ON THE STABILITY OF SOME PROPERLY DEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. Luca Biasco. Luigi Chierchia

DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 9 Number March 3 pp. 33 6 ON THE STABILITY OF SOME PROPERLY DEGENERATE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM Luca Biasco Settore di Analisi Funzionale SISSA/ISAS Via Beirut 4 343 Trieste Italy Luigi Chierchia Dipartimento di Matematica Università Roma Tre Largo S. L. Murialdo 46 Roma Italy Communicated by Antonio Ambrosetti Abstract. Properly degenerate nearly integrable Hamiltonian systems with two degrees of freedom such that the intermediate system depend explicitly upon the angle variable conjugated to the non degenerate action variable are considered and in particular model problems motivated by classical examples of Celestial Mechanics are investigated. Under suitable convexity assumptions on the intermediate Hamiltonian it is proved that in every energy surface the action variables stay forever close to their initial values. In non convex cases stability holds up to a small set where in principle the degenerate action variable might in exponentially long times drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a blow up complex analysis near separatrices KAM techniques and energy conservation arguments.. Introduction and results. As pointed out with particular emphasis by H. Poincaré [8] one of the main problem in Dynamical Systems concerns the stability of action variables in nearly integrable Hamiltonian systems. Notwithstanding the efforts of Poincaré himself and the great success of powerful more modern techniques such as averaging theory KAM and Nekhoroshev theory see [] for general information the action stability problem for general nearly integrable Hamiltonian systems remains essentially open. By action stability problem we mean the following. Consider a real analytic nearly integrable one parameter family of Hamiltonian functions HI ϕ; ε = hi + εfi ϕ where I ϕ are standard symplectic action angle variables in a d dimensional phase space the angles ϕ i are defined modulus π and ε is a small parameter. The problem is then to give upper bounds on the quantity It I where It ϕt denotes the H flow at time t of the initial datum I ϕ and stability means that sup t It I goes to zero when ε goes to zero. 99 Mathematics Subject Classification. 37C75 7H8 7K43 7K45 34D 34C9. Key words and phrases. KAM theory stability nonlinear dynamics Hamiltonian systems proper degeneracies fast averaging action-angle variables. Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations. 33

34 L. BIASCO AND L. CHIERCHIA V.I. Arnold in 963 in one of the fundamental paper on KAM theory [] conjectured that the general feature of nearly integrable Hamiltonian dynamics with more than two degrees of freedom i.e. with phase space of dimension greater than four is action instability. On the other hand KAM theory yields in general metric stability i.e. stability for the majority of initial data and this implies total stability in systems with two degrees of freedom: in such a case under suitable non degeneracy assumptions the three dimensional energy surfaces are separated by a multitude of two dimensional invariant KAM tori and trajectories are trapped in between these tori allowing only for a small with ε variation of the action variables for any time and for any initial data. The non degeneracy KAM assumption at fixed energy is that the unperturbed map between action variables on fixed energy surface and the frequency map viewed in projective space is a diffeomorphysm. The main motivation for Poincaré to look up at the action stability problem came from Celestial Mechanics. Now a typical feature in Celestial Mechanics is that the unperturbed system is properly degenerate i.e. the unperturbed Hamiltonian function does not depend upon all action variables. In such a case the above mentioned non degeneracy condition is obviously strongly violated. However in [] Arnold proved the following result compare also [] Chapter 5 Section 3. Consider a nearly integrable real analytic Hamiltonian system with two degrees of freedom governed by HI ϕ; ε := H I; ε + ε H I ϕ := H I + εh I + ε H I ϕ. I ϕ = I I ϕ ϕ U T where U R and T denotes the standard two dimensional torus R /πz. We say that the perturbation removes the degeneracy 3 on the energy level H E if H I I H I I I H E.. Theorem.. []If in a real analytic properly degenerate system with two degrees of freedom the perturbation removes the degeneracy i.e. condition. holds then for all ε small enough total stability holds i.e. for all initial data on the given energy level the values of the action variables stay forever near their initial values. Remark. i If condition. is violated instability channels may appear as suggested by the following example which is a trivial modification of an example due to N.N. Nekhoroshev [7]. Let H I + εh I := I εi.3 and notice that the first inequality in condition. is violated on each energy level crossing the axis {I = } in particular is violated at E =. Then one Arnold calls it topological instability and the conjecture is the following: near an arbitrary point in phase space there are trajectories along which the action variables undergo a displacement of order one i.e. independently of ε in a finite albeit exponentially long time. This happens for example in three body problems or in the D Alembert planetary model. In Appendix A below the D Alembert planetary model is discussed in detail. 3 Or more precisely that the intermediate term H removes the degeneracy.

ON THE STABILITY OF... 35 can construct a sequence ε j and a sequence of perturbations H j ϕ with H j ϕ uniformly bounded such that sup Imϕ i I ε t := e / ε ε t ε 3/ t ϕ ε t := e / ε ε t ε5/ t.4 is a solution of the Hamilton equation associated to H I + εh I + ε H j ϕ when ε = ε j. In fact it is enough to take ε j := j H jϕ := e j sinϕ jϕ. Notice that a displacement of order one of the action variables I ε t with respect to their initial value I ε = occurs in the exponentially long time exp/ ε j /ε j. ii Condition. is violated at E = also by the convex Hamiltonian H := I + ε I ε >. However in such a case H consists only of one point and exploiting convexity and using energy conservation arguments it is not difficult to show that also on the energy level E = total stability holds for ε > small enough. It is therefore clear that convexity or more in general steepness should play a fundamental role in this business. Properly degenerate systems with two degrees of freedom of the form. are in general more integrable than non degenerate systems as A.I. Nejshtadt proved in 98: Theorem.. [6] Assume that a real analytic properly degenerate system with two degrees of freedom satisfies condition. together with H I. Then the measure of the set of unperturbed tori that disappear when ε > is exponentially small i.e. Oexp const/ε rather than O ε as in general nondegenerate systems. Furthermore the deviation of a perturbed torus from the unperturbed one is of Oε rather than O ε. In this paper we take up the action stability problem for properly degenerate Hamiltonian system with two degrees of freedom allowing the intermediate system H to depend also on the angle ϕ. Thus we shall consider real analytic properly degenerate systems with two degrees of freedom described by nearly integrable real analytic Hamiltonians given by HI ϕ; ε := H I +εh I ϕ +ε a H I ϕ < ε a >..5 The interest for such systems stems again from Celestial Mechanics. For example the planetary D Alembert model describing the motion of a nearly spherical planet subject to the gravitational attraction of a fixed star occupying a focus of a Keplerian nearly circular ellipse along which the centre of mass of the planet revolves is governedup to an exponentially small term by a Hamiltonian of the form H D I I ϕ ϕ ; ε µ := I + ε ĉ I I + ˆd I I cos ϕ + ε a GI I ϕ ϕ ; ε µ.6 where: ε and µ ε c with c > / are perturbation parameters related respectively to the oblateness of the planet and to the eccentricity of the Keplerian orbit; a 3/ ]; ĉ ˆd and G are given real analytic functions uniformly bounded in suitable analytic norms; see Appendix A for a full description of this model.

