7 PERSISTENCE TORI - unimi.it · We shall say that the Hamiltonian (7.2) is in Kolmogorov’s normal form. The suggestion of Kolmogorov is to reduce a Hamiltonian to the normal form

7

PERSISTENCE OF

INVARIANT TORI

This chapter is devoted to the celebrated theorem of Kolmogorov on persistence ofinvariant tori in near to integrable Hamiltonian systems. The relevance of the theoremfor Classical and Celestial Mechanics was emphasized by V.I. Arnold, who wrote in [2]:

“ One of the most remarkable of A.N. Kolmogorov’s mathematical achieve-ments is his work on classical mechanics of 1954. A simple and novel idea,the combination of very classical and essentially modern methods, the so-lution of a 200 year–old problem, a clear geometric picture and a breadthof outlook — these are the merits of the work. ”

Kolmogorov announced his theorem at the International Congress of Mathemati-cians held in Amsterdam in 1954. Then he published a sketch of the proof in theshort note [58]. The note contains in a synthetic form the formal scheme of an iter-ative procedure and a few essential hints on the proof of convergence. Kolmogorovalso professed a series of lectures in Moskow where he gave the complete proof, aswitnessed by some Russian mathematicians (see [19], ch. 11), but it seems that thetext of the lectures has never been published. A proof of a related theorem for mapshas been published by Moser [80]. Two papers with proofs for Hamiltonian systemsthat generalize and extend the ideas of Kolmogorov have been published by Arnoldin [2] and [3]. Proofs based on the original paper of Kolmogorov may be found, e.g.,in [9], [10], [11], [51]. The work of Kolmogorov, Arnold and Moser marked the begin-ning of a wide research field nowadays known as KAM theory. A lot of papers havebeen published on the subject: an exhaustive list would fill several pages.

7.1 The normal form of Kolmogorov

In view of the great relevance of the work of Kolmogorov and of its impact on theprogress of our knowlenge about non integrable systems this section is devoted to ashort exposition of the main ideas. The exposition closely follows the scheme that

178 Chapter 7

can be found in the short note of Kolmogorov [58]. It goes without saying that theexposition in this section reflects the personal experience of the author.

The general framework is still the general problem of dynamics as stated byPoincare (see chapter 4). With a negligible change of notation I shall write the Hamil-tonian as

(7.1) H(p, q) = h(p) + εf(p, q, ε) ,

were p ∈ G ⊂ Rn are action variables in an open set G and q ∈ Tn are anglevariables. The Hamiltonian is assumed to be holomorphic in all variables and in thesmall parameter ε.

We know that the unperturbed system, namely for ε = 0, is characterized byquasi–periodic motions on invariant tori parameterized by the actions p (see sect. 4.1).The classical question is whether such a behaviour persists for the perturbed system,at least for ε small enough. That this can not be true for all tori is stated by Poincare’stheorem on non integrability (see chapter 4). The theorem of Kolmogorov claims thatinvariant tori characterized by strongly non resonant frequencies (e.g., diophantinefrequencies in the sense of sect. 4.2.3) are slightly deformed, but not destroyed by theperturbation; they continue to exist for ε small enough and still carry quasi–periodicorbits.

The first idea is that there is a very simple Hamiltonian which brings immediatelyto evidence the existence of an invariant torus. Precisely, let the Hamiltonian be

(7.2) H(q, p) = 〈ω, p〉+R(q, p)

where ω ∈ Rn and R(q, p) is at least quadratic in the actions p, i.e., R(q, p) = O(p2).Hamilton’s equations read

qj = ωj +∂R

∂pj, pj = −

∂R

∂qj, j = 1, . . . , n .

Let now the initial point be p = 0 with arbitrary q. Since ∂R∂pj

= O(p) and ∂R∂qj

= O(p2)

it is immediate to conclude that the torus p = 0 is invariant for the flow, and carriesa Kronecker flow with frequencies ω. We shall say that the Hamiltonian (7.2) is inKolmogorov’s normal form.

The suggestion of Kolmogorov is to reduce a Hamiltonian to the normal formabove in a neighbourhood of a strongly non resonant unperturbed torus.

7.2 The formal scheme

Referring to the Hamiltonian (7.1), assume h(p) be nondegenerate. Pick a point p∗ ∈ Gsuch that the corresponding unperturbed frequency ω = ∂h

∂p (p∗) satisfies the diophan-

tine condition

(7.3) |〈k, ω〉| ≥ γ|k|τ for all k ∈ Zn , k 6= 0 ,

Persistence of invariant tori 179

with some γ > 0 and τ ≥ n − 1. By expanding the Hamiltonian in Taylor’s series ofthe actions around the point p∗ and forgetting unessential constants we get

H(q, p) = 〈ω, p−p∗〉+εA(q)+ε〈B(q), p−p∗〉+1

2〈C(q)(p− p∗), p− p∗〉+g(q, p−p∗) ,

where g(q, p− p∗) = O(

(p − p∗)3)

. The functions A(q), Bj(q), Cj,k(q) are calculatedas

A(q) = f(q, p∗, ε) ,

Bj(q) =∂f

∂pj(q, p∗, ε) ,

Cj,k(q) =∂2h

∂pj∂pk(p∗) + ε

∂2f

∂pj∂pk(q, p∗, ε) .

We may perform a translation setting the point p∗ as the origin of the actions, whichis tantamount to putting p∗ = 0 in the formulæ above, so that the Hamiltonian takesthe form

(7.4) H(q, p) = 〈ω, p〉+ εA(q) + ε〈B(q), p〉+1

2〈C(q)p, p〉+ g(q, p) ,

where the parameter ε has been added for convenience in the next calculation. IfA(q) = B(q) = 0, then the Hamiltonian is in Kolmogorov’s normal form.

Kolmogorov’s suggestion is to kill the unwanted terms A(q) and 〈B(q), p〉) in (7.4)by applying a canonical transformation of the special form

(7.5)

q′j = qj + εYj(q)

pj = p′j − ε

[ n∑

l=1

p′l∂Yj(q)

∂ql+ ξl +

∂X(q)

∂ql

]

where ξ ∈ Rn and X(q) and Yj(q) are still periodic in the angles q. The effect ofX(q)+ 〈Y (q), p〉 is a deformation of the torus, while 〈ξ, q〉 induces a small translation.It is matter of a moment to check that such a canonical transformation is producedby a generating function (in mixed variables)

S(p′, q) = 〈p′, q〉+ ε[

X(q) + 〈ξ, q〉+ 〈Y (q), p′〉]

.

Exploiting the formalism of Lie series we may avoid inversions and substitutions byusing the generating function

χ(q, p) = X(q) + 〈ξ, q〉+ 〈Y (q), p〉 ,

where the function X(q), the vector function Y (q) and the real vector ξ are determinedso that the transformation kills the unwanted terms A(q) + 〈B(q), p〉.

Let calculate the transformed Hamiltonian H ′ = exp(

εLχ)

H, keeping only termsof order ε. Here the Poisson bracket with the linear term 〈ω, p〉 plays a special role,so let us introduce the notation ∂ω = L〈ω,p〉. We get (recall that the names of the

180 Chapter 7

variables are irrelevant)

H ′ =〈ω, p〉+⟨

C(q)p, p⟩

+ g(p, q) + εLχg(p, q)

+ ε[

A(q)− ∂ωX + 〈ω, ξ〉]

+ ε

[

⟨

B(q), p⟩

+

⟨

C(q)p, ξ +∂X

∂q

⟩

− ∂ω〈Y (q), p〉

]

+O(ε2)

Here the first line contains the part already in normal form (at least quadratic in p,including Lχg). Terms of order O(ε2) are left unhandled, and must be removed later.The second and third line contain the parts that should be cleared. To this end wewrite the equations

(7.6)

A(q)− ∂ωX = 0 ,

Cξ +B +∂X

∂q= 0 ,

B(q) + C(q)

(

ξ +∂X

∂q

)

− ∂ωY = 0 .

Here the overline denotes the average with respect to the angles q, i.e., the termindependent of q in the Fourier expansion of a function. The first and third equation(usually called homological equations) can be solved provided the average of the knownterm is zero (see next section). In the first equation the average is a constant that canbe neglected. The second equation aims at determining the real vector ξ precisely inorder to clear the average of the known term in the third equation. It may be solvedprovided the constant matrix C is not degenerate, which is initially assured by nondegeneracy of h(p). The translation vector ξ keeps the frequency fixed.

Having determined the generating function we may perform the transformationand then rearrange the Hamiltonian in a form similar to (7.4), namely

H ′ = 〈ω, p〉+⟨

C′(q)p, p

⟩

+ ε2A′(q) + ε2⟨

B′(q), p⟩

+ g′(p, q) , g′(p, q) = O(|p|3) ,

with a new symmetric matrix C′(q) which is a small correction of the previous one.

The factor ε2 in front of A(q) and B(p, q) reminds us that these terms have beenproduced by the transformation, but have been left unhandled. Explicit formulæ willbe given later.

Thus, the consistency of the procedure depends on the existence of the solutionof the equations (7.6). If so, then the procedure may be iterated in order to hopefullyreduce the size of the unwanted terms to zero, thus giving the Hamiltonian the normalform of Kolmogorov.

7.2.1 Small divisors and the problem of convergence

The problem of solving the homological equation has been already discussed insect. 4.3.1, in connection with nonexistence of first integrals. However, here thereis a nice difference: the frequencies are constant and non resonant. Let us state theproblem as follows. Given a known function ψ(p, q) with zero average, namely ψ = 0,


find χ such that ∂ωχ = ψ. The actions p here are just parameters. The procedure isquite standard. Expand in Fourier series

ψ(p, q) =∑

0 6=k∈Zn

ψk(p) exp(

i〈k, q〉)

, χ(p, q) =∑

k∈Zn

ck(p) exp(

i〈k, q〉)

,

with coefficients ψk(p) known and ck(p) to be found. Calculate

∂ωχ = i∑

k

〈k, ω〉ck(p) exp(

i〈k, q〉)

.

Therefore, assuming that the frequencies ω are non resonant, we get the formal solutionwith coefficients

ck(p) = −iψk(p)

〈k, ω〉.

The solution can be proved to be holomorphic on the basis of the following con-siderations, already made by Poincare. If ψ(p, q) is holomorphic, then the coeffi-cients ψk(p) decay exponentially, i.e.,

∣

∣ψk(p)∣

∣ ∼ e−|k|σ for some σ. Therefore we get∣

∣ck(p)∣

∣ ∼ |k|τe−|k|σ ∼ e−|k|σ′

with some σ′ < σ. This shows that χ(p, q) is still holo-morphic, thus making every single step of Kolmogorov to be formally consistent.

The problem now is that iterating the procedure we produce an accumulation ofsmall divisors: at every step the coefficients gain a new small divisor, which makesconvergence doubtful. Here comes the second, great idea of Kolmogorov: apply theso called generalized Newton method (in his own terms), based on the work of Kan-torovich [57]. Do not use expansions in a parameter. At every step collect all contri-butions independent of and linear in p in a single pair of functions A(q) and B(p, q).In very rough heuristic terms this is what happens. Starting with functions of size εand forgetting for a moment the contribution of small divisors the procedure reducesstep by step the size of the unwanted terms to ε2, ε4, ε8, . . . that decrease quadrati-cally, as it happens in Newton’s method (as remarked by Kolmogorov himself). Sucha strong decrease compensates the dramatically growing factors due to small divisors,eventually assuring convergence of the procedure. The latter heuristic argument wascommonly used in the past, and it has been often synthetized in the words “quadraticmethod”, “quadratic convergence”, “superconvergence” and so on.

The heuristic considerations above on the fast convergence are too optimistic,because the contribution of the small divisors is ignored. As a matter of fact, in theproofs available in literature it is shown that after r iteration steps the size of the

functions A(q) and B(q) decreases as εr3/2

or some similar power less than two, orgeometrically as cr with some c < 1, or even as an inverse power r−k with some k > 1.Anyway, the procedure can be proven to be actually convergent, and this is indeedthe wonderful result of Kolmogorov.

The matter concerning fast convergence methods deserves a short extra discussionwhich is relevant here because the proof given in this chapter uses classical expansions.Let us quote a sentence from a paper of Helmut Russman [91]: “It has often been saidthat the rapid convergence of the Newton iteration is necessary for compensating theinfluence of small divisors. But a deeper analysis shows that this is not true. (. . .)

182 Chapter 7

Historically, the Newton method was surely necessary to establish the main theoremsof the KAM–theory. But for clarifying the structure of the small divisor problemsthe Newton method is not useful because it compensates not only the influence ofsmall divisors, but also many bad estimates veiling the structure of the problems.”Up to the author’s knowledge the first proof of existence of invariant tori that doesnot make use of the fast convergence assured by the quadratic method has beenpublished by Russman[92]. Other proofs have been published by Ugo Locatelli and theauthor [36] [37] [34].

The main reason for avoiding Newton’s method is the better understanding ofthe mechanism of accumulation of small divisors. A second relevant reason is thatwe may produce a constructive algorithm, that can be implemented with algebraicmanipulation.

