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R 2l M a
t (x 0 )
x 0 (t) g x 0 (0) 0 U t R Ј
0
Fig. 11.7 Since the flows g t j i are canonical, ∂q/∂t j =
p f j , and hence ∂(q
1 , . . . , q l )
1 , . . . , t l )
∂(f 1 , . . . , f l ) ∂(p 1 , . . . , p l )
, (11.104)
which is clearly non-singular. Remark 11.16 Evidently the map g x 0 cannot be a global diffeomorphism, because M a is assumed to be a compact manifold, while R l is not compact. It is worth noting that, because of the local character of this lemma, we made no use of the compactness assumption in the proof. L emma 11.2 The action of R l on M
a defined by g t is transitive, and hence for each pair of points x 1 , x 2 belonging to M a there exists t ∈ R l such that g t (x 1 ) = x
2 . Proof Since M a is a connected manifold, there exists a regular curve γ : [0, 1] → M a joining x 1 and x
2 : γ(0) = x 1 , γ(1) = x 2 . By Lemma 11.1 every point γ(τ ) of 11.5 Analytic mechanics: Hamilton–Jacobi theory and integrability 449
a
t i g t i – t Ј
Ј
x 2
2
) 0 = t 1 t i t i Ј
i+1 )
i+1 t = t N x 1 Fig. 11.8 the curve, 0 ≤ τ ≤ 1, has an open neighbourhood V (τ) restricted to which g t acts as a local diffeomorphism. The family {V (τ)} τ ∈[0,1] is an open covering of the curve γ. By compactness, there exists a finite subcovering {V (τ i
} N i =1 , with τ 1 = 0 and τ N = 1. Consider any sequence of points γ(τ i ) of the curve defined by the conditions γ(τ i ) = V (τ i ) ∩V (τ i +1 ) ∩γ([0, 1]), τ i > τ i (Fig. 11.8), as i = 1, . . . , N − 1 varies. Since g t is a local diffeomorphism between an open set of R l and every open set V (τ i ), there exist t i and t
i such that g t i
i ) = γ(τ
i ) and
g t i γ(τ i +1 ) = γ(τ i ). It follows that γ(τ i +1 ) = g t i −
t i γ(τ i ), and therefore x 2 = g
t x 1 where t = N − 1 i =1 (t i − t i ). The two previous lemmas show that the action of R l on M
a yields a way to construct an atlas of M a whose elements are the local parametrisations defined 450 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.5 by Lemma 11.1. Since the action of R l on M
a is transitive, M a is called a homogeneous space of R l . D efinition 11.3 Given x 0 ∈ M
a , the stationary subgroup of the action g t of
l on M
a at the point x 0 is the subgroup of R l : Γ x 0 = {t ∈ R l |g t x 0 = x 0 }. (11.105) Remark 11.17 It is immediate to verify that Γ x 0 is a subgroup of R l . Indeed 0 ∈ Γ x 0 , and
if t ∈ Γ x 0 then g −t x 0 = g −t g t x 0 = x 0 , and therefore −t ∈ Γ x 0 . In addition, if t and s both belong to Γ x 0 , g
t+s x 0 = g t g s x 0 = g t x 0 = x
0 , and hence t + s ∈ Γ
0 . L emma 11.3 The stationary subgroup Γ x 0 is independent of x 0 (we shall henceforth denote it simply by Γ ). Proof It is enough to prove that if t ∈ Γ
0 then g
t x = x for every x ∈ M a
action is transitive, there exists s ∈ R
l such that x = g s x
. From this it follows that g
t x = g
t g s x 0 = g s g t x 0 = g s x 0 = x. D efinition 11.4 A subgroup of R l is called discrete if it has no accumulation point. Thus a subgroup Γ is discrete if each of its points is isolated in R l , and hence it is the centre of a ball which contains no other point of Γ . L emma 11.4 The stationary subgroup Γ is discrete. Proof Since g
t is a local diffeomorphism, the origin 0 ∈ Γ
hence it has a neighbourhood U ⊂ R
l such that Γ ∩ U = {0}. Suppose that t = / 0 is an accumulation point of Γ . Then
t + U = {s + t|s ∈ U} is a neighbourhood of t, and hence there exists s ∈ (t + U) ∩ Γ , s =
/ t. But then s − t = / 0 and s − t ∈
Γ ∩ U, contradicting the hypothesis Γ ∩ U = {0}. The following lemma yields a classification of all discrete subgroups of R l . L emma 11.5 Every discrete subgroup G of R l is isomorphic to Z k , where k ∈ {0, . . . , l}. Hence there exist k linearly independent vectors e 1 , . . . , e k in R
l such
that G =
{m 1 e 1 + · · · + m k e k |m = (m 1 , . . . , m k ) ∈ Z k }. (11.106) The vectors e 1 , . . . , e k are called generators (or periods or bases) of G. 11.5 Analytic mechanics: Hamilton–Jacobi theory and integrability 451 Proof