36 L. BIASCO AND L. CHIERCHIA Remark. i In the above D Alembert model the intermediate system is given by H := I + εĉ I I. It turns out that in physically interesting phase space regions 4 H is non convex. For this reason below we shall consider also non convex models. ii The planetary D Alembert model motivated in [4] new investigations about action instability Arnold Diffusion. For such studies of course exponentially small terms cannot be disregarded. In relation with the full D Alembert problem the results presented here go in the direction of giving action stability bounds for exponentially long times. Such bounds would not immediately follow from standard Nekhoroshev techniques because of the strong degeneracies of the model 5. iii The dependence of H upon the angle ϕ that is on the angle conjugated to the non degenerate action I besides being motivated by classical examples is the only significative angle dependence one wants to take into account in connection with the problems considered here. In general in fact a Hamiltonian function of the form H I + εh I ϕ + ε H I ϕ will be trivially unstable as the following example shows. Let H = I + cos ϕ and H =. Then one has sup t I t I = for any ε > and for any motion with I ϕ belonging to the open separatrix of the pendulum H. Moreover these hyperbolic motions would be persistent under non vanishing perturbations H. The Hamiltonian H I ϕ ; ε := H I + εh I ϕ.7 regarded as a one degree of freedom system in the I ϕ variables is still integrable exhibiting in general the typical features of a one degree of freedom dimensional system phase space regions foliated by invariant circles of possibly different homotopy stable/unstable equilibria separatrices etc.. A natural approach which we shall in fact follow is to introduce action angle variables for the one degree of freedom Hamiltonian H I ϕ ; ε regarding I as a dumb parameter and then to apply KAM techniques trying to confine all motions among KAM tori as in the non degenerate case. The problem with this approach is that the action angle variable for the I ϕ system are singular in any neighbourhood of the separatrix and stable equilibria and is exactly near separatrices where one expects the motion to become chaotic and where in principle drift of order one in the I variable is conceivable 6 even in the two degrees of freedom properly degenerate case considered here. Therefore a careful analysis near these singular phase space regions is needed and arguments different from KAM theory have to be used to control the displacement of the action variable in such singular regions. Clearly as discussed in Remark regions where the non degeneracy assumption fails need a separate discussion: in fact in such zones and in the non convex case we can not exclude a possibly non chaotic drift of the I action. 4 Such regions correspond to unperturbed situations in which the spin axis of the planet is nearly orthogonal to ecliptic plane i.e. to the plane containing the Keplerian ellipse: this is the observed situation for most planets in the Solar system. 5 For other investigations on exponential Nekhoroshev stability in Celestial Mechanics we refer also to [3]. 6 Better: compatible with energy conservation.

ON THE STABILITY OF... 37 To avoid extra technical difficulties we shall consider in this paper model problems namely we shall let H := I H := H σ := σ I + cos ϕ.8 with σ equal either + or ; the phase space will be taken to be M R := BR T where BR denotes a ball of radius R around the origin. Remark 3. These model problems are intended to capture the main features of general properly degenerate systems with two degrees of freedom and in particular the features of the exponential approximation.6 to the D Alembert Hamiltonian. This is the reason for considering both the convex and the non convex case in.8 corresponding respectively to σ = and σ = compare also point i of Remark. We can now state our main results. Denote as above by It ϕt := φ t H I ϕ the time t evolution of the initial data I ϕ := I ϕ governed by the Hamiltonian H. We shall prove the following Theorem.3. Let H σ I ϕ; ε := HI ϕ; ε and M R be as in.5.8. Assume a > 3/ and choose { < R < R and < b < min 4 a a 3 4 3 }..9 Then there exists ε > such that for all < ε < ε the φ t H evolution It ϕt of an initial datum I ϕ satisfies It < R It I < ε b t R. where in the case σ = I ϕ is an arbitrary point in the phase space M R while in the non convex case σ = I ϕ belongs to M R \N N being an open region whose measure does not exceed ε /3. This theorem will be a simple corollary of the following result which describes the distribution and density of KAM tori. Let H p denote the pendulum Hamiltonian 7 H p := H p I ϕ ; ε := I ε + cos ϕ.. Theorem.4. Let the hypotheses and choices of Theorem.3 hold and let M σ := M R \N σ where the sets N σ := N σ ε b are defined by { N := I ϕ : H p < ε +b or H p < ε + ε +b} {I ϕ : I < Rε b} N := {I ϕ : c ε 3 + 4 3 b < H p < ε } 3 c N := N N. < c < being a suitable constant. Fix q such that < q < a 3 3b..3 7 H p is a standard mathematical pendulum having the stable equilibrium in with energy ε the unstable equilibrium in ±π with energy hence the separatrix as well has energy.

38 L. BIASCO AND L. CHIERCHIA Then there exists ε > such that for all < ε < ε the following holds. Apart from a small dense subset of measure Oexp /ε q the region M σ is filled up by two dimensional real analytic H σ invariant tori; each of these tori is Oexp /ε q close to an unperturbed torus {I ϕ : H p = E} {I ϕ s.t. I =const} in M σ. Furthermore for any motion It ϕt in M σ the displacement of It from its initial value I is bounded for all times t by ε. Remark 4. i By simple energy conservation argument one sees immediately that I t I < const ε for any motion It ϕt in M R ; thus the stability statement in. concerns actually only the I action variable. ii The discarded region N σ is a elementary set small with ε. If we replace N σ by a small set of order one say {I ϕ : H p < δ} {I ϕ : I < δ} for a fixed < δ then the displacement of It from its initial value I is bounded by ε. iii In the two degrees of freedom case considered here as mentioned above the -dimensional KAM tori constructed in Theorem.4 which fill up to an exponentially small set the region M σ separate the three dimensional energy levels. Thus the topological trapping argument may be applied leading to stability for all times of the action variables in M σ. Then an elementary energy conservation argument implies action stability in M R or in M R \N according to whether σ = or σ =. iv In the case a = one can take any < b < /6 and q < 3b. v Theorem.3 and Theorem.4 may be viewed as extensions in the model cases considered here of respectively Theorem. and.. We close this introduction with a list of problems. Generalise Theorem.3 and.4 to the Hamiltonian H D see Appendix A for a full description of H D and deduce exponential stability estimates for the full D Alembert planetary Hamiltonian. Find general conditions on H under which Theorems.3 and.4 hold. 3 Extend the example in Remark to ε independent perturbations H. Extend the example in Remark to H dependent also on ϕ. 4 The examples in 3 may indicate a possible route to O drift of action variables in properly degenerate systems different from Arnold Diffusion. The paper is organized as follows. In we list the technical tools we need in order to prove Theorems.3 and.4 namely: a quantitative accurate discussion of the real analytic extension of action angle variable for the pendulum with particular care to singular regions; an averaging or normal form lemma standard in Nekhoroshev theory; a quantitative iso energetically KAM theorem. In 3 the proofs of the Theorem.3 and.4 are given. In Appendix A we discuss the D Alembert planetary model and show how averaging theory may be used to reduce it up to an exponentially small term to the form in.6. In Appendix B the lengthy but elementary details for the construction of the real analytic action angle variables for the pendulum are provided. Acknowledgements. In an early stage of this work LC benefited of thorough and enlightening discussions with Carlangelo Liverani. We thank Alessandra Celletti and Enrico Valdinoci for their interest in our work and for various comments. LC gratefully acknowledges pleasant visits during the academic year / at the SISSA/ISAS of Trieste where part of this work has been carried out.

