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(0)
is independent of H 0 . Indeed, if I (0)
depends on H 0 , i.e. I (0)
= g 0 (H 0 ), (12.74) then if I 0 is a first integral I 0 − g
0 (H) is a first integral too. Moreover from equations (12.73) and (12.74) it follows that I 0 − g 0 (H) = I (0) 0 + εI (1) 0 + O(ε 2 ) − g 0 (H 0 + εF )
= I (0)
0 + εI
(1) 0 − g 0 (H 0 ) − εg
0 (H 0 )F + O(ε
2 ) = ε[I (1) 0 − g 0 (H 0 )F ] + O(ε
2 ); hence setting I 1 = I 0 − g 0 (H)
ε , (12.75) 514 Analytical mechanics: canonical perturbation theory 12.4 I
is a new analytic first integral that is independent of H (because by hypothesis I 0 cannot be expressed as a function of H) and I 1 = I (0)
1 + εI
(1) 1 + ε 2 I (2) 1 + · · · (12.76) is its expansion in powers of ε. The coefficients I (j) 1
the coefficients I (k)
0 and from the Taylor series expansion of g 0 (H):
∞ j =0 I (j)
1 ε j = 1 ε I (0)
0 + ∞ k =1 I (k) 0 ε k − g
0 (H 0 + εF ) = 1 ε I (0) 0 + ∞ k =1 I (k) 0 ε k − g
0 (H 0 ) − ∞ k =1 ε k k! g (k) 0 (H 0 )F k = ∞ k =1 I (k) 0 − 1 k! g (k) 0 (H 0 )F k ε k −1 , where g (k)
0 is the kth-order derivative of g 0 . Therefore we have I (j)
1 = I
(j+1) 0 − 1 (j + 1)!
g (j+1)
0 (H 0 )(H − H
0 ) j +1 , (12.77) for every j ≥ 0. We can again ask if I (0) 1
0 or not. In the first case, the proof is finished: I = I 1 . If however I (0)
1 = g
1 (H 0 ), by repeating the previous argument, setting I 2
I 1 − g 1 (H)
ε , I 2 is a new analytic first integral that is independent of H. If I (0) 2
again on H 0 , we must iterate this procedure. But after a finite number n of iterations we necessarily obtain an integral I n for which I (0) n does not depend on H 0 . Indeed, if otherwise I (0) n = g n (H 0 ) for every n ≥ 0, since (cf. (12.77)) g n
0 ) = I
(0) n = I (1) n −1 − g n −1 (H 0 )(H − H 0 ) = I (2)
n −2 − 1 2! g n −2 (H 0 )(H
− H 0 ) 2 − g
n −1 (H 0 )(H
− H 0 ) = . . . = I (n)
0 − n −1 k =0 1 (n − k)! g (n−k)
k (H 0 )(H − H
0 ) n −k ,
12.4 Analytical mechanics: canonical perturbation theory 515 we would find I (n)
0 = g
n (H 0 ) + n −1 k =0 1 (n − k)!
g (n−k)
k (H 0 )(H − H
0 ) n −k , for every n, and hence I 0 would depend on H, contradicting the hypothesis. We now prove the theorem of Poincar´ e. Proof of Theorem 12.8 Let I be an analytic first integral of the motion. By Proposition 12.3, I (0)
is only a function of the action variables. Expanding in Fourier series I (1) and F
in equation (12.71) for n = 1 we therefore find the equation im · ω(J)ˆI (1) m (J) = i(m · ∇ J I (0) (J)) ˆ
F m (J), for every m ∈ Z
l . Hence, for ˆ I (1)
m (J) to be well defined, m · ω(J) must vanish for every value of J annihilating the right-hand side. By the hypothesis of genericity of the Fourier series expansion of F , there is no loss of generality in assuming that ˆ
F m (J) = / 0 (otherwise, there certainly exists a vector m parallel to m for which ˆ
F m (J) = / 0; but m is parallel to m only if there exists an integer k such that m = km). Therefore m · ∇ J
(0) (J) must vanish every time that m · ω(J) is zero (and vice versa). For a fixed resonant vector ω ∈
Ω r , consider the associated resonance module M ω (see Definition 11.7). The condition that m · ∇ J I (0) (J) and m · ω(J) are both zero is equivalent to imposing that ω and ∇ J
(0) are both orthogonal to M ω
M ω is equal to l − 1, the orthogonal complement of M ω has dimension 1 and ω and ∇ J
(0) are parallel. By the non-degeneracy hypothesis, the correspondence between ω and J is bijective and continuous, and hence the set A r of the values of J ∈ A corresponding to resonant frequencies ω(J) is dense. 1 It follows that ω(J) = ∇ J H 0 (J) and
∇ J I (0) (J) must be parallel as J varies in a dense set in R l . By continuity there must then exist a scalar function α(J) such that ∇ J H 0 (J) = α(J) ∇ J I (0) (J),
for every J ∈ R
l . Hence there exists a function A : R → R such that α(J) = (A (H
0 (J)))
−1 and I