If l = 1 every discrete subgroup G of R is either trivial, G = {0}, or else it is of the form G = {me
1 , m
∈ Z}, where e 1 = min x ∈G\{0}
|x|. Indeed, since G is discrete, e 1 is a non-zero element of G and every other element x of G must be an integer multiple of it, otherwise the remainder r of the division of |x| by e
1 would give another element of G, 0 < r < e 1 , which contradicts the definition of e 1 . For l ≥ 2 the proof that G is isomorphic to Z k with 0
≤ k ≤ l can be obtained by induction on l, by projecting G onto R l −1
1 of G \{0} of minimum norm. Since the projection of G is again a discrete subgroup, this yields the proof. The representation (11.106) of the discrete subgroup G is not unique. If e 1 , . . . , e k generates G, evidently it also true that e 1 + e
2 , e
2 , . . . , e k gener-
ate G, and so on. However, it is possible to characterise uniquely all possible choices of the generators of a discrete subgroup, as shown by the following. L emma 11.6 e 1 , . . . , e k and e
1 , . . . , e k are two k-tuples of generators of the same discrete subgroup G of R l if and only if there exists a k ×k matrix A, with integer coefficients and with determinant equal to ±1 (i.e. A ∈ GL(k, Z)), such that for every i = 1, . . . , k we have e i
k j =1 A ij e i . (11.107) Proof Evidently if e 1 , . . . , e k generates G, and A ∈ GL(k, Z), the k-tuple e 1 , . . . , e k defined by (11.107) generates a discrete subgroup G of R l . In addition G ⊂ G, for if t
∈ G , t = k i =1 m i e i = k i =1 k j =1 m i A ij e j = k j =1 m j e j , where m j = k i =1 m i A ij ∈ Z. Since det A = ±1 the inverse matrix A −1 also has integer coefficients and det A −1 = ±1; therefore A −1 ∈ GL(k, Z) and it can be shown immediately that G ⊂ G . Conversely, if e 1 , . . . , e k also generates G, let A be the k × k matrix defined by (11.107) which transforms e 1 , . . . , e k in e
1 , . . . , e k . The
coefficients of A are integers, as e i ∈ G for every i = 1, . . . , k. Applying the same reasoning to A −1 we see that the latter must also have integer coefficients. It follows that there exist two integers m and n such that det(A) = m, det(A −1 ) = n. But det(A) det(A −1 ) = 1, and therefore m = n = ±1 and A ∈ GL(k, Z). We can finally prove the theorem of Arnol’d. Proof of Theorem 11.5 Since the stationary subgroup Γ of the action of R l on M
a is discrete, there exists k ∈ {0, . . . , l} such that Γ is isomorphic to Z k . Therefore, there exist k linearly independent vectors e 1 , . . . , e k in R
l that generate Γ . Let ˜
e k +1 , . . . , ˜ e l be l − k vectors in R l chosen arbitrarily in such a way that e 1 , . . . , e k , ˜
e k +1 , . . . , ˜ e l is a basis of R l . Setting then t = ψ 1 2π e 1 + · · · +
ψ k 2π e k + t k +1 ˜ e k +1 + · · · + t l ˜
l , (11.108) 452 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.5 when (ψ
1 , . . . , ψ k , t
k +1 , . . . , t l ) ∈ R l vary, the action g t defines a parametrisation of M a . By Lemmas 11.1, 11.2, and 11.4 (recalling also Example 1.39) the manifold M a is diffeomorphic to T k × R
l − k . But M a is compact, and hence k = l. We conclude this section by proving the following. P roposition 11.1 Under the hypotheses of the theorem of Arnol’d, there exists a neighbourhood U ⊂ R
2l of M
a that is diffeomorphic to the direct product of a neighbourhood of an open set V ⊂ R
l with a torus T l :
l . Proof The idea is to prove that the functions f 1 , . . . , f l and the angles ψ 1 , . . . , ψ l constructed in the proof of Theorem 11.5 give a regular parametrisation of a neighbourhood U of M
a . Indeed, in a neighbourhood of any point P of M a the
functions f (p, q) can be inverted with respect to (x i 1 , . . . , x i l ), and hence to l of the variables x = (p, q). This is due to the independence of the f i which
ensures that the condition (11.7) is always satisfied. Hence we determine a regular submanifold N of dimension l, implicitly defined by x i
= ˆ x i k (f ),
(11.109) where k = 1, . . . , l and f varies in a neighbourhood V of a in R l .
