Besides, the torsion is
with τ̄Quad =
Proof. We carry out the same computation as in the proof of Lemma 10.3, now up to the order 2 in the r̃j’s. The truncated expression is a trigonometric polynomial in the angles φ̃ j of degree ≤ 4. Eliminating nonresonant monomials, that is, functions of k ⋅ φ with k ⋅ αQuad(0), is a classical matter. Two kinds of terms cannot be eliminated by averaging:
Such monomials actually cannot occur in the expansion, due to the invariance by rotations (theywould not satisfy the d’Alembert relation [24, 64]).33Adirect computation leads to the given expression of the torsion τQuad.
Note that the torsion τQuad, as a function of Λ1 and Λ2, extends analytically outside ℒ(2) (as often do first-order normal forms). This allows us to define the quadrupolar frequency map
a first-order approximation of the normal frequencies.
Proposition 10.6. The first quadrupolar frequency map has constant rank 3 and, in restriction to the symplectic submanifold obtained by fixing vertically the direction of the angular momentum, is a local diffeomorphism.
Proof. For this lemma, we denote by C = (Cx , Cy , Cz) ∈ ℝ3 the angular momentum of the first two planets. The submanifold ? of vertical angular momentum has equation Cx = Cy = 0. It is a symplectic, codimension-2 submanifold, transverse to the Hamiltonian vector fields X Cx and X Cy of Cx and Cy. Since it is invariant by the flow of Quad, its tangent space has equations in the coordinates (xj , yj)j=1,...,4 of the proof of Lemma 10.3, x4 = y4 = 0. So, the upper-left 3 × 3 submatrix τ̃Quad of τQuad is the Hessian of the restriction of Quad to ?. In order to conclude, one merely needs to notice that the determinant of the torsion τ̃Quad:
is nonzero.
So, Quad (adequately truncated) has a nondegenerate quasiperiodic dynamics in the three degrees of freedom corresponding to coordinates (ψj , si)j=1,2,3.
End of the proof of Theorem 10.1. We would like to prove the persistence of some ofthe invariant tori of our normal form, which have frequencies of the following order (assuming a1 = O(1) and a2 →∞):
The conclusion thus follows from any of the four arguments below:
–the first item of Theorem 8.2 (using Proposition 10.6)
–the second item of Theorem 8.2 (using again Proposition 10.6), which yields not only the precedingly found Diophantine tori but also resonant tori (which induce Diophantine tori after reduction by the symmetry of rotation)
–the third item of Theorem 8.2 (using Proposition 10.4, for which the computation of the torsion is not needed, at the expense of deteriorating measure estimates)
–the fourth item of Theorem 8.2 (using again Proposition 10.4).
In three cases, the existence of Diophantine invariant tori of Kep + Quad is proved, either at the partially reduced level or at the fully reduced one. Locally, they will havepositive measure provided 
Theorem 8.2 applies with a perturbation ofthe size |H − Ko| = O(γN) for some N (Remark 7.9). Thus, the theorem really applies to the perturbation of the normal form of the Hamiltonian of order ∼ 7N/2 in 1 /a2.
Bibliographical comments.
–The discovery of the eccentricity vector is often wrongly attributed to Runge and Lenz [1].
–For an anachronistic proof of Bertrand’s theorem using Kolmogorov’s theorem, see [41].
–Lemma 10.2 is obvious in the plane problem, where the analogous reduction leads to a 2D reduced secular space, with coordinates (g1, G1) (see [35, 63]). This is less so in space. Harrington noticed it only after having carried out the computation [49]. Lidov–Ziglin [62] called this a “happy coincidence,” and indeed this invariance allowed them to study the bifurcation diagram of the quadrupolar Hamiltonian Quad. This was also crucial in various studies [53, 103, 104].
–Among the many accounts of the work of Lagrange and Laplace (comprising Lemma 10.3), we refer to [42, 58, 96].
