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Physics + Uspekhi 50 (9) 939 + 964 (2007) METHODOLOGICAL NOTES

# 2007 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences PACS numbers: 05.45. л a, 05.45.Ac

Dynamical chaos: systems of classical mechanics
A Loskutov
DOI: 10.1070/PU2007v050n09ABEH006341

Contents
1. Introduction 2. Background 3. Hamiltonian mechanics 4. Nonlinear resonance
3.1 Integrable systems; 3.2 Perturbed motion 4.1 Small denominators; 4.2 Universal Hamiltonian; 4.3 Width of the separatrix; 4.4 Internal resonances; 4.5 Overlapping of resonances; 4.6 Higher-order resonances 5.1 The Kolmogorov theorem; 5.2 The Arnol'd diffusion 6.1 Twist map; 6.2 Fixed-point theorem; 6.3 Elliptic and hyperbolic points; 6.4 Splitting of separatrices. Homoclinic tangles 7.1 The Mel'nikov function; 7.2 The Duffing oscillator and nonlinear pendulum 8.1 Ergodicity and mixing; 8.2 Unpredictability and irreversibility; 8.3 Decay of correlations 9.1 The Lorentz gas; 9.2 Scattering billiards with oscillating boundaries. The Fermi acceleration; 9.3 Focusing billiards with oscillating boundaries. Particle deceleration

939 941 942 943 947 949 952 954 956 960 961

5. Elements of the Kolmogorov + Arnol'd + Moser theory 6. The nature of chaos

7. The Mel'nikov method

8. Principal properties of chaotic systems 9. Billiards

10. Conclusion References

Abstract. This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov + Arnol'd + Moser theory, В the Poincare + Birkhoff fixed-point theorem, and the Mel'nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems Р unpredictability, irreversibility, and decay of temporal correlations Р are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years Р billiards with oscillating boundaries Р are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitaA Loskutov Physics Department, Lomonosov Moscow State University, Vorob'evy gory, 119992 Moscow, Russian Federation Tel. (7-495) 939 51 56. Fax (7-495) 939 29 88 E-mail: Loskutov@chaos.phys.msu.ru Received 31 January 2007, revised 25 April 2007 Uspekhi Fizicheskikh Nauk 177 (9) 989 + 1015 (2007) Translated by A V Getling; edited by A M Semikhatov

tions of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.

1. Introduction
For a long time, the concept of chaos was associated with the assumption that, at least, the excitation of an extremely high number of degrees of freedom is necessary in the system. The formation of this idea seems to have been influenced by the concepts of statistical mechanics, in which the motion of an individual gas particle can be predicted in principle but the behavior of a system consisting of a huge number of particles is extremely complex, and therefore a detailed dynamic appr oach is meaningless. This dictated the need for a statistical analysis. But extensive studies have demonstrated that the validity o f t he statis tical l aws a nd s tatistical description is not restricted to highly complex systems with a large number of the degrees of freedom. Random behavior can also be exhibited by entirely deterministic systems with a moderate number of the degrees of freedom. Here, the point is not the complexity of the system or the presence of external noise but the emergence of an exponential instability of motion at certain values of the parameters. The dynamics of systems subject to such an instability is called dynamic stochasticity, or deterministic (dynamic) chaos. Investiga-