Lemma 11.1 and construct a local parametrisation of the corresponding manifold M f : x = ˆ
x(f , t). (11.110)
The parametrisation (11.110) is by construction differentiable, and invertible with respect to both f and t. It follows that we have a local diffeomorphism between a neighbourhood U P ⊂ R 2l of any point of M a and a domain V × W , where W is a neighbourhood of 0 in R l . Since by Lemma 11.2 the action of R l on M
a is transitive, considering any other point P in M a , there exists t ∈ R l such that g t P = P . It is immediate to verify that g t
U P ) is a neighbourhood of P that is diffeomorphic to U P and hence also to V × W via the parametrisation x = ˆx(f, t + t ). Since P is arbitrary, we conclude that there exists a neighbourhood U of M a
coordinates (f , t) ∈ V × R
l through a differentiable function x = ˆ x(f , t). However this function is not invertible, because the stationary subgroup Γ of the action of R l used to construct it is isomorphic to Z l . If e
1 , . . . , e l generates Γ , although g e
P = P , the map from g e i to N generates a new submanifold N = g e i (N ) containing P but distinct from N . The existence of a local parametrisation ˆ x(f , t) in a neighbourhood of M a ensures that for every point P ∈ N (determined uniquely by the corresponding value f ) there exist l differentiable functions τ i (f ), i = 1, . . . , l, and a point P ∈ N such that τ i (a) = 0 and g τ i (f ) P = P for every i = 1, . . . , l. Hence g e
+τ i (f ) P = P , and in a neighbourhood of P we can construct generators 11.6 Analytic mechanics: Hamilton–Jacobi theory and integrability 453 e
(f ), with regular dependence on f and which on every level manifold M f determine the stationary subgroup of the action of R l . On each manifold M f we
1 , . . . , ψ l providing a global parametrisation, and thus obtain a regular parametrisation x = ˜
x(f , ψ),
(11.111) of a neighbourhood of M a through coordinates (f , ψ) ∈ V × T l . Remark 11.18 In general, the coordinates (f , ψ) constructed in the course of the previous proof are not canonical (recall the analogous discussion for the case l = 1, Section 11.3). Liouville’s theorem guarantees the existence of l coordinates β 1 , . . . , β l , canonic- ally conjugate to f 1 , . . . , f l , but the variables β 1 , . . . , β l are not angles, as required by the previous proposition. In the next section we show how to overcome this difficulty, by introducing the action-angle variables. Remark 11.19 The previous proposition is sufficient to prove that the phase space of an autonomous Hamiltonian system having as many first integrals independent and in involution as degrees of freedom is foliated in invariant tori, provided all traject- ories are bounded. In that case the invariant tori in the family {M a } a∈V
depend regularly on a. This is an important geometric characterisation of integrable Hamiltonian systems, which will be discussed in depth in the next section. 11.6
Integrable systems with several degrees of freedom: action-angle variables In Section 11.3 we introduced action-angle variables for one-dimensional systems. We started from the observation that, for example for oscillatory motions, every phase curve is diffeomorphic to a circle enclosing the same area. In the case of an autonomous Hamiltonian system with l degrees of freedom, which admits l integrals that are independent and in involution, the analogous observation is that the level manifold of the first integrals, M a , when is compact, it is diffeomorphic to an l-dimensional torus (Theorem 11.5). Starting from this, we try to extend the construction of the action-angle variables to systems with several degrees of freedom. D efinition 11.5 An autonomous Hamiltonian system, with Hamiltonian H(p, q) having l degrees of freedom, is called completely canonically integrable if there exists a completely canonical transformation p = ˆ p(J,
χ), q = ˆ
q(J, χ) (11.112) (where the dependence of ˆ p and ˆ
q on each variable χ i is 2π-periodic) to new vari- ables (J, χ) ∈ R
l ×T l , called action-angle variables, such that the new Hamiltonian 454 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.6 K is only a function of the actions J: K = H(ˆ p(J,
χ), ˆq(J, χ)) = K(J). (11.113)
If a system is completely canonically integrable, from (11.113) it follows that Hamilton’s equations can be written as ˙J = −∇ χ
˙ χ = ∇
J K ≡ ω(J). (11.114) The system (11.114) can be immediately integrated: J(t) = J(0), χ(t) = χ(0) + ω(J(0))t, (11.115) for every t ∈ R. The actions are therefore a system of l integrals that are independent and in involution, while each angle variable χ i , by hypothesis defined mod 2π, has a time period T i = 2π ω i (J)
. (11.116)
Since the dependence of (p, q) on (J, χ) is regular and 2π-periodic with respect to each angle, it follows that the motions of a completely canonically integrable system are bounded and quasi-periodic (see Section 11.7). We aim to prove the following theorem. T heorem 11.6 Let H(p, q) be a Hamiltonian system with l degrees of freedom which admits l first integrals f 1 (p, q), . . . , f l (p, q) that are independent and in involution. Assume that for a certain fixed value a ∈ R
l the level manifold M a
transformation of the variables (p, q) ∈ U to action-angle variables (J, χ) ∈ V ×T l (where V is an open subset of R l ). The system is therefore completely canonically integrable. Theorem 11.5 implies that it is possible to parametrise M a through l angles (ψ 1 , . . . , ψ l ). This fact is essential in the proof of Theorem 11.6. More pre- cisely, we refer to the conclusion of Proposition 11.1, that in a neighbourhood of M
a in R
2l one can introduce the generalised coordinates (not canonical) (f 1
l , ψ
1 , . . . , ψ l ). For fixed f = a and varying between 0 and 2π only one of the angles ψ i , we obtain a cycle γ i ⊂ M
a (corresponding to one of the generators of its fundamental group). Hence we can construct l cycles γ 1 , . . . , γ l that are
and not continuously reducible to one another (hence not homotopic). It is now possible to introduce the action variables, in analogy with Definition 11.2. 11.6 Analytic mechanics: Hamilton–Jacobi theory and integrability 455 D
1 , . . . , J l ) defined as J i = 1 2π γ i l j =1 p j dq j , (11.117) where i = 1, . . . , l. Apparently the definition we have just given of action variables has some degree of arbitrariness, due to the indetermination of the cycles γ i , i.e. the arbitrariness in the choice of the variables ψ j , j = i. However, the invariant manifolds M a are Lagrangian (see Remark 11.9), and this can be used to show that the above definition determines the action variables uniquely. More precisely, one has the following. P roposition 11.2 The action variables J i do not depend on the choice of the cycles γ i inside the same class of homotopy: if γ i is a new cycle obtained by a continuous deformation of γ i we have γ i l j =1 p j dq j = γ i l j =1 p j dq j . (11.118) The action variables depend only on the integrals f 1 (p, q), . . . , f l (p, q), and are independent and in involution. Proof
The independence of the choice of γ i in the same class of homotopy is an immediate consequence of (11.69) and of Stokes’ theorem (see Appendix 4). On the other hand, by (11.117) every action variable J i is independent of ψ i and cannot depend on the other angles ψ j , j =
/ i, either, since as ψ j varies, the cycle γ i is continuously deformed and the integral (11.117) does not change. Hence the actions are only functions of the integrals f 1 , . . . , f l . They are also in involution, as {J i , J k } = l m,n
=1 ∂J i ∂f m ∂J k ∂f n {f m , f n } = 0.
