On ‘Arnold’s theorem’ on the stability of the solar system

Arnold's theorem on the planetary problem states that, assuming that 
 the masses of $n$ planets are small enough, there exists in the 
 phase space a set of initial conditions of positive Lebesgue 
 measure, leading to quasiperiodic motions with $3n-1$ 
 frequencies. Arnold's initial proof is complete only for the plane 
 $2$-planet problem. Arnold had missed a resonance later discovered 
 by Herman. The first complete proof, by Herman-Fejoz, relies on the 
 weak non-degeneracy condition of Arnold-Pyartli. A second proof, by 
 Chierchia-Pinzari, is closer to Arnold's initial idea and shows the 
 strong non-degeneracy of the problem after suitable reduction by 
 (part of) the symmetry of rotation. We review and compare these 
 proofs. In an appendix, we define the Poincare coordinates and prove 
 their symplectic nature through the shortest possible computation.

We think of the body 0 as the Sun and the other bodies as n planets revolving around the Sun. In our Solar System, the mass of Jupiter is 1/1000 that of the Sun, which justifies that we consider small values of ǫ. The equations have a limit when ǫ → 0, for which the Sun is still (in an appropriate frame of reference) and each planet undergoes the only attraction of the Sun. If their energies are negative, planets describe Keplerian ellipses, with some fixed semi major axes and excentricities. As a whole, the system is quasiperiodic with n frequencies. For a generic Hamiltonian system with 3n degrees of freedom, one would expect 3n frequencies; due to this dynamical degeneracy of the Newtonian potential, we are dealing with a singular perturbation problem.
Theorem 1. For every m 0 , m 1 , ..., m n > 0 and for every 0 < a 1 < ... < a n there exists ǫ 0 > 0 such that for every 0 < ǫ < ǫ 0 , in the phase space in the neighborhood of circular and coplanar Keplerian motions with semi major axes a 1 , ..., a n , there is a subset of positive Lebesgue measure of initial conditions leading to quasiperiodic motions with 3n − 1 frequencies.
The proof of this theorem is rendered difficult by the multitudinous degeneracies of the planetary system. Arnold's initial proof does not fully describe these degeneracies. Actually Arnold's strategy fails, in its straightest form, due to a resonance he had not forseen (Herman's resonance). In 1998, in a series of lectures M. Herman sketched a complete and more conceptual proof of this theorem, showing some weak non-degeneracy property of the planetary problem, and then concluding by calling upon a theorem of Diophantine approximation of Arnold-Pyartli [7]. Later, Chierchia and Pinzari strengthened the result qualitatively and quantitatively with a slightly different proofn closer to Arnold initial strategy, showing in particular the strong non-degeneracy property of the planetary problem [5,6]. We now review some ideas of these proofs.
2 Reduction by translations Switch to the symplectic heliocentric coordinates: The conservation of the total linear momentum Y 0 and the invariance by translation of Newton's equations allow us to focus on the subspace Y 0 = 0 without loss of generality, and to ignore the variable X 0 . The equations now readẊ the 'fictitious' masses µ j ∼ m j and M j ∼ m 0 are functions of the m j 's and of ǫ, defined by 1 ǫµ j = 1 m 0 + 1 ǫm j and M j = m 0 + ǫm j .
3 The Keplerian part and the perturbing function The Hamiltonian splits into H = H Kep + ǫH per , the sum of a Keplerian Hamiltonian and an ǫ-small perturbing function. Let (λ j , Λ j , ξ j , η j , p j , q j ) ∈ T 1 ×]0, +∞[×R × R × R × R be the Poincaré coordinates of the j-th planet. Those coordinates are analytic and symplectic on a neighborhood of the union of circular, direct, horizontal (with respect to some given plane in space) Keplerian ellipses: -the angle λ j parameterizes the ellipse proportionally to the area swept by the position vector, -Λ j = µ j M j a j , -ξ j and η j determine the eccentricity and orientation of the ellipse within its plane, -p j and q j determine the position of the plane of the ellipse. Those coordinates straighten the (degenerate) Keplerian dynamics, since H Kep now only depends on Λ j 's: be the so-called mean motions (a weird, old name for the Keplerian frequencies); Kepler's third law follows from this expression of ν j .
