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Analytical mechanics: canonical perturbation theory 12.5 completely canonical transformation transforming (12.82) to a Hamiltonian that is independent of the new angle variables. In general the series (12.86) and (12.87) diverge, and hence perturbations of harmonic oscillators do not give rise to integrable problems. The divergence of the Birkhoff series can be easily illustrated by an example, as is shown in Problem 7 of Section 12.8. In addition, there holds a theorem analogous to Theorem 12.7 (see Siegel 1941, 1954), which we simply state. Consider the set H of the Hamiltonians h : R 2l → R which are analytic and of the form (12.82). We can associate with every Hamiltonian h its power series expansion h(p, q) = k,n∈N
l h k,n p k 1 1 . . . p
k l l q n 1 1 . . . q
n l l . (12.95)
Comparing with (12.82) it follows that for every r ≥ 3 we have f r
|k|+|n|=r h k,n p k 1 1 . . . p
k l l q n 1 1 . . . q
n l l , (12.96)
where |k| = k
1 + · · · + k l . D efinition 12.8 Let h ∗ ∈ H. A neighbourhood of h ∗ in H is given by the set of all Hamiltonians h ∈ H such that for every k, n ∈ N l we have
|h k,n
− h ∗ k,n | < ε k,n
, (12.97)
where {ε k,n } k,n∈N
l is an arbitrary fixed sequence of positive numbers such that ε k,n
→ 0 for |k| + |n| → ∞. Two Hamiltonians are therefore close if all the coefficients of the corresponding power series expansions are close. T heorem 12.10 (Siegel) In every neighbourhood of a Hamiltonian h ∗ ∈ H there exists a Hamiltonian h such that the corresponding flow does not admit a first integral of the motion which is analytic and independent of h. Systems which are not (completely canonically) integrable are therefore dense in H, and hence the set of Hamiltonians for which the Birkhoff series diverge is also dense. Remark 12.9 Siegel’s theorem also shows how, in general, the Hamilton–Jacobi equation does not admit a complete integral near a point of linearly stable equilib- rium (see Remark 11.2). Indeed, moving the equilibrium point into the origin, the Hamiltonian has the form (12.82), and therefore it belongs to H. If the Hamilton–Jacobi equation admitted a complete integral, the system would have l first integrals of the motion, independent of h. By Theorem 12.10 this is not the case for any h in a dense subset of H. Birkhoff series, although divergent, are very important in practice, for the qualitative study of degenerate Hamiltonian systems, and for the study of the 12.5 Analytical mechanics: canonical perturbation theory 521 stability of the Hamiltonian flow for finite but long time. Indeed, there holds the following remarkable result (see Nekhoroshev 1977, Gallavotti 1984). T heorem 12.11 Consider a Hamiltonian quasi-integrable system, degenerate and of the form (12.78), and assume that: (1) the Hamiltonian (12.78) is analytic with respect to J, χ and ε for |ε| ≤ 1; (2) the frequency vector ω satisfies a diophantine condition (12.79). Then there exist two constants ε 0 > 0 and ρ 0 > 0 and a completely canonical transformation, analytic and near the identity: J = J + εA(J , χ , ε), χ = χ + εB(J , χ , ε), (12.98) defined for |ε| ≤ ε 0
J ≤ ρ
0 , such that the transformed Hamiltonian H (J , χ , ε) is of the form H (J , χ , ε) = ω · J + εK (J , ε) + ε ε
exp −(l + 3)
ε ε 0 1/(l+3) R(J ,
χ , ε), (12.99)
where K and R are analytic functions of their arguments and K (J , 0) = 0, R(J , χ , 0) = 0. An interesting consequence is the following. C orollary 12.1 There exist two constants C 1 > 0 and C 2 > 0 such that if (J(t), χ(t)) is the solution of Hamilton’s equations for the Hamiltonian (12.78) with initial data (J(0), χ(0)), for every time t such that |t| ≤ C 1
ε ε 0 1/(l+3) , (12.100) we have |J(t) − J(0)| ≤ C 2 ε
0 . (12.101) Proof (sketch) From equation (12.99) it follows that ˙J = − ε
0 exp
−(l + 3) ε ε 0 1/(l+3)
∇ χ R(J , χ , ε), (12.102)
and therefore, if t is chosen as in (12.100), then |J (t) − J (0)| ≤ C 3 ε
0 , (12.103) 522 Analytical mechanics: canonical perturbation theory 12.6 where J (0) is the initial condition corresponding to J(0), C 3 = max |∇ χ R(J , χ , ε) |, (12.104)
and the maximum is taken as J varies on the sphere of radius ρ 0 , while χ ∈ T l and ε ∈ [−ε 0 , ε 0 ]. The inequality (12.101) follows from the remark that the canonical transformation (12.98) is near the identity and from the inequality |J(t) − J(0)| ≤ |J(t) − J (t)| + |J (t) − J (0)| + |J (0) − J(0)|. (12.105) It is not difficult to convince oneself, by a careful inspection of (12.100) as the ratio ε/ε 0 varies in [ −1, 1], that the order of magnitude of the time over which the previous corollary ensures the validity of (12.101) can be very large. As an example, in the applications to celestial mechanics (see Giorgilli et al. 1989) one can obtain stability results for the restricted three-body problem for times of the order of billions of years, and hence comparable with the age of the Solar System.