The independence of the actions can be proved by showing that det
∂J i ∂f j = / 0, (11.119) and then using the independence of the integrals f i . The proof is simplified when the variables are separable, which is the most interesting case in practice. Indeed, in this case the set M a is the Cartesian product of curves in each subspace (p i , q
i ), identifiable with the cycles. Following the procedure of separation of variables we obtain that J i depends only on f 1 , . . . , f i so that the Jacobian matrix is 456 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.6 triangular. Therefore (11.119) amounts to showing that ∂J i /∂f
i = 0, i = 1, . . . , l, which follows by repeating in each subspace the same argument used in the one-dimensional case. For simplicity, we limit the proof to the separable case. Proof of Theorem 11.6 By Proposition 11.2 the action variables are a set of independent integrals which are also in involution. By Liouville’s theorem (see in particular (11.78) and (11.80)) the function W (J, q) = q q 0 l j =1 p j dq j M J (11.120)
is the generating function of a completely canonical transformation to new vari- ables (J, χ), and the new Hamiltonian K is a function only of the action variables J. To complete the proof it is then sufficient to show that the new coordinates χ are angles defined mod 2π. By (11.117), the increment of the function W (J, q) when integrating along a cycle γ i is
i W = 2πJ
i , (11.121) and hence the increment of each variable χ k along the same cycle is ∆ i χ k = ∆ i ∂W ∂J k = ∂ ∂J k ∆ i W = 2πδ
ik . (11.122) It follows that χ ∈ T
l . Remark 11.20 The action-angle variables are evidently not unique. The construction of action- angle variables depends on the choice of the homotopy classes of the cycles γ 1
l generating the fundamental group of the torus (see Singer and Thorpe 1980, and Dubrovin et al. 1991b), and a different choice (of other cycles not homo- topic) produces a different determination of the variables (Fig. 11.9). Because of this arbitrariness it is possible to have completely canonical transformations to new action-angle variables. P roposition 11.3 Let J, χ be action-angle variables. The variables ˜J, ˜χ obtained through any of the following completely canonical transformations are still action-angle variables. (1) Translations of the actions: for fixed a ∈ R
l we have
˜ J = J + a, ˜ χ = χ.
(11.123) (2) Translation of the origin of the angles on each torus: let δ : R l → R be an arbitrary regular function, then ˜ J = J, ˜ χ = χ + ∇ J δ(J).
(11.124) 11.6 Analytic mechanics: Hamilton–Jacobi theory and integrability 457
1
2
2
2
2
2
2
1
2
2
1
1
2
1 = g 1 Ј
1 = g 1 Ј
1 = J 1 Ј
2 = J 1 J 2
2 = g 1 g 2
1 Ј
2 Fig. 11.9 (3) Linear transformations of the torus onto itself: let A be a matrix in GL(l, Z) (hence an l × l matrix with integer entries and |det A| = 1), then ˜ J = (A T ) − 1 J, ˜ χ = Aχ. (11.125)
Proof The verification that these transformations are completely canonical is left to the reader. Evidently the transformations (11.123) and (11.124) are canonical and leave invariant the property of being action-angle variables. We remark that the function W (J, q) in (11.120) is defined up to an arbitrary function δ(J), which reflects precisely the transformation (11.124). As for (11.125) it is sufficient to note that ˜ K = K(A
T ˜ J) and the variables ˜ χ are still defined mod 2π. We observe that A
−1 also has integer entries, thanks to | det A| = 1, which also preserves the measure of the torus. Example 11.14 Consider a system of l harmonic oscillators: H(p, q) = l i =1 p 2 i + m
2 i ω 2 i q 2 i 2m i . (11.126) 458 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.7 The cycles γ 1 , . . . , γ l are given by γ i = {(p i , q i ) |p 2 i + m 2 i ω 2 i q 2 i = 2m i f i }, (11.127)
where f 1 , . . . , f l are the l integrals in involution (see (11.84)). It is immediate to verify that the actions are given by J i = 1 2π γ i p i dq i = f i ω i , (11.128) and that the function W (q, J) = l i =1 q i 0 ± 2m
i ω i J i − m 2 i ω 2 i ξ 2 i dξ i (11.129)
generates the transformation to action-angle variables (J, χ):
p i = 2m i ω i J i cos χ i , q i = 2J i m i ω i sin χ i , (11.130) where i = 1, . . . , l. 11.7 Quasi-periodic motions and functions The analysis of the previous sections yields the conclusion that the integrable completely canonical Hamiltonian systems are characterised by the fact that they admit l independent integrals in involution, and the phase space is foliated in invariant tori. On these tori, the motion is governed by the equations ˙ χ = ω.