ON THE STABILITY OF... 39. Preliminaries. The construction of KAM tori in M σ is based on the following three lemmata: the first lemma provides real analytic action angle variable for the pendulum slightly away from the separatrix and the stable equilibrium; the second lemma is a normal form lemma ; the third lemma is a iso energetic KAM theorem. For general information about normal forms KAM theory etc we refer to e.g. [] and references therein. In the following we shall use the following notations: if A R d and r > we denote by A r the subset of points in C d at distance less than r from A; T d s denotes the complex set {z C d : Imz j < s for all j} thought of as a complex neighborhood of T d. If fi ϕ is a real analytic function on A r T d s we let f rs denote the following norm 8 f rs := sup ˆf k I e k s.4 I A r k Z d ˆf k I being the Fourier coefficients of the periodic function ϕ fi ϕ. Lemma. Real analytic action angle variables for the pendulum. Let D := [ R R ] let E := H p R = R/ let < η < ε/3 and define { } M + p := M + p η ε := I ϕ D T : I > η < H p I ϕ < E ε { } M p := M p η ε := I ϕ D T : ε + η < H p I ϕ < η..5 Then for all r < R / and s positive there exist positive numbers r and s closed intervals D ± R symplectic transformations φ ± real analytic on D ± T and functions h ± real analytic on D ± such that φ ± : Î ˆϕ D ± r T s φ ± Î ˆϕ D r T s.6 φ ± D ± T = M ± η ε.7 H p φ ± Î ˆϕ = h ± Î Î ˆϕ D ± r T s..8 The analyticity radii r and s may be taken to be r := c r η ε s := c s lnε/η.9 where < c < is a suitable universal constant. Furthermore the functions h ± satisfy for all Î D ± r the following bounds η Re h + Î E ε ε + η Re h Î η. dh ± Î = α± dî π ±. d h ± Î = ±β ± π± dî π ±. 3 8 The specific choice of norm will play no role in the sequel; obviously if f is a real analytic function on T d s f s stands for k Z d ˆf k e k s ˆfk being the Fourier coefficients of f while if f is a real analytic function on A r then f r = sup I Ar fi.

4 L. BIASCO AND L. CHIERCHIA where π ± := π ± Î := ε ln + ε Re h ± Î π ± := π ± Î := Re h ± Î.3 Re h ± Î + ε α ± := α ± Î β ± := β ± Î are real analytic functions such that d Re α± R Re β ± d ; Imα± R Imβ ± d.4 for suitable universal constants < d < d and < d < d /. An identical statement holds if in the definition of M + p one replaces I > with 9 I <. The next two lemmata are typical statements from KAM theory see [] for generalities. The first one is a normal form lemma common also in averaging theory. The second one is an iso energetic KAM theorem i.e. a KAM theorem on fixed energy levels. The only technical difference in the statement of the normal form lemma is that we have to allow different radii of analyticity in the action variables a fact that is convenient for our application of the KAM theorem; see also point ii of Remark 6 below. For notational simplicity we state the normal form lemma for d = which suffices for our applications. Lemma. Normal forms. Let D and D be two subsets of R and consider a Hamiltonian function HÎ ˆϕ := hî + fî ˆϕ real analytic on Ŵˆr ˆr ŝ := Dˆr D ˆr T ŝ for some ˆr ˆr > and ŝ >. Assume that there exist K 6/ŝ and α > such that ωî k α k Z < k K Î Dˆr D ˆr.5 where ωî := hî. Assume also that f ˆrˆr ŝ αˆr 8 K..6 Then there exist a real analytic symplectic transformation such that with Φ : J ψ Ŵˆr /ˆr /ŝ/6 ΦJ ψ Ŵˆr ˆr ŝ H ΦJ ψ = hj + gj + f J ψ.7 g f ˆr/ˆr / 6 K αˆr f ˆrˆr ŝ 4 f ˆr ˆr ŝ f ˆr/ˆr /ŝ/6 f ˆrˆr ŝ exp Kŝ/6 ΦJ ψ J ψ ˆr/ˆr /ŝ/6 ĉ f ˆrˆr ŝ.8 9 By symmetry the interval D + in the case I < is just the opposite of the interval D + in the case I >. Adapt the norms in.4 and in the footnote 8 in the obvious way replacing A r by D r D r and replacing the subscript r in the norms by ˆr ˆr. f is the zero Fourier coefficient of f i.e. the average of fî ˆϕ over T.

ON THE STABILITY OF... 4 where ĉ > is a suitable constant. Let as above ωj denote the gradient hj let h J denote the Hessian matrix of h. We recall that a vector ω R d is said to be γ τ Diophantine if ω k γ k τ k Zd \{}.9 for some γ > and τ >. Lemma.3 Iso energetic KAM theorem. Let D R d be a bounded domain and consider a Hamiltonian HJ ψ := hj+fj ψ real analytic on the domain W rs := D r T d s for some r > and s >. Assume that h r > and that the d + d + matrix h U := J ωj.3 ωj is invertible on D r. Given E R such that h E and given denote D = < γ < min ij D ω ij and τ d.3 { J D : hj = E } and ωj is γ τ Diophantine..3 Then if f rs is small enough for each J D there exists a unique d dimensional real analytic invariant torus T H E which is a graph over the angle ψ which is close to the torus {J} T d and on which the H flow is analytically conjugated to the translation θ θ + ωj + κt κ being a small real number. More precisely let A F and G be positive numbers such that { } A h r F A f rs γ G max A U r.33 let < s < s and let { γ s s } c C := max F := C c4 F ln F c A r ln F c3 s s c5 Gc6 c7.34 where the c i > are suitable constants depending only upon τ and d. If F then for each J D there exists a unique invariant torus T H E satisfying the following properties: } i T = {J ψ ψ : ψ T d with J real analytic on T d s and J ψ J r F for all ψ T d s; ii there exist real analytic functions on T d s u v and a smooth function κ : D r C real for real J such that max{r v J s u s κ } F ; the map θ T d s vθ θ + uθ is a real analytic embedding whose real image { } is the torus T : T = vθ θ + uθ θ T d ; on the torus T the H flow φ t linearizes: denoting ω := + κj ωj one has φ t vθ θ + uθ = vθ + ω t θ + ω t + uθ + ω t ; Necessarily τ d by a theorem of Liouville. Also.9 with k = implies that γ min i ω i.

