Ergodic Problems of Classical Mechanics (The Mathematical physics monograph series)

->-00

(or +

00;

CPt m

Figure 13.9

This property will be proved general for C-systems. REMARK

13.10

The previous construction is quite general. Let (V,
where v ( V, 4

S

(

[0, 1],
M, we mean a completely integrable field of k-planes M. The connected complete integral manifolds are called the sheets. They

By a foliated manifold

over

are k-dimensional submanifolds.

60

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

§14. Geodesic Flows on Compact Riemannian Manifolds with Negative Curvature

We turn next to an important example of C-system. DEFINITION 14.1. MANIFOLDS WITH NEGA TIVE CURVATURE 5

Let v be a point of a Riemannian manifold V and TVv the tangent vector space at v. Two noncollinear vectors e 1 , e 2 of TVv define a twoplane (e 1 , e 2 ). The geodesics of V emanating from v and tangent to (e 1 , e 2 ) generate a surface

~.

~

The Gaussian curvature of

is a Riemannian submanifold of V. ~

at

v

is called the sectional curvature

p(e 1 , e 2 ) of V in the two-plane (e 1 , e 2 ). If the sectional curvature is

negative for every (e 1 , e 2 ), V is called a manifold of negative curvature. Then, if V is compact, the continuity of (e 1 , e 2 ) implies there exists a constant _k2 which bounds the sectional curvature from above. Appendix 20 provides an example of a manifold with negative curvature. INSTABILITY OF THE GEODESICS 14.2

The geodesic flow ¢ t of a Riemannian manifold V describes the movement of a material point on the frictionless manifold II without external forces [(see (1. 7)], If V has negative curvature the geodesics are very un-

6

stable: if v, Vo ( Tl V , the distance l¢tV, ¢tVol increases exponentially with t. To be precise, we. have the following result: LOBA TCHEWSK Y - HADAMARD THEOREM 7 14.3

Let V· be a compact, connected Riemannian manifold of negative curvature. Then, the geodesic flow on the unitary tangent bundle M

=

Tl V

is a C-flow. ~ppendix

21 details the proof we sketch here (see Figure 14.4).

For further infonnation see S. Helgason[I], Chapter 1. 6 .

V=

universal coyering of

V.

This theorem is "due to Lobatchewsky for surfac:.es of constant negative curvature. Hadamard

[1]

extended it to surfaces of arbitrary negative curvature.

61

UNSTABLE SYSTEMS'

y'

positive asymptotes negative asymptotes Figure 14.4

Let y (u, t)

= y (t) = y

etrized by arc length t.

be a geodesic emanating from u ( T1 V and paramLet x be a point of V. There exists a geodesic

Y1 . issuing from x and passing through- y (t1)' As t1 .... + 00, Y1 converges to a limit which is a geodesic y '(u', t) emanating from u' ( TVx ' For a suitable choice of the origin of y, it can be proved that:

(14.5)

distance (y (t), y '(t»

5. b· e->'"

t ,

t ::::

0,

where the constants b and A are positive and independent for y, y', t.

Geodesics such as y' are known as the positive asymptotes to y. They can be proved to be orthogonal trajectories to (n -1) -dimensional submanifolds (n = dim V): the so-called positive horospheres S+. Let us denote by S+ (u) the horosphere emanating from the origin of u ( T1 V and which is orthogonal to the positive asymptotes of y (u, t). This horosphere can be interpreted as an (n -1) -dimensional submanifold of T1 V: S+(u) is the union of its. orthogonal unitary vectors oriented as u. The tangent plane at u of S+(u) C T1 V is an (n -1) -plane Yu of T( T1 V).

62

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Exchanging the role of t and - t, we define negative asymptotes and negative horospheres S- in the same way. The tangent plane at u of S-(u) C Tl V is an (n -1) -plane Xu of T (T1 Vu). From the very definition,

we have:

where' Zu is the one-dimensional space generated by the velocity-vector of the geodesic flow. That is the condition (1) of C-flows (Definition 13.3). Condition (2) comes from (14.5). Observe that the fields Xu and

¥u are completely integrable. Their

integral manifolds are the horospheres S+ and S-. Both of these foliations are invariant under the geodesic flow, for the horospheres are orthogonal trajectories of (n-l) -parameter families of geodesics of V . . We turn to prove that general C-systems admit two invariant foliations.

§15. The Two Foliations of a C-System Let(M, ¢) be a C-diffeomorphfsm; Xm (resp. Ym ) denotes the k-di-

mensional dilating space at m ( M (resp. the IGdimensional contracting space). A Riemannian. metric on M is definitively selected. Hence, Xm and Ym are:Euclidean su~spaces of TMm' SINA(THEOREM 8

15.1

Let (M, ¢) be a C-diffeomorphism, then: (1) There exist two foliations

X and

~ that are invariant under¢

and that are respectively tangent to the dilating field Xm and the contracting field Ym. Hence, these fields are always integrable. (2) Every diffeomorphism ¢': M

-+

M, C 2 -cIose enough to ¢, is a

C-dilleomorphism. The dilating and contracting foliations

X, and

~' of ¢'

depend (;Ontinu6usly on ¢'. 8 This w~s proved essentially in the paper by.V. I. Arnold and Y. Sinai [6]; al~hough

their discussion was concerned with the particular case of small perturba-

tions of automorphisms of a two-dimensional torus, the proof extend to the general case.

63

UNSTABLE SYSTEMS

Appendix 22 completes the proof we sketch here. CONSTRUCTION

15.2

The. space K of the fields p of the tangent k-planes to M inherits a natural metric Ip1 - P2 1 which makes it into a complete metric space. Let

p be such a field, and p (m) the k-plane of TM m' The diffeomorphism cp induces a mapping cp**: K ~ K: cp**(m)

=

cp*p(cp- 1 m).

where cp * is the differential of cp which maps a k-plane of TM m onto a k-plane of TM cp(m) The dilating and contracting fields X and Y of cp are clearly fixed points of cp**. It can be proved (see Appendix 22) that the axioms of Csystems imply that cp ** (or a positive integer power (cp **)n) is contracting in a neighborhood of the dilating field X:

(15.3) for

IX -P 1 1 < 0, IX -P 2 1 < 0,

°

small enough. Of course, (15.3)

and

still holds for any diffeomorphism cp'. C 2 -c1ose enough to cp, since cp'* is C 1-close to cp *. We deduce from the contracting mapping theorem that a mapping verifying (15.3) admits a fixed point. The fixed point of the mapping cp is X, but for cp' it is another field p'. Clearly:

p'

(cp ,**)n X ,

lim

=

n=~

cp'*p'(m)

=

p'(cp'm) ,

and the field p' is dilating for cp'. A similar study of (M, cp -1) leads to the contracting field of cp' that is close to Y. Thus cp' is a C.diffeomorphism. THE INVARIANT FOLIATIONS

15:4

First assume that (M, cp) possesses two invariant foliations

!

and

tangent, respectively, to X and Y. Then, the same property holds for (M, cp '). In fact, the invariant field p' of cp' is obtained as:

p'

=

lim (cp ,**)n X n=oo

y,

64

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

But (¢ ,**)n X is the field of the i
:X:'.

This completes the proof of the part (2) of Theorem (15.1). As a matter of fact, the previous argument proves that there exists a foliation

:X:.

Let us

cover the compact manifolds M by a finite number of local charts (C j , !/Jj); each C j is a neighborhood of a point mj and !/J j is the mapping !/Jj: C j Rn

(n

=

->

dim M).

Consider a foliation

:X:?1

of C.1 such that 'fJ ./..I CX?) be the foliation of I

!/J.( C.); which consists in planes parailel to !/J~I (X ml ). If the local charts I I

are selected small enough, then the tangent plane XO(m) of I

:X:?I

at m

f

C.I

is close enough to the dilating plane Xm ' for all' m. Obviously there are

se~eral planes.

X?(m) passing through m if m belongs to several C I.. I

Consider the foliations

:X:7

=

¢n:x: jO of ¢nC i' These foliations cover

M and (15.3) implies that their fields of tangent planes con verge to X as

n

->

+ 00. We deduce readily that there exists a limit foliation :x: tangent to

X. This concludes the proof. REMARK

15.5

Each sheet of :x: is C1-differentiable. But the foliation :x: is not required to be smooth: the normal derivative to t!le sheets may not exist. If k = I = 1, the field X is C1-differentiable (see Arnold and Sinai [6]). It

seems probable that :x: is not smooth in the general case. Anosov concocted an example where :x: and 'Yare not C 2 -differentiable. REMARK

15.6

The preceding proof extends to C-flows. r

§16. Structural Stability of C-Systems In this section we prove that C-systems are structurally stable. DEFINITION

16.1'.

STRUCTURAL STABILITY

(A) Diffeomorphisms

Let M be a compact, smooth manifold and ¢: M able diffeomorphism...

-+

M a CT-differenti_

6S

UNSTABLE SYSTEMS

By definition, ¢ is structurally stable if given a neighborhood V (Id M )

of the identity Id M (in the CO-topology 9) there exists a neighborhood W(¢) of ¢ (in the Cr-topology) such that if

t/I (

W(¢), then there exists an ho-

meomorphism k ( V(IdM ) making the following diagram commutative:

that is: k.· ¢

=

onto the orbits (B) Flows

t/I. k. In of It/ln I n

other words, k maps the orbits of (

I¢" In ( ZI

ZI. \

Let M be a compact smooth manifold and X a Cr-differentiable vector field which generates a flow ¢ t:. X(m) =

d.,l., m I at 'f't t=

°,

m (M.

By definition, ¢t is structurally stable if given a neighborhood V(IdM ) of the identity IdM (CO-topology) there exists a neighborhood W(X) of X (C r_ topology) such that if Y (W(X), then there exists an homeomorphism k ( V(IdM) that maps each orbit of X onto an orbit of Y. In the future we assume r S 2. REMARK

16.2

One might restrict k to be a diffeomorphisjJlJather than an homecimor~ phism. Then, consider in R2 the system: x = y,

y = -x-Ky,

where K is a positive constant. The orbits are spirals focusing to the singular point (0,0) (see Figure 16.3). But, according to the Poincare theory 10 of proper values, K is a continuous invariant of diffeomorphisms.

Two mappings (or two vector fields) are C'-close if their derivatives of order inferior to r+ 1 are close. 9

10 See, for instance, K. Coddington and Levinson

[1].

66

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Consequently, two systems with distinct values of K would not be topologically conjugate and the focus (0,0) would not be structurally stable. y

x

focus

Figure 16.3

In the continuous case, one could be tempted to propose a definition quite similar to that of the discrete case: there exists an homeomorphism k ( V (Id M) that makes the following diagram commutative for all t:

where rjJ t is the flow generated by Y. But, in such a definition, a limit cycle (see Figure 15.4) would not be structurally stable since its period is a continuous invariant (see footnote 10, p. 65). Two problems arise naturally: (1) What are the structurally stable phase-portraits? (2) Given a manifold, are the structurally stable vector fields generic (or dense) in the space of the vector fields?

UNSTABLE SYSTEMS

67

Figure 16.4

Andronov and Pontrjagin [1] gave an affirmative answer if M is the sphere S2. The other two-dimensional manifolds were studied by Peixoto

[1].

1£ the dimension is greater than two, the situation is involved. For instance, the system (13.1) is structurally stable but very complicated 11 (ergodic, the cycles are everywhere dense, and so on). On the other hand, Smale [2] constructed an example (M, ¢) such that: dim M ~ 3, every diffeomorphism close enough to ¢ is not structurally stable (see Appendix 24). This invalidates the genericity of structurally stable systems. ANOSOV'S THEOREM 12

16.5

C-Systems are structurally stable. Sketch of the proof (see Appendix 25). Let (M, ¢) be a C-diffeomorphism, and W(¢) a neighborhood of ¢ (C 2 -topology) in the space of the diffeomorphisms of M. We prove next that given ¢'

f

W(¢), there exists a small and well-defined homeomor-

phism k: M .... M such that:

11

See S. Smale [1].

12

See Anos ov

[11.

68

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

¢'==k.¢.r 1

sup d[k {m), m] < £ , 'm£M

where d ( , ) is the Riemannian distance. We prove next that £, which depends on ¢', is bOlmded from above: sup

¢£ W(¢)

£ < £1 .

Hence the structural stability holds good. In fact, we make correspond to £1 a neighborhood W(¢) such that given ¢'

£

W(¢) there exists a

homeomorphism k: M ...• M satisfying:

sup d[k (m), m] < f:: 1

m£M

Let ¢'

£

W(¢) be a diffeomorphism, C 2 -close to

¢. We already know that

¢ , is· a C-diffeomorphism and ¢ and ¢ 'have invariant dilating foliations X. X' and invariant contracting foliations Y. Y' (Sinai's theorem, §15.1). If there exists an £-homeomorphism k: M ... M such that ¢' ==k· ¢. k- 1 , setting m' == km, we have: (16.6) for any n

£

Z.

From the very definition of C-systems we see that there exists at most one point m' satisfying (16.6). In fact, for any

in one or the other case n ... +

00

~.f,

0, we have:

(or -(0). The uniqueness of the homeo-

morphism k Qeing proved, if k exists we obtain it by making correspond to each point m the only point m' that satisfies (16.6) for all n. Theproblem is reduced to·proving that such a point m' exists. The construction of m' is as follows. The images ¢ 'n f3 (n > 0) of each dilating sheet

f3

£

X', close to m,

clearly have some points neighboring ¢n m (see Lemma A, Appendix 25

69

UNSTABLE SYSTEMS

for the precise meaning). It can be proved (Lemma B, Appendix 25) that there exists, among these sheets, a unique sheet (j(m) such that its images

cp '" (j(m) are still close t~ cp"(m) for all n


The same device proves that there exists a unique contracting sheet 8(m) (

'Y'

whose images cp '"8(m) are close to cp"(m) for all n (' Z. The

foliations ~' and

'Y'

are transverse; thus (3(m) and 8(m) intersect in a

unique point m' in the neighborhood of m. It is eas y to see that cp '(m ') =

(cp(m)) , and that m' varies continuously with m; thus the mapping k:

m ... m' is the desired homeomorphism.

C-FLOWS 16.7

Anosov's theorem extends to C-flow. ~(m)

m

Figure 16.8

A similar construction (Figure 16.8) gives two sheets, (3(m) and 8(m). They are formed with orbits of cp; (respectively for t which are

a~ymptotiC

->

+ 00 and t ... -(0)

to an orbit of cp; that is their intersection. As a

geometric (nonparametrized) curve, this orbit is close to the orbit cp/m. The desired homeomorphism k is obtained by making correspond to m the nearest point 0f (3(m) n 8(m) in the. Riemannian sense.

70

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

§17. Ergodic Properties of C-Systems Throughout this section we study C-systems that possess a positive and invariant measure 11. Thus, in contrast with the preceding sections, we 8!e concerned with classical dynamical systems (see Definition 1.1). Let us begin with ergodic properties of the automorphism (13.1): M is the torus I(x, y) (mod 1) I; the invariant measure is dll = dx dy; the automorphism is:

¢: ( ; ) -THEOREM

G~ )(;)

(mod 1) .

17.1

(M, 11, ¢) is a K-system

With a view to proving this theorem, we construct a subalgebra (f of THE SUBALGEBRA

(1'. 17.2

The matrix:

(~ ~) y

o

x Figure 17.3

f.

71

UNSTABLE SYSTEMS

°

has two real proper values A1, A2 : < A2 < 1 < A1 with corresponding proper directions X and Y. We split the unit square of (x, y) into a square

(3) and four triangles (1, 2, 1 " 2 '), with sides parallel to X and Y (see Figure 17.3). By pairwise identification of the opposite sides of the unit square, 1 and 1', 2 and 2', and 3 give rise to three disjoint parallelograms P1,P2, P3 on the torus M. Again split P1, P2 , and P3 into parallelograms the sides of which are parallel toX and Y and whose lengths are inferior

to a constant A. We get a partition {3 of M and we set:

a

~

V (p-n{3. n~O

The subalgebra

ct

is the closure of the algebra m(a) generated by a (see

Appendix 18).

17.4

LEMMA

If the constant A is small enough, then: (1) The atoms of

ct

are segments that are parallel to the contract-

ing direction Y. (2) Let I be a given positive number. The measure (in the sense of dxdy) of the union of those atoms whose length is inferior to I isbound-

ed from above by C· I, where C is an absolute constant. In particular, almost every atom of

ct.. is a

segment and does not reduce to a point.

Proof:

We argue in the covering plane (x, y). An element of {3 V ¢ -1 {3 is of the form: B1 n¢-l B2 ;

B 1, B2

'f

{3.

¢ -1 B 2 is a paraUelogram with sides parallel to X and Y and whose lengths are, respectively, inferior to sections B1 ¢-l B2

Ai- 1 . A

and A;l. A. On M, the inter-

n ¢ -1 B2 results from the intersection on

with six parallelograms B 1, TB 1, l

'that are deduced ,from B I C

(x, y) ~

M of

1B 1, tB1' tTB1' tT- 1B I ,

[0: 1] x [0, 1] by the translations

t ~ (-1,0). The projections of these six parallelograms 1nto

T ~ (1,0),

72

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

x

it

c (X, y) =

are six segments, the lengths of which are inferior to A and whose mutual distances are greater than K - A, where:

K as Ik 1 + 1 k'i

I<

=

inf 1 k cos e-k' sin e 1

'

0; k = -1,0,1; k' = - 2, -1, 0, 1, 2; and

e is the angle

of X with Oy. Consequently, if 2A is inferior to K, ¢ -1 B 2 intp.rsects at most one of those six parallelograms. Thus, on M, Bin ¢-1 B2 is a connected set: it is a parallelogram with sides parallel to X and y, the lengths of which are, respectively, inferior to A~1 • A and A. Exchanging {3 and (3 V ¢-1{3, the above argument applies for Max(A

1 1 A, A)

::; A.

TJlUs, the elements of:

are parallelograms with sides parallel to X and Y

and the lengths of

which are, respectively, inferior to A~2 A and A. By induction on n, we see that the c~lements of {3V¢-I{3V···V¢-n{3 are parallelograms with sides parallel to X and Y and the lengths of which are, respectively, inferior to A1- n A and A. As n

-+

+ 00, we obtain the first

part of Lemma (17.4), Al > 1. Let L be the sum of the lengths of the sides of the elements of {3 that are parallel to X. The analogous sum for

¢P{3, p ( Z, is

At· L.

Thus, the analogous sum for a is bounded from

above by. def

L

I-A -1

C.

1

Some of the elements of a have a side parallel to Y with length inferior to 1. Consider the set of these elements and dellote by m the measure of their union. From the preceding formula we deduce at once that m < C· 1 ; this proves part (2).

73

UNSTABLE SYSTEMS

THE CONTRACTING FOLIATION 17.5 Consider a nonnull constant vector field on M whose vectors are parallel to the contra(.ting direction Y. The integral lines are called the con-

tracting sheets (see §15). This foliation is called ergodic if any union of sheets, with positive measure, coincides with M up to a set of measure zero. LEMMA

17.6

The contracting foliation is invariant under ¢ and ergodic. Proof: The invariance follows from the invariance of Y. The contracting sheets are the orbits of a translation group of M. To prove ergodicity it is sufficient to prove that each sheet is everywhere dense (Appendix 11) cr, equivalently, is not closed (Jacobi theorem; Appendix 1).

Assume

there exists a closed contracting sheet F with length f. The invariance of the foliation implies that ¢n F is again a closed sheet with length

A; .f.

Since 0 < 1..2 < 1, we have

A;' f

.... 0 as n ... + 00.

But this con-

tradicts the obvious fact that the length of any sheet has to be greater than 1.

Proof of Theorem 17.1: According to the very definition of U': U' C ¢U' this is the condition (a) of Definition (11.1). Let us prove the condition (c): 00

V ¢nU'

~

~

1.

n~O

It is sufficient to prove that the atoms of 00

V ¢na n~O

have arbitrarily small length. According to Lemma (17.4) the atoms of

a V .. · V ¢na are of the form: Aon¢A1n .•• n¢nA n ;

AO, ... ,An (a,

74

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

where the A;'s are segments parallel to Y and with length inferior to A. Since A is small enough, then

Ao n ¢Al n ... n ¢n An

is a connected

segment (see argument of Lemma (17.4)). The length of this segment is inferior to A2 • A, for ¢ is contracting in the direction Y As n

-+

+ 00, we have A2 A

-+

(0 < A2 < 1).

0; this proves the condition (c).

We now turn to the condition (b). Let H be an element of positive measure of

-00

H is the union of atoms of (j', the union of atoms of ¢-1(j', ¢-2(j', and so

on. Let E be an arbitrary positive number inferior to 1. Take a number I such that: C·I

<

E· p.(H),

where C is the constant of Lemma (17.4). Up to a set E of measure inferior to E· p.(H), the atoms of (j' have a side parallel to Y whose length is greater than I. Since ¢-1 is measure-preserving and dilating in the direction Y with ratio A;l, one concludes that, up to a set ¢-nE of measure inferior to €. p.(H), the atoms of ¢-n(j' have a side parallel to Y whose length is greater than

~-n

. 1. As n

-+

+ 00, we obtain-up to a set

of measure inferior to E· p.(H) -H is the union of contracting sheets. From the arbitrariness of E and from the ergodicity of the contracting foliation (Lemma (17.6)), one deduces at once that p.(H)

=

1. Whence:

00

(Q. E. D.)

The above arguments extend to general C-diffeomorphisms. The delicate point consists in constructing a partition similar to

f3.

A theorem due to

Anosov [2] overcomes this difficulty. First we need a definition. Consider a foliation of an n-dimensional Riemannian manifold Minto p-dimensional sheets.

Let us take any small element of an (n-p)-

75

UNSTABLE SYSTEMS

dimensional mc....,ifold II transversal to the sheets. Assume that near II there is another such manifold II' and that each sheet passing through m f

II intersects II' in a point m' near m in the induced Riemannian metric

of the sheet. Consider the map f: II.... II' taking m to m '. DEFINITION

17.7

If for any such pair of neighboring manifolds the map f has a continuous generalized Jacobian, and if for smaIl deformations of II' this Jacobian varies continuously, then the foliation is caIled absolutely continuous.

For both of the two foliations of a C-system (see §15), Anosov proved: THEOREM

17.8

The foliations ~ and Yare absolutely continuous.

We cannot give here the proof of Theorem (17.8), which is too long. We only announce the final results obtained with its help. The statement of Theorem (17.1) extends and leads to the following theorems (Anosov [2]). THEOREM

17.9

Every C-system (C-diffeomorphism or C-fIow) is ergodic. THEOREM

17.10,

(SEE SINAI

[11])

Every C-diffeomorphism is a K-system.

For C-flows the situation is more complicated. The C-flow ample (15.5) has nonconstant proper functions. Consequently,

1>t of Ex1>t does not

have Lebesgue spectrum (see §1O) and cannot be a K-flow (Theorem 11.5). The following theorem proves that, in some sense, Example (15.5) is the only one exception. THEOREM

Let

1>t

17. 11 be a C-fIow on a compact, n-dimensional manifold M, then:

(1) either (2)

1>t

1>t

is a K-system; or

has nonconstant proper functions.

In case (2), such a proper function is continuous and there exists a compact, (n -1) -dimensional submanifold V of M and a C-diffeomorphism

76

1>:

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

V

->

V such that

1>/

is the one obtained from

1>

by means of the con-

struction described in (13.10), up to a change of the time scale

t (t ...

C· t, C = constant). COROLLARY 17.12

The geodesic flow on the unitary tangent bundle of a Riemannian manifold 'V of negative curvature is a K-system. Proof: According to Theorem (14.3), the geodesic flow from Theorem (17.9),

1>/

1>/

is a C-flow. Hence,

is ergodic.

On the other hand, by passing to the double covering, we may assume V orientable. If V is two-dimensional, it is not diffeomorphic to· the torus

T2 since the Gauss-Bonnet formula implies that the ·E;uler-Poincare characteristic is negative. Consequently, according to Corollary (A 16.10) of Appendix 16, the only continuous proper functions of

1>/

are constants.

The second case of Theorem (17.11) cannot occur and this theorem proves that

1>/

is a K-flow. 13

Thus,

1>/

has positive entropy (see (12.31)), has denumerably multiple

Lebesgue spectrum 14 (see (11.5)), is mixing 15 (see (10.4)), and ergodic 16 (see (8.4)).

§18. Boltzmann-Gibbs Conjecture The methods and ideas of the preceding sections are applicable to cere tain problems of Classical Mechanics, for ex-ample to the Boltzmann-Gibbs model of a gas. This model consists of hard spheres contained in a parallelepipedic box with rigid walls. Collisions between spheres, or spheres

13 This result is due to Smai

[10] for compact surfaces of negative curvature.

14 This result is due to Gelfand and Fomin [2] for arb~trary dimension and constant negative curvature. 15 This result

1S

due to E. Hopf

16 This result is due to Hedlund

[1l in the case of constant negative curvature. [1l and E. Hopf h] for surfaces of negative

curvature and manifolds of constant negative curvature.

77

UNSTABLE SYSTEMS

and wall, are supposed to be perfectly elastic. Sinai has proved ergodicity17 of this system on each manifold T

=

constant ~ O.

Ergodicity derives from the collisions. As the simplest model let us consider the motion of two perfectly elastic circular particles on the surface of a

two~dimensional

torus having a Euclidean metric. For simplicity

we consider one of the particles as fixed. The second particle (which can now be regarded as a point) moves on a "torus billiard table" (Figure 18.1), being reflected from

t~e

fixed circumference according to the law "the

angle of incidence a is equal to the an gle of reflection

f3."

Let us, at the same time, consider an elliptic billiard table (Figure

18.2). The ellipse can be regarded as an oblate ellipsoid on which the I

I

I I I I

I I

-;------------I

Figure 18.1

17

This result was conjectured as the "quasi-ergodic hypothesis." At the beginning of Ergodic Theory this conjecture occasioned sharp discussions. But the problem was, perhaps, overvalued: statistical mechanics deals with asymptotic behavior as N -> +00 (N = number of parhcles) and not as I'" +00 for fixed N.

78

ERGODIC PROBLEMS OF CLASSIC' AL MECHANICS

point moves along a geodesic, passing at each reflection from one side to the other. In the same way a torus billiard table can be regarded as a twosided torus with a hole on which the point moves along a geodesic. But if the two-sided ellipse is a'n oblate ellipsoid, the two-sided torus with a hole will be an oblate surface of genus 2. Thus, motion on our tor~s billiard table is a limiting case of geodesic flow on a surface of genus 2.

Figure 18.2

We now turn to our billiard tables and consider the curvature to which Figures 18.1 and 18.2 correspond. The ellipsoid has a positive curvature whose integral is equal to 411 (Gauss-Bonnet formula). On flattening the ellipsoid, all the curvature is accumuiated along the boundary of an ellipse. For a surface of genus 2 the integral of the t:urvature is equal to -411. Thus, a two-sided torus billiard table can be regarded as an oblate surface with negative curvature everywhere: on flattening, all the curvature is accumulated along the circumference. The preceding arguments (Arnold [4] p. 184) are not, of course, a proof of ergodicity, not even in the simplest cases we considered. Nevertheless, using methods and notions dealing with C-systems (asymptotic orbits, transverse foliations), Sinai [4], [5] succeeded in proving that the BoltzmannGibbs model is ergodic on each submanifold T more, is a K-system.

=

constant

f.

0 and, even

UNSTABLE SYSTEMS

79

The proofs of these results require hundreds of pages and contain a theory of generalized C-systems with discontinuous foliations. We only mention that the general case reduces to a "billiard table" problem in the configuration space: the colliSion-hypersurface of the configuration space plays the part of our circumference.

General References for Chapter 3 Anosov, D. V., Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature, Trudy Instituta Steklova 90 (1967). Hopf, E., Statistik der geodatischen Lin~en in Mannigfaltigkeiten negativer Kriimmung, Ber. Verh. Sachs. Akad. Wiss., Leipzig 91 (1939) pp. 261-

304. Sinai, Va, Dynamical Systems with Countably Multiple Lebesgue Spectra

II. lsvestia Math. Nauk.30 No.1 (1966) pp. 15-68.

CHAPTER 4 STABLE SYSTEMS Many dynamical systems are known, the orbits of which, with remarkable stability, fail to fill up the "energy level"

H =, Ct -ergodically

and remain (to the end) in their particular corner of phase-space. The systems that are close to an "integrable" s'ystem, and the systems to which the Theory of Perturbations of Celestial Mechanics applies, belong to this class. Let us mention, for instance, the three-body problem, th: fast rotations of a heavy-rigid body, the motion of a free point in a geodesic on a convex surface, and the adiabatic invariants. Only in recent times, "beginning with the work of C. L. Siegel [1] (1942) and A. N. Kolmogorov [5] (1954), has some progress been made in studying these systems. This chapter reports on the present status of this problem. We refer to V. Arnold [4], [5] for further details. Let us begin with an example. §19. The SWing and the Corresponding Canonical Mapping EQUATIONS OF THE MOTION 19.1

Pendulum equations are: (19.2)

q = p,

• 2· P = -w Sin q,

where w is the "proper frequency" that depends on the length of the penduillm. A swing is a pendulum whose length I periodically varies (Figure 19.3). Its equations can be written as

81

82

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Figure 19.3

(19.4)

q = p,

p

=

_w 2 (t) sin q,

where w(t+ r) '" ,w(t). Appendix 5 studies (19.2) by using the phase-plan.e. Equations (19.4) contain the time variable t explicitly.

Thus, we deal

with the study of a vector field in the three-dimensional space p, q, t (see Figure 19.5).

Figure 19.5

83

STABLE SYSTEMS THE MAPPING T 19.6

The initial data p(O) = Po, q(0) = qo define an orbit p = p(t), q

=

q(t). The time periodicity of Equations (19.4) allows one to identify the surfaces t

=0

and t

=

the space R1 x Sl x Sl T of the surface

r. Thus, (19.4) can be regarded as equations in Ip, q mod217, t mod

=

~ (t= 0)

rI. This defines a mapping

into itself:

Clearly: p (nr), q (nr) = Tn(po' qo) ,

and the study of p (t), q (t) as t ... + oc reduces to the study of the iterates Tn, n ( Z. The mapping T is canonical, for the Equations (19.4) are canonical. Therefore, T preserves the area dp 1\ dq. The equilibrium positions p

= 0,

q

=

k· 17 (k= 0, 1) are solutions of

(19.4). Thus, the points p = 0, q = krr are fixed points of T. 19.7

THE INTEGRABLE CASE

Let us

be~in

"inte~rable

with the

case" (,)

=

constant, to get acquaint-

ed with the mapping T. In this case, the system is conservative; thus, the energy is invariant. In other words, on the surface ~,the curves (Figure AS.l, Appendix S):

r:

'hp2 -

(,)2

cos q

=

h

are invariant under T. L~t us consider the part of ~ that is interior to the separatrix (h

< (,)2).

We use "action-angle variables 1, ¢" to study T. It can be proved (Appendix 26) that there exists a canonical transformation p, q ... 1, ¢ such that the equation 1

=

Ct defines an invariant curve

nate ¢ (mod 217) is the angular coordinate of

r[

r

=

r[.