7.2.2 Statement and proof of the theorem

In order to reduce the technical troubles to a minimum, let us restrict our attentionto a rather simple model. Consider a system of coupled rotators as described by theHamiltonian H(p, q) = H0(p) + εH1(p, q), where

(7.7) H0(p) =1

2

n∑

j=1

p2j , H1(p, q) =∑

|k|≤K

ck(p)ei〈k,q〉 , p ∈ R

n , q ∈ Tn ,

with a fixed K > 0 and coefficients ck(p) that are polynomials of degree at most 2. Thechoice is made in order to reduce technicalities to a minimum (though there remainenough), but all the crucial difficulties of the problem are accounted for. The extensionto the general case is matter of not being scared by long and boring calculations. Somehints on how to deal with the general case are provided later, in sect. 7.4.

The following statement is adapted to the particular case we are considering.

Theorem 7.1: On the phase space Tn × Rn consider the Hamiltonian (7.7) whereH1(q, p, ε) is a polynomial of degree at most 2 in the action variables p and is atrigonometric polynomial. For ε = 0, let p∗ be an unperturbed torus with frequenciesω = p∗ satisfying a diophantine condition (7.3). Then there exists a positive ε∗ suchthat for every |ε| < ε∗ the Hamiltonian (7.7) possesses an invariant torus εa–close top∗ for some positive a < 1. The flow on the torus is quasi periodic with frequencies ω.

7.2.3 The formal constructive algorithm

According to the procedure outlined in sect. 7.1, and considering the Hamilto-nian (7.7), we select an unperturbed torus p∗ such that the corresponding unperturbedfrequency ω = p∗ satisfies the diophantine condition (7.3). By translating the originof the action variables to p∗, we write the Hamiltonian as

(7.8) H(q, p) = 〈ω, p〉+1

2〈p, p〉+ ε

[

A1(q) +B1(p, q) + C1(p, q)]

This is rather obvious, because H1 is supposed to be a polynomial of degree 2.


The aim is to construct an infinite sequence H(0)(p, q), H(1)(p, q), H(2)(p, q), . . .of Hamiltonians, with H(0) coinciding with H in (7.8), which after r steps of normal-ization turn out to be written in the general form

(7.9) H(r) = ω · p+

r∑

s=0

εshs(p, q) +∑

s>r

εs[

A(r)s (q) +B(r)

s (p, q) + C(r)s (p, q)

]

,

where h0(p) = 12 〈p, p〉. The Hamiltonian H(r)(p, q) is in Kolmogorov’s normal form

up to order r, in formal sense. The following properties will be kept along the wholeprocedure:

(i) h1(p, q), . . . , hr(p, q) are quadratic in p, so that they are in normal form, anddo not change after step r;

(ii) A(r)s (q) is independent of p;

(iii) B(r)s (q) is linear in p.

(iv) C(r)s (q) is a quadratic polynomial in p;

(iv) A(r)s (q), B

(r)s (q) and C

(r)s (q) are trigonometric polynomials of degree sK, where

K is the degree of H1 in the original Hamiltonian.

The properties appear to be quite strange, are precisely the characteristics that simpli-fies the proof, allowing us to concentrate on the crucial problems — e.g., the impact ofsmall divisors on convergence. The original Hamiltonian (7.8) obviously satifies theseproperties.

The normalization process is worked out with a minor recasting of the method ofKolmogorov. Assuming that r−1 steps have been performed, so that the HamiltonianH(r−1)(p, q) has the wanted form (7.9) with r−1 in place of r, we construct in sequencethe new Hamiltonians

H(r) = exp(

εrLχr,1

)

H(r−1) , H(r) = exp(

εrLχr,2

)

H(r) ,

the first one being an intermediate Hamiltonian, and the second one being in normalform up to order r. At every step r we apply a first canonical transformation withgenerating function χr,1(q) = Xr(q)+〈ξr, q〉, followed by a second transformation withgenerating function χr,2(p, q) = 〈Y (r)(q), p〉. The explicit constructive algorithm for asingle step is summarized in table 7.2. A reader who has devolped some expertise inperturbation methods will probably be able to check the table by himself. However,some help on how to construct the table may be welcome; hence, a detailed (very closeto pedantic) explanation is included here.

We calculate the Hamiltonian H(r) by applying exp(Lχr,1) to every term ofH(r−1).

It may be useful to represent the Lie triangle, splitting it in two parts. The part that

184 Chapter 7

involves terms of order ε0 and εr is the following:

ε0 : 〈ω, p〉 h0

εr : Lχr,1〈ω, p〉 Lχr,1

h0 A(r−1)r B(r−1)

r C(r−1)r

ε2r : 0 12L2χr,1

h0 0 Lχr,1B(r−1)r Lχr,1

C(r−1)r

ε3r : 0 0 0 0 12L

2χr,1

C(r−1)r

Since the generating function is of order εr the rows of the triangle proceed by powersof εr. The second line contains all terms that are involved in the process of removingthe unwanted functions.

Extracting from the second line the two contributions independent of p and clear-ing them we get

12Lχr,1

〈ω, p〉+ A(r−1)r = −∂ωXr +A(r−1)

r − ∂ω〈ξ, q〉 = 0 .

Ignoring the last term ∂ω〈ξ, q〉, which is a constant that can be ignored, we get thefirst homological equation in table 7.2 that determines the function Xr(q). Forgetting

also the average of A(r−1)r (q), which is a constant, the equation is solved as explained

in sect. 7.2.1.

Extracting from the second line the contributions linear in p we get

12Lχr,1

〈p, p〉+B(r−1)r = −

⟨∂Xr

∂q, p⟩

− 〈ξr, p〉+B(r−1)r .

Here we remark that the average of B(r−1)r (q) might be non zero, so that it would

introduce an unwanted correction of the frequencies ω in H(r). We avoid it by de-termining ξr from the second equation in table 7.2, which is a trivial one in view of

h0 = 12〈p, p〉. Having removed the average, what is left is the function B

(r)r in the

second line of table 7.2.

The remaining term C(r−1)r is quadratic in p, and is left unchanged. Similarly the

term 12L

2χr,1

h0, of order ε2r and linear in p, is included in B

(r)r ; the terms Lχr,1

B(r−1)r

of order ε2r, and 12L2χr,1

C(r−1)r , of order ε3r, are independent of p, and are included in

A(r)2r and A

(r)3r , respectively. The triangle does not include anything else, due to χr,1

being independent of p. No term of degree higher than 2 in p is generated.

For h1, . . . , hr−1, which are all quadratic functions, we have

εs expLεrχr,1hs = εshs + εr+sLχr,1

hs +ε2r+s

2L2χr,1

hs ,

all the rest of the series being zero. The new term Lχr,1hs is linear in p, and is included

in B(r)r+s in table 7.2. Similarly, ε2r+s

2 L2χr,1

hs is independent of p, and is included in


A(r)r+s . For the functions A

(r−1)s , B

(r−1)s and C

(r−1)s of higher order s > r we get a

small triangle, namely

εs : A(r−1)s B(r−1)

s C(r−1)s

εr+s : 0 Lχr,1B(r−1)s Lχr,1

C(r−1)s

ε2r+s : 0 0 12L

2χr,1

C(r−1)s

The triangle contains all terms that are generated, which are at most quadratic in p,

as wanted. Here, Lχr,1B

(r−1)s and 1

2L2χr,1

C(r−1)s are independent of p, and are included

in A(r)r+s and A

(r)2r+s , respectively. Finally, Lχr,1

C(r−1)s is linear in p and is included in

B(r)r+s . All unchanged functions are also included, as appropriate. The functions C

(r−1)s

for s > r remain unchanged in this step.Having performed the transformation with χr,1 step we have an intermediate

Hamiltonian

H(r) = ω ·p+

r−1∑

s=0

εshs(p, q)+B(r)r +C(r−1)

r +∑

s>r

εs[

A(r)s (q) + B(r)

s (p, q) + C(r)s (p, q)

]

,

where all functions are defined in table 7.2. Remark that there is no term A(r)r , which

has been cleared.We come now to calculating the HamiltonianH(r) = expLχr,2

H(r). We split againthe triangle, first considering terms of order ε0 and εr. We get

ε0 : 〈ω, p〉 h0

εr : Lχr,2〈ω, p〉 Lχr,2

h0 B(r)r C(r−1)

r

ε2r : 12L2χr,2

〈ω, p〉 12L2χr,2

h0 Lχr,2B(r)r Lχr,2

C(r−1)r

ε3r : 13!L

3χr,2

〈ω, p〉 13!L

3χr,2

h012L

2χr,2

B(r)r

12L

2χr,2

C(r−1)r

ε4r : 14!L

4χr,2

〈ω, p〉 14!L

4χr,2

h013!L

3χr,2

B(r)r

13!L

3χr,2

C(r−1)r

......

......

The triangle here is infinite, because χr,2 is linear in p and so it does not change thedegree of a function.

Extracting the linear terms in p from the second line and forcing it to zero we get

Lχr,2〈ω, p〉+ B(r)

r = −∂ωχr,2 + B(r)r = 0 ,

which is the third equation for the generating functions in table 7.2. In order to showthat it may be solved we need a few more considerations. Recall that χr,2 = 〈Yr(q), p〉.

186 Chapter 7

On the other hand, since B(r)r is linear in p, we may write it as, e.g., B

(r)r = 〈Ψ(q), p〉.

Then the equation to be solved splits into the n equations ∂ωYr,j(q) = Ψj(q), whichcan all be solved because the right members have null average (it has been removedby appropriately finding ξr). Having determined χr,2 we can construct all the restof the triangle, and move every element to the appropriate function in 7.2. Only the

functions B(r)kr deserve some more explanation. For k > 1 we observe that in view of

the homological equation we have

1k!Lkχr,2

〈ω, p〉 = − 1k!Lk−1χr,2

∂ωχr,2 = − 1k!Lk−1χr,2

B(r)r .

Thus we calculate

1k!L

kχr,2

〈ω, p〉+ 1(k−1)!L

k−1χr,2

B(r)r =

(

1(k−1)! −

1k!

)

Lk−1χr,2

B(r)r = k−1

k! Lk−1χr,2

B(r)r .

The result is included in B(r)kr in 7.2.

For the functions h1, . . . , hr we have

εm exp(

εrLχ2hm

)

=∑

k≥0

εkr+m

k!Lkχr,2

hm , m = 0, . . . , r − 1 .

For k = 0 we get hm, which is left unchanged. For k > 0 we include every term in the

sum in the corresponding function C(r)kr+m . Finally, for s > r the functions A

(r)s , B

(r)s

and C(r−1)s are transformed as

εs exp εrLχ2A(r)s = εs

∑

k≥0

εr

k!Lkχr,2

A(r)s ,

εs exp εrLχ2B(r)s = εs

∑

k≥0

εr

k!Lkχr,2

B(r)s ,

εs exp εrLχ2C(r−1)s = εs

∑

k≥0

εr

k!Lkχr,2

C(r−1)s .

Every term in the sums should be moved to the appropriate function in table 7.2.It remains to show that also the property (v) is satisfied, namely that every func-

tion of order εs is a trigonometric polynomial of degree sK. The generating functionsXr and χr,2 have degree rK because they are solutions of homological equations withknown members of degree rK. On the other hand, if a function fl has degree lK thenLχr,j

fl clearly has degree (r+ l)K. Just apply this rule to every Lie derivative in thealgorithm. Thus, the justification of the formal algorithm in table 7.2 is complete.

7.3 Quantitative estimates

We come now to the crucial problem that has challenged mathematicians for a coupleof centuries: the accumulation of small divisors, which is indeed the most challengingpart of the proof that the sequence of formal transformations to normal form actuallyconverges to a Hamiltonian possessing the normal form of Kolmogorov. In this section


we construct the scheme of analytic estimates, paying particular attention to smalldivisors.

In view of the form of the Hamiltonian, which is a polynomial in the actions p anda trigonometric polynomial in the angles, we shall consider all functions as analyticin a domain D(,σ) = ∆(0) × Tnσ , where the choice of and σ is rather arbitrary.Thus we pick some values for them, and keep them constant in the whole proof. Ourchoice is intended to make easier to adapt the proof to more generale situation, as willsketched later in sect. 7.4.

In view of our purposes it is convenient to use the weighted Fourier norm, forwhich the estimates of Poisson brackets are provided in sect. 6.7.2, forgetting theobvious fact that the estimate the derivatives of a polynomial of degree two does notrequire the sophisticated tool of Cauchy estimates. An exception is represented by thefunction 〈ξ, q〉 with ξ ∈ Rn, which is part of the generating functions χ1, but is nottrigonometric. However, we shall see that this causes a very little trouble. We simplifya little the notation by writing the norm as, e.g., ‖ · ‖1−d in place of ‖ · ‖(1−d)(,σ) . Inparticular we write ‖ · ‖1 in place of ‖ · ‖,σ .

The strategy of the proof follows the following scheme:(i) Translate the formal algoritm into a sequence of estimates for the norms of all

functions involved.

(ii) Isolate the problem of accumulation of small divisors, and show that they obeysome strict, perhaps surprising rules that make their contribution to grow notfaster than geometrically with the order (no factorials).

(iii) Prove that the norms of the generating functions satisfy the convergence con-dition of proposition 6.11.

Thus the proof of the theorem of Kolmogorov will be complete.

7.3.1 Estimates for the generating functions

The formal algorithm is based on the operations of solving the homological equationsand of calculating Lie derivatives. In order to use the generalized Cauchy estimates ofsect. 6.7 we should make a choice for the restrictions of domains. We need an infinitesequence of restrictions d1 < d2 < d3 < . . . tending to a limit d < 1. We shall actuallyimpose the stronger condition d ≤ 1/6. To this end let us introduce the arbitrarysequence for r > 0

(7.10) δr =1

2π2·1

r2,

∑

r>0

δr =1

12.