4 L. BIASCO AND L. CHIERCHIA iii if τ > d and ˆγ := const. h d r min det U D r γ then { meas H E\ tori satisfying i and ii } ˆγ. Remark 5. As mentioned above in the case of two degrees of freedom d = considered in this paper the above KAM tori separate the three dimensional energy levels forming barriers for the motion; any two KAM tori with equal energy bound an invariant region in corresponding energy level. More precisely let [a b ] [a b ] D with a i < b i. Then because of.3 we can take as coordinates for the three dimensional energy level H E either of the action variables 3 plus the angles ψ. Take first as coordinates J ψ ψ and fix J [a + δ a δ] where δ := max{r F ˆγ} ˆγ measures the complement of the surviving KAM tori and r F the maximal oscillation of the graph of each KAM torus. Then by i and iii in Lemma.3 it follows that there exist two tori T and T so that sup ψ J < J < inf ψ J and < inf ψ J sup ψ J Oδ. The same reasoning applies to J. Hence if Jt ψt := φ t J ψ for any ψ one has that sup t Jt J Oδ. Remark 6. On the proofs of the lemmata i The action angle variables for the pendulum H p I ϕ ; ε = I ε+cos ϕ are produced by the generating function I dϕ where Γε E denotes the ΓεE positively oriented circle H p E the homotopy of Γε E depends on whether E > or ε < E <. The point is that we need a very detailed and quantitative analysis for Γ very close to the separatrix i.e. E close to and for Γ close to the stable equilibrium i.e. E close to ε regions where the action angle variables become singular; very close meaning here at a distance of order ε β with β >. Therefore in such singular regions a careful blow up analysis is needed. Furthermore we also need to study the complex analytic continuation of the action-angle variables since we want to apply a KAM theorem in real-analytic class. To perform this blow up in analytic class a certain amount of straightforward although rather lengthy computations are needed: we provide details in Appendix B. We mention also that for our main purpose i.e. total stability of action variables it would be enough to apply a iso energetic KAM theorem in smooth class since all we need is a topological trapping argument ; however a quantitative version of such a theorem necessary for our task is not available in literature and providing the details for its proof would be certainly much longer and far less elementary than the proof of Lemma.. ii Lemma. as mentioned above is a standard normal form lemma ; a proof may be found e.g. in [9] pag 9. Keeping track of different radii going into the proof in e.g. [9] is routine notice that in the smallness condition.6 there appears the smallest radius. We add only a technical comment: in [9] there appears the condition r α/constk; such a condition is needed to control the 3 Furthermore the map J α J = ω J J J /ω J J J where J hj J J = E is a diffeomorphysm: dα = det U dj ω 3 ; J J J and a completely symmetric statement holds interchanging the indices and. is such that

ON THE STABILITY OF... 43 small divisor bounds on complex domains. Since we are assuming the small divisor bounds directly on complex domains such a condition is not needed in our case. iii Also Lemma.3 is by now rather standard. In fact it is easy under an extra nondegeneracy condition satisfied in our application 4 to derive the iso energetic KAM theorem directly by the standard one by means of a standard Implicit Function Theorem. Alternatively one can find a very detailed version e.g. in [5]. For these reasons we shall omit the proof of Lemma.3. In our application the exact values of the constants c i are not needed; however we can prove Lemma.3 with the following constants: c = τ + c = 6 τ+ c 3 = c c 4 = d c 5 = τ + c 6 = c 7 = τ +. Also in our case it will be C =. 3. Proofs of the Theorems. We first prove Theorem.4 Theorem.3 will be a simple corollary of it. Since most of the arguments are identical for both models σ = and σ = we shall usually do not indicate explicitely the dependence upon σ. The only point where the two models differ is in the estimates regarding the iso energetical non degeneracy see Lemma 3. below. Proof of Theorem.4 The first step is to use Lemma. to put H in.7 into action angle variables. Let R be as in.9 and assume that H in.5 is analytic on B r T s where B denotes here BR and < r < R/ s >. Since in our case H p is an entire function we can choose in Lemma. the parameters Let b and q be as in.9 and respectively.3 let and let q be a number such that r := r s := s. 3.35 λ = + b 3.36 q < q < a 3 3b. 3.37 Notice that with such choices the following relations hold: We also set λ > < b < λ b + λ + q + < a. 3.38 so that r and s in Lemma. become r = c r ε λ / s = c λ Let D D ± and φ ± be as in Lemma. and let D := [ R R ] D Now define D ± σ R as follows: η := ε λ 3.39 ln ε s. 3.4 R := R + R. 3.4 D σ := D D + := D + D + T := φ + M + 3.4 4 Namely the invertibility of the Hessian h on D r which is the usual nondegeneracy condition in the standard KAM theorem.

44 L. BIASCO AND L. CHIERCHIA where M + := M + η ε := M + p \R := M + p \ {I ϕ : c ε 3 + 4 3 b H p ε } 3 3.43 c with a suitable small positive constant c to be fixed later. Denoting J = J J ψ = ψ ψ Î = Î Î ˆϕ = ˆϕ ˆϕ then by Lemma. we have ˆφ ± : Î ˆϕ D± σ r D r T s I ϕ B r T s where I ϕ := φ ± Î ˆϕ I ϕ := Î ˆϕ. 3.44 In the symplectic coordinates Î ˆϕ the Hamiltonian H in.5 takes the form H ± Î ˆϕ; ε := H ˆφ ± Î ˆϕ = h± Î + εσ Î + εa H ± Î ˆϕ; ε 3.45 where h ± is as in Lemma. and H ± := H ˆφ ± ; hence H ± r r s H rs. 3.46 The second step is to apply the normal form lemma Lemma. in a suitable phase space region to the Hamiltonian H ± : in such a way we shall be able to to put H ± in a normal form of the type appearing in.7.8 to meet the stringent KAM condition F in the KAM theorem Lemma.3 and to give a good estimates on the measure of the KAM tori. We therefore set 5 hî := h± Î + εσ Î f := εa H ± ˆr := r = cr ε λ ˆr := R εb ŝ := s = c λ D := D ± σ D := {Î R : R ε b Î R } Ŵˆrˆr ŝ := {Î D ± σˆr D ˆr ln ε s } T ŝ. 3.47 Notice that the second relation in 3.38 implies that ˆr ˆr for ε small. Define also K := 3.48 ε q ln ε ± where q is as in 3.38. Let us now estimate α in.5. Denote by ω Î := h ± Î εσî. Then for any k Z \{} with k K by..4 and the choice of η we find ε ± ω Î k h ± εr K κ R ln ε if k R εb+ if k = 3.49 for a suitable constant 6 κ and provided ε > is small enough. We can therefore take α := R ε+b. 3.5 We can now check.6. Since by 3.46 f ˆrˆr ŝ := ε a H ± ˆr ˆr ŝ ε a H rs 3.5 5 I = is a singularity resonance: we therefore have to stay a bit away from it. 6 From here on κ i denote suitable constants depending possibly on λ a b c q i s and r.