(0) (J) = A(H 0 (J)). By Lemma 4.1 the integral I is then a function of H. The ‘negative’ results proved in this section apparently leave only two possib- ilities of establishing the existence of a regular solution of equation (12.13): (a) consider only degenerate Hamiltonian systems—for example systems that are linear in the action variables, as in the case of harmonic oscillators; (b) admit that the solution does not have a regular dependence on the actions. 1 It is not difficult to check that the subset of Ω r made of the vectors ω whose resonance module
M ω has dimension l − 1 is dense in R l : it is enough to observe that dim M ω = l − 1 if and only if there exists ν ∈ R and m ∈ Z l , such that ω = νm. 516 Analytical mechanics: canonical perturbation theory 12.5 Both cases are possible and lead to physically significant results. In the next section, we examine briefly the first possibility, then we survey the important developments related to the second. 12.5 Birkhoff series: perturbations of harmonic oscillators While not eternity, this is a considerable slice of it. 2 In the previous section we showed (Theorem 12.6) that it is not possible to find a regular solution of the fundamental equation of the canonical theory of perturbations for non-degenerate Hamiltonians. In this section we restrict our analysis only to degenerate quasi-integrable Hamiltonian systems: H(J,
χ, ε) = ω · J + εF (J, χ). (12.78)
In this case, the frequencies ω are fixed constants that are independent of the actions. The condition of non-resonance for the frequencies does not imply any restriction on the action variables, as opposed to what happens in the case of non-degenerate systems, and Theorem 12.7 ensures the existence of a formal solution of equation (12.13) for all J ∈ A and χ ∈ T l . We shall indeed show that it is possible to prove a result analogous to Theorem 12.4: if the frequen- cies satisfy a diophantine condition, the formal solution (12.63) gives rise to a convergent Fourier series and the fundamental equation of the canonical theory of perturbations admits a regular solution for J ∈ A and χ ∈ T l . D efinition 12.7 Fix l > 1. A vector ω ∈ R l satisfies a diophantine condition (of constant γ > 0 and exponent µ ≥ l − 1), and we write ω ∈ C γ,µ , if for every m ∈ Z
l , m =
/ 0, we have |m · ω| ≥ γ|m| −µ ,
where |m| = |m
1 | + · · · + |m l |.
It is not difficult to show, generalising Theorem 12.3 (of Dirichlet), that the condition µ ≥ l − 1 is necessary, and hence that if µ < l − 1 there does not exist a vector ω ∈ R l
/ 0. In addition, it can be proved—using an argument slightly more sophisticated than the one used in Section 12.3—that for every fixed µ > l − 1 the Lebesgue measure of C γ,µ ∩ [0, 1]
l satisfies the inequality |C γ,µ
∩ [0, 1] l | ≥ 1 − aγζ(µ + 2 − l) > 0, (12.80) where a is a constant depending only on l. Note that if l = 2 we again find (12.50) (and a = 4). Hence for almost every ω ∈ [0, 1] l there exists γ > 0 such that ω ∈ C
γ,µ , for fixed µ > l − 1. 2
12.5 Analytical mechanics: canonical perturbation theory 517 Remark 12.8 It is possible to prove the following generalisation of the theorem of Liouville referred to in Remark 12.5. Suppose that (ω 1 , . . . , ω l ) is a basis on Q of a field of algebraic real numbers. Then ω = (ω
1 , . . . , ω l ) satisfies the diophantine condition (12.79) with µ = l −1 (see Meyer 1972, proposition 2, p. 16). Hence, for example, (1, √
√ 3, √ 6) and (1, 2 1/3
, 2 2/3
) satisfy equation (12.79) with µ = l − 1 and
l = 4, l = 3, respectively. There also exists a generalisation of the theorem of Roth known as the subspace theorem, see Schmidt (1991). T heorem 12.9 Consider ω ∈ C γ,µ , and let A be an open subset of R l and
F : A × T
l → R, F = F (J, χ), a function of class C ∞ . The Fourier series W (1)
(J , χ) =
m∈Z l m=0 − ˆ F m (J )
im · ω
e im·χ
(12.81) converges uniformly for (J , χ) ∈ K × T l , where K is any compact subset of A. Proof The proof is analogous to that of Theorem 12.4. Indeed, exploiting the diophantine condition on ω, we find m∈Z l
− ˆ F m (J )
im · ω
e im·χ
≤ m∈Z
l m=0
ˆ F m γ |m|
µ , where ˆ F m = max J ∈K