Then set

(7.11) d0 = 0 , dr = 2(δ1 + . . .+ δr) , r > 0 ,

which satisfies our request. Our aim is to find recurrent estimates for H(r) in thedomain D(1−dr−1−δr)(,σ) and for H(r) in the domain D(1−dr)(,σ) .

Let us begin with an estimate of the solution of a homological equation. Havinggiven a non resonant frequency vector ω ∈ Rn we introduce the real, non increasing

188 Chapter 7

Table 7.2. The formal constructive algorithm for Kolmogorov’s normal form.

• Equations for the generating functions χr,1 = Xr + 〈ξr, q〉 and χr,2 = 〈Yr(q), p〉:

∂ωXr = A(r−1)r ,

〈ξr, p〉 = B(r−1)r ,

∂ωχr,2 = B(r)r ,

B(r)r = B(r−1)

r −B(r−1)r −

⟨

∂Xr

∂q, p

⟩

.

• Intermediate Hamiltonian H(r) = exp(

Lχr,1

)

H(r−1):

A(r)r = 0

A(r)s =

A(r−1)s , r < s < 2r ;

1

2L2χr,1

hs−2r + Lχr,1B

(r−1)s−r + A(r−1)

s , 2r ≤ s < 3r ;

1

2L2χr,1

C(r−1)s−2r + Lχr,1

B(r−1)s−r +A(r−1)

s , s ≥ 3r .

B(r)s =

Lχr,1hs−r +B(r−1)

s , r < s < 2r ;

Lχr,1C

(r−1)s−r +B(r−1)

s , s ≥ 2r .

• Transformed Hamiltonian H(r) = exp(

Lχr,2H(r)

)

(set k = ⌊s/r⌋, m = s (mod r),s = kr +m):

hr = Lχr,2h0 + C(r−1)

r .

A(r)s =

k−1∑

j=0

1

j!Ljχr,2

A(r)s−jr , s > r .

B(r)s =

k − 1

k!Lk−1χr,2

B(r)r +

k−2∑

j=0

1

j!Ljχr,2

B(r)s−jr , k ≥ 2 , m = 0 ;

k−1∑

j=0

1

j!Ljχr,2

B(r)s−jr , k ≥ 1 , m 6= 0 .

C(r)s =

1

k!Lkχr,2

hm +

k−1∑

j=0

1

j!Ljχr,2

C(r)s−jr , s > r .


Table 7.3. Quantitative estimates for the normalization scheme.

• Generating functions χr,1 = Xr + 〈ξr, q〉 and χr,2 = 〈Yr(q), p〉:

‖Xr‖1−dr−1≤

1

αr‖A(r−1)

r ‖1−dr−1,

∣

∣ξr,j∣

∣ ≤2

∥

∥

∥B

(r−1)r

∥

∥

∥

1−dr−1

‖χr,2‖1−dr−1−δr≤

1

αr‖B(r)

r ‖1−dr−1−δr

• Intermediate Hamiltonian H(r) = exp(

Lχr,1

)

H(r−1). Set

Gr,1 =e

σ

(

‖A(r−1)r ‖1−dr−1

+2αrδrσ

∥

∥

∥B

(r−1)r

∥

∥

∥

1−dr−1

)

.

For r < s < 2r , 2r ≤ s < 3r and s ≥ 3r , respectively, get

‖A(r)s ‖1−dr−1−δr

≤

‖A(r−1)s ‖1−dr−1

;

(

Gr,1δ2rαr

)2

‖hs−2r‖1−ds−2r+Gr,1δ2rαr

‖B(r−1)s−r ‖

1−dr−1+ ‖A(r−1)

s ‖1−dr−1;

(

Gr,1δ2rαr

)2

‖C(r−1)s−2r ‖

1−ds−2r+Gr,1δ2rαr

‖B(r−1)s−r ‖

1−dr−1+ ‖A(r−1)

s ‖1−dr−1;

For r < s < 2r and s ≥ 2r , respectively, get

‖B(r)s ‖1−dr−1−δr

≤

Gr,1δ2rαr

‖hs−r‖1−ds−r+ ‖B(r−1)

s ‖1−dr−1;

Gr,1δ2rαr

‖C(r−1)s−r ‖

1−ds−r+ ‖B(r−1)

s ‖1−dr−1.

• New Hamiltonian H(r) = exp(

Lχr,2H(r)

)

. Set Gr,2 = eσ‖B

(r)r ‖(1−dr−1−δr)

. Fors ≥ r get

‖hr‖1−dr ≤Gr,2δ2rαr

‖h0‖1 + ‖C(r)r ‖1−dr−1

.

‖A(r)s ‖1−dr≤

k−1∑

j=0

(

Gr,2δ2rαr

)j

‖A(r)s−jr‖1−dr−1−δr

;

‖B(r)s ‖1−dr≤

k−1∑

j=0

(

Gr,2δ2rαr

)j

‖B(r)s−jr‖1−dr−1−δr

;

‖C(r)s ‖1−dr≤

(

Gr,2δ2rαr

)k

‖hs−kr‖1−ds−kr+k−1∑

j=0

(

Gr,2δ2rαr

)j

‖C(r)s−jr‖1−dr−1−δr

.

190 Chapter 7

sequence {αr}r≥0 defined as

(7.12) α0 = 1 , αr = min(

1, min0<|k|≤rK

∣

∣〈k, ω〉∣

∣

)

.

That is, αr is the smallest divisor that may appear in the solution of the homologicalequation for the generating functions χr,1 and χr,2 at step r of the normalizationprocess. If the frequencies are non resonant then the sequence has zero limit for r → ∞.A further conditions that characterizes strong non resonance will be found later.

Lemma 7.2: Let ψs(p, q) be a trigonometric polynomial of degree s, analytic andbounded in a domain D,σ . Let also ω ∈ Rn be a non resonant vector. Then thehomological equation ∂ωχ = ψs possesses a solution with zero average satisfying

(7.13) ‖χ‖,σ ≤‖ψ‖,σαs

with αs defined by (7.12) .

Proof. The solution is constructed as explained in sect. 7.2.1: the coefficient ck(p)of the Fourier expansion of χ are

ck(p) = −iψs,k(p)

〈k, ω〉,

where ψs,k(p) are the coefficients of ψs(p, q). Hence we estimate |ck| ≤|ψs,k|αs

. Thenwe apply the definition (6.52) of the weighted Fourier norm. Q.E.D.

The lemma is used in order to produce the estimates of the generating functionsχr,1 and χr,2 in table 7.3. The estimates of Xr and χr,2 are a direct applicationof lemma 7.2. The estimate of |ξr| requires a couple of elementary considerations.

The function B(r−1)r is linear in p. On the other hand, in view of the special form of

h0(p) =12 〈p, p〉 the equation for ξr in table 7.2 has the very simple form 〈ξ, p〉 = B

(r−1)r .

Therefore, by Cauchy’s estimate on the center of the disk ∆(1−d)(0) and in view ofthe inequality d < 1/2 that we shall use we get

(7.14) |ξr,j| ≤1

(1− d)

∥

∥B(r−1)r

∥

∥

1−d≤

2

∥

∥B(r−1)r

∥

∥

1−d.

Lemma 7.3: Let ξ ∈ Rn. If f(p, q) is analytic and bounded in D1 then

∥

∥L〈ξ,q〉f∥

∥

(1−d)≤

1

d|ξ| ‖f‖1 , |ξ| = |ξ1|+ . . .+ |ξn| .

Proof. An almost verbatim repetition of the proof of lemma 6.14. The angles q playthe role of parameters. For any (p, q) ∈ D1 the function F (τ) = f(p+τξ, q) is analytic

and bounded by ‖f‖1 in the disk |τ | < d|ξ|. Use L〈ξ,q〉f = dF (τ)

dτ

∣

∣

∣

τ=0, and apply the

Cauchy estimate. Q.E.D.


7.3.2 The scheme of estimates

We are now ready to construct a recurrent scheme of estimates, which is collected intable 7.3. Constructing the table is a tedious but straightforward operation. Here aresome hints that the reader may find useful.

The estimates of the generating functions Xr and χr,2 follow from lemma 7.2. Theestimate for the components ξr,j of ξr are a copy of (7.14). However, we shall actuallyuse lemma 7.3 for the Poisson bracket. If we know the norm ‖f‖1−dr−1

of a function

f(p, q) (which is precisely what we need) then a straightforward calculation gives

(7.15)

∥

∥Lχr,1f∥

∥

1−dr−1−δr≤

‖Xr‖1−dr−1

δ2r‖f‖1−dr−1

+2∥

∥B(r−1)r

∥

∥

1−dr−1

δr2‖f‖1−dr−1

≤

(

‖A(r−1)r ‖1−dr−1

+2αrδrσ

∥

∥B(r−1)r

∥

∥

1−dr−1

)

‖f‖1−dr−1

δ2rαrσ.

That is: the quantity between parentheses plays the role of the norm ‖χr,1‖1−dr−1.

Remark the extra factor αrδr which will be used in order to control the accumulationof small divisors. It is matter of a moment to realize that the estimates (6.56) oflemma 6.15 for powers of Lχr,1

remain true, because they are based on the estimateof a single Lie derivative.

All estimates of the functions A(r)s and B

(r)s that enter the expansion of H(r) are

nothing but an application of the estimates of multiple Lie derivatives of lemma 6.15using (7.15) for the estimate of χr,1. All functions C(r−1) remain unchanged, so thatthere is no need of estimating them: just keep their norms. The constant Gr,1 is definedso that in all estimates we collect all constants together and bring into evidence thedivisor δ2rαr which represents the real trouble in the proof of convergence.1

The estimates of hr and of all functions A(r)s ,B(r and C

(r)s that enter the expansion

of H(r) do not require any further comment, since it is only matter of applying againlemma 6.15 with the norm of χ2 estimated at the beginning of the table. The functionsh1, . . . , hr−1 remain unchanged, and so are their norms.

The fact to be remarked and exploited is that all estimates exhibit a commonstructure: all estimates are sums of different contributions obtained by multiplying afactor

Gr,1

δ2rαror

Gr,2

δ2rαr(or a power of it) by the known norm of some function.

In view of the latter remark we are led to consider the quantities βr = δ2rαr as thesmall divisors to be put under observation. It will be convenient also to define β0 = 1.

1 The reader will remark that the role of the small divisors is played not only by thequantities αr that depend on the frequencies, but also by the restrictions δr of thedomain that are requested by Cauchy estimates of Lie derivatives. By comparison, thereader my observe that in the case of formal expansion of first integrals of chapter 5 thesources of divergence are both the divisors αr and the factors coming from the exponentsof polynomials, due to derivatives.

192 Chapter 7

7.3.3 The game of small divisors

The accumulation of small divisors can be analyzed paying particular attention to theremark that we have just made. First, note that the mechanism of accumulation hasnothing to do with the actual value of divisors: it is rather matter of indices. Consider

any function hs, A(r−1)s , B

(r−1)s and C

(r−1)s in the expansion of H(r−1). In view of the

recurrent scheme of estimates of table 7.3 the norm of every function is estimated bya sum of terms with general form c(βj1 · · ·βjm)−1, where c is a positive number and mis the number of divisors that have been generated through the scheme of estimates.

E.g., we construct the collection of terms to be added up in the estimate for ‖A(r)s ‖

(the domain is not relevant here) by multiplying each term in ‖hs‖ by(Gr,1

βr

)2, then

multiplying every term in ‖B(r−1)s−r ‖ by

Gr,1

βr, then taking the collection of terms in

‖A(r−1)s ‖ as is, and making the union of the three collections so constructed without

applying any algebraic simplification. This applies to every estimate in the table, andit reflects precisely the scheme of estimates.

Let us now say that a function f owns a list of indices I = {j1, . . . , jk} if itsestimate contains a term with the divisor βj1 · · ·βjk . The following considerations areimmediate.

(i) If ψr (trigonometric polynomial of degree r) owns a list I, then the solution χrof the homological equation ∂ωχr = ψr owns the list {r}∪I, the union meaningconcatenation of lists (repeated indices are kept in the list).

(ii) If fs (of order s) owns a list I ′ and χr is as at point (i) then Lχrfs owns the

list {r} ∪ I ∪ I ′.(iii) For the multiple Lie derivative the following scheme applies:

fs owns I ′ ;

Lχrf owns I1 = {r} ∪ I ∪ I ′

L2χrf owns I2 = {r} ∪ I ∪ I1

L3χrf owns I3 = {r} ∪ I ∪ I2

and so on, so that we may proceed by recurrence.Hence the points (i) and (ii) contain all the relevant information about the mechanismof accumulation of divisors.

7.3.4 The kindness of small divisors

Let us open a long parenthesis. Forget for a moment the technicalities of the theorem ofKolmogorov, and focus attention on the propagation of lists of indices in the simplestcase.

We call list of indices a collection2 {j1, . . . , js} of non negative integers, with lengths ≥ 0. The empty list {} of length 0 is allowed, as well as repeated indices. The index0 is allowed, too, and will be used in order to pad a short list to the wanted length,when needed. The lists of indices provide a full characterization of the products of

2 The name list is used in order to emphasize that repeated elements are allowed.


Table 7.4. The special lists I∗s for 0 ≤ s ≤ 16.

s I∗s1 {}2 {1}3 {1 , 1}4 {1 , 1 , 2}5 {1 , 1 , 1 , 2}6 {1 , 1 , 1 , 2 , 3}7 {1 , 1 , 1 , 1 , 2 , 3}8 {1 , 1 , 1 , 1 , 2 , 2 , 4}9 {1 , 1 , 1 , 1 , 1 , 2 , 3 , 4}10 {1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 5}11 {1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 5}12 {1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 4 , 6}13 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 4 , 6}14 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 7}15 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 3 , 5 , 7}16 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 5 , 8}

small divisors: to the list {j1, . . . , js} we associate the product {βj1 , . . . , βjs}. Addingany number of zeros to a list of indices is harmless, for we have set β0 = 1.