(11.131) In what follows we ignore the trivial case ω = 0. If l = 1 the motions are periodic. In the more general case that l ≥ 2, the motions are not necessarily periodic. Before starting a more detailed analysis, we consider the case l = 2. In this case, the solution of equation (11.131) can be written as χ 1 (t) = χ 1 (0) + ω 1 t, χ 2 (t) = χ
2 (0) + ω
2 t. (11.132) Eliminating time t, the orbit is given by the line ω 2 (χ 1 − χ 1 (0))
− ω 1 (χ 2 − χ
2 (0)) = 0. (11.133) We can therefore assume without loss of generality that χ 1 (0) = χ
2 (0) = 0, so that the line passes through the origin (otherwise, it is sufficient to translate the origin to (χ 1 (0), χ
2 (0))). Since (χ 1 , χ
2 ) ∈ T 2 , and T
2 = R
2 /(2πZ)
2 , it is
11.7 Analytic mechanics: Hamilton–Jacobi theory and integrability 459 2p n = 2 n = 4 n = 1 n = 3 n = 5 n = 0 0
1
2 2p Fig. 11.10 clear that the line must be represented in the square [0, 2π] 2 with opposite sides identified with each other, according to the rule χ ≡ χ + 2πm, where m ∈ Z 2 .
If ω 1 = / 0, the sequence of intersections of the orbit with the vertical segment [0, 2π] on the χ 2 -axis is given by {n (mod 2π)} ∞ n =0 , where
= 2π ω 2 ω 1 , (11.134) while if ω 1 = 0 all trajectories are clearly periodic. We thus obtain a map of T 1 onto itself defined by a rotation of angle . T heorem 11.7 The sequence {n (mod 2π)} ∞ n =0 on the circle T 1 is periodic if and only if /2π
∈ Q. Else if /2π is irrational, the sequence is dense in T 1 . Proof A necessary and sufficient condition for the sequence to be periodic is that there exists an integer s > 0 such that s (mod 2π) = 0, and hence that there exists an integer r such that s = 2rπ, from which it follows that /2π = r/s. If /2π is irrational, all points of the sequence are distinct. Since the circle is compact, for every ε > 0 there exist integers r, s such that |(r − s )(mod 2π)| < ε. Setting j = |r − s|, the subsequence {nj (mod 2π)} [2π/ε] n
subdivides the circle into adjacent intervals of length less than ε, and hence every point of the circle is at a distance less than ε from a point of the sequence. Since ε is arbitrary, the sequence is dense. An obvious corollary of this proposition is the following.
460 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.7 C
2 is periodic if and only if ω 2
1 is rational or ω 1 = 0, otherwise it is dense on the torus. If l > 2, the solution of (11.131) is still given by χ i (t) = χ i (0) + ω i t, (11.135) where i = 1, . . . , l. Eliminating time, we find that the orbit is still represented by a line in T l = R
l /(2πZ)
l , and hence with the points χ ≡ χ + 2πm identified, where m
∈ Z l , and can therefore be represented in the hypercube [0, 2π] l with
opposite faces identified. To study the periodicity of the orbit it will be useful to introduce the following. D efinition 11.7 Choose ω ∈ R l . The resonance module M ω
vector ω is the subset 2 of Z
l given by
M ω = {m ∈ Z l |m · ω = 0}. (11.136) The dimension of the resonance module M ω
pendent resonance relations m · ω = 0 satisfied by ω. It is also called resonance multiplicity. Since we excluded the case ω = 0 we have 3 0
ω ≤ l − 1.
If l = 2 only the extreme cases dim M ω = 0, and hence M ω = {0}, and
dim M ω = l − 1 are possible, and are called, respectively, non-resonance and complete resonance. Corollary 11.1 implies that in this case the orbit (11.132) is periodic for complete resonance and dense for non-resonance. We can indeed prove the following generalisation of Theorem 11.7. T heorem 11.8 Let M ω be the resonance module associated with the frequency vector ω of the motions (11.135). Then (1) the orbit is periodic if and only if dim M ω = l − 1 (complete resonance); (2) if dim M ω = 0 the orbit is dense on the whole torus T l ; (3) if 0 < d < l − 1, with d = dim M ω , the orbit is dense on a torus of dimension l − d embedded into T l .
We subdivide the proof of Theorem 11.8 into a series of partial results, of some interest by themselves. Consider an arbitrary invertible linear transformation of coordinates of the torus T
l which preserves orientation. By Lemma 11.6 its general form is χ = Mχ, (11.137)
2 Evidently M ω
Z l . 3 Indeed if ω = 0 we have dim M ω = l, but then all the points of the torus T l are fixed. 11.7 Analytic mechanics: Hamilton–Jacobi theory and integrability 461 where M
∈ SG(l, Z). The system of equations (11.131) is transformed by (11.137) into
˙ χ = ω ,
(11.138) where
ω = Mω. (11.139)
L emma 11.7 Let M ω be the resonance module corresponding to ω. There exists a coordinate transformation (11.137) of the torus T l such that ω l − d +1 = · · · = ω l = 0,
where d = dim M ω . Proof
First of all we note that a collection of l vectors of Z l (e 1 , . . . , e l ) is a basis of Z l if and only if the parallelepiped with sides e 1 , . . . , e l has volume 1. Indeed, the canonical basis e 1 = (1, 0, . . . , 0), e 2 = (0, 1, 0, . . . , 0), . . . generates a cube of side 1, and by Lemma 11.6 every other basis in Z l is related to the canonical one by a volume-preserving linear transformation. We now try to complete an arbitrary basis (m 1 , . . . , m d ) of
M ω with l − d lin- early independent vectors of Z l , (
µ 1 , . . . , µ l − d ), in such a way that ( µ 1
µ l − d , m 1 , . . . , m d ) is a basis of Z l . If this is possible, then the lemma is proved by constructing the matrix M whose rows are the components of the vectors ( µ 1 , . . . , µ l
d , m
1 , . . . , m d ); indeed ω l − d +j = ω · m j = 0 for every j = 1, . . . , d. The matrix M has integer components and determinant equal to ±1 by the previous remark, and hence it induces an invertible coordinate transformation on the torus T l which satisfies the statement. On the other hand, it is immediate to prove that such a choice of ( µ 1 , . . . , µ l − d ) is possible. Let ( µ 1 , . . . , µ l − d ) be linearly independent vectors of M ⊥ ω = {µ ∈ Z
l |µ · m = 0 for every m ∈ M ω }.