The coordi-

and, in coordinates 1, ¢'

the mapping T becomes: T: 1, ¢ ... 1, ¢ + AU) • ~his

means that the curve

r[

rotates through an angle A(I) that is con-

84

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

stant along each curve 11 (if ¢ is selected as parameter), but depends on the curve. It is readily seen that lim A(I) to the separatrix V2p2 - cu 2 cos q

=

=0

as the curve 11 converges

cu 2 , and that 1 lim A([)

=

CUT

as 11

converges to (0,0). Hence, some curves I~ are rotated through an angle

A([) commensurate with 277, and others through an angle incommensurate with 277. Let us consider the iterates of T. If x curve 1 such that

A=

277 ~ , then: Tnx

=

belongs to a

= (p, q)

x .. Thus, each point o~ 1 is

a fixed point of Tn, and the orbit of x consists in a finite number of points (Figure 19.8). If the angle A (I) that corresponds to 11 is

incommen~urate

Figure 19.8

with 277, then the points

Finally, observe that the equilibrium position p

= q =

IXol = jp~ + q~ IT" xol remains small for any

n ( Z.

is small enough, then

1

T" x are everywhere dense on 11. (Appendix 1).

The linear part of T at zero (p, q«

1): It

= p,

fJ

= _CU 2 q

0 is stable: if

is a rotation with

frequency CU (see Appendix 5). This gives rise to a rotation with angle II)T after time

T.

85

STABLE SYSTEMS

NONINTEGRABLE CASES 19.9

Suppose that

is a nonconstant periodic function. We assume addi-

W

tionally that w(t) is close to a constant w O' for instance:

o< The mapping TE , which corresponds to

T. This mapping

~

WE'

E

«1,

v = 27T/r.

is close to the above mapping

preserves the area dp /\ dq and the point (0,0), but

neither the energy nor the curves

rl

are preserved. The main purpose of

the Theory of Perturbations is to study the behavior of the iterates

n

-+

+ 00 and E« (1) E

«

I,

T; as

1. There are two ways to consider the problem: -00

Inl

(2) E« I,


00

(Theory of Stability); (Asymptotic Theory of the k-th approximation).

The main result of the Theory of Stability is due to Kolmogorov (see §21). In our example, it reads as follows: THEOREM 19.10

If E is small enough,

curves

~

t~an

the mapping TE has invariant analytical

that are close to the invariant curves

small enough, these curves trices (lhp2 -

wg

cos q

:S

~

wg),

r

of T. Besides, for E

fill up the domain interior to the separaup to a set whose Lebesgue measure is

small with E . Roughly speaking, for E small enough, the invariant curves

r I , whose angles

1I.(I) are "incommensurate enough" with

27T, do

not collapse but are only slightly deformed. The images TEnx of x are all contained in

t ~

~.

To an invariant curve ~ of ~ corresponds a torus 2 ,of the space p, q, t (q mod 27T, t mod r) that is invariant under (19.4). These tori divide

the space p, q, t (see Figure 19.11). Therefore a trajectory of (19.4) peginning in a gap between two invariant tori cannot pass out of this gap. Thus, Theorem (19.10) allows us to reach conclusions regarding the stability of motion. 2 "That is a conditionally periodic motion of the swing.

86

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Figure 19.11

Theorem (19.10) is a direct corollary of Theorein (21.11) whose proof can be found in Appendix 34.

§20. Fixed Points and Periodic Motions Let us study the fixed points of the mapping

~

and its iterates, to

understand better the structure of the gaps between two invariant curves

r;.

These points correspond to the periodic motions of the swing.

THE ELLIPTIC AND HYPERBOLIC POINTS 20.1

In the neighborhood of a fixed point, a mapping reduces to its linear part, that is, its differential, up to terms of ~econd order. The differential of a canonical mapping is a linear canonical mapping. Linear canonical mappings are studied in Appendix 27. If this linear mapping is hyperbolic (resp. hyperbolic with reflection, resp. elliptic), then the fixed point is called hyperbolic (resp. hyperbolic with reflection, resp. elliptic). Hyperbolic fixed points are easily proved to be unstable, not only for the linear mapping, but also for the nonlinear mapping (Hadamard).

The

problem of stability of elliptic points is known as the Birkhoff problem. Points of elliptic type are, in general, stable in the two-dimensional case (see Appendix 28).

STABLE SYSTEMS STABILITY OF THE LOWEST POSITION OF EQUILIBRIUM 20.2

Now, consider the fixed point p

If E

=

0, then the mapping Te

=T

= q =

0 of the mapping ~ of §19.

is elliptic at the origin; in fact, the

differential of T is an elliptic rotation with angle A = CUT Thus, for E small enough and A f, krr (k

=

= 21T(CU/V).

1,2, ... ), the mapping Te' is

elliptic" too. In other words, stability can change only for: (20.3)

v

2cu 2cu 2cu = -1, 2' 3 ' ....

A sharper computation proves that, for these values of v (called values of "parametric resonance"; see Appendix 29), the mapping Te is hyperbolic at p

= q =

0 (see Figure 20.4). In other words, the equilibrium of the

swing alters (and the swing starts oscillating) if one deflects during an integral number of half-periods of the proper oscillations. This result!s wellknown empirically. 3

E

values of parametric resonance

••• 2w

k

20) ,Figure 20.4

3 Besides, observe that the amplitude of the oscillations of system (19.4) remains small if e is small enough. For, according to Theorem (19.10), the orbit remains between two curves ~. This follows from the nonlinearity of our system: the frequency A(1) depends on the amphtude; condition (20.3) is no longer fulfilled since the oscillations do not have tIme enough to amphfy.

88

ERGuDIC PROBLEMS OF CLASSICAL MECHANICS

Te: EXISTENCE 20.5

THE FIXED POINTS OF THE ITERA TES OF

Now, consider the fixed points of 1~n. Let

r

be the invariant curve of

T formed with the fixed points of T" (see §19). We set: A([)

=

dA / dT r O.

217 E!.. , n

r

On an n-fold iteration of T each point of

returns to its original position.

This property of T is not retained for a small perturbation (T ... Te ). But Poincare proved, for E small enough, that Ten has 2kn fixed points close to the curve

r.

In fact, let us consider two curves that are invariant under

T and close to r: namely, the curves r+ and r-, with angles of rota-

tion A+ > A> A- (Figure 20.6). Thus, the mapping T" rotates r+ posi-

r-

tivelyand

negatively.

Figure 20.6

This property still holds for Ten if E is small enough. Thus, on each radius ¢

=

constant, there exists a point d¢, E) which moves under Ten

along the radius

Moreover, if E is small enough, the points r(¢, E) (O:S ¢ clos~d

1'" n

analytical curve Re close to

r.

:s 277)

form a

Now, remember that the mapping

is canonical and so is area-preserving.

Consequently, the image

89

STABLE SYSTEMS

TenRe cannot be surrounded by Re and, conversely, Re cannot be surrounded by TenRe' Thus, Re and ~nRe intersect (Figure 20.7).

The

points of intersection are fixed points of Ten, for Ten moves each point of

Re along its /adius ¢ ~n

=

Ct. This proves the existence of fixed points of

in the neighborliood of

r.

Figure 20.7 FIXED POINTS OF THE ITERATES OF

Te: CLASSIFICATION 20.8

What is the type of these fixed points? Are they elliptic or hyperbolic?

If € = 0, then they are all parabolic with proper value '\12 = 1. Consequently, for € small enough,. '\12 "" 1 and the hyperbolic case with reflection is impossible. On the other hand, consider the "radial displacement":

90

ERGODIC PROBLEMS'OF CLASSICAL MECHANICS

~n.

The function D.(¢) vanishes at the fixed points of case" these zeros have multiplicity one (~' zeros at which

~

= d~/d¢

In the "generic

f, 0). Hence, the

'> 0 separate the others, and the number of fixed points,

is even. Let us make correspond to any point x the vector whose extremity is

Te n x (see Figure 20.7). It is readily seen that the index (Appendix 27) of this vector field at a fixed point is: Ind

=

sign of ( dA • d~ \ dl d¢)

Thus, half of the fixed points have index + 1 and the others have index -1This means that half of these points are elliptic and half of hyperbolic type (an elliptie point has index + 1, an hyperbolic point has index -1). Elliptic and hyperbolic points are illustrated in Figure 20.7. Now, consider an elliptic~ fixed point: Ten x

=

x. The orbit of x is

x, Te x, ... , Ten-lx, therE!'fore:

All points of the orbit of x are fixed points of Ten and are elliptic since they l1ave the same proper value. Hence, the set of the elliptic fixed points splits into orbits consisting of bits, then there are

£.0

~

points. Let 1c be the number of such or-

elliptic points. This gives us 2kn fixed points

(elliptic and hyperbolic), as promised in §20.5.

ZONES OF INSTABILITY 20.9

We now turn to the neighborhoods of the above elliptic and hyperbolic fixed points. According to V. Arnold [7] (see also Appendix 28), each "generic" elliptic point is surrounded by closed curves that are invariant under Ten. These curves form" islands" (see Figure 20.10). Each island repeats in miniature the whole structure, with its curves C'. islands be-

91

STABLE SYSTEMS

tween these curves, and so on. Between these islands and curves

~

, re-

main zones around the hyperbolic points. In fact,4 the separatrices of hyperbolic fixed points of the Ten,s intersecting each other create an intricate network, as depicted in Figure 20.10. On discovering this, Poincart: wrote ([2], V.3, Chap. 33, p. 389): "One is struck by the complexity of thIs figure that I am not even attempting to draw. N othmg can give us a better idea of the complexity of the three-body problem and of all problems of dynamICS where there is no holomorphlc integral and Bohlin's series diverge."

Figure 20.10

The ergodic properties of the motion in zones of instability are unknown. There probably exist systems with singular spectrum and K-systems among the ergodic components.

4 See Poincare [21. V. Melnikov [1].

92

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

REMARK

20.11

The above argument does not prove there exist infinitely many elliptic islands for a given €« 1. Poincare's last geometrical theoremS proves that there exist infinitely many fixed points of ~n (n

->

+ 00) with index

0.1

o -0.1

-0.4 -0.3 -0.2-0.1

0 0.1

0.2 0.3 0.4 0.5 0.6 Y

Figure 20.12 Yr-~-.-r-.--~~~-r-.--r-~-r~-.r-~

.5 .4

"-,3

..

"

...: "

-.5-.4-.3-.2-.1

,/

/

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Y Figure 20.13.

5 See Poincare [3], G. D.'Birkhoff [1].

93

STABLE SYSTEMS

+ 1 inside the annulus located between the invariant curves

~

(Theorem

19.10). Perhaps some of these points are not elliptic but hyperbolic with reflection. Numerical computations 6 seem to support this conclusion. y 0.8 0.6

0.2

o -0.2 -0.04 -0.6 -0.8 -0.8 -0.6 -0.04-0.2

0

0.2 0.04 Q6 0.8 X

Flgure 20.14

We have borrowed Figures 20.12-14 from the work of M. Henon and

C. Heiles [1]. They depict the orbits of a mapping of type ~ computed with an electronic computer. All the points, exterior to the curves, belong to one and the same orbit!

§21. Invariant Tori and Quasi-Periodic Motions

The example we considered in §19 and §20 is a particular case of a situation which occurs for each system close enough to an "integrable" system. 6

See Gelfand, Graev. Sueva. Mlchai1ova. Morosov M. Henon. C. Helles [1].

b);

[3];

Ochozimskl. Sarychev •...

94

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

(A) INTEGRABLE SYSYEMS 21.1

If one takes a look at the "integrable" problems of Classical Mechanics,7 one finds that, for all of these problems, bounded orbits are either periodic or quasi-periodic. In other words, the phase-space is stratified

i~to

invariant tori supporting quasi-periodic motions.

E'XAMPLE 21.2 Assume the phase-space Q is the product of a bounded domain Bn of

Rn by the torus 1"'. Let P

=

(P l '

, .. , Pn )

be coordinates on Bn and q

=

(ql' "" qn) (mod 277) coordinates on Tn. The Hamiltonian equations, with

Hamiltonian function H = Ho(p), read: (21.3) The motion is quasi-periodic on the invariant tori P = Ct, with frequencies w (p). Frequen«ies depend on the torus: if

f 0, "

then, on each neighborhood o(the torus p = Ct , there are invariant tpri un which frequencies are independent and orbits everywhere dense (see Appendix 1). There exist other tori on which frequencies are commensurate; they are exceptional, that is they form a set of measure zero. Coordinates (p, q) of B n x 1'" are called "action-angle" coordinates. For all integrable systems, it can be shown (Appendix 26) that a certain (2n - 1)-dimensional hypersurface divides the phase-space into invariant domains each of which is stratified into invariant n-dimensional manifolds. If the domain is bounded, these manifolds are tori supporting quasiperiodic motions. The action-angle coordinates can be introduced into such a domain, thus, the system can be described by (21.3).

7 For instance, the motion of a free point along a geodesic on the surface of a triaxial ellipsoid or a torus (see (1.7) and Appendix 2), a heavy solid body (Euler, Lagrange, and Kovalewskaia cases),

95

STABLE SYSTEMS

(B) SYSTEMS CLOSE TO INTEGRABLE SYSTEMS 21.4 Now, we assume that the Hamiltonian function is perturbed:

the "perturbation" HI being" small enough." The Hamiltonian equations are then:

(21.5)

oH I oq

---,

p

q

For most initial data, A. N. Kolmogorov [6] proved that the motion remains quasi-periodic (see Theorem 21. 7). Consequently, (21.5) is not ergodic on the "energy surface" H = Ct and, among the ergodic components, there are components with discrete spectrum, the complement of which has small Lebesgue measure as H I is small. Assume that the function H (p) is anal ytic in a c0cnplex domain [0] of the phase-space: O(~p,

I~ pi

'R q, (0,

< p,

I~ ql

< p) .

Assume also that the unperturbed system is nondegenerate: Det

(21.6)

(U I~ 0

=

lo2H

Det __0_

op

op2

I

Select an incommensurate 8 frequency-vector w

oJ- O.

=

w*. The equations of

the invariant torus To(w*) of the unperturbed system (21.3) are p = p*, where wo(P*) = w*. Thus, the system (21.3) has frequencies w* on

To(w*). THEOR.EM

21.7

If HI is small enough, then for almost 9 all w*, there exists an invari-

ant torus T(w*) of the perturbed system (21.5) and T(w*) is close to To (w*). To be precise: That is, (w, k) ,;, 0 for all integers k. All, except for a set of Lebesgue measure zero.

96

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

For all K> 0 there exist E> 0 and a mapping p = p(Q), q = q(Q) of an abstract torus T =

I Q (mod

21T)} into

n such that,

according to (21.5)

Q = w*, and:

Ip(Q)-p*1
Iq(Q)-QI
if:

holds in [n]. Moreover, the tori T (w*) form a set of positive measure whose complement set has a measure that tends to zero as

IH 11

->

O. Proof of Theorem

(21. 7) is found in Arnold [5]. The behavior of the orbits emanating from this complement is not well known. If our system has two degrees of freedom (n = 2) then the phasespace

n

has dimension four, and the two-dimensional invariant tori that

are found divide the three-dimensional manifold H = constant.

The do-

mains of the complement set are toric annuli betwee~ these invariant tori (see Figure 19.11).

For n > 2, the n-dimensional invariant tori do not di-

vide the (2n -1) -dimensional manifold H

=

constant and those orbits that

do not belong to the tori T (w*) can travel very far along H

=

h (see §23).

(C) ApPLICATIONS AND GENERALIZATIONS 21.8 Theorem (21. 7) applies to the motion of a free point along a geodesic on a convex surface close to an ellipsoid or '8 surface of revolution. This theorem allows one to prove the stability in the plane restricted circular

three-body problem. 10

One can also deduce the stability of the fast rota-

tions of a heavy asymmetric solid body. 11 But this theorem does not apply if the unperturbed motion has fewer frequencies than the perturbed motion (degenerate case) for, in this case,

10 A. N. Kolmogorov [7]. 11 V. I. Arnold [5].

97

STABLE SYSTEMS

condition (21.6) does not hold: Det

- O.

The cases of "limiting degeneracy" of the oscillation theory (points of equilibrium, periodic motions) also require a particular study. In that direction we mention some results that generalize Theorem (21.7).

V. I. Arnold [7] proved the stability of positions of equilibrium and of periodic motions of systems with two degrees of freedom in the general elliptic case. As a corollary, A. M. Leontovich [1] deduced the stability of

the Lagrange periodic solutions for the reduced problem of the three-body (plane and circular).

V. Arnold [8], [9], [10], studied the generation ~f new frequencies from the perturbation of degenerate systems. As a corollary, one obtains the

perpetual adiabatic in variance of the action for a slow periodic variation of the parameters of a nonlinear oscillatory system with one degree of freedom, and also that a "magnetic trap" with an axial-symmetric magnetic

field can perpetually retain charged particles. Finally, quasi-periodic motions in the n-body problem have been found. If the masses of n planets..are small enough in comparison with the mass of the central body, the motion is quasi-periodic for the majority of initial conditions for which the eccentricities and inclinations of the Kepler ellipses are small. Further, the major semiaxes perpetually remain close to their original values, and the eccentricities and inclinations remain small (see V. Arnold [4]). On the other hand,

J. Moser [1], [5] generalized Theorem (21.7). Moser

abandons the requirement of analyticity of the Hamiltonian and substitutes

instead the requirement that several hundred derivatives exist.

For in-

stance, for systems with two degrees of freedom, it is sufficient (hat 333 derivatives exist! (D) INVARIANT TORI OF CANONICAL MAPPINGS 21.9 Theorem (21.7) can be reformulated by using the construction of the

98

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

"surfaces of section" of Poincare-Birkhoff. Assume that, in Equations (21.3), the first component w l of w is nonvanishing.

Consider a sub-

manifold ~2n-2 of the phase-space n2n whose equations are: ql

=

0,

H = h = constant. The orbit x (t) of (21.5) through a point x on ~2n-2 will, as t increases from zero, return to ~2n-2 and will cut ~2n-2 in a uniquely determined point Ax (Figure 19.11). If tile perturbation H 1 is small enough and w 1 (p)

0, the mapping A: ~2n-2

f.

-->

~2n-2 is well

defined in a neighborhood of the (n -1) -dimensional torus: p = Ct,

ql = O.

Since

then P2' ... , Pn; q2' ... , qn (mod 211) are "action-angle" coordinates in this neighborhood. The mapping A is canonical (see Appendix 31). Now, consider the unperturbed system (H 1

=

0). According to (21.3),

the map A may be written as follows: (k = 2, ... , n).

(21.10)

In other words, each torus p

=

Ct is invariant and rotates through w(p)

under the mapping A.

If the perturbation H 1 is small, then the corresponding canonical map1,2n-2

-->

~2n-2 is close to (21. l-O). The (n -1) -dimensional

invariant tori of

d'

are, obviously, similar to the n-dimensional invariant

ping

d':

tori of (21.5) and there is a theorem, similar to Theorem (21.7), for mappings (see Theorem 21.11). Let

n

again be the phase-space p, q:

Assume that B: p, q

-->

p '(p, q), q '(p, q) is a global canonical mapping,

that is:

¢Pdq=~Pdq, Y

for any closed curve y of

n

By

(see Appendix 33). Then, assume that the

99

STABLE SYSTEMS

functions p '(p, q), q'(p, q) - q are analytic in the complex neighborhood

[a] of a:

Let A: p, q

-+

p, q + w (p) be the canonical mapping defined by an analyt-

ic function w(p) in [a], and To(w*) the torus p = p*, w(p*) = w* that is invariant under A.. THEOREM

21.11

If B is close enough to the identity, then, for almost

12

all w *, there

exists a torus T (w*) that is invariant under BA and close to To(w*). To be precise, to any K> 0 corresponds an S> 0 and a mapping 0:

T

-+

A, p = p(Q), q = q(Q), of an abstract torus T = !Q (mod 27T)1 into

A, such that: O(Q+w*)

A

=

8·A·0(Q),

B

01-1 Q - Q+w*, and: Ip(Q)

-p*1 < K,

Iq(Q)-QI
provided that:

holds in [01. Moreover the tori T (w*) form a set of positive measure whose complement set has measure tending to zero as

Ip' - pi

+ Iq' -

ql

-+

O.

Theorem (19.10) is a direct corollary of Theorem (21.11): n = 1. Theorem (21.11) has been known since 1954, though its proof was never published.

J.

Moser [1] gave a proof for mappings of the plane (n = 1). This

proof makes use of the topology H2. Appendix 34 gives a proof for arbitrary n: the topological part is reducl'd to the technique of generating functions of global canonical transformations (see Appendix 33). 12 All, except for a set of Lebesgue measure zero.

100

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

(E) COMPARISON OF THEOREM (21.7) WITH THEOREM (21.11) 21.12 Whether each analytic canonical transformation close to A can be constructed by using a section of a suitable Hamiltonian system is unknown. Thus, Theorem (21.11) cannot be deduced from Theorem (21. 7). Even if one restricts to the Hamiltonian system (21.5), Theorems ('U.7) and (21.11) are nonequivalent. In fact, according to (21.6) and (21.10), the nondegeneracy conditions of Theorems (21.7) and (21.11): Det

I ~; I

Det

ji 0 ,

I aw I

ji 0

ap

may be written in terms of the unperturbed Hamiltonian function H 0 as follows:

(21.13)

Det

a2H 0 ap2

ji 0,

Det

a2H0

aHo

ap2

ap

aHo ap

0

.;, 0

These conditions (21.13) are clearly independent. Each of them is sufficient to emmre that there exist invariant tori. The second one ensures, additionally, that there exists such tori on each "energy surface"; this implies the stability (see Figure 19.11) for systems with two degrees of freedom (n = 2 for Theorem 21.7, n = 1 for Theorem 21.11). In most applications, both of the conditions (21.13) are simultaneously fulfilled or invalid.

§ 22. Perturbation Theory We turn next to the asymptotic theory, that is we restrict our study to the behavior of orbits for 0

< t < liE,

E being the magnitude of the per-

turbation. In contrast, non-Hamiltonian systems can be considered.

101

STABLE SYSTEMS

(A) AVERAGING METHOD 13 22.1

Let Tk = !¢ = (¢1' ... , ¢) (mod 277) 1 be the k-dimensional t~rus and B1 =

II

)1

= ([1' .. ,1 1

a bounded domain of Rl. In the phase-space

n

=

Tk x B1, consider the unperturbed system:

(22.2) It is, obviously, a generalization of system (21.3): each torus I = con-

stant is invariant and, if the frequencies ware incommensurate on this torus T, then the orbits ¢(t) are everywhere dense on T. In such a case, the motion (22.2) is called quasi-periodic on the torus T. If the frequencies are commensurate, then the closure of an orbit is a k-dimensional torus (k < n) (resonance). Consider next the perturbed system that generalizes (21.5): (22.3)

{~ I

=

w([)+ Et(I,¢)

=

EF(I, ¢)

where {

f(I, ¢ + 277) := f([, ¢) F([,

1> + 277)

Of course, for t '" 1, the eVQluti,on \Ht) -1(0)\ '" E

«

:=

F([, ¢),

E« 1.

1. Notable ef-

fects, of order 1, of the evolution appear only after a long enough time:

t '" 1/ E . Perturbation Theory proceeds to study the perturbed system as follows. Let

fi(n

be the mean:

One considers the "averaged system," or "system of evolution":

j = E·fi(]).

(22.4) For E

«

1, one expects that:

(22.5)

IHi) -j(t)1 « 1 for 0 < t < 1

E

where [(t), ¢(t) is a solution of (22.3) and j (t) is the solution of (22.4) with initial data: j (0) =

Ho).

13 Thls method goes back to Lagrange, Laplace, and Gauss, who used It in Celestia I Mechanics.

102

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Now the problem arises as follows: what relations exist, for 0 between the perturbed motion [(t) and the "motion of evolution"

< t < 1.,

I (t)

E

?

Does the inequality (22.5) 'hold? For the simplest periodic motions (k = 1) it is readily proved -(see Appendix 30 and Bogolubov and Mitropolski [1]) that if

< C·E, for 0 <

I[(t) 71(t)1

,

But the situation

be~mes

t

(U

~ 0 then:

< 1.E

more complicated as the number k of frequen-

cies increases, even for k = 2.

(B) A COUNTER- EXAMPLE 22.6 Assume k = 1= 2, a> 1 and consider the system: ¢1 = 11 ,

¢2 = 12 ,

11 = E,

i2

= Eacos(¢1-¢2)·

Of course, the system of evolution is:

11 = E,

12 = 0

(corresponding to small arrows on Figure 22.7). Consider the following initial data: 11 =12 =1 1 "'1 2 = 1, 12 .J2

¢1

0,

1 ¢2 = arcos a 1 (t)

-----

~

--....,. ~ ~ ~

J

~

--....,.

--....,.

~

~

---+

~

---+ ~

(t)

-+ ~

11 , J 1 Figure 22.7

STABLE SYSTEMS

103

Then:

1. Thus,

I[(l/s) - ](l/s)1 In other words, after the interval of time

=

lis,

1. the averaged motion loses

any relation with the real motion which remains locked in by the resonance wI = w 2 •

(C) MATHEMATICAL FOUNDATIONS OF THE AVERAGING METHOD 22.8

There exist, at least, four distinct approaches to the problem of the mathematical foundations of averaging method. All four lead to rather modest results. (1) The neighborhoods of particular solutions (for example, positions of equilibrium

F

=

0) of the averaged system can be fairly well studied.

For instance, there exist attracting tori of (22.3) which correspond to the

< t < (0) obviously holds in the neighborhood of such a torus. N. N. Bogolubov [2], J. Moser attracting points of system (22.4). Stability (for 0

[2], [5], and 1._ Kupka [1] proved that attracting tori still exist for perturbed systems. This approach does not apply to Hamiltonian systems because attracting points do not exist according to the Liouville theorem (see 1.10).' (2) One can study the relations between I (t) and] (t) for most (in the sense of measure theory) initial data, neglecting points that correspond to resonance. For instance, Anosov [3] and Kasuga [1] proved theorems of the following type: Let R ( s, p) be the subset of

n

of the initial data such that

I[(t) -](t)1

for certain 0

>P

< t < lis. Then, lim measure R (S, p) e

-+

=

0 for all p> O.

0

This approach allows one to obtain similar results for systems much more general than (22.3); whence its weakness: estimates of the measure

104

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

of R (f;., p) are not realistic and one has no information concerning the motion in R(E,p). (3) One can study passages through states of resonance, (.4) One restricts oneself to Hamiltonian systems to obtain more information. (D) PASSAGE THROUGH STATES OF RESONANCE 22.9

Let us begin with an example: ¢1 = II + 12,

1>2 = 12,

il

= E,

i2

= E cos(¢1 -¢2)·

The averaged equations are (see Figure 22.10):

j2=0.

J 1 =E,

-- -- ---- -

-~ ,~

~

-

-

----.

-

--

-----

Figure

--

•

=

¢2(O)

J

-

22.10

Consider the initial data that correspond to resonance wI ¢1(O)

I (t)

= 11 (0)

=

12 (0)-1

=

= w2:

O.

The system is easily integrated: lI(t)-j(t}1

=

1/2(t}-11

=

fiEjTcosx2.dX, o

T=..jE/2t.

l?

105

STABLE SYSTEMS

For t

=

liE, obviously,

I/( t) - J( t) I =

C· YE.

Thus, the passage through the resonance w 1 = w 2 disperses the bundle of orbits I(t), cp (t), which in the beginning differ only by phases

cp(O). The scattering of 12 after going through the resonance is of the order of

Vc (see

quencies (k

=

Figure 22.10). For a general system (22.3) with two fre-

2), one obtains 14 the following theorem:

If the quantity:

does not vanish in {l, then we have the estimate: for all 0

(22.11)

< t < !. E

Condition A ;, 0 means that the system cannot remain locked in at any resonance: (22.3) implies

In example (22.6) condition A

changes sign at 11 A

t.

=

t.

0 is violated:

12 if a> 1. This example shows that condition

0 cannot be replaced by an analogous condition for the averaged sys-

tem. The idea used in proving (22.11) is that the scattering produced by each resonance is of the order C YEand that, among the infinitely many resonances

w/ w2

=

min, only the greatest (m, n

produces notable effects. 14 V. I. Arnold [12].

< In !. ) E

106

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Passage through resonance!': for systems with more than two frequencies (k > 2) has not been studied. (E) EVOLUTION OF HAMILTONIAN SYSTEMS 22.12

Next, apply the. averaging . method to Hamiltonian systems (21.5). If condition (21. 6) of nondegeneracy holds, then most of the unperturbed orbits are ergodic on tori p = constant. Thus, it is satisfactory to write this system in the form (22.3), with I

= p, ¢ = q, k

=

I

= n:

aH 1 al

, ¢

w(I) +' E - -

~, i

- E--

aH 1 a¢

where

w The averaged system is ] = 0, for

o. In other words, there is no evo}ution for nondegenerate Hamiltonian systerns: ] = constant. Theorem (21.7) of conservation of quasi-periodic motions rigorously establishes this conclusion. In fact, Theorem (21. 7) implies that:

IHt) -](t}1 < (for all initial data if n

K for all t ( Rand

2, and

a2 H o aP

E<

Eo(K)

107

STABLE SYSTEMS

and for most initial data in the general case). This illuminates the part that conditions of conservation play in Theorem (21.7): they prevent evo1ution. ls In the same way, evolution is prohibited in Theorem (21.11), for the mapping is globally canonical. On the other hand, one also understands the part that the condition of nondegeneracy plays. In fact, in case of degeneracy, the generic orbit of the unperturbed system is ergodic on k-dimensiona1 tori (k strictly inferior to n). In such a case, the algorithm of perturbation theory allows one to predict the averaging on the k-dimensiona1 torus. Hence evolution becomes possible, even for canonical systems. IS This particular feature of canonical systems already comes out from simple examples. Let us consider the following perturbations of a center (Figure 22.13):

{t

y

-x - Ey

y

y

x.

x

Figure 22.13 The first perturbation, which is canonical, moves the orbits along a direction which is orthogonal to the perturbation, and does not give rise to evolution. The second perturbation, which is not canonical, gives rise to an evolution to zero. Volume-preserving examples with evolution can b~ constructed in the four-dimens~onal

space x, y, z, u:

x = y,

y

-X-E Y,

.i

u,

ti

-Z+EU.

108

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

EXAMPLE

22.14

Consider the Hamiltonian system:

1<
=

a2H o

-f, O.

a1 2 This system has the form (22.3), with k

< n,

2n - k, and the averaged

I

system is:

J0

=

(j l' ... , Jk)'
J

=

= (jk+1' ... ,

(d 277)

J)

where

If this averaged system is either integrable (e.g. the plane three-body problem) or close to an integrable system (e.g. the planetary many-body problem) the!! there exist 16 quasi-periodic solutions corresponding to the initial system. These quasi-periodic motions have k "fast" frequencies (w 1' ... ,

W k) ,....,

1 that come from the unperturbed system, and I = n - k

"slow" frequencies (w k+ l' ... , Wn ) ,...., E that arise from the averaging system.