4 The averaged Hamiltonian From now on, restrict to the open set, diffeomorphic to T n × R 5n , over which Keplerian ellipses do not meet. Up to renumbering, we can assume that 0 < a 1 < · · · < a n . Away from the boundary and in a neighborhood of circular Keplerian ellipses, the perturbing function is uniformly ǫ-small. Thus a change of coordinates ǫ-close to the identity transforms H into the averaged Hamiltonian up to terms of order 2 in ǫ, along the Cantor set of Diophantine Keplerian frequencies.
(Technically, this means that the new Hamiltonian equals the above expression, plus a term which is C ∞ -flat on the Cantor set, which plays no role and which we will ignore here.) For the averaged Hamiltonian, the momenta Λ j are first integrals (this is the first stability theorem of Lagrange and Laplace). It descends to the quotient by the Keplerian action of T n and induces a Hamiltonian system on the space, diffeomorphic to R 4n = {(ξ j , η j , p j , q j ) j∈{1,...,n} }, of Keplerian tori with fixed semi major axes. This Hamiltonian, up to a constant, is H per ; it is called the secular Hamiltonian and its phase space the secular space. This system describes the slow variations of excentricity and orientation in space of the Keplerian ellipses of the planets, under the influence of mutual attraction, at the first order in ǫ, outside resonances in mean motions. Contrary to its analogue in the Lunar problem (see [8]), it seems not integrable.
5 The elliptic secular singularity By symmetry, the origin ξ = η = p = q = 0 of the secular space is an equilibrium point. It turns out to be an elliptic critical point of the averaged Hamiltonian (this is the second stability theorem of Laplace), having the following remarkable expansion: where the "horizontal" and "vertical" quadratic forms Q h and Q v are of the form coefficients C j are real analytic with respect to the semi major axes, and can be expressed in terms of the Laplace coefficients.
6 The secular frequencies Let ρ h and ρ v be orthogonal diagonalizing transformations of Q h and Q v (depending analytically on the masses and semi major axes): In the full phase space, the map is symplectic and, the problem thus boils down to studying a Hamiltonian of the form where remainders do not depend on λ. The Hamiltonian obtained by neglecting those remainders is integrable and its integral curves (outside the elliptic singularity) are quasiperiodic, with the 3n frequencies given by α o := (ν 1 , ..., ν n , ǫσ 1 , ..., ǫσ n , ǫς 1 , ..., ǫς n ) .
The arithmetic properties of the frequency vector α o play a deciding role in the existence of higher order normal forms; α o depends on the masses and the semi major axes, and can also be thought of as a constant function of the actions ξ 2 j + η 2 j and p 2 j + q 2 j (when one moves away from the elliptic secular singularity).
7 Properties of the frequency vector KAM theory asserts that there is a perturbed frequency vector α, not explicitely known but tending to α o when ǫ, ξ 2 j +η 2 j and p 2 j +q 2 j tend to zero, such that whenever α is Diophantine there is a corresponding invariant torus.
It is customary in this theory to measure the abundance of invariant tori given by the KAM theorem, with fixed masses. J. Moser justified this habit with the argument that one cannot change the masses of the planets. We will let the reader decide form himself whether it is easier to 'change' semi major axes.
-If α is seen as a function of a = (a 1 , ..., a n ) alone (implying that we look for tori very close to circular coplanar Keplerian ellipses), the best we can hope for is the weak non-degeneracy property of Arnold-Pyartli i.e., the local image of α does not lie in a hyperplane. In this case, the theory of Diophantine approximations shows that the set of parameters a such that the frequency vector is Diophantine has positive Lebesgue measure.
-If α is seen as a function of both a and the actions ξ 2 j +η 2 j 's and p 2 j +q 2 j 's, one can hope for the strong non-degeneracy property of Kolmogorov i.e., the map α is a local diffeomorphism, thus obviously reaching Diophantine vectors for a set of parameters of positive measure.
There are difficulties in both cases, which are removed when one takes advantage of the invariance of the system by rotations. This symmetry allows us to decrease the number of degrees of freedom by two units and get rid of annoying resonances.
8 Weak non-degeneracy In the plane (p = q = 0), the application of KAM theory is quite straightforward, due to the following fact.
