Global Kolmogorov tori in the planetary N-body problem. Announcement of result

We improve a result in [L. Chierchia and G. Pinzari, Invent. Math. 2011] by proving the existence of a positive measure set of $(3n-2)$--dimensional quasi--periodic motions in the spacial, planetary $(1+n)$--body problem away from co--planar, circular motions. We also prove that such quasi--periodic motions reach with continuity corresponding $(2n-1)$--dimensional ones of the planar problem, once the mutual inclinations go to zero (this is related to a speculation in [V. I. Arnold. Russ. Math. Surv. 1963]). The main tool is a full reduction of the SO(3)--symmetry, which, in particular, retains symmetry by reflections and highlights a quasi--integrable structure, with a small remainder, independently of eccentricities and inclinations.


Set Up and Background
In [1], V. I. Arnold, partly solving, but undoubtedly clarifying important mathematical settings of the more than centennial question (going back to the investigations by Sir Isaac Newton, in the XVII century) on the motions of the planetary system, asserted his "Theorem on the stability of planetary motions" as follows.
Theorem 1.1 ([1, 19, 26, 15, 10, 22, 8]) In the many-body problem there exists a set of initial conditions having a positive Lebesgue measure and such that, if the initial positions and velocities belong to this set, the distances of the bodies from each other will remain perpetually bounded, provided the masses of the planets, eccentricities and inclinations are sufficiently small.
In this paper, we announce an improvement (Theorem 2.1 in the next section) of Theorem 1.1. To present it, we devote this section to a short survey of related techniques, referring the reader to the aforementioned literature, or to the review papers [11,3,6] or, finally, to the introduction of [23] for more details.
Consider (1 + n) masses in the configuration space E 3 = R 3 interacting through gravity. Let such masses be denoted as m 0 , µm 1 , · · · , µm n , where m 0 is a leading mass ("sun", of "order one"), while µm 1 , · · · , µm n are n smaller masses ("planets", of "order µ", with µ a very small number). This problem, a sub-problem (usually referred to as "planetary" system) of the more general Nbody problem, emulates the solar system; hence, the study of it has a relevant physical meaning. It is very natural to regard this system (which is Hamiltonian 1 ) as a small perturbation of the leading dynamical problem consisting into the gravitational interaction of the sun separately with each planet. This corresponds to what follows. After letting the system free of the invariance by translations (i.e. , eliminating the motion of the sun), one can write the 3n-degrees of freedom Hamiltonian governing the motions of the planets as where x (i) = (x 3 ) their generalized conjugated momenta and m i := m0mi m0+µmi , M i := m 0 + µm i the "reduced masses". In order to exploit the integrability of the "two-body terms" h (i) a natural approach is to put the system in Delaunay 2 coordinates. This is a system of canonical action-angle variables ((Λ, Γ, H, ℓ, g, h) ∈ R 3n × T 3n ), whose rôle is the one of transforming (via the Liouville-Arnold Theorem) h (i) 2B into "Kepler form" , i.e. , a function of actions only. It is well known that, due to the too many integrals of h 1 I. e., its motions are described by equations of the form     ẏ 3 are canonical coordinates of the point-mass i, and H 3+3n is a suitable (3 + 3n)-degrees of freedom Hamilton function, depending on (p, q) = (p (0) , · · · , p (n) , q (0) , · · · , q (n) ). 2 Delaunay and (see below) Poincaré coordinates are widely described in the literature. A definition may be found, e.g. , in [7,12]. Note that (H, h) ∈ R n × T n are denoted as (Θ, θ) in [7]. where the system (1) expressed in Delaunay coordinates. The purpose is to determine a positive measure set of quasi-periodic motions for this system.
In 1954 A.N. Kolmogorov [18] discovered a breakthrough property of quasi-integrable dynamical systems: for a regular, slightly perturbed system open, a great number of quasi-periodic motions (I 0 , ϕ 0 ) → (I 0 , ϕ 0 + ∂ I h(I 0 )t) of the unperturbed system h may be continued in the dynamics of the perturbed system, provided the Hessian ∂ 2 I h(I) does not vanish identically in A. Due to the proper degeneracy, for the planetary system expressed in Delaunay variables (3), taking I := (Λ, Γ, H) and ϕ := (ℓ, g, h), Kolmogorov's non-degeneracy assumption is clearly violated. Despite of this fact, the perturbing function has good parity properties: Arnold noticed that such parities help in determining a quasi-integrable structure in all the variables for the planetary system, as now we explain.
