Exponential stability for the resonant D’Alembert model of Celestial Mechanics∗ Luca Biasco
Luigi Chierchia
Settore di Analisi Funzionale
Dipartimento di Matematica
SISSA/ISAS
Universit` a “Roma Tre”
Via Beirut 2–4, 34013 Trieste (Italy)
Largo S. L. Murialdo 1, 00146 Roma (Italy)
(
[email protected])
(
[email protected])
June 3, 2002
Contents 1 Introduction and Result
2
2 Proof of Theorem 1.1 in the case (p, q) 6= (1, 1), (2, 1) 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 2.2 Step 1 (linear change of variables) . . . . . . . . . . . 2.3 Step 2 (time averaging) . . . . . . . . . . . . . . . . . 2.4 Step 3 (averaging over ϕˆ1 ) . . . . . . . . . . . . . . . 2.5 Step 4 (energy conservation) . . . . . . . . . . . . . .
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8 9 10 11 12 14
3 Complex action–angle variables
17
4 Proof of Theorem 1.1 in the case (p, q) = (1, 1), (2, 1)
30
∗
Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.
1
1
Introduction and Result Che fai tu, luna, in ciel? dimmi, che fai, Silenziosa luna? (G. Leopardi)
1.1 Perturbative techniques have been (and still are) a basic tool for a deep understanding of the long time behavior of conservative Dynamical Systems. Such techniques, whose main representatives are the so–called KAM and Nekhoroshev theories (compare [1] for generalities), have, by now, reached a high degree of sophistication and have been applied to an enormous corpus of different situations (including infinite dimensional systems and PDE’s). KAM and Nekhoroshev techniques work under certain non–degeneracy assumptions (e.g, invertibility of the frequency map for KAM or quasi–convexity or steepness for Nekhoroshev: see [1]); however such assumptions are strongly violated exactly in typical examples of Celestial Mechanics, which – as well known – were the main motivation for the Dynamical System investigations of Poincar´e, Birkhoff, Siegel, Kolmogorov, Arnold, Moser,... 1.2 In this paper we shall consider the resonant “day/year” (or “spin/orbit”) planetary D’Alembert model, which, together with the three–body problem, may be considered among the oldest and most intriguing problem in Celestial Mechanics (compare, e.g., [10]); in particular we shall address the problem of the (exponentially) long time stability for such a model and will show how to obtain results a l`a Nekhoroshev for it. The D’Alembert planetary model is a Hamiltonian model for a rotational planet (or satellite) with polar radius slightly smaller than the equatorial radius, whose center of mass revolves periodically on a given Keplerian ellipse of small eccentricity around a fixed star (or “major body”) occupying one of the foci of the ellipse; the planet is subject only to the gravitational attraction of the major body. The “resonant model” deals with phase space regions around resonances, i.e., regions where the ratio between the periods of rotation and revolution is rational: such a case is particularly common in our Solar System (as our Moon testifies). This system is modelled by a Hamiltonian system of two and a half degrees of freedom depending on two action–variables J1 and J2 (corresponding, respectively, to the absolute value of the angular momentum of the planet and its projection onto the unit normal to the ecliptic plane1 ), on the two conjugated angle–variables and (explicitly and periodically) on time. The smallness parameters are the oblateness of the planet, say ε, and the eccentricity of the Keplerian orbit, say µ. This system (in suitable, natural variables) turns out to be properly degenerate, i.e., in the integrable limit (ε = µ = 0) the Hamiltonian depends only upon J1 (the absolute value of the angular momentum); of course, it is easy to see that such a variable is stable (i.e., stay close to its initial value) for the full Hamiltonian, for all times for which the action 1
The “ecliptic plane” is the (fixed) plane containing the Keplerian ellipse described by the periodic motion of the center of mass of the planet.
2
variables stay in their domain of definition. The problem is then to study the stability of J2 , the projection of the angular momentum onto the unit normal to the ecliptic plane (which measures the inclination of the spin axis on the ecliptic plane). We shall prove that for small oblateness and eccentricity and for any motion starting near an exact spin/orbit resonance, the Hamiltonian evolution J(t) of the action variables stay close to their initial values for exponentially long times. This result was announced in the note [2]. Before giving a more precise formulation of this stability result, let us briefly mention its connection with Arnold diffusion2 . In [6], it was claimed that the planetary D’Alembert model, near a resonance (2:1), has an instability region where the variable J2 undergoes a variation of order one (i.e., independent of the perturbative parameters) in finite time, provided ε and µ = εc (for a suitable c > 1) are positive and small enough. The proof of this claim proposed in [6] contained an algebraic error (see the Erratum in [6]) and, even though such error has been corrected ([7]) and several technical progresses, in such direction, have been obtained (see, e.g., [8], [9], [13]), a complete proof of the above claim is still missing. As well known, proving Arnold diffusion for analytic problems is an extremely difficult problem; for the D’Alembert model an extra (technical) problem (related to the degeneracies present in the model) might come from the particular slowness of the expected instability: compare comment 1.4.1 below. 1.3 Let us proceed to formulate, in a more precise way, our main result. It is a classical fact (essentially due to Andoyer) that the planetary D’Alembert model near an exact (p:q) resonance may be described (in suitable physical units) by a a real–analytic Hamiltonian of the form3 Hε,µ :=
I12 + ω(pI1 − qI2 + qI3 ) + εF0 (I1 , I2 , ϕ1 , ϕ2 ) + εµF1 (I1 , I2 , ϕ1 , ϕ2 , ϕ3 ; µ) , 2
(1)
where: • (I, ϕ) ∈ A × T3 are standard symplectic coordinates; the domain A ⊂ R3 is given by n o A := |I1 | < rε` , |I2 − J¯2 | < r , I3 ∈ R , (2) with 0 < ` < 1/2, r > 0. In the terminology of the preceding item 1.2, I1 := J1 − J¯1 , I2 := J2 ; J¯ being a fixed “reference datum” corresponding to the exact (p:q) resonance; p and q are two positive co–prime integers, which identify the spin– orbit resonance (the planet, in the unperturbed regime, revolves q times around the major body and p times around its spin axis): J¯1 = pω and the period of the Keplerian orbit is 2π/(qω). The action I1 measures the displacement from the exact resonance, while I3 is an artificially introduced variable canonically conjugated to ϕ3 (the “mean anomaly”), which is proportional to time.
2
“Arnold diffusion” refers, usually, to the action instability conjectured by Arnold according to whom nearby any point in the phase space of a nearly–integrable Hamiltonian system, in general, there pass trajectories whose action variables undergo a variation of order one in finite time. 3 Compare, e.g., [6], [4]
3
• 0 ≤ ε, µ < 1 are – as in the preceding item – two small parameters (measuring, respectively, the oblateness of the planet and the eccentricity of the Keplerian ellipse). • The functions Fi are real–analytic functions in all their arguments, and may be computed (via Legendre expansions in the eccentricity µ) from the Lagrangian expression of the gravitational (Newtonian) potential; for explicit computations, see, e.g., [6]. While the explicit form of F1 is not important in the sequel, and, in fact, our result holds for any function F1 real–analytic and bounded on A, the explicit form of F0 plays a major rˆole in the following analysis. The function F0 is a trigonometric polynomial given by F0 (I1 , I2 , ϕ1 , ϕ2 ) =
X
cj cos(jϕ1 ) + dj cos(jϕ1 + 2ϕ2 ) ,
(3)
j∈Z |j|≤2
where cj and dj are suitable functions of J = (J¯1 + I1 , I2 ) which may be described as follows. Let L κ1 := κ1 (I1 ) := ¯ , J1 + I 1 ν1 := ν1 (I1 ) :=
q
1 − κ21 ,
I2 κ2 := κ2 (I1 , I2 ) := ¯ , J1 + I 1 ν2 := ν2 (I1 , I2 ) :=
q
1 − κ22 ;
where L is a real parameter (L corresponds to the projection of the angular momentum of the planet onto the polar axis of the planet and, since the planet is rotational, it turns out to be a constant of the motion). The parameters J¯i , L and the constant r are assumed to satisfy L + 3rε` < J¯1 ,
|J¯2 | + 3r(ε` + 1) < J¯1 ,
(4)
so that 0 < κi < 1 and the νi ’s are well defined on the domain A. Then, the functions cj and dj are defined by 1 2 2 ν2 2κ1 ν2 + ν12 (1 + κ22 ) , d0 (I1 , I2 ) := − 2 (2κ21 − ν12 ) , 4 4 (1 ± κ2 )κ1 ν1 ν2 κ1 κ2 ν 1 ν 2 , d±1 (I1 , I2 ) := ∓ , c±1 (I1 , I2 ) := 2 2 ν2ν2 ν 2 (1 ± κ2 )2 c±2 (I1 , I2 ) := − 1 2 , d±2 (I1 , I2 ) := − 1 . 8 8
c0 (I1 , I2 ) :=
(5)
With the above positions, for the motions governed by the (p:q)–resonant D’Alembert Hamiltonian Hε,µ , there holds the following Theorem 1.1 Let c > 0, 0 < ` < 1/2 and 0 < γ0 < min{c, `}. Assume that (4) holds and that 2 ν1 (0) 6= (6) 3
4
(which is equivalent to J¯1 = 6 and 0 ≤ µ ≤ εc , then
√
3L). Then, there exist ε0 , γi > 0 such that, if 0 ≤ ε ≤ ε0
|I(t) − I(0)| < γ3 r εγ1 ,
∀ |t| < T (ε) :=
γ γ5 2 exp , ω ε γ4 ε γ0
(7)
where (I(t), ϕ(t)) denotes the Hε,µ –evolution of an initial datum (I(0), ϕ(0)) ∈ A × T3 . 1.4 Let us make a few comments on the above result. 1.4.1 Standard Nekhoroshev estimates for a system with d degrees of freedom yield a Nekhoroshev exponent (i.e., the exponent of ε in the exponential part of T (ε), which, in our case is γ0 ) 1/(2d); compare, e.g., [12]. In our case, taking ` close to 1/2 and c ≥ `, we see that γ0 ∼ 1/2, getting a better Nekhoroshev exponent with respect to the general non–degenerate case (“better” means that the stability time is longer, for small ε). This fact (as mentioned above) makes, a priori, Arnold diffusion particularly slow in the present example. This phenomenon is due to the appearance of three well separated time scales, as discussed in the next item. 1.4.2 After a natural, symplectic linear change of variables (I, ϕ) → (I 0 , ϕ0 ), (I10 := I1 , I30 := pI1 − qI2 + qI3 ), the unperturbed part, H0,µ , becomes simply I10 2 /2 + ωI30 , showing, in a more clear way, the proper–degeneracy of the system (H0,µ , in these variables, does not depend upon I20 = I2 , which is the physically interesting variable). The appearance of three (well separated) time scales for the evolution of the angles ϕ0 ’s is also more evident in these variables: in fact (for ε small and µ < 1) √ ϕ˙ 02 = O(ε) , ϕ03 = ω = O(1) . (8) ϕ˙ 01 = O(ε` ) O( ε) , The “non–degeneracy” assumption (6) is made in order to really have ϕ˙ 02 = O(ε), i.e., in order to have, 1ε ∂I20 Hε,µ |I10 =0,I20 =J¯2 6= 0. 1.4.3 An immediate consequence of the above different time scales is the following. Having only one frequency (namely ω) of order one suggests to perform averaging over ϕ 03 ; this can be done at any order of ε and, in fact, using “normal form” theory, we shall do it at an exponential order. This implies immediately that the quantity |I30 (t) − I30 (0)| is exponentially small with (a suitable power of) 1/ε for times which are exponentially long with (a suitable power of) 1/ε (at least, as long as the other action variables remain in their domain of definition). This shows that, up to such exponentially long times, the resonant D’Alembert Hamiltonian behaves effectively as a two–degree–of freedom system (see, also, [4]). Furthermore, besides the just mentioned “super stability” of I30 (which, after all, has a limited physical interest), we see immediately, by (approximate) energy conservation4 , that also I10 is stable (again: provided I20 stays in its domain √ of definition); in this case, in fact, one checks immediately that |I1 (t)−I1 (0)| ≤ const. ε. This comment shows that – as mentioned above – the only non–trivial stability is related to I20 = J2 , which (not by chance) is the most relevant physical quantity. 4
“Approximate” because, for the obtained two–degree–of freedom system, energy conservation holds only up to exponentially long times.