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tions in this area are of fundamental importance because they disclose the nature of randomness by supplementing the hypothesi s of molecular chaos with the hypothesis of dynamic stochasticity. В H Poincare [1] was the first to note the relation between statistics and instabilities. However, a statistical approach to the description of systems with numerous degrees of freedom was previously suggested by L Boltzmann [2], who conjectured that the motion of particles in a rarefied gas should be regarded as random and that the entire energetically allowed phase-space domain is accessible to each particle. Such a view of the properties of multiparticle systems, known as the ergodic hypothesis [2 + 4], became the basis of classical statistical mechanics. A rigorous substantiation of this hypothesis could not be found for a long time, however. Some progress in this direction was achieved due to studies by P Ehrenfest [5, 6] (see also Refs [7, 8]). In particular, they allowed establishing applicability limits for the laws of statistical mechanics. However, a well-known study by E Fermi, J Pasta, and S Ulam [9] (see Refs [10, 11] for a more detailed exposition), who made the first attempt to verify the ergodicity hypothesis, put the problem of substantiation of statistical physics in the forefront again. This problem can be partially resolved based on studies by В Poincare (see Ref. [12]), who concluded that the motion of a system is extremely complex in the neighborhood of unstable fixed points in phase space. This was the earliest indication of the possible chaotic properties of nonlinear dynamical systems. Later, G D Birkhoff [13] showed that if the ratio of the frequencies is rational (i.e., at a resonance), stable and unstable fixed points appear in phase space. Higher-order resonances change the topology of phase trajectories and lead to the formation of a chain of `islands.' It turned out that the regular perturbation theory fails to describe such resonances, because the solutions are strongly perturbed near the resonances, and therefore small denominators emerge in the expansion and the series diverge. N S Krylov [14] was the first to deeply investigate the nature of statistical laws. He showed that the property of mixing and the related local instability of nearly all trajectories of the corresponding dynamical systems underlie this nature. In view of this, M Born [15] (see also [16]) suggested that the behavior of the systems is not predictable in classical mechanics. Later, the dynamics due to such instabilities in the systems came to be known as dynamic stochasticity, or deterministic (dynamic) chaos. The word `chaos,' in this meaning, seems to have been introduced by J A Yorke [17] (see Ref. [18], p. 338). But as noted by Ya G Sinai, the word combination `deterministic chaos' was first used by B Chirikov and G Ford in the 1960s. Physically, due to unavoidable fluctuations (i.e., small perturbations of the initial conditions), the initial state of the system is to be specified by some distribution. The problem is in predicting the evolution of the system based on this initial distribution. If the system is stable, such that small perturbations do not increase exponentially with time, its behavior is predictable. In contrast, if the system is subject to exponential instability (which is expressed by saying that the system has a sensitive dependence on the initial conditions), the process allows only a probabilistic description. In essence, precisely these considerations formed the basis of the modern views of dynamical chaos. The discovery that it is chaos, rather than external noise, that mainly determines the behavior of the system was unexpected (see Ref. [20] for a review).

The schools of A N Kolmogorov and A A Andronov, to which a brilliant group of outstanding contemporary mathematicians belongs, have deeply influenced the development of the theory of dynamical chaos. In particular, Kolmogorov's theorem of the preservation of almost periodic motion in weakly perturbed Hamiltonian systems, proved by V I Arnol'd and J Moser and known as the Kolmogorov + Arnol'd + Moser (KAM) theorem (see Refs [21 + 25]), became a keystone in understanding the origin of chaotic behavior. In their early studies, D V Anosov [26] and Sinai [27, 28] showed that dynamical chaos is a widespread phenomenon. In his pioneering investigations of the bifurcations of a saddle + focus separatrix [29, 30], L P Shil'nikov developed, among other things, a special technique for the analysis of the dynamics of systems near saddle-type trajectories and uncovered the extreme complexity of the structures that develop as homoclinic trajectories appear. It was demonstrated that the behavior of systems must be complex in the full neighborhood of the parametric values at which a homoclinic orbit exists. Later, Shil'nikov, L M Lerman, N K Gavrilov, I M Ovsyannikov, D V Turaev, and others developed new methods that allowed describing a finite number of bifurcations that lead to chaotic dynamics (see Refs [31, 32] and the references therein). A new stage in explaining chaotic behavior and its origin in deterministic systems was initiated by Kolmogorov's and Sinai's studies [33 + 35], where the concept of entropy was introduced for dynamical systems. These studies laid the foundations of a consistent theory of chaotic dynamical systems. Various abstract mathematical constructions have played an important role in the development of the theory of deterministic chaos. In particular, S Smale [36], to disprove the hypothesis of the density of systems that exhibit only a periodic-type behavior, constructed a notable example, currently known as the `Smale horseshoe.' This example implies that there exist systems that have both an infinite number of periodic orbits with different periods and an infinite number of aperiodic trajectories [18, 36, 37]. Subsequent to the Smale horseshoe, Anosov's C-systems were found [26, 38], which are characterized by the most pronounced mixing properties. Such systems were generalized by introducing Smale's `Axiom A' [37] (see also Refs [39 + 41] and the references therein) and hyperbolic sets [18, 37, 40 + 42]; these generalizations specified an important class of dynamical systems that have the property of the exponential instability of trajectories (see Ref. [43] for a review). At nearly the same time, mathematical studies began appearing in which attempts were made to substantiate statistical mechanics based on the analysis of billiard systems [27, 28]. Originally, billiards were introduced as simplified models appropriate for the consideration of certain problems of statistical physics [13] (see also the references in Refs [44, 45]). A billiard on a plane is a dynamical system that describes bodies (balls) moving inertially inside a bounded domain, in accordance with the law of equality of the incidence and reflection angles. In essence, planar mathematical billiards are the usual billiards without friction, although with an arbitrary configuration of the table and without pockets. Krylov's problem of mixing in a system of elastic balls [14] was first solved using billiard systems. Furthermore, it was shown that systems corresponding to billiards with scattering