This means that no resonance is identically verified and, by Pyartli's theorem [13], that the perturbed frequency vector passes through Diophantine vectors in positive measure. Arnold's theorem in the plane follows. The proof of the proposition goes along these lines.
For n = 2 planets, there is an explicit asymptotics of α in the 'wellspaced regime' where a 1 ≪ a 2 . Then the conclusion of the proposition is readily checked. Complexifying the semi major axes and using the analytic continuation of Q h , one can deduce the conclusion of the proposition with n = 2. In the case of n + 1 planets, in the well-spaced regime the spectrum of Q h splits into a very small eigenvalue and the spectrum corresponding to n planets. By induction over the number n of planets the conclusion follows.
In space, the situation is more intricate. The proof of the proposition goes along the same lines as for the plane problem.
Unsurpisingly, the first resonance is due to the invariance of the system by rotations about any horizontal axis (hence the linearized secular vector field has two vanishing eigenvalues), so that one of the spatial secular frequencies, say ς n , vanishes. Thus this resonance disappears when one restricts to a fixed direction, say vertical, of the angular momentum (a codimension-2, hence (6n − 2)-dimensional, symplectic submanifold).
The second resonance is mysterious and does not seem to be associated with any symmetry. Although it was known to astronomers in particular cases (n = 2), it was discovered in its full generality by Herman. It happens to disappear when one completes the reduction by rotations, by fixing the angular momentum vector and quotienting by rotations around the fixed direction of the angular momentum. For n = 2 planets, this is checked by carrying out the classical reduction of the node of Jacobi; Arnold did so without noticing that he did get rid of a resonance in the process.
The difficulty when n ≥ 3 is that we do not know of simple coordinates adapted to the reduction by rotations. Arnold thus suggested to merely fix the direction of the angular momentum, not being aware of Herman's resonance. In order to check without much computation that Herman's resonance does not exist at the fully reduced level, one can use a trick (also used by Poincaré in order to find some periodic orbits), consisting in adding to the Hamiltonian a term in δ C , where δ is small real number, and C = C z is the length of the angular momentum. This amounts to swithcing to a rotating frame of coordinates. The KAM theorem applies to show the existence of Diophantine Lagrangian tori of dimension 3n − 2, invariant for the modified Hamiltonian, for a positive measure of values of δ (one value of δ would suffice). Then, by an argument of Lagrangian intersection, these tori must be invariant by the flow of the initial Hamiltonian. Adding back the frequency corresponding to the rigid rotation of the system, one gets tori of dimension 3n − 1, which are sometimes ergodic, and sometimes foliated in ergodic invariant tori of dimension 3n − 2. This proves Arnold's theorem.
9 Strong non-degeneracy Chierchia and Pinzari have strengthened and completed the previous result by investigating the part of the secular Hamiltonian which is quartic with respect to the secular Poincaré coordinates ξ, η, p, q, as in Arnold's original strategy. As above, the invariance by rotations about a horizontal axis prevents the non-reduced secular system from being non-degenerate, this time in the sense of Kolmogorov. But this obstruction vanishes in restriction to the codimension-2 submanifold obtained by fixing the direction of the angular momentum.
It happens that Herman's resonance does not create any resonant term at the fourth order (nor until the 10-th order [9]). So it does not prevent to compute the torsion in the partially reduced system (fixed direction of the angular momentum, but norm of the angular momentum not fixed and no quotient by rotation around the angular momentum). This partial reduction by rotations, which already played a key role above for proving the weak nondegeneracy condition, can be understood as follows. Topologically, the group SO 3 is an S 1 -bundle over S 2 . The action of the maximal torus S 1 = SO(2) carries all the non-trivial dynamical information, while all directions of the angular momentum are equivalent to each other.
The fourth order terms are formidable to compute. Chierchia-Pinzari use some regularized version of the Deprit coordinates. Those coordinates do not reduce to coordinates per planet. The Deprit coordinates were independently rediscovered by G. Pinzari during her PhD [11], before she realized they matched Deprit's coordinates. Interestingly, Deprit believed that nobody would ever make anything useful with these coordinates. Pinzari managed to compute asymptotics, as before in the well-spaced regime, of the 4-th order terms of the averaged Hamiltonian. Chierchia-Pinzari can conclude triumphantly, probably as much as possible in the spirit of what Arnold had in mind in 1963: Theorem 4. The torsion of the planetary problem at the elliptic singularity of the averaged system is non-degenerate, both in restriction to the submanifold obtained by fixing the direction of the angular momentum, and in the system fully reduced by rotations.
