ's as in (10) and Let φ rps denote the map Remark 2.1 The coordinates (73) have been constructed as follows.First of all, we look for a linear and canonical transformation which replaces Ψ i , Γ i , Λ i with with the conventions in (74).To find the coordinates α i , β i , λ i respectively conjugated to I i , J i , Λ i we impose the conservation of the standard 1-form: with β 0 := 0, β n+1 := 0. This provides the following relations These equations may be solved recursively, and give Note that λ i , α i , β i are in fact angles as the linear combinations at right hand sides of (76) have integer coefficients.As a second step, one defines and obtains (73) .The transformations (77) are well known to be canonical.
The main point is that where the eccentricities e j and and the angles i * j in (72) are allowed to be zero.In particular, • e j = 0 corresponds to the rps coordinates η j = 0 = ξ j ; • i * j = 0 corresponds to the the rps coordinates p j = 0 = q j .
From the definitions (73)-(74) it follows that the variables are integrals (as they are defined only in terms of the integral C), hence, cyclic for the Hamiltonian (4).Therefore, if H rps denotes the planetary Hamiltonian expressed in rps variables, we have that where H is as in ( 4) and φ rps as in (75) has 3n − 1 degrees of freedom, as it depends on Λ, λ, z, where z = (η, p, ξ, q) with p = (p 1 , . . ., p n−1 ) We denote as a i = 1

Mi
Λi µi 2 the semi-major axis associated to Λ i .The next result solves the problem of the construction of the Birkhoff normal form for the Hamiltonian (4), mentioned in Section 1.1.

Theorem 2.1 ([5, 4]) For any s ∈ N there exists an open set
, a positive number ε and a symplectic map ("Birkhoff transformation") which carries the Hamiltonian (79) into where the average f av b (Λ, w) := T n f b dl is in BNF of order s: P s being homogeneous polynomial in r of order s, parameterized by Λ.Furthermore, the normal form (81)-( 82) is non-degenerate, in the sense that, if s ≥ 4, the (2n − 1) × (2n − 1) matrix τ (Λ) of the coefficients of the monomial with degree 2 in P s (r) is non singular, for all Λ ∈ A.
Denote by B ε = B 2n2 ε = {y ∈ R 2n2 : |y| < ε} the 2n 2 -ball of radius ε and let The second ingredient is a KAM theorem for properly-degenerate Hamiltonian systems.This has been stated and proved (with a proof of about 100 pages) by Arnold in [1], who named it the Fundamental Theorem.Here we present a refined version appeared in [2].
Then, there exist positive numbers ε * , µ * , C * and b such that, for one can find a set T ⊂ P formed by the union of H-invariant (n 1 + n 2 )-dimensional tori, on which the H-motion is analytically conjugated to linear Diophantine quasi-periodic motions.The set T is of positive Liouville-Lebesgue measure and satisfies An application of Theorem 2.2 with n 0 = n, n 1 = 2n − 1 to the system in (81) with s = 4 now leads to the proof of Theorem 1.1.