On the set of lists of indices we introduce a partial ordering as follows. Let I, I ′

be lists with the same length s. We say that I ⊳ I ′ in case there is a permutationof the indices such that the relations j1 ≤ j′1, . . . , js ≤ j′s hold true. If the lists havedifferent lenghts then we pad the shorter one with zeros, and apply the criterion. Thecomparison is made easy by just ordering all elements of every list and comparingelement by element. The order is clearly partial: e.g., the criterion does not apply tothe lists {1, 3} and {1, 1}. This will not harm us, however.

A central role will be played by special lists of indices that we denote by I∗s , withs ≥ 0. We define

(7.16) I∗s =

{⌊

s

s

⌋

,

⌊

s

s− 1

⌋

, . . . ,

⌊

s

2

⌋}

.

In table 7.4 we give examples of the special lists just defined.

Lemma 7.4: For the sets of indices I∗s = {j1, . . . , js} the following statements holdtrue:

(i) the maximal index is jmax =⌊

s2

⌋

;

(ii) for every k ∈ {1, . . . , jmax} the index k appears exactly⌊

sk

⌋

−⌊

sk+1

⌋

times;

(iii) for 0 < r ≤ s one has(

{r} ∪ I∗r ∪ I∗s

)

⊳ I∗r+s .

Proof. The claim (i) is a trivial consequence of the definition.

194 Chapter 7

(ii) For each fixed value of s > 0 and 1 ≤ k ≤ ⌊s/2⌋ , we should determine thecardinality of the set Mk,s = {m ∈ N : 2 ≤ m ≤ s , ⌊s/m⌋ = k}. For this purpose, weuse the trivial inequalities

⌊

s

⌊s/k⌋

⌋

≥ k and

⌊

s

⌊s/k⌋+ 1

⌋

< k .

After having rewritten the same relations with k + 1 in place of k , one immediatelyrealizes that a index m ∈ Mk,s if and only if m ≥ ⌊s/(k + 1)⌋ + 1 and m ≤ ⌊s/k⌋.Therefore #Mk,s =

⌊

sk

⌋

−⌊

sk+1

⌋

, as claimed.

(iii) Since r ≤ s , the definition in (7.16) implies that neither {r} ∪ I∗r ∪ I∗s nor I∗r+scan include any index exceeding

⌊

(r + s)/2⌋

. Let us define some finite sequences ofnon-negative integers as follows:

Rk = #{

j ∈ I∗r : j ≤ k}

, Sk = #{

j ∈ I∗s : j ≤ k}

,

Mk = #{

j ∈ {r} ∪ I∗r ∪ I∗s : j ≤ k}

, Nk = #{

j ∈ I∗r+s : j ≤ k}

,

where 1 ≤ k ≤ ⌊(r+s)/2⌋ . When k < r , the property (ii) of the present lemma allowsus to write

Rk = r −⌊ r

k + 1

⌋

, Sk = s−⌊ s

k + 1

⌋

, Nk = r + s−⌊ r + s

k + 1

⌋

;

using the elementary estimate ⌊x⌋+⌊y⌋ ≤ ⌊x+y⌋ , from the equations above it followsthat Mk ≥ Nk for 1 ≤ k < r . In the remaining cases, i.e., when r ≤ k ≤ ⌊(r + s)/2⌋ ,we have that

Rk = r − 1 , Sk = s−⌊ s

k + 1

⌋

, Nk = r + s−⌊ r + s

k + 1

⌋

;

therefore, Mk = 1 + Rk + Sk ≥ Nk . Since we have just shown that Mk ≥ Nk ∀ 1 ≤k ≤ ⌊(r + s)/2⌋ , it is now an easy matter to complete the proof. Let us first reorderboth the set of indices {r}∪I∗r ∪I

∗s and I∗r+s in increasing order; moreover, let us recall

that #(

{r} ∪ I∗r ∪ I∗s

)

= #I∗r+s = r+ s− 1 , because of the definition in (7.16). Thus,since M1 ≥ N1 , every element equal to 1 in {r} ∪ I∗r ∪ I∗s has a corresponding indexin I∗r+s the value of which is at least 1 . Analogously, since M2 ≥ N2 , every index 2 in{r} ∪ I∗r ∪ I∗s has a corresponding index in I∗r+s which is at least 2 , and so on up tok = ⌊(r + s)/2⌋ . We conclude that {r} ∪ I∗r ∪ I∗s ⊳ I

∗r+s . Q.E.D.

We shall also need the following

Lemma 7.5: Let Ir and Is be lists of length r − 1 and s − 1, respectively, with1 ≤ r ≤ s (pad them with zeros, if needed). If Ir ⊳ I

∗r and Is ⊳ I

∗s then we have

r ∪ Ir ∪ Is ⊳ I∗r+s .

The proof is elementary: just reorder the indices.

We come now to exploit the relation between lists of indices and products of smalldivisors. Let 1 = α0 ≥ α2 ≥ α3 be a sequence of positive numbers; so is the sequence


of small divisors, with the additional property that the sequence tends to zero. To alist I associate the quantity Q(I) =

∏

j∈I1αj

. The following property is obvious:

if I ⊳ I ′ then Q(I) > Q(I ′) .

Let us consider the special sequence

(7.17) Q∗s =

∏

j∈I∗s

1

αj;

We look for a sufficient condition assuring that the sequence has a finite limit. In viewof lemma 7.4 we may evaluate

(7.18) lnQ∗s = ln

∏

j∈I∗s

1

αj≤ −

s∑

k=1

(⌊ s

k

⌋

−⌊ s

k + 1

⌋)

αk ≤ −s∑

k≥1

lnαkk(k + 1)

.

We are thus led to introduceCondition τ: The sequence {αr}r≥0 satisfies

(7.19) −∑

r≥1

lnαrr(r + 1)

= Γ <∞ .

If so, then we also have the estimate Q∗s < esΓ, that is, it grows not faster than

geometrically.Taking the sequence of small divisors associated to the frequencies ω ∈ Rn as

in (7.12) condition τ will provide the strong non resonance condition for the validityof the theorem of Kolmogorov.

In his original proof Kolmogorov used the diophantine condition. More recentlyother conditions have been introduced with the aim of finding the optimal one. Themost known is the so called Bruno condition, introduced by Alexander Bruno.For theproblem of Schroder–Siegel (a map of the complex plane with the origin a fixed point)the optimality of the condition of Bruno has been proved by Jean Christophe Yoccoz.For the problem of Kolmogorov the question is still open.

Example 7.1: Comparison with other conditions Here are a few notes that allow thereader to compare condition τ with other commonly used conditions that are foundin literature.(i) The diophantine condition introduced by Siegel says αr = r−k with k > 1 (an

innocuous multiplicative constant is omitted). This gives

−∑

r≥1

lnαrr(r + 1)

= k∑

r≥1

ln r

r(r + 1)<∞ .

This shows that diophantine frequencies satisfy also contidion τ. By the way, thisalso shows that if the sequence αr satisfies condition τ then so does the sequenceαr = δ2rαr with δr ∼ r−2 that appears in our estimates for the case of Kolmogorov,for it just adds to the estimate (7.18) a convergent series.

196 Chapter 7

(ii) Condition τ is weaker than the diophantine one. E.g., if αr = e−r/ ln2 r then

−∑

r≥1

lnαrr(r + 1)

=∑

r≥1

1

(r + 1) ln2 r<∞ .

(iii) There are ω’s that violate condition τ. For instance, if αr = e−r then

−∑

r≥1

lnαrr(r + 1)

=∑

r≥1

1

(r + 1)= ∞.

(iv) The condition of Bruno writes

−∑

r≥1

lnα2r−1

2r= B <∞ .

It is equivalent to condition τ, for one gets Γ < B < 2Γ. The proof is left to thereader.

7.3.5 Small divisors in the algorithm of Kolmogorov

We can now go back to the estimates for the normal form of Kolmogorov, and focus onthe accumulation of small divisors. The problem is to identify the the worst possibleproduct of divisors in every coefficient of every function. To this end, in view of thediscussion of the previous section, we concentrate on the indices, and look for twoinformations, namely

(i) the number of divisors βj ;

(ii) a selection rule that specifies which lists of coefficiens may really show up.

Definition 7.6: For all integers r ≥ 0 and s > 0 , we introduce the collection of lists

(7.20) Jr,s ={

I = {j1, . . . , js−1} : 0 ≤ jm ≤ min{r, ⌊s/2⌋} , I ⊳ I∗s}

,

Lemma 7.7: The following properties hold true: for r′ < r we have

(7.21) Jr′,s ⊂ Jr,s for r′ < r ;

for 0 ≤ r ≤ s we have

(7.22) {r} ∪ Jr−1,r ∪ Jr,s ⊂ Jr,r+s .

Proof. The first property is obvious. For the second one check that: the number ofindices in the two members is the same; the maximum value of the indices is respected;the selection rule is respected in view of property (iii) of lemma 7.4. Q.E.D.

Our goal now is to identify the list of divisors owned by every function, withparticular care for the generating functions. This will be made later with lemma 7.8.However, before stating the lemma we need some considerations concerning the normof the vector ξr , for a straightforward estimate will add a divisor that apparently


breaks the toy constructed in sect. 7.3.4. This may be seen by looking at the expressionof Gr,1 in table 7.3, namely

Gr,1 =e

σ

(

‖A(r−1)r ‖1−dr−1

+2αrδrσ

∥

∥

∥B

(r−1)r

∥

∥

∥

1−dr−1

)

.

The immediate inequality (the domains are not relevant here)∥

∥B(r−1)r

∥

∥ ≤∥

∥B(r−1)r

∥

∥

suggests that the second term should contain a divisor δ2rαr in addition to those in the

first term A(r−1)r . That divisor is only partially compensated by the factor δrαr. The

point is that a careful estimate of∥

∥B(r)s

∥

∥ contains an extra factor δr which removesthe extra divisors. Regrettably, the proof requires some tedious calculations.

We proceed by induction, as usual. The problem does not show up for Gr,1, be-

cause there are no small divisors in B(0)s . For r > 1 we should take into consideration

the expressions of B(r)s and B

(r)s in table 7.2. Let us recall them:

(7.23) B(r)s =

{

Lχr,1hs−r +B(r−1)

s , r < s < 2r ;

Lχr,1C

(r−1)s−r +B(r−1)

s , s ≥ 2r .

and, recalling that k = ⌊s/r⌋,

(7.24) B(r)s =

k − 1

k!Lk−1χr,2

B(r)r +

k−2∑

j=0

1

j!Ljχr,2

B(r)s−jr , k ≥ 2 , m = 0 ;

k−1∑

j=0

1

j!Ljχr,2

B(r)s−jr , k ≥ 1 , m 6= 0 .

In the expressions for B in (7.23) the estimate of B(r−1)s is assumed to contain the

wanted extra factor δr by the inductive hypothesis. On the other hand, the two ex-pressions are much the same, the irrelevant difference being in the functions hs−rand C

(r−1)s−r . Hence let us work out the estimate only for the second one. Denoting by

xk ∈ Cn and by ck(p) the coefficients of the Fourier expansions of Xr and C(r−1)s ,

respectively, we obviously have

LXrC

(r−1)s−r = −i

∑

k∈Zn

n∑

l=1

klxk∂c−k∂pl

, L〈ξ,q〉C(r−1)s−r =

n∑

l=1

ξr,l∂c0∂pl

.

Hence, taking into account that ck(p) is a homogeneous polynomial of degree 2, weestimate the first expression as

∥

∥

∥LXr

C(r−1)s−r

∥

∥

∥

1−dr−1

≤ 2∑

k

|k| |xk|∣

∣c−k∣

∣

(1−dr−1)

≤ 2 ‖C(r−1)r−1 ‖

1−dr−1

∑

k

|xk| e(1−dr−1)|k|σ |k| e−δr|k|σ

≤2

eδrσ‖Xr‖1−dr−1

∥

∥C(r−1)s−r

∥

∥

1−dr−1

198 Chapter 7

(remark that no restriction of the domain is required). Similarly (and simpler), thesecond expression is estimated as

∥

∥

∥L〈ξ,q〉C

(r−1)s−r

∥

∥

∥

1−dr−1

≤n∑

l=1

|ξr,l|∣

∣

∣

∂c0∂pl

∣

∣

∣≤ 2|ξr|

∥

∥C(r−1)s−r

∥

∥

1−ds−r.

In both cases the general estimate of the Lie derivative would contain a divisor δ2rσdue to the restriction of domains. Hence in both estimates we gain an extra factor δr,

which is precisely what we need. Coming to the second expression for B(r)s in (7.24),

let us separate the term j = 0, thus getting

(7.25) B(r)s = B(r)

s + Lχr,2Ψ , Ψ =

s∑

j=0

1

j!Lj−1χr,2

B(r)s−jr ,

where Ψ is linear in p. Then, denoting by bk and ψk the coefficients of the Fourierexpansion of χr,2 and Ψ, respectively, we calculate

Lχr,2Ψ = i

∑

k

n∑

l=1

kl

(

ψk∂b−k∂pl

+ b−k∂ψk∂pl

)

.