Littlewood (1959a,b), who first thought of a ‘rigorous’ application of Birkhoff series to the three-body problem, wrote that, ‘while not eternity, this is a considerable slice of it.’ 12.6
The Kolmogorov–Arnol’d–Moser theorem In Section 12.4 we saw that, under fairly general hypotheses, the fundamental equation of perturbation theory does not admit regular solutions. In Section 12.5 we studied a special case, which does not satisfy the assumptions of Theorems 12.7 and 12.8 of Poincar´ e. Under appropriate hypotheses of non-resonance, for these systems it is possible to write formally the series of the canonical theory of perturbations to all orders. However, these series are in general divergent (see Theorem 12.10). It would therefore seem impossible to prove the existence of quasi-periodic motions for Hamiltonian quasi-integrable systems, and the theory of perturb- ations seems, from this point of view, bound to fail. (It can still yield interesting information about the stability problem, though. This is shown by Theorem 12.11.) Consider a quasi-integrable Hamiltonian system. If ε = 0 the system is integ- rable and all motions are bounded and quasi-periodic. When ε = / 0, instead of requiring that this property is preserved, and hence that the system is still integrable, as we did so far, we can ask if at least some of these quasi-periodic unperturbed motions persist in the perturbed version. We shall not therefore seek a regular foliation of the phase space in invariant tori, but simply try to prove the existence, for values ε = / 0, of ‘some’ invariant tori, without requiring that their dependence on the action J is regular as J varies in an open subset A of R l
12.6 Analytical mechanics: canonical perturbation theory 523 The Kolmogorov–Arnol’d–Moser (KAM) theorem gives a positive answer to this question: for sufficiently small values of ε the ‘majority’ (in a sense to be clarified shortly) of invariant tori corresponding to diophantine frequencies ω are conserved, and are slightly deformed by the perturbation. The motions on these tori are quasi-periodic with the same frequency ω which characterises them for ε = 0. To be able to state the KAM theorem precisely, we must first give a meaning to the statement that ‘the invariant tori are slightly deformed’ under the action of a perturbation. Let H(J,
χ, ε) = H 0 (J) + εF (J, χ) (12.106)
be a quasi-integrable Hamiltonian system. Suppose that, for fixed ε 0 > 0, H : A ×T l ×(−ε 0 , ε 0 ) → R is an analytic function and that H 0 is non-degenerate (cf. Definition 12.4). Every invariant l-dimensional unperturbed torus T 0 = {J 0 } × T l ⊂ A×T l is uniquely characterised by the vector ω 0 = ω(J 0 ) of the frequencies of the quasi-periodic motions that stay on it. D efinition 12.9 Let ε 0 > 0 be fixed. A one-parameter family {T ε
ε ∈(−ε
0 ,ε 0 ) of l-dimensional submanifolds of R 2l is an analytic deformation of a torus T 0
{J 0 } × T l if, for every ε ∈ (−ε 0
0 ), T ε has parametric equations J = J 0
χ = ψ + εB(ψ, ε), (12.107)
where ψ ∈ T
l , A : T
l × [−ε
0 , ε
0 ] → R l and B : T l × [−ε
0 , ε
0 ] → T l are analytic functions. Note that setting ε = 0 in (12.107) we again find the torus T 0
{J 0 } × T l . Remark 12.10 The function B in (12.107) has the additional property that its average ˆ B 0 on the torus T l is zero. Indeed, εB = χ − ψ, and since χ and ψ are both coordinates on a torus T l we have
ε (2π)
l T l B(ψ, ε) d l ψ = 1 (2π)
l T l χ d l χ − T l ψ d l ψ = 0.