Following Poincaré, one switches from Delaunay coordinates to a new set of canonical coordinates (Λ, λ, η, ξ, p, q). These are not in action-angle form, but are in mixed action-angle (the couples (Λ, λ)) and rectangular form (the z := (η, ξ, p, q)). The variables (Λ, λ) have roughly the same meaning of the (Λ, ℓ); the z are defined in a neighborhood of z = 0 ∈ R 4n and the vanishing of (η i , ξ i ) or of (p i , q i ) corresponds to the vanishing of the i th eccentricity, inclination, respectively.

Let us denote as
the system (1) expressed in Poincaré variables.
Since the perturbation f hel in (1) does not change under reflection (y and rotation transformations (y and due to the fact that the transformations (respectively, reflections with respect to the coordinate planes and rotation about the k-axis) where R (3) g := cos g − sin g sin g cos g g ∈ T have a nice expression in Poincaré variables, respectively, one then sees that the averaged ("secular") perturbation f av P (Λ, η, ξ, p, q) := 1 (2π) n T n f P (Λ, λ, η, ξ, p, q)dλ enjoys the following symmetries. If we denote the Taylor expansion of f av P in powers of t, t * , we then have Proposition 1.1 (D'Alembert rules) where |a| 1 := n i=1 a i . By D'Alembert rules one has that the expansion of f av P around z = 0 contains only even monomials and starts with ij (Λ) are suitable coefficients, expressed in terms of Laplace coefficients, computed in [19,15,10]. This expansion shows that the point z = (η, ξ, p, q) = 0 is an elliptic equilibrium point for f av P (Λ, η, ξ, p, q). A natural question is wether, from here, it is also possible to transform f av P intoH P (Λ, λ, z) = h K (Λ) + µf P (Λ, λ, z) wheref av P is in "Birkhoff normal form" (hereafter, BNF) of a suitable order (say, of order three). This means where σ i (Λ), ς i (Λ) are the eigenvalues of Q (h) (Λ), Q (v) (Λ) and, for 1 ≤ i ≤ n, w i := Then Arnold aims to solve the problem of the proper degeneracy (and hence to prove Theorem 1.1) by obtaining Kolmogorov full-dimensional tori bifurcating from the elliptic equilibrium z = 0, via the following abstract result.
Then, for any κ > 0 one can find a number ε 0 = ε 0 (κ) such that, if 0 < ε < ε 0 and 0 < µ < ε 8 , the set F ε := A × T ν × B 2ℓ ε (0) may be decomposed into a set F * ε which is invariant for the motions of H and a set f ε the measure of which is smaller than κ. More precisely, F * ε foliates into (ν + ℓ)-dimensional invariant manifolds {T ω } ω close to where the motion is analytically conjugated to the linear flow Despite this brilliant strategy, Arnold applied Theorem 1.2 to the case of the planar three-body problem only, by explicitly checking assumptions (i)-(iii). For the general case, he was aware of some extra-difficulties, about which he gave just some vague 3 indications. A first problem is represented by the so-called "secular degeneracies" : the "first order Birkhoff invariants" σ 1 , · · · , σ n , ς 1 , · · · , ς n satisfy, identically 4 , Such relations are in contrast with usual non-resonance requirements in order to construct BNF [16]. But the problem is only apparent. Indeed, it has been recently understood [21,8] that, by the symmetry R g in (9), only resonances are really important for the construction of BNF, while resonances (13) do not belong to this class. Moreover, in [10] it has been proved that they are the only ones to be identically satisfied; result next improved in [8], where, by direct computation, it has been seen that they are the only ones to be satisfied in an open set: compare item (v)-(a) of Theorem 1.3.
A much more serious problem is the following 5 Proposition 1.2 (Rotational degeneracy [7]) For the system (5), BNF can be constructed up to any prefixed 6 order p but all the coefficient τ i1···ir (Λ) of the generic monomial w i1 · · · w ir with some of the i k 's equal to 2n vanish identically.
for which, in particular, the "torsion" matrix (the matrix of the second-order coefficients) τ = (τ ij ) has an identically vanishing row and column, hence, This violates assumption (iii) of Theorem 1.2. However, such negative result, understood only "a posteriori", is just the counterpart of Theorem 1.3 below.
which are related to Poincaré coordinates (Λ, λ, η, ξ, p, q) by where U (Λ) is a n × n unitary matrix, i.e. , verifying U (Λ)U t (Λ) = id and ϕ 1 , ϕ 2 are suitable functions defined in a global neighborhood of z = 0, such that (i) (p n , q n ) are integrals for f RPS .