In the general case, when the averaged system is not integrable, the relation between the solutions of the perturbed and the averaged systems is still unknown even for 0 < t

< liE. The only known results arise from

approaches 2 and 3 (22.8). Moreover, observe that, even for nondegenerate systems, we need a study of the motion in the zones of instabilify 16 V. Arnold.[101. [4].

109

STABLE SYSTEMS

(complement set of the invariant tori) for n> 2, at least for t '" 1/E: (or t '" liE: m).

In such a zone, one can probably find} 7 (n -1) -dimensional

invariant tori of "elliptic" or "hyperbolic" type that generalize, in arbitrary dimension, periodic motions of §20. If n > 2, recall that n~dimen­ sional invariant tori do not divide the (2n -1) -dimensional energy level H

=

constant. Consequently, the "separatrices" of the "hyperbolic"

tori can travel very far along H

=

constant, producing instability.

The

next section is devoted to the study of a similar mechanism of instability.

§23. Topological Instability and Whiskered Tori We give next an example 18 (see 23.10) of an Hamiltonian system that satisfies conditions of Theorems (21. 7) and (21.11), but that is topologi•cally unstable: I[(t) - ] (t) I is unbounded for

-00


00.

According to

Theorems (21.7) and (21.11), this system is stable for most initial data (the corresponding motions are quasi-periodic). The secular changes of I (t) have the velocity exp (-1/ {E.) and consequently cannot be dealt

with by any approximation of the classical theory of perturbations. We first introduce some definitions. (A)

THE WHISKERED. TORI

23.1

Assume that in the phase-space of the dynamical system there is an invariant torus T and on it a quasi-periodic motion with everywhere dense orbits. We shall call T a whiskered torus if T is a connected component of the intersection of two invariant open manifolds: T

= M+

n M-,

where

17 One can find motivation in V. Arnold [I4]. Since this was written, the proof was given independently by V. K. Melnikov [2],

J.

Moser [5], and G. A. Krasinskii

[1]. 18 Example (23.10) is rather artificial, but we believe that the mechanism of "transition chains" which guarantees that instability in our example is also applicable to the generic case (for instance, to the three-body problem).

110

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

all the orbits on arriving whisker M- approach T as t

departing whisker M+ all the orbits approach T as t lim t ......

Ix{t) - TI

-+

-+

+ 00, and on the

-00:

0 for x(O) ( M+

-00

lim Ix{t) - TI + 00

0 for x (0) ( M-.

t ......

For instance, the torus Tk: x=y=z=

x=

(23.2)

A· x,

in the system:

°z =

= - f.L • y,

y

if> =

0,

(U

(A, f.L > 0, cp(mod 211) ( Tk, (U incommensurate) defined in the space R/+ x R/- x RIo x Tk has a (/+ + k)-dimensional whisker

W

(y = z = 0) and

a (/_ +'k)-dimensional whisker M- (x = z = 0). (B) THE TRANSITION TORI 23.3

Let M be a smooth submanifold of space X. We shall say that the subset

n

C X obstructs the manifold M at the point x ( M if every manifold

n.

N that is transverse to M at x is intersected by

n which winds onto a limit cycle

spiral

For instance, 19 a

M obstructs M (see Figure 16.4,

Chapter 3). If the whisker~ torus T has the property that the images of an arbitrary neighborhood V of an arbitrary point ~ of one of its arriving whiskers M- obstruct the departing ~hisker M + at an arbitrary point TJ of M+, then the torus will be saip to be a transition torus (see Figure 23.4). LEMMA 23.5

The torus x

= y =

z

0 in system (23.2) is a trafsition torus.

Proof: We set ~ = (0, Yo, 0, cpo),

TJ = (Xl' 0, 0, CPl).

mensurate, there exists a sequence ti' ti from

CPo + (Uti

-+

The (U's being incom-

+00, such that the distance

to CPl tends to zero.

Consider the part V of V whose equation is y = yo· By

19 Articles by Sitnikov

[1]

and A. Leontovich

[11 are

n

=

based on this fact.

U 00

V(t)

111

STABLE SYSTEMS

we denote the set of all points of all the orbits emanating from U. Then

n

contains the set of all the images gtlV, where gt is the transformations

I-------....,..T)

Figure 23.4

group' defined by (23.2). For t; large enough, these images gt, V intersect the neighborhood of TJ (because ,\ > 0). The intersections have equations: Y = Y·, J

Thus

Y

J

=

e -Ilt; • Yo ... 0 •

n contains the set of all the surfaces

g tl V that are parallel to M +

and converge to M+. These surfaces already obstruct M+ at TJ; this proves Lemma (23.S). (C) THE TRANSITION CHAINS 23.6

Assume that the dynamical system has transition tori T1, T2, ... , Ts' These tori will be said to form a transition chain if the departing whisker

M7 of every preceding torus T; is transverse to the arriving whisker M;-::'l of the following torus Ti+ 1

at some point of their intersection (see

112

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Figure 23.7):

Mi nM;

;, 0, M; nM3- ;, 0, ... , M;_l nM; ;, 0.

M; M;_l Figure 23.7

LEMMA 23.8

Let T1 , T2 ,

..• ,

Ts be a transition chain. Then, an arbitrary neighbor-

hood V of an arbitrary point ~

f

M;- is connected with an orbit (t) to

an arbitrary neighborhood V of an arbitrary point TJ

(0)

f

V,

(t)

f

f

M; :

V for a certain t.

Proof: Consider the future U ~ torus, then

Uobstructs Mi

Ut > 0 V(t) at ~1 ~

the open set U. Let ~; be a point of

of V. Since Tl is a transiti~n

Mi nM;.

Thus, M; intersects

M; nu, then there exists a neigh-

borhood V 1 of ~; that belongs to U. The future of V 1 belongs to U and it is sufficient to perform the same argument s times to prove that U obstructs

M;

at TJ.

(Q. E. D.)

(D) AN UNSTABLE SYSTEM 23.9

Let U ~ R2 x T3 be the five-dimensional space 20 11' 12 ;
20

It is easy to construct a conservative system with the Hamiltonian (23.10).

113

STABLE SYSTEMS

H

=

Yi/~ + Ii) + ErOS (
B

=

sin
In other words, we consider the system of differential equations: (23.10)

where J.L« E THEOREM

«

1.

23.11

< A < B. For every E> 0 there exists a J.Lo = J.Lo(A, B, E,) > 0 such that for 0 < J.L < J.Lo the system (23.10) has a solution satisfying: 12 (0) < A, 12 (t) > B for a certair) t. Assume 0

To prove Theorem (23.11) it is sufficient, in view of Lemma (23.8), to find a transition chain T1, ... , Ts such that: 12

< A on T1 and 12 > B on

Ts' LEMMA

23.12

Each manifold Tw defined by the equations 11 =
(23.10). In fact: (1) Tw is clearly an invariant torus of (23.10);

(2) for J.L = 0, the three-dimensional whiskers have equations:

11 (3) for J.L

t-

=+ -

2"jEsin .p1 , 2

0 and small enough, the whiskers still exist and can be found

by the Hadamard method (see Chapter 3, §15). The argument of Lemma (23.5) proves that the tori Tw are also transition tori. finally, making use of variation formulas of the whiskers for J.L small enough, 21 the following lemma is proved: 21 See Poincare [2] and V. K. Mel;',kov

[tl.

114

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

LEMMA 23.13 Assume A < w < B.

Then the departing whisker M~ of the torus Tw

intersects with the arriving whiskers M -, of all tori

w

ficiently close (provided that

Iw - w'l < K,

T

w ,which are suf-

where K = K(s, 11, A, B)).

Proof of this lemma requires certain computations that will be found in V. Arnold [13]. These computations show also that K ~ 11 exp (

-1/ Vs).

Lemmas (23.12) and (23.13) imply that the whiskered tori T , ... , T Wi

(w. irrational, I

Iw. 1

(J.).

1+

11

< K,

-

w1 < A, ws > B)

Ws

form a transition. chain.

Application of Lemma (23.8) to this chain implies Theorem (23.11).

General References for Chapter 4 Arnold, V., Small Denominators I, Izvestia Akad. Nauk., Math. Series 25 1 (1961) pp. 21-86. [Trans!. Am. Math. Soc. 46(1965) pp. 213-284.] Small Denominators II, Usp. Math. Nauk. No.5 (1963) pp. 13 -40. [Russian Math. Surveys no. 5 (1963) pp. 9-36.] Small Denominators III, Usp. Math. Nauk. No. 6(1963) pp. 89-192. [Russian Math. Surveys No. 6(1963) pp. 85-193.] Birkhoff, G. D., Dynamical Systems, New York (1927). Moser,

J., On Invariant Curves of Area-Preserving Mappings of an Annulus,

Gottingen Nachr. No.1 (1962). Poincare, H., Les methodes nouvelles de la mecanique celeste, I, II, III, Gauthier-Villars, Paris (1892, 1893, 1899). Siegel, C. L., Vorlesungen tiber Himmelsmechanik, Springer, Berlin (1956).

APPENDIX]

THE JACOBI THEOREM (See Example 1.2) Let S 1 = (mod 1),

W f

! x (mod

1)1 be the circle and ¢ be the translation: x ... x + w

R. Each orbit of ¢ is ever:ywhere dense if, and only if, w

is irrational. Proof: Assume w rational:

Let w = pi q where p, q .f Z, g > O. Here ¢ q is the identity transformation. Since every point of our circle is left fixed by ¢ q, every orbit is closed and consists of a finite set of points. Assume w irrational: Let x be an arbitrary point of 51. The ¢n x's are distinct for

implies that (n -.m)w

f

Z and then n = m. Thus each orbit consists in an

infinite number of distinct points. Since S1 is compact, this orbit has a limit point. Consequently, for any E > 0 there are distinct integers n and m such that:

Setting

In - ml

p and observing that ¢ is length-preserving, we have:

Thus·, ¢Px, ¢2 p x, ... , ¢kp x, ... partition S1 into segments of length less than E. Our theorem is proved, for E is arbitrary. An N-dimensional extension of this theorem reads as follows:

115

116

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Let Tn x

->

=

Rn /Zn

be the n-dimensional torus and

x';' w (mod 1), w ( Rn. Each orbit of


be the translation:

only if, k· w ( Z and k ( Zn imply k = O.

In the continuous case we have: Let x+ tw (mod 1), t ( R. w (R n . Each orbit of

APPENDIX 2

GEODESIC FLOW OF THE TORUS (See Example 1. 7) V is the two-dimensional torus, that is, the surface of revolution of a

circle of radius r about a line Oz in the plane of the circle at the distance 1 (> r) from the center of the circle. Equations of V in geographical coordinates are: x ~ (1 + r cos

t/J)

cos

if>

y ~ (1 + r cos

t/J)

sin

if>

z where

~

t/J,

r sin

if> is the latitude and t/J is the longitude.

-", \ \

\ \

,

Flgure A2.1

117

118

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Figure A2.2

Conservation of energy and conservation of angular momentum about Oz give equations of the geodesics: r2

¢, 2 + (1 + r cos rjI)2 " ¢ 2 =

h = constant.

¢" . (1 + r cos rjI) 2 = k = constant. If h = 1 we obtain the geodesic flow on M = Tl V. This flow is invariant

under rotations ¢

-+

¢ + constant. Thus, in order to study the geodesics

it is sufficient to study those of them with initial point on a prescribed meridian. Figures A2.1 and A2. 2 depict the generic cases Yl and Y3 that are separated by

Y2"

APPENDIX 3

THE EULER-POINSOT MOTION (See Example 1. 7) It is the motion of a heavy rigid body fixed at its center of gravity.

The rigid body is a system with three degrees of freedom and a six-dimensional phase space. There exist four independent single-valued first integrals: the energy T and three components of the angular momentum vector m. These four functions of position in the phase space do not change their values for a given motion. The points of the six-dimensional phase space for which the four functions have given values form, in general, a two-dimensional manifold M. These manifolds M(T, m) are tori. In fact,

M is invariant, and so the phase-velocity vector at each point of M is tangent to M; consequently M admits a vector field without singular point. It is evident that M is orientable and compact. The only compact two-

dimensional orient able manifold admitting a tangential vector field without Singular point is, as is well-known, the torus. On the other hand, M being invariant under the dynamical flow ¢ t' admits an invariant measure /1 (Liouville theorem). Thus (M, /1, ¢ t) is a classical system. A canonical transformation makes (M, /1, ¢ t) into a system of the form: x

= 1, ;. = -a (Example 1.2; see Appendix 26). Thus the Euler-Poinsot

motion is, in general, quasi-periodic and the orbits are dense on M. Its two periodic components are called, respectively, precession and rotation (Euler [1]).

119

APPENDIX 4 GEODESIC FLOWS OF LIE GROUPS (See Example 1. 7) The geodesic flow of a Lie group carrying a left-invariant (or rightinvariant) metric has important applications: The geodesic flow of the connected component SO(3} of the group of the rotations of the three-dimensional Euclidean space corresponds to the rotations of a heavy solid moving around a fixed point. Each orbit corresponds to a motion. The homotheties of positive ratio and the translations of the n-dimensional space Rn form a group that generates the geodesic flow of the (n+ l}-dimensional space of constant negative curvature.

Then, let us consider the group S Diff(~) of the measure-preserving diffeomorphisms of a compact Riemannian domain~. T~e corresp.onding algebra consists in divergence-free vector fields V or. ~: div V = O. The energy

< V,

V

>

=

Ii) v2 dx

is a positive definite quadratic form on the

space of such vector fields, and defines some right-invariant Riemannian metric on the group S Diff(~). The geodesics of this metric are just. flows of perfect (incompressible and unviscous) fluid in~. The Rief!1annian curvature of this infinite dimensional "manifold" S Diff(~} can be computed. For example, if ~ T2 is the torus· lx, y mod 2171 with its usual f!1etric, then the sectional

curvature is nonpositive in every section containing the lamihar flow: Yx

=

cosy,!... Vy

=

O.

See ~lso V, I. Arnold [11 and L. Auslander, L. Green, F. Hahn [1].

120

=

APPENDIX 5

THE PENDULUM (See 1.13) The equation of the pendulum is:

q+ k

sin q

=

0, where k is a posi-

tive constant; it is equivalent to the system:

j

q ~ p

~

p=

-k sin q.

The Hamiltonian function is H = (p2/2) -k· cos q and Figure (AS. 1) depicts the orbits. The system is invariant under the symmetry p'" - p and the translations: (q, p) ... (q + 2K7T, p),

K

l

Z.

p

q

Figure AS.!

121

122

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

The points (kl7,O) are critical points: points (217k, 0) are centers (stable equilibrium) and points ((2k + 1)17,0) are saddle points (unstablE! equilibrium). The orbits split into three types: the orbits 1 (small oscillations); the separatrices 2 joining two saddle points; and the orbits 3 (complete rotation around the hanging point). The natural phase space is no longer the plane (p, q) but it is rather the cylinder (q mod 17, pl. This is an example of global Hamiltonian flow.

APPENDIX 6

MEASURE SPACE (See Chapter 1, Section 2) An a-algebra

93

defined on M is a class of subsets of M that is closed

under the formation of complement and countable union. A nonnegative (possibly infinite) countably additive set function /1 defined on a measure (M,

93, 11)

is called a measure space;

93

'B

is called

is the family of the

measurable subsets of M and 11 is the measure. (See Halmos [2] and Rohlin [3j for these concepts.) In fact, the object of interest is not (M,

93, 11)

but equivalence classes

(M,I1) of measure spaces. Let us make this point clearer. Let A and B

be elements of

93,

we set A ~ B (mod 0) if I1(A U B - A

n B)

~ O. This ;.

relation is an equivalence relation and the equivalence class of the empty set consists in the sets of measure zero. We denote th~ quotient of der this equivalence relation by

93

93

un-

(mod 0); it is a Boolean algebra, for

the following properties are readily proved: B2 (mod 0)

imply

M - A1

~

M - A2 (mod 0) .

If A ~ B (mod 0), then I1(A) ~ I1(B) and 11 can be regarded as a function defined on

93

(mod 0).

To study abstract dynamical systems one neglects sets of measure zero. This means that one replaces the study of (M,

123

93, 11)

by the study of

124

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

(M, ~ (mod 0), p.), which we still denote unambiguously by (M, p.), because

the function p. defines completely ~ (mod 0) (but not ~). In other words, we identify (M,~, p.) and (M, ~ " p.') if their measurable a-algebras ~ (mod 0) and ~' (mod 0) coincide. Let (M, p.) and (M', p.') be two measure spaces. A mapping ¢: M

-+

M'

is called an homomorphism modulo zero if: (a) ¢ is defined on M -I, where / is a set of measure zero (possiblyempty); (b) /'

=

M' -¢ (M -I) has measure zero, that is ¢ is onto, up to a

subset I' of M' of measure zero; (c) ¢ is measure-preserving, that is each equivalence class of ~' (mod 0) contains a representative, say A', such that ¢-l(A') exists and belongs to ~ with:

Thus, ¢ induces an homomorphism of Boolean measurable algebras: ¢-l:

:B ' (mod

0)

-+

:B

(mod 0).

If (M, p.) = (M', p.') then ¢ is called an endomorphism (mod 0). If both of the mappings ¢ and ¢-l are homomorphisms (mod 0), then ¢ is called an isomorphism (mod 0); if, additionally, (M, p.) and (M', p.') coincide, then ¢ is called an automorphism (mod 0).

APPENDIX 7

ISOMORPHISM OF THE BAKER'S TRANSFORMATION AND B(Y:z, Y:z) (See Example 4.5) We need to construct an isomorphism (mod 0) 1 making the following diagram commutative:

1

De/inition 01 I. Let m f(m)

=

a_ 1 , aO' a 1 ,

...

be a point of Z2Z. We set

(x, y), where 00

(A7.1)

= ... ,

~

x=

i= 0

a -,.

00

y=

2i+1

~ i= 1

The mapping 1 is a bijection, except on the elements (x, y) of T2 for which x or y is a dyadic fraction. Such elements are denumerable and so constitute a set of measure zero.

1 is measure-preserving. It is sufficient to prove it for a generator A!, 1m I a i = iI of the measure ~lgebra of Z2Z : the set

f(Aj)

=

,

~( k;O 2:~ k~1 ;~ )1 a, 125

j

~

126

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

consists in 21il rectangles, the sides of which are 1 and 1/2I i l+1. Thus, we get:

The diagram is commutative. Let x and y be given by formula (A 7.1), we have:

where

ai

al _ 1, 00

t

k=O

that is to say:

f (2x, 'hY)

t

if a o

=

0, i.e., 0 ~ x

(2x, Wy+ 1)) if a o = 1, i.e., 'h

Consequently: I¢ 1- 1 = ¢'.

< 'h

~ x < 1. (Q. E.. D)

APPENDIX 8

LACK OF COINCIDENCE EVERYWHERE OF SPACE MEAN AND TIME MEAN (See Remark 6.5) Consider again the dynamical system of Example (1.16): M is the torus {(x, y) mod 11. the measure is dx dy and the automorphism

¢ is:

¢(x,y) = (x + y, x+ 2y) (mod 1).

This automorphism ¢ induces a linear mapping

M=

of

YH.

{(x,

¢

¢

of the covering plane

The matux

has two proper values: 0 X =

<

"2 < 1 < \.

The line

s

{

y = ("2 - 1)s,

s (

R

projects onto a curve y of M under the natural projection

rii

->

M.

This

curve y is invariant under ¢ and y is dense on M, for .:\2 - 1 is irrational Qacobi's theorem, Appendix 1). Let m

=

(x, y) be a point of y. Of course,

we have:

and 0

<

"2 < 1 implies: lim ¢"(m) = (0, 0) . n=oo

127

128

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Consider the analytic function f(x, y) = e 2TTix • We have N-l

N-l

1N ~

f(rpnm) =

~ ~

n=O

e 2TTixA,f.

n=O

Usual convergence implies Cesaro convergence and lim x· A;

0, there-

n=oo

fore: N-l

*

lim

f(m)

1

N-++oo N

~

f(rpn m)

1 .

n= 0

On the other hand:

Thus, whatever the point m of the dense subset y be, we have: [*(m) though f is analytic and

rp

is classical.

f T,

APPENDIX 9

THE THEOREM OF EQUIPARTITION MODULO I (See 6.6) We prove here 1 the Theorem of Equipartition Modulo 1 due to Bohl, Sierpinskii, and Weyl: II ¢ is a rotation 01 the circle M through an angle

incommensurate with 21T: M

=

I z ( C, \z\

=

I\, ¢ (z)

=

8· z, 8 =

e2lTiW ,

w is irrational

and I is a Riemannian integrable lunction, then the time mean 01 I exists everywhere and coincides with the space mean.

Prool. lsi ca~e: i(z) = zP, P (

N-I

~ ~

N-I

~ ~

I(¢n z ) ,=

n=O

z.

We get:

(8 n z)P =

{

1 1 N-·

n=O

Since w is irrational'we have eP -1 ~ 0 and

-

I

2nd case:

*

'" I(z)

=

0

zP •

eN P - l 8p -1

\8PN -1\ < 2,

0

if p = if p ~ 0 •

I is a trigonometrical polynomial, that is l(z) '" ~ a zP, P

1

{I

if p = 0

p (Z,

Compare to G. Polya and G. Szego [I] p. 73.

129

Z (

M,

ifp~O.

so we get:

~RGODIC PROBLEMS OF CLASSICAL MECHANICS

130 in which

B

p

=

0, except for a finite number of them. From the first case

one deduces at once:

* i(z)

=

BO

=

r.

3rd case: f is realovalued and Riemannian-integrable. To every E > 0 correspond two trigonometrical polynomials ~- and ~+ such that: Pe -(z)

< fez) < Pe + (z) for every z ( M

and j(Pe+(z)-Pe-(Z))d ll

<

E.

M

From the second case we deduce: (A 9.1)

N-l

~ lim sup 1 N ~ N~oo

f(¢n z)

0

~

!

~ + • dll

•

M

< E. Since E is arbitrary we have fez), which exists everywhere. Relation (A9.1)

Consequently, lim sup - lim inf lim sup = lim inf = lim = implies that

fez) is constant, whence:

* i(z)

-

= f

(Q. E. D.)

Extension to translations of the torus Tn is obvious: the time mean and the space mean of a Riemannil\n-integrable function coincide everywhere

if, and only if, the orbits are everywhere dense.

APPENDIX 10

SOME APPLICATIONS OF ERGODIC THEORY TO DIFFERENTIAL GEOMETRY The Birkhoff theorem was used by A. Avez [1] to prove the fOliowlng: Let V be a compact n-dimensional Riemannian manifold without conjugate point, then the proper values of the operator

tl-~

n-1

are nonnegative (tl is the Laplacian _Va Va' R is the scalar curvature). In particular (L. W. Green):

131

APPENDIX 11

ERGODIC TRANSLATIONS OF TORI (See Example 7.8) We prove that translations of tori (Examples 1.2 and 1.15) are ergodic if and only if, their orbits are everywhere dense (or if. and only if, the time mean and the space mean of a continuous function coincide everywhere). Let M be the n-dimensional torus le 277ix l x ( Rn \, where x = (xl' ... , X )

n

and e 277ix means (e277ixl, ... , e277ixn). The measure of M is the usual

product measure /l. The translation is:

THEOREM.

(M, /l, ep) is ergodic it, and only il k· 0

(

Z and k ( Zn imply k = O.

Prool: Let ( be a measurable invariant function. Its Fourier coefficients are: ak =

f

e- 277ik

· x • I(x)d/l

M-

The Fourier coefficients of I(cf>x) are: bk =

1

e- 277ik (x -0), f(x)d/l = e 277ik ' 0 . a k .

M

The invariance of I is equivalent to b k = a k for any k, that is a k = 0 or k·

0

(

Z. 132

133

APPENDIX 11

If the

w;'s

are integrally independent, the second case occurs only for

k = O. Thus a o is the only Fourier coefficient possibly nonzero, f is constant, and (M, /L, ¢) is ergodic (see 7.2).

If a k f- 0 exists such that k· w ( Z, then f (x)

=

e 21Tik · x is a non-

constant invariant function and (M, /L, ¢) is not ergodic. Remark. In the continuous case (M, /L, ¢t)' where

¢ t:

e 21Tix

->

e 21Ti (X + tw) ,

we have a similar result: (M, /L, ¢t) is ergodic, if, and only if, k ( Zn and k·w = 0 imply k = 0 (or if, and only if, the orbits are everywhere dense; see Jacobi's theorem).

APPENDIX 12

THE TIME MEAN OF SOJOURN (See Chapter 2, Section 7) TH~OREM

A12.1

, An abstract dynamical systeJll (M, /l, ¢ /) is ergodic if, and only if, the sojourn time reT) in an arbitrary measurable set A of an orbit

I¢ /x I 0 $. t s; TI is asymptotically proportional to the measure of A:

(A12.2)

.

dT)

T->oo

T

11m - -

=

It (A),

for all measurable A and almost every initial point x ( M. Proof:

* Assume (M, /l, ¢/) is ergodic and A is measurable. We have f(x) for every f ( Ll (M, It) and for almost every x ( M (see 7.1). Take f

=

T XA

(characteristic function of the set A), we obtain:

for almost every x. The converse is derived at once: (A12.2) implies ergodicity. It is sufficient to observe that the characteristic functions

XA

generate Ll (M, /l).

Theorem (A12 .1) clearly holds in the discrete case (M, Il. ¢ ). 134

135

APPENDIX 12

EXAMPLES AI2.3: TRANSLATIONS OF TORI Let M be the n-dimensional torus

I e 27Tix I x

£

Rn I, Il the usual mea-

sure, and ¢ the translation:

If k ( Zn artd k· (U

(

Z imply k

=

0, then (M, Il, ¢) is ergodic (see Appen-

dix 11). Thus, relation (AI2.2) holds for almost every initial point. This can be rephrased as follows: denote by r(N, A) the number of elements of the sequence

that belong to A, then: r(N, A) (A) · 11m - - - = Il

(AI2.4)

N~oo

N

for almost every initial point e 27Tix . If A is Jordan-measurable, that is, if ~A is Riemannian-integrable, then (AI2.4) holds for every initial poiRt. To prove it, it is sufficient to use the theorem of Appendix 9 and to take f

= ~A'

Extension to·the continuous case holds good. This result i!l

known as the theorem of equipartition modulo 11 and is due to P. Boh1 [1], W_Sierpinskii, and H. Wey1 [1], [2], [3] ~ It is one of the first ergodic theorems. Historically, it originated from an attempt to solve the Lagrange problem of the mean motion of the perihelion (see Example 3.1 and Appendix 13). Here follow some applications. 2 ApPLICATION AI2.5: DISTRIBUTION OF THE FIRST DIGITS OF 2" (see Example 3.2) The first digit of 2n is equal to k if, and only if: k • lOT $. 2n

<

(k +

P . lOT

1 F. P. Callahan [11 gave an elementary proof. 2 The reader will find further applications to various fields in: Compositio Mathe-

matica. V 16. fascicles 1. 2.

136

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

that is to say: r + Log10k S n Log 10 2 ::; r + Log10(k+ 1) . Set a = Log 10 2 and (n· a) = na - [n· a], where [ ] means the integer part. The above inequality may be written: Log10k S (na)

< Log10(k+

1)

Now, we turn to the dynamical system consisting of the one-dimensional torus M = e 21Tix

->

le 21Tix I x

e 21Ti (X+ a).

(

RI.

the usual measure /l, and the translation ¢:

(M, /l, ¢) is ergodic, for a is irrational (see Exam-

ple 7.8). Thus, the sequence 1(na) I n ( N I is equidistributed. In particular, take A

=

[Log 1ok, Log10(k+ 1)] in relation (A12.4), we have: lim

r(N, A)

-N-

N->oo

=

/l(A)

1

= Log 10(1-+; k-).

But r(N, A) is nothing but the number of elements of the sequence 1, 2, ... ,

2N - 1 , the first digit of which is k. Thus, if we go back to the notation of ExamRle (3.2), we have:

Consequently, the proportion of 7's is greater than the proportion of 8's in the sequel!,ce of thEt~fi,,~t digits of

12n In

=

1,2, ... 1. This is not what one

expects from an inspection of the first terms: 1,2,4,8, 1,3, 6, 1, 2,5, ... This is due to the fact that a = 0, 30103 '" REMARK

is very close to 3/10.

A12.6

Since the sojourn time in a domain A of a point of an ergodic system is proportional to the measure of A, it is natural to ask about the dispersion. Let us mention some results due to Sinai [1]. Let ¢t be the geodesic flow of the unitary tangent bundle Tl Y of a surface .y of constant negative curvature. If A is a domain of Tl Y with piecewise differentiable boundary, then the mean sojourn time of a geodesic ¢tX in this domain has a Gaussian distribution and verifies the central limit theorem:

137

APPENDIX 12

lim !l T-+oo

where

~ x ITT(X) T

TT(x) =

measure

- !l(A)

< ;; yT

I t I ¢tX

f

A, 0

f

=

$nl 217

:s. t :s. Tl

f

Ca e- u2/2 • du

-00

and C is a constant.

APPENDIX 13

THE MEAN MOTION OF THE PERIHELION (See Example 3.1 and Appendix 12) The problem of mean motion arises from the theory of the secular perturbations of the planetary orbits (Lagrange [1]). One asks for the existence and estimate of: (A13.1)

n

n

1 Arg iwkt , ~ ak . e t-++oo k= 1

= lim

where w k ' t ( Rand a k f, 0, ak ( C. In other words, if one considers a plane linkage AoAl ... An consisting of the links A k _ 1 Ak of fixed lengths lakl, moving with constant rotation-velocity w k ' we are interested in the mean rotation-velocity of the vector AO An (see Figure A13.1). THEOREM A13.2 (See H. Weyl, [1]- [5].) Assume that the wk's

(A13.3)

W·

are

integrally independent, that is:

k = 0 and k ( Zn imply k = O.

Then the mean motion 0 exists and is expressed as:

(A13.4) The Pk's depend on the lakl only. Ifl p(ak ; al' ... , ~k' ... , an) is the probability that an (n-l) -linkage with prescribed sides al' ... , ~k' ... , an spans a distance inferior to ak (see formula A13.12), then: 1

A

means cancellation.

138

139

APPENDIX 13

y

o =AO Figure A13.1.

x

.

Case n '" 3: initial position of the linkage .

(A13.S) In particular, for n '" 3, if there exists a triangle the sides of which &re

lall, la2 1, la3 1 and the

angles of which are Al , A 2 , A 3 , then (Bohl's

formula): (A13.6)

n '"

_A_l_w..;.1_+_A...;:2:....w.....;2'--.+_A_3:....w_3;:.... 17

The case in which no triangle can be constructed was investigated by Lagrange [1]. In the general case A. Wintner [1] found the expression:

in terms of the Bessel functions ]0 and ]1· Relation ~ Pk an "addition" theorem for these functions.