5
1.4.4 It is remarkable (particularly in view of the special occurrence, in our Solar System, of the resonances (1:1) and (2:1)) that the analysis of the resonant D’Alembert Hamiltonian is different according to whether the resonance (p:q) is considered with (p, q) = (1, 1), (2, 1) or with (p, q) 6= (1, 1), (2, 1). In fact, the case (p, q) = (1, 1), (2, 1) will turn out to be significantly more difficult, at a technical level, than the other cases, as briefly explained in the next item 1.4.5. To see why there appears a difference between the case (p, q) = (1, 1), (2, 1) and the cases (p, q) 6= (1, 1), (2, 1), we observe that, after the above exponential averaging in ϕ 03 (calling Iˆ and ϕˆ the new averaged symplectic variables, which are close to the variables I 0 and ϕ0 ), the D’Alembert Hamiltonian takes the form Iˆ12 (1) + ω Iˆ3 + εH 01 (Iˆ2 , ϕˆ1 ) + εa H1 (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε) + O exp(1/εconst ) , 2
(9)
(1)
with a := 1 + min{c, `}, H1 real–analytic and bounded by a constant and the “effective Hamiltonian” H 01 given by 1 H 01 (Iˆ2 , ϕˆ1 ) := 2π
Z
0
2π
F0 (0, Iˆ2 , ϕˆ1 + pϕˆ3 , ϕˆ2 − q ϕˆ3 )dϕˆ3 .
(10)
A straightforward computation of this average using the explicit expression for F0 given in (3), yields
H 01 (Iˆ2 , ϕˆ1 ) =
c0 (0, Iˆ2 ) ,
if (p, q) 6= (1, 1), (2, 1)
c0 (0, Iˆ2 ) + djp (0, Iˆ2 ) cos(jp ϕˆ1 ) ,
(11)
if (p, q) = (1, 1) or (2, 1)
with j1 := 2 and j2 := 1: thus, in the case (p, q) 6= (1, 1), (2, 1) the effective Hamiltonian H 01 is independent of ϕ, ˆ while in the cases (p, q) = (1, 1) or (2, 1), it depends explicitly upon ϕˆ1 . 1.4.5 In the case H 01 in (9) is independent of ϕ, ˆ one can exploit the separation of scales ˙ ˙ between ϕˆ1 and ϕˆ2 (which remain of the same order of ϕ˙ 01 and ϕ˙ 02 ): in a suitable ε– ˆ dependent I–domain (i.e., excluding an ε–dependent small neighborhood of {Iˆ1 = 0}) the variable ϕˆ1 is fast with respect to the variable ϕˆ2 and, therefore, the dependence upon ϕˆ1 (again, by means of “averaging” or “normal form” theory) can be removed up to exponentially small terms. Optimizing the various choices and using energy conservation arguments will allow us to obtain the stability claim in all action variables for the case (p, q) 6= (1, 1) or (2, 1). To carry out an analogous strategy in the case H 01 in (9) does depend upon ϕˆ1 , one Iˆ2 has to, first, put the two–dimensional integrable system 21 + εH 01 (Iˆ2 , ϕˆ1 ) into action– angle variables. In principle, this is a simple exercise, but, for our purposes, we need detailed information on the analyticity domains of this symplectic change of variables, which becomes singular as ε → 0. Notwithstanding the enormous literature on elliptic integrals, such information does not seem to be available. Therefore we have to perform
6
an elementary (but lengthy) complex analysis of the symplectic transformation casting Iˆ12 + εH 01 (Iˆ2 , ϕˆ1 ) into action–angle variables. This analysis is carried out in § 3 below. 2 1.4.6 Apart from the complex analysis technicalities mentioned in the previous item, our approach consists in a careful joint use of normal form theory and energy conservation arguments fully exploiting the separation of scales described in item 1.4.2. We remark Iˆ2 that the integrable part of the resonant D’Alembert Hamiltonian, i.e., the term 21 + ω Iˆ3 + ˆ compare [3] and [4] where it is shown εH 01 (Iˆ2 , ϕˆ1 ) in (9), is a non convex function of I; that the coefficient of Iˆ22 in c0 (0, Iˆ2 ), in typical regions of phase space, is negative. This explains the reason why the arguments that we use, in order to obtain action-confinement, are somewhat more related to the original approach of Nekhoroshev [11] rather than to simpler arguments based on convexity, as introduced in [5] (and used, also, in [12]). 1.4.7 For the D’Alembert model, a different approach to exponential stability is, in principle, possible. Namely, one may use KAM theory to establish the existence of maximal tori for the two–dimensional system in (9) obtained disregarding the exponentially small terms (and the term ω Iˆ3 ) concluding the entrapment of the Iˆ2 variable up to exponentially long times (for which the dynamics of the two–dimensional approximation coincides, essentially, with the full dynamics). Furthermore, using smooth, iso–energetic KAM theory, this approach might avoid the delicate complex analysis mentioned in item 1.4.5. We decided not to follow this approach since in order to apply smooth KAM one needs to control (in the two–dimensional case) the C r –norm of the Hamiltonian with r > 4, a fact which would involve to have a precise control, as ε → 0, of the first five derivatives of Iˆ2 the symplectic diffeomorphism casting 21 + εH 01 (Iˆ2 , ϕˆ1 ) into action–angle variables: these computations might not be simpler than the computations performed in § 3 below. A second reason is that the two–dimensional Hamiltonian obtained from (9) is in general not a convex function of Iˆ and iso–energetic KAM does not hold in the whole four–dimensional phase space. Indeed, instability channels (along which the action variables may drift away by a quantity of order one in exponentially long times – compare [3]) may, in general, appear. We believe that, in the D’Alembert case, such channels do not appear: it would be interesting to prove this fact, since the presence of channels might lead to a topological instability for the D’Alembert problem based on a mechanism entirely different from Arnold diffusion. 1.4.8 Finally, we mention that the estimates provided in the proof of Theorem 1.1 allow to compute explicitly all the constants involved. For example, let us indicate, here, for the case (p, q) 6= (1, 1), (2, 1), the exact expression of the constants γi and ε0 involved in the Theorem. Let a := 1 + min{c, `} and choose 1/2 < b ≤ (a − γ0 )/2. Then the time of stability can be taken to be5 T (ε) =
eSr
1
(2) 576M 2
εa−2b
(2)
exp(C2 ε−γ0 ).
(12)
where S, C2 , M 2 are defined, respectively, in (20),(34),(40) below. Furthermore, for any 5
Here e is the Neper constant e :=
P
k (k!)
−1
.
7
time |t| ≤ T (ε), we obtain 5 5 |I1 (t) − I1 (0)| ≤ rεb and |I1 (t)| ≤ rε` ; 4 4 √ defining γ1 := (2b − 1)/2, γ6 := C3 + 2−6 , with C3 is defined in (47) below, we obtain √ r |I2 (0)| ≤ γ6 rεγ1 =⇒ |I2 (t) − I2 (0)| ≤ (1 + 2)γ6 rεγ1 ≤ 8 2 2 γ r r |I2 (0)| > γ6 rεγ1 =⇒ |I2 (t) − I2 (0)| ≤ 6 ε2b−1 ≤ . 8 |Iˆ2 (0)|
(13) also (14) (15)
An explicit expression for ε0 may be immediately obtained by looking at the conditions (26), (27), (34), (38), (48), (52) below. 1.5 We close this Introduction by collecting some open questions mentioned above and some other related problems. 1.5.1 The major open problem in this context is certainly to prove Arnold diffusion and, eventually, to compare the diffusion time with the stability times given in Theorem 1.1. 1.5.2 A related question is the following. We saw that, taking ` ∼ 1/2 and c ≥ `, one can √ take γ0 ∼ 1/2 so that the stability time is O exp(1/ ε) . Prove (or disprove) that this bound cannot be improved. 1.5.3 Provide a simpler proof of Theorem 1.1 (in particular avoiding the use of the action– angle variables mentioned in item 1.4.5). 1.5.4 Prove (or disprove) the non–existence of “instability channels” (i.e., straight lines in action space supporting drifting orbits) for the two–degrees–of–freedom system obtained by disregarding the exponentially small terms in (9). 1.5.5 Refine the techniques used in the proof of Theorem 1.1 so as to apply the stability result to some concrete problem arising, for example, in the Solar System using astronomical data and compare the obtained result with the existing estimates on the life time for the Solar System. 1.5.6 Generalize the results of this paper to, e.g., the case of a non–rotational planet (in which case, the system has one more degree of freedom), the case of a non–fixed star, etc.
2
Proof of Theorem 1.1 in the case (p, q) 6= (1, 1), (2, 1)
In this section we present the proof of Theorem 1.1 in the case (p, q) 6= (1, 1) or (2, 1). In such a case the function H 01 (Iˆ2 , ϕˆ1 ) is simply c0 (0, Iˆ2 ). Bare in mind, however, that subsection 2.1, 2.2, and 2.3 hold for the general case and, therefore, in these subsections we maintain the general notation H 01 (Iˆ2 , ϕˆ1 ).
8
2.1
Preliminaries
We shall use the following notations: if ∅ 6= D ⊂ Rd and ρ := (ρ1 , ρ2 , . . . , ρd ) with 0 < ρj ≤ ∞ for 1 ≤ j ≤ d, we denote Dρ := {I = (I1 , . . . , Id ) ∈ Cd : |Ij − I¯j | < ρj , j = 1, . . . , d, for some I¯ ∈ D} ; Tdσ denotes the complex set {z ∈ Cd : |Imzj | < σ, j = 1, . . . , d} (thought of as a complex neighborhood of Td ). We shall work in the Banach space HR (Dρ × Tdσ ) of functions f real–analytic on Dρ × Tdσ having finite norm kf kρ,σ :=
X
sup |fˆk (I)|e|k|σ ,
k∈Zd I∈Dρ
fˆk (I) being the Fourier coefficients of the periodic function ϕ → f (I, ϕ). Notice that if f ∈ HR (Dρ × Tdσ ) and ρj = ∞ for some 1 ≤ j ≤ d, then, by Liouville Theorem, f does not depend on Ij . We shall use the following standard result from normal form theory; see [12] for the proof with ρ1 = · · · = ρd ; (for the trivial modifications in the case of different analyticity radii, see [3] or [4]). Lemma 2.1 (Normal Form Lemma) Let H := H(I, ϕ) := h(I) + f (I, ϕ) be a real– analytic Hamiltonian on D × Td belonging to HR (Dρ × Tdσ ) for certain ρ := (ρ1 , ρ2 , . . . , ρd ) with 0 < ρj ≤ ∞, 1 ≤ j ≤ d, and σ > 0. Let ρ0 := min1≤j≤d ρj . Let Λ a sub–lattice of Zd , α > 0, K ∈ N with Kσ ≥ 6. Suppose that ∀ I ∈ Dρ and ∀ k ∈ Zd \ Λ, |k| ≤ K, we have |h0 (I) · k| ≥ α and that the following condition is satisfied: kf kρ,σ =: η ≤
αρ0 . 210 K
(16)
Then, there exist a real–analytic symplectic transformation φ : (J, ψ) ∈ Dρ/2 × Tdσ/6 7→ (I, ϕ) = φ(J, ψ) ∈ Dρ × Tdσ and real–analytic functions f∗ := f∗ (J, ψ), g := g(J, ψ) := HR (Dρ/2 × Tdσ/6 ) such that the following properties hold: (i) H ◦ φ(J, ψ) = h(J) + (ii) kgkρ/2,σ/6 ≤
X
k∈Λ,|k|≤K
P
k∈Λ
gk (J)eik·ψ belonging to
fk (J)eik·ψ + g(J, ψ) + f∗ (J, ψ);
211 2 2 1 η ≤ η ≤ η; αρ0 σ Kσ 3
(iii) kf∗ kρ/2,σ/6 ≤ ηe−Kσ/6 ; (iv) |I − J| ≤
25 ρ0 η≤ 7 , ασ 2
|ϕ − ψ| ≤
σ 26 η≤ 5 , αρ0 2
9
∀ (J, ψ) ∈ Dρ/2 × Tdσ/6 .
ˆ 1 , . . . , Id−1 ) + ωId and f := f (I1 , . . . , Id−1 , ϕ), Remark 2.1 If we have h := h(I) := h(I then the symplectic transformation φ preserves the form of the Hamiltonian and has the form: (
Ij = I˜j (J1 , . . . , Jd−1 , ψ) , ϕj = ϕ˜j (J1 , . . . , Jd−1 , ψ) , Id = Jd + I˜d (J1 , . . . , Jd−1 , ψ) , ϕd = ψd ,
(1 ≤ j ≤ d − 1) ,
and, also, f∗ and g do not depend on Jd . In what follows, we shall assume that, for any 0 ≤ ε ≤ ε¯, the Hamiltonian6 Hε,µ belongs to HR (AR × T3s ), where R := (rε` , r, ∞) , (17)
s > 0 (and 0 < ε¯ < 1). We shall also denote M0 and M1 (ε–independent) upper bounds on, respectively, kF0 kR,s and kF1 kR,s .