(ii) D'Alembert rules (9) are preserved, and correspond to the reflections and the rotation in (8). In particular, denoting as the system (1) expressed in the RPS variables, wherez denotes z deprived of (p n , q n ), then (iii) The pointz = 0 ∈ R 2n−1 , which corresponds to the vanishing of all eccentricities and mutual inclinations, is an elliptic equilibrium point forz → f av RPS (Λ,z). (iv) For any fixed p ∈ N, p ≥ 2, it is possible to conjugate H RPS tȏ 5 Proposition 1.2 answers, in particular, a question raised by M. R. Herman, who, in [15], declared not to know if the planetary torsion might vanish identically. More in general, Proposition 1.2 generalizes Laplace resonance in (13) to any order of BNF. 6 Namely, with 3 replaced by p and O(z 7 ) by O(z 2p+1 ) in (11). 7 RPS stands for "Regular", "Planetary" and "Symplectic".

Result
Clearly, the elliptic equilibrium point of the secular perturbation at the origin plays a fundamental rôle in order to determine a quasi-integrable structure in the problem with respect to all of its degrees of freedom. As remarked, such equilibrium is determined by the symmetries of the system (i.e. , relations (8)). Once the system is put in a set of coordinates such that SO(3)-symmetry is completely reduced, hence R g will not play its symmetrizing rôle anymore, the elliptic equilibrium, in general disappears. On the other hand, reducing completely the number of degrees of freedom has its advantages, since it clarifies the structure of phase space and lets the system free of extra-integrals. It is then natural to ask what is the destine of Kolmogorov tori, in such case. In order to clarify this and other related questions, let us add some more comments.
• The variables (14) realize a partial reduction of the SO(3)-invariance: in such variables, the system has (3n − 1) degrees of freedom, one over the minimum. As said, this is useful in order to describe with regularity the co-inclined, co-circular configuration and to keep the elliptic equilibrium forz = 0. On the other hand, the fact of having one more degree of freedom than needed implies that possible (3n− 1)-dimensional resonant tori corresponding to rotations in the invariable plane of non-resonant (3n − 2)-dimensional tori are missed, with subsequent under-estimate (∼ ε 4n−2 instead of ∼ ε 4n−4 ) of the measure of the invariant set F * ε mentioned in Theorem 1.2.
• In [8] a construction is shown that allows to switch to a "full reduction" to (3n − 2) degrees of freedom. Such procedure is a bit involved, but allows, at the end to reduce completely 8 Substantially, switching from Poincaré to RPS variables corresponds to replace the n inclinations of the planets with respect to a prefixed frame (i, j, k), with (n − 1) mutual inclinations among te planets plus the negligible inclination of the invariable plane with respect to k. Recall that the invariable plane is the plane orthogonal to the total angular momentum C. 9 The variables (Λ, Γ, Ψ, ℓ, γ, ψ), in such "planetary form", have been rediscovered by the author during her PhD. Note that, apart for few cases [20,13] of application to the three-body problem, where they reduce to the variables of Jacobi reduction, Deprit variables seem to have remained un-noticed by most. See also [5] for the proof of the symplecticity of (Λ, Γ, Ψ, ℓ, γ, ψ) found in [22]. the number of degrees of freedom and, simultaneously, to deal with one only singularity. It generalizes the analogue singularity of Jacobi variables for n = 2, for which, the planar configuration is not allowed. Therefore, one has one has to discard a positive measure set in order to stay away from it. The measure of the invariant set F * ε is therefore estimated as ∼ (ε 4n−4 − ε 4n−4 0 ) with an arbitrary 0 < ε 0 < ε.
• The completely reduced variables that are obtained via the full reduction of the previous item for the n = 2 case are analogue Jacobi's variables (they are not the same) and lead to the same BNF studied in [26]. Differently from what happens for the above discussed case n = 2, for n ≥ 3, the full reduction studied in [8] looses (besides the R g -symmetry in (10)) also reflection symmetries and hence the elliptic equilibrium. Such equilibrium needs to be restored via an Implicit Function Theorem procedure, that is successful in the range of small eccentricities and inclinations.