Global Kolmogorov tori
The quasi-periodic motions of Theorem 1.1 provide almost circular and almost planar orbits.This is because the normal form of Theorem 2.1 is constructed around the strip M 6n−2 0 , and the origin corresponds to zero eccentricities and zero mutual inclinations.The question whether similar motions may exist outside such regime is therefore natural and important from the physical point of view.To this end, one has to understand that the Birkhoff normal form (assumption (A2) of Theorem 2.2) is used in the proof only to construct a reasonable integrable approximation for the whole Hamiltonian, in fact given by Therefore, a possible construction of full dimensional quasi-periodic motions outside the small eccentricities and small inclinations regime should start from a different integrable approximation.
In this section we describe an approach in such direction, where we look at the first terms of the series expansion of the -averaged f with respect to a small parameter.The small parameter will be taken to be the inverse distance between the planets (the idea goes back to S. Harrington [9]).In addition, the use of the coordinates P will allow to construct (3n − 2)-dimensional quasiperiodic motions without singularities when the inclinations become zero.Recall that the tori of Theorem 1.1 may be reduced to (3n − 2) frequencies (as shown in [5]), in a almost co-planar, co-centric configuration, but away from it, due to singularities.
Here we discuss the following result.
Theorem 2.3 (Global Kolmogorov tori in the planetary problem, [14]) Fix numbers 0 < e i < e i < 0.6627 . .., i = 1, • • • , n.There exists a number N depending only on n and a number α 0 depending on e i , e i , and n such that, if α < α 0 , µ ≤ α N , in a domain of planetary motions where the semi-major axes a n < a n−1 < • • • < a 1 are spaced as follows there exists a positive measure set K µ,α , the density of which in phase space can be bounded below as consisting of quasi-periodic motions with 3n − 2 frequencies where the planets' eccentricities e i verify e i ≤ e i ≤ e i .
Let us consider a general set of coordinates C = (Λ, , u, v) which puts the Kepler Hamiltonians (6) into integrated form and hence carries the Hamiltonian (4) to where so that For such any C one always has, as a consequence of the motion equations of ( 6), the following identities with a j the semi-major axes.Consider now the average f C (Λ, u, v) in (89) with respect to .Due to the fact that y j has zero-average, one has that only the Newtonian part contributes to f C (Λ, u, v): We now consider any of the contributions to this sum and expand any such terms Then the formulae in (90) imply that the two first terms of this expansion are given by Namely, whatever is the map C that is used, the first non-trivial term is the double average of the second order term, which is given by Using Jacobi coordinates, S. Harrington noticed that depends on one only angle: the perihelion argument of the inner planet, hence is integrable.
When n = 2, Lemma 2.2 provides an effective good starting point to construct quasi-periodic motions without the constraint of small eccentricities and inclinations, because in that case one can take, as initial approximation, The motions of H Harr have indeed widely studied in the literature, after [9].When n > 2, the argument does not seem to have an immediate extension using Deprit coordinates (which, as said, are the natural extension of Jacobi reduction).The generalization of (92) for such a case is It turns out that, even looking at the nearest neighbors interactions with 1 ≤ i ≤ n − 2 depend on two angles: γ i and ψ i−1 , so the effective study of the unperturbed motions of (93) is involved.Using the P-coordinates (94) one has that the terms f i,i+1 with 1 ≤ i ≤ n − 2 depend on 3 angles: κ i−1 , ϑ i and ϑ i+1 , but the dependence upon κ i−1 and ϑ i is at a higher order term.This is shown by the following formula, discussed in [14]: where We denote as H P (X P , ) = h 0 fast (Λ) + µf P (X P , ) where the (3n − 2)-dimensional Hamiltonian (1) expressed in P-coordinates.The proof of Theorem 2.3 is based on three steps: in step 0 we compute the holomorphy domain of H P ; in the step 1 the Hamiltonian is transformed to a similar one, but with a much smaller remainder.In step 2, a well fitted KAM theory is applied.Note that, as the terms of the unperturbed part are smaller and smaller as and when the distance from the sun increases, such KAM theory will be required to take such different scales into account.
Step 0: Choice of the holomorphy domain A typical practice, in order to use perturbation theory techniques, is to extend Hamiltonians governing dynamical systems to the complex field, and then to study their holomorphy properties.
It can be proven that a domain of holomorphy for the perturbing function f P in (96), regarded as a function of complex coordinates can be chosen as , where, for given positive numbers 30 with χ −1 := 0, χ 0 := Θ 1 , and with s ∈ (0, 1) arbitrary, D i , C * i , C * i depending only on m 0 , . .., m n , a ± i as in (88).
Step 1: Normal Form Theory We call m-scale Diphantine set, and denote it as D γ1,...,γm,τ , the set of , with k j ∈ Z νj , the following inequalities hold: The set D γ1,...,γm,τ reduces to the usual diophantine set taking γ j = γ ∀ j.The first multi-scale Diophantine set was proposed by Arnold in [1] with m = 2. Proposition 2.1 Let µ j , M j be as in (2) and m j : There exists a number c, depending only on n, m 0 , • • • , m n , a ± 1 , e j , e j , and a number 0 < c < 1, depending only on n such that, for any fixed positive numbers γ < 1 < K, α > 0 verifying (101) and a real-analytic transformation where f exp (p, q, χ, Λ, κ, ) is independent of κ n−1 , and the following holds.