=

1 provides

140

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Proof of Theorem A13.2 THE CORRESPONDING DYNAMICAL SYSTEM

A13.7

Let us consider the dynamical system (M, p., ¢>t)' where M

=

Tn

= Izlz = (zl, ... ,zn)\'

is the n-dimensional torus, p. is the usual measure, and ¢>t is the translation group:

The phase space of the n-linkage is M, and ¢>t depicts the movement. Let us define a function a on M by: n

(A13.B)

a(z) = Arg

I

lakl

Zk'

O:s a < .211 •

k=l This function is discontinuous over the slit l:

= I z I a(z) = 01,

and is not

defined on the so-called singular manifold S = Izll:~lak~zk = 01 which consists of all possible states of a closed n-linkage with the prescribed sides lakl. Nevertheless, the function:

(A13.9) is analytic outside of S. The limit (k13.1), if it exists, is nothing but the time mean ~ of {3:

(A13.10) where

THE SPACE MEAN

A13.11

The system (M, p., ¢>t) is ergodic, for the

(Uk's

are integrally indepen-

dent (Appendix 11). If the function {3 were Riemannian-integrable, then,

141

APPENDIX 13

according to the theorem of Equipartition modulo 1 (Appendix 9), the time mean

f1*

~

n

n ~ i3

the Birkhoff theorem implies that

f1

is Lebes-gue-integrable. Thus,

for almost every initial phase.

This suggests the study of the space mean shows that

f1

f1.

would be equal to the space mean

We only know (see A. Wintner [1]) that

i3.

Relation (A13.9)

depends linearly on the (U/s. Therefore,

f1

depends lin-

early on the (Uk's:

To compute Pl (for instance) we set: (Ul

~

217,

~

(U2

•••

~

(Un

~

0 •

We have:

1 -

217

f1 (217,0, ... ,0)

,

where

Relation (AI3.8) allows one to carry out the integration over (Jl:

if Ila2Ie217i(J2 + ••• + Ian Ie 217i (Jn I < lall if Ila:ile217i(J2 + ••• + Iani e 217i (Jn I > lall Thus we obtain:

where

This proves relation (AI3.S). Relation ~ Pk ~ 1 is derived easily by setting (Ul

~

•••

~

(Un

~

2" •

142

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

A13.13

EXISTENCE OF THE TIME MEAN

Hence, formulas (A13.4) and (A13.s) are proved for almost every initial phase Arg a k . To prove them for all initial phases we use a special device inaugurated by Bohl [1] for n n

=

3 and improved by Weyl [4], [5] for

> 3. We define a function on the torus M by: N (z)

=

algebraic number of the points of intersection of the curve

I¢/z, -271 < t < 01 with the slit ~. We count + 1 a point of intersection z. for which (3 (z.) > 0 and -1 if J

(3(z.) J

J

< O. (See Figure AI3.1s.) It can be proved that

N(z) is bounded.

N=O Fii:Ure

~13.1S

Thus, according to (AI3.10), the following relation holds uniformly over

m: (~13.14)

Since the function N is piecewise nian-integrable, the time mean

N*

continuo~s

and, in particular, Rieman-

exists everywhere and is equal to N

(Appendix 9). From (AI3.14) one deduces that {3* where and is constant.

=

*=

N

N exists every-

(Q. E. D.)

APPENDIX 14

EXAMPLE OF A MIXING ENDOMORPHISM Let us consider the transformation: 1

cf>: (x, y) ... (2x, 2y) (mod 1) of the torus M

= l(x, y) mod 11

carrying the usual measure dx dy. cp-' A

M

I'

~

~ ~

~

@ ~

~

@

~

~

~

@

G

~

~

~

Figure A14.l

~--------------------~

Which is called "multiplication of loaves" since Figure (A14.1) shows the solution of a well-known historical problem.

143

144

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

To be more explicit, we write: ( (2x, 2y)

¢ (x, y)

~ (2x,2y-l)

t

(2x-1,2y)

if 0 $ x,

y<\6

if 0 $ x < \6, \6$ y < 1 if \6 $ x < 1,

O$y
, (2x-1,2y-l) if \6 $ x, y < 1. The application ¢ is not one-to-one, in fact it is everywhere four-to-one.

If E is a square with dyadically rational vertices, then ¢-1 E istheunion of four similar squares (see Figure). Thus, in such a case, p.(¢-1E) = p. (E) and from there it follows easily that

¢ is measure-preserving. We

have here an example of a measvre-preserving mapping that is not invertible. The transformation ¢ is mixing, that is: (A14.2) for any measurable sets A and B. To prove it, it is sufficient to consider the souares B: I, m ( Z+ .

If N::: p, then B contains 4 N -

p

an inverse image has measure

4- N ° p.(A),

inverse images of A under ¢-No Such therefore:

Thus (A14.2) holds good because p is arbitrary. Similar arguments show that the mappings: ¢k: (x, y)

->

(kx, ky)(mod 1),

k ( Z+

are measure-preserving and mixing. They satisfy: ¢ko¢r = ¢kr for k,r (Z+, i.e., !¢k1k (Z+! .

is a mixing s~igroup under composition. 2

2 This semigroup can be interpreted in terms of Tchebyschev polynomials (R. Adler and T. Rivlin [1])

0

APPENDIX I))

SKEW-PRODUCTS (See Definition 9.5) Let (M, p.) and (M', p.')

b~.two

Lebesgue spaces. We denote by (M x M',

p.x p.') their direct product: the measure algebra

1M , A' ( ~; 1M , A' ( 1M "

erated by the AxA', A ( p.(A)·p.(A') for any A

f

I Mx M'

of Mx M' is gen-

and we set (p.xp.')(AxA')

=

Assume that (M, p., ¢) is a dynamical system, and make correspond to every m

f

M an automorphism .pm: M' .. M' such that (m, m') ... .pm(m')

!s measurable for every m (M, m' ( M'. Then

(¢ x l.pl): M x M' ... M x M', defined by

is measurable and measure-preserving. In fact, for every measurable set F

f

IMxM '

whose characteristic

function is XF' we have: (p.xp.')«¢ x I.pD-IF) =

f

X F [(¢ x I.pD(m, m')]d(p.xp.')

MxM'

=

£[£,x

F(¢m,

rp mm ')dp.

J

Since .pm is p. '-measure preserving, this may be written (p.x p. ')«¢ x l.pl)-I F)

=

£[ 145

{,X F(¢m, m ')dp.

J

dp.

dp.

146

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Apply Fubini's theorem and observe that ¢ is measure-preserving: (Il X Il ')«¢x !t/JO-1 F) =

=

tX~ XF(¢m, m ')dll] dll' ~{LXF(m,m')djdll '

=

(Il X Il')F

The dynamical system (MxM', IlXIl', ¢x!t/JI) ~s called the skew-product of (M, Il' ¢) and (M', Il" IjJ m).

A 15.1

EXAMPLE

If t/J m

=

¢' is constant, that is, if ¢ x !t/J 1 is defined by (¢ x ¢ ')(m, m ') = (¢m, ¢ 'm ') ,

then the skew-product is the direct product of (M, Il' ¢) and (M', Il', ¢ ').

EXAMPLE

A 15.2

Tflke M

=

51

=

! x (mod 1) 1 with the usual measure dx and ¢: 51

->

51

an ergod'ic translation: ¢x = x + w (mod 1), w irrational. Now, let n be an integer.

with the usual measure dy and we make correspond to each x ( M translation t/J x,n : M'

->

1)1

We take M' = 51 = !y (mod =

51 a

M' defined by:

t/J x,n (y) = y +

(lX

(mod 1) •

Thus, to any integer n corresponds the skew-product (51 x 6 1, dxx dy, ¢x!t/J x

nO.

Anzai [1] proved that the ergodic measure-preserving automorphisms ¢x!t/J K,n 1 and ¢x!t/J K,p l, with w irrational and

Inl p Ipl,

are not isomor-

phic although they have the same spectral type and the same vanishing entropy.

APPENDIX 16

DISCRETE SPECTRUM OF CLASSICAL SYSTEMS (See 9.13) Let (M, 11, ¢ t) be a classical flow and Ut the one-parameter group of unitary transformations induced by ¢ t

The discrete component of the spec-

trum of Ut is called the discrete spectrrim. The dynamical systems constructed in the second part of the discrete spectrum theorem (see 9.13) are classical systems if the rank 1 of the abelian group of the proper values of Ut is finite. All known examples of ergodic classical systems have discrete spectrum with finite rank and this rank is bounded from above by the uimension of the space. 2 Thus, the finiteness of this rank is a natural conjecture. The proper functions of a classical system can be everywhere discontinuous (see an example due to Kolmogorov [1]), but if all of them are continuous, then the rank of the discrete spectrum is bounded from above by the first Betti number b l THEOREM

=

dim H 1 (M, R).

A 16.1

u,t

If the proper functions of the induced unitary group

of an ergodic

classical system (M, 11, ¢t) are continuous, then: rank of the discrete spectrum

S. b l

.

This theorem is an obvious corollary of the following one:

I. e., the maximum number of independent generators.

2 This fact is general for systems with continuous proper functions (Avez [2]).

147

148

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

THEOREM A 16.2

Let (M, p., ¢t)

be an ergodic classical system. Denote by C the sub-

group of the discrete spectrum corresponding to continuous proper functions, then: rank

C :s

b1 ·

Before proving this theorem we first introduce the winding n':lmbers. WINDING NUMBERS A

16.3

Let (M, p., ¢t) be an ergodic classical system.

The first homology

group H 1(M, R) has finite integral base: Y1"'" Yb l ' Each Yk is a closed curve which can be assumed differentiable. Let a

:s T I

=

I¢tx\ x ( M,

O:s t

be an arc of an orbit of ¢t. We join the endpoints ¢Tx and x with

a geodesic arc {3 (geodesic in the sense of some Riemannian metric). Thus, y(T) = a{3 is a piecewise differentiable closed curve (see Figure A 16.3 ') and there exist integers nk(T) such that: y(T) = n 1 (T) ·Y1 + ••• +.nb1(T). Yb 1

.

Fieure A 16.3'

Let (w k ) be the dual base of '(Yk) in the first cohomology group H1(M, Z), that is the closed one-forms satisfying: if i = k ifi,tk.

149

APPENDIX 16

We obtain:

which can be rewritten:

(A 16.4)

(y, w k ) is the value of the one-form w k for the infinitesimal generator y of ¢t at point ¢tx. Since length {3 < diameter M, we have:

where

(A 16.5)

!. (Wi<

lim

o

1{3

T

T-+oo

On the other hand, ergodicity implies (see 7.1): (A 16.6) for almost every initial point x, and the limit does not depend on x. Finally, from (A 16.4), (AI6.5), and (AI6.6) we deduce that: .

(A 16.7)

nk(T)

hm - T

T-+oo

=

f.

,

(y, wk)dJL

M

de!

=

JLk

exists for almost every x and does not depend on x. The numbers a k

= e2TTil-'lc,

k

=

1, ... , b l , ~enerate a sub~roup ~ of

the circle ~roup; ~ is called the ~roup of the windin~ numbers. It is read-

ily seen that ~ does not depend on the base (Yk). Thus, ¢t defines a r:al homology class:

the winding numbers JLk define the "homological position" of a generic orbit. In other words, they define how an "average" orbit w-anders around M. 'This concept was first introduced by H. Poincare [1] for flows on the torus T2. Further investi~ations of systems:

x=

F(x, y),

y

=

G(x, y)

150

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

on the torus T2 are due to A. Denjoy [1] and C. L. Siegel. Proof of Theorem A 16.2 3

From its very construction, the rank of the winding numbers group is bounded from above by b 1 , Therefore, Theorem (A 16.2) is a corollary of the following lemma. LEMMA A

16.8

The subgroup C of the discrete spectrum corresponding to continuous proper functions is a subgroup of the winding numbers group. Proof:

Let i(x) be a nonvanishing continuous proper function of ¢ I: f (¢ IX) = e217iAI. f (x) •

The function f is continuously differentiable with respect to the flow ¢I' that is: 217iA' e217iAI • i(x) ,

which may be written:

(y,

(A 16.9)

for t

=

df(x))

=

217iA·f(x)

O. But ergodicity implies (see Theorem 9.12) If(x)1

=

constant

a.e.,

that is to say li(x)1 = constant';' 0 everywhere,

because f is continuous. Thus, up to a constant, we have: f(x) = e217iljl (X) ,

where 1jI: M

-+

S1 is continuous and (A 16.9) reads:

(A 16.10) Therefore dljl defines a closed current of degree 1 in De Rham's sense. 4 3 See also I. M. Gelfand and Shapiro-Piatetski

[1].

S. Schwartzman

4 G. De Rham. Varieles differenliables. Hermann (Paris) 1955.

[1].

151

APPENDIX 16

As it is well-known, such a current is homologous to a smooth closed oneform [dy,]: dy,

=

[dy,] + dh .

Assume that M is metrized with a Riemannian metric whose volume element is dfL. Since "if> t is fL-measure-preserving, the infinitesimal generator yof

¢; is co-closed (oy = 0)

Then, relation (A 16.10) implies:

f

f

(y, [dy,]) dfL = (y, dy,) dfL = A • M M Now it is sufficient, according to (A 16.3), to prove that [dy,] has integral periods. Let u: [0, 1] .... M be an arbitrary smooth loop of M. Since dy, and [dy,] are homologous, we have:

f u

[dy,]

=

f

dy,

= Y, [u (1)] -

y, [u (0)] ( Z .

u

(Q. E. D.) COROLLAR-V' A 16.11 (Arnold [2], [3]).

Let V be a compact orientable Riemannian manifo: { which is not. a torus and the dimension of which is greater than !'ne. If the geodesic flow on the unitary tangent bundle M

=

Tl V is ergodic, then the continuous proper

functions reduce to constants. Proof; According to Lemma (A 16.8) it is sufficient to prove that every winding number vanishes. Under our topological assumptions, Gysin [1] proved that every closed one-form w, which is not homologous to zero, is the lift of a closed one-form of V, say again w. On the other hand, a winding number of the flow has the form (see A 16.7):

where 71 (resp. a) is the volume element of V (resp. the fiber Sn-l).

152

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

From

we deduce:

f (1 V

S"- 1

(y, w) a)1J =

Jrv

(1

S"-1

ya, w)1J

o• (0. E. D.)

APPENDIX 17

SPECTRA OF K-SYSTEMS (See Theorem 11.5) §A. Subalgebras of Measurable Sets Let (M, /L) be a measure space. We denote by 1 the algebra of all the measurable sets and by DEFINITION

6 the

algebra of the subsets of measure 0 or 1.

A 17.1

A subalgebra (f of measurable sets is a subset of

f

which is closed

under the formation of complement and the denumerable union, and which contains M. INCLUSION A

17.2

If (1'0 and (fl are subalgebras of

1,

then (fo C (fl means that (fo is

a subalgebra of (tl' that is, that every element of (to is an element of (fl. The relation C is a reflexive partial ordering on the family of sub algebras of 1. INTERSECTION A 17.3

Let

«(1'); ([

be a family of subalgebras of 1. We denote by

n

(f

I

;(/

the largest subalgebra of SUM

f

which belongs· to each (fr

A 17.4

Likewise, we denote by 153

154

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

V G'; ; ( I

the sum of the (1.'s, that is, the smallest subalgebra of 1

f

which contains

every (J;. THE SPACE

L 2 (G'). A17.5

Let (J be a subalgebra of 1. We denote by L 2 (G') the subspace of

G'.

L 2 (M, /L) generated by the characteristic funCtions of the elements of

The following preperties are readily verified:

G' c 93 L2(

implies L 2 (G') C L 2 (93) ,

n G';)

;(1

L 2 (6)

".

n L 2(G';)

,

;(1

". H 0' one-dimensional space of the constants.

§ B. Spectra of K-Systems

We next prove the following theorem (see Theorem 11.5). THEOREM

A 17.6

A K-system (M, /L, ep) has denumerably multiple Lebesaue spectrum. Recall that there exists (see Definition 11.1) a subalgebra

G'

of

f

such that: (A 17.7)

n

6 '=

¢nG' .•• c ¢-1G' c G' c ¢G' c··· c

V··· ¢nG'

1.

n=-oo

The proof breaks up into several lemmas. LEMMA A

17.8

Let U be the unitary operator induced by ¢. If H ". L 2 «(t) then:

155

APPENDIX 17

n""

HO =

UnH

c···c UH

Let us denote by H

U

C H C···C

n=-oo

UnH = L 2(M, /l).

0=-00

e H0

H' the ·orthocomplement of H 0 in H; this may

=

be written again:

101

"" n

=

UnH'C"'CUH'CH'CU-1H'C'"

0==-00

"" U

C

UnH' = L; = L 2 (M, /l)

e Ho

.

0=-00

Proof:

Let A be .an element of

(1, and

~ A its characteristic function. Then:

if ¢ (x) =

{

I

A

if ¢ (x) ( A ,

that is, {

U~A(x) Thus,

uXA

=

0 if x

I ¢-lA

1 if x ( ¢-lA .

== ~¢-lA and, according to the definition of L 2 «f), we have:

UL 2«(f) = L2(¢-1(f) •

Now, the lemma is a direct corollary of properties (A 17.5). L~MMA

A17.9

U has Lebesgue spectrum, the multiplicity of which is equal to

dim (H

e

U H) .

Proof:

Let

Ih.1I be a complete orthonormal basis of

H' e UH '. Denote by J{.I

the closure of the subspace spanned by hi' Uh i ,.... From their very construction, the Uih I.'s, and so the J{.'s are orthogonal to each other. From I Lemma (A 17.8) we have plete system of H ':

n""

n=-~

UnH' =

101.

therefore lUih.! is a comI

156

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

(A 17.10t On the other hand, the relation:

can be rewritten: oJ

U

v-nH'

L~.

n= 0

This relation and (A 17.10) yield: 00

EB

~ i

U

V-n}{i

n=O

Let us set: (A 17.11)

H,

U V- n }{., , n=O

this can be rewritten L~ =

(A 17.12)

EB

~ Hi i

According to (A 17.11), from the basis Ih i' Vh i' ... 1 of }{ i we obtain a complete orthonormal basis of Hi' namely:

Furthermore:

Ve ,,] .. for every i and j. Together with (A 17.12), this proves that V has Lebesgue spectrum, the multiplicity of which is equal to the cardinality of IH ii, that is dim (H' e VH') = dim (H

e

uR) •

157

APPENDIX 17 LEMMA

A 17.13

e

dim(H

uR)

00

•

Proal: Lemma (A 17.8). implies: ••• $. dim UH $. dim H $. dim U-1H $. •••• This proves dim H dimUnH

=

=

=

dimU n+1H

Similarly, dim UH function I ( H

e

=

and UH ,;, H, since dim H < 00 would imply

00

00.

dimlOI for n large enough, that is H

Since UH ~ H, there exists some non-vanishing

UH. Let us denote by F the support of I:

F =

1m I m (

We have F ( (1 and p. (F) > 0 for L =

Ig·:tFlg

Lo

i(

01 •

L2 «(1), and the space

=

00.

F)

Similarly, as p. (F) > 0 and dim UH

=

19·:tFI g ( UHI has infinite dimension. Let us set L e L1 and take g.:t F ( Lo and h ( UH, then we have:

the space L1 =

M,/(m) ,;,

(:t F : characteristic function of

(HI

has infinite dimension for dim H 00,

= O.

=

= = O.

Thus Lo C H

e UH,

and it is sufficient to prove that dim Lo

AS,[,l is infinite-dimensional and

=

00.

p. (F) > 0, there exists a sequence

of bounded real-valued functions: h 1, h2' ... ( UH such that the XF~h1'

:t ~2'

... ( L1 are linearly independent. Since

I does not vanish in F,

we find that the Ih 1, Ih2' ... ( L are linearly independent. But they belong to Lo because h ( UH implies:

(h k • h ( UH is ortJlOgonal to

n.

Thus, there exist infinitely many linear-

ly independent functions in Lo.

(Q. E. D.) The preceding lemmas prove Theorem (A 17.6).

APPENDIX 18

CONDITIONAL ENTROPY OF A PARTITION a WITH RESPECT TO A PARTITION {3 (See Section 12, Chapter 2)

§ A. Measurable Partitions DEFINITION

A 18.1

Let (M, /1) be a measure space. A partition a : IA)j ([ of M is a col-

lection of nonempty, nonintersecting measurable sets that cover M: ,,(A.I nA.): 0 if i 1= j, ,,(M-U A.) : J r . I

r

O.

I

A partition a is said to be measurable if there exists a countable system

lB.J LJ l J

of measurable sets such that:

(1) each B j is a sum of elements of a; (2) for any two elements A., A., of a there exists a Bk such that eiJ

I

ther Aj C B k , Aj

rt.

B k , or Aj

rt.

.

B k , Aj C B k •

A finite or countable partition is clearly measurable. DEFINITION A 18.2

From the very definition, it follows that we can remove the elements of measure zero from a partition.

Mo~e

generally, two partitions a and {3 will

be identified: a: {3 (mod 0) if their elements coincide up to some sets of measure zero.

In the future we delete (mod 0). DEFINITION A 18.3

A partition {3 is said to be a refinement of a partition a: a:S {3 if 158

159

APPENDIX 18

every element B of {3 is a subset of some element A of a: p. (B - B n A) =

o.

DEFINITION A 18.4

Let

la;\iEI

be a family of measurable partitions. We define their sum: a =

V

ai

iff

as the smallest partztion which contains every ai • In other words:

n A·IJ A.J

a ={

f

a. for all

jff

The operation

J

i}.

V is commutative and associative and a :S. a',

{3:S. {3' imply a V {3 :S. a' V (3' •

DEFINITION A 18.5

Given an arbitrary measurable partition a, we denote by m(a) the sub-

al~ebra of the algebra

f that consists of the sets

that are sums of elements

of a. The al~ebra m(a) is called the algebra ~enerated by a. It turns out that for every subalgebra (j of

i, there exists a measur-

able partition a such that:

One verifies at once: a

= (3 <

a :S. (3

'l]j (

V

>m(a)

= m({3)

< >m(a)

C ')J{(f3}

ai )

=

iEI

V

m(a)

iEI

(See A. N. Kolmogorov [3] and V. Rohlin [3]).

§ 8. Entropy of a Given {3

Let a

=

IAi I i

= 1, ... , d and {3 =

IB j 1 i

= 1, ... , 81 be- two finite mea-

surable partitions. We can assume, without losing generality, that any element Ai or B j has positive measure (see A 18.2).

160

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

DEFINITION

A 18.6

Let z (t) be the function over [0, 1] defined by:

~

=

z(t)

- t log t if 0 < t

o

:s 1 •

ift=O.

f3

The conditional entropy of a with respect to

is:

h(alf3} = ~ Il(B j ) ~ z(Il(A/B j )), i where

Il(A./B.) 1

]

=

Il(A. nB.) 1

]

Il(B.) ]

is the conditional measure of Ai relative to Bj" We tum next to the proof of Theorem (12.5) which we reformulate. THEOREM

Let a

12.5

=

f3 = IBjl,

1Ai\'

=-ICkl

y

be finite measurable partitions.

Then: h (al f3) ~ 0 with equality if, and only if a

(12.6)

:s f3;

h(aVf3.(y) = h(aly) + h(f3/aVy);

(12.7)

:s f3

(12.8)

a

(12.9)

f3:s

(12.10)

y

> Maly)

:s h (f3/y) ;

> h(a/y)

~ h(alf3};

h(aVf3/y)

:s h(aly) + h(f3/y)

•

Proof: Proof of (12.6) is left to the reader as an easy exercIse. The elements of aV f3 and aVy are, respectively, of the form: Ai B j and Ai

n Ck ·

Therefore

h (aV f3/y)

But we have

n

161

APPENDIX 18

jl(A,nBjnC k )

=

jl(Aj~Ck)

jl(AjnBjnC k )

jl(C k )

jl(A,nC k )

jl(C k )

= jl(A/CkhdB/ A( n'C k )

and we deduce relation (12.7): h(aV{3/y)

= -

!

jl(AjnBjnC k ) Log jl(A/C k )

i,j,k

= -

!

jl(A j

n C k ) Log

jl(A/C k ) -

i,k

h(aly) + h({3/aVy)

Let us prove relation (12.8): If a :::. {3, then aV {3

(3 and relations

=

(12.6) and (12.7) imply: h({3/y)

=

h(a/y) + h({3/aVy) ~ h(aly) •

Let us prove relation (12.9): Since lk jl(CkIB j ) = 1 and jl(C/B j ) ~

0, the concavity of z (t) implies:

Since {3 :::. y, each B j is the disjoint union of some Ck's; therefore we have:

where the sum extends over those C k,'s be longing to B j

.

We deduce:

162

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

!

z(p.(A/Ck))·p.(CkIB j ) $. z[p.(A/B;)].

k

Multiplying both members by p. (B j) and summing over i and j yields (12.9). Finally, (12.10) is a consequence of (12.7) and (12.9): a V y ::: y implies h «(3/ aVy) $. h «(3/y)

and h(aV(3/y)

=

h(a/y) + h«(3/aVy) $. h(aly) + h«(3/y).

The preceding definitions and properties extend to denumerable measurable partitions (see Rohlin and Sinai [5]).

APPENDIX 19 ENTROPY OF AN AUTOMORPHISM (See Theorem 12.26) The purpose of this appendix is to prove the following theorem dl.e to Kolmogorov. THEOREM

1

A 19.1

If ¢ possesses a generator a, then h (¢) = h (a, ¢):

The proof breaks into several lemmas. Denote by F the set d all finite measurable partitions. Given a, {3

f

F we write

Ia, {31

= h (a

I (3) +

h ({31 a). LEMMA

19.2

Ia, {31 is a distance on F. Proof: It is clear that

Ia, {31

~

O. From formula (12.6) of Chapter 2 we de-

duce:

Ia, {31

=

0

> h (a I (3)

It is also evident that

= h ({3

la, (31

I a)

=

='

0

>a $. {3 and {3 $. a

>a = {3.

1{3, al. According to (12.11), (12.12), and

(12.9) we have: h(a/y) = h(aVy) -h(y) $. h(aV{3Vy) -h({3Vy) + h(j3vy) -h(y) = h(aI{3vy)+h({3/y) ~ h(a/{3)+h({3!y)

1 The proof follows Rohlin [4].

163

164

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

and symmetrically: h(y/ a)

5 h({3/a) + h(y/{3) •

Addition yields:

la, yl ~ la, {31 + 1{3, yl • LEMMA A

19.3

Given ¢' h (a, ¢) is a continuous function on F in' its argument a. More precisely:

Proof:

Given a, {3 ( F, we set:

an = aV¢a •.• V¢n-l a ; {3n = (3V··· V¢n-l{3 • From (12.11) of Chapter 2 follows: h({3n/an) -h(an /{3n) = [h(an V{3n) -h(an)]-[h(an V{3n)-h({3n)] = h ({3n) -h (an) •

Since h (

I) ~

0, we deduce:

On the other hand, from (12.10) of Chapter 2 follows: .h(an /(3 )n = h(aV···V ¢n- 1 al{3n ) < h(a/{3 n ) + ••• + h(¢n-l a/{3 n ). .

Similarly, from (12.9) and because {3, ... , ¢n-l {3

Symmetrically: Addition yields:

513

fT ,

'we have:

165

APPENDIX 19 _

Dividing both sides of this inequality by n and passing to the limit as n ->

00,

we obtain Lemma (A 19.3).

LEMMA A 19.4

If aI' a 2 , ... is a sequence of finite partitions such that

00

V n~l

m(an )

~

~

1 .

then the set B of partitions {3 ( F, such that {3

~ an

for at least one

value of n, is everywhere dense in F. Proof: We-rreed to prove that for every finite partition a and every 0 > 0 there exist an n and a {3 ( B such that:

{3

s.

an'

la, {31 < 0 .

Let AI" .. , Am be the elements of a.

is dense in

( m(an)

1,

Since

for every 0' > 0 there eXist an n and subsets AI',···, A~_l

such that: i~1 .... ,m-1.

Let us denote by {3 the partition of M into sets B 1 •

It is clear that {3

s.

an. On the other hand:

la. (31 ~

r hi (3) + h ({31 a)

...•

Bm defined by:

166

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

k

IL(A)

!.

Il(B k nA j )

j

Il(B k )

- ! k

=

Il(A j n B k )

Il(A) !

- !

j

Log [ Il (A I. n B k) Il (A j) Log

]

[1l(B k nA.h I

Il(B k )

Il (B k)

J

- 2 ! IL(AjnB k ) Log IL(AjnB k ) + ! IL(A) Log IL(A j ) i

i,k

These formulas show that and vanishes when

Al

=

la, ~I

depends continuously on

A l , ... , A~_l

=

Ai, ... , A~_l

A m _ 1 . Therefore, if 8' is small

enough, then la,~1 < 8. Proof of Kolmogorov theorem:

Assume that ¢ possesses a generator a. We set, for i\

l

F and q =

0,1, ... :

We have:

From Lemma (A 19.4) it follows that the set B' of partitions that ~

~

~ l

F such

$. an for at least one value of n is everywhere .dense in F. Let

be an element of· B '. Clearly:

Therefore, from (12.12) of Chapter 2 follows: h (~m) $. h (iin+m_ l

)

167

APPENDIX 19

n + m -1 m

Now, observe that:

q

q

h(>.V ••• V ¢2 q -2>..)

2q-1

2q-1 Thus, passing to

~he

... 2h (A, ¢), as q ...

00

•

q

limit as m ... + 00, we obtain h ({3, ¢)

s

h (a, ¢) .

Recall that B' is everywhere dense in F and that h ({3, ¢) is continuous in {3 (Lemma A 19.3), then: h (a, ¢) that is:

~

sup h ({3, ¢) B'

sup h({3, ¢) F

~ h(¢) ,

·\PPENDIX 20

EXA\IPLES OF RIEMANNIAN \IANIFOLDS

\"\ITII NEGATIVE CURVATURE (See 14.1, Chapter 3) Consider the proper affine group G of the real line

I tit

( R I. An ele-

ment g' of G has the form:

g:

t

-+

yt +

x, y (R,

x,

y > 0,

and can be denoted by (x, y). Given g'

we obtain:

= (x', y'),

g'(g(t))

y'(yt+x)+x'

=

=

y'yt+y'x+x'.

Therefore, if we denote the group operation by .L this may be written: (x', y').L (x, y) = (y'x + x', y'y) •

The neutral element is e = (0, 1) an~ the inverse of (x, y) Both .L and g

-+

IS

(_xy-l, y-l).

g-l are smooth operations. Thus, G is a L"ie group that

is diffeomo.rphic to. the upper half-plane I (x, y) I y >

01.

to. a Riemannian manifo.ld. THEOREM A 20.1.

THE RIEMANNIAN METRIC OF G

The leFt-invariant metric of G which reduces to

at the neutral element e

=

(0, 1) is: dx 2 + di

y2

168

No.W we turn G in-

169

APPENDIX 20

Proof: To any element X = (x, y) of G corresponds the left translation LX· L x(U)

=

x 1. U,

where U

=

(u, v)

l

G .