For fixed ε ∈ (−ε
0 , ε
0 ), equations (12.107) establish a correspondence of every point ψ 0
l with the point T ε
J = J 0 + εA(ψ 0 , ε),
χ = ψ 0 + εB(ψ 0 , ε).
(12.108) Denote by (J(t, ψ 0 ), χ(t, ψ 0 )) the solution of the Hamilton equations associated with (12.106) and passing through the point of coordinates (12.108) at time t = 0.
524 Analytical mechanics: canonical perturbation theory 12.6 D
ε } ε ∈(−ε 0 ,ε 0 ) of T 0 is a deformation of T 0 into invariant tori for the quasi-integrable system (12.106) if, for fixed ε ∈ (−ε
0 , ε
0 ), and every choice of ψ 0 ∈ T
l the Hamiltonian flow (J(t, ψ 0 ),
0 )) can be obtained from equations (12.107) by setting ψ = ψ 0 + ω(J 0 )t: J(t, ψ 0 ) = J 0 + εA(ψ
0 + ω(J 0 )t, ε),
χ(t, ψ 0 ) = ψ 0 + ω(J 0 )t + εB(ψ 0 +
0 )t, ε).
(12.109) It follows that (J(t, ψ 0 ),
0 )) belongs to T ε
∈ R. Remark 12.11 The motions on T ε are quasi-periodic with the same frequency vector ω 0 of the motions on T 0
We now show how it is possible to carry out, by means of a perturbative approach, the computation of the functions A and B. Setting ω
= ω(J
0 ) and ψ = ψ 0 +
0 t, from equation (12.109) it follows that ˙J = ε dA
(ψ, ε) = ε ω 0 · ∇ ψ A(ψ, ε), ˙ χ = ω
0 + ε
dB dt (ψ, ε) = ω 0 + ε ω 0 · ∇ ψ B(ψ, ε),
(12.110) to be compared with Hamilton’s equations associated with (12.106) and computed along the flow (12.109): ˙J = −ε∇
χ F (J(t, ψ 0 ),
0 )) =
−ε∇ χ F (J 0 + εA(ψ, ε), ψ + εB(ψ, ε)), ˙ χ = ω(J(t, ψ 0 )) + ε
∇ J F (J(t, ψ 0 ), χ(t, ψ 0 )) = ω(J 0 + εA(ψ, ε)) + ε ∇ J F (J 0 + εA(ψ, ε), ψ + εB(ψ, ε)). (12.111) Expanding A and B in power series in ε (the so-called Lindstedt series): A(ψ, ε) = ∞ k =0 ε k A (k)
(ψ) = A (0)
(ψ) + εA (1)
(ψ) + · · · ,
B(ψ, ε) = ∞ k =0 ε k B (k)
(ψ) = B (0)
(ψ) + εB (1)
(ψ) + · · · ,
(12.112) and
ω in Taylor series around J 0 : ω(J 0 + εA(ψ, ε)) = ω 0 + ε ∇ J ω(J 0 ) · A(ψ, ε) + · · · , (12.113) and then comparing (12.111) and (12.110) to first order in ε, we find ε ω
· ∇ ψ A (0) (ψ) =
−ε∇ χ F (J 0 , ψ),
(12.114) ε ω 0 · ∇
ψ B (0) (ψ) = ε ∇ J ω(J 0 ) · A (0)
(ψ) + ε ∇ J F (J 0 , ψ). (12.115) 12.6 Analytical mechanics: canonical perturbation theory 525 Equation (12.114) can be solved immediately by expanding A (0) and F in
Fourier series: setting A (0) (ψ) = m∈Z
l ˆ A (0) m e im·ψ , (12.116) since ∇ χ F (J 0 , ψ) = m∈Z l im ˆ F m (J 0 )e im·ψ , (12.117)
by the uniqueness of Fourier series we have for all m ∈ Z
l that
im · ω
0 ˆ A (0) m = −im ˆ F m (J 0 ). (12.118) The solution, if ω 0
ω(J 0 ) is non-resonant, is given by ˆ A (0) m = − m ˆ F m (J 0 ) m · ω
0 , (12.119) for m = / 0, while for the time being the average ˆ A (0)
0 of A on the torus T l is
Substituting the solution (12.116), (12.119) into the expression (12.115), and expanding in turn B (0) in Fourier series, we similarly find the coefficients ˆ B (0)