1.
The function h f ast,sec (p, q, χ, Λ) is a sum where, if then h f ast and h sec are given by where the functions h i sec have an analytic extension on D n and verify 2. The function f exp satisfies and, moreover, it satisfies (100), with m = 2n − 1, τ = τ > 2, and 4. The mentioned constants are Step 2: KAM theory Theorem 2.4 (Multi-scale KAM Theorem, [14]) s+s , where h(p, q, I) depends on (p, q) only via Assume that ω 0 := ∂ J(p,q,I) h is a diffeomorphism of A with non singular Hessian matrix U 1 := ∂ 2 (J(p,q,I) h and let U k denote the , the matrix with entries where log + a := max{1, log a} Then one can find two numbers ĉν > c ν depending only on ν such that, if the perturbation f is so small that the following "KAM condition" holds for any ω ∈ Ω * := ω 0 (D) ∩ D γ1,••• ,γm,τ * , one can find a unique real-analytic embedding where r := c ν Ê ρ such that T ω := φ ω (T ν ) is a real-analytic ν-dimensional H-invariant torus, on which the H-flow is analytically conjugated to ϑ → ϑ + ω t.Furthermore, the map is Lipschitz and one-to-one and the invariant set K := ω∈Ω * T ω satisfies the following measure estimate Finally, on T ν ×Ω * , the following uniform estimates hold Theorem 2.4 generalizes Theorem 3 in [2] and hence the Fundamental Theorem of [1], to which Theorem 3 in [2] is inspired.
where c is as in ( 103) and c will be fixed later.We aim to apply Theorem 2.4 to the Hamiltonian H n of Proposition 2.1, with these choices of γ and K. To this end, we take where K := max{K, K}.The number 1 γ 2 (an−1) 3 (a − n ) 3 can be bounded by 1 α N for a sufficiently large N depending only on n.Hence, if c < c N and α < c 6 , we have Ê < 1 and the theorem is proved.

