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8 sin(2χ
2 − χ
1 )] + J
3/2 2 [F 9 sin χ
2 + F
10 sin 3χ
2 ],
540 Analytical mechanics: canonical perturbation theory 12.10 where F
1 , . . . , F 10 are constants depending on m, ω 1 , ω
2 , a
30 , a
21 , a
12 , a
03 (for
example: F 1 = 3 4 (2/mω 1 ) 3/2 a 30 , F 2 = − 1 4 (2/mω 1 ) 3/2 a 30 , etc.). Setting H 0 = ω 1 J 1 +ω 2 J 2 and I
(2) = ω
1 J 1 −ω 2 J 2 we must impose the condition {H, I} = {H 0 + F, I (2) + I
(3) } = {H 0 , I (2) } + {H
0 , I
(3) } + {F, I (2) } + {F, I (3) } = 0,
neglecting terms of degree ≥ 4 in J
1/2 1 , J 1/2 2 and in their products. Since {H 0 , I (2) } = 0 and {F, I (3) } is of degree 4 we arrive at the equation {H 0 , I (3) } = −{F, I (2) };
ω 1 ∂I (3) ∂χ 1 + ω 2 ∂I (3) ∂χ 2 = ω 1 ∂F ∂χ 1 − ω 2 ∂F ∂χ 2 , from which it follows that to each term A sin(mχ 1 + nχ
2 ) appearing in F there corresponds a term ω 1 m − ω
2 n ω 1 m + ω
2 n A sin(mχ 1 + nχ
2 ) in I (3) . Therefore the required integral is I (3)
(J, χ) =J 3/2
1 [F 1 sin χ 1 + F 2 sin 3χ
1 ] + J
1 J 1/2 2 − F
3 sin χ
2 + 2ω 1 − ω
2 2ω 1 + ω 2 F 4 sin(2χ
1 + χ
2 ) +
2ω 1 + ω 2 2ω 1 − ω 2 F 5 sin(2χ
1 − χ
2 ) + J 1/2 1 J 2 F 6 sin χ 1 + ω 1 − 2ω 2 ω 1 + 2ω 2 F 7 sin(2χ
2 + χ
1 ) + ω 1 + 2ω 2 ω 1 − 2ω 2 F 8 sin(2χ
2 − χ
1 ) + J
3/2 2 [ −F 9 sin χ 2 − F
10 sin 3χ
2 ]. The procedure followed is justified as long as 2ω 1 ± ω 2 = / 0, ω 1 ± 2ω 2 = / 0. 12.10 Analytical mechanics: canonical perturbation theory 541 Problem 3 Consider the motion of a ball of mass m bouncing elastically between two walls that are slowly moving towards one another, and prove that the action is an adiabatic invariant. Solution
Consider the motion of a ball of mass m bouncing elastically between two fixed walls at a distance d (see Percival and Richards 1986). Although this system cannot be described by a regular Hamiltonian (because the speed v of the ball varies discontinuously from v → −v at each hit), the motion can be studied simply and the system is integrable. Let E = 1 2
2 be the energy of the ball, q be its position and p = mv be the momentum. The phase curves are rectangles (Fig. 12.3), the action J is given by J = 1
(area of the rectangle) = 1 2π (2mvd) = d π √ 2mE,
and the energy, as a function of the action, is given by E =
1 2m πJ d 2 . p mv O –mv q Fig. 12.3 542 Analytical mechanics: canonical perturbation theory 12.10 Suppose now that one of the two walls moves towards the other with a velocity ε, such that at time t the distance between the walls is x(t) = d − εt. Let ε ω
= π 2 md 2 J = π d v, and indicate by v n the velocity of the ball just before the nth collision with the moving wall, and by v n +1 the velocity immediately after the collision (Fig. 12.4). Evidently v n
= v n + 2ε, from which it follows that v n = v 0 + 2nε. If x n is the distance between the planes at the moment corresponding to the nth collision, and ∆ t n measures the time interval between the (n + 1)th and the nth collisions, we have ∆ t n = x n +1 + x n v n +1 = x n − x
n +1 ε , from which we deduce x n
= v n +1 − ε
v n +1 + ε x n , ∆ t n = 2x n v n +1 + ε
. (12.154)
« v n v n + 1 x n + 1 x = 0 x n v n + 1 Fig. 12.4 12.10 Analytical mechanics: canonical perturbation theory 543 Table 12.1 x 0 = 1, v
0 = 1, m = 1, ε = 0.01. n x
v n t n E n π · J n 0 1 1.0
0 0.5
1 10 0.835 1.2 16.5
0.72 1.002
50 0.502
2.0 49.8
2.0 1.005
100 0.336
3.0 66.4
4.5 1.007
500 0.092
11.0 90.8
60.5 1.009
1000 0.048
21.0 95.2