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boundaries have much in common with geodesic flows in negative-curvature spaces, i.e., with Anosov flows. Somewhat later, the class of billiard systems capable of exhibiting chaotic properties was substantially extended (see Refs [45 + 47] and the references therein). It was shown based on a generalization of such systems (a modified two-dimensional Lorentz gas) that motion in purely deterministic systems can be similar to Brownian motion [44, 45]. This result became the first rigorous confirmation of chaotic properties exhibited by dynamical systems (not involving any random mechanism). Further investigations of nonlinear systems, both theoretical and experimental, have shown how typical the chaotic behavior is in systems with few degrees of freedom. It became obvious that chaotic properties can be manifested by a variety of nonlinear systems; if chaos is not revealed, this can merely result from the fact that the development of chaos is restricted either to very small domains in the parameter space or to physically unrealizable domains. How does chaotic motion originate? What is the nature of chaos? Seemingly, there should be many paths toward the onset of chaos. But it became known that the scenarios of chaotization are far from numerous. Moreover, some of them obey universal laws and are independent of the nature of the system. The same development scenarios for chaos are inherent in a variety of objects. The universal behavior resembles the usual second-order phase transitions, and invoking renormalization-group and scaling techniques known in statistical mechanics opens new avenues in investigating chaotic dynamics. This article presents the foundations of the theory of dynamical chaos. We describe the principal results obtained in the field that belong to classical mechanics, such as elements of the theory of nonlinear resonance and the В KAM theory; the Poincare + Birkhoff fixed-point theorem, which is important for understanding the sources of chaotic behavior; and the Mel'nikov method, which allowed obtaining a criterion for the origin of chaos analytically in some cases. Particular attention is given to the nature of chaos. Specifically, we detail the factors that lead perturbed systems to manifest self-similarity, to the splitting of separatrices, and to homoclinic and heteroclinic tangles. We also show that unpredictability, irreversibility, and decay of temporal correlations occur in systems in which such phenomena are observed. In Section 9, we describe models of nonequilibrium classical statistical mechanics with chaotic properties, highly popular today: billiards with oscillating boundaries. The Lorentz gas and stadium-type billiards are considered in detail. An interesting result is presented: the analytic form of billiard-particle acceleration law, i.e., a proof of the presence of Fermi acceleration in billiards with well-developed chaos. But if a billiard system is near an integrable system, such that the curvature of the billiard boundary is not large, small oscillations of the boundary lead to a new phenomenon. A specific, billiard version of Maxwell's demon originates: a billiard particle either accelerates or decelerates depending on the initial conditions. In other words, perturbation of the boundaries of such billiards results in a velocity stratification of the particle ensemble. The modern mathematical techniques used to analyze the chaotic properties of dynamical systems are fairly complex. But in this article, we pursue the aim of giving a general idea of the origin of the phenomenon of deterministic chaos and expose the fundamental concepts underlying the currently

known approaches to chaotic dynamics. Therefore, our presentation is mainly based on geometric techniques and a qualitative approach. Although most of the results described here have been known for a rather long time, we present them in a form appropriate for a nonexpert to comprehend the origins of chaotic dynamics. Therefore, in particular, this article is methodological.