Evidently ( µ 1 , . . . , µ l − d , m 1 , . . . , m d ) is a basis of R l . If the volume of the par- allelepiped they generate is equal to 1, it is also a basis of Z l and the proof is finished. Otherwise, since the volume is a positive integer, there exists a non-zero vector v
∈ Z l inside the parallelepiped: v = λ 1 µ 1 + · · · + λ l − d µ l − d + λ
l − d +1 m 1 + · · · + λ l m
, (11.141)
with 0 ≤ λ
j < 1 and λ j a suitable rational, for every j = 1, . . . , l. Since the subspace of R l generated by M ω does not contain any point of Z l different from those of M ω , the vector v cannot belong to M ω (which has no vectors inside the parallelepiped), and therefore it is not restrictive to assume that λ 1 =
replacing v by µ 1 , we find a new l-tuple of linearly independent vectors of R l 462 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.7 such that det ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ v µ 2 .. . µ l − d m 1 .. . m d ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ≤ det
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ µ 1 µ 2 .. . µ l − d m 1 .. . m d ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ − 1.
The volume of the parallelepiped generated by the basis is therefore diminished by at least one unit. If it is not equal to 1, by repeating this procedure a sufficient number of times, we find the basis sought. Example 11.15 Consider a system of three independent harmonic oscillators; the Hamiltonian of the system in action-angle variables is given by (see Example 11.14) K(J) = ω 1 J 1 + ω
2 J 2 + ω 3 J 3 . (11.142) Suppose that the frequencies satisfy the resonance relations ω 1 + 2ω 2 − 4ω 3 = 0,
ω 1 − ω 2 = 0,
(11.143) so that
M ω has dimension 2 and a basis for it is clearly given by m 1 = (1, 2,
−4), m 2 = (1, −1, 0). In this case the canonical linear transformation ˜ J = (M
T ) − 1 J, ˜ χ = Mχ, (11.144)
where M is the following matrix of SL(3, Z): M =
⎛ ⎝ 1 2 −4 1 −1 0 0 −1 1 ⎞ ⎠ , (11.145)
transforms the Hamiltonian (11.142) into ˜ K(˜ J) = ˜ J · Mω = (ω 3 − ω
2 ) ˜
J 3 , (11.146) and hence in the new variables two frequencies vanish (˜ ω 1
ω 2 = 0, ˜ ω 3 = ω 3 − ω 2 ). D efinition 11.8 A continuous function φ: R → R is called quasi-periodic if there exist a continuous function f : T l → R and a vector ω ∈ R l such that φ(t) = f (ω 1 t, . . . , ω l t).
(11.147) 11.7 Analytic mechanics: Hamilton–Jacobi theory and integrability 463 The time average φ T of a quasi-periodic function is given by φ T
T →∞ 1 T T 0 φ(t) dt = lim T →∞ 1 T T 0 f (ω
1 t, . . . , ω l t) dt,
(11.148) as long as the limit exists. Evidently, the kinetic momenta and the coordinates (p, q) of a completely integrable Hamiltonian system are examples of quasi-periodic functions. More generally, if f is any continuous function defined on the torus T l , then if we consider the values f ( χ(0) + ωt) that the function takes along the flow (11.135) we find a quasi-periodic function, for which it is meaningful to consider the time average (11.148) (this is a function of the orbit considered, parametrised by the initial data χ(0)) and also the phase average, i.e. the average on the torus T l :
1 (2π)
l T l f ( χ) d
l χ. (11.149) The comparison of the time average with the phase average allows us to establish whether the motion on T l is dense. Indeed, we have the following. T heorem 11.9 Let f: T l → R be a continuous function, and consider the quasi-periodic function obtained by composing f with the flow (11.35): φ(t) = f (
χ(0) + ωt). If the frequencies ω are not resonant, i.e. if dim M ω = 0, the time average φ T ( χ(0)) exists everywhere, it is constant on T l and coincides with the phase average (11.49). Proof
First of all we prove the theorem in the special case that f is a trigonometric polynomial and hence can be written as f ( χ) =
m∈ F ˆ f m e im·χ , (11.150) where F ⊂ Z
l is a finite set of indices. If F is made of only one index m, then if m = 0 the function is constant and φ T = f 0 = f . Otherwise, if m = / 0 it is immediate to check that the phase average is zero and the time average is given by φ T
im·χ(0) lim
T →∞ 1 T T 0 e im·ωt
dt = e im·χ(0) im · ω
lim T →∞ e im·ωT
− 1 T = 0, for any χ(0) ∈ T
l . If
F has a finite number of indices, one can use the linearity of the time average and phase average operators to show that the averages in phase and time coincide. Now let f be a generic continuous function. By Weierstrass’s theorem (see Giusti 1989) for every ε > 0 there exists a trigonometric polynomial P ε
464 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.7 approximating f uniformly on T l up to ε/2: max χ∈T
l |f(χ) − P ε (
(11.151) Setting P − = P
ε − ε/2 and P + = P
ε + ε/2, we have P − ≤ f ≤ P
+ and
1 (2π)
l T l (P + (χ) − P − (χ)) d l χ ≤ ε. Therefore for every ε > 0 there exist two trigonometric polynomials P − and P + such that f − P
− ≤ ε,