2.2
Step 1 (linear change of variables)
Let φ0 be the following linear symplectic map: p 1 φ0 (I 0 , ϕ0 ) := (I10 , I20 , − I10 + I20 + I30 ), (ϕ01 + pϕ03 , ϕ02 − qϕ03 , qϕ03 ) . q q
(18)
Then, φ0 casts the Hamiltonian Hε,µ into the form H (0) (I 0 , ϕ0 ; ε, µ) := Hε,µ ◦ φ0 (I 0 , ϕ0 ) :=
I 0 21 2
(19)
+ ωI30 + εG0 (I10 , I20 , ϕ01 , ϕ02 , ϕ03 ) + εµG1 (I10 , I20 , ϕ01 , ϕ02 , ϕ03 ; µ) ,
which belongs to HR (AR × TS ) with S := c0 s ,
c0 := min{1/(1 + p), 1/(1 + q)} ,
and kG0 kR,S ≤ M0 ,
Moreover:
with
e (I 0 , I 0 , ϕ0 , ϕ0 , ϕ0 ) G0 (I10 , I20 , ϕ01 , ϕ02 , ϕ03 ) := H01 (I10 , I20 , ϕ01 ) + G 0 1 2 1 2 3 Z
0
and H01 (I10 , I20 , ϕ01 ) 6
kG1 kR,S ≤ M1 .
:=
2π
e (I 0 , I 0 , ϕ0 , ϕ0 , ϕ0 )dϕ0 = 0 , G 0 1 2 1 2 3 3
c0 (I10 , I20 ) ,
if (p, q) 6= (1, 1), (2, 1) ,
c0 (I10 , I20 ) + djp (I10 , I20 ) cos(jp ϕ01 ) ,
if (p, q) = (1, 1), (2, 1) ,
Recall the definitions of Hε,µ and A given in 1.3.
10
(20)
with j1 := 2 and j2 := 1. Obviously, since φ0 depends upon p and q, also the functions Gi and H01 depend upon p and q, but we shall not indicate such dependence in the notation. We remark that, in general, φ0 is not a diffeomorphism of R3 × T3 (since the induced map on T3 has determinant equal to q); this fact, however, does not affect the following analysis. If a := 1 + min{c, `}, using the fact that |I10 | < 2rε` and µ ≤ εc , one see that H (0) has the following form: I 0 21 (0) + ωI30 + εH 01 (I20 , ϕ01 ) + εG0 (I20 , ϕ01 , ϕ02 , ϕ03 ) + εa H2 (I10 , I20 , ϕ01 , ϕ02 , ϕ03 ; ε) 2 e (0, I 0 , ϕ0 ), where G0 (I20 , ϕ) := G 0 2 Z
2π
0
G0 (I20 , ϕ01 , ϕ02 , ϕ03 )dϕ03 = 0 ,
(21)
and (recall (5)) H 01 (I20 , ϕ01 )
:=
H01 (0, I20 , ϕ01 )
where c0 (I20 )
:=
I 0 22 := c00 + c02 , 2
c0 (I20 ) ,
if (p, q) 6= (1, 1), (2, 1) ,
c0 (I20 ) + djp (I20 ) cos(jp ϕ01 ) ,
if (p, q) = (1, 1), (2, 1) , (22)
c00 :=
and
1 2 − ν 21 , 4
1 3 c02 := ¯2 ν 21 − 1 J1 2
(23)
v u
u 1 I 02 I 02 d1 (I20 ) := d1 (0, I20 ) := − κ1 ν 1 t1 − ¯22 1 + ¯ , 2 J1 J1 1 I 0 2 2 d2 (I20 ) := d2 (0, I20 ) := − ν 21 1 + ¯ , 8 J1
where
L κ1 := κ1 (0) := ¯ , J1
ν 1 := ν1 (0) :=
q
1 − κ21 .
(24)
(25)
(0)
The function H2 (I10 , I20 , ϕ01 , ϕ02 , ϕ03 ; ε) belongs to HR (AR × T3S ) and (0)
kεH 01 + εG0 + εa H2 kR,S = kεG0 + ε1+c G1 kR,S ≤ ε(M0 + εc M1 ) .
2.3
Step 2 (time averaging)
Here, we shall remove, up to exponentially small terms, the (fast) dependence upon ϕ03 . To do this, we shall apply the Normal Form Lemma with d := 3, (I, ϕ) := (I 0 , ϕ0 ), H := H (0) ,
11
(0)
h := I 0 21 /2 + ωI30 , f := εH 01 + εG0 + εa H2 = εG0 + ε1+c G1 , D := A, ρ := R, ρ0 := rε` , σ := S, Λ := {(k1 , k2 , 0) s.t. k1 , k2 ∈ Z}, α := ω/2, K := ω/(4rε`). The condition Kσ ≥ 6 is implied by ε ≤ (ωS/24r)1/` . (26) Condition (16) becomes ε(M0 + εc M1 ) ≤ 2−9 r 2 ε2`
which is verified, for example, if
ε≤
1/(1−2`) r2 . 29 (M0 + M1 )
(27)
Hence, for ε small enough, we can apply the Normal Form Lemma, finding a real–analytic symplectic transformation ˆ ϕ) φ1 : (I, ˆ ∈ AR/2 × T3S/6 7→ (I 0 , ϕ0 ) ∈ AR × T3S , such that ˆ ϕ; ˆ ϕ; H (1) (I, ˆ ε, µ) := H (0) ◦ φ1 (I, ˆ ε) 2 Iˆ (1) ˆ ε) , := 1 + ω Iˆ3 + εH 01 (Iˆ2 , ϕˆ1 ) + εa H1 (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε) + H∗(1) (Iˆ1 , Iˆ2 , ϕ; 2
(28)
ˆ ϕ) and such that the following bounds hold. For any (I, ˆ ∈ AR/2 × T3S/6 , 6 r 1 ˆ ≤ 2 ε(M0 + εc M1 ) ≤ rε2` ≤ 7 rε` , |I 0 − I| ωS 8Sω 2
(29)
and (1)
(1)
kH1 kR/2,S/6 ≤ M1 , where (1)
kH∗(1) kR/2,S/6 ≤ M∗(1) := ε(M0 + εc M1 ) exp(−C1 ε−` ) ,
M1 := M1 +
2.4
8r (M0 + M1 ) , ωS
C1 :=
ωS . 24r
(30)
(31)
Step 3 (averaging over ϕˆ1 )
Here, we shall exploit the fact that, for (p, q) different from (1, 1) and from (2, 1), the Hamiltonian H 01 (Iˆ2 , ϕˆ1 ) is independent of the angles, allowing to treat the angle ϕˆ1 as a ˆ Consider, therefore, the Hamiltonian “fast” angle in a suitable domain A.
and let
ˆ2 ˆ (1) (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε) := I1 + εc0 (Iˆ2 ) + εa H1(1) (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε) , H 2
(32)
5 1 1 5 5 Aˆ := {Iˆ1 ∈ (− rε` , − rεb ) ∪ ( rεb , rε` ), |Iˆ2 − J¯2 | < r} . 4 2 2 4 4
(33)
12
In order to apply the Normal Form Lemma, we let b, γ0 be as in 1.4.8 and let: d := 2, (1) ˆ ρ := (rεb /4, r/4), (I, ϕ) := (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ), h := Iˆ12 /2 + εc0 (Iˆ2 ), f := εa H1 , D := A, (1) ρ0 := rεb /4, σ := S/6, Λ := {(0, k2 ) s.t. k2 ∈ Z}, α := rεb /8, K := r 2 (215 M1 εγ0 )−1 . With such positions, we see that we can apply Lemma 2.1, provided ε ≤ min
nh
(1)
i1/(1−b−γ0 ) 212 M1 , |c02 |r(J¯2 + 2r)
1/γ0
C2
o
,
C2 :=
r2 S (1)
9 · 217 M1
.
(34)
Under such condition, we can find a real–analytic symplectic transformation φˆ2 : (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ) ∈ Aˆ(rεb /8,r/8) × T2S/36 7→ (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ) ∈ Aˆ(rεb /4,r/4) × T2S/6 such that ˆ (2) (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) := H (1) ◦ φˆ2 (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) H I˜2 (2) ˆ (2) (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) := 1 + εc0 (I˜2 ) + εa H1 (I˜1 , I˜2 , ϕ˜2 ; ε) + H ∗ 2
(35)
with 4 (1) (2) (2) kH1 k(rεb /8,r/8),S/36 ≤ M1 := M1 , 3 (2) (2) ˆ k(rεb /8,r/8),S/36 ≤ M ˆ := εa M1(1) exp(−C2 ε−γ0 ), kH ∗ ∗ and (1)
3 · 29 M1 a−b rεb ε ≤ 9, |Iˆ1 − I˜1 |, |Iˆ2 − I˜2 | ≤ Sr 2
∀ (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ) ∈ Aˆ(rεb /8,r/8) × T2S/36 . (36)
Extend such symplectic transformation on Aˆ(rεb /8,r/8) × C × T3S/36 by setting φ2 (I˜1 , I˜2 , I˜3 , ϕ˜1 , ϕ˜2 , ϕ˜3 ; ε) := (φˆ2 (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε), I˜3 , ϕ˜3 ) . ˜ ϕ) In this way, denoting (I, ˜ = (I˜1 , I˜2 , I˜3 , ϕ˜1 , ϕ˜2 , ϕ˜3 ), we see that ˜ ϕ; ˜ ϕ; H (2) (I, ˜ ε) := H (1) ◦ φ2 (I, ˜ ε) 2 I˜ (2) := 1 + ω I˜3 + εc0 (I˜2 ) + εa H1 (I˜1 , I˜2 , ϕ˜2 ; ε) + H∗(2) (I˜1 , I˜2 , ϕ; ˜ ε) 2 with
(37)
ˆ (2) + M (1) . kH∗(2) k(rεb /8,r/8,∞),S/36 ≤ M∗(2) := M ∗ ∗
In order to simplify the calculus of the constants we assume that ε ≤ min
n
o C1 1/(`−γ0 ) , (`C1 )2/` . 2C2
13
(38)
Using (38) it is simple to prove that ε−1 exp(C1 ε−` ) ≥ ε−a exp(C2 ε−γ0 ).
(39)
In fact, using (38) and the fact that a < 3/2, it is sufficient to prove that exp(C1 ε−` ) ≥ ε−1 which is guaranteed7 again by (38). By (39) we, also, obtain (2)
(2)
M∗(2) ≤ M 2 εa exp(−C2 ε−γ0 ),
2.5
M 2 := M1 + 1 +
8r (M0 + M1 ) . ωS
(40)
Step 4 (energy conservation)
We are, now, in the position of concluding the proof of Theorem 1.1 for the case (p, q) 6= (1, 1), (2, 1). The arguments we shall use, here, are based on energy conservation. However, such arguments, are not completely straightforward because we have to keep track of domains (recall that the variables (I˜1 , I˜2 ) are not defined in a neighborhood of the origin) and also because we shall freely use different sets of variables. • (Energy conservation for the Hamiltonians H (1) and H (2) ) ˆ ϕ(t)) ˜ ϕ(t)) Denote by zˆ(t) := (I(t), ˆ and z˜(t) := (I(t), ˜ the solutions of the Hamilton equations associated, respectively, to the Hamiltonians H (1) in (28) and H (2) in (37), ˆ ˜ with respective initial data zˆ(0) := (I(0), ϕ(0)) ˆ and z˜(0) := (I(0), ϕ(0)). ˜ Furthermore, if ˜ ˆ ˜ F = F (I, ϕ), ˜ denote ∆t F := F (ˆ z (t)) − F (ˆ z (0)) and ∆t F := F (˜ z (t)) − F (˜ z (0)). Then, 8 conservation of energy for the Hamiltonians in (28) and (37) yields : i h i h ˆ t Iˆ2 )2 + Iˆ1 (0)∆ ˆ t Iˆ1 + 1 (∆ ˆ t Iˆ1 )2 ˆ t Iˆ2 + 1 (∆ εc02 Iˆ2 (0)∆ 2 2 (1) a ˆ t Iˆ3 + ε ∆ ˆ t H1 + ∆ ˆ t H (1) = 0 , +ω ∆ ∗
(41)
and h i h i ˜ t I˜2 + 1 (∆ ˜ t I˜2 )2 + I˜1 (0)∆ ˜ t I˜1 + 1 (∆ ˜ t I˜1 )2 εc02 I˜2 (0)∆ 2 2 (2) a˜ ˜ ˜ ˜ +ω ∆t I3 + ε ∆t H1 + ∆t H∗(2) = 0.