| ˆ F m (J ) |. Since F is of class C ∞ , for any r > µ+l, there exists a constant M > 0 depending only on r and K such that ˆ F m ≤ M|m|
−r (see Appendix 7), and therefore m∈Z l
ˆ F m γ |m|
µ ≤ M γ m∈Z
l m=0
|m| µ −r < + ∞. Consider now the Hamiltonian systems h(p, q) = l j =1 p 2 j 2 + ω 2 j q 2 j 2 + ∞ r =3 f r (p, q),
(12.82) where f
r is a homogeneous trigonometric polynomial of degree r in the variables (q, p). The Hamiltonian (12.82) represents the perturbation of a system of l
518 Analytical mechanics: canonical perturbation theory 12.5 harmonic oscillators. In this problem the perturbation parameter ε does not appear explicitly, as in (12.4), but an analogous role is played by the distance in phase space from the linearly stable equilibrium position corresponding to the origin (p, q) = (0, 0). Indeed, consider the set B ε = J ∈ R l |J i = p 2 i + ω
2 i q 2 i 2ω i < ε, for every i = 1, . . . , l , (12.83) and suppose for simplicity that the sum in (12.82) is extended only to odd indices r (this assumption guarantees that only integer powers of ε will appear in the series expansion (12.84)). Then if we perform a change of scale of the actions J → J/ε such that B ε is transformed to B 1 , and a change of time scale t → εt and of the Hamiltonian H → H/ε (recall that t and H are canonically conjugate variables, see Remark 10.21), we find H(J, χ) = ω · J + ∞ r =1 ε r F r (J,
χ), (12.84)
where we have introduced the action-angle variables (J, χ) of the unper- turbed harmonic oscillators, and the functions F r are homogeneous trigonometric polynomials of degree 2(r + 1): F r (J, χ) =
m∈Z l |m|=2(r+1) ˆ F (r) m (J)e
im·χ . (12.85) Suppose that the frequency ω is not resonant (hence that the condition (12.60) is satisfied). In the series expansion in powers of ε of the perturbation, the corresponding term F r has to all orders a finite number of Fourier components. We now show how it is possible, at least formally, to construct the series of the canonical theory of perturbations, to all orders ε r , r
≥ 1. Denote by W the generating function of the canonical transformation near the identity that transforms the Hamiltonian (12.84) into a new Hamiltonian H , depending only on the new action variables J : H (J , ε) = ∞ r =0 ε r H r (J ). (12.86) Expanding W = W (J , χ, ε) in a series of powers of ε: W (J ,
χ, ε) = J · χ + ∞ r =1 ε r W (r)
(J , χ),
(12.87) and substituting the transformation induced by equation (12.87) J = J + ∞
=1 ε r ∇ χ W (r) (J ,
χ) (12.88)
12.5 Analytical mechanics: canonical perturbation theory 519 into the Hamilton–Jacobi equation for the Hamiltonian (12.87), we find the equation ω · J +
∞ r =1 ε r ∇ χ W (r) + ∞ r =1 ε r F r J + ∞ r =1 ε r ∇ χ W (r) , χ =
∞ r =0 ε r H r (J ).
(12.89) Expanding in Taylor series the second term: F r
∞ r =1 ε r ∇ χ W (r) , χ = F
r + ∇ J F r · ∞ r =1 ε r ∇ χ W (r) + · · · + 1 k! l m 1 ,...,m k =1 ∂ k F r ∂J m 1 . . . ∂J
m k ∞ n =k ε n × j 1 +···+j
k =n ∂W (j 1 ) ∂χ m1 · · · ∂W (j k ) ∂χ mk + · · · ,
(12.90) equation (12.89) can be written as ( ω · J − H 0 ) + ε(
ω · ∇ χ W (1) + F
1 − H
1 ) + ε 2 ( ω · ∇ χ W (2) + F 2 + ∇ J F 1 · ∇
χ W (1) − H 2 ) + · · · + ε
r ( ω · ∇ χ W (r) + F r + ∇ J F r −1 · ∇ χ W (1) + · · · − H r ) +
· · · = 0. (12.91)
To all orders in ε we must solve the fundamental equation of the theory of perturbations: ω · ∇ χ
(r) (J ,
χ) + F (r)
(J , χ) = H
r (J ),
(12.92) where
F (r)
= F r + r −1 n =1 n k =1 1 k! l m 1 ,...,m k =1 ∂ k F r −n ∂J m 1 · · · ∂J m k j 1 +···+j
k =n ∂W (j 1 ) ∂χ m1 · · · ∂W (j k ) ∂χ mk (12.93) depends only on F 1 , . . . , F r and on W
(1) , . . . , W (r−1) . Therefore H r is determined by the average of F (r) on T l : H r (J ) = ˆ F (r)
0 (J ),
(12.94) while W
(r) is a homogenous trigonometric polynomial of degree 2(r + 1). If the series (12.86) and (12.87), called Birkhoff series, converge for |ε| < ε
0 in the domain A × T l
l , the Hamiltonian (12.80) would be completely canonically integrable. Indeed, we would have a perturb- ative solution of the Hamilton–Jacobi equation (12.89); W would generate a |