2. Background
T h e s u b j e c t o f o u r a n a l ys i s i s t h e s y s t e m s d e sc r ib e d b y ordinary differential equations x vxY a Y 1

wher e v fv1 Y v2 Y F F F Y vl g is a v ector functi on (typical ly assumed to be smooth), a symbolizes the set of parameters, and x fx1 Y x2 Y F F F Y xl g i s an l-di m ensi onal vector wi t h components x1 Y x2 Y F F F Y xl that characterizes the state of a dynamical system. If the substitution of a function FtY ci , ci const, in Eqn (1) turns them into identities, this function is called a solution. Specifying initial conditions x0 x0 uniquely determines the solution at any instant t, xt FtY x0 X 2

Any solution xt of system of equations (1) can be geometrically represented by a curve in the l-dimensional space of variables x1 Y F F F Y xl . This l-dimensional space is called the phase space of the system (we let M denote this space). Each state of the dynamical system corresponds to a point in M, and each point in M corresponds to a unique state of the system. Changes in the state of the system can be interpreted as the motion of a point (called the representation point) in the phase space. The trajectory of this representation point, i.e., the set of its consecutive positions in the phase space M, is called the phase trajectory. We assume that system (1), whose phase space is M, was in the state x0 at an instant t0 . Then, generally, it is in another state at an instant t T t0 . We let F t x0 denote this new state. Thus, for any t, we define the evolution operator, or the shift map F t X M 3 M of the phase space M into itself. The map F t transforms the system from its state at t0 into the state at t. In other words, solution (2) of Eqns (1) establishes a correspondence between the point xi t0 , i 1Y 2Y F F F Y l, of the phase space M at the instant t0 and a certain phase-space point xi t at an instant t: F t x0 xt X 3

Therefore, in the general case, any phase-space domain O0 transforms in time t into another domain, Ot F t O0 , under the action of the map F t . The map F t X M 3 M is also called the phase flow, and the function vx is the vector field of the given dynamical system whose phase space is M. If the phase flow F t has a secant S, i.e., a certain codimension-1 hypersurface that is transversally intersected by phase curves, then a map F can be defined on S as follows: to any point p of S we associate the closest (next to p) point, p H , of intersection of the phase curve with the same hypersurface S. Then the analysis of the dynamics of the original system reduces to analyzing the properties of the map F, В which is called the succession function, or the Poincare map В (sometimes, the Poincare return map).


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Systems for which div v 0 are called conservative. Most objects considered in classical mechanics belong to this class. Investigating the evolution of conservative systems is of fundamental importance because it is related to a number of problems such as the substantiation of the Boltzmann ergodic hypothesis, planetary motion and the N-body problem, the dynamics of charged particles and plasma heating, etc. We begin our analysis with the Hamiltonian approach, whose advantage is not in its formalism but in a deeper interpretation of the physical essence of the phenomenon in question. In fact, the Hamiltonian approach is a geometrical method of analysis. It has some advantages and allows obtaining solutions to problems that are not amenable to other techniques of analysis.

where j is the angle of deviation from the vertical and g is the acceleration of gravity. The equations of motion of the pendulum are p Аmgl sin j, j paml 2 , or j o2 sin j 0 Y 0 6

3. Hamiltonian mechanics
The Hamiltonian formalism is based on the well-known Ha milt o n e qu at ion s, wh ic h ar e give n b y th e s yste m o f ordinary differential equations qi qH Y qpi pi А qH qq i 4

p where o0 gal is the oscillation frequency. Hamiltonians of th is type a re ty p ic a l of many p r obl e ms and pl a y a fundamental role in classical mechanics. If the full energy of the pendulum, H E, exceeds the maximum value of the potential energy, E Erot b mgl, the momentum p is always different from zero, which leads to an infinite increase in the angle j, i.e., to the rotation of the pendulum. Energies E Eosc ` mgl correspond to oscillations of the pendulum. If E % Es mgl, the oscillation period tends to infinity and the motion follows the separatrix between the two types of motion, oscillation and rotation (Fig. 1a). The equation of the separatrix can be written as В Г ps Ж2o0 ml 2 cos js a2, js 4 arctan exp o0 t А p, where the plus and minus signs respectively correspond to the upper and lower branches.
p E Es 0 j a
1

(where i 1Y 2Y F F F Y n), and for which the initial conditions at t t0 are specified as qi t0 qi0 , pi t0 pi0 . Such systems are rich in a variety of motions, from completely integrable dynamics to quasi-periodicity and chaos. Among the fundamental properties of Hamiltonian systems is the conservation of the volume of an arbitrary phase-space domain, i.e., the validity of the Liouville theorem, dq0 dp0 dq dp Y
D0 D
t

E Erot

a

b

E Eosc Аp

O

1

p

J

O
2

2 2

J

1

a

Figure 1. (a) Phase portrait of a nonlinear pendulum and (b) a visual representation of an integrable Hamiltonian system with two degrees of freedom in angle + action variables.

where D0 Y Dt & M.