(42)
• (A–priori exponential estimates for the drift of I˜1 , Iˆ3 and I˜3 ) For9 0 ≤ t ≤ T (ε) we have directly by Hamilton equations, (39) and Cauchy estimates10 (1)
˜ t I˜3 | = |∆ ˆ t Iˆ3 | ≤ sup |∂ϕˆ H2 (ˆ z (τ ); ε)|t ≤ |∆ 3 0≤τ ≤t
7
M0 + M1 2b 6 (1) rε , M2 t ≤ (2) eS 96M 2
(43)
Setting x := ε−` and y := 1/`C1 we have to prove that ex ≥ xy . This is obvious if y ≤ 1; if y > 1 it is true if, for example, x ≥ y 2 . 2 2 8 Recall (22) and observe that for any numbers x, y, one has x2 − y2 = 21 (x − y)2 + y(x − y). 9 We shall consider only positive times since negative times are treated in a completely analogous way. 10 “Cauchy estimates” allow to bound derivatives of analytic functions in terms of their sup–norm on larger domains; with our choice of norms, Cauchy estimates take the following form. Consid-
14
˜ t I˜1 | ≤ |∆ •
(2)
sup |∂ϕ˜1 H2 (˜ z (τ ); ε)|t ≤
0≤τ ≤t
1 36 (2) M2 t ≤ rε2b . eS 16
(44)
Consider, now, (real) initial positions (Iˆ1 (0), Iˆ2 (0)) ∈ {|Iˆ1 | ≤ (1 + 2−7 )rε` } × {|Iˆ2 − J¯2 | ≤ (1 + 2−7 )r} ,
and let us consider, separately, two cases: 1 (i) |Iˆ1 (t)| < rεb , 2
∀ 0 ≤ t ≤ T (ε);
1 (ii) ∃ 0 ≤ t∗ < T (ε) s.t. |Iˆ1 (t)| < rεb 2
∀ 0 ≤ t < t∗
1 and |Iˆ1 (t∗ )| ≥ rεb . 2
• (Case (i) and stability of Iˆ2 ) Consider case (i): by (43) and (41), we see that, until |Iˆ2 (t) − J¯2 | ≤ 3r/2, we have where we can take11
ˆ ˆ t Iˆ2 2I2 (0)∆
(45)
ˆ t Iˆ2 )2 ≤ C3 r 2 ε2b−1 + (∆ (1)
(46)
(2)
M 1 1 M0 + M1 ω 16 M1 + + + 4 22 . C3 := (2) 2 |c02 | 4 3 r r 48M 2 r
(47)
We need, at this point, an elementary estimate (whose trivial proof is left to the reader): Lemma 2.2 Let y, y0 ∈ R and C > 0 and suppose that |2y0 y + y 2 | ≤ C 2 . Then: (1) if |y0 | ≤ C then |y| ≤ |y0 | +
q
y02 + C 2 ≤ (1 +
(2) if |y0 | > C then12 |y| ≤ C 2 |y0 |−1 ≤ C.
√ 2)C,
P P ik·ϕ |k|s er a 2π–periodic function f (ϕ) := , analytic on Ts with kf ks := , then k∈Z fk e k |fk |e 1 maxϕ∈T |∂ϕ f (ϕ)| ≤ es kf ks . In fact for all 0 < σ < s we have maxϕ∈T |∂ϕ f (ϕ)| ≤ k∂ϕ f ks−σ = P P 1 −|k|σ |k|s |fk |e|k|s ≤ eσ and taking the sup over σ < s of the right hand side, we have k |k|e k |fk |e the thesis. 11 Recall (6), which implies c02 6= 0. √ √ 12 We set x := C 2 y0−2 and we have used that 1 + x − 1 ≤ x/2 and 1 − 1 − x ≤ x for 0 ≤ x ≤ 1.
15
Let us now assume that ε≤
1 √
20 C3
2/(2b−1)
and let us apply the estimates of Lemma 2.2 to (46) with C := ˆ Iˆ2 . Then: y := ∆ |Iˆ2 (0)| ≤ |Iˆ2 (0)| >
q
q
C3 rεγ1
=⇒
C3 rεγ1
=⇒
(48) √ C3 rεγ1 , y0 := Iˆ2 (0) and
1 √ q 1 − 8 r , (49) 2) C3 rεγ1 ≤ 8 2 2 1 1 C3 r 2b−1 (50) − 8 r, ε ≤ |Iˆ2 (t) − Iˆ2 (0)| ≤ 8 2 |Iˆ2 (0)|
|Iˆ2 (t) − Iˆ2 (0)| ≤ (1 +
which in particular imply (45). • (Case (ii) and and stability of Iˆ1 ) If (ii) occurs, then, by (50), we have that (1 + 2−7 ) ≥ |Iˆ1 (t∗ )| ≥ rεb /2 ,
|Iˆ2 (t∗ ) − J¯2 | ≤ 5r/4 .
Then, by (36), we can find ˆ I˜∗ , I˜∗ , ϕ˜∗ , ϕ˜∗ ) = (Iˆ1 (t∗ ), Iˆ2 (t∗ ), ϕˆ1 (t∗ ), ϕˆ2 (t∗ )) . (I˜1∗ , I˜2∗ , ϕ˜∗1 , ϕ˜∗2 ) ∈ Aˆ( rεb , r ) × T2 s.t. φ( 1 2 1 2 29
29
Now, as in (44), we have hence, using (36),
|I˜1 (t) − I˜1 (t∗ )| ≤ rε2b /16
(51)
|Iˆ1 (t) − Iˆ1 (0)| ≤ |Iˆ1 (t) − I˜1 (t)| + |I˜1 (t) − I˜1 (t∗ )| + |I˜1 (t∗ ) − I˜1 (0)| + |I˜1 (0) − Iˆ1 (0)| 1 1 + 8 )rεb ≤ (1 + 16 2 1 1 + 6 )rε` . |Iˆ1 (t)| ≤ (1 + 16 2 Finally, using (29), we have (13), provided the following condition is, also, satisfied ε≤
1/1−b r . 210 ωS(M0 + M1 )
(52)
• (Stability of I˜2 ) In order to prove stability for the I2 -variable, we can apply (42) until 1 1 − 9 rεb |I˜1 (t) − I˜1∗ | ≤ 8 2
and
1 1 − 9 r |I˜2 (t) − I˜2∗ | ≤ 8 2
(53)
obtaining again (as for (46)) ˜∗ ˜ 2I2 (I2 (t) − I˜2∗ ) + (I˜2 (t) − I˜2∗ )2
16
≤ C3 r 2 ε2b−1 .
(54)
We prove the √ first inequality in (53) using (36) and (51). As in case (i) we use Lemma 2.2 with C := C3 rεγ1 , y0 := I˜2∗ and y := I˜2 (t) − I˜2∗ . Using again (48) we have that |I˜2∗ | ≤ |I˜2∗ | >
q
q
C3 rεγ1
=⇒
C3 rεγ1
=⇒
1 √ q 1 − 8 r, 2) C3 rεγ1 ≤ 8 2 2 1 1 C r 3 − |I˜2 (t) − I˜2∗ | ≤ ∗ ε2b−1 ≤ r, ˜ 8 28 |I 2 |
|I˜2 (t) − I˜2∗ | ≤ (1 +
(55) (56)
which, in particular, imply the second condition in (53). • (Conclusion) Finally (49),(50),(55),(56),(29),(36) imply (14) and (15), concluding the proof of Theorem 1.1 in the case (p, q) 6= (1, 1), (2, 1).
3
Complex action–angle variables
We now turn to the special cases (p, q) = (1, 1) or (2, 1), in which case the Hamiltonian H 01 (Iˆ2 , ϕˆ1 ) depends explicitly on ϕˆ1 ; see (22). In order to carry out the analogous of step 3 above, we have, first, to introduce action–angle variables for the two–dimensional inteIˆ2 grable system 21 + εH 01 (Iˆ2 , ϕˆ1 ), which may be viewed as a “suspended pendulum” (with potential cos ϕˆ1 or cos 2ϕˆ1 ) having a small gravity varying with a second action–variable. Of course such action–angle variables will be singular near the separatrices and near elliptic periodic orbits (corresponding to the equilibria of the pendulum) and therefore a careful blow–up analysis near the singularities, as ε → 0, is needed. The results are collected in the following proposition13 . Throughout this section, we shall denote z1 and z2 , respectively, the real and imaginary part of a complex number z = z1 + iz2 . Proposition 3.1 Let k(I) real–analytic on ∆0 := [α, β] for α < β with analytic extension on ∆0r2 for r2 > 0. Let k¯ := max k(I¯1 ), I¯1 ∈∆0
kˆ := min k(I¯1 ), I¯1 ∈∆0
|k 0 (I)|, k¯ 0 := max 0 I∈∆r2
and suppose that kˆ > 0. Let ε > 0, η > 0, R0 ≥ 2r1 > 0, 0 < s1 ≤ 1, s2 > 0, D 0 := [−R0 , R0 ], E(p, I, q) := p2 /2 − εk(I)(1 + cos q) and14
M+ :=
M− :=
13
n
n
(p, I, q, ϕ) ∈ [0, R0 ] × ∆0 × T2 : η ≤ E(p, I, q) ≤ R02 /2
o
o
(p, I, q, ϕ) ∈ D 0 × ∆0 × T2 : −2εk(I) + η ≤ E(p, I, q) ≤ −η .
For the simpler case k ≡ 1, see Lemma 2.1 and Appendix B of [3]. An analogous statement holds for p ∈ [−R0 , 0]. Of course, we are assuming that the various parameters are chosen so that M± 6= ∅. 14
17
¯ ˆ ¯0 Then, there exist positive constants c1 , c2 (sufficiently large and depending only √ √ on k,k,k ) and c3 ,c4 ,c5 ,c6 ,ε0 (sufficiently small) such that if ε ≤ ε0 , R0 ≥ c1 ε, r1 ≥ c2 ε, η ≤ c3 ε and η s1 s2 η ρ2 := min{c5 , r2 }, σ1 := c6 , σ2 := , ρ1 := c4 √ , ε ε ln(ε/η) ln(ε/η) 2 then the following holds. There exist two real–analytic symplectic transformations φ ± and 2 two real–analytic functions E ± : Ω± (ρ1 ,ρ2 ) → C so that 0 0 φ± : (P, J, Q, ψ) ∈ Ω± (ρ1 ,ρ2 ) × Tσ1 × Tσ2 → (p, I, q, ϕ) ∈ Dr1 × ∆r2 × Ts1 × Ts2 ,
p = p± (P, J, Q), I = J , q = q ± (P, J, Q) , ϕ = ϕ± (P, J, Q, ψ) , E ± (P, J) = E(p± (P, J, Q), J, q ± (P, J, Q)) , φ± (Ω± × T2 ) = M± , where:
n
Ω± := (P1 , I1 ) ∈ R2
s.t.
(57)
o
P1 ∈ D ± (I1 ), I1 ∈ ∆0 ,
D + (I1 ) := P + (η, I1 ), P + (R02 /2, I1 ) , D − (I1 ) := P − (−2εk(I1 ) + η, I1 ), P − (−η, I1 ) , √ Z q √ Z 2 π 2 2 ψ0 (E,I) q + − P := g(E, I, θ)dθ, P := g(E, I, θ)dθ, π 0 π 0 and ψ0 (E, I) := arccos(−1 − E/εk(I)).
g(E, I, θ) := E + εk(I)(1 + cos θ),
Moreover, the following estimates hold for (P, I) ∈ Ω± (ρ1 ,ρ2 ) : q
q
εk1 (I) 1 1 εkˆ r r , ≥ √ ≥ εk1 (I) ˆ 4π 2 2π ln 1 + ln 1 + εηk |E1± (P,I)| q √ 4 6π|k 0 (I)|ε r ∂I E ± (P, I) ≤ . (I) ln 1 + |Eεk±1(P,I)|
∂P E ± (P, I)
(58)
(59)
1
Finally, since ∂I E ± (P, I) = −∂I P ± (E ± (P, I), I)[∂E P ± (E ± (P, I), I)]−1 , we can write, for real P and I, ∂I E ± (P, I) = −εk(I)[1 + Y ± (P, I)]
|Y ± (P, I)| < 1
with
(60)
where
Y ± (P, I) :=
Z
ψ± 0
q
1 g(E ± (P, I), I, θ)
with ψ + := π and ψ − := ψ0 (E − (P, I), I).