We have:

!:) ,

( -u-x y-' y the tangent mapping of which is:

(A 20.2) Define a metric over the Lie algebra TG e by setting:

This defines a left-invariant metric at each point X:

Therefore, if X = (x, y), (A 20.2) implies:

(~1)2 + (~2)2

In other words, the mejric is: (A20.3)

DEFINITION A 20.,+

The upper half-plane G endowed with the metric (A 20.3) is called the Lobatchewsky-Poincare plane. It can be useful to represent a point (x, y) of G by the complex number z = x + iy. THEOREM A 20.S.

THE ISOMETRIES OF G

The symmetry (x, y)

-+

(-x, y) and the homographies:

170

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

(A20.6)

az + b cz + d ;

z ... z

a, b, c, d (

R. ad-bc

1

preserve the metric (A 20.3).

Proof: Proof is purely computational and easy if one observes that: ds 2

=

-4dzdz h - - - - , were z (z_z)2

=

.

X-ly.

THEOREM A20.7. ANGLES

The angles of metric (A20.3) coincide with Euclidean angles. Consequently, words such as "orthogonal," and so on can be used unambiguously.

Proof: i's proportional to dx 2 + di .

THEOREM A 20.8. GEODESICS

The geodesics of (A 20.3) are the straight lines: x

=

constant, y > 0

and the upper half-circles centered on ox. In particular, there exists one, and only one, geodesic passing through two given distinct points. Proof: Let ab be a segment of x

0, y > O. For any arc y joining a and

b we have:

f

ds 2

•

ab

This proves that x = 0, y > 0 is a geodesic. An image of this geodesic under any isometry (A20.6) is still a geodesic. We obtain so all the upper half-circles centered on ox and the halfstraight lines x

=

constant, y > O. In fact we obtained all the geodesies

for, given a vector u ( Tl G, there exists a half-circle centered on ox (ora parallel to oy) which is tangent to u.

171

APPENDIX 20 THEOREM

A20.9.

CURVATURE

The Gaussian curvature of (A 20.3) is equal to -1. Proof: The Gaussian curvature K is constant,

tor the metric is invariant

under a transitive group of isometries. The Gauss-Bonnet formula applied to a geodesic triangle L\

=0

A+ B+ C ':"

ABC gives:

1T

+

ff

K· da

=

1T

+ K· area L\ •

t" ~

~

The particular case of Figure (A 20.10) gives A

B

C

=0

O. As the

element of area is da = (dxdy)/y2, we obtain: area L\ We conclude that K

=

1T •

-1.

y

----O~=-B---------/------~~==~C~--~x r Fi~re

A20.10

172

ERGODIC PROBLEMS OF CLASSICAL MECHANICS ;:

THEOREM A20.11. ASYMPTOTIC GEODESICS

Let y (u, t) = y (t) be a geodesic parametrized by arc length t, and g

G. The geodesic passing through g and

f

as t1

->

}J!J.- has a

limit position

+ 00 (resp. -00). This limit position is the geodesic passing

through g and the intersection y(+oo) (resp. y(-oo)) of y with ox. Geodesics emanating from y(+oo) (resp. y(-oo)) are called the positive (resp. negative) asymptotes to y. Proof: Let y(t 1) be a point of y. The geodesic passing through g and y(t 1) is a circle centered on ox, possibly reduced to a straight line (A 20.8). From the very definition of the metric (1\20.3), y(tl) runs to-ox as tI

-+

+ 00 (resp. - 00), that is y (t 1) converges to the, intersection y (+ 00)

(resp. y(-oo)) of y with ox (see Figure A20.12). Thus our geodesic has a limit position, namely the upper half-circle centered on ox and passing through g and y (+ 00) (resp. y (-00)). Consequently, this limit position is a geodesic. y

'Y(-m)

Figure A 20.12

x

173

APPENDIX 20

DEFINITION A 20.13. HOROCYCLES

1

The orthogonal trajectories of the positive (resp. negative) asymptotes to yare called the positive (resp. negative) horocycles of y. THEOREM A20.14.

The positive (resp. negative) horc.::ycles of yare the Euclidean circle of G which are tangent to y

=0

at y(+oo) (resp. y(-oo)). In particular,

the straight lines y = C > 0 are horocycles. They are positive horocycl/ of the axis oy (y

-+

00).

Proof:

The 'positive (resp. negative) asymptotes to y form the upper part of the pencil of circles that are orthogonal to y

=

0 at y(+oo) (resp. y(-oo)).

Theorem (A 20.14) follows at once from the elementary properties of conjuii!,ate pencils of circles. The points y(+oo) and y(-oo), which do not belong to G, have to be removed. THEOREM A20.1S. RIEMANNIAN CIRCLES

The Riemannian circles of (A 20.1) centered at m form the upper part of the pencil of circles whose radical axis is ox and whose Poncelet points consist in m and the symmetric m' of m with respect to ox. Proof:

The Riemannian circles centered at m are the orthogonal trajectories of the geodesics emanating from m. This family of geodesics is

nothi~g

but the upper part of the pencil of circles passing through m and m'. (Q. E. D.) In particular, the power of any point d of ox with respect to one of these Riemannian circles centered at m is:

(see Figure A 20.17). 1 Notion due to Lobatchewsky (in Greek, "horos" = horizon).

174

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

THEOREM

A 20.16

Horocycles are Riemannian circles the radii of which are infinite and the centers of which are at infinity (on y

=

0). '/

Proof: Consider the Riemannian circle passing through a fixed point n of a geodesic y and centered at m ( y (see Figure A 20.17). If m moves to infinity along y, that is, if m converges to ox, then mm' ... O. Therefore, the power of any point of ox with respect to our circle tends to zero. Thus our circle has a limit position which is the circle tangent to ox at y (+ 00) and which passes through n. Theorem (A 20.16) shows that this limit position is an horocycle. Conversely, any horocycle is obtained from the (Q. E. D.)

above construction.

'Y(+OO )

m' Fi\t:ure A20.17

175

APPENDIX 20 THEOREM

A 20.18

Let y(u, t) and y'(u', t) be two geodesics which are positively (to fix the idea) asymptotic one to the other. We denote their arc length counted from their origins nand n' by t. Then, after a suitable selection of nand n " we have: d (y (t), y '(t)) $. nn 'e- t ,

t 2. 0,

where d means the Rierrftmnian distance, and nn' is the arc-length of the horocycIe. Proof: Origins nand n' are selected on the same horocycle 1 (Figure A 20.19). Denote by m and m' the intersections of y and y' with another horocycle 2. Arcs

nm

and

0

'm' are equal, for 1 and 2 are parallel curves: n 'm' =

nm =

Let us compute the arc

mm'

t.

that belongs to 2. Horocycle 2 has the equa-

tion: x

=

r sin u, y

= r + r cos

u.

Thus, with obvious notations: m' -l-+-d -Cu-o-s-u

{

Symmetrically, on horocycle 1: nn

,

=

un'

tg -

2

U

- tg ~

2

A straightforward computation with y and y' leads to: t =

t

~

om =

n'm'

Log 1tg

I

Log tg

~n I -

Log I tg u;

Uri -

Log tg

I

I'

U; 'I .

176

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

y

r

x

Figure A 20.19

Consequently:

tg et

u _n_

2 u

tg .2!.. 2

tg

u

~

2--u

,

tg .2!.. 2

mm

I

=

u

tg ~ - tg 2

u

tg

,

~

- tg

2

I-t

nn • e

Theorem (A 20.18) follows from d(m, m ')

:s.

mm' .

u

-!!.

2

u .2!!_

2

nn

mm

177

APPENDIX 20 GENERALIZATION A 20.20

The manifold V is the upper space xn > 0 of Rn endowed with the metric: (dx )2 + •.. + (dx )2 1

n

(x )2 n

V is the Lobatchewsky space of constant curvature - 1. The horocycles

are (n -1) -dimensional manifolds, namely the planes xn

constant and

the Euclidean spheres of V which are tangent to the plane xn

=

o.

APPENDIX 21

PROOF OF TilE LOBATCHEWSKY-IIADAMARD THEOREM (See 14.3, Chapter 3) § A. Manifolds of Neglltive Curvature Fa:>,

of negative

,t us recall some classical properties of Riemannian manifolds '-M

'me.

THEOREM A 21.1

Let V be a complete, simply connected Riemannian manifold of negative curvature. Then: (1) There exists one, and only one, geodesic passing through two

dis'tinct given points; (2) V is diffeomorphic to the Euclidean space;

(3) let ABC be a geodesic triangle whose angles are A, B, C 'and whose sides are a, b, c. Then:

Proof will be found in S. Helgason [1], A direct consequence is the following corollary; COROLLARY A 21.2

Under the above assumptions, Riemannian spheres of V are convex, that is, a geodesic has at most two common points with a sphere.

178

179

APPENDIX 21

§ B. Asymptotes to a Given Geodesic As usual, y (x,

II,

t)

= y (t) = y

denotes a geodesic emanating from x,

with initial velocity-vector u and arc length t. The point of y corresponding to t is denoted also by y (t). The Riemannian distance of two points a and b is denoted by

la, bl.

Denote a complete, simply connected Rie-

mannian manifold of negative curvature by V. THEOREM A

21.3

Let v' be a point of V. The geodesic joini!lg v' to a point y (t) ( y converges to a limit as t

->

+00 (resp. t

->

-(0).

This limit is a geodesic.

froof: (See Figure A 21.4.)

v' Y(v',u',t)

----------------~------------~----~--~Y

Fi~re

A21.4

The points v' and y (t 1 ) define one, and only one, geodesic y (v', u l' t ). We set

s1 =

Iv: y(t 1)1.

Take t2 > t1 and apply relation (3) of Theorem

(A21.1) to the geodesic triangle v.', y(t 1), y(t 2 ); With obvious notations

we have:

On the other hand, the triangular inequality applied to v, v' y (t 1) gives:

t1 whence:

lv, v'l

~ s1 ~ t1 +

lv, v'l '

180

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Similarly:

We deduce: 1, that is to say:

Thus, according to Cauchy, u 1 converges to a limit u' as t1

->

+

00.,

The geodesic y (v', u ',t) is the limit position of y (v', u l' t), for the exponential mapping Expv' is continuous.

y(v', u', t) is called a posi-

tive asymptote to y. Negative asymptotes are defined in the same way (t 1

->

-(0).

REMARK

A 21.5

It is readily proved that the positive asymptote to y emanating from a

given point of the positive asymptote y (v', u', t) is nothing but y (geometrically). Therefore, we may speak of a positive asymptote to y without referring to a definite point v'. Furthermore, the set of the positive asymptotes to y is a (dim V -1)- parameter family of geodesics. § C. The Horospheres 1 of V

The Riemannian manifold V is again complete, simply connected, and of negative curvature. Let y (v,

11,

t )., =' y (t) be a geodesic and v' an ar-

bitrary point of V. LEMMA

A21.6

converges to a finite limit L (v': y, v) as t differentiable function of v' and v. 1

See A. Grant

[1].

->

+ 00, and this limit is a C 1 .

181

APPENDIX 21

Proof: Take t2 > t l' The triangular inequality applied to v, Y (t l)' Y(t 2 ) gives: ¢(t2)

lv', y(t2)1 -Iv, y(t2)1 ::: lv', y(tl)1 + Iy(tl)' y(t2)1 -Iv, y(t2)1 =

Iv,' yUl)1 -Iv, y(tl)1

=

¢(tl) .

Therefore, ¢ (t) decreases monotonicall::::. On the other hand, ¢ (t) is bounded, for the triangular inequality applied to v, v', y (t) gives:

This proves the existence of: ¢(t) = L(v'; y, v) .

lim t ..... +

00

The second assertion follows from the inequality:

that is

Obviously: L(v'; y, v) - L(v'; y, Vl) = vVl

(A 21. 7)

where

vVl

is the algebraic measure of

vVl

'

on the oriented geodesic y .

DEFINITION A 21.8

The locus of the points x for which L (x; y,O) = 0 is called the

pos~

itive horosphere through 0 of y and will be denoted by H+(y, 0). According to Lemma (A21.7), H+(y,O) is a Cl-differentiable submanifold. of dimension (dim V-I). Let vt be an arbitrary point of y. Relation (A 21. 7) shows that H +(y, 0) has equation:

L(x; y,vt )

=

OVl

Now we obtain the horospheres as spheres with center at infinity and radius

182

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

infinite. The Riemannian sphere, the center of which is a and passing through b, will be denoted by L (a, b). LEMMA

A21.9

L (y(t),O) converges to H+(y,O)

as t -+ +00.

Proof:

Let x be a point of H+(y, 0), we have: ¢(t) '" Ix,y(t)I-IO,y(t)1

On the other hand, ¢ (t)

~

-+

0 as t

-+

+00.

O. Therefore, L (y (t), 0) intersects the geo-

desic segment xy (t) at a point b(t) (see Figure A 21.10).

-----+--------~--------+-------~y

o

Figure A21.10

We have: lx, b(t)1 = Ix,y(t)I-ly(t), b(t)1

Ix,y(t)I-IO,y(t)1

-+

0 as t

-+

+00.

This means that every point of H+(y, 0) is a limit point of the spheres L(y(t),O) as t

-+

+00. Conversely, we prove that SUGh a limit point be-

longs to H+(y,O). Let b(t) be a point of L (y(t), 0) and x

=

lim b(t). t -+ +00

The triangular inequality gives:

Ilx, y(t)1 -10, y(t)11 < I lx, y(t)1 -Ib(t), y(t)11 + Ilb(t), y(t)1 -10, y(t) II Ix, b( t) I

-+

0 as

t

-+

+ 00 •

183

APPENDIX 21

Therefore: L (x; y, 0)

~

0, that is, x

f

H+(y,O).

COROLLARY A21.11

Horospheres are convex, and strictly convex if the curvature of V

IS

boundoo from above by a negative constant. Proof: H+(y,O) is the limit of the balls passing through 0 and the center of

which goes to infinity along y, and these balls are convex (see A 21.2). LEMMA

A21.12

Let H+(y, 0) and H+(y, 0') be two horospheres of y. If a (H+(y, 0) and a' ( H+(y, 0'), then la, a'i :: 10,0'1. Proof:

Assume la, a' I

< 10, 0' I. From (A 21. 9) we conclude that to each

corresponds a point a (t)

and a point a'(t)

f

~ (y (t),

f

~ir(t),

0) such that:

lim a(t) ~ a, t ... + 00 0') such that: lim

t ... + 00

a'(t)

=

a'.

Thus, for t large enough, we have: la(t),a'(t)1

< 10,0'1.

To fix the ideas assume that the point 0' lies between the points 0 and y (t). We obtain the following contradiction: laCt), y(t)1 ::; laCt), a'(t)1 + la'(t), y(t)1 ~ 10,y(t)1

LEMMA

=

< 10,0'1 +

la(t),y(t)I·

la'(t), y(t)1

(Q. E. D.)

A21.13

Two positive horospheres H+(y, 0) and H+(y, 0 ') cut off an arc of length 10,0' I on every positive asymptote to y.

184

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Proof:

Y(a',Uo,t) -----+--------~------------~-----------Y ·0

FIgure A 21.14

Let y(a', u, t) be a positive asymptote to y that intersects H+(y, 0') at a '. The points y (t) and a' define a geodesic on which we select a point a(t) such that la(t), a'i = -L(a'; y,O) = 10,0'1 and a' lies between a (t) and y (t) (see Figure A 21.14). Since the exponential mappinl Exp a' is continuous, we obtain: . a(t) = a

lim

f

y(a',u',t) and la,a'i

-L(a'; y,O).

We deduce: I la,y(t)I-IO,y(t)1 I :; Ila,a(t)1 + la(t),y(t)I-IO,y(t)11 =

I Ia, a (t) I + Ia', y (t ) I -I

°"

y (t) II

->

0 as t ... + 00 •

Thus, a ( H+(y, 0). (Q. E. D.) THEOREM A

21.15

The positive asymptotes to yare the orthogonal trajectories of the positive horospheres of y. Proof: DireCt consequence of (A 21.12) and (A 21.13). Finally, observe that negative horospheres H-(y, O) can be defined as above from the negative asymptotes (t

->

-00).

185

APPENDIX 21

§ D. The Horospheres of Tl V

The unitary tangent bundle of V is denoted by Tl V and p: Tl V

-+

V

is the canonical projection. Let u be a point of Tl V; u defines a geodesic y (pu, u, t)

= y (u, t)

y (t) .the lift of which, in Tl V, is denoted again by y (t). From § B we know there exist two horospheres H +(y, pu)

=

H +(u) and H -(y, pu) =

H -(u) passing through pu. The set of the unitary vectors orthogonal to H+(u) (resp. H-(u)) along H+(u) (resp. H-(u)) and oriented like u is a

(dim V-I)-dimensional submanifold }(+(u) (resp. }(-(u)) of Tl V. The }{'s are called the horospheres of Tl V. THEOREM

A 21.16

(1) The y(u, t)'s and the }{+(u)'s, }{-(u)'s are the sheets of three

foliations of Tl v. (2) At each point u ( Tl V these foliations are transverse, that is:

T(T1 V)u where

X:

(resp.

X;;,

=

X;

E9

X;; Zu

Zu) is the tangent space of }(+(u) (resp. }(-~u),

y (u, t) ) at u.

(3) These foliations are invariant under the geodesic flow ¢t:

Proof: (1) Follows from the very construction of the sheets. (2) Follows from the strict convexity of H + (resp. H -, see A 21.11). (3) Follows from Theorem (A 21.15). The invariance of the foliations reduces the study of the differential

¢; to the study of its restriction to }(+(u) (resp. }(-(u) ) and y(u). we assume definitively that V is the universal covering

Iv

Now,

of a compact

. Riemannian manifold W of negative curvature. In particular, 'the curvature of V is bounded from above by a negative constant _ k 2 .

186

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

LEMMA

A21.17

Let rs(t) be a one-parameter family (s> 0) of numerical, C"-differentiable functions. Assume that: 2 r. s > - k ·r s

(k = constant

> 0)

for every s,t 2: 0, and rs(O) > 0, rs(s) = O. Then:

<

r (t) S

cosh [k (s -~, for 0 S t S s . cosh [ks]

r (0)· s

Assume, additionally, that: lim s-++oo

Then, for s large enough: [i)t)[
sinh [k (s - t)]

,forOStS4.

cosh (ks) Proof:

The function: r s(O)

rs(t) -

• cosh[k(s-t)]

cosh (ks) verifies:

i'

.s

(t)

> k2·1 (t), s

Thus, Isis concave between 0 and s and vanishes for t

=

0, s, conse-

quently I s(t) S 0 for 0 S t S s. This proves the first part. This proves also that is increases between 0 and s; thus: i s(t) S is(s) for 0 S t

S

s. On the other hand,

in particular,

is (t)
rs(s)

... 0 as s ... + 00

•

187

APPENDIX 21 THEOREM

A 21.18

Let ¢>t be the geodesic flow of Tl V. Then, for any positive number

1I¢>;(il S..b'e-kt·II(II, II¢>; (;1

~

a'ektll(ll,

The positive constants

IlcP~t(11 ~a.ektll(11 if(lX:,

11¢>~lll S

a and

b

b'e-ktll(11

are independent of

if ( l X~ t and

~,

and

!I

denotes the length of a vector of Tl V equipped with its natural Riemannian metric. Proof: We p.rove the first inequality, the others can be proved in the same way. Let y (0, u, t)

=

y (t)

=

y be a geodesic of V, and let x be a point of

H+(y,O), close enough to O. There is a well-defined geodesic y s(x, us' t) = y s(t) passing through x and y (s)

the Riemannian distance of y(t) and

l

y. Our first purpos"e is to compute

ys(t), regarded as ~lements of

Tl V.

Let r s(t) be the Riemannian distance of their projections y(t) and y s(t) on V. To compute r s(t) '·we consider a Jacobi field 2 1/1 (t) -along y, 'that is orthogonal to y and vanishes for t

=

s. By definition:

where R ( , ) is the curvature tensor and V is the covariant derivative along y. By definition the sectional curvature in the two-plane (y,l/I) is:

< R (y, I/I)y, 1/1 > :11/1 112 We know that p(y,l/I)

On the other hand,

2 See

J.

Milnor

[1].

S

_k 2 , consequently:

188

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

V

=

YlV 2 \11/I112

=

Yl d2211 1/1112 , dt

I\VI/I 112 ? (~ 111/1 11)2 Therefore, the length ls(t). of I/I(t) verifies:

that is

rs

> k 2 • I s , and I s (0) > 0, -

ls(s)

=

0 .

Lemma (A 21.17) and the classical possibility to select the Jacobi field

1/1 such that: r s(t) = I s(t) + 0(1)

if x is close enough to y imply: r s(t) < r s(O) •

(A21.19)

cosh[k(s-t)] , for 0 ::;, t ::;, s . cosh (ks)

Now it is readily seen that the angle of y and y s at y (s) converges zero as s

->

+ 00. Thus,

i)s)

->

0 as s

->

~o

+ 00, and Lemma (A 21.17) im-

plies again:

I; s(t)1 <

(A 21.20)

As s

->

k· r (0) sinh [k (s - t)] s

, for 0 ::;, t ::;, s .

sinh (ks)

+00, ys(t) converges to a point y'(t) of the positive asymptote

y'(x, u', t) to y, and

ys(t)

converges to V(t). If r(t) denotes the dis-

tance of y (t) to y '(t), then the inequalities (A 21.19) and (A 21.20) imply (s

->

+(0):

IHOI < kr(O) e- kt ,

for t > 0 .

189

APPENDIX 21

Thus, the Riemannian distance of y(t), V(t)

Tl V verifies:

f

We easily deduce th.e first inequality of Theorem (A 21.18).

.

.

Due to this theorem, the sheets J{+(u) (resp. J{ -(u) ) are called the ~

"contracting" (resp. "dilating") sheets of Tl V.

§ E. Proof of the Lobatchewsky-Hadamard theorem 3 THEOREM

A 21.21

Let W be a compact, connected Riemannian manifold of negative curvature, then the geodesic flow on Tl W is a C-flow. Proof: Let V =

IV

be the universal covering of W equipped with the inverse

image of the Riemannian metric of W under the canonical projection":

IV . . W. V

satisfies the assumptions of preceding sections. Thus the geo-

desic flow on Tl V verifies the conditions of C-flows: condition (0) is trivially fulfilled; condition (1) follows from Theorem (A 21.16); condition (2) follows from Theorem (A 21.18). We finish the proof by proving that" is compatible with the three foliations of V

=

IV

and Tl IV. The first ho-

motopy group "l (W) is isomorphic to a group of automorphisms of

IV,

for

W is connected. The group "l(W) acts also as a group of automorphisms

of Tl IV: if u', J{ ±.(u

U" f

Tl

IV

are congruent mod" 1 (W), then J{ ±.(u ') and

") are themselves congruent mod" 1 (W).

REMARK

A 21.22

The horospheres of a compact, n-dimensional manifold Ware diffeomorphic to Rn-l. In fact, let us consider the horosphere J{ +.

It is a

See J. Hadamard [l]. Proofs of SectlOns Band C are mainly due to H. Busemann: Metric Methods in Finsler Spaces and Geometry. Ann. Math. Study. No.8.

3

Princeton University Press.

190

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

paracompact manifold. Let S be a compact subset of J{+. Then 1>tS is covered by a disk D of

1> t J{ +

(take t large enough). The counterimage

1>; 1D i~

a disk which covers S in J{ +. Therefore, J{ + is diffeomorphic

to Rn -

according to the following lemma of Brown (Proc. Amer. Math.

1,

Soc., 12(1961), 812-814) and Stallings (Proc. Cambridge Philos. Soc., 58 (1962), 481-488): Let M be a paracompact manifold such that every compact subset is contained in an open set diffeomorphic to Euclidean space. Then M itself is diffeomorphic to Euclidean space. This result

~oes

not hold for noncompact manifolds.

Consider the

space I(x, y) I y > 0, x-(mod 1)1 endowed with the metric:

The Gaussian curvature is equal to - 1 and the universal covering space is the Lobatchewsky plane (see Appendix 20). The curve y horocycle homeomorphic to S1.

FIgure A 21.23

=

1 is an

APPENnIX 22 PROOF OF THE SINAI THEOREM (See Section 15, Chapter 3) Let (M, ¢) be a C-diffeomorphism and X m {resp. Ym ) the k-dimensional dilating space at m Crespo the I-dimensional contracting space). A Rie-

mannian metric is definitively selected on M. Thus, Xm and Ym are Euclidean subspaces of TMm' THE METRIC SPACE OF THE FIELDS OF TANGENT k-PLANES

A22.1 The tangent space TMm is the direct sum Xm

III

Ym . Therefore, the

equation of a k-plane V m C TM m' transverse to fm' is:

where x ( Xm , y (fm , and P(Um): X m ... fm is a linear mapping. We define a metric in accordance with the norm of the linear mappings P(V): if V m and U:n are two k-planes of TM m ' then we set:

IVm -V'[ m

=

IIP(U m )-P(U')II m

=

sup X(Xm

.lx[
IP(v)x-P(U')xl m

m

We turn the set K of the fIelds of the tangent k-planes transverse to Ym into a metric space by setting:

IV-V'I

= sup

m(M

IV m -

V'

m

I.

V, V' ( K.

The distance of Inequality (15.3) of Section 15 has to be understood in this sense. Since M is compact, then K is a compact and complete metric space.

191

192

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

LEMMA A22.2 Let RI and R2 be two n-dimensional Euclidean spaces (n Assume that R j (i

=

Rj =

Let A: RI

->

=

dim M).

1,2) is the direct sum of two subspaces Xj and Yj:

Xj

III

lj,

dim Xj

=

k,

dim Y.I

=

I:

R2 be a linear mapping such that: X2 ,

AX I

{

Y2 ,

Ixl

for x ( Xl

IIAyl < alxl

for y ( YI

IIAxl

(A 22.3)

AYI =

~ fL

where fL and a are constants. Let us denote by

Ci'

the operator induced by A, which makes corre-

spond to the k-planes of RI the k-planes of R 2 . If V and V' are transverse to YI , then:

Proof: By definition:

lCi'v - Ci'v'l

=

sup IP(Ci'U)x - P(Ci'v')x1

Ixl < I

X(X 2

=

sup

Ixl

IA[P(U)A-Ix] - A'[P(U')A-Ix]1

<1

X(X 2

According to (A 22.3), Ixl < 1 and x ( X2 imply A-Ix ( Xl and IA-Ixl

:s.

fL- l . Therefore:

:s.

sup

Ixl
sup

Izl
X(X 2

,

X(X I

and

:Ci'v-Ci'v'l:s.

fL- l

.

sup

!z < 1 Z(

XI

IA[P(U)-P(U')]z!

193

APPENDIX 22

Since [p(U) - p(U ')]z ( Yl , (A 22.3) Implies: IA[P(U) - p(U ')]z

I s:.

a • I[p(U) - p(U ')]z

~ a'

I

sup I[P(U)-P(U')]zl

Izl < 1

a'IU-U'1 .

Z(X I

Finally we have: IAU-AU'I ~ /I-laIU-U'1 •

(Q. E. D.) INEQUALITY

(15.3) OF SECTION 15. A 22.4

The mapping I ¢** (or a positive integer power ¢un is contracting in a neighborhood of the dilating field X:

for

that is, for Ul and U2 transverse to Y. Proof: We apply the preceding lemma and we set:

.

y

The linear mapping A is the differential (¢n) * of ¢n. Since the dilating and the contracting fields X and Yare invariant under ¢, AX 1 = X 2 and

AYI = Y2 are verified. Inequalities (A 22.3) are consequences of the axioms of C-systems:

For n large enough we have

e= I

In fact

¢**

a- l b·e- 2An < 1.

is the extension of the mappIng

¢**

to the k-planes.

(Q. E. D.)

APPENDIX 23

SMALE CONSTRUCTION OF c-OIFFEo\10RPHISMS (See Section 12.3, Chapter 3) Smale [3] has proved that there do exist nontoral C-diffeomorphisms. We give next an example of his construction. The Space M.

Let G be the nilpotent Lie group of the 6x 6 matrices:

C: ;J f1

g

where x, y, z, X, y, Z ( R. The group G is diffeomorphic to R6. Let us denote by Q(v'3)

Ip + qy'3l p, q (

=

zi the number field of YJ x = p - qv'3 the nontriv-

adjoined to the rationals, and by x = p + qy'3 ~

ial Galois automorphism. We consider the subgroup

r

of G, the elements

of which satisfy x, y, z ( Q( v'3) ,

X It is readily proved that

=

r

x,

Y

=

y,

Z

=Z

is discrete and that the right coset space M =

lar! = G/r is compact. Of course, the first homotopy group of M is isomorphic to

r

and, con-

sequently, is a nonabelian· nilpotent group. Therefore, M is nontoral. The Diffeomorphism ¢: M

~

M.

Let us identify an element

a(G

mapping ¢: G • G by:

194

with (x, y, z, X, y,Z). We define a

APPENDIX 23

1> (x,

195

y, z, X, Y, Z) = (Ax, p.y, vz,

AX, ii Y, i7 Z)

,

where:

A

1> 1>

=

2 + V3,

v = (2-V3)2,

P. = Av = 2-y3,

is an automorphism of G, because p. = Av. Therefore, ¢;f

f, and

defines a diffeomorphism ¢ of M by:

(M, ¢) is a C-Diffeomorphismo

An element of the Lie algebra TG e of G is of the form

C:iO;g) The metric

of TG e defines a right mvariant metric on G and, consequently, a Riemannian metric on M = Golf. The Lie algebra TG e splits into the sum X + Y, where the elements of X (respo Y) are of the form:

(respo)

Next, by right translations, the splittmg TG ~ of every point

IS

imposed on the tangent space

g of G:

TG~ = X~ + ~ Thus, the tangent space TMm at m ( M splits into: ™m

= Xm+ Ym

It is ea,sily checked that the linear tangent mapping" d9 and contracting on Ymo

IS

dilating on X m

APPENDIX 24

SMALE '5 EXAMPLE (See Section 16, Chapter 3) Smale [2] proved the following theorem, which gives a negative answer to the "problem of structural stability": are the structurally stable diffeomorphisms dense in the C 1-topology? THEOREM

A24.1

There exists a diffeomorphism phism

1/1',

C 1-c1ose to

1/1,

1/1:

T3 ... T3 such that no diffeomor-

is structurally stable.

We tum to the construction of

1/1.

§ A. The Auxiliary Diffeomorphism ¢> Let T2 be the torus I(x, y) mod 1\. We define a diffeomorphism ¢>1 of

T2 x Iz 1-1 ~ z ~ 1\ onto itself by setting:

{

¢>. 1·

(p . . q P (P

(mod 1)

z ... ~z

Let By, be the ball (Figure A 24.2) of T2 x R with center (0, 0, 2) and ra-

dius

1/2: x2

We defioe a diffeomo",hism

+;+ (z_2)2

~; ¢>1:

{Of

:v: i~~

~ ~ .

