**4.2** **Elimination of the angles: first step**

The canonical transformation eliminating the angles to first order in the perturbation can be looked for in the form of the flow of a certain Hamiltonian. More precisely we set

C_{ε}^{−1}=Φ^{ε}_{G}_{1}=Φ^{1}_{εG}_{1} ^{,} ^{(4.7)}

where G1 is an unknown Hamiltonian. By defining the Lie derivative

L_{1}:= { ,G1} , (4.8)

and recalling formula (1.21) of Proposition 1.5, one gets
H ◦C_{ε}^{−1} = H ◦Φ^{ε}_{G}_{1}= e^{εL}^{1}H =

*= (1 + εL*1*+ ε*^{2}*. . . )(h + εP*1*+ ε*^{2}. . . ) =

*= h + ε(L*1h + P1*) + ε*^{2}. . . .

*Requiring that the last row above be equal to h + εZ*1*+ ε*^{2}, one has to solve the so-called
homological equation of perturbation theory, namely

L_{1}h + P1= Z1, (4.9)

with unknowns G_{1}and Z_{1}, and the requirement that the latter quantity be independent of the
angles. Since h depends only on the actions, equation (4.9) explicitly reads

*ω(J) ·∂G*1(*θ, J)*

*∂θ* = P1(*θ, J) − Z*1(J), (4.10)

where the frequency vector

*ω(J) :=∂h(J)*

*∂J* (4.11)

has been defined. The average of equation (4.10) on the torusT^{n}yields
Z_{1}(J) = 〈P1〉 := 1

(2*π)*^{n}
Z

T^{n}P_{1}(*θ, J) d*^{n}*θ .* (4.12)
Thus, a first result is that if the problem admits a solution, then the angles to first order are
removed by averaging the perturbation over the torus. The homological equation is then easily
solved by means of the Fourier transform. We recall that any smooth function f :T^{n}→ R
admits the Fourier series expansion

f (*θ) =* X

k∈Z^{n}

fˆ_{k}e* ^{ik·θ}* , (4.13)

where the Fourier coefficient ˆf_{k}is given by
fˆ_{k}= 1

(2*π)*^{n}
Z

T^{n}e* ^{−ik·θ}*f (

*θ) d*

^{n}

*θ .*(4.14)

We recall that the velocity of convergence of the Fourier series, i.e. the rate of decay of the
Fourier coefficients ˆf_{k} with |k| is linked to the regularity of the function f . More precisely,
one has that if f ∈ C^{s}(T^{n}), then there exists a constant A_{s}> 0 such that | ˆfk| ≤ As/|k|_{1}^{s}, where

|k|1:=Pn

j=1|kj*|; moreover, if there exists ρ > 0 such that f is analytic in the complex extension*
of the torusT^{n}* _{ρ}* := {z ∈ C

^{n}: Re(z

_{j}) ∈ T ; |Im(zj

*)| ≤ ρ ; j = 1,..., n}, then there exists a constant*B

_{s}> 0 such that | ˆfk| ≤ Bse

^{−|k|}

^{1}

*. A proof of such statements is reported in Appendix B. Here we make use of the Fourier series at a formal level, in order to solve the homological equation (4.10). Indeed, by inserting the Fourier series expansions*

^{ρ}G_{1}(*θ, J) =* X

k∈Z^{n}

Gb_{1,k}(J)e* ^{ik·θ}* (4.15)

and

P_{1}(*θ, J) =* X

k∈Z^{n}

Pb_{1,k}(J)e* ^{ik·θ}* (4.16)

into (4.10), and observing that

Pb_{1,0}(J) = 〈P1〉 (J) = Z1(J),
one gets

*ik · ω(J) b*G_{1,k}(J) = bP_{1,k}*(J) − δ*k,0Pb_{1,0}(J), (4.17)
to be solved for any k ∈ Z^{n}and any J ∈ B. The formal solution of (4.17), for k 6= 0, reads

Gb_{1,k}(J) = Pb_{1,k}(J)

*ik · ω(J)* , (4.18)

and immediately points out the major problem of Hamiltonian perturbation theory, i.e.
*deal-ing with the possible small divisors k · ω(J). Notice that resonant frequencies, namely those*
frequency vectors*ω(J) such that there exists k ∈ Z*^{n}\ {*0} satisfying k · ω(J) = 0, are just the*
worst possibility. Indeed, one has to recall that in order to get the homological equation one
has expanded the operator e^{εL}^{1}= e^{ε{·,G}^{1}^{}}. The latter expansion is meaningful if, roughly
speak-ing G_{1}is of order one (with respect to the small parameter*ε). In particular, any of its Fourier*
coefficients must be of order one, otherwise its contribution to the expansion would cause
*prob-lems. Obviously, a denominator k · ω(J) = O(ε** ^{α}*), with

*α ≥ 1, makes the perturbative expansion*meaningless. At a deeper level than the one considered here, one finds out that, in general,

*dangerous small denominators are those satisfying k · ω(J) = O(*p

*ε).*

The problem of small divisors immediately calls for a partition of the integrable systems to be perturbed into the following two classes:

*• isochronous systems, with h(J) = ω*0· J, i.e. a constant (independent of J) frequency map
*ω(J) = ∂h/∂J = ω*0;

• non-isochronous systems, with h(J) a nonlinear function of J, i.e. a non constant
fre-quency map*ω(J).*

4.2. ELIMINATION OF THE ANGLES: FIRST STEP 71
Of course, small divisors in isochronous systems are easily treated, at least in principle.
In-deed, either *ω*0 is resonant or is not. In the latter case equation (4.17) admits the formal
solution (4.18), and one is left with the problem of convergence of the series (4.15). In the
for-mer case, one has to characterize the resonance module M_{ω}_{0}:= {k ∈ Z^{n}*: k · ω*0= 0}. Obviously,
for those vectors k ∈ M*ω*0 the ratio on the right hand side of (4.18) is not defined, unless the
numerator Pb_{1,k}(J) = 0 for any J, which, though not generically, may happen in practice. In
such a resonant, isochronous case, one cannot simply remove all the angles, but only part of
them. More precisely, in this case the transformed/simplified Hamiltonian to first order reads

H(Φ_{G}^{ε}_{1}^{(}*θ, J)) = h(J) + ε* X
with arbitrary Fourier coefficients g_{k}(J) on the resonance modulus.