P + − f ≤ ε, (11.152) and for every T > 0 we have 1 T
0 P − ( χ(0) + ωt) dt ≤ 1 T
0 f (
χ(0) + ωt) dt ≤ 1 T T 0 P + ( χ(0) + ωt) dt. (11.153) However, by the previous remarks, for every ε > 0 there exists T (ε) > 0 such that for every T > T (ε) one has P ± − 1 T T 0 P ± ( χ(0) + ωt) dt ≤ ε. (11.154) Combining (11.152)–(11.154) we find that for every ε > 0 and for every T > T (ε) we have f − 1 T T 0 f (
χ(0) + ωt) dt ≤ 2ε, (11.155)
and the theorem is proved. It is not difficult now to prove Theorem 11.8. Proof of Theorem 11.8 Statement (1) is of immediate verification, and it is left to the reader. Suppose now that dim M ω = 0. If there exist a point χ ∈ T
l and an open neighbourhood U not visited by the orbit, take any continuous function f : T l → R with the following properties: (a) f = 1; (b) f ( χ) = 0 for every χ ∈ U. The function f would then have zero time average, different from the phase average, contradicting Theorem 11.9. 11.7 Analytic mechanics: Hamilton–Jacobi theory and integrability 465 Finally, if dim M ω = d and 0 < d < l − 1, by Lemma 11.7 there exists a coordinate transformation on T l which annihilates the last d frequencies. It is therefore sufficient to repeat the previous argument restricted to the torus T l − d with points (χ 1 , . . . , χ l − d , χ l − d +1 (0), . . . , χ l (0)).
Example 11.16 Apply Theorem 11.9 to solve a celebrated problem proposed by Arnol’d. Consider the sequence constructed by taking the first digit of 2 n for n ≥ 0: 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, . . . and compute the frequency with which each integer i appears in the sequence. The first digit of 2 n is equal to i if and only if log 10 i ≤ {n log 10 2 } < log 10 (i + 1), 1 ≤ i ≤ 9,
where {x} denotes the fractional part of x: {x} = x(mod 1). On the other hand, log 10
{n log 10 2 } is dense on the interval [0, 1]. The frequency ν i with which the integer i appears in the sequence is given by ν i = lim N →+∞ card( {{n log
10 2 } ∈ [log 10 i, log
10 (i + 1))
|0 ≤ n ≤ N − 1}) N , (11.156) where card(A) indicates the cardinality of the set A. Evidently (11.156) coincides with the time average χ i T
of the function χ i : [0, 1] → R given by χ i
1, if x
∈ [log 10 i, log 10 (i + 1)), 0, otherwise, (11.157) computed for the sequence {n log 10
}: ν i = χ i T
= lim
N →+∞
1 N N −1 j =0 χ i ( {j log 10 2 }). (11.158)
It is not difficult to prove, by adapting the proof of Theorem 11.9, 4 that, although the function χ i is not continuous, the conclusions of the theorem still hold, and in particular, that the average χ i T
is constant and equal to the average of χ i on the interval [0, 1]: ν i = χ i T = 1 0 χ i (x) dx = log 10 (i + 1) − log 10 i. (11.159) Hence the frequency of 1, 2, . . . , 9 in the sequence of the first digit of 2 n is
respectively. 4 Note that the function χ i can be approximated by trigonometric polynomials, although the convergence occurs only pointwise. 466 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.8 Considering only the first 40 terms of the sequence, it would appear that the sequence is periodic, with period 10: 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, etc. The number 7 appears for the first time for n = 46, and 9 for n = 53. This behaviour illustrates how the convergence to the limit (11.158) is possibly very slow, and in our case it can be explained by observing that log 10 2 = 0.301029996 . . . , while an irrational number is very close to 3/10, which would produce the sequence {3n/10} that is periodic with period 10. It is interesting to compare the behaviour of the sequence {n log 10
} with {n(
√ 5 − 1)/2}. 11.8 Action-angle variables for the Kepler problem. Canonical elements, Delaunay and Poincar´ e variables The Hamiltonian of the Kepler problem in spherical coordinates is given by H(p
r , p
θ , p
ϕ , r, θ, ϕ) = 1 2m
2 r + p 2 θ r 2 + p 2 ϕ r 2 sin 2 θ − k r . (11.160) The moment p ϕ canonically conjugate to the azimuthal angle ϕ coincides with the component along the z-axis (normal to the ecliptic) of the angular momentum, as can be immediately verified from the definitions. In addition we have |L| 2
|mr × ˙r| 2 = m 2 r 4 ( ˙ θ 2 + sin 2 θ ˙ ϕ 2 ) = p 2 θ + p 2 ϕ sin 2 θ , (11.161)
and |L| = constant (areas constant). If i indicates the angle of inclination of the orbit with respect to the ecliptic z = 0, evidently p ϕ
|L| cos i. (11.162)
The angle ϕ is cyclic, and hence L z = p ϕ is a first integral of the motion. It is very easy to check that all assumptions of Arnol’d’s theorem are satisfied. Two angular coordinates ϕ, θ are immediately available to obtain the respective cycles γ ϕ , γ θ . The first action variable for the Kepler problem therefore coincides with p ϕ
J ϕ = 1 2π γ ϕ p ϕ dϕ = p ϕ (11.163) (here γ ϕ is the cycle obtained by varying ϕ ∈ S 1 and keeping r, θ, p r , p
θ , p
ϕ constant). The second action variable is given by J θ = 1 2π γ θ p θ dθ.