T2 x Iz 1

y ... ~y

z ... 2z-2 . 196

°~ z ~

3\ by setting

APPENDIX 24

197

Now, the torus T3 is T2 x S1, where S1 is [-3,3] with endpoints identified.

z

x: dilating direction Y: contracting direction

o

t

t Figure A 24.2

The following lemma is easily proved.

x

198

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

LEMMA

A24.3

There exists a diffeomorphism ¢: T3 ... T3 such that: (1) its restriction to T2 x Iz 1-1 S; z S; 1\ is ¢l'(2) its restriction to By, is

¢i ,-

(3) ¢ leaves {(O, 0, z) 10 < z S; 2\ invariant with no fixed point. PROPERTIES OF

¢. A 24.4

T2 x 10 \ is obviously an invariant torus under ¢. The restriction of ¢ (or ¢l) to T 2 x 10\ is nothing but the diffeomorphism of Example (13.1):

(~)

(A 24.5)

. . 0 0(;)

(mod 1).

Let us recall some properties of this diffeomorphism: There exist two foliations

X and

~ on T2 x 10\. They correspond, respectively, to the dilat-

ing and the contracting eigenspaces Xm and Ym of (A 24.5). Every sheet of

X (or

~) is everywhere dense in T2 x 10 \. The periodic points 1 of ¢

are dense in T2 x 10\. This fact can be proved by observing that every rational point (pi q, p 'I q) ( T2 is periodic. Now we pass to the diffeomorphism ¢: T3 ... that the periodic points of

cp,

in T2 x I z

I-

those of (A 24.5), as do also the foliations

1 S; z

r. s:.

It is easy to see

1 \, coincide with

and ~ in T2 x 10\. The foli-

X

ation ~ generates an invariant contracting foliation of T2 x 1z S; 1\, whose sheets are the "planes" of the form Y x Izl-1

I-

1 S; z

s:. z S; 11,

where

Y is some sheet of ~. § B. The Diffeomorphism ifJ

The diffeomorphism ifJ is obtained by perturbing. cp_ Let Go be the ball of T3 with radius d and center (0, 0, %): x 2 + ; + (z _ %)2 We set G =

d2 .

cp-l Go' ¢ tx, y, z) = (x', y: z '),

1 That is, the points'; ( integer N.

s:.

T2x

10\

such that

¢N';

and we observe that d can

= .; for some nonvanishing

199

APPENDIX 24

be chosen small enough for ¢G

nG =0 .

We define. our desired diffeomorphism tjJ(x, y, z)

=

~

¢ (x, y, z)

r/J by setting: = (x', y', z ')

outside G

(x' + 1/1I>(x, y, z), y', z ') on G,

where II> is a nonnegative COO function with compact support in G and nondegenerate maximum value + 1 at ¢-1(0, 0, %), and finally 1/

>

°

is

small enough so that tjJ is a diffeomorphism. PROPERTIES OF tjJ. A24.6

" z)\ Now the curve tjJI(O,.O,

°s.

z

This bump lies in the region T2 x I z ¢

=

S. 21 has a bump B (see Figure A24.2). \-1 $. z S. 11 where tjJ coincides with

¢l' This region is foliated into contracting planes (see A 24. 4). Let

~xla) + (ylb) =

1 be the equation of such a contracting plane in the chart

(x, y, z). Among these planes intersecting the bump B, we select the plane ~, for which a is maximum (see Figure A 24.2). Either j= contains a peri-

odic point of tjJ, or it does not. In the first case, the bump is called periodic and in the second case, nonperiodic. LEMMA A24.7 tjJ is not structurally stable.

This follows from two remarks: (1) An arbitrarily small change of 1/ in the definition of tjJ gives

rise to a diffeomorphism tjJ" arbitrarily C1-close to tjJ and similar to tjJ. The density of the periodic points (see A 24.4) implies that we can suppose the bump of tjJ is periodic and the bump of tjJ" is nonperiodic, and vice versa. (2) If tjJ and tjJ" are in the opposite cases there is no homeomorphism h: T3

->

T3 close to the identity" such that tjJ".

h establishes a one-to-one correspondence planes, and periodic points of tjJ and tjJ ".

b~tween

h=

h· tjJ. In fact,

bumps, contracting

200

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

LEMMA

A24.8

Every diffeomorphism

!/J',

Ci·close to

!/J,

possesses an invariant tom.

similar to T2 x 101, a bump, a contracting sheet similar to

1,

and so on:

Complete. proof of this lemma is announced by Smale [2]. Now Theorem (A 24.1) follows readily from Lemma (A 24.7) and (A 24.8).

APPENDIX 25

PROOF-OF THE LEMMAS OF THE ANOSOV THEOREM (See Section 16, Chapter 3) Lemma A

Let (M, ¢) be a C-diffeomorphism. We select definitively a Riemannian metric on M. Since M is compact, there exists a number d > 0 such that, whatever be the ball B (p ; d) C TM with radius d and center p ( M, . p the restriction Exppl B(p; d)

of the exponential mapping at p is a diffeomorphism. Let l¢nm I n ( Z! be an orbit of ¢' A chart of a neighborhood of this orbit is (B, 1jJ-1), where B is the sum of the balls B

n

= B(¢nm, d)

C TM,J..n o..p

m.

and the restriction IjJ I-B is Exp ,J..n' . Let us denote by X the dilating n "fJ m n k-space X (¢n m) of TM¢nm' and by Yn the contracting I-space y(¢nm), Tke invariant dilating and contracting foliations rand new foliations on B:

Finally, ¢ induces a mapping:

201

Y

of M induce

202

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

such that the restriction ¢1 I

Bn

maps B

n

into B

n+

l' with obvious restric.

tions concerning the range of ¢1' Assume d small enough, then the sheets of the foliations

X1

and '!:J 1 can be regarded as sheets of the Euclidean

space XnED Yn of origin 0 = ¢nm, In which their equations are, respectively: y

= y(O)

+ in(x, y(O)) and

x

= x(O)

+ gn(Y' x(O))

where x ( X n , Y ( Yn and where in' gn' and their first derivatives can bl: made arbitrarily small by a suitable choice of d. a n of '!:J n which passes through the center 0 of

Consider the sheet

The mapping:

e: whose restriction

eIY

n:

I Yn I n Yn

->

( ZI an

->

I an In (

is defined for

ZI

,

y ( Yn n B n

by

is a diffeomorphism. Therefore, y ( Yn can be regarded as coordinates on

Figure A2S.1

203

APPENDIX 25

The diffeomorphism ¢1 maps an into a n+ 1 (see Figure A 25.1). the coordinates y, this defines a mapping ¢2:

In

and

ASSERTION A 25.2 It follows from the very definition of C-systems that the restriction

is contracting:

(A 25.3) where

e is a constant.

REMARK A 25.4 To be precise, (A25.3) holds for a certain iteration ¢~ of ¢2: we must "kill" the constant b in the definition of C-systems. For simplicity, we assume that (A 25.3) already holds for v = 1. Now let ¢' be a diffeo· morphism C 2.close to ¢' Then ¢' is a C-diffeomorphism (Sinai theorem, Section 15) and the foliations :t~ = 1j1- 1:t.' 'lJ; =rljl-1'lJ' and the mapping ¢; induced by ¢': ¢;: B ... B.

¢; = 1j1-1¢'1jI,

¢;

IBn:

Bn ... B n+ 1 ,

are defined as above. If ¢' is C 2·close enough to ¢. then the sheets of

:t~ are close to those of :t1 and transverse to the sheet an' Therefore, there exists a projection IT:

which makes correspond to each point a

l

Bn' the intersection ITa of an with

the sheet of:t; passing through a (see Fig~re A 25.5). Now let us consider the mapping (Figure A 25.5):

¢;

=

e- 1 IT¢;e,

204

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

cp'

1

Figure A 25.5

ASSERTION A25.6

If ¢' is C 2-close enough to ¢, If

¢;. is C-c1ose to ¢2: to any

> 0 corresponds a positive 0 su~: Lnat 11¢;'y-¢:yll <

(A 25.7) where

I II c 2

If

II¢'-¢ I c 2 < 0

for any y ( Yn

n

implies:

Bn ,

is the C 2 -norm.

Proof: ¢{ is close to ¢1' II: ¢; an -+ an+ 1 is small for ¢{ an '" ¢1 an = a n + 1 , and the sheets of !; are transverse to an +l' Now denote the sheet of!; passing through the center 0 '" m of Bo by {3, LEMMA A. A 25.8 If ¢' is C 2 -close enough to ¢, then the sheet ¢'n{3 is close to ¢nm

for any n (A25.9)

where

~

0,- To be precise:

IW;mll

e, is defined at (A 25.3).

<

If

1-e

205

APPENDIX 25

Proof:

According to (A 25.6), given ce > 0 there exists 0 > 0 such that

11¢'-¢ll c 2 < 0

implies:

From (A25.3) and (A25.7) we deduce:

II¢; y I < 8 Ily II Set €

=

IIYII <

ce/(l- 8); therefore, if

+ ce .

€, then

II¢; y I <

€ and

Iwi y II

< £, and so on. But since 11m II < €, inequality (A 25.9) is proved. REMARK

A25.10

Ily!1 <

From (A 25.3) and (A 25.7) one deduces also that

IW2n y I ::; In fact

IIY I <

c and c

if c ? 1 ~ 8

c

c implies:

.

? ce/(1- 8) imply:

II¢; y II <

Ily II

8·

+ ce

< 8c + ce < c . (Q. E. D.)

Lemma B Now we consider the sheet Yn of Yn1

:x:

C B n be the corresponding sheet of

Yn

1

=

which passes through ¢n m. Let

:x: 1:

',1,-1 'f'

Yn

'

The equation of Yn1 is y = fn(x,O). We define a mapping

11,

similar to

e,

by setting:

11: lXnln 11

l

Z!

lYn1 1n

->

is a diffeomorphism and x

l

l

11l xn :

ZI,

Xn

->

Yn1

X n can be regarded as coordinates on Yn1.

The diffeomorphism ¢1 maps Yn1 into Y;+1 ~ In the coordinates x, this defines a mapping: ¢3 = ~-11:D:

lXnln

Obviously ¢3 (0) = 0,

l

Z!

->

lXnln

l

ZI,

206

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

ASSERTION A 25.11 It follows from the very definition of C-systems that ¢31x : X n

n

~

Xn+ J

is dilating: (A25.12)

REMARK

A 25.13

In fact, (A 25.12) holds for a certain iteration of ¢3. For simplicity we assume that (A 25.12) already holds for ¢3. Now let ¢' be a diffeo-

f3 n =

morphism C 2 -close to ¢. Consider the sheet foliation

¢'In

f3

C Bn

X; = rjJ-J'X' which passes through e¢';(O) (n::. 0).

of the

(See Figure

A25.14.) According to Lemma A, this sheet is close to the center 0 of Bn.

Let y = hn(x), (x ( X n ), be the equation of

enough to ¢' then we can choose x ( Xn

n Bn

f3 n .

If ¢' is close

as local coordinates of

f3 n :

the mapping E

which is defined by x ~ (x, hn(x)) for x ( Xn

Ixn :

1

..

FIgure A 25.14

f3 n '

n Bn is a diffeomorphism. Yn+l

q>'

Xn ~

207

APPENDIX 25

From the very construction of the (3n's, we see that ¢;

maps (3n into

(3n+1' Therefore, this defines a diffeomorphism:

ASSERTION A 25.15

If ¢ and ¢' are C 2 -close enough, then ¢3 and ¢; are C 1-close: To any ce > 0 corresponds a positive 0 such that 11¢-¢'ll c 2 < 0 implies:

(A25.16)

n 8 n , n ~ O. This is a direct consequence of the construction of the Yn , (3n (see Sinai theorem, Section 15), and of the

for any x, xl' x2 ( Xn

fact that the (3n's are C 1-close to the Yn's.

LEMMA B. A25.17 II ¢' is C 2 -close enough to ¢' then there exists a well-defined sheet

o(

'Y'

such that ¢,n

o is close to

¢nm for any n ~ O. To be precise,

there exists one and only one point Xo ( Xo such that 11¢;n xoll

<

E for

any n-::: O. First we need a sublemma.

LEMMA A 25.18 Let R be a union of equi-dimensional Euclidean spaces R n , n Let T

=

K + L: R

->

R, T IR n : R n

->

~

O.

Rn+ 1 be diffeomorphisms such that:

(1) K(O) = 0,

IIK(x) -K(y)11 > 0 Ilx - y II,

(2) IlL II $. E,

IIL(x) - L(y)11 <

Eilx - y II,

0 > 1, 0 - E > 1,

for any x, y (R. Then, there exists one and only one point x ( Ro such that' the sequence Tnx is bounded, and (A 25.19)

IiTnx II $. _ E _ 0-1

for any n ~ O.

208

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Proof:

The mapping T-11 R n : R n

->

R n- 1 (n > 1) is obviously a diffeomor-

phism. On the other hand: II (Kx - Ky) + (Lx - Ly) II

ilTx- Tyll

2:. IIKx-Kyll-IILx-Lyll 2:. (8- E)llx-yll; therefore: (A 25.20) Let bn(c) be the ball Ilx II ::; c of Rn' Since IITxl1 = IIKx + Lxii 2:. IIKxll-IILxll > 811xll- E , we have: (A25.21) As:;ume c large enough, that is: 8c - E 2:. c.

(A25.22)

Then (A25.21) implies Tbn(c) ) bn+1(c), therefore T-1bn+1(c) C bn(c) ,

consequently T-1b1(c) ) T- 2 b 2(c) ) •.. ) T-nbn(c) ) ....

But, according to (A 25 .20), we have: diam~er

T-nb n (c) ::; 2c(8'- E)-n

->

0 as n

->

+

00

•

Therefore nn> 0 T-nbn(C) reduces to a unique point x (bo(c)'

This

finishes the proof if one observes that c = E/(8 -1) verifies (A 25.22) . . Proof of Lemma B. A2S.23

According to (A 25.11) and (A 25.15), the mapping ¢; verifies the conditions of the preceding lemma. It is sufficient to set K

=

¢3' L

=

¢~ - ¢3' and change E into ce in condition (2). If we take ce = E(8 -1)

in (A25.19), we obtain 11¢;nxoll < E. This proves Lemma B.

209

APPENDIX 25

To summarize, we found a contracting sheet 0 (

'l:J' which remains

close to the orbit ¢nm for n ::: 0 (in the sense of Lemma 8). If ¢' is close enough to ¢'

°

is close to ¢nm, even for n

< O. To prove it, it

is sufficient to apply Lemma A t~ ¢ -1. The foliation

'!:I'

is the dilating

foliation of ¢,-1 and the sheet 0 is close to m. Consequently, according to Remark (A2S.10), the sheets ¢mo, (n

< 0) stay in the neighborhood

of the orbit ¢nm (in the sense of Lemma A):

Therefore, Lemmas A and 8 imply the following assertion: ASSERTION A2S.24 If ¢' is C 2-close enough to ¢' then there exists a sheet ;5 C Bo of

the foliation '!:I~ such that the sheets

¢t 8 C Bn'

(-00

< n < 00), stay

inside an E-neighborhood of the center of Bn. Using the same argument for ¢,-1, we find a sheet

f3 8

erties. Since the sheets

C Bo of the foliation ~; with similar prop-

and

fj are transverse in B o' there exists one 8 n fj in an E-neighborhood of the

and only one point of intersection z =

center of Bo. The desired homeomorphism k of the Anosov theorem is defined by setting k (m)

=

y,z. One easily checks that all the preceding con-

structions depend continuously on m. This proves that k is an homeomor: phism. The relation ¢'k to the identity.

= k¢

is obvious, as is the fact that k is E-close

APPENDIX 26

INTEGRABLE SYSTEMS (See Section 19, Chapter 4)

J. Liouville

.

(A26.1)

p

proved that if, in the system with n degrees of freepom:

aH aq

= --,

q "" ap aH ,

p = (Pl" .. , p ), q = (ql'···' q ), n

n

n first integrals in involution 1

(A26.2) are known, then the system is integrable by quadratures. Many examples of integrable problems of classical mechanics are known. In all these examples the integrals (A 26.2) can be found. It was pointed out long ago that, in these examples, the manifolds specified by the equations F,

f; "" constant turn out to be tori, and motion along

=

them is quasi-periodic (compare with Example 1.2). We shall prove that such a situation is unavoidable in any problem admitting single-valued integrals (A 26.2). The proof is based on simple topological arguments. THEOREM

A 26.3

Assume that the equations F;

=

f;

=

constant, i

=

1, ... , n, define an

n-dimensional compact, connected manifold M = Mf such that: (1) at each point of M the gradients grad F; (i

=

1, ... , n) are linearly

independent; 1 Two funchons F(p, q) and G(p, q) are in involution if their Poisson bracket vanishes idenhcally: (F, G) = aF aG ap aq

_ aF aG aq ap

210

== 0 •

211

APPENDIX 26

(2) a Jacobian Det lal/af I, which is defined below (A 26.7) does not

vanish identicaIIy. Then: (1) M is an n-dimensional torus and the neighborhood of M is the direct product Tn

Rn;

X

(2) this neighborhood admits action-angle coordinates (I, ¢», (I ( B n C Rn , ¢> (mod 2") ( Tn), such that the mapping I, ¢> ... p, q is canonical 2 and Fi

=

F/I).

Thus, Equations (A 26.1) may be written: I

=

1>

0,

=

w (I), where w (I)

=

aH , aI

and the motion on M is quasiperiodic since H = Fl = H(I) and Equations (A 26.1), in action-angle coordinates, are Hamiltonian equations 2 with corresponding Hamiltonian function H(J).

Proof: NOTATIONS A

26.4

,

We use the following notations. Let x

= (p, q)

be a point of the phase

space R2n; we shall denote by grad F the vector gradient F

Xl

, ... , F

x2n

of

a function F(x). The Hamiltonian equations (A 26.1) then take the form: (A26.5)

x=

1=(O-E) E-O

I grad H,

where E is the unit matrix of order n. We introduce in R2n the skewscalar product of two vectors x, y:

[x, y] = (Ix, y) = -[y, x], where ( , ) is the usual scalar product. As can be easily verified, [x, y] expresses the sum of the areas of the projections of the parallelogram with sides x, y onto the coordinate planes Piqi (i = 1, ... , n). Linear transformations S, which preserve the skew-scalar product [Sx, Sy}

--------------------See Appendix 32.

=

[x,

Jd

for all x, y,

212

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

are called symplectic. For instance. the transformation with matrix I is symplectic. The skew-scalar product of the gradients [grad F, grad G] is called the Poisson bracket (F, G) of the functions F, G. Obviously. F is a first integral of the system (A 26.5) if and only if its Poisson bracket (F, H) with the Hamiltonian vanishes identically. If the Poisson bracket

of two functions vanishes identically. the functions are said to be in invo-

lution. THE CONSTRUCTION. A 26.6

Consider the n vector fields: ~,~ I grad ~, (i ~ 1 •...• n). On account of the nondegeneracy of I and the linear independence of the grad F;'s. the vectors

~;

are linearly independent at each point of M.

Let us consider the system (A 26.5) with Hamiltonian F;. Since (F;, F.) ~ O. all the functions F. are first integrals. and every orbit lies wholJ

J

lyon M. Therefore the velocity field Finally. the fields

fJ

and

f,

~; ~ I

grad F, is tangent to M.

commute, for their Lie bracket is nothin~

but 3 the velocity field of the system (A 26.5) with Hamiltonian (F., F.) ~ O. , J Thus, M is a connected. compact orbit of the group Rn acting smooth· ly and transitively; therefore we proved that M ~ Tn. Besides. M being specified by the equations F;

~

f; ~ constant. the fields grad F; define a

structure of direct product in the neighborhood of M. Now. let us choose the torus Mf: F

~

f in the neighborhood of M and

consider the n integrals (A26.7) over the basic cycles y /f) of the torus Mf" Since the l;
3

The Lie bracket of the Hamiltonian vector fields I grad F

and I grad G

IS

an

Hamiltonian vector field with Hamiltonian function - (F, G). We shift the proof to Appendix 32.

APPENDIX 26

S([, q)

(A26.8)

jq

=

213

pdq

'lo

where the path of integration lies on M([) (therefore, p

=

p ([, q) ).

The

many-valued function S is the generating function (see Appendix 32) of the canonical transformation f,

1> ->

p, q, which defines action-angle coordi-

nates: p

(A26.9)

LEMMA

as

as oq

af

A26.10

The one-form pdq of M([) is closed. Proof:

It is sufficient to prove that the integral of pdq along infinitely small

parallelograms lying in M([) vanishes. If D is a parallelogram with sides ~,.,." then ~ pdq (i.e., the sum of the areas of the projections of D onto

the coordinate planes P; q;, i = 1, ... , n) is the skew-product [~,.,.,] of ~ and .,.,. Suppose now that ~ and .,., touch M([) at a certain point. In accordance with (A 26.6) any vector tangential to M ([) is a linear combination of the n vectors f grad F;. But these vectors are skew-orthogonal since, in accordance with (A 26.2), [grad F , grad F.]

0 ,

J

1

and thus, since f is symplectic,

[! grad F;, f grad Fj J

=

O.

Therefore [~,.,.,] ~O, as required. The integral (A 26.8) can therefore be regarded as a many-valued function S and Equations (A 26.9) define, locally, a canonical transformation f, ACTION-ANGLE VARIABLES A

1>

->

p, q.

26.11

In fact formulas (A 26.9) define a global canonical mapping in which p and q have period 211 with respect to

1>.

To prove it, we observe that, for

214

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

every I, the differential of S([, q) is a global one-form on M([). fore d¢, as defined by (A 26.9), is also a global

There-

one~form.

Let us compute the periods of the one-forms d¢, over the basic cycles of the torus Mf' According to (A 26.7) we have:

f

Yi

d¢i

=

f d(aS) al. Yj'

=

~ d/.

f

'Yj

dS

=

~(2771.) = 2770 .. d / . } · '} ,

Therefore the variables ¢, are angular coordinates on the torus M(l) and our theorem is proved.

APPENDIX 27

SYMPLECTIC LINEAR MAPPINGS OF PLANE (See Section 20, Chapter 4) Let A be a symplectic linear mapping of the plane (p, q). The mapping A preserves the area-element dp 1\ dq, therefore Det A

=

1. Consequent-

ly, the product of the proper values Al and A2 of A is equal to 1. Besides, Al and A2 are roots of the real polynomial Det (A -AE). Therefore, either Al and A2 are both real, or they are complex conjugate: A2 = Xl •. In the first case, we have:

(A27.1)

p

q

Figure A 27.3

215

216

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

In the second case, we have: (A 27.2)

and the roots belong to the unit circle (see Figure A 27.3). The third and last possible proper value coniiAuration is:

q

p

hyperbolic rotation

p

hyperbolic rotation with reflection

Figure A27.5

217

APPENDIX 27 EXAMPLE

A27.4

The hyperbolic rotation: p, q

2p, Yz q ,

-+

or the hyperbolic rotation with reElection: p, q In both cases the orbit Tnx of x

2p, - Yz q (see Figure A 27.5).

-+ -

(p, q) belongs to the hyperbola pq

=

=

constant. Of course, the fixed point 0 is unstable. From classical theorems of linear algebra it follows that every mapping A of the first type (Ai of, A2 ; Ai' A2 ( R) is an hyperbolic rotafion, possibly with reflection. This means that, up to a suitable change of variables, A may be written

under the form: P, Q

EXAMPLE

->

1 Q• AP, X

A27.6

A r9tation through an angle a belongs to the second class (Ai A2 = ei~:

p, q

-+

p cos a- q sin a, p sin a + q cos a .

q

x p

Figure A27.7

=

e- ia,

218

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

This rotation transforms into an "elliptic rotation" (see Figure A27.7) under a linear change of ,,:ariables. In this case the orbit r'x of x = (p, q) belongs to an ellipse centered at O. The fixed point 0 is obviously stable. Classical theorems of linear algebra show that every mapping A of the second type (1'\11 = 1'\21 = 1, '\1';' '\2) is an elliptic rotation. In the first case (A 27.1), the fixed point 0 is called an hyperbolic point and one says that A is hyperbolic at O. In the second case (A 27.2)

the fixed point 0 is called an eIIiptic point and one says that A is eIIiptic at O. Finally, the third case (,\ 2 = 1) is called the parabolic case. REMARK

A27.8

Every canonical mapping A ~ close enough to an eIIiptic mapping A, is eIIiptic. In fact, the roots '\1 and '\2 depend continuously on A and are

restricted to lie either on the real axis or on the unit circle (see Figure A 27.3). Therefore, these roots cannot leave the unit circle, except at points ,\ = :!::. 1 which correspond to the parabolic case. Finally, we define the topological index of a vector field at a fixed point. Let us consider a vector field ';(x) of the plane p, q, with an isolated fixed point ';(0) This defines a mapping of the unit circle x2

= p2

+

l

=

O.

= 1 onto itself:

If E is small enough, then the topological degree of this mapping does not depend on E and is called the index of .; at 0, or the index of O. Now, consider the vector field ';(x) = Ax - x.

If the mapping 1\ is

nonparabolic, then 0 is an isolated fixed point of ';(x). THEOREM

A27.9

An eIIiptic point, or an hyperbolic point with reflection, has index + 1. An hyperbolic point has index - 1.

Proof consists in a mere inspection of Figures (A27.S) and CA27.7).

APPENDIX 28

STABILITY OF THE FIXED POINTS (See Section 20, Chapter 4) Consider an analytical canonical mapping A of the plane p, q, with

(0,0). Assume that 0 is elliptic, that is, that the differential of A at zero has proper values A1 = e- 1a , A2 ~ e ia . It has been fixed point 0

=

known since G. D. Birkhoff's time 1 that, if al2rr is irrational, to every s > 0 corresponds a canonical mapping B = B(s) of a neighborhood of 0

B: p, q

->

P, Q,

B (0)

=

0,

which reduces A to a "normal form":

that is as follows. Let I, ¢ be the canonical polar coordinates: 21 = p2 + Q2, 2/'= p'2 + Q'2,

¢

=

arctg(P/Q)

¢'

=

arctg(P'/Q')

then: I'-I

(A28.1)

¢'-¢

=

The coefficients a, a 1 ,

= O(ls+1)

a+

a1 i+ ai 2 + .•. + a/ s +

...

do not depend on the mapping R (~) by which A

is reduced to the form A '. If a

~ 2rrm/n

0([S+1).

and if there is a nonvanishing

coefficient a 1 , a 2 , ... , Birkhoff says that A is of "generic elliptic type." 1 Dynamical Systems, Chapter 3.

719

220

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

THEOREM A 28.2

(See Arnold [7]).

The fixed point of a

~eneric

elliptic mapping is stable.

The proof consists in applying the construction of Theorem (21.11) of Chapter 4 (see Appendix 34) to the mapping (A 28.1): for 1« I, 0(18+1) is regarded as a perturbation of the mapping l'

=

I,

Similar theorems are obtained concerning the stability of equilibrium positions and elliptic periodic solutions of Hamiltonian systems with two degrees of freedom (see Arnold [7]).

J. Moser [1]

obtained the strongest re-

sult in that way: MOSER'S THEOREM A28.3

The fixed point of an elliptic canonical mapping A of the plane is stable provided that: (1) a f, 217

j!- ,

217 :

(2) a 1 f, 0 ;

(3) A is C333 -differentiable. (As it is pointed out in a recent paper of Moser [6], this number of derivatives can be fairly well reduced.) A complete proof will be found in

J. Moser [1].

REMARK A 28.4

If a

=

Civita [1].

217m/3, th~n the fixed point can be unstable, as shown by Levi-

APPENDIX 29

PARAMETRIC RESONANCES (See Section 20, Chapter 4) The analysis of the stability of the fixed point (0,0) of a linear mapping of the plane is due to Poincare and Lyapounov. Only in recent times (1950), were these results extended by M. G. Krein [11. [2] to systems with many degrees of freedom. Krein's investigations have been enlarged by Jacoubovich [11. Gelfand and Lidskii [41. and so on. J. Moser [3] published a report of Krein's theorem. Let A be a linear symplectic mapping 1 of the canonical space R2n. We say that A is stable if the sequence An is bvunded. We say that A is parametrically stable if every symplectic mapping, close to A, is

st~ble.

We proved in Appendix 27 (and used it in Section 20, Chapter 4) that every elliptic mapping of R2 is parametrically stable. M. G. Krein displayed all the parametrically stable mappings of R2n. LEMMA A 29.1 (Poincare-Lyapounov) Suppose A is a symplectic mapping and ,\, is a proper value of A. Then 1/,\"

1

X, and I/X

are proper values of A.

A preserves the skew-scalar product

product and 1 =

(~=~), E

[e-,." 1 = (1

e-, .,,), where (

, ) is the inner

= unit matrix of order n. Therefore, we have:

[Ae-. A."l

=

[e-. ."l 221

and

A iA

= 1.

222

ERGODIC PROBLEMS OF CLASSICA.L MECHANICS

Prool: It is sufficient to prove that the characteristic polynomial of' A is real

and reciprocal. In fact, we have: p(..\)

= Det(A-..\E) =

Det(-lA,-1 1 + ..\p)

= Ded-A '-1 + ..\E) = Det(-A- 1 + ..\E) = Ded-E + ..\A) = ..\2n. Det (A_..\-1 E) = ..\2n. p(..\-I)

•

From thi~ lemma the following corollary is readily deduced; COROLLARY A29.2

The proper values of A divide into couples and "quadruples." Couples are lormed by ..\

~nd

..\-1, ..\ belonging to the real axis or the unit circle:

1..\1 = 1. Quadruples are formed by ..\,X,..\-I, and X-I (see Figure A 29.3) .

• 1.3

I

I I

I

\ I

\I

~.l

\I'

,.

'~'I

;\3

\ \

Flgure A 29.3

223

APPENDIX 29

COROLLAR Y

A 29.4

If the proper values are simple and lie on the unit circle IAI = 1, then A is parametricalIy stable, because if all the proper values are simple and lie on IAI

= 1,

then:

(1) A is stable (for obvious reasons of normal form); (2) all the proper values of a symplectic mapping A', close enough

to A, lie on IAI = 1. In fact, assume the contrary, then A' would have' two proper values A and X-I close to a unique isolated proper value of A. (see Figure A 29.3). Let us now assume definitively that :i 1 are not proper values of A. Krein classified the proper values belonging to the unit circle IAI = 1: they split into positive and negative proper values. First assume that all the proper values are simple; we prove the following lemma: LEMMA A29.5

Let ~1 and ~2 be the proper vectors with corresponding proper values

Al and A2 · Then, either AIA2

=

1, or [~1' ~2] = O.