18
−1
dθ
Z
ψ± 0
q
cos θ g(E ± (P, I), I, θ)
dθ
Proof • (First step: estimates on the action domains) Let k¯1 := supI∈∆0ρ |k1 (I)|, k¯2 := supI∈∆0ρ |k2 (I)|, a ¯1 := k¯1 (1 + cosh s1 ) + k¯2 sinh s1 and 2 2 a ¯2 := k¯2 (1 + cosh s1 ) + k¯1 sinh s1 . q For suitable c7 , c8 > 0 small enough, we define E2∗ (E1 ) := c7 η ln−1 (1 + |Eε1 | ), E := c8 r1 (R0 + r1 ) + R2 /2, and, for I¯1 ∈ ∆0 , we define the following domains15 0
E + := E + (I¯1 ) := {E1 + iE2 , s.t. η/2 ≤ E1 ≤ E ; |E2 | ≤ E2∗ (E1 )}, E − := E − (I¯1 ) := {E1 + iE2 , s.t. − 2εk(I¯1 ) + η/2 ≤ E1 ≤ −η/2 ; |E2 | ≤ E2∗ (E1 )}. Let F (p) := p2 /2. We claim that |I − I¯1 | ≤ ρ2 , I¯1 ∈ ∆0 , E ∈ E + ∪ E − (I¯1 ), θ ∈ Ts1
=⇒
ˆ where It is immediate to see that F (Dr01 ) ⊇ E,
g(E, I, θ) ∈ F (Dr01 ).
(61)
q
Eˆ := {−r12 ≤ 2E1 ≤ R02 − r12 , |E2 | ≤ r1 2E1 + r12 } [ {R2 − r 2 ≤ 2E1 ≤ (R0 + r1 )2 , |E2 | ≤ Eˆ2 (E1 )} 0
1
and Eˆ2 (E1 ) := [−E1 + (R0 + r1 )2 /2]R0 /(R0 + r1 ). Next, we define Ee
q
:= {−2εk¯1 ≤ E1 ≤ 2ε, |E2 | ≤ E20 } ∪ {2ε < E1 ≤ E, |E2 | ≤ 2c7 η E1 /ε} ,
where E20 := max{E2∗ (2ε), E2∗ (2εk¯1 )}, and E¯ := E¯(1) ∪ E¯(2) ∪ E¯(3) with:
E¯(1) := {−2εk¯1 − ε¯ a1 ≤ E1 ≤ 2ε − ε¯ a1 , |E2 | ≤ E20 + ε¯ a2 }, q a2 , |E2 | ≤ 2c7 η (E1 + ε¯ a1 )/ε + ε¯ a1 }, E¯(2) := {2ε − ε¯ a1 < E1 ≤ E − ε¯ q
a1 < E1 ≤ E + ε¯ a1 , |E2 | ≤ 2c7 η E/ε + ε¯ a2 }. E¯(3) := {E − ε¯
¯ moreover (recalling the definitions of a We observe that obviously Ee ⊆ E; ¯1 , a ¯2 ) e θ∈T |I − I¯1 | ≤ ρ2 , I¯1 ∈ ∆0 , E ∈ E, s1 q
=⇒
q
¯ g(E, I, θ) ∈ E.
We observe that E1 ≥ 2ε implies ln−1 (1 + ε/E1 ) ≤ 2 E1 /ε and hence Ee ⊇ E + ∪ E − (I¯1 ). Now we prove E¯ ⊆ Eˆ which will imply (61), since F (Dr01 ) ⊇ Eˆ ⊇ E¯ ⊇ Ee ⊇ E + ∪ E − (I¯1 ). It is simple to see that E¯ ⊆ Eˆ is implied by the following conditions: q
ˆ which is implied by r1 2(−2εk¯1 − ε¯ (1) (−2εk¯1 − ε¯ a1 ) + i(E20 + ε¯ a2 ) ∈ E, a1 ) + r12 ≥ E20 + ε¯ a2 15
In the positive energy case E + does not really depend on I¯1 .
19
q
q
ˆ2 (E + ε¯ (2) (E + ε¯ a1 ) + i(2c7 η E/ε + ε¯ a2 ), which is implied by E a1 ) ≥ 2c7 η E/ε + ε¯ a2 q
q
(3) if (R02 − r12 )/2 > 2ε − a ¯1 then r1 2E1 + r12 ≥ 2c7 η E/ε + ε¯ a2 for all 2ε − a ¯ 1 ≤ E1 ≤ 2 2 a1 }. min{(R0 − r1 )/2, E − ε¯ Defining k˜1 := max{1, k¯1 }, one sees that (1) holds provided q q √ c2 ≥ max{ 4c7 c3 k˜1 + 2¯ a2 , 2 2k¯1 + a ¯1 }. ˆ a1 ) ≥ R0 (R0 r1 /2 − ε¯ a1 )/(R0 + r1 ) and hence (2) holds If c8 ≤ 1/2, we have √ that E2 (E + ε¯ provided c2 ≥ 16 2c7 c3 √ and c1 c2 ≥ 4¯ a1 + 8¯ a2 . Finally, conditions c2 ≥ 2¯ a2 and c2 ≥ 2c7 c3 imply (3). • In the following we will choose the positive branch of the square root i.e. if z = |z|eiα q √ with α ∈ (−π, π), then we define the analytic function z := |z|eiα/2 . We also define √ ln z := ln |z| + iα and arccos z := −i ln(z + i 1 − z 2 ). We need the following elementary lemma (whose obvious proof is left to the reader): √ Lemma 3.1 Let x1 , x2 ≥ 0. Then x1 ± ix2 = w1 ± iw2 and (x1 ± ix2 )−1/2 = y1 ∓ iy2 where := √12 q
r
q
x21
x1 + + √ 2 2 x1 + x1 +x2 y1 (x1 , x2 ) := √ √ 2 2 , w1 (x1 , x2 )
2
x22
,
w2 (x1 , x2 )
:= √12 q
y2 (x1 , x2 ) :=
x1 +x2
r
q
−x1 + x21 + x22 , √ −x1 + x21 +x22 √ √ 2 2 . 2
x1 +x2
We observe that, for x1 fixed, w1 (resp. √ y1 ) is increasing (resp. decreasing) for x2 ≥ 0; y2 is also increasing but only for x2 ≤ 3x1 . Moreover if x1 ≥ x2 the following estimates hold √ √ √ √ 1 √x2 ≤ w2 ≤ √12 √xx21 , x1 ≤ w 1 ≤ x1 + x 2 ≤ 2 x1 , 3 x1 √ 1 1 2 1 x2 1 1 x2 √ √ ≤√ √ ≤ y ≤ . ≤ y1 ≤ √ , 2 3/2 x1 + x 2 x1 4 x3/2 2 x1 2 x 1 1 • (Second step: estimates on the action derivatives in the positive energy case) In the following we put := εk1 (I). We observe that, for θ ∈ [0, π], 2˜ g /π 2 ≤ g1 ≤ g˜, where g˜(E1 , I; θ) := E1 + εk1 (I)(π − θ)2 . The following estimates hold16 1 √ ln 1 +
16
s
E1
!
≤
Z
π
0
dψ
q
g˜(ψ)
=
Z
π
0
√
dψ 1 Za dy √ √ = 2 0 E1 + ψ 1 + y2 s
!
2π 1 ; = √ arcsinh (a) ≤ √ ln 1 + E1 Z π Z π 1 dψ dψ √ ≤ = (62) 3/2 g (ψ)) E1 E1 + 0 (˜ 0 (E1 + ψ 2 )3/2 dy π 1 Za π √ √ = 3/2 q ; = ≤ 2 3/2 E1 0 (1 + y ) E1 E1 + E1 1 + π 2 /E1
Use ln(1 + t) ≤ arcsinh (t) = ln(t +
√ 1 + t2 ) ≤ 2 ln(1 + t)
20
q
where a := π /E1 . Next, we prove that ∀ |I − I¯1 | ≤ ρ2 , I¯1 ∈ ∆0 , E ∈ E + g1 (E, I, θ) ≥ 2|g2 (E, I, θ)|. In fact we first have that if c5 ≤ 1/8k¯0 then g1 (E, I, θ) ≥ E1 /2 and, hence, we have only to prove that E1 /4 ≥ E2∗ (E1 ) + 2k¯0 c5 η ln−1 (ε/η). Taking c5 ≤ c7 /4k¯0 we have only to verify that |E1 | ≥ 6E2∗ (E1 ) .
(63)
It is easy to see that the previous inequality is verified for c7 ≤ 1/36. Consider, now, I = I¯1 ∈ ∆0 real and g2 = E2 ≥ 0. Using Lemma 3.1, g1 ≥ g˜ and (62) we have √ √ ! s Z π 1 dψ 2 2 + ¯ q E2 E2 √ ln 1 + ≥ . (64) P2 (E, I1 ) ≥ 3π 3π E1 0 g˜(ψ)
Using Lemma 3.1, 2˜ g /π 2 ≤ g1 ≤ g˜ and (62) we have 1 √ ln 1 + 2π
s
E1
!
≤
∂E P1+ (E, I)
√ ! s 2π ≤ √ ln 1 + . E1
(65)
Using Lemma 3.1, 2˜ g /π 2 ≤ g1 , (62) and the fact that |g2 | ≤ |E2 | + 2εk¯2 we have π4 1 √ |∂E P2+ (E, I)| ≤ (|E2 | + 2εk¯2 ) √ . 4 2 E1 E1 +
(66)
We observe that17 q q c7 η |E2 | c7 q q ≤ ≤ c7 ln(1 + ε/E1 ) ≤ ln(1 + /E1 ) . E1 E1 ln(1 + ε/E1 ) k1 (I)
(67)
Let us proceed by proving that
εk¯0 ρ2 π 3 = k¯ 0 c5 π 3 η ≤ η ln(1 +
q
2/η) ≤ 2E1 ln(1 +
q
q
/E1 ) .
(68)
In fact, last inequality holds because the function E1 ln(1 + /E1 ) is increasing and attains minimum for E1 = η/2; the 3 ≤ 1/8 and using q first inequality is proved, if c√ √ ˆ which is verified if c5 ≤ (π 3 2k¯ 0 )−1 min{1, k}. ˆ ˆ k1 (I) ≥ k/2, if k¯0 c5 π 3 ≤ ln(1 + 2 2 k), Using (65), (66), (67), (68) we have √ ! s 2π 2 . (69) |∂E P + (E, I)| ≤ √ ln 1 + E1 It remains to estimate +
|∂I P (E, I)| = 17
We use the fact that x ln2 (1 +
p
√
2 0 ε|k (I)|| 2π
Z
0
π
1 + cos θ
q
g(E, I; θ)
ε/E1 ) ≥ η if x ≥ η/2 and c3 ≤ 1/8.
21
dθ| .
We observe that |
Z
π
1 + cos θ
q
0
where F1 (x) :=
dθ| = |
g(E, I; θ)
Z
√
2 0
√
q
x
2 − x E + εk(I)x
dx| ≤
Z
2 0
F1 (x)dx ,
√ √ √ x( 2 − x E1 + x)−1 . In order to estimate last integral we split it as Z
2
F1 (x)dx =
0
Z
1 0
F1 (x)dx +
Z
2 1
F1 (x)dx.
We have Z
1 0
which implies
√ Z 1 1 x x √ √ dx ≤ dx = √ , F1 (x)dx ≤ E1 + x E1 x + x E1 + x 0 0 √ √ Z 2 Z 2 2 2 2 1 √ , F1 (x)dx ≤ √ dx = √ E1 + 1 E1 + 1 2−x Z
1
√
√ √ |k 0 (I)| √ ε 2 2 0 ε. |k (I)| √ |∂I P (E, I)| ≤ ≤ 2 6q π E1 + k1 (I) +
We, now, prove that
P + (E + ) ⊇ (D + (I¯1 ))2ρ1 √ R 2 π π 0
∀ I¯1 ∈ ∆0 .
(70)
(71)
w1 (E1 + εk(I¯1 )(1 + cos θ), E2 )dθ, we have by Lemma 3.1 Since P1+ (E1 + iE2 , I¯1 ) = + that P1 is an increasing function for E2 ≥ 0. Hence, in order to prove (71), we have to prove the following estimates, ∀ E1 + iE2 ∈ E + , I¯1 ∈ ∆0 : (i) (ii) (iii) (iv) If c4 ≤
√ 2 c 6π 7
P2+ (E1 + iE2∗ (E1 ), I¯1 ) ≥ 2ρ1 , P1+ (η/2 + iE2∗ (η/2), I¯1 ) ≤ P1+ (3η/4, I¯1 ), P1+ (η, I¯1 ) − P1+ (3η/4, I¯1 ) ≥ 2ρ1 , P1+ (E, I¯1 ) − P1+ (R02 /2, I¯1 ) ≥ 2ρ1 .
min{1, k¯1 } we obtain (i), since, from (64),
q q √ √ ¯ ln(1 + k( I ) ε/E1 ) 2η 1 2η 1 + ∗ q P2 (E1 + iE2 (E1 ), I¯1 ) ≥ c7 √ q ≥ c7 √ min{1, ¯ }. 3π ε k(I¯1 ) ln(1 + ε/E1 ) 3π ε k
Inequality (ii) follows from P1+ (η/2
+
iE2∗ (η/2), I¯1 )
Z πq √ η/2 + E2∗ (E1 ) + εk(I¯1 )(1 + cos θ)dθ ≤ ( 2/π) 0 Z πq √ 3η/4 + εk(I¯1 )(1 + cos θ)dθ = P1+ (3η/4, I¯1 ). ≤ ( 2/π) 0
22
Using (65) and the fact that η ≤ ε/8, we have18 P1+ (η, I¯1 )
−
P1+ (3η/4, I¯1 )
q
i 1 η h 1 ¯ ≥ 1 √η min{1, 1 }, √ √ ln(1 + 2 k) ≥ 8π ε k¯ 8π ε k¯
which implies (iii), provided c4 ≤ Again, from (65), we have
1 16π
min{1, k1¯ }.