(11.164) 11.8 Analytic mechanics: Hamilton–Jacobi theory and integrability 467 The equation of the cycle γ θ is indeed (11.161) from which J θ
1 2π γ θ |L|
2 − J 2 ϕ sin 2 θ dθ. (11.165) On the other hand, J ϕ =
π/2 + i while r and ϕ remain constant, and hence we have J θ = − 4 |L| 2π π/ 2 − i π/ 2 1 sin θ sin
2 i − cos 2 θ dθ =
2 |L|
π sin
2 i π/ 2 0 cos 2 ψ 1 − sin 2 i sin 2 ψ dψ, where we have substituted cos θ = sin i sin ψ. Setting now u = tan ψ we finally find
J θ = 2 |L|
π +∞ 0 du 1 + u
2 − cos
2 i du 1 + u 2 cos 2 i = 2 |L|
π π 2 − π 2 cos i = |L|(1 − cos i), from which, since |L| cos i = J ϕ we can deduce that |L| = J θ + J ϕ . The third and last action variable J r is given by J r = 1 2π γ r p r dr = 1 2π γ r 2m E + k r − (J θ + J ϕ ) 2 r 2 dr. (11.166)
Note that, because of equations (11.160), (11.161), the cycle γ r in the plane (p r , r) has precisely the equation p 2 r = 2m E +
k r − |L| 2 r 2 , (11.167) from which we immediately find the extreme values of r: r ± = a 1 ± 1 −
|L| 2 mka , with a =
−k/2E > 0. The interpretation of J ϕ , J θ and J
r is clear in terms of the areas of the cycles depicted in Fig. 11.11.
468 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.8
–
+
2pJ w 2pJ r O O 2p p 2pJ u 2
2
2 – i + i w u Fig. 11.11 The integral (11.166) can be computed by elementary means and the final result is J r
−(J θ + J ϕ ) + k
m −2E
, from which H = E = −
2 2(J
r + J
θ + J
ϕ ) 2 . (11.168)
Differentiating with respect to the action variables we find the frequencies ω =
∂H ∂J r = ∂H ∂J θ = ∂H ∂J ϕ = mk 2 (J r + J
θ + J
ϕ ) 3 = mk 2 −2E mk 2 3/2 . (11.169)
Since the frequencies are all equal, the problem is completely resonant and all orbits are periodic with period T = 2π
= 2π mk 2 mk 2 −2E 3/2
. (11.170)
11.8 Analytic mechanics: Hamilton–Jacobi theory and integrability 469 From the relation a = −k/2E linking the major semi-axis with the energy, one derives Kepler’s third law (see (5.43) and recall that k/m is independent of m): a 3
2 = k 4π 2 m . The so-called Delaunay elements, which can be interpreted as orbital elements, are defined through the linear canonical transformation of the kind (11.125), naturally suggested by the physical meaning of J ϕ , J
ϕ + J
θ , J
ϕ + J
θ + J
r : L = J θ + J
ϕ + J
r , G = J ϕ + J
θ , H = J ϕ ; l = χ r , g = χ θ − χ
r , h = χ ϕ − χ
θ , (11.171) where (χ r , χ θ , χ
ϕ ) are the angle variables conjugate to (J r , J
θ , J
ϕ ). Relation (11.171) annihilates two frequencies (see Lemma 11.7) and the Hamiltonian in the new variables is written H = −
2 2 L 2 . (11.172) It follows that the only non-constant element is l. On the other hand, the first three elements are combinations of constants, while the constancy of g and h is a consequence of complete resonance. It is not difficult to see that l is the mean anomaly, g is the perihelion argument and h is the ascending node longitude (Fig. 11.12). Here L, G and H are related to the semi-major axis a, the eccentricity e and the inclination i of the orbit by L =
√ mka,
G = |L| = L 1 − e 2 , H = |L| cos i. (11.173)
Although appropriate to the complete resonance of the Kepler problem, the Delaunay variables are not particularly convenient to describe the orbits of the planets of the Solar System. This is due to the fact that these variables become singular in correspondence to circular orbits (e = 0, therefore L = G and the argument of the perihelion g is not defined) and to horizontal orbits (i = 0 or i = π, therefore G = H and the ascending node longitude h is not defined). All the planets of the Solar System have almost circular orbits (except Mercury, Mars and Pluto) and small inclinations (see Table 11.1, taken from Danby (1988)). 470 Analytic mechanics: Hamilton–Jacobi theory and integrability 11.8 Pericentre P (x, y, z) u w g y x z Download 10.87 Mb. Do'stlaringiz bilan baham: 
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