• From the two previous items one has that, while a "continuity" (letting the inclinations to zero) between (3n − 1)-dimensional Lagrangian tori of the partially reduced problem in space (whose existence has been discussed in [10,8]) and (2n)-dimensional Lagrangian tori of the unreduced planar problem follows from [8], instead, an analogous continuity between (3n − 2)-dimensional Lagrangian tori of the fully reduced problem in space (again discussed in [10,8]) and (2n − 1)-dimensional Lagrangian tori of the fully planar problem (discussed in [7]) once inclinations go to zero is naturally expected but, up to now, remains unproved.
Compare also the arguments in [26,10] on this issue. As mentioned in the previous section, we recall, at this respect, that a controversial (indeed, erroneous) continuity argument between the planar Delaunay coordinates and the spacial planetary coordinates obtained via Jacobi reduction of the nodes was argued by Arnold [1] in order to infer non-degeneracy of BNF of the spacial three-body problem.
• Recall the definitions of F ε , F * ε in Theorem 1.2. In both the cases discussed above (partial and full reduction), the "density" of F * ε inside of F ε , i.e. , the ratio d := meas F * ε meas F ε goes to one as ε → 0. That is, one has to keep more and more close to the co-inclined, cocircular configuration, in order to encounter more and more tori. In [4] it has been proved that one can take d = 1 − √ ε .
Note in fact that the perturbative technique which leads to Theorem 1.2 (or to its improvement discussed in [4]) is developed with respect to ε, rather than with respect to the initial parameter µ appearing in (1). This circumstance is an intrinsic consequence of the fact that the tori obtained via Theorem 1.2 bifurcate from the elliptic equilibrium and that, in general, the Birkhoff series (11) diverges.
• In [1] Arnold realized that, in the case of the planar three-body problem the series (11) is instead convergent (in this case f av P is integrable). This allows him to prove where χ(µ) → 0 as µ → 0. For this particular case, the tori do not bifurcate from the elliptic equilibrium, but a different quasi-integrable structure is exploited in [1] (besides also a different perturbative technique at the place of Theorem 1.2). In [23], a slightly weaker result has been proved for the case of the spacial three-body problem and the planar general problem: where α denotes the maximum semi axes ration and χ(µ, α) → 0 as (µ, α) → 0. Note that for such cases f av P is not integrable.
• From the astronomical point of view, the investigation mentioned in the two last items is motivated by the fact that, for example, Asteroids or trans-Neptunian planets exhibit relatively large inclinations or eccentricities. From the theoretical point of view, the question is to understand wether it is possible to find different quasi-integrable structures in the planetary N-body problem besides the one determined by the elliptic equilibrium.
We prove the following result.
Theorem 2.1 Assume that the semi-major axes of the planets are suitably spaced; let α denote the maximum of such ratios. If α is small enough and the mass ratio µ is small with respect to some power of α, one can find a number ε 0 and a positive measure set F * α,µ ⊂ F := F ε0 of Lagrangian, (3n − 2)-dimensional, Diophantine tori, the density of which in F goes to one as (α, µ) → (0, 0). Letting the maximum of the mutual inclinations going to zero, such (3n − 2)dimensional tori are closer and closer to Lagrangian, (2n − 1)-dimensional, Diophantine tori of the corresponding planar problem.
In the next sections we provide the main ideas behind the proof of Theorem 2.1. Note that we shall not enter into the (technical) details of the estimate of the density of F * α,µ , for length reasons.

Tools and Sketch of Proof
The proof of Theorem 2.1 relies upon four tools.
Note that • The variables (17) are very different from the planetary Deprit variables (Λ, Γ, Ψ, ℓ, γ, ψ) mentioned in the previous section. For example, they do not provide the Jacobi reduction of the nodes when n = 2. Indeed, the definition of (17) is based on 2n nodes (16), the nodes between the mutual planes orthogonal to S (j) and P (j) and P (j) and S (j+1) . Deprit's reduction is instead based on n nodes, the nodes among the planes orthogonal to the S (j) 's. Let us incidentally mention that, for the three-body case (n = 2), the variables (18) are trickily related to certain canonical variables introduced in §2.2 of [23]. This relation will be explained elsewhere.