Exploiting the fact that χr,2 and Ψ are both linear in p we get

(7.26)

∥

∥

∥Lχr,2Ψ

∥

∥

∥

1−dr−δr≤ 2

∑

k

|k|∣

∣b−k∣

∣

(1−dr−1−δr)

∣

∣ψk∣

∣

(1−dr)

≤2

eδrσ‖χr,2‖1−dr−δr‖Ψ‖1−dr .

The estimate of the first expression in (7.24) goes the same way, in view of the obviousinequality k−1

k!< 1

(k−1)!. Thus we gain again a factor δr with respect to the general

estimate of Lie derivative. Since the estimate of B(r)s contains the same factor, as we

have shown, we conclude that for all r ≥ 1 also the estimate of∥

∥B(r)s

∥

∥ contains an

extra factor δr. This is true in particular for B(r−1)r in the equation for ξr. Using this

information in the expression of Gr,1 we see that both addends own the same list ofdivisors, in the worst case. This was our claim, indeed.

We can now state the

Lemma 7.8: Every function in the sequence H(r) of Hamiltonians owns only listsof indices as specified in table 7.5.

Proof. For r = 0 there are no divisors, therefore every function in H(0) owns anempty list, namely {}. Therefore, by padding every list with an appropriate numberof zeros, the table is correct, because J0,s is a list of s − 1 zeros. For r > 0 we mayproceed by induction, supposing that the table applies up to step r − 1. We followthe path of the formal algorithm of table 7.2, taking into account that new lists ofindices are generated according to the considerations made at points (i) and (ii) in


Table 7.5. The number of indices and the selection rule for the function hr for

r ≥ 0 and for the functions hr , A(r)s , B

(r)s and C

(r)s for 1 ≤ r < s.

function #of indices selection rule bounded by

h0 0 {} T0,0 ;

A(r)s 2s− 2 Jr,s ∪ Jr,s T 2

r,s ;

B(r)s 2s− 1 {r} ∪ Jr,s ∪ Jr,s

1

βrT 2r,s ;

hr, C(r)s 2s {r} ∪ {r} ∪ Jr,s ∪ Jr,s

1

β2r

T 2r,s ;

sect. 7.3.3. The part concerning the generating functions gives (the verb “owns” isused as a shortening of the expression “owns only lists in the set”)

A(r−1)r owns Jr−1,r ∪ Jr−1,r ;

δrB(r−1)r owns Jr−1,r ∪ Jr−1,r ;

Gr,1 owns Jr−1,r ∪ Jr−1,r ;

χr,1 owns {r} ∪ Jr−1,r ∪ Jr−1,r ;

B(1)1 , Gr,2 owns {r} ∪ Jr−1,r ∪ Jr−1,r ;

χr,2 owns {r} ∪ {r} ∪ Jr−1,r ∪ Jr−1,r .

Use the information on χr,1 and select the worst case among all terms in the sum; get

A(r)s owns 2×

(

{r} ∪ Jr−1,r

)

∪ Jr,s−2r+1 ⊂ Jr,s ;

B(r)s owns {r} ∪ Jr−1,r ∪ Jr−1,s−r ⊂ {r} ∪ Jr,s .

Here the multiplication by 2 means the union of identical lists. The inclusion relationsfollow from lemma 7.7.

Coming to the last part of table 7.2, use the information on χr,2 ; for all allowed valuesof j in the sums get

hr owns {r} ∪ {r} ∪ Jr−1,r ∪ Jr−1,r ⊂ {r} ∪ {r} ∪ Jr,r ;

A(r)s owns 2j ×

(

{r} ∪ Jr−1,r

)

∪ Jr,s−jr ⊂ {r} ∪ Jr,s ;

B(r)s owns 2j ×

(

{r} ∪ Jr−1,r

)

∪ {r} ∪ Jr,s−jr ⊂ {r} ∪ {r} ∪ Jr,r ;

C(r)s owns 2j ×

(

{r} ∪ Jr−1,r

)

∪ {r} ∪ {r} ∪ Jr,s−jr ⊂ {r} ∪ {r} ∪ Jr,r .

Here too the inclusion relations follow from lemma 7.7, and hold true for every termin the sums. This completes the induction. Q.E.D.

Having settled the selection rules for indices we associate to the collections of lists

200 Chapter 7

Jr,s introduced in definition 7.6 the double indexed sequence of positive numbers

(7.27) Tr,s =∏

j∈Ir,s

β−1j , r ≥ 0 , s > 0 .

From lemma 7.7 we immediately get the inequalities

(7.28)

Tr′,s ≤ Tr,s for 0 ≤ r′ < r ;

1

βrTr−1,rTr,s ≤ Tr,r+s for 0 ≤ r ≤ s

The bounds for the products of small divisors in table 7.5 are straighforward conse-quences of (7.27) and (7.28).

7.3.6 Recurrent estimates

Let us look again at table 7.3 and take into account the control of divisors that wehave found in the previous section. Our goal is now to obtain a manageable estimate ofthe generating sequences. We shall prove that the norms are bounded geometrically. Itis worth to recall also that we have actually made an expansion in a small parameterε, so that a geometrical growth of the generating sequences χr,1 and χr,2 as , e.g., M r

will only mean that the threshold ε∗ for the validity of the theorem is ε∗ ∼M−1.

Lemma 7.9: Let

(7.29) E = max(

‖h0‖1, ‖A(0)1 ‖, ‖B

(0)1 ‖, ‖C

(0)1 ‖

)

, M = max

(

1,eE

σ(1 + 2Eσ)

)

.

Let the double indexed sequences νr,s and νr,s be defined as

(7.30)

ν0,s = 1 for s ≥ 0 ,

νr,s =

min(⌊s/r⌋,2)∑

j=0

νjr−1,rνr−1,s−jr for r ≥ 1 , s ≥ 0 ,

νr,s =

⌊s/r⌋∑

j=0

νjr−1,rνr,s−jr for r ≥ 1 , s ≥ 0 .


Then for 1 ≤ r < s the following estimates hold true

‖A(r)s ‖1−dr ≤ νr,sM

s−2T 2r,sE ,

‖B(r)s ‖1−dr ≤ νr,sM

s−1 1

βrT 2r,sE ,

‖hr‖1−dr ≤ νr−1,rMr 1

β2r

T 2r,rE ,

‖C(r)s ‖1−dr ≤ νr,sM

s 1

β2r

T 2r,sE ,

Gr,1 ≤ νr−1,rT2r−1,rM

r−1 ,

Gr,2 ≤ νr−1,rMr 1

βrT 2r−1,r .

with dr, δr defined by (7.10) and (7.11) and with Tr,s defined by (7.27).

Proof. The estimates are clearly true for r = 0 in view of T0,s = 1 and ν0,s = 1. Forr ≥ 1 we use induction. Assuming that the claim is true up to r − 1 we have

δrαr∥

∥B(r−1)r

∥

∥

1−dr−1≤ δ2rαr νr−1,rT

2r−1,rM

r−2E2 .

Remark that here the estimate, including the extra factor δr, comes from the estimateof the average, in particular from (7.26), as explained in sect. 7.3.4. Thus we get

Gr,1 ≤ νr−1,rT2r−1,rM

r−2 eE

σ(1 + 2Eσ) ≤ νr−1,rT

2r−1,rM

r−1 .

Now we estimate the intermediate Hamiltonian H(r). For A(r)s , remarking that there

is no relevant difference between the estimates of hr and of C(r−1)s , we get

‖A(r)s ‖1−dr−1−δr

≤ ν2r−1,r

( 1

βrT 2r−1,rM

r−1)2

× νr−1,s−2rT2r−1,s−2rM

s−2rE

+ νr−1,r1

βrT 2r−1,rM

r−1 × νr−1,s−rT2r−1,s−rM

s−r−1E

+ νr−1,sTr−1,sMs−2E

≤ νr,sTr,sMs−2E ,

where the definition (7.30) of νr,s has been used. Next we have

‖B(r)s ‖1−dr−1−δr

≤1

βrT 2r−1,rM

r−1νr−1,r ×1

βrνr−1,s−rT

2r−1,s−rM

s−rE

+ νr−1,sT2r−1,sM

s−1E

≤1

βrT 2r−1,sM

s−1Eνr,s .

Coming to the transformed Hamiltonian H(r) we first have

Gr,2 ≤ νr−1,r1

βrT 2r−1,rM

r ≤ νr−1,r1

βrT 2r−1,rM

r .

202 Chapter 7

Then we estimate the functions. For hr we get

‖hr‖1−dr ≤1

β2r

T 2r−1,sνr−1,rM

rE + νr−1,rT2r−1,sM

rE

≤1

β2r

νr,rT2r−1,sM

rE

For A(r)s , recalling that k = ⌊s/r⌋, we get

‖A(r)s ‖1−dr ≤

k−1∑

j=0

(

1

βrT 2r−1,sνr,sM

r

)j

× νr,sTr,s−jrMs−jr−2E

≤ νr,sTr,sMs−2E ,

where the definition of νr,s has been used. For B(r)s , still recalling k = ⌊s/r⌋, we get

‖B(r)s ‖1−dr ≤

k−1∑

j=0

(

1

βrT 2r−1,sνr,sM

r

)j

×1

βrνr,sTr,s−jrM

s−jr−2E

≤1

βrνr,sTr,sM

s−1E .

For C(r)s , recalling once more that k = ⌊s/r⌋, we get

‖C(r)s ‖1−dr ≤

(

1

βrT 2r−1,rνr,sM

r

)k

× Tr,s−krMs−krE

+k−1∑

j=0

(

1

βrT 2r−1,rνr,sM

r

)j

×1

β2r

νr,s−jrTr,s−jrMs−jrE

≤1

β2r

νr,sTr,sMsE .

Q.E.D.

7.3.7 Completion of the proof

In order to complete the proof we must show that the generating functions χr,1 andχr,2 satisfy the convergence condition of proposition 6.11. To this end let us look atthe scheme of estimates of table 7.3, from which we see that we may use the estimate

‖χr,1‖1−dr−1≤Gr,1αr

, ‖χr,2‖1−dr−1≤Gr,2αr

Therefore from the estimates of lemma 7.9 we have(7.31)

‖χr,1‖1−dr−1≤

1

αrνr−1,rT

2r−1,rM

r−1 , ‖χr,2‖1−dr−1≤

1

αrβrT 2r−1,rνr−1,rM

r .

Thus the problem is to show that the quantities νr,s and Tr,s can be bounded geo-metrically. This would be enough because we should not forget that the generating


functions are actually εrχr,1 and εrχr,2 . In the discussion below, coherently with theattitude of the present notes, there are also explicit estimates of the constants. How-ever one should keep in mind that also the evaluation of the constants is made withestimates that are typically far from being optimal. The suggestion is only that actualvalues can be found, and possibly (certainly) be improved with respect to the onesgiven here.

Lemma 7.10: Let the sequence {αr}r≥1, introduced by (7.12), satisfy condition τand the sequence {δr}r≥1 be defined as in (7.11). Then, the sequence {Tr,s}r≥0 , s≥0

defined by (7.27) is bounded by

Tr,s ≤1

asδ2sTr,s ≤

(

215eΓ)s

for r ≥ 1 , s ≥ 1 .

Proof. Since asδ2s < 1 (see (7.12) and (7.11)), it is enough to prove the second part

of the inequality stated in the lemma, i.e., Tr,s/(asδ2s) ≤ AsesΓ for r ≥ 1 , s ≥ 1 .

Starting from the definition (7.27), using properties (i) and (ii) of lemma 7.4, theselection rule in (7.20) and the decreasing character of the sequence {asδ

2s}s≥1, we get

Tr,sasδ2s

=1

asδ2smaxI∈Js,s

∏

j∈I , j≥1

1

ajδ2j≤

∏

j∈{s}∪I∗s , j≥1

1

ajδ2j.

Starting from the estimate above, we have

(7.32)

logTr,sasδ2s

≤ − log(asδ2s)−

⌊s/2⌋∑

k=1

(⌊ s

k

⌋

−⌊ s

k + 1

⌋)

log(akδ2k)

≤ −s

∑

k=1

(⌊ s

k

⌋

−⌊ s

k + 1

⌋)

(log ak + 2 log δk)

≤ −s∑

k≥1

log ak + 2 log δkk(k + 1)

= s

(

Γ−∑

k≥1

2 log δkk(k + 1)

)

,

where we used properties (i) and (ii) of lemma 7.4, the fact that the sequence {asδ2s}s≥1

is decreasing and the condition τ in (7.19). A numerical evaluation is done as follows.Recalling the definition of δr we get

−∑

k≥1

2 log δkk(k + 1)

= 2∑

k≥1

log 8π2

3 + 2 log k

k(k + 1)

< 2 log8π2

3+ 4

(

log 2

6+

∫ ∞

2

log x dx

x2

)

< 15 log 2 ,

where the known relation∑

k≥1 1/[k(k + 1)] = 1 is used. Putting the estimate aboveinto (7.32), we conclude the proof. Q.E.D.

204 Chapter 7

It remains to verify the sequence νr−1,r is bounded geometrically. This requiressome calculations that we work out in the next section where we prove the inequalityνr−1,r < 4r−1.