220.5 1.010
Table 12.2 x 0 = 1, v 0 = 1, m = 1, ε = 0.1. n x
v n t n E n π · J n 0 1 1 0 0.5 1 10 0.355 3 6.45 4.5 1.065
50 0.099
11 9.01
60.5 1.089
100 0.052
21 9.48
220.5 1.092
500 0.01088
101 9.89
5100.5 1.099
1000 0.0055
201 9.95
20200.5 1.099
Since v j +1 − ε = v j + ε, iterating equation (12.154) we find x n +1 = x 0 n j =0 v j + ε
v j +1 + ε = v 0 + ε
v n +1 + ε x 0 = v 0 + ε v 0 + (2n + 3)ε x 0 , (12.155)
and hence the (n + 1)th collision happens at the instant t n +1 − t
0 = n j =0 ∆ t j = x 0 − x n +1 ε = 2(n + 1)ε v 0
. The action between two successive collisions is given by J n
m π v n x n (the system keeps the memory of the last hit); therefore by equation (12.155) we have
J n = m π (v 0 + 2nε)
v 0 + ε v 0 + (2n + 1)ε x 0 = J 0 + εx
0 m π 2nε v 0 + (2n + 1)ε , (12.156) while E n = 1 2 mv 2 n = 1 2 m(v 0 + 2nε) 2 = E
0 + 2εv
0 nm + 2mn
2 ε 2 . 544 Analytical mechanics: canonical perturbation theory 12.10 It is immediate to check that from (12.156) it follows that |J n − J 0 | ≤
mεx 0 π , for all n ∈ N. The action is therefore an adiabatic invariant, while the energy is not, because E n
0 = 2mεn(v
0 + nε),
and therefore E n − E 0 = O(1) if n = O (1/ε). Tables 11.1 and 11.2 show the values of E, J, x and t for n = 10, 50, 100, 500 and 1000, corresponding to x 0 = 1, v 0 = 1 and ε = 0.01 and ε = 0.1, respectively. 13 ANALYTICAL MECHANICS: AN INTRODUCTION TO ERGODIC THEORY AND TO CHAOTIC MOTION The completely canonically integrable systems are the mechanical model for the study of systems with an orderly and regular behaviour. The main idea in all studies in the nineteenth century has been to reduce the study of mechanical systems to the study of integrable systems, both exactly by using canonical transformations and the Hamilton–Jacobi equations, and approximately using the canonical theory of perturbations. Poincar´
e proved however that this is not always possible, and that Hamiltonian systems may exhibit a behaviour that is totally different from the behaviour of integrable systems, exhibiting disorderly and chaotic orbits. The appropriate language for the study of these systems connects the study of dynamical sys- tems to probability theory (of which we recall the main introductory notions). This is the point of view underlying ergodic theory, which we introduce in this chapter. We start by introducing the notions of measure and a measurable dynamical system. 13.1
The concept of measure D efinition 13.1 Let X be a non-empty set. A non-empty family A of subsets of X is a σ-algebra on X if it satisfies the following properties: (1) A
∈ A ⇒ A c ∈ A; (2) for every sequence {A i }, i ∈ N of elements in A we have ∪ i ∈N A i ∈ A. Any family of subsets of X for which (1) and (2) are valid for finite sequences is called an algebra. It is immediate to verify that any σ-algebra is also an algebra. In particular if A, B ∈ A then A ∪ B ∈ A. Example 13.1 For a given set X we can obtain trivial examples of σ-algebras by choosing (a) the family of all subsets of X; (b) the pair {∅, X}.
546 Analytical mechanics 13.1 Remark 13.1 If A is a σ-algebra on X it is easy to prove that the following properties hold. (i) Imposing (2) is equivalent to imposing that ∩ i ∈N A i ∈ A (it is enough to note that ∩ i ∈N A i = ( ∪ i ∈N A c i ) c and A c i ∈ A); in particular A ∩ B ∈ A for every A, B ∈ A. (ii)
∅ ∈ A, X ∈ A (indeed A ∈ A ⇒ X = A ∪ A c ∈ A, ∅ = A ∩ A c ∈ A).
(iii) A, B ∈ A ⇒ A \ B ∈ A (indeed A \ B = A ∩ B c ).