• While, in the case of the variables (Λ, Γ, Ψ, ℓ, γ, ψ), inclinations among the S (j) 's cannot be let to zero, it is not so for the variables (18), where the planar configuration can be reached with regularity. And in fact, in the planar case, the change between planar Delaunay variables (Λ, Γ, ℓ, g) and the planar version (Λ, χ, ℓ, κ) of (18) reduces to with g 0 ≡ 0. Note incidentally that the variables (18) are instead singular in correspondence of the vanishing of the inclinations about P (j) and S (j) or S (j+1) and P (j) ; configurations with no physical meaning.
• The variables (17) have in common with the variables (Λ, Γ, Ψ, ℓ, γ, ψ) and the Delaunay variables (Λ, Γ, H, ℓ, g, h) the fact of being singular for zero eccentricities (since in this case the perihelia are not defined). We however give up any attempt to regularize such vanishing eccentricities. The reason is that the Euclidean lengths of the C (j) 's are not 10 actions (apart for χ n−1 = |C (n) |) and hence the regularization does not seem 11 to be (if existing) easy. Note that, since we are interested to high eccentricities motions, we shall have to stay away from these singularities.
• Another remarkable property of the variables (17), besides the one of being regular for zero inclinations is that they retain the symmetry by reflections, as explained in Proposition 3.1 below. This does not happen for the variables (Λ, Γ, Ψ, ℓ, γ, ψ). As we shall explain better in the next section, such symmetry property plays a rôle in order to highlight a global 12 quasiintegrable structure of H χ0 in (19) below and, especially, to have an explicit expression of it. 10 Indeed, for 1 ≤ j ≤ n − 1, 11 Recall that e (j) = 0 corresponds to |C (j) | = Λ j . 12 With a remainder independent of eccentricities and inclinations; compare Proposition 3.3. (17) are canonical. Moreover, letting H χ0 the system (1) in these variables, (Θ 0 , ϑ 0 , χ 0 ) are integrals of motion for H χ0 , which so takes the form

An integrability property
The second tool is an integrability property of the planetary system. To describe it, we generalize a bit the situation, introducing the concept of Kepler map.
• The following classical relations then hold for (not necessarily canonical) Kepler maps • Given a canonical Kepler map φ, put H φ := H hel • φ, where H hel is as in (1). Then where h K is as in (4) and is the perturbing function (1) expressed in the variables (Λ, ℓ, P, Q). Imposing a suitable restriction of the the domain so as to exclude orbit collision, one has that the secular φ-perturbing function, i.e. , the average (f φ ) av (X) := 1 (2π) n T n f φ (X, ℓ 1 , · · · , ℓ n )dℓ 1 · · · dℓ n is well defined. Due to (22), the "indirect" part of the perturbing function, i.e. , the term φ (X, ℓ j )/m 0 has zero average and hence (f φ ) av is just the average of the Newtonian (or "direct") part: • If we consider the expansion φ (X, ℓ j )| ε=0 we have that, in this expansion, the two first terms depend only on Λ j . More precisely, due to (23), Therefore, the term (f av carries the first non-trivial information. In the case of the map φ = φ P * , we have denotes the minimum of a and b.

Note that
• The main point of Proposition 3.2 is that the action χ n−1 = |C (n) | is an integral for (f av . Clearly, this is general: whatever is φ, av . This fact has been observed firstly, for the case of the three-body problem, in [14], using Jacobi reduction of the nodes. In that case Harrington observed that (f (12) φJac ) (2) av depends only on (Λ 1 , Λ 2 , Γ 1 , Γ 2 , G, γ 1 ) and the integrability is exhibited via the couple (Γ 1 , γ 1 ). As we already observed, in such case the planetary Deprit variables and the variables obtained by Jacobi reduction of the nodes are the same.
• An important issue that is used in the proof of Theorem 2.1 (precisely, in order to check certain non-degeneracy assumptions involved in Theorem 3.1 below) is the effective integration of (f av . Clearly, in principle, this could be achieved using any of the sets of variables mentioned in the two previous items: planetary Deprit variables (Λ, Γ, Ψ, ℓ, γ, ψ) or the variables (17). However, the integration using planetary Deprit variables carries considerable analytic difficulties and has been performed only qualitatively [20,13]. Using the variables (17), such integration can be achieved by a suitable convergent Birkhoff series, exploiting the equilibrium points in (21). Compare also Proposition 3.3 and the comments below.
Here are some comments of the proof of Proposition 3.3.

Multi-scale KAM theory
The fourth tool is a multi-scale KAM Theorem. To quote it, let us fix the following notations.