Finally, we have shown that all terms in the estimates (7.31) for the generatingfunctions are bounded geometrically. Therefore, setting d = 2

∑

r δr as in (7.11), thereexists a positive ε∗ such that for ε < ε∗ we have

1

d2σ

∑

r

εr‖χr,1‖1−d <1

2e,

1

d2σ

∑

r

εr‖χr,2‖1−d <1

2e,

namely the convergence condition of proposition 6.11. Having chosen d = 1/6, thecanonical transformation is holomrphic in a domain, e.g., D(,σ)/2, and the Sequence

of Hamiltonians H(r) converges to a holomorphic Hamiltonian

H(∞) = 〈ω〉+O(p2)

in Kolmogorov normal form. This concludes the proof that there exists a perturbedinvariant torus carrying quasiperiodic motions with frequencies ω, which is close tothe unperturbed one.

*** spiegare meglio la deformazione ***

7.3.8 Estimate of the sequence ν

With a minor change, let us replace the sequence with the majorant one (let k = ⌊s/r⌋)

(7.33)

ν0,s = 1 for s ≥ 0 ,

νr,s =

k∑

j=0

νjr−1,rνr−1,s−jr for r ≥ 1 , s ≥ 0 ,

νr,s =

k∑

j=0

νjr−1,rνr,s−jr for r ≥ 1 , s ≥ 0 .

Remark that we are interested in an estimate of the norms of the generating functionsχr,1 and χr,2 ; therefore it is enough to have an estimate for the elements νr−1,r .

Two remarks are immediate. The first one is that νr,s ≥ νr,s , for the right memberis the sole term j = 0 in the sum in the second formula. The second is that for r > swe have k = ⌊s/r⌋ = 0; hence the sums reduce to the sole term j = 0. Thus we get

(7.34) νr,s = νr,s = νr−1,s = . . . = νs,s .

For r ≤ s we may simplify the sequences by calculating

νr,s = νr−1,s + νr−1,r

k−1∑

m=0

νmr−1,rνr−1,(s−r)−mr= νr−1,s + νr−1,rνr,s−r ,

νr,s = νr−1,s + νr−1,r

k−1∑

m=0

νmr−1,rνr,(s−r)−mr = νr−1,s + νr−1,rνr,s−r .


As a matter of fact, we may discard the first formula defining νr,s , for from the secondformula we get

νr,s ≤ νr−1,s + νr−1,rνr,s−r .

Putting this formula together with (7.34) we find a majorant of the sequence νr,s,namely νr,s ≤ ηr,s by setting

(7.35)

η0,0 = 0 , η0,s = 1 for s ≥ 0 ,

ηr,r = ηr,r−1 = . . . = ηr,0 for r ≥ 1 ,

ηr,s = ηr−1,s + ηr−1,rηr,s−r for r ≤ s .

We show now that we have ηr−1,r ≤ λr where {λr}r≥1 is the Catalan’s sequence,defined as

(7.36) λ1 = 1 , λr =

r−1∑

j−1

λjλr−j for r > 1 .

To this end, let us apply r − 1 times the definition (7.35), so that we get in sequence

ηr−1,r = ηr−2,r + ηr−2,r−1ηr−1,1 ,

ηr−2,r = ηr−3,r+ ηr−3,r−2ηr−2,2 ,

. . . . . . . . . . . . . . .

η2,r =η1,r + η1,2η2,r−2 ,

η1,r = η0,r + η0,1η1,r−1 .

Then we get

(7.37)

ηr−1,r = η0,1η1,r−1 + η1,2η2,r−2 + . . .+ ηr−3,r−2ηr−2,2 + ηr−2,r−1ηr−1,1

=r−1∑

j=1

ηj−1,jηj,r−j .

Now, from (7.35) we also have, trivially, ηr−1,s ≤ ηr,s . If j < r − j then we have thechain of inequalities

ηj,r−j < ηj+1,r−j < . . . < ηr−j−1,r−j ;

on the other hand, if j ≥ r−j then, always from (7.27), we have the chain of equalities

ηj,r−j = ηj−1,r−j = . . . = ηr−j−1,r−j .

Thus we may replace (7.37) with

ηr−1,r ≤

r−1∑

j=1

ηj−1,jηr−j−1,r−j .

We conclude νr−1,r ≤ ηr−1 ≤ µr .

206 Chapter 7

The last step is to show that the Catalan’s sequence is bounded geometrically. Itis known that

(7.38) λr =2r−1(2r − 3)!!

r!≤ 4r−1 ,

where the usual notation of the semifactorial (2n + 1)!! = 1 · 3 · . . . · (2n + 1) hasbeen used. Let us see how the formula can be proved, using the method of generatingfunction due to Gauss. Let the function g(z) be defined as g(z) =

∑

r≥1 λrzr, so that

λr = g(r)(0)/r!. Then it is immediate to check that the recurrent definition (7.36) isequivalent to the equation g = z+g2. By repeated differentiation of the latter equationone readily finds

g′ =1

1− 2g, . . . , g(r) =

2r−1(2r − 3)!!

(1− 2g)2r−1

(check by induction). From this, (7.38) follows. The inequality is just a rough estimate:we do not need to do better for our purposes.

7.4 A general scheme

The scheme of proof of the previous section may be extended to a Hamiltonian in theform H(p, q) = H0(p) + εH1(p, q) of the general problem of dynamics. There are twodifferences to be taken into account, namely:

(i) In the neighbourhood of an unperturbed torus the Hamiltonian must be ex-panded in power series of the actions, convergent in a polydisk ∆′(0) for some′ > 0.

(ii) The perturbationH1(p, q) is not restricted to be a trigonometric polynomial: theFourier expansion may actually contain infinite terms, but it must be convergentin a complexified torus Tnσ′ for some positive σ′.

The difference at point (i) is not really harmful. The immediate consequence is that thearbitrary parameter must satisfy 0 < < ′, so that the norms of the functions arebounded, as requested by all estimates. The complexity of the algorithm will increase,because also the polynomial degree must be taken into account, but this is just atechnical matter.

The difference at point (ii) may be puzzling, for two good reasons. The first reasonis that an algorithm using infinite expansions can hardly be considered as constructive.The second, more severe reason is that the control of accumulation of divisors makesstrong use of the fact that all functions of order O(εr) are trigonometric polynomialsof degree rK. An infinite Fourier expansion seems to vanify all the scheme of controlof divisors.

The way out from the second problem has already been indicated by Poincare:3

3 The problem raised by Poincare was concerned with the calculation of orbits for theplanetery system. Since the Fourier expansion of the Hamiltonian in that case clearlycontains infinite terms, how can we assure that truncating the expansion does not badlyaffect our conclusions?


see [87], Ch. XIII, § 147. We should exploit the fact that the size of the coefficientsof the Fourier expansion of a holomorphic function decreases exponentially with thedegree. Therefore we may choose a truncation parameter K > 0 and expand theHamiltonian as a series

H(p, q, ε) = H0(p) +H1(p, q, ε) +H2(p, q, ε) + . . .

by putting in H1 all Fourier components with 0 < |k| ≤ K, in H2 all Fourier com-ponents with K < |k| ≤ 2K, and so on. That way the original Hamiltonian satisfiesthe properties that at every order s we deal only with trigonometric polynomials ofdegree sK. Then we may proceed with the power expansion in p around a stronglynon resonant unperturbed torus.

An attentive reader will perhaps raise two objections, namely: (i) the minimaldegree at every order will be lost during the normalization process; (ii) the expansionin powers of ε is lost, because terms with low power of ε but high |k| will be moved athigher orders. The answer to the first question is direct: the only thing we really needis that the maximal degree sK is preserved at every normalization step; this remainstrue. The answer to the second question is: well, relax the requests concerning theexpansion in ε and let the lower indices, s say, denote the order. The smallness ofevery term will be assured by its norm.

There is however a point that should be made clear (although it may be difficultto understand it unless one works out the proof in detail). A naive argument would bethe following: since the coefficients decrease as e−|k|σ with some σ, choose K so thate−Kσ ∼ ε. This leads to setting K ∼ − log ε. The drawback is that for ε → 0 one islead to let K → ∞. This, again, has a strong negative impact on the effectiveness ofthe control of small divisors. Moreover, the analyticity of the normal form in ε seemsto be destroyed, because K changes by steps for varying ε. Actually, the normal formis still analytic in ε, but one needs to give a proof.

It is remarkable indeed that the best choice is to set K to a constant independentof ε, which does not need to be a large one.4 A proof for the general case that exploitsthe suggestions made here may be found in [36].

7.5 An alternate proof using fast convergence

It would be a sin to close this chapter without a report on the original method ofproof of Kolmogorov, based on Newton’s method of fast convergence. The presentsection is intended to fill the gap. The purpose is to provide a a proof of the theorem,in a general setting, that closely follows the method proposed by Kolmogorov in hisoriginal short note. The only minor difference is that the method of Lie series willreplace the traditional method of generating functions in mixed variables in writingthe canonical transformations.

4 In most practical application the choice ofK is dictated by reasons that are more relevantthan the actual value of ε. E.g., by the requirement to include since the beginning of theexpansion some relevant resonances.

208 Chapter 7

For the sake of definiteness, I am going to reformulate the theorem of Kolmogorovin a version that is fully equivalent to the statement in theorem 7.1.

Theorem 7.11: On the phase space Tn×Rn consider the following quadratic Hamil-tonian:

(7.39) H(q, p) = 〈ω, p〉+ A(q) + 〈B(q), p〉+1

2〈C(q)p, p〉 .

Assume that ω is diophantine and that all functions A, Bj and Cj,k are analytic onTnσ for a fixed σ > 0. If the perturbing terms A, B1 , . . . , Bn are small enough andthe matrix C =

(

Ci,j)

i,j=1,...,nis nondegenerate, then the Hamiltonian possesses an

invariant torus that is close to {(q, p) : q ∈ Tn , p = 0}. Moreover, the flow on thattorus is quasi periodic with frequencies ω.

A few comments are in order.A more quantitative statement, including a detailed definition of the threshold on thesmallness of the perturbation is given at the end of this chapter.It must be emphasized that there is not any need to preliminarly assume the non-resonant (diophantine) condition since the very beginning, as it has been stressedalso in the statement of theorem 7.1. Indeed, starting from a Hamiltonian of typeH(q, p) = 1

2〈Jp, p〉 + εf(q, p, ε) where J is a nondegenerate symmetric matrix, nearlyall values of p∗ (in the sense of the Lebesgue measure) are such that the correspondingfrequencies ω = Jp∗ satisfy a diophantine condition (7.3). As it has been outlined atthe beginning of sect. 7.5.1, by translating the origin of the action variables to p∗, theHamiltonian can be rewritten as in form (7.39), provided that also f is quadratic withrespect to the actions p.In order to avoid misunderstandings, a special attention is devoted to keep separateas much as possible the following proof, based on a fast convergence algorithm, withrespect to the previous one. Therefore, the rest of the present chapter attempts to berather self-contained; as an unavoidable consequence of this choice, a few repetitionswill be introduced.

7.5.1 Formal algorithm based on the fast convergence method

Let me start by stating the formal algorithm in an explicit algebraic form. I collect hereall formulæ that we need in order to perform a single step. The complete procedurerequires iterating such formulæ. The algorithm will be justified immediately after thisstatement. Let me emphasize here that the following approach basically implementsthe fast convergence method, as it has been sketched in section 7.2.1.

To the Hamiltonian in the form (7.39) we apply two near to identity canonicaltransformations with generating functions

(7.40) χ1(q) = X(q) + 〈ξ, q〉 , χ2(q, p) = 〈Y (q), p〉 .

The function X(q), the real vector ξ and the vector function Y (q) are determined bythe equations

∂ωX + A = 0 ,(7.41)


Cξ +B + C∂X

∂q= 0 ,(7.42)

∂ωY +B + C

(

∂X

∂q+ ξ

)

= 0 .(7.43)

The transformed Hamiltonian is computed as

(7.44) H ′(q, p) = exp(

Lχ2

)

◦ exp(

Lχ1

)

H(q, p) ,

and has again the form (7.39) with a function A′(q), a vector function B′(q) and amatrix C′(q) given by

A′ = exp(L〈Y,p〉)A ,(7.45)

〈B′, p〉 =∑

j≥1

j

(j + 1)!Lj〈Y,p〉〈B, p〉 ,(7.46)

〈C′p, p〉 = 〈Cp, p〉+∑

j≥1

1

j!Lj〈Y,p〉〈Cp, p〉 ,(7.47)

A =1

2

⟨

C

(

∂X

∂q+ ξ

)

,

(

∂X

∂q+ ξ

)⟩

+

⟨

B,

(

∂X

∂q+ ξ

)⟩

,(7.48)

B = B + C

(

∂X

∂q+ ξ

)

.(7.49)

The canonical transformation is explicitly written as

(7.50)q = exp

(

L〈Y,p〉

)

q′

p = exp(

L〈Y,p〉

)

◦ exp(

LX+〈ξ,q〉

)

p′

Remark that exp(

LX+〈ξ,q〉

)

q′ = q′, which justifies the first line.5

We now come to justify the formal algorithm. This is just matter of playing alittle with the Lie series expansions.

Let us write explicitly the first transformation with generating function χ1(q) =X(q) + 〈ξ, q〉, considering the function X(q) and the real vector ξ as unknowns. De-

5 Some confusion could arise from the fact that the new Hamiltonian H ′ as been expressedabove as function of q, p instead of the new variables q′, p′. This, however, turns out tobe natural if one recalls that the substitution of variables is automatically performedby applying the exponential operator of the Lie series to functions, the names of thevariables being unrelevant.