Proof: Since A ~1

Al ~1 and A~2 = A2~2' we have:

[A~I' A~2]

=

Al A2 [~1' ~2]

=

[~1' ~2]

(Q. E. D.) COROLLAR y A

Let

(J

29.6

be a plane, invariant under A and correspollding to conjugate

proper values AI' A2, IAll = IA21 = 1. Then: (1) (J is skew-orthogonal to every proper vector ~3 corresponding to another proper value A3 ; (2) the skew-product [~,

1)]

of noncolinear vectors ~ and

1)

of

(J

IS

nonvanishing. Assertion (1) is a direct consequence of AIA3 ~. 1, A/\3 ~ 1: in view

224

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

of Lemma (A 29.5) we have [~l' ~3] We have also [~1' ~1]

= 0,

=

[~2' ~3] = O. Suppose [~1' ~2] = O.

and assertion (1) implies [~1' ~3]

= 0 for

every

~3' Therefore [~1' 71] = 0 for every 71, which is impossible. Consequent-

ly [~1' ~2] f, 0 and assertion (2) holds good. DEFINITION

A29.7

A proper value A, such that IAI

=

1, A2 f, 1, is called a positive (resp.

negative) proper value of A if: [A~,~] > 0 (resp.

< 0) for every

~ of the

real invariant plane a corresponding to the proper values A and X • This definition is correct. Indeed, the vectors

A~

and

~

of a are non-

co:·near because A2 f. O. Therefore, in view of Corollary (A 29.6), [A~,~]

f. 0 on

Consequently [A~,~] has constant sign for every ~ (a.

a.

REMARK A

\

29.8

The sign of a proper value lias a simple geometrical meaning. The plane a

admits a canonical orientation, for [~, 71] f. 0 if ~ is nonparallel to 71'

Therefore, oile may speak of positive (or negative) rotations. The restriction of A to a is an elliptic rotation through an angle a, 0

< lal < 7T. The

proper value A is positive (resp. negative) if A rotates a through a posi-

tive (resp. negative) angle. Krein's main result is: collision of two proper values with identical

signs on the unit circle IAI

=

1 does not provoke instability. In contrast,

two proper values with opposite signs can leave the unit circle after they have collided, so forming a "quadruple" with their conjugates (see Figure

A 29.3). To be p~ecise, let A(t) be a symplectic mapping -.yliich depends continuouslyon a parameter t, and the proper values of which are different from :. 1 if It I < A are simple collide for t THEOREM

an~

=

T.

Suppose that, for t < 0, all the proper values Ak of

lie on the unit circle, while certain of these proper values

O.

A 29.9

If all the proper values that collide have identical sign, then they re-

225

APPENDIX 29

main on the unit circle after the collision and the mappinl1 A remains sta-

ble for t < E, E > O. We shall prove the theorem in the simplest case in which all the proper values A, 1A > 0 collide. The general case can be reduced to this c'ase by selecting a canonical subspace R21(t) corresponding to th'e I colliding proper values and their conjugates. 1'0 fix the ideas, suppose that the proper values, Ak are positive:

[Ae-. e-] > 0 for e- ( Uk ' where Uk is the plane generated by e-k , ~k (Ae-k = .Ake-k).

Proof of the Theorem. A29.10 Consider the quadratic form [A e-. e-]; its polar bilinear form is nondegenerate. We have, indeed:

[Ae-. 1/] + [A1/. e-] Suppose [(A-A- 1 )e-. 1/]

(A 2 - E) Ae-

=

=

[Ae-. 1/] - [A -1 e-. 1/]

=

=

[(A - A-I) g. 1/] •

0 for every 1/, then (A-A- 1 )e-

=

0 and

O. Thus, 1 would be a propervalue of A2. which contr"dicts

the condition of Theorem (A 29.9) (A

del1enerate for It

I
i' :t 1). Therefore [A(t} e-, e-] is non-

and, in particular, for t

=

O. On the other hand,

this form is positive definite for t < O. In fact, every vector 1/ is equal to the sum of its projections 1/k into the invariant planes Uk corresponding to the proper val,ues Ak

,

Xk

.

According to Lemma (A 29. 6) these planes

Uk are skew-orthogonal, therefore: .

[A1/. 1/]

=I

[A1/k' 1/1]

k,l

because

But [A1/k' 1/k] > 0 for Ak is a positive proper value, hence:

[A1/. 1/] >

t) .-

So, the form [A (t ) e-, e-] is positive definite for t < 0 and nondel1enerate

226

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

for t = O. Therefore, this form is positive definite for t ly, for t

< E, E> O. But [A An~,

An~] = [A~. ~l for

Thus the orbit An ~ lJelongs to the ellipsoid [A~.~] A(t) is stable for t

=

0 and, consequent.

An is symplectic. =

< E.

constant, that is

(Q. E. D.)

RE:'IARK A 29.11

The above argument proves the criterion of parametric stability:

The symplectic mapping A is parametrically stable if and only if all the proper values

>'k lie

on the unit circle

1.\1

=

1,

>.;

t

1. Besides, the

quadratic form [A~.~] is definite on every invariant subspace correspg.nd. ing ." the multiple proper values t.. k

,

Xk

.

APPENDIX 30

THE AVERAGING METHOD FOR PERIODIC SYSTEMS (See Section 22, Chapter 4) Let

n=

8 1 )( SI be the phase space, where 8 1 =

an open bounded subset of RI and SI sider ¢-periodic smooth functions

F:

n ...

RI,

f:

CI)

II =

([1' ... ,11)\ is

I¢ (mod 21T)} is a circle. We con-

=

(J), F (I, ¢), HI, ¢):

n ...

RI,

CI):

8 1 ... RI ,

and finally, E« I denotes a small parameter. THEOREM

A30.I

n:

We consider the systems defined in (A 30.2)

, ~ =

CI) ( [ )

1I

E· F ([, ¢)

=

+ Et([, ¢)

and

j

(A30.3)

= E·

F(j), where F(j)

=

---.L 21T

If

CI) ( [ )

,;,

0 in

n,

f

21TF (j, ¢) d¢ .

0

then the solutions I (t) and ] (t) of (A 30.2) and (A 30.3),

with equal initial data [(0) 1[(t)-](t)1

= ]

(0), satisfy 1 :

< C·E for every t, O:s: t ::: liE,

where C is a constant which does not depend on E,

1

We suppose that J(I) ( 8' for every t, 0::: t

227

:S

l/f .

228

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Proof:

Let us improve (A 30.2) by using a new variable: (A 30.4)

P ~ P(I, ¢) ~ 1+ Eg(J, ¢),

From (A 30.2) and (A 30.4) follows: (A 30.5)

In order to cancel the terms of order E, we set: (A30.6)

g(I,¢) ~

I

¢ -

F(P)-F(P,¢) d¢; cu (P)

o

this expression is well-defined since cu(p) .;, 0,

f

2TT

(ii - F)d¢

=

0,

o

and therefore g(¢ + 2TT) ~ g(¢). Now, our system (A30.S) may be written: (A 30.7)

Let P(t) be the solution of (A 30.7) with initial data P(O) ~ J(O) (A30.8)

= /(O):

p(t) " P([(t), ¢(t».

Obviously, (A 30.7) implies: (A 30.9)

IP(t)-J(t)1

< Cl'

E

for every t, 0 $. t $. liE.

Finally, from (A 30.4), (A30.6) and (A 30.8) follows: (A30.10)

IP(t}-J(t}!

< C2 •

E

for every t.

Inequalities (A30.9) and (A 30.10) conclude the proof. They prove also that the motion decomposes into the averaged motion and fast small oscillations (see Figure A30.U).

229

APPENDIX 30

I (t )

"-="_- J(t)

t=o

Figure A30.1l

APPENDIX 31

SURFACES OF SECTION (See Section 21. 9, Chapter 4) Let H(p, q) be the Hamiltonian function of a system with n degrees of freedom (therefore the phase space is 2n-dimensional). Let L: H = h, q1 =

°

be a (2n - 2) -dimensional submanifold of the "level of energy"

H = h. If, in a certain domain Lo of L, P = (P2' ... , Pn)' Q = (q2' ... , qn) forrt: a local chart and q1 10 0, L is called a surface of section (see Figure A31.1). Assume that an orbit of the Hamiltonian system, through a

------;...... y

,I

, I

I

----__\r-"

I

x

Figure A 31.1

point x of a point

L o' returns to Lo' Then, in view of q1 10,0, the orbit through

x: on Lo sufficiently close to x, will, as 230

t increases, return .to

APPENDIX 31

231

Lo and will cut Lo in a uniquely determined point Ax '. In this manner, we define a mapping A:

THEOREM 1

A 31.2

The mapping A is canonical, that is for every closed curve y of L l , we have:

f

(A 31.3)

PdQ =

Y

where PdQ

=

f

PdQ,

Ay

P2 dq2 + ... + Pn dqn .

Proof: Consider the orbits emanating from y in the (2n + 1) -dimensional space I(p, q, t)l. The curves y and Ay of the space I(p, q)l are the projections of

two closed curves y'and Ay'of !(p, q,Ol which are formed respectively, q

p Figure A 31.4

by the initial points (t

=

0) and the end points of tbe above orbits (see Fig-

ure A31.4). Therefore, we have by the 1

Poinc~re-Cartan

theorem:

A proof of this well-known theorem has apparently never been published.

232

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

L pdq- Hdt

(A 31.5)

~,

where pdq

=

f

pdq- Hdt ,

Ay'

PI dql + .•• + Pn dqn· But H

=

=

constant along y' and Ay'

then:

.! Hdt ~y'

o.

Besides. we have:

f

pdq

Ay'

f

pdq.

Ay

constant on 1. we also have:

In view of ql

Thus. finally.

f

f

,pdq-Hdt = jPd Q,

pdq-Hdt

=

Ay'

Y

y

1.

PdQ.

Ay

and (A 31.5) implies (A 31.3). This proves the theorem. EXAMPLE A

31.6

Consider the problem of the "convex billiard table" (Birkhoff). Let

r

be a closed convex curve of the plane E2. Suppose that a material point

o

N Figure A31.7

233

APPENDIX 31

M moves inside r and collides with r according to the law "the angle of

incidence is equal to the angle of reflection" (see Figure A 31. 7).

The

states of M, immediately before aAd immediately after a reflectim, are determined by the angle of incidence a, 0 :::; a :::; 217, and the point of incidence. The point of incidence A is defined by the algebraic length q2 of the arc OA of r (0 is an arbitrary origin). In other words, the set of the states of M, immediately before and immediately after a reflection, form a torus

r2

=

la(mod 217), q2 (mod L)! in the phase space (L is the length

r2

of r). We obtain naturally a mapping A of a subset of

info another

·one: the state which immediately follows a reflection is transformed into the state immediately preceding the next reflection. THEOREM A 31.8 (G. D. Birkhoff)

I

=

sin a· d q2 /\ da is invariant under the mapping A .

Proof:

Between two reflections, the motion of M is determined by Hamiltonian equations in the corresponding four-dimensional phase-space. In the neighborhood of the above torus

r2, let us select special coordinates. Our point

M is well-defined by coordinates (ql' q2)' where ql

=

MN is the distance

from M to rand q2 is the algebraic length of the arc ON. Coordinates ql and q2 (mod L) .are clearly Lagrangian coordinates in a neighborhood of r. Let Pl and P2 be the corresponding momenta (the mass of M is supposed to be 1). On r, P l and P2 coincide obviously with the components of the velocity-vector v: Pl

= Ivl·

sin a,

P2

= Ivl·

cos a.

The Hamiltonian function H is the kinetic energy: H

v2

=""2

In the four-dimensional space I(P l , P2 ' ql' q2)~' consider the surface ~ whose equation is: 1, M ( r) :

234

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

From one reflection to the next, the motion defines a mapping :\: L

->

L.

The coordinates P2' q2 are local coordinates of L (a 1= 0) which is a Surface of section. In view of Theorem (A 31.2), the mapping t\ is canonical and therefore preserves the two-form: dP 2 /\ dq2 = sin a' dq2 /\ da . (Q. E. D.) An elementary proof of this theorem, due to G. D. Birkhoff [1], requires ex~ tensive computations.

APPENDIX 32 FUNCTIO~S

TilE GENERATI:\G

OF

"APPI~GS

CANONICAL

(See Section 21, Chapter 4) The following results are due to Hamilton and Jacobi. §A. Finite Canonical Mappings

Let x ~ (p, q), (p ~ (Pi' ... , Pn)' q ~ (ql' ... , qn))' be a point of the .canonical space R2n. The differentiable mapping:

is

call~d

canonical if A preserves the Poincare integral-invariant:

f

(A 32.1)

pdq

y

1.

~

pdq,

Ay

fOf- any closed curve y. Let a be an arbitrary two-chain. Relation (A 32.1) lmplies that A preserves the sum of the areas of the projections of a 'into the coordinate planes Pj' qj: (A 32.2)

[(a)

~

ff

a

dp 1\ dq

~

ff

dp 1\ dq

~

[(Aa) .

Aa

In other words, the two forms dp 1\ dq and dP 1\ dQ coincide: (A 32.3)

dp 1\ dq = dP 1\ dQ, where P = P(p, q), Q = Q(p, q) .

If the domain of A is simply connected, then conditions (A 32.1) and (A 32.2) are equivalent. Relation (A 32.3) shows that:

235

236

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

=

pdq + QdP, where P

P(p, q),

Q

=

is a closed form of R2n (since dp 1\ dq + dQ 1\ dP

A(x)

=

Q(p, q), =

0).

Therefore:

f

x pdq + QdP, where P = P(p, q), Q = Q(p, q), Xo

defines, locally, a function on R2n. Suppose that ql'· .. , qn; PI'···' Pn , form a local chart in some neighborhood of the point x, that is: Det(

~:) ~

O.

Then, ~(x) can be regarded as a function of P, q, defined in the neighborhood of the point P, q: (A 32.4)

A(P, q)

=

f

(P, q)

pdq + QdP, where p

=

p(P, q),

Q

=

Q(P, q).

DEFINITION A 32.5

The function A(P, q) is called the generating function of the canonical mapping A. Of course, A is only defined locally and up to a constant. From (A32.4) follows: (A32.6)

LEMMA

Q,

aA aq

= p.

A32.7

Let A (P, q) be a function such that:

in the

nei~hborhood

of a point (P, q). Then, Equations (A 32.6) can be

solved locally with respect to P and Q:

P

= P(p,

q), Q

= Q(p, q),

and the fuactions P, Q determine a canonical

mappin~

A.

237

APPENDIX 32

In fact, pdq + QdP is a closed form on R2n; thus dp 1\ dq = dP 1\ dQ . (Q. E. D.) Unfortunately, the generating fur-:-tion A is not a geometric object: A not only depends on the mapping A, but also on the coordinates p, q of R2n.

According to (A 32.6), the generating function of the identity 1 is Pq. Thus, every canonical mapping, close enough to the identity, has a generating function close to Pq. §R. Infinitesimal Canonieal Mappings

Consider a family of canonical mappings SE' the generating functions Pq + ES(P, q; E) of which depend smoothly on a parameter E «1. The

mapping

Se is close to the identity, if E is small. According to (A 32. 6),

the Taylor expansions of P(p, q) and Q(p, q) with respect to E are: (A 32.8)

where S

=

S(p, q; E) ,

By definition, the infinitesimal canonical mapping Se is a class of equivalent families Se: two families Se and SE' are equivalent if ISE- Se' I

=

O(E 2 ), DEFINITION

A32.9

The function S(p, q) on the phase-space is called the generating function of the infinitesimal mapping SE (or Hamiltonian function). Of course, S is defined up to a constant. Now we prove that the function S is a geometric object: S neither depends on the canonical coordinates p, q, nor on the choice of a representant SE in the class of equivalence: it is a mapping S: R 2n ... Rl. In fact, let y be a curve of R2n 1 ThiS is a way to memorize (A 32.6).

238

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

joining x and y:

ay = y - x. We set

Ye = Se Y , and we denote by a(E) the strip form-

ed by the curves Ye" 0 < E' < E, and oriented in such a way that

aoe

=

x

Y - Ye + ••• (see Figure A32.10).

Figure A32.10

Let us set: (A 32.11)

![a(E)]

11

=

dp/\dq.

a(e)

According to (A 32.2), this integral does not depend on the canonical coordinates and, according to (A 32.1), does not depend on the curve Y, but only on x and y. LEMMA A32.12 The generating function S. of the infinitesimal canonical mapping Se

is given by: (A32.13)

~

S(y) - S(x) =

![a(E)]1

dE

' e=O

and does not depend on the choice of the canonical coordinates p, q. Proof:

.

Let us set Se x (A32.14)

X

=

ox

= (op,oq).

E(S(y) -S(x)) = E

According to (A 32.8), we have:

f~ yap

~

f

dp +

~

dq

aq

(oqdp-opdq) + 9(E2) •

Y

On the other hand, according to (A32.11), the integral of dp /\ dq along a(E) is;

(A32.1S)

239

APPENDIX 32

Formulas (A32.14) and (A32.1S) imply (A32.13).

(Q. E. D.)

One can express the invariance of the generating function S in another form. Let -\ be a finite canonical mapping and SE an infinitesimal canonical mapping. The canonical mapping T E = AS E A-I is clearly infinitesimal. LEMMA

A32.16

The generating functions Sand T of the infinitesimal mappings SE and TE are related by: T(Ax)

(A 32.17)

S (x) + constant.

Proof; Let YE and a (E) be the curve and the surface of Lemma (A 32.12). The curve y' = Ay joins the points Ax and Ay. Besides, the curves

TE ,y', 0

~ E' ~ E,

form a strip T(E), which is nothir.g but:

(A 32.18) From (A 32.13) follows: (A32.19)

S(y)-S(x) = ~ J[a(E)), dE

T(Ay)-T(Ax) = ~[[r(E)]. dE

But the mapping A is canonical. Thus, according to (A32.2) and (A 32.18), we have: [[o(E)] = [[r(E)].

Comparison with (A 32.19) yields (A 32.17).

(Q. E. D.)

COROLLARY A32.20

Let '\ and CE be infinitesimal canonical mappings with corresponding generating functions Band C, and let ,\ be a finite canonical mapping .• Then, the infinitesimal canonical mapping: (A 32.21)

has the following generating function:

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

240 (A 32.22)

B '(x) =

c (x)

+ B (x) - C (A -1 X) + constant.

In fact, (A 32.8) implies that the generating function of the product of two infinitesimal mappings is the sum of their generating functions, and also that the generating function of the inverse mapping C;-1 is - C. Relation

(A32.22) is easily derived from these remarks and from Lemma (A 32.16). §C. Lie Commutators and Poisson Brackets Given two infinitesimal canonical mappings Ae and Be' there exists one and only one infinitesimal canonical mapping Ce such that:

(A32.23)

AaBbA_aB_b

Cab + O(a 2 ) + O(b 2 ); a, b ... O.

=

The mapping Ce is called the Lie commutator of Ae and Be . LEMMA

A 32.24.

C

The generating function

of

Ce

is equal, up to a sign, to the Pois-

son bracket of the generating functions A and B of Ae and Be :

(A32.2S)

'lie

=

-

[V A, V Bl.

V

=

gradient.

We use the notation [x, y] = (lx, y) as in Appendixes 26 and 27.

Proof:

Again let Y be a curve joining x and y:

ay

=

y - x. We consider the

five-sided prism (see Figure A 32.26) formed by four strips: a1

BeY' -b < E < 0,

aa 1

Y - Y1 + ''',

a2

\y~. -a<E
oa2

Y1 - Y2 + "',

a3

BeY2'

·0 < E < b,

aa3

Y2 - Y3 + ''',

a4

.'\ Y3'

0< E < a,

aa4

Y3- Y4+"',

and closed by a fifth strip as' formed by the

~egments

joining the corre-

sponding points of Y and Y4' aas = Y4 - Y· ... Finally, we denote by T x and T y the bases of our prism, such that the two-chain a 1 +a2+a3+a4+Us+Ty+Tx

=

~

241

APPENDIX 32

y

Figure A 32.26

forms a two-cycle homologous to zero. Since dp /\ dq is a closed form we have, with notations of (A 32.2): [(Ul) + [(u 2 ) + [(u3 ) + [(u 4 ) + [(us)

(A 32.27)

+ [(7 y) + [(7)

=

[(~)

0 .

But Lemma (A32.12) implies:

(A32.28)

1(ul )

- b[B(y) - B(x)] + 0(b 2 ),

[(u2 ) [(u 3 )

- a[A(Yl)-A(x l )] + 0(a 2 ) , b[B(Y2)-B(x2 )] + 0(b 2 ),

[(u 4 )

a[A(Y3)-A(x 3 )] + 0(a2 ) ,

[(us)

-ab[C(y)-C(x)] + 0(a 2 ) + 0(b 2 ) •

On the other hand, 1y - y 41 along the surfaces (A 32.29)

{

=

0 (ab); therefore, the integrals of ~ dp " dq

are given by:

7

[(7 )

=

--ab[V B(y), VA(y)] + 0(a2 ) + 0(b 2 ),

y

[(7) = + ab[V B(x), V A (x)] + 0(a2 ) + 0(b 2 ) •

Finally, from (A 32.8) itO follows that the vector fields corresponding to Ae and Be are 1 V A and 1 V B. Consequently, up to terms of order 0 (a 2 + b2 ), we Have: A(Y3)-A(Yl) = (VA, Y3- Yl) = (VA, Y3- Y2) + (VA, Y2- Yl). =

{V A, 1 V B)b - (V A, 1 VA)a =

= {VB, VA]b -

[VA, VA]a = [VB, VA]b.

242

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Similarly:

- [V A, V B]a . Thus, we deduce from (A 32.28): (A 32.30) Comparison of (A 32.27), (A 32.29), and (A 32.30) yields (A 32.25). (Q. E. D.)

APPENDIX 33

GLOBAL

CANO~ICAL

\lAPPINGS

(See Section 21, Chapter 4) This appendix gives the topological reasons for the existence of periodic orbits for Hamiltonian systems with n degrees of freedom. §A. Generating Functions

n = B n x Tn be the canonical space, p: n B n C Rn and q: n T" = j(ql, ... ,qn )(mod21T)! thecoordi~ates p

Let p)1 n

->

->

and q

= q(x)

(n.

of the point x

=

/Pl' ... ,

= p(x)

By definition, the mapping A:

n

->

n

is globally canonical if it is homotopic to the identity and satisfies

f

(A33.1)

pdq = Y

f

pdq

Ay

for anyone-cycle y (even nonhomologous to zero). In conformity to Appendix 32, the mapping A is locally determined by a generating function Pq + A (P, q), provided that

Det (~: )

~

0,

(A33.2) p

=P

+

aA ,

aq

Q

aA

= q+ - ,

ap

Thus, locally, the function A (P, q) verifies: (A33.3)

A (P, q) =

j

(P, q)

(Q- q)dP + (p-P)dq.

243

E~GOD/C PROBLEMS OF CLASSICAL MECHANICS

244

Let us set: A(x) = A(P(x), q(x)) ,

where x

LEMMA

=

(p(x), q(x)) (

n.

A33.4

The mapping (A 33.2) is globally canonicai if and only if the function A (x), defined by (A 33.3), is single-valued on

n.

Proof:

Let

y be a

closed curve of

(A33.S)

n.

Let us prove that:

f
o.

In fact, (A 33.1) is equivalent to:

f

(A33.6)

pdq =

y

f

PdQ:

y

Thus, we obtain:

f

(Q- q)dP + (p-P)dq

y

=

f

QdP + PdQ-(qdP + Pdq)

=

fd

[P(Q-:- q)].

Y

y

But the increment of P (Q - q) along y is equal to zero, because A is homotopic to zero: (A33.7)

f

d[P'(Q-q)] = O. Y

Conversely, (A33.S) and (A33.7) yield (A33.6).

(Q. E. D.)

§B. A Topological Lemma

Now, let A be a globally canonical diffeomorphism, T the torus, p = 0, and AT the image of T by A.

APPENDIX 33

LEMMA

245

A33.8

The tori T and AT have at least 2n common points (counted with their order 01 multiplicity) provided that the equation 01 AT (A33.9)

IS:

p = p(q),

Besides. the number 01 geometrically different points 01 intersection is, at feast, n + 1. Prool: Consider the following function on AT:

f

I(x) =

(A33.10)

x

p(x)dq(x) ,

Kij

where x

=

(p(x), q(x))

f

AT, and where the path of integration lies on AT.

Function I(x) is well-defia.ed since the integral (A33.10) does not depend on the path. In fact, let y be a closed curve of AT. then:

f

pdq =

y

1.

pdq = 0,

A-'y

because A- 1 'is globallycanonical, A- 1 y C T, and p

=

0 on T. The

function f(x) is a smooth function on the n-dimensional torus Tn. Thus, by the Morse' inequalities, the number of critical points is at least 2n (and, following the Lusternic-Schnirelman theorem, the number of geometrically distinct points of intersection is at least the Lusternic-Schnirelman category of Tn, which is n + 1). From (A 33.10) follows dl = p (x) . dq(x) on AT. The function p (x) vanishes at the intersection of AT and T. Thus, the points of intersection of T and AT are critical points of I on AT. Conversely, in view of condition (1\ 33.9), we have pdq of I on AT; therefore p(x)

=

=

0, for any dq, at every critical point x

0 and the critical point x belongs to the in-

tersection of AT and T. ,

See Milnor [1]; 2 n

= ~o

(Q. E. D.) bl' b,

= i-th

Beth number of 'r'.

246

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

COROLLARY

A33.11

The tori T and AT have at least 2 n common points, provided that their equations are:

p = p'(q), p =

(A33.12)

p"(q)(I~:'1

<

00, I~'I

<

00)

and that dp 1\ dq vanishes on T.

In fact, if dp 1\ dq

:=

0 on T, then the mapping p, q

-+

p- p'(q), q

is a canonical diffeomorphism. This diffeomorphism reduces (A 33.12) to (A 33.9) with p(q)

REMARK

If n

=

p"(q).-p'(q).

A33.13 1 (mappings of annuli), Lemma (A 33.8) still holds without con-

dition (A33.9). The proof makes use of Jordan's theorem and does not extend for n > 1. Whether T and AT intersect, for n> 1, if condition (A33.9) is not fulfilled, is an open question.

If condition (A 33.9) can be relaxed from Lemma (A 33.8), we ob1ain many "recurrence theorems" of the following type: Assume that the initial values a"

b i of the axis of the Kepler ellipses,

in the plane many-body problem, are such that the ellipses do not intersect. Then, whatever

T

be, there exist initial phases 2 1., g. such that the axis

.

"

of the ellipses return to their initial values after a time REMARK

T.

A 33.14

If we drop condition (A 33.9), Lemma (A 33.8) cannot hold without assuming that i\ is a diffeomorphism, because (even for n

=

1) regular and

globally canonical mappings can be constructed such that T and AT do not intersect. 2 Phases 1/. ~/ are angles (mod 271); ~/ determines the position of the ellipses and 1/ determmes the position of the planets on these ellipses.

247

APPENDIX 33

§C. Fixed Points

Now, let A be a global canonical mapping of the following particular type: (A33.1S)

A:

p., q -+ p, q <+

W

(p)

Assum~ that; on the torus p = Po' all the frequencie~ are commensurate:

(A33.16) and that the twisting is nondegenerate: (A33.17)

THEOREM

Det (

aw ) ~. 0 ap Po

•

A33.18

Every globally canonical mapping B, C1.close enough to A, has at least 2" points 3 with period N in the neighborhood of the torus p = Po: BNx =

x.

Proof: In view of (A33.1S), (A33.16), and (A 33.17) the mapping AN can be written under the form: (A33.19) A": p, q

-+

p, q + a(p), where a(po)

in the neighborhood of the torus p

a(p)

=

=

=

0,

Det

(aa) ap

Po

Po' It is sufficient to put:

N[w(p)-w(po)] •

The neighboring mapping B" can be written under the form: (A33.20)

BN : (p, q

-+

p+

f3 1(p, q),

3 Counted with their multiplicity.

q + a(p) +

f3 2 (p, q))

=

(P, Q) •

~ 0

2
ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Let us consider the points which move along the radii: Q

= q,

that is:

(A33.21) From the implicit function theorem we deduce: (1) equation (A 33.21) determines a torus T, close to p wh~ch

=

Po and

moves along the radii; (2) the tori T and BN T have equations of the form:·

(A 33.22)

P

I I

= P '(q), P = P "(q), where d: d ' <

00

IddPq " I < -,~ .

,

The mapping BN , being globally canonical, is given by a generating function of the form Pq + B (P, q). In view of Lemma (A 33.4), the function B(x) = B(P(x), q(x)) is s-ingle valued in

a.

Now, consider the restriction

of B (x) to the torus T. This is a smooth function on an n-dimensionaltorus, thus B has at least 2" critical points (compare to Lemma A 33.S). Let us prove that these critical points belong to the intersection of T and BNT. Formula (A33.3) yields: dB

=

(Q-q)dP + (p-P)dq, where B: x

=

(p, q)

->

(P(x), Q(x)).

But, according to (A 33.20) and (A 33.21), we have Q - q T. Therefore (p - P)dq

=

=

0 on our torus

0 at the critical points of B on T. But, in view

of (A 33.22), this implies P

=

p.

(Q. E. D.) REMARK

A 33.23

This theorem is not a corollary of Lemma (A 33.S): the manifold Q- q

o need

not verify dp 1\ dq

JhJapping proves:

;r

0, as the following example of canonical

APPENDIX 34

PROOF OF THE THEOREM ON THE CONSERVATION OF INVARlA"T TORI UNDER SMALL PERTURBATIONS OF THE CANONICAL \It\PPING (See Theorem 21.11. Section 21, Chapter 4) The construction of the ipvariant tori is pertormed in Sections E - H of this appendix. Lemmas of Sections B - D are used. Proof is based upon a method of successive approximations of Newtonian type suggested by A. N Kolmogorov [6].

§A. Newton s Method Newton's approximation for the construction. of a zero x of a function f, with a prescribed approximation, consists in replacing the curve y = f(x) by its tangent at ~xo' f(x o))' where Xo is an approximation of x. If

Ix-xol < E . the substitution gives an error of order E2. Hence, the linearized equation f(x o ) + f'(xo)(x ~ x o) = 0

admlts a solution Xl whose deviation from x will be of the order E2 (see Figure A 34.1).

Iteration of this process leads to an accelerated

conver~

gent approximation: (A 34.2)

I' x n+ l-xl
Hence, the deviation of the n tl :approximatlOn from the solution will be: I

Ix -

x n 1 .....JE 249

2n-1

250

ERGODIC PROBLEMS OF CLASSICAL MECHA,NICS

Figure A 34.1

This method extends easily to equations in Banach spaces 1. But, in analysis, one is most often concerned with poly-Banachic spaces. and the operator ['(xO)-l maps a space into another one. This situation is illustrated by'the following lemma, which offers a characteristic example, and where (A 34.4) plays the role of (A 34.2). LEMM.A

A34.3 2

Let L be an operator which maps the functions £(z), which are analytic in each complex domain G, into the functions Lf(z) which are analytic in the domain 3 G-o and such that, for every 0 (A344)

°

< 0 <00 :

ILfIG_o < Ifl5 . o-v ,

where v > 0, 0 0 > are absolute constants. Then, for every 0' > 0, the series ~sIT,Sf! converges in G-o', provided that IflG < M = M(o') is small enough. 1

See KantoroVlch [ I

J.