Hence KAM theory can be applied to the planetary problem, with full strength. Chierchia-Pinzari's theorem gives some additional information on the secular system and allows Chierchia-Pinzari to give better measure estimates than what the weak non-degeneracy would give. (Un)fortunately, the torsion is undefinite, which prevents from using variational methods to (easily?) find invariant sets from Mather theory...

A Poincaré coordinates
For the Kepler Hamiltonian we will first define some symplectic action-angle coordinates, attributed to Delaunay, which blow up circular or horizontal Keplerian motions; blowing down the singularity will yield the symplectic analytic Poincaré coordinates. For elementary facts about K, we refer to [3]. For other ways to introduce those coordinates, see [15,Chap. vii], or [4,10,12,14]. Notations in the plane Kepler problem with negative energy: v = true anomaly = argument of q, measured from the pericenter a = semi major axis g = argument of the pericenter from Ox ǫ = eccentricity.
Restrict to the set {(v, a, g, ǫ), 0 < ǫ < 1} ≡ T 2 × R 2 of non-degenerate, non-circular, elliptical, Keplerian motions. Define coordinate t as the time from the pericenter; it is defined modulo the period T of the orbit. Define ℓ, the mean anomaly, as the angle obtained by rescaling time: ℓ := 2πt/T (mod 2π). Now, if we want an action coordinate L(K) conjugate to ℓ: dt ∧ dK = dℓ ∧ dL, we see that Conventionally choosing L = 0 at infinity where a = +∞, we get We now wish to define coordinates on the space of non-circular, nondegenerate, prograde Keplerian ellipses in the plane with fixed L. The angular momentum is a first integral of K and thus descends to the space of Keplerian orbits. Its Hamiltonian flow acts by 2π-periodic rotations around the origin. Define g as the angle, modulo 2π, measuring time along X G -orbits, and vanishing when the pericenter meets the Ox-semi-axis. Hence it suffices to show that {ℓ, g} = 0 in restriction to the section {ℓ = g = 0 (mod π)} of the L-and G-flows. We may thus assume that the body is on the major axis and that the major axis itself is the x-axis. But then the partial derivatives of ℓ and g with respect of x or p y are zero, and In the 3-dimensional Kepler problem, choose R 2 × {0} ⊂ R 3 as a reference plane, called horizontal. Temporarily restrict to non-horizontal, noncircular, non-degenerate, prograde elliptic Keplerian motions.
Let C = q × p be the angular momentum vector and Θ be its projection on the vertical axis. The flow of X Θ consists of 2π-periodic rotations in the horizontal plane (diagonally for positions and impulsions), leaving the horizontal 4-plane invariant.
Each Keplerian oriented plane meets the horizontal plane along a half axis, the ascending line of the node. Let θ be the angle measuring time along X Θ -orbits, vanishing when the line of the node is the Ox-semi-axis.
-The three Poisson brackets between angles can be checked to vanish as above in the plane. Indeed, on the submanifold {ℓ = g = θ = 0 (mod π)}, the partial derivatives of any of these angles with respect to x, p y or p z vanish. Now, define the Poincaré coordinates (λ, Λ, ζ, z) by the following formulas (several sign conventions exist): Knowing that the Delaunay coordinates are symplectic, it is straightforward to check that the Poincaré coordinates are symplectic too. From the above formulas, one checks that the Poincaré coordinates extend to continuous coordinates at direct circular coplanar motions (ζ = z = 0), since ζ = Λ/2 ǫ (1 + O(ǫ 2 )) e −ig z = Λ/2 ι 1 + O(ǫ 2 ) + O(ι 2 ) e −iθ , cos ι = Θ G (in particular, g and ι need not be defined at the singularity). In fact, their extension is analytic, as one can see by expressing the coordinates as explicit analytic functions of analytic first integrals; see [1] for an elegant choice of first integrals.
Thank you to A. Albouy, A. Chenciner, L. Chierchia, S. Kuksin and G. Pinzari for their enthusiasm and useful remarks.