ON THE STABILITY OF... 45 because of the choices of α ˆr ŝ and K see 3.5 3.4 we find.9 3.39 and 3.48 αˆr 8 K = c R r ε b+λ+q+/ ln ε 9. 3.5 Thus in view of the choice of the various parameters made in 3.38.6 is satisfied for ε > small enough. Thus by Lemma. there exist a real analytic symplectic transformation such that Φ ± : J ψ Ŵˆr /ˆr /ŝ/6 Φ ± J ψ Ŵˆr ˆr ŝ 3.53 H ± Φ ± J ψ = h ± J + εσ J + g± J + H ± J ψ 3.54 with recall.8 3.5 3.48 g ± ε a H ± ˆr /ˆr / 4 εa H rs H ± κ ˆr/ˆr /ŝ/6 f ˆrˆr ŝ exp Kŝ/6 H rs exp ε q ln ε Φ ± J ψ J ψ ˆr/ˆr /ŝ/6 ĉ ε a H rs 3.55 for a suitable κ > and ε small enough. Thus if we pick a q so that we have that for all ε > small enough H ± ˆr/ˆr /ŝ/6 H rs q < q < q 3.56 exp. 3.57 ε q Third step. In order to apply the KAM theorem Lemma.3 we set: hj = h ± J + εσ J + g± J := h ± J fj ψ = H ± J ψ r = κ 3 r ε λ s = κ3 s ln ε s = s D = D ± σ D W rs = D r T s 3.58 where κ 3 is a suitable constant such that 7 r ˆr 4 s ŝ 6. Obviously the norm relative to the domain W rs will again be denoted rs but beware that the sup norms in the action variables are taken on different domains according to whether σ = or σ = recall 3.4 and 3.43: in the case σ = the set R has to be discarded. The estimates on h ± and on 8 U require 7 Recall 3.53 and that ˆr < ˆr. The factor /4 is included in order to bound derivatives of g ± and hence of h via Cauchy estimates. We recall the statement concerning Cauchy estimates in our context: if gj is a function analytic on D r D r then for any integers p p and for any < c < p +p g J p J p const. p!p crcr! g rr r p r p 8 Recall the definition of the matrix U in Lemma.3. c p +p.

46 L. BIASCO AND L. CHIERCHIA some computations which we collect in the following lemma. 3.36 and.9 there follows that b and λ satisfy b < 4 λ < a + Recall that from. 3.59 Lemma 3.. There exists 9 C > such that for all ε > small enough h ± r C ε λ ln ε 3 U C r ε λ 3.6 ln ε where r denotes the sup norm on D r defined in 3.58 3.4 3.43 3.47. Proof. First we need estimates on the derivatives of g ±. From 3.55 there follows g ± ˆr/ˆr / 5 4 εa H rs ; whence by Cauchy estimates g± r H rs κ 5 ε a λ+ g ± r H rs κ 5 ε a b J r J r g ± H rs κ 5 ε a λ+ g ± H rs κ 5 ε a b r r J r g ± r H rs κ 5 ε a λ b+ 3.6 J J r J with a suitable constant κ 5 >. By 3.38 and 3.59 one has a λ + > a b > 3 a λ + > a b > 5 4 a λ b + >. 3.6 The symmetric matrix U has the form U = where recall 3.58 3.54 and.8 u = h ± J u 3 = h± J u 3 = h± J = h ± + g ± u u u 3 u u u 3 u 3 u 3 J = h± J + g± J u = h ± J Since recall the estimates in Lemma. r 3.63 u = h ± J J = g ± J J = εσ + g ± J = εσj + g± J. 3.64 C ε λ ln ε 3 π± 3 π ± C C π ± C ln ε ε C ε π ± C ε λ+ 3.65 9 From here on C i denote suitable constants depending possibly on λ n a b q i E R r and H r s. It is exactly in order to get the estimates 3.6 that we kept track of the different complex extension sizes in the variables J and J.

ON THE STABILITY OF... 47 for suitable constants C i > by 3.6 and 3.6 we see that there exists a q > such that for all J D r the following asymptotics hold u = ±β ± π± π ± + Oε q u = Oε + q u = εσ + Oε q 3 + Oε q u 3 = εσj + Oε q. 3.66 u 3 = α± π ± From these relations there follows immediately that h r Let us now write the matrix U as follows where U = δ C 3 ε λ ln ε 3. 3.67 u u u 3 + u 4 u u 5 + u 6 u 3 + u 4 u 5 + u 6 u 7 + u 8 δ := u u u 3 + u u u 3 u := 3 u 3 u3 u 3 u := u 3 u 3 u 3 := u u 3 u 4 := u u 3 u 3 u 6 := u u u 7 := u8 := u u u 3 u 3 u 3 Observe that from the above asymptotics 3.66 it follows u u = ± β± 3 α ± π± + Oε q 3.68 u 5 := u u 3 u 3. 3.69 3.7 for some q > ; we also recall that π ± = E ± ε + E ± E ± := Re h ± J 3.7 where ε + ε λ E ε λ ε λ E + E. Notice that from 3.65 and 3.66 it follows also that C 3 sup u i ij D r ε λ. 3.7 ln ε Thus it remains to estimate / δ. From 3.69 3.66 and 3.7 one sees that there exist a complex number z := z +iz with z > and z < z / such that δ = ε ± zπ ± ε Re J + σ + Oε q3 Re δ = ε ± z π ± ε Re J + σ + Oε q3 3.73 for a suitable q 3 >. Let us consider the two different signs separately. In the plus case we have to distinguish whether σ = or σ =. When σ = since z π + ε Re J > one has δ Re δ = ε z π + ε Re J + + Oε q3 ε 3.74 Obviously x = Oε c means that there exists a positive constant d such that for all ε small enough x dε c. Use also that for J D r ImJ / Re J const. ε λ b < ε by 3.38.