On the co-existence of stable and whiskered tori
In this section we discuss how the use two different sets of coordinates may lead to prove the co-existence of stable and unstable motions.Specifically, we deal with the following situation, which we shall refer to as outer, retrograde configuration (orc): Two planets describe almost co-planar orbits, revolving around their common sun, in opposite sense.The outer planet has a lower angular momentum and retrograde motion, as seen from the total angular momentum of the system.
We aim to discuss the following Theorem 2.5 1.There exists a 8-dimensional region D s in the phase space almost completely filled with a positive measure set of five-dimensional kam tori, in orc configuration; 2. There exists a 8-dimensional region D u in the phase space including a 6-dimensional, hyperbolic invariant region D 0 u consisting of co-planar, retrograde motions for the outer planet.3. D s and D 0 u have a non-empty intersection.
Theorem 2.5 leads to the following conjecture, which is likely to be proved somewhere.
Conjecture 2.1 Full dimensional quasi-periodic motions and hyperbolic 3-dimensional tori coexist in D s .
The proof of statements 1. and 2. in Theorem 2.5 relies on the use of two different sets of coordinates for the Hamiltonian (4) with n = 2: Proof of 1.We consider the coordinates (71) with n = 2.It will turn to be useful to work with regularizing complex coordinates, which we denote as and define via the formulae We also define, for later need, η 1 , η 2 , p, ξ 1 , ξ 2 , q via Observe that M π := (Λ, λ, t, t * ) : (t, t * ) = (0, 0) (109) corresponds to co-circular, co-planar orbits for the two planets, with the outer planet in retrograde motion.
We denote as the expression of the Hamiltonian (105) using the coordinates rps C π in (106), which, similarly to the prograde case, H rps C π is independent of (T, T * ).Abusively, we shall continue calling rps C π the coordinates (106) deprived of (T, T * ).We now define a domain where letting the rps C π coordinates vary.First of all, we observe that orc configuration can be realized only if the planetary masses are tuned with the semi-major axes.More precisely, that, if we denote as "2" and "1" the inner5 , outer planet; as a 2 , a 1 , the semi-major axes of their respective instantaneous orbits around the sun; α − , α + , with 0 < α − < α + < 1, two numbers such that the semi-axes ratio α := a2 a1 verifies then the following inequality needs to be satisfied Indeed, since the motions are almost-circular, the lenghths of the angular momenta of the planets, C 1 , C 2 are arbitrarily close to the action coordinates Λ 1 , Λ 2 related to their semi-major axes, which in turn are related to the semi-axes and the mass ratio via where µ i , M i are as in (5).This inequality does not make conflict with (111) if one assumes that whence the necessity of (112).
We then fix the domain as follows.The coordinates Λ 1 , Λ 2 will be taken to vary in the set with k ± as in (113), and 0 < Λ − < Λ + to be chosen later.
The coordinates λ = (λ 1 , λ 2 ) will be taken to run in the torus T 2 .As for the coordinates (t, t ), we take a domain of the form The domain for rps C π will then be The following statement is a more precise version of statement 1. in Theorem 2.5.
Proof The eigenvalues of σ can be explicitly computed: Since tr σ = 1 Λ2 − 1 Λ1 s is real, we have to check that the discriminant is positive.Recalling that the Laplace coefficients verify (see Ref. [8] for a proof), one has and we have the assertion.
The formulae in (120)-( 121) show that, as in the prograde case, the eigenvalues of σ(Λ) and the number ς(Λ) verify, identically By analogy with the latter identity in (24), we shall refer to (124) as Herman resonance.The asymptotic values of the eigenvalues σ 1 , σ 2 and ς in the well-spaced regime (114) can be computed directly from ( 122)-( 123), or from the corresponding ones in [8,5] applying the transformation (118).In any case, the result is It shows that there is no other resonance besides Herman resonance in (124), provided the semiaxes are well spaced.Recall the definition of L in (114).
Lemma 2.4 For any K > 0, there exist Λ ± , α ± such that the triple with some N ∈ Z.
At first sight, Lemma 2.4 might seem an obstruction towards the construction of the Birkhoff normal form for the Hamiltonian (110).However, as in the prograde case, the conservation of the angular momentum lenghth is of great help.Indeed, by the commutation of f rps C π and C, it turns out that, in the Taylor expansion (119), only monomials with literal part t a t * a * verifying appear.In [4] it is shown that, because of (127), then (125) is sufficient for constructing a Birkhoff normal form (i.e., Theorem 2.1 with n = 2) for the Hamiltonian (110).Moreover, the torsion matrix (i.e., the matrix τ (Λ) defined via (83)) for this case can be computed from the analogue one from the prograde case again applying (118) to the torsion of the prograde problem.The computation is omitted (see [13] for the details), apart for stating that it is non-singular.An application of Theorem 2.2 then leads to the proof of Theorem 2.6.
Proof of 2. As a second set of coordinates, we use the P-coordinates defined in Section 1.6.
In the case n = 2, they reduce to We denote as the four-degrees-of-freedom Hamiltonian (105) written using P-coordinates, which is independent of Z, ζ and κ 2 .The manifols corresponds to retrograde motions.It is invariant as f P has an equilibrium on it and includes, in particular, the manifold M π in (109).We establish a suitable domain (including D 0 u ) for the coordinates P where H P is regular.We check below that the following domain is suited to the scope: where with L is as in (114), while c is an arbitrarily fixed number in (0, 1).We need to establish two kinds of conditions.
a) existence of the perihelia We need that the planets' eccentricities e 1 , e 2 stay strictly confined in (0, 1).Namely, that the following inequalities are satisfied: with C 2 := |C 2 |, C 2 as in (25).The expression of C 2 using P is We observe that C 2 may vanish only for (Θ 2 , ϑ 2 ) = (0, π).Since we deal with the equilibrium (128), the occurrence of this equality is automatically excluded, limiting the values of the coordinates (Θ 2 , ϑ 2 ) in the set B in (130) since in this case Moreover, the two right inequalities in (131) are satisfied taking where we have used the triangular inequality b) non-collision conditions We have to exclude possible encounters of the planets with the sun and each other.Collisions of the inner planet with the sun are excluded by (130).Indeed, using (132), whence the minimum distance of the inner planet with the sun a 2 (1 − e 2 ) is positive.In order to avoid planetary collisions, it is typical to ensure the following inequality: with 0 < c < 1.A sufficient condition for it is Indeed, if this inequality is satisfied, one has The hyperbolic equilibrium [13] By the formulae (94)-( 95) with n = 2, the of H P is given by