–Resonance (10.6) of order 3 was known to Clairaut, noticed by Delaunay as un résultat singulier [26], and discovered by Herman in the general n-planet problem.
–In the proof of Lemma 10.5, it is a happy coincidence that resonant terms associated with the second resonance actually do not occur at our order of truncation. Malige has computed that higher degree resonant monomials occur, starting at order 10 [64].
–The strategy in Fejoz [37] for proving Arnold’s theorem corresponds to the third argument given above at the end of the proof of Theorem 10.1. The strategy of Chierchia–Pinzari [25] corresponds to the first and second arguments.
Let (X, ω) be a symplectic manifold, (φt)a symplectic flow, and T aminimal quasiperiodic invariant embedded torus for (φt).
Lemma A.1. If ω is exact, T is isotropic.
Proof. We may assume that φt(θ) = θ+tα (t ∈ ℝ, θ ∈ ?n) for some nonresonant vector α. Let
be the 2-form induced by ω on T. Since (φt) preserves ω, for all t, we have
Since the flow on T is minimal, νij is constant. By integrating with respect to θi and θj, this constant must be zero (more learnedly: according to the Hodge theorem, the zero 2-form is the unique harmonic representative of the cohomology class).
If ω is not exact, the conclusion may be wrong. M. Herman has even constructed codimension-2 minimal invariant tori, in such a robust manner that this disproved the quasi-ergodic hypothesis [99].
The following lemma is used in two instances in the proof of Lemma 2.1, as well as in the proof of Kolmogorov’s Theorem 6.5.
Lemma B.1 (Cohomological equation). Let s and σ be given in ]0, 1]. If 
there exists a unique function f ∈ ?(υ) of 0-average such that
and there exists a Cc = Cc(n, τ) such that, for any s, σ:
Proof. Up to substituting g − ∫?n g, we may assume that g has zero average. Then, let
be the Fourier expansion of g. The unique formal solutionto the equation Lαf = g is given by
Since g is analytic, its Fourier coefficients decay exponentially; thus, we find
by shifting the torus of integration to a torus Im θj = ±(s + σ) (the sign depending on the sign of kj). Using this estimate and replacing the small denominators k ⋅ α by its Diophantine lower bound, we get
where, as a change of variable and a rough approximation show, the latter sum is bounded by
Hence, f belongs to 
and satisfies the wanted estimate.
Bibliographical comments. The estimate has been obtained by bounding the terms of Fourier series one by one. In a more careful estimate, one should take into account the fact that if |k ⋅ α| is small, then k′ ⋅ α is not so small for neighboring k′’s. This makes it possible to find the optimal exponent of σ, uniformly with respect to the dimension [69, 81].
We have also used the following inverse function theorem. Recall that we have set 
Proposition B.2. Let 
The map id 
induces a map 
whose restriction 
has a unique right inverse 
Furthermore,
and, provided 2σ−1|υ|s+2σ ≤ 1,
Proof. Let 
be a continuous lift of id +υ and k ∈ Mn(ℤ), k(l) :=Φ(x + l) − Φ(x). Denote by 
the universal covering of
(i)Injectivity of 
Suppose that 
and Φ(x) = Φ(x̂). By the mean value theorem,
hence, x = x̂.
(ii)Surjectivity of 
For any given 
the contraction
has a unique fixed point, which is apreimage of y by Φ.
(iii)Injectivity of 
Suppose that 
and φ(px) = φ(px̂), that is, Φ(x) = Φ(x̂) + κ for some κ ∈ ℤn. That k be in GL(n, ℤ), follows from the invertibility of Φ. Hence, Φ(x − k−1(κ)) = Φ(x̂), and, due to the injectivity of Φ, px = px̂.
(iv)Surjectivity of 
This is a trivial consequence of that of Φ.
(v)Estimate on 
Note that the wanted estimate on ψ follows from the corresponding estimate for the lifted map 
But, if 
,
hence, |Ψ − id |s ≤ |υ|s+σ.