48 L. BIASCO AND L. CHIERCHIA for ε > small enough. Let now σ = and notice that 3 Choose π + E + π + ε 3 /c c ε E + ε 3 c π + E + π + cε 3 + 4 3 b cε +b E + cε 3 + 4 3 b. 3.75 c := 6 min { z R Thus in the region E + ε /3 /c one has δ Re δ ε 4z cr + Oε q3 ε ; in the region E + cε 3 + 4 3 b one has z R δ Re δ 8c } z R. 3.76 + Oε q3 ε. Let us turn now to the minus sign case and notice that ε λ E ε and π κ 6/ε 3/ with a suitable κ 6 >. Hence recalling 3.66 and the assumption b < /4 δ C 4 π ε+b C 5 ε C 4 κ 6 ε +b C 5 ε C 6 ε +b 3.77 where C 4 C 5 and C 6 are suitable positive constants. Thus since + b < we see that in all cases δ C 7 ε 3.78 with a suitable C 7 >. This bound together with 3.7 leads to the estimates on U given in 3.6 completing the proof of the lemma. We proceed to estimating the parameters appearing in the statement of Lemma.3. From 3.64 3.66 and.3 there follows that h± ε := u3 C 8 J ln ε h± := u3 C 8 ε +b 3.79 J h for a suitable C 8 > so that min ± ij J i C8 ε +b. We next choose γ C 8 ε +b. Since the norm of H ± is exponentially small with ε we can choose also γ exponentially small with ε: we let in fact for a suitable γ > γ := γ exp with q < q < q. 3.8 ε q Therefore in view of 3.57 3.6 and 3.8 we can take 4 A := C ε λ ln ε 3 F := exp C 9 G := ε q ε λ ln ε 4 3.8 3 B y 3.7 π + is a decreasing function of E +. Recall also 3.43 that c < and that ε is small. 4 Recall the definitions of F and G given in.33.

ON THE STABILITY OF... 49 for a suitable C 9 >. Next we show that C in.34 is one in our case. By 3.58 3.6 3.8 and 3.8 we see that for a suitable C > γs s ε c c3q exp c A r ln F = C ε q c3 ln ε c ; which implies that C = for ε small enough. Therefore recalling the definition.34 of F we can take for a suitable C > see 3.8 and 3.58 and for ε > small enough F C exp 3.8 ε q which obviously will be smaller than one for any ε > small enough. Thus under conditions 3.38 3.59 and 3.8 Lemma.3 can be applied to the Hamiltonian 3.54 showing the existence of KAM tori in each energy level of W rs apart from a small set of measure bounded by 5 Oˆγ Oexp /ε q. Thus recall Remark 5 the motions starting in W rs have action variables Oexp /q close to their initial values for all times. In the original coordinates I ϕ the measure of the complementary of the KAM tori is again bounded by Oexp /q; the KAM tori fill up the region M σ with the exception of a set of measure Oexp /q. In view of 3.55 the displacement of the KAM tori from the corresponding unperturbed ones is Oε a while the oscillation of the graph of the tori may be bounded by O ε. Repeating the argument in Remark 5 we find that denoting It ϕt the φ t evolution of I ϕ with 6 I ϕ M σ provided ε > is small enough. This concludes the proof of Theorem.4. Proof of Theorem.3 It I < C ε t 3.83 We proceed to show that Theorem.4 and energy conservation imply. in M R when σ = and in M R \N when σ = recall the definition of N in.. In view of the oscillations of the KAM tori in the region M σ we shall consider slightly smaller sets M σ M σ. To define such sets we let Ñ := N and: N := {I ϕ : c ε 3 + 4 3 b < H p < ε } 3 { c Ñ := I ϕ : H p < ε +b or H p < ε + ε +b} { Ñ := I ϕ : I < Rε b} M := M R \Ñ Ñ M := M \Ñ. 3.84 Remark 7. Because of Theorem.4 and hence because of the confinement due to the presence of two dimensional KAM tori in three dimensional energy levels the smaller sets M σ have the property that t R φt σ M σ M σ where φ t σ denotes the H σ flow. In particular in the case σ = a trajectory cannot cross 5 Provided τ is choosen striclty larger than one; the constant ˆγ is defined in iii of Lemma.3 and in view of Lemma 3. is related to γ by a power of ε. 6 Recall that R < R < R and that ε will be small compared also to R R.

5 L. BIASCO AND L. CHIERCHIA the region N a fact that could also be checked directly by energy conservation since 3 + 4 3 b <. Denote by zt := It ϕt the motion with initial data z := I ϕ governed by H σ in M R if σ = or M R \N if σ =. Let us consider the different cases which may occur. i If z M σ then as remarked above zt doest not leave M σ where 3.83 and hence. holds. ii If zt Ñ for t < T for some T > then by energy conservation. holds 7 for t < T. iii If zt Ñ for t < T then. trivially holds for t < T. iv By ii and iii. holds until zt Ñ Ñ. But if zt leaves Ñ Ñ and enters the region M σ then by i. holds again. Appendix A. The Planetary D Alembert Hamiltonian. In this appendix we revisit briefly the Hamiltonian version of the planetary D Alembert model as presented in [4] and discuss a connection with the result presented in this paper. In [4] Section it is shown that the motion of a planet modelled by a rotational ellipsoid with flatness ε > whose center of mass revolves on a Keplerian ellipse of eccentricity µ > subject to the gravitational attraction of a fixed star occupying one of the foci of the ellipse is governed in suitable units by a Hamiltonian function given by H εµ J ψ = J + J J + ωj 3 J + εf J J ψ ψ + εµf J J ψ ψ ψ 3 A. where: J ψ A T 3 are standard symplectic coordinate; the domain A R 3 is given by { A := J < cε l J J } < d J 3 R A. with l < < cε l d J J fixed reference data verifying certain assumptions spelled out below; π/ω is the period of the Keplerian motion year of the planet ; the functions F i are trigonometric polynomial given by F = c j cosjψ + d j cosjψ + ψ j Z j F = j Z j 3c j cosjψ + ψ 3 + dj { } cosjψ + ψ + ψ 3 7 cosjψ + ψ ψ 3 7 In fact calling E pt = H pi t ϕ t if zt Ñ for t < T then E pt E p Oε λ for all t T recall that λ = + b and that a > λ. Thus by energy conservation there follows that I t I Ept Ep + = Oε a ε for all t < T. Therefore I t I = Oε λ and. follows.