2 This lemma was.already known by H. Cartan [1]. whose' paper [I] is one of the first appearances of Newton's approximatlOns m theoretical analysis 3 Here and further,

G - 5 denotes the set of the pomts of G which are at a dis-

tance > 5 from the complement of G. Example of an operator which verifies (A 34.4): Lf = f·:Ii.. dz

251

APPENDIX 34

Prool: Let ~

U

2v + 1

1

and u~ 2

~ 3/2 = u1 ,

M13/2 . .... Ms+1

M2

=

... , U~ s+1

Ms3/2,

hence M

s

= 02v+l s

Then, if

we have:

Now, let G 1 = G, Then,

IIIG < Ml

IIIG s < M

S

G2=Gl-ol.···'Gs+l~Gs-oS:

implies

ILsII Gs <

Ms because, in view of (A 34.4),

implies:

But G s => G-o'forany s,because LOs< 0', Therefore,wehave:

inG-o',

(Q. E. D.) §B. Small Denominators

Let I(q) be a function on a torus Tn, I(q) =

~ k~O

I

q = (ql' .... , qn) (mod 217): 'ei(k,q)

k

'

where (k, q) = k 1 ql +'''+ knqn' and let W = (wI' ... ,w n ) be a vector with irrational components, such that (k, w) ,;, ko for nonvanishing integers

252

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

k, k o ' Consider the equation g(q+ w) -g(q) = f(q) ,

(A34.5)

where g is an unknown 271 periodic function. This equation admits a "formal" solution: (A 34.6)

$

g(q) =

gk' e,(k,q), ei(k,c.» _ 1

k~O

The following lemma asserts the convergence of (A 34.6): LEMMA A34.7 Assume that the function f(q) is analytic and If(q)1 for 11m q I

<

<

M always holds

p. Then, for almost every vector w (except for a set of Lebes-

gue measure zero), the function g (q), defined by (A 34.6), is analytic and, for Ilmql Here 00

< p-o,

we have: Ig(q)1

< Mo-II, II

=

2n+4, if 0<0<00.

> 0 is an absolute constant (independent of n).

Proof of this lemma 4 is based on elementary results from th~ Theory'of Diophantine approximations. In fact, Lemma (A 34.7) holds, with 00 0o(n, K) and for some K

> 0, for any element w of the set no(K) (defined

below). Let us denote by

n (K)

(A34.8)

lei(k,c.» - 11

for any w, Iw'-wol

=

< K~II, II

the set of the wo satisfying:

=

> KN-l/

n+2, and for any k, Ikl

<

N. Let us de-

II

= n+2.

note by no(K) the set of the Wo verifying: (A34.9) Of course,

le,(k,c.>o) - 11

n (K) c

> K

Ik I-II,

no(K) •

LEMMA A34.10 Almost every (in the Lebesgue measure sense) point Wo belongs to

n (K)

for some K> 0 (hence, Wo ( no(K)).

4 The technique of evaluating small denominators was extensively worked out by C. L. Siegel [2],

[3],

in connectIOn with similar problems.

APPENDIX 34

253

Proof:

Let

n

be a bounded domain of the space

< d for some

lk,d = Iwollei(k,Cil) - 11

Then, clearly: only on.

n.

mea~ (lk, d

n m :s.

Iwol.

Let:

w, Iw-wol

< dl .

C· d, where the constant C depends

Relation (A 34.8) holds outside of Uk Ik,Klkl- v ' But we have

since

~

!k I-v <

for v

00,

=

n+2 .

k

Therefore: meas

n n (K)

o.

k"'O

(Q. E. D.) Now, let ~ fk

f(q)

0

ei(k,q)

k

be an analytic function. LEMMA

A34.11

(A) Iffor IImql < p we have If(q)1 < M, then Ifkl < Me-plkl. (B) If Ifk(q)! < Me-P !kl, then for 11m ql < p-o (where 0 < 0 < 0 0 ):

(C) If for 11m ql

< P we have

If(q)1

<

M, then for 11m q I

< P -0,

0<0 < 0 0 :

Here, 0 0 and v are absolute constants, which depend only on n, and RN is: RNf =

~ f n . ei(k,q) . ikl>N

254

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

To prove (A) we have to

s~ift

the contour of integration in the formula:

fk =

to

f

f· e-i(k, q) dq

±. ip. Proofs of (B) and (C) consist in mere summations of geometrical series. Lemma (A34.7) follows at once from Lemmas (A34.10) and (A 34.11):

take Wo ( Do(K) and take into account (A34.6), (A34.9), (A) and (B) from Lemma (A34.11), and the elementary inequality:

<

e-lkI5'lklv

C(v)o-V •

Then, to obtain Lemma (A 34.7) it is sufficient to take 00

<

K/C(v). We

refer to Arnold [11] for further details. REMARK

A34.12

Suppose that Wo ( D(K) and f k

=

0 for Ik

1

>

N. Then, (A 34.6) re-

duces to a finite sum, which depends continuously on w. Besides, Lemma (A34.7) still holds, with the same 00 Iw-w'l

<

=

0o(K, n), for any w' such that

KN- v • Because if w ( D(K), then, in view of Definition (A 34.8),

every w', Iw-w'l

<

KN- v verifies (A34.9) for Ik 1 ~ N. But the proof

of Lemma (A 34.7) makes use of (A 34.9) only for Ik

1

~

N if [k = 0 for

Ikl > N. §c. Sketch of the Proof

Now, let us recall notations of Theorem (21.11) (see Chapter 4). The set

n

~ Bn

x Tn is a domain of the canonical space, a point x of

n

is

denoted by x = (p, q), where p = (Pl' ... , Pn ) is a point of the Euclidean ball 8 n and q = (ql' ... , qn) (mod 217) is a point of the t
invari~

255

APPENDIX 34

Following the idea of the Perturbation Theory (Appendix 30), we try to "kill" the perturbation B by an appropriate canonical coordinate transformation C, with generating function Pq + C(P, q). In coordinate Cx, the mapping B A can be written:

C(BA)C- 1 where B'

=

C B A C- 1 A-I.

=

B'A •

Therefore, taking into account Corollary

(A 32.20), the generating function of B' is:

Hence, killing B reduces to solving, with respect to C, C(x) + B(x) - C(A- 1 x) = O.

Now, observe that this equation, for each fixed p, has precisely the form of Equation (A34.5) (with w

= w(p),

f = -B,

g

=

C). Now, construct-

ing successive approximations to the invariant tori reduces to obtaining the inequalities: (A34.13) in a domain

n' c n

(but not "too small"). Then, the convergence is car-

ried out as in Lemma (A 34.3); Inequalities (A 34.13) are proved with the help of Lemma (A 34.7): they hold "far from resonances," that is for w(p) ( n(K).

To perform, along Sections E - H, the above program, we shall make use of several devices. First of all, instead of "killing" B, we only kill a truncation of its Fourier series: the remainder term RNB can be regarded as part of the "terms of higher order" 0(B 2 ) and 0(C 2 ). Hence, every app.roximation deals with only a finite number of resonances and small denominators in (A 34.6)5.

On the other hand, to use Lemma (A 34.7), we

5 This process was already used by Bogolubov. Mltropolski [1]. One could do without it by replacing the frequencIes cu(p) with constants cu· ( Q K in the small denominators.

256

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

need first to eliminate the constant term B 0 (averaged value of B with respect to q): we add Bo to the "unperturbed" mapping A (vahation of the frequencies, Section E).

J. Moser [4], the I\olmogorov

Finally, observe that, as pointed out by

method still works in the case of differentiable mappings: in this case, one can devise a smoothing process like a suitable truncation of the Fourier series at the Nth frequency.

(21.11) in the case n

~

Indeed, Moser [1] improved Theorem

1 (mappings of the plane) by abandoning the re-

quirement of analyticity and substituting instead the requirement that 333 derivatives exist! Recently Moser [6] gave a proof which requires only a temperate number of derivatives. §O. Canonical Mappings Close to the Identity In this section we use the following notation: Let F(o, M) be an

arbi~

trary function and a a proposition involving 0, M, F. We shall say that

"a is true and

IF I h M"

if there exist absolute constants 6 VI' v 2 > 0

and 00 > 0, such that: a is true and

(A 34.14) provided that 0 < LEMMA

°< °

0,

M<

IF I :s.

MO- v 2

,

oV I •

A34.15

Let G be a complex domain and f(z) an analytic function in G, which satisfies Il(z)1 < M. ..Then, for z ( G-o, the k-th derivative of f lytic and

IS

ana-

< n M.

6 That is that these constants depend on the d,mens,ons of the domains dealing with the proposition a, on the number of depivatives, etc., but neither on the functions nor on the domains.

257

APPENDIX 34

Proof: From the Cauchy formula dkf

f

k!

F«()d(

277i . Y «(_z)k+l

we deduce:

.E kf

\ dz k

I

:s.

1 k! MB- k •

Hence, we obtain (A34.14) with:

1 k'

o. (Q.~. D.)

Here: F (0, M) =

sup f.G.z (G-O

dkf I , Idz k

where sup ranges over all the bounded domains G, all the analytic tions f,

If I

func~

< M in G, and all the points z of G - B.

Observe that: CM B- v ~ M. and that the inequalities F ~ M and G ~ M imply F + G ~ M, FG ~ M2

Then. if

F(B, M) ~ M.

we have F (C 0, M) ~ M (here, C and

f.J

are absolute positive constants).

Now, let us make clearer how the global canonical mappings S, close to the identity. are related to their generating functions Pq + S (P, q). Let

o

=

B n x Tn, B n = !p lip I < y, p ( Rn

Tn

= !q (mod 277), q = (ql' ... , qn)l ,

I,

and [0) be the complex domain of 0, given by Ipi < y, IImql < p, where 0 < Y < 1.0 < P < 1.

LEMMA

A34.16

Let S (p, q) be an analytic function in [0], satisfying: IS(p, q)1

< M.

258

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Then; the formulas:

Q

(A34.17)

=

as , q+-

s

ap

S (p, q)

determine a global canonical diffeomorphism S:

P

=

Q

P (p, q),

Q(p, q) ;

=

S: [0] - 20

->

[0] - 8 ,

and the folJowing inequalities hold in [0] - 28:

I(P LEMMA

p) - as (p, q)

ap

I <

q) I < I(Q- q) + as(p, aq n

M2

n'

M2.

A34.18

= P (p, q), Q = Q(p, q) is a global analytic canonical mapping, which satisfies IP - pi < M, IQ - ql < M in [0], then this mapping is deIf P

fined in [0] -8 by formulas IS I

LEMMA

~

M,

(A 34.17),

IS(p, q) -

f

where S is analytic and verifies:

(p,q)

(Q- q)dp - (P-p)dql

~

M2

A34.19-

Let S (p, q) an.d T(p, q) be two analytic functions, which satisfy IS I

< M, IT I < M in [0]. Then the product R

=

ST of the corresponding

canonical mappings is a global canonical diffeomorphism, which maps [0] -38 infO [0] - 28, and which is defined in [0] - 8

by an analytic gen"

erating function R satisfyinB:

Now let. 'A: [0]

->

[0 '] be an analytic global canonical diffeomorphism

r

([0 is 'given by Ip I < y', 11m q I < p', 0 < y', p' < 1). Let a-1ly - xi < lA
259

APPENDIX 34 LEMMA

A 34.20

The formula T of [n'] - 38 into

=

A S A-I defines a global canonical diffeomorphism

to'] - 28.

which is given in [n'] - 8 by an analytic gen-

erating function T satisfying 7 :

Proofs of the preceding lemmas reproduce those of Appendix 32. making use of Lemma (A 34.15) to compute the terms of order 0 (c: 2). For further details we refer to Arnold [4]. [5].

§E. Variation of the Frequencies

Let us now begin the construction of the invariant tori of the mapping B A (see Theorem 21.11. Chapter 4). To understand the estimates performed in Sections E - F. it is useful to keep in mind that the positive numbers {3. y. 8. M and p. K. 0 satisfy:

0,< M « 8 and that vi'

Ci

are absolute

«

«

y

(3

«

constants~ v

p. K. 0- 1

<1 •

> 1 > c.

CONSTRUCTION OF THE VARIATION OF THE FREQUENCIES.

A 34.21

Let A and B be two global canonical mappings:

!\: P. q

-+

P. q + w( q) •

r. f

and B have generating function Pq + B(P. q). We put:

ii(p)

= (21T)-n

w 1(p)

B (P. q)dq l '" dqn •

aii ap

= w(p) + -

and we consider the following canonical mappings (see

Figu~

7 In thiS lemma, the constant 50' which enters mto the defmihon of also on a.

A 34.22):

~

, depends

260

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Figure A 34.22 Of course, BA = B'A 1 · LEMMA CONCERNING THE VARIATION OF THE FREQUENCIES. A34.23

Suppose that: O-l\dp\

< \dw\ < O\dp\ and \B(p, q)\ < M

hold in the domain \p - p*\ < y, \Im q \ < p. Then the global canonical mapping B' possesses a generating function Pq + B '(P, q) which satisHes 8 :

where

in the domain \p-p*\ <

y-o, \Imq\

<

p-o.

Proof: Application of Lemma (A34.19) to the mappings Band AA1l: p, q

-->

p,q-(aB/ap) yields \B'-(B-B)\ ~ M2 (Q. E. D.)

§F. Fundamental Lemma We now make use of the estimates of the small denominators (Section B) to obtain inequallties of the form \B 1\ ~ M2, after a suitable change of coordinates C. 8 The constant 50' m the relation

~

eventually depends on

e.

261

APPENDIX 34

A34.24

FUNDAMENTAL CONSTRUCTION.

Let A and B be global canonical mappings:

A: p, q

-+

p, q + wI (p) ,

and B defined by a generating function Pq + B (P, q), B (P, q)

~ B k(P)ei(k, q)

B(P) +

=

.

k,t 0

We put: -

~

C (P, q)

Ck(P)ei(k,q) ,

O
(A 34.25) Bk(P)

1

e-i(k,Ull (P)) _

where N is a positive integer. We denote by C the global canonical mapping, whose generating function is Pq + C (P, q), and we consider the global canonical mapping Bl

C B A C- 1 A-I. Of course:

=

BIA (see Figure A 34.26).

=

C(BA)C- 1

Observe also that:

A »

B ""'C » A

B1 . B

•

cl • ----

A

Bl

F,gure A34.26 FUNDAMENTAL LEMMA

We suppose that wI (p),

in

/

A34.27

the domain

Ip - p*1 < y, 11m qI <

B (p, q) are analytic and satisfy: IB(p,

,

q)1 <

M,

p, the functions

IB(p)1 < Ii

262

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

We suppose also that w * ~ wI (p *) belongs to the set U(K) of Lemma (A34.10), then: (1) in the domain IP-p*1

< y, IImql < p-5 the function C(P,q)

is analytic and verifies:

(2) in the domain, IP - p*1 < y -5, 11m q I < p -{3, the generating function Pq + B 1 (P, q) of B1 satisfies: (A34.28) provided that [for (1) and (2) l:

Proof: From Idwl < OldPI and y < C4 N-(n+2), C 4 < K/O, it follows that all the points w 1 (p) of the domain Ip-p*1 < y ,:erify Iw-w*1 < KN-(n+2). Consequently, according to the last remark of Section 8", the assertion of Lemma (A 34.7) holds"for the trigonometric sum C. Let 50 ~ 5 0(n, K) be t~e constant of Lem~a (A 34.7). If C 3 < 00' then we have 5 < 50; therefore Lemma (A 34.7) implies that IC I < M5- vI for jIm q I < p -8. This is

precisely the (1) of our lemma. Observe next that if C 2 (in y < C 2 (3) is small enough, then we have 11m w(p) I < Oy < (3. Under this condition, the mappings A and A-I are diffeomorphism.s: !p, q\lp-p*1 < y', IImql < p'l .. !p, q\lp-p*1 < y", IImql < p"l, where' p" < p' +

Or < p.

This allows one to apply Lemma (A 34.20) to

9 That is. tpey depend only on the dimension n and the constants K,

e.

The fun-

damental lemma states that there exist constants vI' v2 (large enough) and C l' C 2 , C 3 , C 4 (small en0':1gh) such that (1) and (2) hold.

263

APPENDIX 34

A C- 1 A-1. Besides, if v 1 and v 2 (in M < 8 V2 ) are large enough and C 3 (in 8 < (3 < C 3 ) small enough, the relations ~ of Lemmas (A 34.16),

(A34.18), (A34.19), and (A34.20) are of the form: < M8- vl . Hence, we deduce from these lemmas that, under the hypotheses of the fundamental lemma, with suitable constants C, v, the mapping B1

=

C BAC- 1 A- 1 has a generating function Pq + B 1 (P, q) such that in the domain IP-p*1 < y-8, IImql < p-(3, the function B 1(P, q) is analytic and:

(A34.30) But from (A34.2S) follows:

(A34.31) If 11m q I < P - (3 < P - 8, then assertion (C) of Lemma (A 34.11) implies: IRNB

(A34.32) Since 181

I < Me- 13N . (3-v1

•

< ii, formulas (A 34.30), (A34.31), and (A 34.32) imply (A 34.28). (Q. E. D.) §G. The Inductive Lemma

The construction of the invariant tori makes use of an iterative process, each step being based on the following construction. THE INDUCTIVE CONSTRUCTION.

A 34.33

Let A and B be two canonical mappings: ,,; p, q

-+

p, q + w(p) ,

B given by a generating function Pq + B (P, q), and N a positive integer. Performing the variation of the frequencies (Section E) we obtain: "1; p, q

-+

p, q + w 1 (p) ,

and B', which is given by a generating function Pq + B '(P, q) and such that B"

=

B'

"1'

We now apply the fundamental construction of Section F

to the mappings "1' B '. We obtain a canonical mapping C, and B1 =

264

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

C B'A 1C- 1A 1 ; hence (see Figure A34.34) we have:

1

B1A1

C(BA) C- 1

A

B

c ._ _ _ _ _ _ ---.:}

Figure A 34.34 THE INDUCTIVE LEMMA

A 34.35

We suppose that in the domain Ip - p*1

IB(p, q)1 < M, Let us define

w(p*)

< y, 11m q I < P, we have:

=

w*

f

O(K) .

P:

by w 1(P:) = w*, and let Pq+ B 1(P, q) and Pq+ C(P, q) be the generating functions of B, and C. Then: (1) in the domain IP-p*1 < y, IImql < p-o, the function C(P, q)

is analytic and

IC I <

M o-Vt ;

(2) the domain IP - p:1 < Y1·IIm q I < PI

=

P -{3 belongs to the do-

main IP - p*1 < y, 11m q I < p, and in this domain the function B1 is analytic and

provided that:

where the constants

1/1,1/2,1/3' 1/4

> 1 > C 1 , C 2 , C 3 , C4 , C s > 0 are ab-

solute constants, that is depend only on the dimension n and on the constants

eo'

K, but not on B, w,

e,

M, and so on.

265

APPENDIX 34

Proof:

Proof of the inductive lemma follows at once from the two preceding lemmas. The only two novelties are:

P: :

(1) The existence of

This existence and the inequality

Ip: I <

CsY follow from the inequal-

ities:

IdWI dp (see Section E) and from M8- v

<

<

_ dw dp

Ip - p*1 < Y-8

belongs to

<

M8- v

C 6 • y (which follows from M8- v

C 6 • Y for v 2 large enough). Taking into account Yl

Ip - p:1 < Y1

I <

< 8 v2- v

C4 y, the domain

because 8 < Cly·

(2) The estimate of B 1 :

From Section E it follows that averaging B' gives:

~ ow

replace M in relation (A 34.28) of the fundamental lemma, we obtain: M28-V1 +

M<

M28- V4 •

(Q. E. D.) §H. Proof of Theorem (21.11) (see Chapter 4) CONSTRUCTION A34.36

The invariant torus T (w *) of the mapping B A, which corresponds to the frequencies w *, is constructed by making use of the process of the preceding section. This process depends on a sequence 0 < N 1 < N 2 < ...

< Ns <

00',

Ns

-+

+ 00 which will be specified later.

After the sequence N s is selected, the construction goes as follows. We put Al ~ A. B1 = B. N 1 = N, and we make use of the inductive con~ struction of Section G. This construction determines canonical mappings, namely C 1 • A2

•

H2

;

we have: B2A2

C1(R1 A1)C 1- 1 .

266

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

We again apply the sam~ construction with A2 = A. B2 = B. N2 = N; we obtain A3 • B3 • C2 , and so on. If As' Bs are constructed, the construction of Section G, with A = As' B = Bs' N = N s leads to Cs ' As+ 1 '

Bs+l (see Figure A 34.37): 1 Cs (B s A s )Cs

000 As

Figure A 34.37

This construction also determines the points p*, ws(p*) • s s Ts(w*) denotes the torus p

=

p:. The mapping

torus, is the translation defined by w *. We put

T:

be the torus p

=

P:,

Now

As' restricted to this

p:

lim

=

p

s' and let

s~""

A"" the mapping:

A : T*

""

= w*.

""

~ T* ,

""

A""q

=

q +w

*•

Finally, we put:

os --

C 1-1 C2-1

...

C s-1 -1

•

0

=

1·1m

n

Us

s~""

The invariant torus of B A is given by: T(w*) = DT*

""

We now prove that the above limits exist and that D:\""

B -\ D on T (w * ). (Q. E. D.)

267

APPENDIX 34

A 34.38

CONVERGENCE,

In view of dw/dp

~ 0, we can suppose that:

0-11 dp I

< Idw(p)1 < Oldp I

holds in [n]. Let us suppose that w * (O(K). According to Lemma (A 34.10), almost every w* belongs to O(K) for some K> O. We put ()o

=

2(),

and we define a sequence of constants by:

Let us put:

oY,s '

f3 s

Y '/.o(n+2) s

-Y,(n+2)

Ys

'

then we have:

f3 s+ 1 = f3 s3/2 '

N s+l

=

To define all these numbers, we just need to choose

o.

Ys+l

=

y3/2

s'

Ns3 / 2

•

Denote by a a

positive constant, if a is taken large enough we have:

Let C, 0 < C < 1/10, be an absolute constant, that is, C depends only on n, K, 00' a and on the constants tive lemma. If 0 < C,' then,

\!k'

C k which enter into the

induc~

obviou~:;ly:

(1)

(A34.39) (2)

(A 34.40) (3) for any s

=

1,2, ... , we have:

(A 34.41)

where C 1 · C 2 , C 3 , C 4 , C s are the constants of the inductive lemma, which

268

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

depend on the constants K and

eo

we introduced above;

(4) (A 34.42) Our number 8 is nowoselected to satisfy: 0 < 8 < C. Hence, the numbers N s' on which our construction depends, are determined.

Suppose now that our mapping B1

=

B has a generating function

Pq+ B 1 (P, q} which verifies IB11 < M1 = 8 1a in Ip-p~1 < Y1' 11m ql < P1 = Y:zp. According to Lemma (A 34.18) this inequality holds if E, under

the conditions of Theorem (21.11), is small enough. Taking into account inequalities (A 34.41), the hypotheses -of the inductive lemma are satisfied A1 , B = B1 (because a > 1J 2 )· Therefore we obtain .'\2' 8 2 , C 1 · and so on from the inductive lemma.

by A

p;,

=

Let us prove that the mappings B2 • A2

again satisfy all the condi-

tions of the inductive lemma in the domain Ip - p;1

< Y2' 11m q I < P2

=

P1 - (31' In fact, (A34.39~ implies P2 > 0; (A 34.40) and the third part of the inductive lemma show that "2 satisfies the inequalities:

Finally, the second part of the inductive lemma, (A34.42), and a> a >

1J 1

21J 4

+ 1,

+2, imply:

IB21 < Mf8 1Y4 +

M 1 e- iV'I1

·8- vl < 8;a-v4 + 813/2 (at-1) < 813012

•

In other words:

Hence, B2 and A2 fulfill all the conditions of the inductive lemma. Iteration of this argument leads to: IB s I < Ms

=

8 a , for Ip-p:1 < Ys ' s

IImql < Ps '

Let G s be the domain Ip-p!1 < Y s ' IImql < P s ' Then,thediffeomor 1 map G 1 into G and, in the C 1.norm, we have: phisms Cs s+ s

o

APPENDIX 34

269

E = identity,

(A 34.43) in G s+I' The point

p:

is the intersection of the balls: S

...

00

•

'. From the estimate (A 34.43), follows at once the convergence of the ns

on

the torus

r*

00

nG s>1

The inequalities

on the torus

IB sl < 8~

s

.

imply;

r*. 00

Finally, from IWs+l -

wsl < 8 s

follows the convergence of the map-

pings As:

Thus'

'\

•

00'

on

r*

00

(Q. E. D.)

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Adler. R. L.. and Rivlin. T. V. [1] Ergodic and Mixing Properties of Chebyschev Polynomials, Proc. Amer. Math. Soc. 15 No.5 (1964) pp. 794-796.

Andronov. A. A.• and Pontrjagin. L. S. [1] Rough Systems, Dokl. Akad. Nauk. 14 (1937) p. 247-250 .

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Arnold. V. I. [1] Sur la geomtHrie des groupes de dif£eomorphismes et ses applications en hydrodynamique des fluides parfaits, Ann. lnst. Fourier (1966). [2] Remarks on Rotation Numbers, Sibirski Math. Zh. 2 No.6 (1961) pp. 807-813. [3] Some Remarks on Flows of Line Elements and Frames, Dokl. Akad. Nauk. 138 No.2 (1961) pp. 255-257. [SOy. Math. Dokl. 2 (1961) pp. 562-564.]

271

272

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

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ERGODIC PROBLEMS OF CLASSICAL MECHANICS

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INDEX

Bernoulli scheme, 7, 11, 21, 30, 32, 39,42 Birkhoff problem, 86 Birkhoff theorem, 16 Boltzmann-Gibbs model, 34, 76

A- Entropy, 48 Abstract dynamical system, 7

Action, adiabatic invariance, 97 Action-angle coordinates, 94, 211, 213 Algebra of measurable sets, lS3 inclusion, 153 intersection, 153 subalgebra, 153 sum, 153 Anosov theorem, 67, 201 Asymptote, 172, 179 negative, 61, 180 positive, 61, 180 Asymptotic theory of the k-th approximation, 85 Automorphism, 124 Averaged system (see System of evolution) Averaging method, 101, 103, 227

C-Flow, 55, 69, 189 C-Systems, 53, 191 cr-Topology, 65 Canonical mapping, 231, 235 global, 99, 243 infinitesimal, 237 Classical systems, 1 Compact manifold of negative curvature, 60, 168, 178 Conservation of energy, 4 Contracting space, 55 Convex billiard table, 232 Decomposable system, 17 Density point, 18 Di1ati~g space, 55 Discrete spectrum, 25, 48, 95, 147

Baker's transformation, 8, 12, 33, 125 283

284

INDEX

Discrete spectrum theorem, 27

Geodesic flow of the torus, 3, 12, 18, 94, 117

Dynamical system, 1 Elliptic point, 218 Endomorphism, 124, 143 Entropy, 24, 35 of an automorphism, 40, 163 of a classical system, 46 of a flow, 45 of partition, 36 of a partition with respect to an automorphism, 38 of a partition a with respect to a partition (3, 37, 158 Equipartition, 129, 135 Ergodic system, 12, 26, 134 classical, 147 Ergodicity, 16, 20, 24 EuleroPoinsot motion, 119 Fibre bundle, 57 Foliation, 59, 185 absolutely continuous, 75 co.ntracting, 73 dilating, 73 ergodic, 73 Generating function, 99, 235, 243 Generator, 41, 163 Generic elliptic type, 219 Generic property, 12 Geodesic flow, 3, 18, 151 of the ellipsoid, 3, 94, 96 of the Lie groups, 3, 120 of the manifolds of negative curo vature, 34, 60, 76, 136

Hadamard method, 113 Hamiltonian flow, 3, 12, 18, 104, 106 global,S, 122 Hamiltonian function (see Generating function) Homomorphism, 124 Horocycle, 173 Horocyclic flow, 45, 49 Horosphere negative, 6~, 181 positive, 61, 180 Hyperbolic point, 218 Hyperbolic rotation, 216 with reflection, 216 Induced operator, 22, 147 Instability (zones of), 90, 108 Integrable system, 94, 108, 210 Invariant torus, 93, 97 Involution (integrals in), 210, 212 Isometry, 21 Isomorphism, 11, 27, 124 Jacobi field, 187 Jacobi theorem, 2, 115

KoSystem, 32, 43, 70, 91 Kolmogorov theorem, 41, 166 Kouchnirenko theorem, 6 Krein theorem, 221 Lagrange problem, 9, 135, 138 Lebesgue space, 7

285

INDEX

Lebesgue spectrum, 28, 34, 154 multiplicity, 29 simple, 29

Periodic point, 198 Periodical apporximation speed, 49 Perturbation (theory of), 85, 100

Levi-Civita theorem, 220

Poincare invariant, 235 Poincare-Lyapounov lemma, 221 Poisson bracket, 210, 212 Proper function, 26, 147 Proper value, 24, 26 Properly continuous spectrum, 25 Properly discrete spectrum, 25

Lie commutator, 240 Linear oscillation, 4 Linear symplectic mapping, negative proper value of, 224 parametrically stable, 221 positive proper value of, 224 stable, 221 Liouville theorem, 3, 4, 119 (see also Integrable Systems) Lobatchewsky-Hadamard theorem, 60, 178 LobatchewskY-Poincare plane, 169

Mean motion (see Lagrange problem) Mean sojourn time, 10 Measure, 123 Measure space, 123 Mixing, 12, 19, 25, 26, 30 n··fold mixing, 22 semigroup, 144 Moser theorem, 97, 220, 256

Quasi-periodic motion, 1, 93, 97, 119, 210

Resonance, 101, 104 Rotation of a solid, 96, 120 (see also Euler-Poinsot motion)

Sheets, 59, 185 contracting, 189 dilating, 189 Sinai theorem, 62, 191, 207 Singular spectrum, 91 Skew-product, 24, 145 Skew-scalar product, 211, 223 Smale example, 56

Parabolic point, 218

Smale nontoral C-system, 194 Smale theorem, 196 Small denominator, 251 Space mean, 15, 127

Parametric resonance, 87, 221 Partition, 158 algebra generated, 159 refinement, 158 sum, 158 Pendulum, 81, 121

Spectral invariant, 22 Spectral measure, 24 Spectral multiplicity, 24 Stability (theory of), 85 Stability of fixed point, 219 Stable system, 81

Newton method, 249

286

INDEX

Structural stability, 64 of C-system, 64 problem (see Smale theorem) Surface of section, 98, 230 Swing, 81 Symplectic manifold,S mapping, 212, 215, 221 (see also Linear symplectic mapping) System of evolution, 101, 108

Three-body problem, 12, 96, 97 108 TimE' mean, 15, 127,134 Topological index, 218 Topological instability, 109 Torus automorphism,S, 11, 21, 28, 30, 34, 43, 53, 62, 70, 127

Torus translation, I, 5, 11, 16, 18, 21, 115, 130, 132, 135 Transition chain, 111 Transition torus, 110

Unsta~le

system, 53

Variation of frequency, 259 •

Weak mixing, 21, 25 Weyl theorem (see Equipartition) Whisker arriving, 110 departing, 110 Whiskered torus, 109 Winding number, 148 group of, 149