P1+ (E, I¯1 ) − P1+ (R02 /2, I¯1 ) ≥
c8 1 r1 (R0 + r1 ) √ π E
v u u E . min 1, 2t εk(I¯1 )
Distinguishing the two√cases for the minimum, we see that (iv) holds, provided c8 c2 ≥ √ ¯ 4 c3 . 2πc4 c3 and c8 c2 c1 ≥ kπc
• (Third step: estimates on the action derivatives in the negative energy case) ˜ := E + 2εk(I) and using the substitution ξ := 1 + εk(I)(cos θ − 1)/E ˜ we Defining E obtain √ Z √ 2 2 1 E˜ ξ − q dξ, P (E, I) = π 0 √1 − ξ Eξ ˜ −E √ Z 2 1 1 − q ∂E P (E, I) = dξ, √ √ π 0 ˜ ξ 1 − ξ Eξ − E q √ 0 Z 1 ˜ −E Eξ 2k (I) √ √ ∂I P − (E, I) = dξ. πk(I) 0 ξ 1−ξ
˜ − E = x1 + ix2 where x1 := 2ξ − E1 (1 − ξ) and x2 := 2εk2 (I)ξ − E2 (1 − ξ). We define Eξ ˆ k¯0 ), we have k1 (I) ≥ |k2 (I)|, we obtain that Using that |E1 | ≥ |E2 | and that, if c5 c3 ≤ k/(2 x1 ≥ |x2 |. We observe also that in order to perform the previous change of variables θ = arccos(−1+ ˜ (ξ − 1)E/εk(I)) we have to verify that the argument of arccos is well defined19 . For any I ∈ ∆0ρ2 , we take I¯1 ∈ ∆0 with |I − I¯1 | ≤ ρ2 . We have −E1 ≤ 2εk(I¯1 ) − η/2. We have to prove that, defining y := k22 (I)/k12 (I) with 0 ≤ y ≤ 1, if E2 = E1 k2 (I)/k1 (I) then [2ε(k(I¯1 ) + k2 (I)) − η/2]x + [−2εk1 + η/2] ≥ 0, which is verified provided c5 ≤ 1/(4k¯0 ). In the following, in order to estimate the derivatives of P − , we set b := 2/|E1 | ≥ 1 and we will use Lemma 3.1. For I = I¯1 ∈ ∆0 , we have √ Z √ √ √ Z 1 1 2 ξ 2E ξ 2 2 − √ (E˜1 y2 + E2 y1 ) √ P2 (E, I¯1 ) = dξ ≥ dξ 3/2 π 0 1−ξ 1−ξ 0 π(2ξ + |E1 |(1 − ξ)) √ s Z b ξ E2 1 E2 dξ ≥ ln 1 + . (72) ≥ √ √ 2π µ 0 (2ξ + |E1 |)3/2 6π µ |E1 | 18
19
Use ln(1 + 2x)/x ≥ min{1, 1/x}. We define arccos(z1 + iz2 ) in the complementary of the set {z1 ∈ (−∞, −1] ∪ [1 + ∞)}
23
Furthermore,20 ∂E P1− (E, I)
√ Z 2Z1 1 1 2 1 y1 (x1 , x2 ) 2Z1 q √ √ √ √ dξ dξ + = dξ ≤ √ π 0 π 0 π 0 ξ 1−ξ ξ ξ 2ξ + |E1 |/2 √ Z √ s 1 2 b 2 12 √√ = . (73) dt + √ ≤ √ ln 1 + √ π µ 0 t 1+t π µ π µ |E1 |
On the other hand, ∂E P1− (E, I) ≥
Z b/2 1 Z 1/2 1 1 1 q √ √√ dt dξ = √ √ π 0 t 1+t 2π µ 0 ξ 2ξ + |E1 |
1 ≥ √ ln 1 + 2π µ
s
. |E1 |
(74)
Using the fact that |x2 | ≤ 2εk¯2 + |E2 | and the estimate |∂E P2− (E, I)|
6 2 |x2 | 1Z1 dξ ≤ ≤ √ + 3/2 , √ √ 3/2 π 0 π|E1 | µ ξ 1 − ξ x1
we obtain that, as in the positive energy case, √ ! s 2 2π − . (75) |∂E P (E, I)| ≤ √ ln 1 + E1 q √ √ Since x1 ≥ |x2 |, one has |x1 + ix2 | ≤ 2 x1 . From the previous inequality we conclude that √ Z √ |k 0 (I)| √ x 2|k 0 (I)| 1 − √ √ 1 |∂I P (E, I)| ≤ ε. (76) dξ ≤ 2 6 q π|k1 (I)| 0 ξ 1−ξ k1 (I) Finally differentiating the equality E ± (P ± (E, I), I) = E with respect to E and I we obtain respectively ∂P E ± (P, I) = [∂E P ± (E ± (P, I), I)]−1 and ∂I E ± (P, I) = −∂I P ± (E ± (P, I), I)[∂E P ± (E ± (P, I), I)]−1 , which, by (65), (69), (70), (74), (75), (76), imply (58) and (59) . Next, we prove that P − (E − (I¯1 )) ⊇ (D − (I¯1 ))2ρ1
∀ I¯1 ∈ ∆0 .
(77)
q √ R Since P1− (E1 + iE2 , I¯1 ) = 2 π 2 01 (E˜1 y1 − E2 y2 ) ξ/1 − ξdξ, we have by Lemma 3.1 that (being y1 , −y2 decreasing) P1− is a decreasing function (for E2 ≥ 0). Hence, in order to prove (77), it is enough to prove the following estimates, ∀ I¯1 ∈ ∆0 , E1 + iE2 ∈ E − (I¯1 ):
20
Use
Rb 0
(i) P2− (E1 + iE2∗ (E1 ), I¯1 ) ≥ 2ρ1 , (ii) P1− (−η/2 + iE2∗ (η/2), I¯1 ) − P1− (−η, I¯1 ) ≥ 2ρ1 , (iii) P1− (−2εk(I¯1 ) + η, I¯1 ) − P1− (−2εk(I¯1 ) + η/2, I¯1 ) ≥ 2ρ1 .
1 √ √ t 1+t
dt ≤ 4 ln(1 +
√
b).
24
If c4 ≤
1 c 12π 7
min{1, k1¯ } we obtain (i) because of:
P2− (E1 + iE2∗ (E1 ), I¯1 ) ≥ c7
ln(1 +
q
q
k(I¯1 ) ε/E1 )
1 η η √ q q ≥ c7 √ min{1, ¯ }. 6π ε k(I¯1 ) ln(1 + ε/E1 ) 6π ε k
Since P1− (−η/2 + iE2∗ (η/2), I¯1 ) − P1− (−η, I¯1 ) ≥ P1− (−η/2, I¯1 ) − P1− (−η, I¯1 ) − |P1− (−η/2 + iE2∗ (η/2), I¯1 ) − P1− (−η/2, I¯1 )|, using (74) and (75), if c7 ≤ 1/(32π 2 ), we have P1− (−η/2
+
iE2∗ (η/2), I¯1 )
−
P1− (−η, I¯1 )
η 1 ≥ − c7 4π √ ln 1 + 4π 1 1 η √ min{1, ¯ }, ≥ 8π ε k
s
E1
!
which (exactly as in the positive energy case) yields (ii). From (74) we have P1− (−2εk(I¯1 ) + η, I¯1 ) − P1− (−2εk(I¯1 ) + η/2, I¯1 ) ≥
√ η 1 √ ln(1 + 1/ 2) ≥ √ √ , 4π µ 8 2π k¯
√ √ ¯ which yields (iii), provided c4 ≤ 1/(16 2π k).
• (Fourth step: construction of the symplectic transformation) We will find our symplectic transformation using the generating function S(E, I, q) := √ Rq q 2 0 g(E, I, θ) dθ. We note that in order to well define S we have to take into account the presence of the square root. In particular we are interested in the definition of the functions ∂E S(E, I, q) ∂I S(E, I, q) χ± (E, I, q) := , ξ ± (E, I, q) := . ± ∂E P (E, I) ∂I P ± (E, I) Let T + := C and T − := {q ∈ C s.t. |q1 | < π} and define21 ˜ ± (E, I) := {q ∈ T ± s.t. g(E, I, q) ∈ D / (−∞, 0]}. For any E ∈ E ± , I ∈ ∆0ρ2 , the functions S(E, I, q) ,
∂E S(E, I, q) ,
∂I S(E, I, q) ,
21
χ± (E, I, q) ,
ξ ± (E, I, q) ,
If a, b ∈ C we denote (a, b) := {z = a + t(b − a), with t ∈ (0, 1)} (and, analogously, for [a, b), (a, b], [a, b]); symbols like (a, α∞), with α ∈ C and |α| = 1, (or [a, α∞), (α∞, β∞), etc) denote lines: (a, α∞) := {z = a + αt, with t > 0}.
25
˜ ± (E, J). So the functions χ± (E(p, I, q), I, q) and ξ ± (E(p, I, q), I, q) are analytic in q on D are analytic in p, q on the disconnected set22 ˜ ± := D ˜ ± (I) := {(p, q) ∈ C × T ± s.t. g(E(p, I, q), I, q) ∈ D / (−∞, 0]} ± 2 = {(p, q) ∈ C × T s.t. p /2 ∈ / (−∞, 0]} = {(p, q) ∈ C × T ± s.t. p1 6= 0} . Our next step will be to define both χ¯± (p, I, q) := χ± (E(p, I, q), I, q) and ξ¯± (p, I, q) := ξ ± (E(p, I, q), I, q) for all (p, q) ∈ C × T ± . We set
χ ˜± (p, I, q) :=
and
ξ˜± (p, I, q) :=
√
1 ± 2 ∂E P (E(p, I, q), I)
π−√
√
Z
dθ
q
q
0
g(E(p, I, q), I, θ)
,
if p1 > 0,
Z q dθ 1 q , 2 ∂E P ± (E(p, I, q), I) 0 g(E(p, I, q), I, θ)
Z q 1 + cos θ q dθ , 2 ∂I P ± (E(p, I, q), I) 0 g(E(p, I, q), I, θ)
π−√
2 ∂I P ± (E(p, I, q), I)
Z
q 0
q
1 + cos θ g(E(p, I, q), I, θ)
if p1 < 0,
if p1 > 0,
dθ ,
if p1 < 0,
˜ ± . Notice that, in the positive energy which are well defined and analytic for (p, q) ∈ D case, there are no problems with the definition of χ¯+ and ξ¯+ , and we note that χ¯+ (p, I, q + 2π) = χ ¯+ (p, I, q) + 2π
and ξ¯+ (p, I, q + 2π) = ξ¯+ (p, I, q) + 2π .
(78)
In the negative energy case we proceed differently. We define FE (E, I, p) := −
Z
p 0
q
1
gˆ(E, I, z)
FI (E, I, p) := −
dz ,
Z
p 0
−E + z 2
q
gˆ(E, I, z)
dz ,
ˆ − (E, I) := where gˆ(E, I, p) := 1−(−1−E/+p2 /2)2 is analytic on the complex domain D {p ∈ C s.t. gˆ(E, I, p) ∈ / (−∞, 0]}. Then, FE (E(p, I, q), I, p) and FI (E(p, I, q), I, p) are well defined and analytic on23 ˆ − := D ˆ − (I) := {(p, q) ∈ C × T − s.t. gˆ(E(p, I, q), I, p) ∈ D / (−∞, 0]} − 2 = {(p, q) ∈ C × T s.t. 1 − cos q ∈ / (−∞, 0]} = {(p, q) ∈ C × T − s.t. q1 6= 0} . 22
23
˜ ± does not really depend on I. We see that D ˆ − does not really depend on I. Also in this case D
26
R
R
R
We now split the integral in the definition of S, ∂E S and ∂I S as 0q = 0ψ0 + ψq0 and in the second integral we perform the change of variable θ = arccos(−1 − E/ + z 2 /2). Then, defining24 −
χˆ (p, I, q) :=
and ξˆ− (p, I, q) :=
π/2 + ∂E P − (E(p, I, q), I)
−1
−π/2 − ∂E P − (E(p, I, q), I)
π/2 + ∂I P − (E(p, I, q), I)
˜− ∩ D ˆ− we have ∀ I ∈ ∆0ρ2 , ∀ (p, q) ∈ D χ˜− (p, I, q) ≡ χˆ− (p, I, q) mod 2π
−1
−1
−π/2 − ∂I P − (E(p, I, q), I)
FE (E(p, I, q), I, p) , FE (E(p, I, q), I, p) ,
FI (E(p, I, q), I, p) ,
−1
FI (E(p, I, q), I, p) ,
if q1 > 0, if q1 < 0,
if q1 > 0, if q1 < 0,
and ξ˜− (p, I, q) ≡ ξˆ− (p, I, q) mod 2π .