ON THE STABILITY OF... 5 where c j and d j are functions of J J listed in the following item; let κ := κ J := ν := ν J := L κ := κ J J := J J + J J + J κ ν := ν J J := κ ; where L is a real parameter; the parameters J i L ε and the constants c and d are assumed to satisfy J + d + cε l < J < L + cε l < J A.3 so that < κ i < and the ν i s are well defined. Then the functions c j and d j are defined by c J J := κ 4 ν + ν + κ d J J := ν 4 κ ν c ± J J := κ κ ν ν c ± J J := ν ν 8 d ± J J := ± κ κ ν ν d ± J J := ν ± κ 8. A.4 Remark 8. i We recall that actually the above model is a first order µ truncation of the full D Alembert model which in place of F + µf has a series j µj F j with F j trigonometric polynomials. ii Since J 3 appears only linearly with coefficient ω the angle ψ 3 corresponds to time t and H εµ is actually a two degrees of freedom Hamiltonian depending explicitly on time in a periodic way with period π/ω. iii The physical interpretation of the action variables J J and the parameter L is the following. The action variable J + J is in suitable units the absolute value of the angular momentum of the planet; the variable J is the absolute value of the projection of the angular momentum of the planet onto the direction orthogonal to the ecliptic plane i.e. the plane containing the Keplerian ellipse and L is the absolute value of the projection of the angular momentum of the planet in the direction of the polar axis of the planet and is a constant of the motion. iv Under our assumptions i.e. that < cε l d the average over the angle of H ε is given by J + J J + ωj 3 J + ε { ν 4 3 ν J J } + Od A.5 where ν := L/ J. By iii we see that ν corresponds to rotations of the planet with spin axis nearly orthogonal to the ecliptic plane a case common for example in the Solar System. In such a case the average over the angle of H ε is not a convex function of the action variable J. This lack of convexity for the effective Hamiltonian is quite a common feature in Celestial Mechanics and is exhibited for example also in three body problems. This is the reason why in our model problem we considered also non convex cases corresponding above to σ =. We proceed now to show how the D Alembert model relates to the model.5.8 investigated in this paper.

5 L. BIASCO AND L. CHIERCHIA We are interested as in [4] to reference data corresponding to day/year resonances as the one often observed in the Solar system. We let therefore J = ω A.6 corresponding to a : day/year resonance. Then the linear symplectic change of variables ˆΦ : Ĵ ˆψ J ψ given by J = Ĵ Ĵ + Ĵ Ĵ + Ĵ3 ψ = ˆψ ˆψ + ˆψ 3 ˆψ ˆψ 3 ˆψ 3 A.7 casts the Hamiltonian H εµ into the form ĤεµĴ ˆψ := H εµ ˆΦĴ ˆψ with where Ĥ εµ Ĵ ˆψ := Ĵ + ωĵ3 + ε F Ĵ Ĵ ˆψ + εµ F Ĵ Ĵ ˆψ A.8 F Ĵ Ĵ ˆψ := F Ĵ Ĵ + Ĵ ˆψ ˆψ + ˆψ 3 ˆψ ˆψ 3 F Ĵ Ĵ ˆψ := F Ĵ Ĵ + Ĵ ˆψ ˆψ + ˆψ 3 ˆψ ˆψ 3 ˆψ 3. A.9 For < ε the angle ˆψ 3 i.e. the time is a fast variable 8 and we may apply averaging theory or normal form theory. We shall apply the resonant version in three degrees of freedom of Lemma. which for the sake of clarity we reformulate 9 : Lemma A.. Let D R and D R and consider a Hamiltonian HĴ ˆψ := hĵ + fĵ ˆψ real analytic on Ŵˆr ˆr ŝ := Dˆr D ˆr T 3 ŝ for some ˆr ˆr > and ŝ >. Assume that there exist K 6/ŝ and α > such that ωĵ k α k Z3 k K k 3 Ĵ Dˆr D ˆr A. where ωĵ := hĵ. Assume also that f ˆrˆr ŝ αˆr 8 K. A. Then there exist a real analytic symplectic transformation such that with g π π Φ : I ϕ Ŵˆr /ˆr /ŝ/6 ΦI ϕ Ŵˆr ˆr ŝ H ΦI ϕ = hi + gi ϕ ϕ + f I ϕ f K I ϕdϕ 3 ˆr/ˆr /ŝ/6 6 K αˆr f ˆr/ˆr /ŝ/6 f ˆrˆr ŝ exp Kŝ/6 ΦI ϕ I ϕ ˆr/ˆr /ŝ/6 ĉ f ˆrˆr ŝ A. f ˆrˆr ŝ 4 f ˆr ˆr ŝ A.3 where f K I ϕ := k K ˆf k I expik ϕ and ĉ > is a suitable constant. 8 In fact when ε = d/dt ˆψ = Oε l d/dt ˆψ = while d/dt ˆψ 3 = ω. 9 The proof is given in [9] and the same comments in the Remark 6 ii apply word by word to the present situation.

ON THE STABILITY OF... 53 Let c = < l < and ε small in particular ε l d; let also J and L be so that A.3 is abundantly verified. If we choose also ˆr := ε l ˆr := d A.4 we see that the functions ν i and κ i and hence the functions c i d i F i are analytic and bounded for any ŝ > and any R > in the domain Dˆr D ˆr T 3 ŝ where D := [ ε l ε l ] D := {Ĵ : Ĵ J d} { Ĵ3 R}. We can now apply Lemma A. to the Hamiltonian ĤεµĴ ˆψ = HĴ ˆψ with hĵ := Ĵ + ωĵ3 fĵ ˆψ := F Ĵ Ĵ ˆψ + µ F Ĵ Ĵ ˆψ. Under the above position for µ we have that f ˆrˆr ŝ const ε. Thus letting α = ω/ and K := ω/4ε l we see that A. is satisfied for any l < /. Now observe that π π F Ĵ Ĵ ˆψd ˆψ 3 = ĉ + ˆd cos ˆψ where ĉ := ĉ Ĵ Ĵ and ˆd := ˆd Ĵ Ĵ are defined as A.5 ĉ Ĵ Ĵ := c Ĵ Ĵ + Ĵ ˆd Ĵ Ĵ := d Ĵ Ĵ + Ĵ. A.6 Thus by Lemma A. and A.5 we find that Ĥεµ ΦI ϕ has the form I + ωi 3 + ε ĉ I I + ˆd I I cos ϕ + gi I ϕ ϕ ; ε µ + f I I ϕ; ε µ A.7 where if g is as in the Lemma g := g ε[ĉ I I + ˆd I I cos ϕ ]. The function f is exponentially small f ˆr/ˆr /ŝ/6 f ˆrˆr ŝ exp Kŝ/6 const ε exp ωŝ 4 ε l A.8 and in view of A.3 the definition of g and A.5 the function g satisfies the bound g ˆr/ˆr /ŝ/6 const ε l + εµ. A.9 Thus assuming µ ε c with c > / and < l < /4 in the above region of phase space the D Alembert Hamiltonian is described up to the exponentially small term in A.8 by the Hamiltonian 3 where H D I I ϕ ϕ ; ε µ = I + ε ĉ I I + ˆd I I cos ϕ + ε a GI I ϕ ϕ ; ε µ A. a := min{ l + c } > 3 G := g ε a G ˆr /ˆr /ŝ/6 const. A. 3 If we disregard f then ωi 3 becomes a constant which we may drop.