210 Chapter 7

noting by H the transformed Hamiltonian we get6

H = exp(

Lχ1

)

H =〈ω, p〉+1

2〈Cp, p〉

+ A+ LX〈ω, p〉+ L〈ξ,q〉〈ω, p〉

+ 〈B, p〉+1

2LX〈Cp, p〉+

1

2L〈ξ,q〉〈Cp, p〉

+1

4L2χ1〈Cp, p〉+ Lχ1

〈B, p〉 .

We now kill the unwanted term A by determining X via the equation ∂ωX + A = 0,namely (7.41). Since a constant in the Hamiltonian function is unrelevant, we canalways assume that A(q) has zero average, i.e., A = 0; therefore a formal solution ofequation (7.41) exists. In view of this, and letting ξ still undetermined, we can writethe transformed Hamiltonian H as

(7.51) H(q, p) = 〈ω, p〉+ A(q) + 〈B(q), p〉+1

2〈C(q)p, p〉 ,

where A and B are given by (7.48) and (7.49). The expressions for A and B are easilydetermined by collecting all terms independent of p and linear in p, respectively,in the expression above for H, and calculating the Poisson brackets. Remark thatL〈ξ,q〉〈ω, p〉 = 〈ω, ξ〉 has been omitted, being an irrelevant constant.

We now perform the second transformation with generating function χ2(q, p) =〈Y (q), p〉, and write the transformed Hamiltonian H ′ as

(7.52)

H ′ = exp(

L〈Y,p〉

)

H

= 〈ω, p〉+ exp(

L〈Y,p〉

)

A

+ 〈B, p〉+ L〈Y,p〉〈ω, p〉+∑

j≥2

1

j!Lj〈Y,p〉〈ω, p〉+

∑

j≥1

1

j!Lj〈Y,p〉〈B, p〉

+1

2〈Cp, p〉+

1

2

∑

j≥1

1

j!Lj〈Y,p〉〈Cp, p〉 .

We now kill the unwanted term 〈B, p〉 by imposing

(7.53) 〈B, p〉+ L〈Y,p〉〈ω, p〉 = 0 .

Using (7.49), this gives the equation

∂ωY +B + C

(

∂X

∂q+ ξ

)

= 0 ,

namely (7.43). However, the latter equation can be solved only if the average of theknown term is zero. Therefore, we kill the unwanted average part by determining the

6 Remark that, due to χ1 being independent of the actions p, most terms in the Lie seriesdo actually vanish. The easy remark is that the Poisson bracket with χ1 decreases thedegree in p by one, so that, e.g., Lχ1

A = 0.


still unknown vector ξ from the equation

Cξ +B + C∂X

∂q= 0 ,

namely (7.42). This admits a solution because C is non degenerate. Hence (7.53) canbe solved, too.

We come now to determining the explicit form of the transformed HamiltonianH ′,which is tantamount to justifying (7.45)–(7.47). Since exp(L〈Y,p〉)A is the only termappearing in (7.52) which does not depend on the actions p, then (7.45) is obviouslyverified. Replacing (7.53) in (7.52) we find

〈B′, p〉 =∑

j≥2

1

j!Lj〈Y,p〉〈ω, p〉+

∑

j≥1

1

j!Lj〈Y,p〉〈B, p〉

=∑

j≥1

1

(j + 1)!Lj〈Y,p〉

(

L〈Y,p〉〈ω, p〉+ (j + 1)〈B, p〉)

;

using again (7.53) we get (7.46). Finally, (7.47) is nothing but the last line of (7.52).This concludes the calculation of the transformed Hamiltonian.

Having defined the generating functions, the canonical transformation of variablesis defined, too, and takes the form (7.50). This concludes the justification of the formalalgorithm.

7.5.2 Quantitative estimates

In this section we find quantitative estimates on all functions involved by the formalalgorithm above. To this end, we first need some analytic setting.

7.5.2.1 Analytic setting

Let me first introduce norms on functions and vectors, taking into account that allfunctions involved here are actually homogeneous polynomials of degree at most 2in the action variables p. For the sake of clarity, in the rest of the present chapter,I will reuse the symbol ‖ · ‖(,σ) to denote the weighted Fourier norm of a functionin a domain of type D(,σ) = ∆(0) × Tnσ . Such an abuse of notation slightly differswith respect to what was done in the previous sections of this chapter, but it is inagreement with the definitions introduced in section 6.7.1.

(i) For real vectors x ∈ Rn I will use the norm

|x| =n∑

j=1

|xj | .

(ii) For an analytic function f(q), where q ∈ Tn are the angle variables, I shall usethe weighted Fourier norm

‖f‖σ =∑

k∈Zn

|fk|e|k|σ ,

212 Chapter 7

where fk are the (complex) Fourier coefficients, and σ is a positive constant.Since f(q) is assumed to be analytic, there is σ such that the series abovedefining the norm converges.

(iii) For a vector–valued function w(q) = (w1(q), . . . , wn(q)) I shall use the norm

‖w‖σ =

n∑

j=1

‖wj‖σ .

I collect here some useful properties of the norms just introduced. For easy referenceI continue the numbering of the definitions.

Lemma 7.12: Consider the domain ∆(0) × Tnσ where ∆(p0) is a polydisk withradius > 0 and center p0 . Let w(q) and v(q) be analytic vector functions and C(q)be a n × n matrix with elements cjk(q) which are analytic functions. The followingproperties hold true.(iv) On a domain ∆(0)× Tnσ, with arbitrary > 0, one has

‖〈w, p〉‖(,σ) ≤ ‖w‖σ .

(v) If ‖〈w, p〉‖(,σ) ≤ D, with some positive D, then ‖w‖σ ≤ nD .(vi) If ‖〈C(q)p, p〉‖(,σ) ≤ D2, then ‖Cj,k‖σ ≤ D .(vii) For vector valued functions w(q) and v(q) one has ‖〈w, v〉‖σ ≤ ‖w‖σ‖v‖σ.(viii) For a function f(q) one has ‖f‖σ ≤ ‖f‖σ and ‖f − f‖σ ≤ ‖f‖σ.

*** Aggiungere la dimostrazione ***

The proof is an easy exercise. I just give an hint for the properties (v) and (vi),which require a straightforward modification of the proof of the inequality (6.53),lemma 6.14. Just use the Cauchy’s inequalities letting the radius of the domain inthe p variables to go to zero and keeping σ unchanged. Since the dependence on p ispolynomial, the derivatives with respect to p are the coefficients.

7.5.2.2 Iterative lemma

The main result of this section is the following iterative

Lemma 7.13: Let H(q, p) be of the form (7.39), and assume:(i) there are positive constants σ and ε such that

max(

‖A‖σ, ‖B‖σ)

≤ ε ;

(ii) there is a positive constant m ≤ 1 such that

m|x| ≤ |Cx| for all x ∈ Rn ;

(iii) for every vector valued function w(q) with bounded norm ‖w‖σ one has

‖Cw‖σ ≤ m−1‖w‖σ ;

(iv) the frequencies ω satisfy the diophantine condition

|〈k, ω〉| ≥ γ|k|−τ for 0 6= k ∈ Zn .


Let d ≤ 1/6 and σ∗ be positive constants satisfying

(1− 3d)σ ≥ σ∗ .

Then there exists a positive constant Λ = Λ(n, τ, γ, σ∗) such that the following holdstrue: if

(7.54)Λε

m6d3τ+4≤ 1

then there exists a canonical transformation (q, p) = C (q′, p′) satisfying

(7.55)|p− p′| ≤

Λε

m3d2τ+3< d ,

|q − q′| ≤Λσε

m3d2τ+1< dσ

for all (q′, p′) ∈ Tn(1−3d)σ×∆(1−3d)(0) which brings the Hamiltonian to the form (7.39)

with the same ω and with new functions A′, B′ and C′ satisfying the hypotheses (i)–(iii) with new positive constants ε′, σ′ and m′ given by

(7.56)

ε′ =Λ

m6d3τ+4ε2 ,

σ′ = (1− 3d) σ ,

m′ = (1− dτ+1)m .

The proof is deferred to sect. 7.3.4.

7.5.2.3 Lemma on small divisors

A major role in the proof of the iterative lemma is played by the control of the effectof the small divisors in the solutions of equations (7.41) and (7.43) for the generatingfunctions.

Lemma 7.14: Let ψ(q, p) be a zero average function, ψ = 0, with bounded norm‖ψ‖(1−δ)(,σ) for some non negative δ < 1, and let ω be diophantine. Let ϕ be theunique zero average solution of the equation ∂ωϕ = ψ. Then for every positive d < 1−δone has

‖ϕ‖(1−δ−d)(,σ) ≤1

γ

( τ

edσ

)τ

‖ψ‖(1−δ)(,σ) ,(7.57)

∥

∥

∥

∥

∂ϕ

∂q

∥

∥

∥

∥

(1−δ−d)(,σ)

≤1

γ

(

τ + 1

edσ

)τ+1

‖ψ‖(1−δ)(,σ)(7.58)

Proof. Recall the elementary inequality7

(7.59) xαe−βx ≤

(

α

eβ

)α

7 Just evaluate the maximum over x of the function on the left.

214 Chapter 7

for all positive α, β and x. Recall also that the solution ϕ is

ϕ = −i∑

k

ψk(p)

〈k, ω〉ei〈k,q〉 ,

where ψk(p) are the known coefficients of the Fourier expansion of ψ. Using the defi-nition of the norm, evaluate

‖ϕ‖(1−δ−d)(,σ) ≤1

γ

∑

k

|k|τ |ψk|e|k|(1−δ−d)σ

≤1

γ

( τ

edσ

)τ ∑

k

|ψk|e|k|(1−δ)σ ,

where use has been made of |k|τe−|k|dσ ≤(

τ/(edσ))τ

in view of the inequality (7.59).The first inequality then follows by the definition of the norm of ψ. As to the secondinequality, remark that we have

∂ϕ

∂qj= −

∑

k

kjψk(p)

〈k, ω〉ei〈k,q〉 .

In view of |k| = |k1|+ . . .+ |kn|, estimate∥

∥

∥

∥

∂ϕ

∂qj

∥

∥

∥

∥

(1−δ−d)(,σ)

≤1

γ

∑

k

|k|τ+1|ψk|e|k|(1−δ−d)σ .

Then (7.58) follows by using again (7.59). Q.E.D.

7.5.2.4 Proof of the iterative lemma

The proof of lemma 7.13 is just matter of a straightforward application of the estimatesgiven by lemma 7.14 and the estimates for Poisson brackets and Lie series.

Let us start with the estimates for the generating functions. By lemma 7.14 thesolution X(q) of (7.41), using hypothesis (i), satisfies

(7.60)

∥

∥

∥

∥

∂X

∂q

∥

∥

∥

∥

(1−d)σ

≤K1

dτ+1ε , K1 =

1

γ

(

τ + 1

eσ∗

)τ+1

.

Using hypotheses (i) and (iii), in (7.42) we have (recall that m, d < 1)

(7.61)

∥

∥

∥

∥

B + C∂X

∂q

∥

∥

∥

∥

(1−d)σ

≤K1 + 1

mdτ+1ε ;

therefore, using also hypothesis (ii), from (7.42) we get

m|ξ| ≤∣

∣Cξ∣

∣ ≤K1 + 1

mdτ+1ε ,

and so also

(7.62) |ξ| ≤K1 + 1

m2dτ+1ε .


By this and (7.61) we estimate B as given by (7.49). Recalling that m ≤ 1 and usingagain hypothesis (iii) we get

(7.63) ‖B‖(1−d)σ ≤2(K1 + 1)

m3dτ+1ε .

Furthermore, by solving (7.43) we get

(7.64) ‖Y ‖(1−2d)σ ≤K2

m3d2τ+1ε , K2 =

2(K1 + 1)

γ

(

τ

eσ∗

)τ

.

This concludes the estimate of the generating functions.

We come now to the estimate of the transformed Hamiltonian. From (7.48), by hy-pothesis (iii) and using (7.60) and (7.62) we get

‖A‖(1−d)σ ≤(2K1 + 1)2

m5d2τ+2ε2 .

Putting this in (7.45), and assuming the convergence condition

(7.65)4eK2

m3d2τ+3σε ≤ 1 ,

we estimate

(7.66) ‖A′‖(1−3d)σ ≤K3

m5d2τ+2ε2 , K3 =

(2e2 + 1)(2K1 + 1)2

2e2.

This requires some comment. By property (iv) at the beginning of this section andby (7.64) the norm of the generating function 〈Y, p〉 is estimated in the domainTn(1−2d)σ ×∆(1−2d)(0), with arbitrary , by

‖〈Y, p〉‖(1−2d)(,σ) ≤K2

m3d2τ+1ε .

On the other hand, by the estimates on Lie series (see lemma 6.15) we know thatunder the convergence condition (7.65), for a function ψ(q, p) analytic in Tn(1−2d)σ ×

∆(1−2d)(0) and with bounded norm ‖ψ‖(1−2d)(,σ) we have

(7.67)

∥

∥expL〈Y,p〉ψ − ψ∥

∥

(1−3d)(,σ)≤

2‖〈Y, p〉‖(1−2d)(,σ)

ed2σ‖ψ‖(1−2d)(,σ)

≤2K2ε

em3d2τ+3σ‖ψ‖(1−2d)(,σ)

≤1

2e2‖ψ‖(1−2d)(,σ) .