m for m =
/ 0. Note that integrating both sides of (12.115) on T l , and taking into account the periodicity of B with respect to ψ, we find ∇ J ω(J 0 ) · ˆ A (0) 0 + ∇ J ˆ F 0 (J 0 ) = 0. (12.120)
Since ∇ J ˆ F 0 (J 0 ) can be non-zero, for equation (12.120) (hence also (12.115)) to have a solution we must require that the matrix ∇ J ω(J 0 ) = ∂ 2 H 0 ∂J i ∂J k (J 0 ) (12.121) be invertible, and hence that the unperturbed Hamiltonian H 0 be non-degenerate in a neighbourhood of J 0 ∈ A. In this case ˆ A (0) 0 = −(∇ J ω(J
0 )) −1 ∇ J ˆ F 0 (J 0 ), (12.122) and this determines the average of A (0)
on T l . This discussion can be summarised in the following proposition. P roposition 12.4 If the Hamiltonian H 0 is non-degenerate on the open set A, for fixed J 0 ∈ A such that ω 0 = ω(J 0 ) is non-resonant, the system (12.114), (12.115) admits a formal solution. We can in fact prove that the argument we have just presented to obtain functions A and B as first-order perturbations can be iterated to all orders.
526 Analytical mechanics: canonical perturbation theory 12.6 Under the hypotheses of non-degeneracy for H 0 and of non-resonance for ω 0
≥ 0 the functions A (k)
and B (k)
in (12.112) through their Fourier series expansions: A (k) (ψ) = m∈Z
l ˆ A (k) m e im·ψ , B (k) (ψ) =
m∈Z l , m=0 ˆ B (k) m e im·ψ , (12.123) at least formally, and hence neglecting the problem of the convergence of the series (12.123). The coefficients ˆ A (k)
m and ˆ
B (k)
m of the series expansions (12.123) can be computed from the solution of a system of the form ω 0 · ∇ ψ A (k) (ψ) =
A (k)
(J 0 , ψ), (12.124) ω 0 · ∇ ψ B (k) (ψ) =
B (k)
(J 0 , ψ), (12.125) where
A (k)
and B (k) depend on A (0)
, . . . , A (k−1)
, B (0)
, . . . , B (k−1)
and on the derivatives of F with respect to J and χ up to order k + 1. Here B (k)
also depends on A (k) and on the derivatives of ω with respect to J up to order k + 1 (hence on the derivatives of H 0 with respect to J up to order k + 2). Note that the structure of equations (12.124) and (12.125) is the same as that of the fundamental equation of perturbation theory (12.13), and it constitutes the natural generalisation of equation (12.46) to the case l > 2. Indeed, Poincar´ e proved in chapter IX of his M´ ethodes Nouvelles (second volume, 1893) that the functions A k and B k appearing on the right-hand side of (12.124) and (12.125) have zero mean on the torus T l , and therefore the formal solvability of the two equations is guaranteed. It follows that we have the following significant extension of Proposition 12.2. P roposition 12.5 If the Hamiltonian H 0 is non-degenerate in the open set A, for any fixed J 0 ∈ A such that ω Download 10.87 Mb. Do'stlaringiz bilan baham: 
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