(79)
˜− ∪ D ˆ − → C/2πZ Using (79), we can finally define25 χ¯− , ξ¯− : (p, q) ∈ D χ¯− (p, I, q) := and ¯−
ξ (p, I, q) := ˜ −, where, on D
χ ˜− (p, I, q),
˜ −, if (p, q) ∈ D
χ ˆ− (p, I, q),
ˆ −, if (p, q) ∈ D
ξ˜− (p, I, q),
˜ −, if (p, q) ∈ D
ξˆ− (p, I, q),
ˆ− , if (p, q) ∈ D
χ− (E(p, I, q), I, q) ≡ χ¯− (p, I, q) and ξ − (E(p, I, q), I, q) ≡ ξ¯− (p, I, q) . Moreover we finally extend by periodicity the definition of χ¯− (p, I, q) and ξ¯− (p, I, q) on all {q ∈ C s.t. q1 6= π + 2kπ, k ∈ Z} = ∪k∈Z (2kπ + T − ) in the following way: if q ∈ 2kπ + T − we define χ¯− (p, I, q) := χ¯− (p, I, q − 2kπ) and ξ¯− (p, I, q) := ξ¯− (p, I, q − 2kπ). Now we are able to construct our symplectic transformation. Since ∂PE± 6= 0, by the Implicit Function Theorem, there exists E ± = E ± (P, J) such that P ± (E ± (P, J), J) ≡ P.
(80)
ˆ−. We note that χ ˆ− and ξˆ− are analytic on D − − ˜ ˆ ˜− ∪ D ˆ − )c = {(p, q) ∈ We observe that D ∪ D is an open set and that its complementary set (D − ˜ C × T , s.t. p1 = 0, q1 = 0} does not interest our analysis. In fact, if (p, q) ∈ (D− ∪ Dˆ − )c , then p = ip2 and q = iq2 and, hence, we have E(ip2 , I, iq2 ) = −p2 /2 − εk(I)(1 + cosh q2 ) and E1 (ip2 , I, iq2 ) = −p22 /2 − εk1 (I)(1 + cosh q2 ) ≤ −2εk1 (I) < −2εk(I¯1 ) + η/2 where |I − I¯1 | ≤ ρ2 (and we have used the fact that c5 ≤ 1/4k¯0 ). We conclude that E(ip2 , I, iq2 ) ∈ / E − (I¯1 ). 24
25
27
Let S ± (P, J, q) := S(E ± (P, J), J, q); we , then, define the following generating functions (which depend on the new actions and on the old angles): G± (P, J, q, ϕ) := Jϕ + S ± (P, J, q). Our symplectic transformation φ± is implicitly defined by
p = ∂q G± = ∂q S ± (P, J, q),
Q = ∂P G± = ∂P S ± (P, J, q),
I = ∂ϕ G± = J,
ψ = ∂J G± = ϕ + ∂J S ± (P, J, q),
We want to express (φ± )−1 as a function of the old variables (p, I, q, ϕ). We immediately have J = I and P = P ± (E(p, I, q), I). Differentiating (80) with respect to J and P we have, respectively, ∂E P ± ∂J E ± + ∂J P ± = 0 and ∂E P ± ∂P E ± = 1. Now, we can express the new angles as functions of the old variables:
Q = Q± (p, I, q) := χ¯± (p, I, q) , h
i
ψ = ψ ± (p, I, q, ϕ) := ϕ − ∂I P ± (E(p, I, q), I) Q± (p, I, q) − ξ¯± (p, I, q) .
We observe that Q− and ψ − are 2π-periodic in q by definition of χ¯− and ξ − ; by (78) we deduce that ψ + is 2π-periodic in q too and Q+ (p, I, q + 2π) = Q+ (p, I, q) + 2π. • (Fifth step: estimate on the angle analyticity radius) We first study the analyticity radius in Q. Fix I ∈ ∆0ρ2 and I¯1 ∈ ∆0 with |I − I¯1 | ≤ ρ2 . We ± ± ± ± must prove that ∀P∗ ∈ Dρ±1 and ∀Q∗ ∈ Tσ1 there exist p± ∗ and q∗ such that Q (p∗ , I, q∗ ) = Q∗ . So it is sufficient to prove that ∀E∗ ∈ E ± (I¯1 ) we have χ± (E∗ , I, Ts1 ) ⊇ Tσ1 . We first consider the positive energy case. We observe that we have χ+ (E, I, 0) = 0, χ+ (E, I, ±π)= ±π. Let us first consider the case I = I¯1 ∈ ∆0 , E = E1 ∈ E + . In such a case χ+ (E1 , I¯1 , (−π, π)) = (−π, π), and where
χ+ (E1 , I¯1 , (0, ±i∞)) = (0, ±is+ (E1 , I¯1 )) ,
χ+ (E1 , I¯1 , (±π, ±π ± iψ0 (E1 , I¯1 ))) = (±π, ±π ± is+ (E1 , I¯1 )), s± (E1 , I¯1 ) :=
√
2(∂E P ± )−1
Z
∞ 0
q
dθ . E1 + εk(I¯1 )(1 + cosh θ)
(81)
In fact, it is χ+ (E1 , I¯1 , D(E1 , I¯1 )) = Ts+ (E1 ,I¯1 ) . We will prove that ˜ + (E1 , I¯1 )) ⊇ Tσ+ (E ,I¯ ,s ) χ+ (E1 , I¯1 , Ts1 ∩ D 1 1 1
(82)
where, for s > 0, σ ± (E, I, s) :=
inf
t∈(−π,π)
χ± 2 (E, I, t + is) .
(83)
Observe that ∂E P + ∈ R and g = g1 − ig2 , where26 g1 = E1 + εk(I¯1 )(1 + cos q1 cosh q2 ) R R R R √ and g2 = εk(I¯1 ) sin q1 sinh q2 . Splitting the integral 0t+is = 0is + isis+t , we have 0is 1/ g = 26
For simmetry reasons we can consider t, q1 ≥ 0.
28
R R R √ √ i 0s 1/ g1 and, using the notation of Lemma 3.1, we obtain Im isis+t 1/ g = 0t y2 (g1 , g2 ), which, since y2 > 0, attains its minimum at t = 0. Collecting all these informations we have that
¯ σ + (E1 , I¯1 , s) = χ+ 2 (E1 , I1 , is) :=
√
2(∂E P + )−1
Z
s
q
0
dθ . E1 + εk(I¯1 )(1 + cosh θ)
It is easy to see that Z
s 0
v
u u dθ εk(I¯1 ) 1 . q ≥ c9 q ln 1 + st E1 + 2εk(I¯1 ) E1 + εk(I¯1 )(1 + cosh θ) εk(I¯1 )
Thus, by (69), we get σ + (E1 , I¯1 , s) ≥
v u u c10 ln 1 + st
v
u u εk(I¯1 ) εk(I¯1 ) t ln−1 1 + , ¯ E1 + 2εk(I1 ) E1
which implies that, ∀ η/2 ≤ E1 ≤ E and I¯1 ∈ ∆0 , σ + (E1 , I¯1 , s) ≥ c11
s . ln(ε/η)
In the general case, using the estimates on χ+ and its derivatives27 , one has that, if s . Taking E = E1 + iE2 ∈ E + and I¯ ∈ ∆0ρ2 , then σ + (E, I, s) ≥ c12 σ + (E1 , I¯1 , s) ≥ c13 ln(ε/η) s := s1 and c6 ≤ c13 , we have the claim concerning the form of σ1 . We now pass to the negative energy case. As before fix E and I and observe that − χ (E, I, 0) = 0 and χ− (E, I, ±ψ0 (E, I)) = ±π/2. Consider first the case I = I¯1 ∈ ∆0 and E = E1 ∈ E − (I¯1 ). We find χ− (E1 , I¯1 , (−ψ0 (E1 , I¯1 ), ψ0 (E1 , I¯1 ))) = (−π/2, π/2) and χ− (E1 , I¯1 , (0, ±i∞)) = (0, ±is− (E1 , I¯1 )), where s− (E1 , I¯1 ) was defined in (81). It is simple to see that we have n
o
n
˜− ∪ D ˆ − , E(p, I¯1 , q) = E1 , |q2 | < s1 ⊇ |Q2 | < σ − (E1 , I¯1 , s1 ) χ ¯− (p, I¯1 , q) s.t. (p, q) ∈ D
o
which is analogous to (82). The estimate on σ1 for the general case E ∈ E − (I¯1 ) and I ∈ ∆0ρ2 with I¯1 ∈ ∆0 , |I − I¯1 | ≤ ρ2 , follows exactly as in the positive energy case.
We now briefly discuss the analyticity radius in the angle ψ. Observing that, as it is simple ± ± ¯± to see, √ |χ¯ |, |ξ | ≤ c14 and remembering (70) and (76), we see that, |ψ (P, I, q, ϕ) − ϕ| ≤ c15 ε. Hence, if ε is sufficiently small, we can take σ2 = s2 /2,. The proof of Proposition 3.1 is now complete. 27
See Appendix B of [3].
29
4
Proof of Theorem 1.1 in the case (p, q) = (1, 1), (2, 1)
We recall that the Hamiltonian (28) has, in the case (p, q) = (1, 1) or (2, 1), the form H (1) = where
Iˆ12 (1) + ω Iˆ3 − εkp (Iˆ2 )(1 + cos jp ϕˆ1 ) + εhp (Iˆ2 ) + εa H1 + H∗(1) , 2
kp (Iˆ2 ) := −djp (Iˆ2 ),
hp (Iˆ2 ) := c0 (Iˆ2 ) + kp (Iˆ2 ),
j1 = 2,
j2 = 1.
(84)
(85)
In this subsection ξi will denote positive (ε–independent) constants and we will take ε as small as we need. Choose 1 < λ ≤ a − γ0 (here λ corresponds to 2b). From (30) we deduce that |Iˆ3 (t) − Iˆ3 (t0 )| ≤ ξ1 ελ ,
∀ 0 ≤ t0 ≤ t ≤ T1 (ε) := ξ3 exp(−ξ2 /ε` ).
(86)
In order to prove stability in the other actions we state the following elementary Lemma concerning the conservation of energy. Lemma 4.1 Let H := H(I, t; µ) := h(I) + µf (t) ∈ R, I, t, µ ∈ R and assume that h is analytic and not identically constant and that |f (t)| ≤ 1 for all t ∈ R. Fix r 0 > 0. Then, there exist 0 < µ0 , v ≤ 1 and c > 0 such that, if for some continous function I(t) := I(t; µ) with |I0 | := |I(0)| ≤ r0 H(I(t), t; µ) ≡ 0, then for all 0 ≤ µ ≤ µ0 we have |I(t) − I0 | ≤ cµv . Proof Being h analytic we have that, if N := {|I| ≤ 2r0 s.t. h0 (I) = 0}, then #N < ∞. Hence, there exists p∗ ∈ N such that ∀I0 ∈ N there exist 1 < p0 ≤ p∗ for which28 h(p0 ) (I0 ) 6= 0 and h(p) (I0 ) = 0 ∀ 1 ≤ p ≤ p0 . ˜ ≥ There exist b0 > 0 and 0 ≤ r00 ≤ r0 such that ∀ I0 ∈ N, |I0 | ≤ r0 we have that |h(p0 ) (I)| b0 , ∀ |I˜ − I0 | ≤ r00 . We claim that the Lemma holds with v ≤ 1/p∗ , c ≤ (2p0 !/b0 )1/p0 and µ0 ≤ (r00 /c)p∗ . In fact, by Taylor’s formula, ∀ |I − I0 | ≤ r00 , ∃ |I∗ − I0 | ≤ r00 such that h(I) − h(I0 ) = h(p0 ) (I∗ )(I − I0 )p0 /p0 ! and hence |I(t) − I0 |p0 ≤ 2p0 !µ/b0 . On the other hand, if |I0 | ≤ r0 with I0 ∈ {|I| ≤ 2r0 , s.t. |I − I1 | ≥ r00 /2, ∀ I1 ∈ N } =: M, then, defining m := minM |h0 | > 0, the Lemma holds with v ≤ 1, c ≤ 2/m, µ0 ≤ r00 m/4, since 2µ ≥ |h(I) − h(I0 )| ≥ m|I − I0 |. We consider first the case (p, q) = (2, 1); the case (p, q) = (1, 1), being analogous, will be considered later. For brevity we will omit the dependence on p = 2 in the formulas. In the Hamiltonian (84) we analyze first the following part, which represents a pendulum with a small gravity depending on a parameter: Iˆ2 E := E(Iˆ1 , Iˆ2 , ϕˆ1 ) := E(Iˆ1 , Iˆ2 , ϕˆ1 ; ε) := 1 − εk(Iˆ2 )(1 + cos ϕˆ1 ) , 2 where k(Iˆ2 ) = k2 (Iˆ2 ). We denote E(t) := E(Iˆ1 (t), Iˆ2 (t), ϕˆ1 (t)). 28
We denote with h(p) the p-th derivative of h with respect to I.