I denotes the inter- section,
I X and hence it is non-empty, the properties (1) and (2) of Definition 13.1 are easily proved). The latter property allows us to generate the smallest σ-algebra on X containing a prescribed family F of subsets of X. D efinition 13.2 Given a family F of subsets of X the σ-algebra on X generated by F is the intersection of all σ-algebras A such that A ⊃ F. The definition is meaningful because there exists at least one σ-algebra A such that A ⊃ F (the σ-algebra of all subsets of X). An important case is the following. D efinition 13.3 Let X = R l . We call a Borel σ-algebra on R l (denoted by B(R l )) the one generated by the family of open subsets of R l . The elements of B(R
l ) are called Borelian sets of R l . More generally, if X is any topolo- gical space, the Borel σ-algebra of X is the σ-algebra generated by the open subsets of X. We can now define the concept of measure. D efinition 13.4 Given a set X and a σ-algebra A on X, a measure is a function µ : A → [0, +∞] such that (1 ) µ(
(2 ) µ i ∈N A i = i ∈N µ(A i ) for every sequence {A i } of disjoint elements of A. Note that the function µ is allowed to take the value + ∞. D
measure µ are called a measure space. A set A
⊂ X has zero measure if there exists A 1 ∈ A such that A ⊂ A 1 and
µ(A 1 ) = 0. Two sets A 1 , A 2 coincide (mod 0) and we write A 1 = A
2 (mod 0) if the symmetric difference A 1 ∆ A 2 has zero measure. 13.1 Analytical mechanics 547 If a property is valid for all points of A ⊂ X except for those in a set of measure zero, we say that the property is true for µ-almost all x ∈ A (written as µ-a.a. x ∈ A). An important case is the case of R and of the Lebesgue measure on B(R) which associates with intervals their lengths, and at the same time the case of the Lebesgue measure on B(R
l ). It can be shown that the Lebesgue measure λ : B(R
l ) → [0, +∞] is the only measure with the property that for every A = (a 1 , b 1 ) × · · · × (a l , b
l ), we have λ(A) = (b 1 − a 1 )(b
2 − a
2 ) · · · (b l − a
l ). Example 13.2 A simple example of measure space is given by a finite set X = {x 1 , . . . , x N } with the σ-algebra A = P(X), the set of parts of X. A measure is defined by assigning to every element x i ∈ X a real number p i ≥ 0. The measure of the subset {x
1 , . . . , x i k
i 1 + . . . + p i k . If N i =1 p i = 1 the measure is called a probability measure. Interesting examples are given by X = {0, 1} or X = {1, 2, 3, 4, 5, 6} with probabilities p 1 = p
2 = 1 2 and p
1 = p
2 = . . . = p 6 =
6 , respectively, which can be chosen to represent the probability spaces associated with the toss of a coin or the roll of a die. Example 13.3 Let (X i
A i , µ i ), i = 1, . . . , l, be measure spaces. The Cartesian product X = X 1
l has a natural structure of a measure space, whose σ-algebra A is the smallest σ-algebra of subsets of X containing the subsets of the form A 1 × . . . × A l , where A i ∈ A
i , i = 1, . . . , l. On these subsets the measure µ is defined by µ(A
1 × . . . × A l ) = µ
1 (A 1 ) . . . µ l (A l ). (13.1) It can be proved (see Lasota and Mackey 1985, theorem 2.2.2, p. 24) that there exists a unique extension of the measure µ defined by (13.1) to the σ-algebra A of X. The space (X, A, µ) thus obtained is called the product space and the measure µ is called the product measure. If X 1
l = {0, 1} or {1, 2, 3, 4, 5, 6} and the measures µ i coincide with the measure defined in the previous example, the product space coincides with the space of finite sequences of tosses of a coin or rolls of a die, and the product measure with the probability associated with each sequence. D efinition 13.6 If µ(X) = 1, a measure µ is called a probability measure and the triple (X, A, µ) is a probability space. In what follows we sometimes denote by M(X) the set of probability measures on a measure space (X, A, µ).
548 Analytical mechanics 13.2 13.2
Measurable functions. Integrability The theory of Lebesgue measurable functions (see Giusti 1989), with its most significant results (the theory of integration, Fatou’s theorems on monotone and dominated convergence, the absolute continuity of the integral, and so on), can be easily extended to the functions f : X → R, where (X, A, µ) is an arbitrary measure space (see Rudin 1974). We recall first of all the notion of an integral of a measurable function. D efinition 13.7 Let f : A → [−∞, +∞] be defined on A ⊂ X belonging to a σ-algebra A on X. The function f is called measurable (with respect to A) if {x ∈ A | f(x) < t} ∈ A, ∀ t ∈ R. It is possible to prove that the inequality f (x) < t can be replaced by one of the following: f (x) ≤ t, f(x) > t, f(x) ≥ t. To define the integral on a measure space (X, A, µ) consider first the so-called simple functions, of the form g =
n i =1 α i χ A i , (13.2) with n finite, α i ≥ 0, A
i ∈ A disjoint and χ A i
i , and hence χ A i (x) = 1, if x ∈ A i , 0, if x
∈ A c i . In this case we define X g dµ =
n i =1 α i µ(A i ). (13.3) In particular X χ A dµ = µ(A), ∀A ∈ A. If f : X
→ [0, +∞], we set X f dµ = sup g ∈ G (f ) X g dµ,
(13.4) where
G(f) is the set of simple functions such that g ≤ f. Finally for a generic f : X → [−∞, +∞] we define X f dµ = X f + dµ − X f − dµ, (13.5) where f
+ (x) = max(0, f (x)), f − (x) = max(0, Download 10.87 Mb. Do'stlaringiz bilan baham: 
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