40
We shall now prove that, restricting the domain (129) a little bit, so that the manifolds (128) are hyperbolic for f 12 P (2) . We fix the following domain where where L is as in (114) and, if C (Λ 2 , C) is the unique positive root of the cubic polynomial Implicitly, we shall prove that We check that the coefficients in front of Θ 2 2 , ϑ 2 2 in the Taylor expansion about (Θ 2 , ϑ 2 ) = (0, 0) have opposite sign in the domain (133), so that the equilibrium manifold (128) is hyperbolic.Indeed, the part of degree 2 in such expansion is where Both Θ 1 → a(Λ 1 , Θ 1 ; C) and Θ 1 → b(Θ 1 ; C), as functions of Θ 1 decrease monotonically from a positive value (respectively, C(5Λ 2 2 − 4C 2 ) and C) to −∞ as Θ 1 increases from Θ 1 = 0 to Θ 1 = +∞.The function a(Λ 1 , Θ 1 ; C) changes its sign for Θ 1 equal to a suitable unique positive value C (Λ 2 , C), while b(Θ 1 ; C) does it for Θ 1 = C.We note that (i) inequality C < min{C + , C } follows immediately from the assumptions (135) (in particular, the two last ones) and (ii), more generally, that C ≤ C is equivalent to Λ 2 ≤ 2C.Since, for our purposes, we have to exclude C = C (otherwise, a(Λ 1 , Θ 1 ; C) and b(Θ 1 ; C) would be simultaneously positive and simultaneously negative, and no hyperbolicity would be possible), we distinguish two cases.Proof of 3.Here we prove that Theorem 2.7 Let α + < 1 16 .There exist universal numbers 1 < k < k such that, if Proof The sets D s in (115) and D 0 u in (128) are expressed with different sets of coordinates.To prove that D s and D 0 u have a non-empty intersection, we need to use the same set for both.We choose to use the coordinates P, so we rewrite D s in terms of P.  It will be enough to check that and where, if C ± are as in (136), L su is defined as Note that (144) is certainly satisfied provided (143) is, since in fact, for (Λ 1 , Λ 2 ) ∈ L s (C)∩L u (C)∩ L su (C), which is well-defined by ( 136)-(137).
On the other hand, in view of the definition of C + in (136), and of C a few lines above, L su in (145) is equivalently defined as Therefore, in view of this definition and the definitions of L s , L u in (135) and (141), one sees that the set on the left hand side in (143) is determined by inequalities We observe that no phase point7 (Λ 1 , Λ 2 ) with Λ 2 − Λ 1 − C < 0 will ever satisfy (147), and that inequality Λ 2 > 2C is implied by Λ 1 > C and (146).Then, we divide such inequalities in three groups, so as to rewrite the set (143) as the intersection of the sets We now aim to choose the parameters Λ ± , k ± and α + so as to find a non-empty intersection of the sets a above.
The last straight line, in the plane (Λ 1 , Λ 2 ), through the origin intersecting C is the tangent line, and it is easy to compute (see below) that such a tangent line has slope has slope k as in (140) (Figure 13).We then conclude that, as soon as we choose k − < k, k + > k, Λ − < Λ 1 , Λ + > Λ 1 , we have the inclusion Let us now turn to L 2 .Since we are assuming α + < 1 16 , we conclude that the strip is all included in the region and this allows to conclude Since the sets L 1 and L 2 have a non-empty intersection, independently of α + (see Figure 14), a fortiori, L 1 and L 2 have one: Observe, in particular, that L 1 ∩ L 2 (hence, L 1 ∩ L 2 ) has non-empty intersection with any strip R × 2C, y , with y > 2C (see Figure 15).
On the other hand, it is immediate to check that L 3 includes the horizontal strip and so we conclude

Figure 4 :
Figure 4: Deprit coordinates Z, C and ζ fix the angular momentum in the initial reference frame (i, j, k).

Lemma 2 . 1 (
[5]) The map φ rps C can be extended to a symplectic diffeomorphism on a set P 6n rps

(a) C > 2 c
√ α + Λ 1 and C + 2 c √ α + Λ 1 < Λ 2 < 2C.In this case C < C. We show that no such G u can exist in this case.In fact, since C < C, in order that the interval (C , C) and the set G have a non-empty intersection, one should have, necessarily,C + = sup G > C , hence, in particular, Λ 2 − C > C .Using the definition of C , this would imply Λ 2 > 2C, which is a contradiction.(b) Λ 2 > max{2C, C + 2 c √ α + Λ 1 }.In this case C < C < Λ 2 − C.In order that the interval (C, C ) and the set G have a non-empty intersection, we needC − < C and C + > C(139)and such intersection will be given by the interval G u as in (135).Note that the definition of C + does not include Λ 2 − C in the brackets because, as noted, C < Λ 2 − C.But (139) are equivalent to (135).

Figure 13 :Figure 14 :
Figure 13: The blue curve is C; the orange line has slope k, the green one has slope k (Mathematica).

Let Λ 1 1 , Λ 2
= kΛ 2 be any straight line through the origin.The straight line intersecting C into the point (Λ 1 , Λ 2 ) = (C, 2C) has k = 2, and intersects this curve, also in the higher point (Λ non-empty.The following values work: