The Planetary N–Body Problem

The planetary N –body problem∗ Luigi Chierchia Dipartimento di Matematica Università “Roma Tre” Largo S. L. Murialdo 1, I-00146 Roma (Italy) [email protected] March 29, 2012 (Preliminary draft) To appear in UNESCO Encyclopedia of Life Support Systems, Vol. 6.119.55 Celestial Mechanics. Eolss Publishers Co Ltd Contents 1 The N –body problem: a continuing mathematical challenge 2 2 The 2.1 2.2 2.3 2.4 classical Hamiltonian structure Newton equations and their Hamiltonian version . . . . The Linear momentum reduction . . . . . . . . . . . . Delaunay variables . . . . . . . . . . . . . . . . . . . . Poincaré variables and the truncated secular dynamics 3 Arnold’s planetary Theorem 3.1 Arnold’s Statements (1963) . . . . . . . . 3.2 Proper degeneracies and the “Fundamental 3.3 Birkhoff normal forms . . . . . . . . . . . 3.4 The planar three–body case (1963) . . . . 3.5 Secular Degeneracies . . . . . . . . . . . . 3.6 Herman–Fejóz proof (2004) . . . . . . . . . 3.7 Chierchia–Pinzari proof (2011) . . . . . . . ∗ . . . . . . Theorem” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 7 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 12 13 15 16 17 18 19 Partially supported by Prin 2009 project “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”. 1 2 Luigi Chierchia 4 Symplectic reduction of rotations 4.1 The Regularized Planetary Symplectic (RPS) variables . . . . . . 4.2 Partial reduction of rotations . . . . . . . . . . . . . . . . . . . . 19 20 23 5 Planetary Birkhoff normal forms and torsion 23 6 Dynamical consequences 6.1 Kolmogorov tori for the planetary problem . . . . . . . . . . . . . 6.2 Conley–Zehnder stable periodic orbits . . . . . . . . . . . . . . . . 25 25 26 References 26 1 The N –body problem: a continuing mathematical challenge The problem of the motion of N ≥ 2 point–masses (i.e., ideal bodies with no physical dimension identified with points in the Euclidean three–dimensional space) interacting only through Newton’s law of mutual gravitational attraction, has been a central issue in astronomy, physics and mathematics since the early developments of modern calculus. When N = 2 the problem has been completely solved (“integrated”) by Kepler: the motion takes place on conics, whose focus is occupied by the center of mass of the two bodies; but for N ≥ 3 a complete understanding of the problem is still far away. While the original impulse, coming from astronomy, has been somehow shaded by the massive use of machines for computing orbits of celestial bodies or satellites, the mathematical richness and beauty of the N –body problem has retained most of its original attraction; for a selection of recent contributions, see, e.g., [6], [15], [17], [5], [16], [11]. Here, we will be concerned with the planetary N –body problem, which, as the name says, deals with the case of one body (the “Sun” or the “Star”) having mass much bigger than the remaining bodies (“planets”). The main question is then to determine “general” conditions under which the planets revolve around the Sun without collisions and in an “regular way” so that, in particular, no planet crashes on another planet or onto the Sun, nor it escapes away from such “solar system”. Despite the efforts of Newton, Euler, d’Alembert, Lagrange, Laplace and, especially, Henri Poincaré and G.D. Birkhoff, such question remained essentially unanswered for centuries. It is only with the astonishing work of a 26–year–old mathematician, V.I. Arnold (1937–2010), that a real breakthrough was achieved. Arnold, continuing and extending fundamental analytical discoveries of his advisor A.N. Kolmogorov on so called “small divisors” (singularities appearing in the 3 The Planetary N –body problem perturbative expansions of orbital trajectories), stated in 1963 [1] a result, which may roughly formulated as follows1 . If the masses of the planets are small enough compared to the mass of the Sun, there exists, in the phase space of the planetary N –body problem, a bounded set of positive Lebesgue measure corresponding to planetary motions with bounded relative distances; such motions are well approximated by Keplerian ellipses with small eccentricities and small relative inclinations. Arnold gave a brilliant proof in a special case, namely, the planar three–body problem (two planets), giving some suggestions on how to generalize his proof to the general case (arbitrary number of planets in space). However, a complete generalization of his proof turned out to be quite a difficult task, which took nearly another fifty years to be completed: the first complete proof, based on work by M.R. Herman, appeared in [14] and a full generalization of Arnold’s approach in [11]. The main reason beyond the difficulties which arise in the general spatial case, is related to the presence of certain “secular degeneracies” which do not allow a tout court application of Arnold’s “fundamental theorem” (see § 3.2 below) to the general planetary case. In this article we shall give a brief account (avoiding computations) of these results trying to explain the main ideas and technical tools needed to overcome the difficulties involved. 2 The classical Hamiltonian structure 2.1 Newton equations and their Hamiltonian version The starting point are the Newton’s equations for 1 + n bodies (point masses), interacting only through gravitational attraction: ü(i) = X 0≤j≤n j6=i mj u(j) − u(i) , |u(i) − u(j) |3 i = 0, 1, ..., n , (1.1) (i) (i) (i)
where u(i) = u1 , u2 , u3 ∈ R3 are the cartesian coordinates of the ith body of pP √ 2 mass mi > 0, |u| = u · u = i ui is the standard Euclidean norm, “dots” over functions denote time derivatives, and the gravitational constant has been set to one (which is possible by rescaling time t). 1 Verbatim formulations are given in § 3.1 below. 4 Luigi Chierchia Equations (1.1) are invariant by change of “inertial frames”, i.e., by change of variables of the form u(i) → u(i) − (a + ct) with fixed a, c ∈ R3 . This allows to restrict the attention to the manifold of “initial data” given by2 n X n X mi u(i) (0) = 0 , i=0 mi u̇(i) (0) = 0 . (1.2) i=0 P The total linear momentum Mtot := ni=0 mi u̇(i) does not change along the flow of (1.1), i.e., Ṁtot = 0 along trajectories; therefore, by (1.2), MtotP (t) vanishes for all times. But, then, also the position of the barycenter B(t) := ni=0 mi u(i) (t) is constant (Ḃ = 0) and, again by (1.2), B(t) ≡ 0. In other words, the manifold of initial data (1.2) is invariant under the flow (1.1). Equations (1.1) may be seen as the Hamiltonian equations associated to the Hamiltonian function3 b := H N n X |U (i) |2 i=0 2mi − X mi mj , (i) − u(j) | |u 0≤i<j≤n (1.3) where (U (i) , u(i) ) are standard symplectic variables (U (i) = mi u̇(i) is the momentum conjugated to u(i) ) and the phase space is the “collisionless” open domain in R6(n+1) given by c := {U (i) , u(i) ∈ R3 : u(i) 6= u(j) , 0 ≤ i 6= j ≤ n} M endowed with the standard symplectic form n X i=0 2 dU (i) ∧ du(i) := X 0≤i≤n 1≤k≤3 (i) (i) dUk ∧ duk . (1.4) Replace the coordinates u(i) by u(i) − (a + ct) with a := m−1 tot n X mi u(i) (0) and c := m−1 tot i=0 n X mi u̇(i) (0) , mtot := i=0 n X mi . i=0 3 We recall that the Hamiltonian equations associated to a Hamiltonian function H(p, q) = H(p1 , ..., pn , q1 , ..., qn ), where Pn (p, q) are standard symplectic variables (i.e., the associated symplectic form is dp ∧ dq = i=1 dpi ∧ dqi ) are given by  ṗ = −∂q H q̇ = ∂p H i.e.  ṗi = −∂qi H , q̇i = ∂pi H , (1 ≤ i ≤ n) . (∗) We shall denote the standard Hamiltonian flow, namely, the solution of (∗) with initial data p0 and q0 , by φtH (p0 , q0 ). For general information, see [2]. 5 The Planetary N –body problem 2.2 The Linear momentum reduction In view of the invariance properties discussed above, it is enough to consider the submanifold n n X X (i) c c M0 := {(U, u) ∈ M : mi u = 0 = U (i) } , i=0 i=0 which corresponds to the manifold described in (1.2). c0 is symplectic, i.e., the restriction of the form (1.4) to M c0 The submanifold M 4 is again a symplectic form . Following Poincaré, one can perform a symplectic reduction (“reduction of the linear momentum”) allowing to lower the number of degrees of freedom5 by three units. Indeed, let φhe : (R, r) → (U, u) be the linear transformation given by  (0) u = r(0) , P u(i) = r(0) + r(i) , (i = 1, ..., n) φhe : (1.5) n U (0) = R(0) − i=1 R(i) , U (i) = R(i) , (i = 1, ..., n) ; such transformation is symplectic6 , i.e., n X dU (i) i=0 Letting ∧ du (i) = mtot := n X i=0 n X dR(i) ∧ dr(i) . (1.6) mi i=0 c0 reads one sees that, in the new variables, M n n X 6(n+1) (0) (0) −1 mi r(i) (R, r) ∈ R : R = 0 , r = −mtot i=1 o and 0 6= r(i) 6= r(j) ∀ 1 ≤ i 6= j ≤ n . c0 is simply The restriction of the 2–form (1.4) on M n X i=1 dR(i) ∧ dr(i) and b ◦ φhe )|M0 HN := (H N n  X X  R(i) · R(j) |R(i) |2 m0 mi  mi mj  + . − (i) = mi − 2 mm00+m |r(i) | m0 |r − r(j) | i i=1 1≤i<j≤n n X dU (i) ∧ du(i)
n X m0 + mi dU (i) ∧ du(i) . m 0 i=1 i=0 5 The number of degree of freedom of an autonomous Hamiltonian system is half of the dimension of the phase space (classically, the dimension of the configuration space). 6 We recall that this means, in particular, that in the new variables the Hailtonian flow is again standard: more precisely, one has that φtHb ◦ φhe = φhe ◦ φtHb ◦φ . 4 Indeed: c0 M = N N 6 Luigi Chierchia b on M c0 is equivalent to the dynamics genThus, the dynamics generated by H N 6n erated by the Hamiltonian (R, r) ∈ R → HN (R, r) on n M0 := (R, r) = (R(1) , ..., R(n) , r(1) , ..., r(n) ) ∈ R6n : o (i) (j) 0 6= r 6= r ∀ 1 ≤ i 6= j ≤ n P with respect to the standard symplectic form ni=1 dR(i) ∧ dr(i) ; to recover the c0 from the dynamics on M0 one will simply set R(0) (t) ≡ 0 full dynamics on M n X (0) −1 and r (t) := −mtot mi r(i) (t). i=1 Since we are interested in the planetary case, we perform the trivial rescaling by a small positive parameter µ: m0 := m0 , mi = µmi (i ≥ 1) , Hplt (X, x) := X (i) := R(i) , x(i) := r(i) , µ 1 H (µX, x) , µ N which leaves unchanged Hamilton’s equations. Explicitly, if m0 mi , and m̄i := m0 + µmi , Mi := m0 + µmi then Hplt (X, x) := n  X |X (i) |2 i=1 (0) 2Mi Mi m̄i − (i) |x |  (1)  X  X (i) · X (j) mi mj +µ − (i) m0 |x − x(j) | 1≤i<j≤n =: Hplt (X, x) + µHplt (X, x) , (1.7) the phase space being M := n (X, x) = (X (1) , ..., X (n) , x(1) , ..., x(n) ) ∈ R6n : o 0 6= x(i) 6= x(j) ∀ 1 ≤ i 6= j ≤ n , (1.8) P endowed with the standard symplectic form ni=1 dX (i) ∧ dx(i) . P Notice that while ni=1 X (i) is obviously not an integral7 P for Hplt , the transformation (1.5) does preserve the total angular momentum ni=0 U (i) × u(i) , “ × ” denoting the standard vector product in R3 , so that the total angular momentum C = (C1 , C2 , C3 ) := n X i=1 7 Ci , Ci := x(i) × X (i) , (1.9) We recall that F (X, x) is an integral for H(X, x) if {F, H} = 0 where {F, G} = FX · Gx − Fx · GX denotes the (standard) Poisson bracket. 7 The Planetary N –body problem k3 C(i) ..... ........ ....... ......... ........... ... .......... .. ... ................ ... ............. . . ... . . . . . . ... . ... .......... ... .......... ... ... ... ... ........................................................... ... ... .............. .. . . . ......... ... ........... ... ..... . . . ...... ....... ... .......... ... ..... . . . ....... .................... ... ...... . . . . ............ ... . . . . . ... . . . . . . ... ... ....... .. .... . . . ...... ... ............. .... ..... . . ........... . .. . . . . . .. ... . . . ......... ....... . . . . . . . . . . . . . . .. . .......... . .... .... . . . . . . . . . . . . . ... ....... ..... .... .... ... ... ........... ..... . ....... .. ... .. ...... ..... . ...... ... ... ... ...... .... . ....... ... ...... ... ... .................. .... . . . . . ...... . . ... . ...... .... ... ................ ... ... ...... ... .. ........... . ..... ...... .. .. ......... ..... .................. ... ............. ..... ........ . . . . . . . . . . . . . . . . . . ........ ................................. ... ..... ...... ............ ........ ..... . ... .. ..... ........ ........ .................. ........... ... ............................. ................................ .............. . ............ ........ . . . . . . . . . . . . ............ ...... .... ............ ...... .......... ........ ...... ............ .. ....... ................... ........ .............. ........ ........................ . . . . ....... ii x(i) • ℓi ν̄i := k3 × C(i) Pi gi O θi k1 ν̄i k2 Figure 1: Spatial Delaunay angle variables. is still a (vector–valued) integral for Hplt . The integrals Ci , however, do not commute (i.e., their Poisson brackets do not vanish): {C1 , C2 } = C3 , {C2 , C3 } = C1 , {C3 , C1 } = C2 , but, for example, |C|2 and C3 are two commuting, independent integrals. 2.3 Delaunay variables (0) The Hamiltonian Hplt in (1.7) governes the motion of n decoupled two–body problems with Hamiltonian (i) h2B = |X (i) |2 Mi m̄i − (i) , 2Mi |x | (X (i) , x(i) ) ∈ R3 × R3∗ := R3 × (R3 \{0}) . (1.10) Such two–body sytems are, as well known, integrable. The expilcit “symplectic integration” is done by means of the Delaunay variables, whose construction we, now, briefly, recall (for full details and proofs, see, e.g., [4, §3.2]). (i) Assume that h2B (X (i) , x(i) ) < 0 so that the Hamiltonian flow φt (i) (X (i) , x(i) ) h2B evolves on a Keplerian ellipse Ei and assume that the eccentricity ei ∈ (0, 1). Let ai , Pi denote, respectively, the semimajor axis and the perihelion of Ei . Let C(i) denote the ith angular momentum C(i) := x(i) × X (i) . Let us, also, introduce the “Delaunay nodes” ν̄i := k (3) × C(i) 1≤i≤n, (1.11) where (k (1) , k (2) , k (3) ) is the standard orthonormal basis in R3 . Finally, for u, v ∈ R3 lying in the plane orthogonal to a non–vanishing vector w, let αw (u, v) denote the positively oriented angle (mod 2π) between u and v (orientation follows the “right hand rule”). 8 Luigi Chierchia The Delaunay action–angle variables (Λi , Γi , Θi , ℓi , gi , θi ) are, then, defined as  √ Λi := Mi m̄i ai ℓi := mean anomaly of x(i) on Ei p  Γi := |C(i) | = Λi 1 − e2i gi := αC(i) (ν̄i , Pi ) (1.12)  Θi := C(i) · k (3) θi := αk(3) (k (1) , ν̄i ) Notice that the Delaunay’s variables are defined on the set where ei ∈ (0, 1) and the nodes ν̄i in (1.11) are well defined; on such set the “Delaunay inclinations” ii defined through the relations cos ii := Θi C(i) · k (3) = , |C(i) | Γi (1.13) are well defined and we choose the branch of cos−1 so that ii ∈ (0, π). The Delaunay variables become singular when C(i) is vertical (the Delaunay node is no more defined) and in the circular limit (the perihelion is not unique). In these cases different variables have to been used (see below). On the set where Delaunay variables are well posed, they define a s symplectic set of action–angle variables, meaning that8 n X i=1 dX (i) ∧ dx (i) = n X i=1 dΛi ∧ dℓi + dΓi ∧ dgi + dΘi ∧ dθi . (1.14) (0) In Delaunay action–angle variables ((Λ, Γ, Θ), (ℓ, g, θ)) the Hamiltonian Hplt takes the form n X Mi3 m̄2i − =: hk (Λ) . (1.15) 2Λ2i i=1 We shall restrict our attention to the phase space Mplt := n (Λ, Γ, Θ) ∈ R3n : Λi > Γi > Θi > 0 , (1.16) Λi √ Mi m̄i endowed with the standard symplectic form n X i=1 8 For a proof, see [4, §3.2]. 6= Λj √ Mj m̄j o , ∀ i 6= j × T3n , dΛi ∧ dℓi + dΓi ∧ dgi + dΘi ∧ dθi . 9 The Planetary N –body problem Notice that the 6n–dimensional phase space Mplt is foliated by 3n–dimensional (0) Hplt –invariant tori {Λ, Γ, Θ} × T3 , which, in turn, are foliated by n–dimensional tori {Λ}×Tn , expressing geometrically the degeneracy of the integrable Keplerian limit of the (1 + n)–body problem. 2.4 Poincaré variables and the truncated secular dynamics A regularization of the Delaunay variables in their singular limit was introduced by Poincaré, in such a way that the set of action–angle variables ((Γ, Θ), (g, θ)) is mapped onto cartesian variables regular near the origin, which corresponds to co–circular and co–planar motions, while the angles conjugated to Λi , which remains invariant, are suitably shifted. More precisely, the Poincaré variables are given by (Λ, λ, z) := (Λ, λ, η, ξ, p, q) ∈ Rn+ × Tn × R4n , with the Λ’s as in (1.12) and λi = ℓi + gi + θi   p ηi = p 2(Λi − Γi ) cos (θi + gi ) ξi = − 2(Λi − Γi ) sin (θi + gi ) p pi = p 2(Γi − Θi ) cos θi qi = − 2(Γi − Θi ) sin θi (1.17) Notice that ei = 0 corresponds to ηi = 0 = ξi , while ii = 0 corresponds to pi = 0 = qi ; compare (1.12) and (1.13). On the domain of definition, the Poincaré variables are symplectic9 n X i=1 dΛi ∧ dℓi + dΓi ∧ dgi + dΘi ∧ dθi = n X i=1 dΛi ∧ dλi + dηi ∧ dξi + dpi ∧ dqi . (1.18) As phase space, we shall consider a collisionless domain around the “secular origin” z = 0 (which correspond to co–planar, co–circular motions) of the form n 4n (Λ, λ, z) ∈ M6n p := A × T × B where A is a set of well separated semimajor axes  A := Λ : aj < aj < aj for 1 ≤ j ≤ n 9 For a proof, see [3, Appendix C]. (1.19) (1.20) 10 Luigi Chierchia where a1 , · · · , an , a1 , · · · , an , are positive numbers verifying aj < aj < aj+1 for any 1 ≤ j ≤ n, an+1 := ∞, and B 4n is a small 4n–dimensional ball around the secular origin z = 0. In Poincaré coordinates, the Hamiltonian Hplt (1.7) takes the form Hp (Λ, λ, z) = hk (Λ) + µfp (Λ, λ, z) , z := (η, p, ξ, q) ∈ R4n (1.21) where the “Kepler” unperturbed term hk is as in (1.15). Because of rotation (with respect the k (3) –axis) and reflection (with respect to the coordinate planes) invariance of the Hamiltonian (1.7), the perturbation fp in (1.21) satisfies well known symmetry relations called d’Alembert rules, namely, fp is invariant under the following transformations10  (Λ, λ, η, ξ, p, q) → (Λ, π − λ, −η, ξ, p, −q)      (Λ, λ, η, ξ, p, q) → (Λ, −λ, η, −ξ, −p, q) (Λ, λ, η, ξ, p, q) → (Λ, λ, η, ξ, −p, −q) (1.22)  π  (Λ, λ, η, ξ, p, q) → (Λ, 2 − λ, −ξ, −η, q, p)    (Λ, λ, z) → Rg (Λ, λ, z) Rg being a symplectic rotation by an angle −g ∈ T, i.e., a symplectic map of the form Rg : (Λ, λ, z) → (Λ′ , λ′ , z′ ) with and Λ′i = Λi , λ′i = λi + g, z′ = S g (z) , (1.23)   S g : z → S g (z) = Sg (z1 ), ..., Sg (z4n ) (1.24) acts as synchronous clock–wise rotation by the angle g in the symplectic zi – planes:   cos g sin g zi . (1.25) Sg (zi ) = − sin g cos g By such symmetries, in particular, the averaged perturbation Z 1 av fp (Λ, λ, z)dλ , fp (Λ, z) := (2π)n Tn (1.26) which is called the secular Hamiltonian, is even around the origin z = 0 and its expansion in powers of z has the form11 fpav = C0 (Λ) + Qh (Λ) · 10 11 u). p2 + q2 η2 + ξ2 + Qv (Λ) · + O(|z|4 ) , 2 2 (1.27) Compare [11, Eq. (6.6)] or [10, Lemma 3.4]. P Here, π − λ means (π − λ1 , ..., π − λn ), etc. Q · u2 denotes the 2–indices contraction i,j Qij ui uj (Qij , ui denoting the entries of Q, 11 The Planetary N –body problem where Qh , Qv are suitable quadratic forms, showing that z = 0 is an elliptic equilibrium for the secular dynamics (i.e, the dynamics generated by fpav ). The explicit expression of such quadratic forms can be found, e.g. , in [14, (36), (37)] (revised version). The truncated averaged Hamiltonian p2 + q2  η 2 + ξ2 + Qv (Λ) · hk + µ C0 (Λ) + Qh (Λ) · 2 2  (1.28) is integrable, with 3n commuting integrals given by Λi , ρi = η i 2 + ξi 2 , 2 ri = pi 2 + qi 2 , 2 (1 ≤ i ≤ n) ; the general trajectory fills a 3n–torus with n fast frequencies ∂Λi hk (Λi ) and 2n slow frequencies given by µΩ = µ(σ, ς) = µ(σ1 , · · · , σn , ς1 , · · · , ςn ) , (1.29) σi and ςi being the real eigenvalues of Qh (Λ) and Qv (Λ), respectively; such tori surround the n–dimensional elliptic tori given by {Λ} × {z = 0}, corresponding to n–coplanar and co–circular planets rotating around the Sun with Keplerian frequencies ∂Λi hk (Λi ). Figure 2: The truncated averaged planetary dynamics 3 Arnold’s planetary Theorem In the following section, we report some of Arnold’s statements concerning the existence of regular quasi–periodic motions for the planetary (1 + n)–body problem. We recall that, in general, a “quasi–periodic” (or “conditionally periodic”) orbit ζ(t) with (rationally independent) frequencies (ω1 , ..., ωd ) = ω ∈ Rd is a solution of the Hamilton equations of the form ζ(t) = Z(ω1 t, ..., ωd t) for a suitable smooth function Z(θ1 , ..., θd ) 2π–periodic in each variable θi . 12 3.1 Luigi Chierchia Arnold’s Statements (1963) At p. 87 of [1] Arnold says: Conditionally periodic motions in the many–body problem have been found. If the masses of n “planets” are sufficiently small in comparison with the mass of the central body, the motion is conditionally periodic for the majority of initial conditions for which the eccentricities and inclinations of the Kepler ellipses are small. Further, the major semiaxis perpetually remain close to their original values and the eccentricities and inclinations remain small. Later, [1, p. 125]: With the help of the fundamental theorem of Chapter IV 12 , we investigate in this chapter the class of “planetary” motions in the three–body and many–body problems. We show that, 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, it follows from our results that 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. Finally, [1, p. 127]: Our basic result is that if the masses, eccentricities and inclinations of the planets are sufficiently small, then for the majority of initial conditions the true motion is conditionally periodic and differs little from Lagrangian motion13 with suitable initial conditions throughout an infinite interval of time −∞ < t < +∞. As mentioned in the introduction, Arnold provides a full detailed proof, checking the applicability (non–degeneracy conditions) of his fundamental theorem, only for the two–planet model (n = 2) in the planar regime. As for generalizations, he states: The plane problem of n > 2 planets. The arguments of §2 and 3 easily carry over to the case of more than two planets. [· · · ] We shall not dwell on the details of the calculations which lead to the results of §1, 4. [1, p. 139]. 12 The “fundamental theorem” is a KAM (Kolmogorov–Arnold–Moser) theorem for properly– degenerate nearly–integrable Hamiltonian systems: it will be discussed in § 3.2 below. For generalities on KAM theory, see, e.g., [2] or [7]. 13 Arnold defines the “Lagrangian motions”, at p. 127 as follows: the Lagrangian motion is conditionally periodic and to the n “rapid” frequencies of the Kepler motion are added n (in the plane problem) or 2n − 1 (in the space problem) “slow” frequencies of the secular motions. This dynamics corrosponds, essentially, to the above “truncated integrable planetary dynamics”; the missing frequency in the space problem is related to the fact that one of the spacial secular frequency, say, ςn vanishes identically; compare § 3.5 below. 13 The Planetary N –body problem As for the spacial general case: The rather lengthy calculations involved in the solution of (3.5.9), the construction of variables satisfying conditions 1)–4), and the verification of non–degeneracy conditions analogous to the arguments of § 4 will not be discussed here. [1, p. 142]. In the next section we shall discuss Arnold’s strategy. 3.2 Proper degeneracies and the “Fundamental Theorem” The main technical tool is a KAM theorem for properly degenerate systems. A nearly–integrable system with Hamiltonian Hµ (I, ϕ) := h(I) + µf (I, ϕ) , (I, ϕ) ∈ Rd × Td , (1.30) for which h does not depend upon all the actions I1 ,...,Id is called properly degenerated. This is the case of the many–body problem since hk (Λ) in (1.15) depends only on n actions Λ1 ,...,Λn , while the number of degrees of freedom is d = 3n. In general, maximal quasi–periodic solutions (i.e., quasi–periodic solutions with d rationally–independent frequencies) for properly degenerate systems do not exist14 but they may exist under further conditions on the perturbation f . In [1, Chapter IV] Arnold overcome for the first time this problem proving the following result which he called “the fundamental theorem”. Let M denote the phase space n o M := (I, ϕ, p, q) : (I, ϕ) ∈ V × Tn and (p, q) ∈ B , where V is an open bounded region in Rn and B is a ball around the origin in R2m ; M is equipped with the standard symplectic form dI ∧ dϕ + dp ∧ dq = n X i=1 dIi ∧ dϕi + m X i=1 dpi ∧ dqi . Let, also, Hµ be a real analytic Hamiltonian on M of the form Hµ (I, ϕ, p, q) := h(I) + µf (I, ϕ, p, q) , and denote by f av the average of f over the “fast angles” ϕ: Z dϕ av . f (I, p, q) := f (I, ϕ, p, q) (2π)n Tn 14 (1.31) (1.32) Trivially, any unperturbed properly–degenerate system on a 2d dimensional phase space with d ≥ 2 will have motions with frequencies not rationally independent over Zd . 14 Luigi Chierchia Theorem 3.1 (Arnold 1963) Assume that f av is of the form f av = f0 (I) + m X 1 Ωj (I)rj + τ (I)r · r + o4 , 2 j=1 rj := p2j + qj2 , 2 (1.33) where τ is a symmetric (m × m)–matrix and lim(p,q)→0 |o4 |/|(p, q)|4 = 0. Assume, also, that I0 ∈ V is such that det h′′ (I0 ) 6= 0 , (1.34) det τ (I0 ) 6= 0 . (1.35) Then, in any neighborhood of {I0 } × Td × {(0, 0)} ⊆ M there exists a positive measure set of phase points belonging to analytic “KAM tori” spanned by maximal quasi–periodic solutions with n + m rationally–independent (Diophantine15 ) frequencies, provided µ is small enough. Let us make some remarks. (i) Actually, Arnold requires that f av is in Birkhoff normal form up to order 6, which means that f av = f0 (I) + m X 1 Ωj (I)rj + τ (I)r · r + P3 (r; I) + o6 2 j=1 (1.37) where P3 is a homogeneous polynomial of degree 3 in the variables ri (with I–dependent coefficients); but such condition can be relaxed and (1.33) is sufficient: compare [8], where Arnold’s properly degenerate KAM theory is revisited and various improvements obtained. (ii) Condition (1.34) is immediately seen to be satisfied in the general planetary problem16 . (iii) Condition (1.35) is a “twist” or “torsion” condition. It is actually possible to develop a weaker KAM theory where no torsion is required. This theory f av is assumed to be is due to Rüssmann [20], Herman and Féjoz [14], where Pm av in Birkhoff normal form up to order 2, f = f0 (I) + j=1 Ωj (I)rj + o2 , and the secular frequency map I → Ω(I) is assumed to be non–planar, meaning that no neighborhood of I0 is mapped into an hyperplane. 15 We recall that ω ∈ Rd is Diophantine if there exist positive constants γ and τ such that |ω · k| ≥ 16 γ , |k|τ ∀ k ∈ Zd \{0} . (1.36) The correspondence with the planetary Hamiltonian in Poincaré variables (1.21) is the following: m = 2n, I = Λ, ϕ = λ, z = (p, q), h = hk , f = fp . 15 The Planetary N –body problem (iv) Indeed, the torsion assumption (1.35) implies stronger results. First, it is possible to give explicit bounds on the measure of the the “Kolmogorov set”, i.e., the set covered by the closure of quasi–periodic motions; see [8]. Furthermore, the quasi–periodic motions found belong to a smooth family of non–degenerate Kolmogorov tori, which means, essentially, that the dynamics can be linearized in a neighborhood of each torus; see § 6.1 for more information. On the base of Theorem 3.1, Arnold’s strategy is to compute the Birkhoff normal form (1.33) of the secular Hamiltonian fpav in (1.26) and to check the non– vanishing of the torsion (1.35). 3.3 Birkhoff normal forms Before proceeding, let us recall a few known and less known facts about the general theory of Birkhoff normal forms. Consider as phase space a 2m ball Bδ2m around the origin in R2m and a real– analytic Hamiltonian of the form H(w) = c0 + Ω · r + o(|w|2 ) , where ( w = (u1 , . . . , um , v1 , . . . , vm ) ∈ R2m r = (r1 , . . . , rm ) , rj = u2j +vj2 2 (1.38) , . The components Ωj of Ω are called the first order Birkhoff invariants. The following is a classical by G.D. Birkhoff. Proposition 3.1 Assume that the first order Birkhoff invariants Ωj verify, for some a > 0 and positive integer s, |Ω · k| ≥ a > 0, m ∀ k ∈ Z : 0 < |k|1 := m X i=1 |kj | ≤ 2s . (1.39) → Then, there exists 0 < δ ′ ≤ δ and a symplectic transformation φ̆ : w̆ ∈ Bδ2m ′ w ∈ Bδ2m which puts H into Birkhoff normal form up to the order 2s, i.e., H ◦ φ̆ = c0 + Ω · r̆ + X Ph (r̆) + o(|w̆|2s )] , (1.40) 2≤h≤s where Ph are homogeneous polynomials in r̆j = |w̆j |2 /2 := (ŭ2j + v̆j2 )/2 of degree h. 16 Luigi Chierchia Less known is that the hypotheses of this theorem may be loosened in the case of rotation invariant Hamiltonians: this fact, for example, has not been used neither in [1] nor in [14]. First, let us generalize the class of Hamiltonian function so as to include the secular Hamiltonian (1.27): let us consider an open, bounded, connected set U ⊆ Rn and consider the phase space D := U × Tn × Bδ2m , endowed with the standard symplectic form dI ∧ dϕ + du ∧ dv. We say that a Hamiltonian H(I, ϕ, w) on D is rotation invariant if H ◦ Rg = H for any g ∈ T, where Rg i the symplectic rotation defined in (1.23) (replacing Λ, λ, z with, respectively, I, ϕ, w). Now, consider a ϕ–independent real–analytic Hamiltonian H : (I, ϕ, w) ∈ D → H(I, w) ∈ R of the form17 H(I, w) = c0 (I) + Ω(I) · r + o(|w|2 ; I) . (1.41) Then, it can be proven the following Proposition 3.2 Assume that H is rotation–invariant and that the first order Birkhoff invariants Ωj verify, for all I ∈ U , for some a > 0 and positive integer s n X m |Ω · k| ≥ a > 0, ∀ 0 6= k ∈ Z : ki = 0 and |k|1 ≤ 2s . (1.42) i=1 Then, there exists 0 < δ ′ ≤ δ and a symplectic transformation φ̆ : (I, ϕ̆, w̆) ∈ D̆ := → (I, ϕ, w) ∈ D which puts H into Birkhoff normal form up to the U × Tn × Bδ2m ′ order 2s as in (1.40) with the coefficients of Ph and the reaminder depending also on I. Furthermore, φ̆ leaves the I–variables fixed, acts as a ϕ̆–indepenent shift on ϕ̆, is ϕ̆–independent on the remaining variables and is such that φ̆ ◦ Rg = Rg ◦ φ̆ . (1.43) We shall call (1.39) the Birkhoff non–resonance condition (up to order s) and (1.42) the “reduced” Birkhoff non–resonance condition. The proof of Proposition 3.2 may be found in [11, §7.2]. 3.4 The planar three–body case (1963) In the planar case the Poincaré variables become simply (Λ, λ, z) := (Λ, λ, η, ξ) ∈ Rn+ × Tn × R2n , with the Λ’s as in (1.12) and λi = ℓi + gi , 17  p 2(Λi − Γi ) cos gi ηi = p ξi = − 2(Λi − Γi ) sin gi f = o(|w|2 ; I) means that f = f (I, w) and |f |/|w|2 → 0 as w → 0. . (1.44) 17 The Planetary N –body problem The planetary, planar Hamiltonian, is then given by Hp,pln (Λ, λ, z) = hk (Λ) + µfp,pln (Λ, λ, z) , z := (η, ξ) ∈ R2n and 1 (2π)n Z Tn av fp,pln =: fp,pln = C0 (Λ) + Qh (Λ) · η2 + ξ2 + O(|z|4 ) . 2 (1.45) (1.46) In [1, p.138, Eq. (3.4.31)], Arnold computed the first and second order Birkhoff invariants finding, in the asymptotics a1 ≪ a2 :   a   a 2 1  3 1 1   1 + O m m Ω = − 1 2 1   4 a2 a2 Λ1 a2 (1.47)  a 2 1   a    3 1 1   Ω2 = − m 1 m 2 1+O 4 a2 a2 Λ2 a2 (1.48) ! 9 3 5     − 4Λ1 Λ2 a2 a1 4 4Λ21 , (1.49) τ = m1 m2 13 1+O 3 9 − − a2 a2 4Λ1 Λ2 Λ22 which shows that the Ωj ’s are non resonant up to any finite order (in a suitable Λ–domain), so that planetary, planar Hamiltonian can be put in Birkhoff normal form up to order 4 and that the second order Birkhoff invariants are non–degenerate in the sense that18 det τ = −(m1 m2 )2 = − a41 117 (1 + o(1)) 16 a62 (Λ1 Λ2 )2 117 1 a31 (1 + o(1)) 6= 0 . 16 m20 a72 (1.50) This allow to apply Theorem 3.1 and to prove Arnold’s planetary theorem in the planar three–body (n = 2) case. An extension of this method to the spatial three–body problem, exploiting Jacobi’s reduction of the nodes and its symplectic realization, is due to P. Robutel [19]. 3.5 Secular Degeneracies In the general spatial case it is custumary to call σi the eigenvalues of Qh (Λ) and ςi the eigenvalues of and Qv (Λ), so that Ω = (σ, ς); compare (1.29). It turns out that such invariants satisfy identically the following two secular resonances n X ςn = 0 , (σi + ςi ) = 0 (1.51) i=1 18 In [1] the τij are defined as 1/2 of the ones defined here. 18 Luigi Chierchia and, actually, it can be shown that these are the only resonances identically satisfied by the first order Birkhoff invariants; compare [14, Proposition 78, p. 1575]. The first resonance was well known to Arnold, while the second one was apparently discovered by M. Herman in the 90’s and is now known as Herman resonance. Notice that: • the first resonance (ςn = 0), which is a resonance of order one, violates the usual Birkhoff non–resonance condition (1.39) for any s ≥ 1 but does not violate (1.42); • Herman resonance is a resonance of order (2n − 1) and violates (1.39) when (2s + 1)/2 ≥ n; while it does not violate (1.42); • combining the two resonances, also (1.42) is violated, for (s + 1)/2 ≥ n, by taking k=   1, ..., 1 , −(2n − 1) . | {z } (1.52) 2n−1 Another serious problem for Arnold’s approach is that the matrix τ indeed is degenerate, as clarified in [10], since   τ̄ 0 (1.53) τ= 0 0 τ̄ being a matrix of order (2n − 1). 3.6 Herman–Fejóz proof (2004) In 2004 J. Fejóz published the first complete proof of a general version of Arnold’s planetary theorem [14]. As mentioned above (remark (ii), §3.2), in order to avoid fourth order computations (and also because M. Herman seemed to suspect the degenaracy of the matrix of the second order Birkhoff invariant19 ), Herman’s approach was to use a first order KAM condition based on the non–planarity of the frequency map. But, the resonances (1.51) show that the frequency map lie in the intersection of two planes, violating the non–planarity condition. To overcome this problem Herman and Féjoz use a trick by Poincarè, consisting in modifying the Hamiltonian by adding a commuting Hamiltonian, so as to remove the degeneracy: by a Lagrangian intersection theory argument, commuting Hamiltonians have the same maximal transitive invariant tori, so that the KAM tori constructed for the modified Hamiltonian are indeed invariant tori also for 19 compare the Remark towards the end of p. 24 in [18]. 19 The Planetary N –body problem the original system. Now, the expression of the vertical component of the total angular momentum C3 has a particular simple expression in Poincaré variables, since n   X 1 C3 := Λj − (η2j + ξ2j + p2j + q2j ) , 2 j=1 so that the modified Hamiltonian Hδ := Hp (Λ, λ, z) + δC3 is easily seen to have a non–planar frequency map (Keplerian frequencies + first order Birkhoff invariants), and the above abstract remark applies20 . 3.7 Chierchia–Pinzari proof (2011) In [11] Arnold’s original strategy is reconsidered and full torsion of the planetary problem is shown by introducing new symplectic variables (called rps– variables21 ), which allow for a symplectic reduction of rotations eliminating one degree of freedom (i.e., lowering by two units the dimension of the phase space). In such reduced setting the first resonance in (1.51) disappears and the question about the torsion is reduced to study the determinant of τ̄ in (1.53), which, in fact, is shown to be non–singular; compare [11, §8] and [10] (where a precise connection is made between the Poincaré and the rps–variables). The rest of this article si devoted to explain the main ideas beyond this approach. 4 Symplectic reduction of rotations We start by describing the new set of symplectic variables, which allow to have a new insight on the symplectic structure of the phase space of the planetary model, or, more in general, of any rotation invariant model. The idea is to start with action–angle variables having, among the actions, two independent commuting integrals related to rotations, for example, the Euclidean length of the total angular momentum C and its vertical component C3 , and then (imitating Poincaré) to regularize around co–circular and co–planar configurations. The variables that do the job are an action–angle version of certain variables introduced by A. Deprit in 1983 [13] (see also [9]), which generalize to an arbitrary number of bodies Jacobi’s reduction of the nodes; the regularization has been done in [11]. 20 Actually, this idea is close to Herman’s original argument, while Fejóz uses a somewhat more abstract argument. 21 Regularized Planetary Symplectic variables; see § 4.1 below. 20 Luigi Chierchia C(i).. Pi := perihelion νi := S (i) × C(i) ν̄i := k(3) × C(i) ..... ........ ......... ... ... ... ... ... ... ... ... ... ... ... .......... ... ......... ... .... . . ... ...................... ................... ... ... .. ................... ... ... ... ................... ... .. .................................... .. . . .. ................ ... .................... .... ... ................... .... .. ................... ... ... .. ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ...... .. ................. ... ... ................................................. .................. ........... ... ... .................. ....... .. .. ............... .... .... ... ..... .. ............ ... .. ... . ... . . . . . . . . . . . . . . . .. ... ... .. ... .......... .......... ... ... ...... .. ......... .......... .... ... ..... ... ........ ............ ... . .. ........ ......... .......... .... ... ... .. ....... ............ . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . .. ..... ..... . ... ..... ....... . ........... ... ... ..... ...... ...... .......... .... ...... ... ... ..... ...... .......... .... ........... ............. .. .. ..... .......... ...... . ..... ........... .. ..... .......... ... . ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... ..... ..... ..... ... .......... ................ .......... ... ...... .. ..... .......... . ............. ... ....... ... ... .......... .. ... .......... ... ... ... ........ .. .......... . . . . . . . . ... ... . . . . . . . . . . . . .. . ... ................... ......... ... ... ......... ....... ... ... .......... ..... ... .............. ............ ....... ............. ................ .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................. ...... .. ................ ... ................ ... ................ .. ................ ................ ... . . . . . . . . . . . . . . . . . ..... . ................ ... ................ ... ................ ... ................ .. ................ . . . . . . . . . . . . . . . . . . ....... ... ........................ .................. S (i) k (3) • νi gi γi • Pi ν̄i Figure 3: The angle γi 4.1 The Regularized Planetary Symplectic (RPS) variables To define Deprit variables, consider the “partial angular momenta22 ” S (i) := i X C (j) , S j=1 (n) = n X C(j) =: C ; (1.54) j=1 and define the “Deprit nodes”   νi+1 := S (i+1) × C(i+1) , ν1 := ν2  νn+1 := k (3) × C =: ν̄ . 1≤i≤n−1 (1.55) The Deprit action–angle variables (Λ, Γ, Ψ, ℓ, γ, ψ) are defined as follows. The variables Λ, Γ and ℓ are in common with the Delaunay variables (1.12), while γi := αC(i) (νi , Pi ) Ψi := ψi :=   |S (i+1) | , C3 := C · k (3) 1≤i≤n−1 i=n αS (i+1) (νi+2 , νi+1 ) ζ := αk(3) (k (1) , ν̄) 1≤i≤n−1 i = n. Define also G := |C| = |S (n) |. 22 Recall the definition of the “individual” and total angular momenta in (1.9). (1.56) 21 The Planetary N –body problem C(i+1) .. ..... ........ ......... ... ... ... ....... ... ....... ... ........ ... .......... ... .... ... ..... .. ... .......... ... .. ... .. ... .... .. . . ... ... ... ... ... ... ... ... .. ... ... .. . . . . . . . . . . . . . . . . ... .. .... ................................ ... . .. ................. ..... .. ................. ..... .... ..... .. .................. ..... ... ....................... ..... . . . . . . . ................. . . . .. ... ... ................. ... ..... ... ... ................. .. ..... ................. ... ... .. ..... ................. . . . . . . . . ................. ... ... .. ... . . ................. . . . . . ...... .. ................. ... . . . . . ................. ...... .. ... . ................. . . . . . . ...... .. ..... ... . . . . . ...... ..... .... . . . . . ..... .... ... . . ..... . . . . . ..... ... .... ... . . . . . . . . . . . . ... .. .... .... ..... ........ ..... ..... ..... ..... ..... ........ ..... ......... ..... . ..... ..... ................ ..................... . . . . . . . . . . ..... ..... ..... ...... .... ..... ..... ..... ..................... ..... ................. .... . . . . ................. .... ................. ..... ................. ................. ..... ................. ..... ................. ..... . . . ................. . ................. ..... ................. (i+1).......... ................. ................. .... ................. ..... . ................. . . . .. ................. ...................... C νi := S (i) × C(i) (i+2) S (i+1) • νi+2 ψi νi+1 πS Figure 4: The angle ψi for 1 ≤ i ≤ n − 2 The “Deprit inclinations” ιi are defined through the relations  (i+1) (i+1) C ·S   , 1≤i≤n−1 ,   (i+1)  |C ||S (i+1) | cos ιi :=   C · k (3)    , i=n. |C| (1.57) Similarly to the case of the Delaunay variables, the Deprit action–angles variables are not defined when the Deprit nodes νi vanish or ei ∈ / (0, 1); on the domain where they are well defined they define a real–analytic set of sympectic variables, i.e., n n X X (i) (i) dX ∧ dx = dΛi ∧ dℓi + dΓi ∧ dγi + dΨi ∧ dψi . (1.58) i=1 i=1 The rps variables are given by (Λ, λ, z) := (Λ, λ, η, ξ, p, q) with (again) the Λ’s as in (1.12) and p
 n ηi = p 2(Λi − Γi ) cos γi + ψi−1 n
λi = ℓi + γi + ψi−1 n ξi = − 2(Λi − Γi ) sin γi + ψi−1 (1.59) ( p pi = p 2(Γi+1 + Ψi−1 − Ψi ) cos ψin qi = − 2(Γi+1 + Ψi−1 − Ψi ) sin ψin where Ψ0 := Γ1 , Γn+1 := 0 , ψ0 := 0 , ψin := X i≤j≤n ψj . (1.60) 22 Luigi Chierchia C. ...... ......... ......... ... ... ... ... ... ... . ... ....... ... ....... ... .......... .. ... ... .... ... ... .... ... ... ....................... ................... (3) ... ... .. ................... ... ... .. ................... ... .. .. .................. . . . . . . . . . . . . . . . . . . . . ... ........................ ... .................... ... ................... ..... .... (n) ................... ... ... . ... ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .. ...... ................. ... ... ... ......... ............ ... ........ ... ... ... ... .. .. .. .. ... ...... ... . . . . ...... ... ... ... ......... .. ... ......... ... ...... ... ... ... .......................................................................................................................... ... . . . ......... ...................................................................................... ... ....... ....................................................................................................................... .. ... ..... ..... .. ..... ..... . .. .... ..... .............................. ..... ... .... ..... ..... ........ ........... .. . . . . . . . . . . . . . . .. .. . .... .... ...... ... ........... .......................... ... ..... ... .............. ........ ..... .... ....... ... ...... ..... ..... .... ... . . . . . . . . . . . .... .. ..... ... .... ..... .. ... ..... ..... ....... ... ................. ..... .. ........................ . . . ..... . . . . . . . . . . . . . . . . . ..... .. ... .... ................ .... ... ..... ................ ..... ..... ... ................ .... .. ..... ................ ..... ................ ..... . ... . . . . . . . . . . . . . . . . . . . . . . . . ..... . ... ..... ................ ... ..... .... ................ ... ..... ..... ................ .... ... ................ ..... ..... ................ . . ... . . . ..... . . . . . . . . . . . . . . . . . . . .......... ... .... ... ................ .... ..... ................ ..... ..... .... ..... (3) ..... ..... . . . . . . . . . . ..... ..... .... ................................................. ..... ................................................................................................................... .................................................................................................................................... C(n) ν̄ := k ×C πC k (3) νn := C × C • k (1) k (2) g ζ ν̄ νn πk Figure 5: The angles ψn−1 =: g and ψn =: ζ The phase space in the rps variables has the same form as in (1.19)–(1.20), namely n 4n (Λ, λ, z) ∈ M6n (1.61) rps := A × T × B with B a 4n–dimensional ball around the origin. The relation between Poincaré variables and the rps variables is rather simple. Indeed, if we denote by φrps : p (Λ, λ, z) → (Λ, λ, z) (1.62) the symplectic trasformation between the rps and the Poincaré variables, one has the following Theorem 4.1 ([10]) The symplectic map φrps in (1.62) has the form p λ = λ + ϕ(Λ, z) z = Z(Λ, z) (1.63) where ϕ(Λ, 0) = 0 and, for any fixed Λ, the map Z(Λ, ·) is 1:1, symplectic23 and its projections verify, for a suitable V = V(Λ) ∈ SO(n), Πη Z = η + O3 , Πξ Z = ξ + O3 , Πp Z = Vp + O3 , Πq Z = Vq + O3 , (1.64) where O3 := O(|z|3 ), 23 I.e., it preserves the two form dη ∧ dξ + dp ∧ dq. 23 The Planetary N –body problem 4.2 Partial reduction of rotations Recalling that Γn+1 = 0 , Ψn−1 = |S (n) | = |C| , one sees that  Ψn = C3 , ψn = αk(3) (k (1) , k3 × C) (1.65) p pn = p 2(|C| − C3 ) cos ψn qn = − 2(|C| − C3 ) sin ψn , (1.66) showing the the conjugated variables pn and qn are both integrals and hence both cyclic for the planetary Hamiltonian, which, therefore, in such variables, will have the form Hrps (Λ, λ, z̄) = hk (Λ) + µfrps (Λ, λ, z̄) , (1.67) where z̄ denote the set of variables
z̄ := (η, ξ, p̄, q̄) := (η1 , . . . , ηn ), (ξ1 , . . . , ξn ), (p1 , . . . , pn−1 ), (q1 , . . . , qn−1 ) . (1.68) In other words, the phase space M6n rps in (1.61) is foliated by (6n−2)–dimensional invariant manifolds Mp6n−2 := M6n (1.69) rps |pn ,qn =const , n ,qn and since the restriction of the standard symplectic form on such manifolds is symply dΛ ∧ dλ + dη ∧ dξ + dp̄ ∧ dq̄ , such manifolds are symplectic and the planetary flow is the standard Hamiltonian flow generated by Hrps in (1.67). We shall call the symplectic, invariant submanifolds Mp6n−2 “symplectic leaves”. They depend upon a particular orienn ,qn tation of the total angular momentum: in particular, the leaf M6n−2 correspond 0 to the total angular momentum parallel to the vertical k3 –axis. Notice, also, that the analytic expression of the planetary Hamltonian Hrps is independent of the leaves. In view of these observations, it is enough to study the planetary flow of Hrps on, say, the vertical leaf M06n−2 . 5 Planetary Birkhoff normal forms and torsion The rps variables share with Poincaré variables the D’Alembert symmetries (i.e. invariance under (1.22)); compare [10, Lemma 3.4]. As for Poincaré variables, this implies that the averaged perturbation Z 1 av frps := frps dλ (2π)n Tn 24 Luigi Chierchia also enjoys D’Alembert rules and thus has an expansion analogue to (1.27), but independent of (pn , qn ): av frps (Λ, z̄) = C0 (Λ) + Qh (Λ) · p̄2 + q̄ 2 η2 + ξ 2 + Q̄v (Λ) · + O(|z̄|4 ) 2 2 (1.70) with Qh of order n and Q̄v of order (n − 1). Notice that the matrix Qh in (1.70) is the same as in (1.27), since, when p = (p̄, pn ) = 0 and q = (q̄, qn ) = 0, Poincaré and rps variables coincide. Using Theorem 4.1, one can also show that   Q̄v 0 Qv := (1.71) 0 0 is conjugated (by a unitary matrix) to Qv in (1.27), so that the eigenvalues ς¯i of Q̄v coincide con (ς1 , ..., ςn−1 ), as one naively would expect. In view of the remark after (1.51), and of the rotation–invariant Birkhoff theory (Proposition 3.2), one sees that, one can construct, in an open neighborhood of av co–planar and co–circular motions, the Birkhoff normal form of frps up to any finite order. More precisely, for ǫ > 0 small enough, denoting Pǫ := A × Tn × Bǫ4n−2 , Bǫ4n−2 := {z̄ ∈ R4n−2 : |z̄| < ǫ} , an ǫ–neighborhood of the co–circular, co–planar region, one can find, for µ small enough, a real–analytic symplectic transformation φµ : (Λ, λ̆, z̆) ∈ Pǫ → (Λ, λ, z̄) ∈ Pǫ such that H̆ := Hrps ◦ φµ = hk (Λ) + µf (Λ, λ̆, z̆) (1.72) with f˘av (Λ, z̆) := 1 (2π)n Z Tn f dλ̆ = C0 (Λ) + Ω · R̆ + 1 τ̄ R̆ · R̆ + O(|z̆|6 ) 2 (1.73) where  Ω = (σ, ς¯)     z̆ := (η̆, ξ, ˘ p̆, q̆) ∈ B 4n−2 , ǫ R̆ = (ρ̆, r̆) , ρ̆ = (ρ̆  1 , · · · , ρ̆n ) ,   η̆i2 +ξ̆i2 p̆2i +q̆i2  ρ̆i := 2 , r̆i = 2 . r̆ = (r̆1 , · · · , r̆n−1 ) , With straightforward (but not trivial) computations, one can then show full torsion for the planetary problem. More precisely, one finds ([11, Proposition 8.1]) 25 The Planetary N –body problem Proposition 5.1 For n ≥ 2 and 0 < δ⋆ < 1 there exist24 µ̄ > 0, 0 < a1 < a1 < · · · < an < an such that, on the set A defined in (1.20) and for 0 < µ < µ̄, the matrix τ̄ = (τij ) is non–singular: det τ̄ = dn (1 + δn ) , and dn = (−1)n−1 6 with |δn | < δ⋆  a 3 Y  1  4 3  45 1 n−1 m2 1 . a 1 5 16 m20 m1 m0 an 2≤k≤n ak (1.74) Dynamical consequences 6.1 Kolmogorov tori for the planetary problem At this point one can apply to the planetary Hamiltonian in normalized variables H̆(Λ, λ̆, z̆) Arnold’s Theorem 3.1 above completing Arnold’s project on the planetary N –body problem. Indeed, by using the refinements of Theorem 3.1 as given in [8], from Proposition 5.1 there follows Theorem 6.1 There exists positive constants ǫ∗ , c∗ and C∗ such that the following holds. If ǫ6 , (1.75) 0 < ǫ < ǫ∗ , 0<µ< (log ǫ−1 )c∗ then each symplectic leaf Mp6n−2 (1.69) contains a positive measure Hrps –invariant n ,qn Kolmogorov set Kpn ,qn , which is actually the suspension of the same Kolmogorov set K ⊆ Pǫ , which is H̆–invariant. Furthermore, K is formed by the union of (3n − 1)–dimensional Lagrangian, real– analytic tori on which the H̆–motion is analytically conjugated to linear Diophantine quasi–periodic motions with frequencies (ω1 , ω2 ) ∈ Rn ×R2n−1 with ω1 = O(1) and ω2 = O(µ). Finally, K satisfies the bound  √ meas Pǫ ≥ meas K ≥ 1 − C∗ ǫ meas Pǫ . In particular, meas K ≃ ǫ4n−2 ≃ meas Pǫ . 24 µ̄ is taken small only to simplify (1.74), but a similar evaluation hold with µ̄ = 1. (1.76) 26 Luigi Chierchia 6.2 Conley–Zehnder stable periodic orbits Indeed, the tori T ∈ K form a (Whitney) smooth family of non–degenerate Kolmogorov tori, which means the following. The tori in K can be parameterized by their frequency ω ∈ R3n−1 (i.e., T = Tω ) and there exists a real–analytic symplectic diffeomorphism ν : (y, x) ∈ B m × Tm → ν(y, x; ω) ∈ Pǫ , m := 3n − 1 , (1.77) uniformly Lipschitz in25 ω such that, for each ω a) H̆ ◦ ν = E + ω · y + Q; (Kolmogorov’s normal form) b) E ∈ R (the energy of the torus); ω ∈ Rm is a Diophantine vector; c) Q = O(|y|2 ) Z ∂yy Q(0, x) dx 6= 0 , d) det (nondegeneracy) Tm e) Tω = ν(0, Tm ). Now, in the first paragraph of [12] Conley and Zehnder, putting KAM theory (and in particular exploiting Kolmogorv’s normal form for KAM tori) together with Birkhoff–Lewis fixed–point theorem show that long–period periodic orbits cumulate densely on Kolmogorov tori so that, in particular, the Lebesgue measure of the closure of the periodic orbits can be bounded below by the measure of the Kolmogorov set. Notwithstanding the proper degeneracy, this remark applies also in the present situation and as a consequence of Theorem 6.1 and of the fact that the tori in K are non–degenerate Kolmogorov tori it follows that in the planetary model the measure of the closure of the periodic orbits in Pǫ can be bounded below by a constant times ǫ4n−2 . References [1] V. I. Arnold. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk, 18(6 (114)):91–192, 1963. English translation in: Russian Math. Surveys, 18(6):85–191, 1963. [2] V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics, volume 3 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, third edition, 2006. [Dynamical systems. III], Translated from the Russian original by E. Khukhro. 25 Even more: C ∞ in the sense of Whitney. The Planetary N –body problem 27 [3] L. Biasco, L. Chierchia, and E. Valdinoci. 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