**Exercise 4.1. Prove the above statement, showing that (4.20) solves the homological equation**
(4.10) withZ_{1}(*θ, J) as defined in (4.19) in place of Z*1(J).

The case of non-isochronous systems is much more complicated, since in that case the small-ness of the denominators on the right hand side of (4.17) varies in the space of the actions, which calls for a major role played by the geometry of the resonances in the problem. Actually, due to the action dependence of the small divisors, non-isochronous systems must be studied by further partitioning them into classes defined by the analytical/geometrical properties of h(J), which in turn establishes precise properties of the frequency map

*ω : B → ω(B) : J 7→ ω(J) =∂h(J)*

*∂J* . (4.21)

By far the easiest class of integrable non-isochronous systems to be studied is the following.

**Definition 4.2. The integrable system defined by h(J), is said to be non-degenerate in B (open**
set in the domain ofh) if there exists a constant c > 0 such that

¯

In such a case the frequency map (4.21) is a local diffeomorphism (i.e. a smooth bijection), so that to any frequency vector having a certain property there corresponds one and only one action. As a consequence, any subset with given metric properties in the frequency space is the image of a subset with the same metric properties in the action space.

**Example 4.4. A dense, Lebesgue-measure zero set**Ω*⊂ ω(B) in the frequency space is the image*
of a dense, Lebesgue-measure zero set*A in the action space. Indeed, consider the set A := ω*^{−1}(Ω^{).}

The smoothness (actually continuity is enough) of the map *ω*^{−1} implies that to close frequency

values there correspond close action values, which implies that*ω*^{−1}preserves density. For what
concerns the measure, observe that

0 = V (Ω) = Z

Ωd^{n}*ω =*
Z

A

¯

¯

¯

¯

*∂ω*

*∂J*

¯

¯

¯

¯

d^{n}J ≥ cV (A) ,
where the non-degeneracy condition (4.22) has been used.

In particular, for non-degenerate systems, one is interested in the sets

Ωr*:= {ω ∈ R*^{n}: dim M* _{ω}*= r} (r = 1,..., n − 1) , (4.23)
namely the sets of resonant frequencies whose resonance modulus is generated by r linearly
independent integer vectors. The setsΩrare denumerable, dense inR

^{n}and have zero relative Lebesgue measure (i.e. for any bounded, measurable set A ⊂ R

^{n}one has V (Ωr∩ A) = 0). The set

*ω(B) \ S*

^{n−1}

_{r=1}Ωrof non resonant frequencies is dense too and of full measure.

**Example 4.5. In the case n = 2 there is only**Ω1, the set of frequency vectors *ω ∈ R*^{2} such that
there exists an integer vectork ∈ Z^{2}satisfyingk_{1}*ω*1+k2*ω*2= 0. In other words,Ω1consists of the
two coordinate axes and of all the straight lines with rational slope. Ω1is denumerable, dense
and has relative measure zero because the rational numbers have the same properties.

**Example 4.6. In the case n = 3,**Ω1is the set of planes orthogonal to integer vectors, whereasΩ2

is the set of straight lines resulting from the intersection of two planes ofΩ1that are orthogonal to two non parallel integer vectors.

By the non-degeneracy condition (4.22) the sets of resonant actions

B_{r}*:= ω*^{−1}(Ωr*∩ ω(B)) = {J ∈ B : ω(J) ∈*Ωr} (4.24)
are denumerable, dense and of zero relative measure; the set of non resonant actions B \
S_{n−1}

r=1B_{r}is dense too and of full measure. The existence of the sets (4.24) in the space of actions
implies that the Fourier coefficients (4.18) are not defined on dense sets. In order to get a sharp
statement of this kind one needs to control the possibility to have zero numerators on the right
hand side of (4.18).

**Definition 4.3. The first order perturbation P**_{1}(*θ, J) is said to be generic in B if for any J ∈ B*
and anyk ∈ Z^{n}there exists q parallel to k such thatPb_{1,q}(J) 6= 0.

The following theorem is due to Poincaré.

**Theorem 4.2 (Unfeasibility of the first step). Suppose that h(J) is non-degenerate and P**_{1}(*θ, J)*
is generic in B. Then, the homological equation (4.17) does not admit solutions on any open
subset ofB.

C PROOF. For any given J ∈ B \S_{n−1}

r=1B_{r} and k ∈ Z^{n}\ {0} equation (4.17) can be solved as in
(4.18). On the other hand, by the density of the sets B_{r}in B, arbitrarily close to such a J
there exists ¯J ∈S_{n−1}

r=1B_{r}, such that ¯*k ·ω( ¯*J) = 0 for at least one ¯k ∈ Z^{n}\ {0}. Moreover, since
the perturbation P_{1} is generic in B, there exists q parallel to ¯k such that P_{1,q}( ¯J) 6= 0 and
*obviously, q · ω( ¯*J) = 0. Thus, equation (4.17) cannot be solved in the dense setS_{n−1}

r=1B_{r}. B