54 L. BIASCO AND L. CHIERCHIA Remark 9. i The form of H D has been for us the main motivation to discuss the dynamics of models described by.5.8. ii The theory developed in this paper cannot be applied directly to H D because of the following two reasons: a The intermediate Hamiltonian ĉ I I + ˆd I I cos ϕ has a more complicate dependence on the action variables than the one considered in this paper and one would need to extend Lemma.. b The second technical reason is that the I domain of analyticity of H D is small with ε while in our model we assumed H analytic in a given ε independent region. iii While problem a needs further investigations problem b may be easily overcome. More precisely the proofs of 3 work also if one allows r i.e. the smaller action radius of analyticity of the perturbation H to depend upon ε say r = const ε l provided a > l+3/. In fact the crucial point where a dependence upon ε of r comes in is in the second step of the proof of Theorem.4 and more precisely in checking the condition.6 for applicability of the normal form lemma.. If r = const ε l for some l > we see by 3.5 that we must have a > l + b + λ + q + = l + 3 + 3b + q A. which is equivalent to require compare.9 and.3 a > l + 3 b < a 3 3 l q < a 3 l 3b. A.3 iv In the case of the D Alembert reduced Hamiltonian H D we have a = l so that A.3 would be satisfied provided < l < /6. Of course the argument in iii is applicable to H D only if the qualitative dependence on ε of the analyticity domain in the analogous of Lemma. remains the same. Appendix B. Real analytic action angle variables. Here we shall give a complete proof of Lemma.. For the purpose of this appendix we shall denote the variables I ϕ with the standard pendulum coordinate names p q and set Ep q := H p p q; ε := H p I ϕ ; ε. We shall also denote the action angle variables for Ep q by P Q which therefore coincide with the variables Î ˆϕ of Lemma.. We shall use the following notation: if A B are two strictly positive functions we shall say A B if there exist positive constants c + c so that c A B c + A pointwise. For example A + B A = B B. A + B + A A + B B. Obviously is transitive. Also if A B and C are strictly positive then A B implies A + C B + C. Let x = x + ix be a complex number with x i R. Since 3 x +x x +x from B. it follows immediately the following trivial lemma 3 x + x / x + x x + x.

ON THE STABILITY OF... 55 Lemma B.. If x then x ± ix = w ± iw ; if x < then x ± ix = w ± iw where w = x + x + x x + x /4 x + x B. w = x + x + x x x + x x /4 x + x B.3 If x then x ± ix / = y iy ; if x < then x ± ix / = y iy where x + x y = + x x + x y x + x = + x x x + x x + x x + x 3/ B.4 If x then x ± ix 3/ = z iz ; if x < then x ± ix 3/ = z iz where z = y y 3y z = y 3y y and y y are as above. Furthermore if y y then y 3y y 3y y y and z x + x z x 3/ x + x 5/ B.5 We divide the proof of Lemma. in several steps considering in particular separately positive and negative pendulum energy E = Ep q. In the following we shall consider energies E = E + ie C such that for a suitable < c <. E c E B.6 First step: action variable positive energy. In such a case as well known the action variable for Ep q is given by π P + E = E + ε + cos ψ dψ B.7 π and denoting by with a dot the derivative with respect to E we have P π + E = dψ π E + ε + cos ψ and in general P + E = 4π d n+ dh n+ P + E = n π n!! π π n dψ E + ε + cos ψ 3/ dψ. E + ε + cos ψ n+/ B.8 Notice that the above functions viewed as functions of E at E fixed have even real part and odd imaginary part. Thus in the following we may consider only the case E. Setting E + ε + cos ψ := E + ε + cos ψ + ie := x ψ + ix with x ψ = E + ε + cos ψ and x = E we get notice that x ψ E > E = x x ψ+x x ψ. Thus P + E = P + E+iP + E P + E = P + E+i P + E P + E = P + E + i P + E.

56 L. BIASCO AND L. CHIERCHIA From B. it follows P + π π/4 π/4 x ψ dψ x ψ dψ E + ε dψ E + ε. Since for ψ π it is x ψ = E + ε + cos ψ E + επ ψ making in the integrals the change of variable y = π ψ ε/e from B.4 we get 3 P + E π Similarly = ε arcsinh dψ π x ψ P + E π From B.4 it follows that ε π π P + E dψ x ψ 3/ = E ε π ε E E dψ E + επ ψ = ε ln + ε E ε π ε E dψ x ψ E ε ln +. ε E π dψ E + επ ψ 3/ dy + y = π 3/ + π ε/e E 3/ and π P + E E dψ x ψ E 3/ E E + ε. Thus using B.4 we find 33 P + E E = π E E ε dψ x ψ E 5/ π ε E Summarizing the following estimates hold π dψ E + επ ψ 5/ dy + y 5/ E E E + ε P + E E + ε P + E E ε ln P + E ε ln + ε E P + E ε + E dy + y P + E E E E + ε E E + ε P + E E E + ε. E E E + ε B.9 B. B. Second step: action variable negative energy. In this case the action variable is given by P E = π ψe E + ε + cos ψ dψ 3 Use arcsinh t = lnt + + t ln + t. 33 In the last estimate we considered separately E ε in which case E + ε ε and π ε E dy +y 5/ and E > ε in which case E + ε E and + y 5/.

ON THE STABILITY OF... 57 where ψ E is the first positive number such that E + ε + cos ψ E =. Differentiating P ψe E = dψ. π E + ε + cos ψ Making the change of variable ψ = arccos Ẽ/ε + ξẽ/ε where Ẽ := E + ε we get Thus and in general d n+ dh n+ P E = P E = π Ẽ ξ dξ ξ Ẽξ E P E = π dξ. ξ ξ Ẽξ E P E = π ξ dξ π ξ Ẽξ E B.3 3/ n!! n ξ n dξ. ξ ξ Ẽξ E n+/ As above for symmetry reasons we may consider only E. Observe that B. Ẽξ E = E ξ+εξ ie ξ = Ẽξ E ie ξ = x ξ ix ξ with x ξ := Ẽξ E and x ξ := E ξ. When ξ it is 34 x ξ x ξ. Let now y ξ y ξ be as in B.4. Then y y x B.4 x x 3/ so that y y provided c is small enough. If z ξ z ξ are as in B.5 then Obviously: z x 3/ = y + iy Ẽξ E z x By B. and B.6 we see that 35 P E = ξ Ẽy E y dξ π ξ x 5/. B.5 Ẽξ E3/ = z + iz. Similarly from B. B.6 and B.4 we get: P E = ξ Ẽy + E y dξ π ξ Ẽ E ξ ξ dξ + Ẽξ E 3/ B.6 Ẽ ξ ξ Ẽ ξ E dξ. B.7 E ξ ξ Ẽ ξ E dξ B.8 34 The function x ξ x ξ is increasing so that x ξ x ξ x x = E E >. 35 Use Ẽy E y Ẽy.