(vi)Estimate on ψ′. We have ψ′ = φ′−1 ∘ φ, where φ′−1(x) stands for the inverse of the map ξ ↦ φ′′(x) ⋅ ξ. Hence,
and, under the assumption that 2σ−1|υ|s+2σ ≤ 1,
CInterpolation of spaces of analytic functions
In this section, we prove some Hadamard interpolation inequalities, which are used in Section 13.4.
Recall that we denote by 
infinite annulus ℂn/2πℤn, by 
thebounded subannulus 
| Im θj| ≤ s, j = 1, . . ., n} and by 
the polydisk {r ∈ ℂn, |rj| ≤ t, j = 1, . . ., n}. The supremum norm of a function f ∈ 
will be denoted by |f|s,t.
Let 0 < s0 ≤ s1 and 0 < t0 ≤ t1 be such that
Proposition C.1. If 
Proof. Let f ̃ be the function of 
constant on 2n-tori of equations (Im θ, |r|) = cst, defined by
(with all possible combinations of signs). Since log |f| is subharmonic (hence upper semicontinuous) and2n is compact, log f ̃ too is upper semicontinuous. Besides, log f ̃ satisfies the mean inequality; hence, it is plurisubharmonic.
By the maximum principle, the restriction of |f| to 
attains its maximum on the distinguished boundary of 
. Due to the symmetry of f ̃:
Now, the function
is well defined on ?s1 , for it is constant with respect to ℜz and, due to the relations imposed on the norm indices, if | Im z| ≤ s1, then |e−(iz+s)t| ≤ es1−s t = t1.
The estimate
trivially holds if Imz = s0 or s1, for, as noted above for j = 1, esj−s t = tj, j = 0, 1. But the left- and right-hand sides, respectively, are suharmonic and harmonic. Hence, the estimate holds whenever s0 ≤ Im z ≤ s1, whence the claim for z = is.
Recall that we have 
and, for a function 
let |f|s = |f|s,s denote its supremum norm on 
nAs in the rest of the paper, we now restrict the discussion to widths of analyticity ≤ 1.
In Section 13.4, we use the equivalent fact that, if 
Proof. In Proposition C.1, consider the following particular case:
Then,
We want to determine max(s0, t0) and max(s1, t1). Let σ1 := s − s0 = s1 − s. Then, t0 = s e−σ1 and t1 = s eσ1 . The expression s + σ − seσ has the sign of σ (in the relevant region 0 ≤ s + σ ≤ 1, 0 ≤ s ≤ 1); by evaluating it at σ = ±σ1, we see that s0 ≤ t0 and s1 ≥ t1.
Therefore, since the norm | ⋅ |s,t is nondecreasing with respect to both s and t,
(thus giving up estimates uniform with respect to small values of s). By further setting σ0 = s − t0 = s (1 − e−σ1), we get the wanted estimate, and the asserted relation between σ0 and σ1 is readily verified.
Bibliographical comments.
–The obtained inequalities generalize the standard Hadamard inequalities. They are optimal and show that the convexity of analytic norms is twisted by the geometry of the phase space. See [72, Chap. 8] for more general but coarser inequalities.
–Interpolation inequalities in the analytic category do not depend on regularizing operators as they do in the Hölder or Sobolev cases. See, for example, [51, Theorem A.5] or [48].
Acknowledgment: These notes are the expanded version of a chapter of the Habilitation memoir [38] and of a subsequent short course given at the workshop Geometric control and related fields, organized by J.-B. Caillau and T. Haberkorn in RICAM (Linz, November 2014). I thank J.-B. Caillau, A. Chenciner, J. Mather, G. Pinzari and J.-C. Yoccoz for their interest or suggestions. Paradoxycally, this work has been partially funded by the ANR project Beyond KAM theory (ANR-15-CE40-0001)
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