30
(87)
We claim that if 0 ≤ t0 ≤ t ≤ T1 (ε) then |E(t) − E(t0 )| ≤ 4ελ
=⇒
√ |Iˆ1 (t) − Iˆ1 (t0 )| ≤ ξ4 ε ,
|Iˆ2 (t) − Iˆ2 (t0 )| ≤ εξ5 .
(88)
√ In fact, if |E(t) − E(t0 )| ≤ 4ελ then the variable Iˆ1 may vary, at most, by order ε and using (86), we can apply the energy conservation to the Hamiltonian (84) obtaining that |h(Iˆ2 (t)) − h(Iˆ2 (t0 ))| ≤ ξ7 ελ−1 . Hence, by the fact that h(·) is a non constant analytic function (as it is immediate to verify), using Lemma 4.1 we get (88). Now we want to apply Proposition 3.1 to the pendulum (87). We set (p, I, q, ϕ) := (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ), (P, J, Q, ψ) = (Iˇ1 , Iˇ2 , ϕˇ1 , ϕˇ2 ), k := k2 , ∆0 := [I¯2 − 45 r, I¯2 + 45 r], r2 := r/4, η := ελ , R0 := 54 rε` , r1 := rε` /4, s1 := s2 := S/6. If ε is sufficiently small we can apply Proposition 3.1 transforming the Hamiltonian ˆ2 ˆ (1) := H ˆ (1) (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε) := I1 − εk(Iˆ2 )(1 + cos ϕˆ1 ) + εh(Iˆ2 ) + εa H1(1) H 2 into the new Hamiltonians ˇ (1)± := H ˆ (1) ◦ φ± (Iˇ1 , Iˇ2 , ϕˇ1 , ϕˇ2 ; ε) = E ± (Iˇ1 , Iˇ2 ) + εh(Iˇ2 ) + εa H1(1)± (Iˇ1 , Iˇ2 , ϕˇ1 , ϕˇ2 ) H 29 which belongs to HR (Ω± (ρ1 ,ρ2 ) × Tσ1 × Tσ2 ), where
ρ1 := ξ8 ελ−1/2 ,
ρ2 := ξ9 ελ−1 ln−1 (1/ε),
σ1 := ξ10 S ln−1 (1/ε),
σ2 := ξ11 S.
We now perform the analogous of the Step 3 in § 2.4. In order to apply the Normal Form Lemma, we take ε sufficiently small and we set30 d := 2, (I, ϕ) := (Iˇ1 , Iˇ2 , ϕˇ1 , ϕˇ2 ), a (1)± h := E ± (Iˇ1 , Iˇ2 ) + εh(Iˇ2 ), f := ε√ H1 , D := Ω± , ρ := (ρ1 , ρ2 ), ρ0 := ρ1 , σ := σ1 , Λ := {(0, k2 ) s.t. k2 ∈ Z}, α := ξ12 ε ln−1 ε−1 , K := ξ13 ε−γ0 . So we find two real–analytic symplectic transformations φˆ± 1 such that ˜ (2)± (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) := H ˇ (1)± ◦ φˆ± (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) := H 2 (2)± ± ˜ ˜ a ˜ 2(2)± (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε) E (I1 , I2 ) + εh(I˜2 ) + ε H1 (I˜1 , I˜2 , ϕ˜2 ; ε) + H belongs to HR (Ω± (ρ1 ,ρ2 )/2 × Tσ1 /6 × Tσ2 /6 ) and |I˜1 − Iˇ1 |, |I˜2 − Iˇ2 | ≤ ρ1 /27 ,
(2)±
˜2 kH
k ≤ ξ14 εa exp(−ξ15 S/εγ0 ) .
Now we complete our two symplectic transformations defining
29 30
˜ ˜ ˜ ˜1 , ϕ˜2 , ϕ˜3 ; ε) := (φˆ± ◦ φ± (I˜1 , I˜2 , ϕ˜1 , ϕ˜2 ; ε), I˜3 , ϕ˜3 ) φ± 2 (I 1 , I 2 , I 3 , ϕ 1
We have ξ8 = c4 , ξ9 = c5 /(λ − 1), ξ10 = c6 /6(λ − 1), ξ11 = 1/12. We can make such a choice of α, using (58) and (59).
31
(89)
so that ˜ ϕ; ˜ ˜ ε) := H (2)± (I, ˜ ε) := H (1) ◦ φ± 2 (I, ϕ; (2)± (2)± E ± (I˜1 , I˜2 ) + εh(I˜2 ) + ω I˜3 + εa H1 (I˜1 , I˜2 , ϕ˜2 ; ε) + H2 (I˜1 , I˜2 , ϕ; ˜ ε).
(90)
belongs to HR (Ω± (ρ1 ,ρ2 )/2 × C × Tσ1 /6 × Tσ2 /6 × TS/6 ) with (2)±
kH2
k ≤ ξ16 εa exp(−ξ17 S/εγ0 ) .
We now perform the analogous of Step 4 in § 2.5. Let
2 ˜ ± := Ω± −7 Ω (1+2 )(ρ1 ,ρ2 ) ∩ R .
˜± From the form of the Hamiltonian (90) we deduce that ∀ (I˜1 (t0 ), I˜2 (t0 )) ∈ Ω |I˜1 (t) − I˜1 (t0 )| ≤ ξ18 εξ19 ,
∀ 0 ≤ t0 ≤ t ≤ T2 (ε) := ξ20 exp(−ξ21 /εγ0 ) < T1 (ε) . (91)
Since I˜3 (t) − I˜3 (t0 ) = Iˆ3 (t) − Iˆ3 (t0 ), from (86) and (91) we deduce, using the energy conservation, that ± ˜ [E (I1 (t0 ), I˜2 (t)) + εh(I˜2 (t))] − [E ± (I˜1 (t0 ), I˜2 (t0 )) + εh(I˜2 (t0 ))]
≤ ξ22 εξ23 .
(92)
We now prove that G± (·) := E ± (I˜1 (t0 ), ·) + εh(·) are non constant analytic functions. From (23) and (85), it follows that h i dG± (y) = ε c02 y − k 0 (y)Y ± (I˜1 (t0 ), y) . dy
(93)
Now we observe that by (85), (24), (25), k(y) is effectively defined and analytic for all |y| < I¯1 and the same is true for Y ± . Thus, from the fact that limy→(−I¯1 )+ k 0 (y) = 0 (as it follows differentiating (24)) and that |Y ± | ≤ 1, by (93) we deduce that lim
y→(−I¯1 )+
dG± (y) = −εI¯1 c02 , dy
which is different from 0 by (23) and the non-degeneracy assumption (6). This proves that G± are non constant analytic functions. Finally, using (92), we can apply Lemma 4.1 and find ξ24 , ξ25 > 0 such that (94) |I˜2 (t) − I˜2 (0)| ≤ ξ25 εξ24 . We remark that, in principle, ξ24 , ξ25 found with Lemma 4.1, depend on I˜1 (t0 ) but, since ˜ ± , we can take them independent on I˜1 (t0 ). we work in compact sets Ω ˜ 0 ), ϕ(t ˜ ± × R × T3 . By (89) We have proved stability for 0 ≤ t0 ≤ t ≤ T2 (ε) and (I(t ˜ 0 )) ∈ Ω ˇ 0 ), ϕ(t this implies stability for (I(t ˇ 0 )) ∈ Ω± × R × T3 . By (57) this is equivalent to prove ˆ 0 ), ϕ(t stability for (I(t ˆ 0 )) ∈ M ± × R × T3 where M + := M − :=
n
n
ˆ ϕ) (I, ˆ s.t. ελ ≤ E(Iˆ1 , Iˆ2 , ϕˆ1 ) ≤ R02 /2, Iˆ2 ∈ ∆0
o
o
ˆ ϕ) (I, ˆ s.t. − 2εk(Iˆ2 ) + ελ ≤ E(Iˆ1 , Iˆ2 , ϕˆ1 ) ≤ −ελ , Iˆ2 ∈ ∆0 .
32
ˆ Using (88) and (86) it is immediate to prove stability for (I(0), ϕ(0)) ˆ ∈ M × R × T3 and 0 ≤ t ≤ T2 (ε), where ˆ ϕ) M := {(I, ˆ s.t.E(Iˆ1 , Iˆ2 , ϕˆ1 ) ≤ R02 /2, Iˆ2 ∈ ∆0 } .
Observing that M ⊃ A 5 R ∩ R3 × T3 , by (29) and the fact that φ0 is linear, we finally 4 obtain (7). This finishes the proof in the case (p, q) = (2, 1). It remains to consider the case (p, q) = (1, 1). The Hamiltonian (84) becomes H (1) =
Iˆ12 − εk(Iˆ2 )(1 + cos 2ϕˆ1 ) + F 2
(95)
(1) (1) ˆ ϕ; where F := F (I, ˆ ε) := ω Iˆ3 +εhp (Iˆ2 )+εa H1 (Iˆ1 , Iˆ2 , ϕˆ1 , ϕˆ2 ; ε)+H∗ (Iˆ1 , Iˆ2 , ϕ; ˆ ε). Next, we ? ? ? ˆ ˆ ˆ ˆ ˆ perform the following linear change of variables I1 := I1 /2, I2 := I2 , I3 := Iˆ3 , ϕˆ?1 := 2ϕˆ1 , ϕˆ?2 := ϕˆ2 , ϕˆ?3 := ϕˆ3 , casting the Hamiltonian (95) into the form
h i 1 H?(1) := H?(1) (Iˆ? , ϕˆ? ; ε) := 4 E(Iˆ1? , Iˆ2? , ϕˆ?1 ; ε) + F (2η1? , ϕˆ?1 /2) 4
(96)
with
k(Iˆ? ) (Iˆ? )2 (97) E(Iˆ1? , Iˆ2? , ϕˆ?1 ; ε) := 1 − ε 2 (1 + cos ϕˆ?1 ) . 2 4 For ease of notation, we have omitted in F the (Iˆ2? , Iˆ3? , ϕˆ?2 , ϕˆ?3 ; ε)–dependence, which, here, plays no rˆole. We now apply31 Proposition 3.1 to E defined in (97), finding two symplectic change of variables Iˆ1? := p± (P, Q), ϕˆ?1 := q ± (P, Q), putting the Hamiltonian (96) in the form (1) H?? := 4[E ± (P ) + F?± (P, Q)] , (98) where F∗± (P, Q) := 41 F (2p± (P, Q), q ± (P, Q)/2). We note that the functions p± are both 2π-periodic in Q. The function q − , and hence F − , is 2π-periodic in Q too; so we can define (Iˇ1 , ϕˇ1 ) := (P, Q) and proceed exactly as for (p, q) = (2, 1), applying the Normal Form Lemma and the subsequent arguments. The positive energy case is different: in fact q + (P, Q + 2π) = q + (P, Q) + 2π so that F∗+ is only 4π-periodic in Q, but in order to apply the Normal Form Lemma we need a 2π-periodic function. We, therefore, define another linear change of variables P0 := 2P, Q0 := Q/2, so that (98) becomes (1) + H??? := 4[E + (P0 /2) + F?? (P0 , Q0 )] .
(99)
+ where F?? (P0 , Q0 ) := F?+ (P0 /2, 2Q0 ) which is 2π-periodic in Q0 . Therefore, we may define (Iˇ1 , ϕˇ1 ) := (P0 , Q0 ) and proceed again as in the case (p, q) = (2, 1). The proof of Theorem 1.1 is, now, complete .
31
Again we will omit the dependence on the variables (Iˆ2? , Iˆ3? , ϕˆ?2 , ϕˆ?3 ; ε).
33
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