Using also the previous estimate for ‖A‖(1−d)σ we readily get (7.66). A similar argu-ment applies to the estimate of 〈B′, p〉 given by (7.46). Indeed, it is enough to remark

216 Chapter 7

that one has

‖〈B′, p〉‖(1−3d)(,σ) ≤∑

j≥1

j

(j + 1)!

∥

∥Lj〈Y,p〉〈B, p〉∥

∥

(1−3d)(,σ)

≤∑

j≥1

1

j!

∥

∥Lj〈Y,p〉〈B, p〉∥

∥

(1−3d)(,σ);

that is: we use the same estimate as for exp(

L〈Y,p〉

)

〈B, p〉 − 〈B, p〉, in particular,by exploiting the limitation provided by the inequality in the second row of (7.67).Using (7.63) in order to estimate B we find

(7.68) ‖〈B′, p〉‖(1−3d)(,σ) ≤K4

2nm6d3τ+4ε2 , K4 =

8n(K1 + 1)K2

eσ∗.

Hence, recalling d ≤ 1/6, and using property (v) at the beginning of this section weget

(7.69) ‖B′‖(1−3d)(,σ) ≤K4

m6d3τ+4ε2 .

We finally come to the estimate for C′. By hypothesis (iii) we have

‖〈Cp, p〉‖(,σ) ≤2

m.

Applying the argument above to (7.47), i.e., using again the upper bound provided bythe inequality in the second row of (7.67), we have

(7.70)∥

∥〈(C′ − C)p, p〉∥

∥

(1−3d)(,σ)≤

K52

4nm4d2τ+3ε , K5 =

8nK2

eσ∗.

Hence, by property (vi) at the beginning of the section, we have

‖C′j,k − Cj,k‖(1−3d)σ ≤

K5

nm4d2τ+3ε .

This allows us to estimate

(7.71)∣

∣C′x∣

∣ ≥∣

∣Cx∣

∣−∣

∣

(

C′ − C)

x∣

∣ ≥

(

m−K5

m4d2τ+3ε

)

|x| ,

provided ε satisfies the condition

(7.72) m := m−K5

m4d2τ+3ε > 0 .

On the other hand, for a vector valued function w(q) we have

(7.73)

‖C′w‖(1−3d)σ ≤ ‖Cw‖(1−3d)σ + ‖(C′ − C)w‖(1−3d)σ

≤

(

1

m+

K5

m4d2τ+3ε

)

‖w‖(1−3d)σ

<1

m‖w‖(1−3d)σ ,


in view of the definition (7.72) of m and of the elementary inequality a−1+b < (a−b)−1

for 0 < b < a < 1. This concludes the estimates for the transformed Hamiltonian.It remains to estimate the canonical transformation (7.50). Here too we proceed in twosteps, first transforming with exp

(

LX+〈ξ,q〉

)

and then transforming with exp(

L〈Y,p〉

)

.The first transformation is explicitly written as8

(7.74) q = q , p = p+ ξ +∂X

∂q

∣

∣

∣

q=q,p=p.

The second transformation is

(7.75) q = exp(

L〈Y,p〉

)

q∣

∣

∣

q=q′, p = exp

(

L〈Y,p〉

)

p∣

∣

∣

q=q′,p=p′.

The estimate is performed by using the fact that the sup norm is bounded by theweighted Fourier norm. By (7.60) and (7.62) the first transformation is estimated by

(7.76)∣

∣pj − pj∣

∣ ≤2K1 + 1

m2dτ+1ε

For the second transformation, by the general estimates on Lie series we get

(7.77)

∣

∣q′j − qj∣

∣ ≤K2

m3d2τ+1ε ,

∣

∣p′j − pj∣

∣ ≤K2

m3d2τ+3σε .

This estimates the canonical transformation, but the first of the previous inequalitiesdeserves some more comments. It can be verified starting from

∣

∣q′j − qj∣

∣ ≤∑

s≥2

1

s!

∥

∥

∥Ls−1〈Y,p〉Yj

∥

∥

∥

(1−3d)(,σ),

then the inequality in (7.64) must be used in jonction with lemma 6.15 in a similarway to what has been done to deduce (7.67).We collect now all the estimates in order to conclude the proof of the lemma. Wedefine Λ by

(7.78) Λ = max

(

2n(2K1 + 1)

,4eK2

σ∗, K3, K4, K5

)

with K1 , K2 , K3 , K4 and K5 given by (7.60), (7.64), (7.66), (7.68) and (7.70). It isseen that Λ depends only on n , τ , γ and σ∗, as claimed. By the way, the definition ofΛ shows that the convergence condition (7.65) is satisfied in view of hypothesis (7.54).By (7.66) and (7.69) one has max

(

‖A′‖σ′ , ‖B′‖σ′

)

≤ ε′, with ε′ given by the firstof (7.56). Coming to the bounds for C

′, use the convergence condition (7.54), andcheck that in (7.72) we have

K5ε

m4d2τ+3≤

Λε

m4d2τ+3≤ mdτ+1 ,

8 Since the generating function is independent of p, the coordinate q is unchanged, andthe series for exp(LX+〈ξ,q〉) reduces to the first term only.

218 Chapter 7

Hence (7.72) and (7.73) are satisfied with m replaced by m′ as given by (7.56), in viewofm′ ≤ m; this gives the bounds for C′. The estimates for the canonical transformationfollow, similarly, by using condition (7.54) and the definition of Λ. This concludes theproof of lemma 7.13.

7.5.3 Conclusion of the proof

By repeated application of the iterative lemma we construct an infinite sequence{C (k)}k≥1 of canonical transformations of the form

(

q(k−1), p(k−1))

= C (k)(

q(k), p(k))

(the upper index labeling the coordinates at the k–th step). This produces a sequence{H(k)}k≥0 of Hamiltonians, where H(0) = H is the original one, satisfying

(7.79)

max(

‖A(k)‖σk, ‖B(k)‖σk

)

≤ εk ,∣

∣C(k)v∣

∣ ≥ mk|v| for all v ∈ Rn ,

∥

∥C(k)w

∥

∥

σk≤

1

mk‖w‖σk

for all vector valued functions w(q), with sequences {εk}k≥0, {σk}k≥0 and {mk}k≥0

defined by ε0 = ε, σ0 = σ, m0 = m and

εk =Λ

m6k−1d

3τ+4k

ε2k−1 ,(7.80)

σk = (1− 3dk)σk−1 ,(7.81)

mk = (1− dτ+1k )mk−1 .(7.82)

These sequences depend on the arbitrary sequence {dk}k≥1. In turn, the latter se-quence must be chosen so that for every k ≥ 1 the conditions dk ≤ 1/6 and

Λεk−1

m6k−1d

3τ+4k

≤ 1 ,(7.83)

(1− 3dk)σk−1 ≥ σ∗ > 0 ,(7.84)

(1− dτ+1k )mk−1 ≥ m∗ > 0(7.85)

are satisfied, with some constants σ∗ and m∗. The canonical transformations satisfy

(7.86)

|p(k) − p(k−1)| ≤Λ

m3k−1d

2τ+3k

εk−1 < dk ,

|q(k) − q(k−1)| ≤Λσk−1

m3k−1d

2τ+1k

εk−1 < dkσk−1 .

The problem now is to show that the sequence {dk}k≥1 can be determined sothat (7.83), (7.84) and (7.85) are satisfied for every k ≥ 1, the sequence {εk}k≥0 con-verges to zero, and moreover that the sequence of canonical transformations convergesto an analytic canonical transformation putting the Hamiltonian in Kolmogorov’s nor-mal form.


The key point is to look at (7.80) as a relation that can be used in order todetermine dk if εk−1 and εk are known, since

dk =

(

Λε2k−1

m6k−1εk

)1/(3τ+4)

.

Therefore, we can regard at {εk}k≥0 as being the arbitrary sequence. Let us make thechoice9

(7.87) εk =ε0

(k + 1)2(3τ+4),

and suppose for a moment that the condition mk ≥ m∗ > 0 is satisfied. Then we get

dk <4

k2

(

Λε0m6

∗

)1/(3τ+4)

,

and so also10

(7.88)∑

k≥1

dk <2π2

3

(

Λε0m6

∗

)1/(3τ+4)

.

Here we use the condition that ε ≡ ε0 be small enough, making the condition quanti-tative by asking

(7.89) 4π2

(

Λε0m6

∗

)1/(3τ+4)

≤ 1 .

This immediately gives

(7.90)∑

k≥1

dk <1

6,

which implies in particular dk < 1/6 for all k ≥ 1. We prove now that (7.84) and (7.85)are satisfied. Starting with (7.84), write

ln∏

k≥1

(1− 3dk) =∑

k≥1

ln(1− 3dk) .

Using the elementary inequality

0 ≥ ln(1− x) ≥ −2x ln 2 for 0 ≤ x ≤ 1/2 ,

evaluate0 ≥

∑

k≥1

ln(1− 3dk) ≥ −6 ln 2∑

k≥1

dk ≥ − ln 2 .

9 This choice is rather arbitrary. The conditions to be fulfilled are εk → 0 for k → +∞ , and∑

kdk ≤ 1/6. It is possible, of course, to make the choice εk ∼ C−k with any constant

C > 1. Some authors prefer to keep the fast convergence by asking, e.g., εk = ε3/2k−1.

10 Recall that∑

k≥1k−2 = π2/6 .

220 Chapter 7

Therefore,∏

k≥1(1− 3dk) ≥ 1/2, so that (7.84) is fulfilled with, e.g.,

(7.91) σ∗ =σ

2.

By the same argument, in view of dτ+1k < dk, we evaluate

∏

k≥1(1− dk) ≥ 1/21/3, sothat (7.85) is fulfilled with, e.g.,

(7.92) m∗ =m0

21/3.

As to condition (7.83), it is clearly satisfied by the choice (7.87) of the sequence εk . Infact, by comparing (7.83) with (7.80), one immediately realizes that it follows fromthe condition that εk ≤ εk−1 , which is clearly fulfilled because of (7.87). Thus, we areleft only with the condition (7.89) on the smallness of ε0 .

It remains to prove that the canonical transformation is well defined on somedomain. To this end, having fixed a positive initial value of the analyticity radius inthe actions 0 = , consider the sequence of domains {Tnσk

×∆k(0)}k≥0 , with σk asin (7.81) and k = (1− 3dk)k−1 . Then the canonical transformation

C (k) : Tnσk×∆k(0) → T

nσk−1

×∆k−1(0)

(q(k), p(k)) 7→ (q(k−1), p(k−1)) = C (k)(q(k), p(k))

is analytic. Therefore, by composition, the transformation

C (k) : Tnσk

×∆k(0) → Tnσ0

×∆0(0)

defined as

C (k) = C (k) ◦ · · · ◦ C (1)

is canonical and analytic. On the other hand, by (7.86) we have

∣

∣q(k) − q(0)∣

∣ <∣

∣q(k) − q(k−1)∣

∣+ . . .+∣

∣q(1) − q(0)∣

∣< σ

k∑

j=1

dj ,

∣

∣p(k) − p(0)∣

∣ <∣

∣p(k) − p(k−1)∣

∣+ . . .+∣

∣p(1) − p(0)∣

∣< k

∑

j=1

dj .

Since∑

j≥1 dj is convergent, the sequence C k converges absolutely to a transformation

C (∞) : Tnσ∗

×∆∗(0) → Tnσ0

×∆0(0)

with, e.g., ∗ = 0/2 in view of (7.91). The absolute convergence implies the uniformconvergence in any compact subset of Tnσ∗

×∆∗(0), and so, by Weierstrass theorem,

C (∞) is also analytic. Finally, denoting by (q(∞), p(∞)) the canonical coordinates in


Tnσ∗×∆∗(0), and combining (7.86) with (7.80), (7.83), (7.87) and (7.88), we have

(7.93)

∣

∣q(∞) − q(0)∣

∣ <2π2

3

(

Λε0m6

∗

)1/(3τ+4)

σ∗ ,

∣

∣p(∞) − p(0)∣

∣ <2π2

3

(

Λε0m6

∗

)1/(3τ+4)

∗ ,

so that it is the identity for ε = 0. By the properties of the Lie series transformationone also has H(k) = H(0) ◦ C (k), so that the sequence H(k) converges to an analyticfunction H(∞) which by construction is in normal form. This concludes the proof ofthe theorem.11

11 The whole proof can now be summarized so to rephrase the statement of theorem 7.11in a more detailed way, from a quantitative point of view, as follows. Let us consider asystem with n degrees of freedom that is described by the quadratic Hamiltonian (7.39).Assume that such a Hamiltonian is analytic on T

nσ ×∆(0) and the conditions (i)–(iv)

of lemma 7.13 are satisfied. Therefore, for all

ε < ε∗ = m6∗/[Λ(4π

2)3τ+4]

the Hamiltonian possesses an invariant torus such that the flow on that torus is quasi pe-riodic with frequencies ω. Moreover, the difference between the original canonical coordi-nates and the normalized ones (such that the invariant torus is corresponding to the zerovalue of the actions vector) are estimated in formula (7.93), where m∗ is given in (7.92),σ∗ = σ/2, ∗ = /2 and Λ can be explicitly calculated using the definitions (7.78), (7.60),(7.64), (7.66), (7.68) and (7.70).

222 Chapter 7

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7 PERSISTENCE TORI - unimi.it · We shall say that the Hamiltonian (7.2) is in Kolmogorov’s normal form. The suggestion of Kolmogorov is to reduce a Hamiltonian to the normal form