3.1 Integrable systems The problem of the integrability of Hamiltonian systems is quite complex. There are a number of rather general (although, naturally, not universal) methods that in some cases allow constructing a solution of Eqns (4) or of an approximation to them. Quite comprehensive reviews and monographs describing these methods in detail are available (see, e.g., Refs [48 + 55] and the references therein). Therefore, we do not consider them here; instead, we only present a geometric analysis of integrable systems. Hamiltonian system (4) is completely integrable (and the Hamiltonian H is integrable) if there exists a canonical transformation to angle + action variables, qY p 3 a Y J. Another definition is based on the Liouville theorem on integrable systems: a Hamiltonian system with n degrees of freedom is integrable if n independent integrals in involution are known for this system. Systems with one degree of freedom n 1 are always integrable because their Hamiltonian HqY p E is an integral of motion. A pendulum model is a highly representative example of such systems. Its Hamiltonian can be written as
H p2 А mgl cos j Y 2ml 2 5

In the neighborhood of the points with the coordinates pY j 0Y 2pk, k 0Y Ж1Y Ж2Y F F F Y the family of phase curves consists of ellipses. Such points are therefore called elliptic. The family of trajectories near the points pY j 0Y p 2pk, k 0Y Ж1Y Ж2Y F F F Y is formed by hyperbolas, and such points are called hyperbolic. For an autonomous system with two degrees of freedom, n 2, the integrable system in the angle + action variables JY a has the topology of a two-dimensional torus (Fig. 1b). In view of the constraints o1 o1 J1 Y J2 and o2 o2 J1 Y J2 (if they are present), the frequencies of gyration in circles O1 and O2 for nonlinear systems can vary from torus to torus. Their ratio can also vary: o1 o1 J 1 Y J2 X o2 o2 J 1 Y J2 7

If quantity (7) is rational, o1 ao2 kam (a resonance), the dynamics of the system are periodic: the phase trajectory closes after k windings over the circle O1 and m windings over the circle O2 . If fraction (7) is irrational, o1 ao2 T kam, the phase trajectory covers the torus everywhere densely, and the motion of the system is called quasi-periodic, of almost periodic. Therefore, because J1 and J2 are arbitrary, the phase space is represented by two-dimensional tori, which can be visualized in ordinary three-dimensional space as a set of tori nested in one another, with the major and minor semiaxes specified by the ratios J1 and J2 (Fig. 2).


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Figure 2. General structure of the phase space of an integrable system with two degrees of freedom in the action + angle variables.

For integrable systems with n degrees of freedom, their phase space is 2n-dimensional; in the angle + action variables, it has the topology of a set of n-dimensional tori. Any possible trajectory lies on one of them. Some trajectories may be closed; others cover the corresponding torus everywhere densely. A torus of dimension n 5 2 with given J1 Y J2 Y F F F Y Jn is called resonant if the relation
n i1

ki oi J1 Y J2 Y F F F Y Jn 0 Y

where ki are nonzero integers, holds for a set of frequencies И Й oi J1 Y J2 Y F F F Y Jn i 1Y 2Y F F F Y n.

3.2 Perturbed motion The overwhelming majority of Hamiltonian equations (for systems with n degrees of freedom) are not integrable. But a Hamiltonian H Ha Y J can in some cases be divided into a n i n t e g r a b l e p a r t H0 H0 J a n d a p a r t t h a t i s n o t integrable but can be represented as a small perturbation H1 of H0 :
Ha Y J H0 eH1 Y 8 where H0 H0 J, H1 H1 a Y J, and e is a small parameter. Systems whose Hamiltonians can be written in form (8) are called nearly integrable systems. In the chosen variables a and J, the canonical equations that follow from Eqns (4) and (8) have the form q H1 Y Ji А e qai ai q H0 qH1 e Y qJi qJi i 1Y 2Y F F F Y n Y 9

fore, the system shortly returns to the neighborhood of the resonance. The so-called phase oscillations thus set in. Resonances can arise not only between the system and external influences but also between different degrees of freedom of the system itself, which corresponds to the autonomy of the Hamiltonian H1 in Eqn (8). This is the case of so-called internal resonances. If the resonance is not isolated, the overlapping of resonances results in a very complex motion in the system. Moreover, the resonances prevent finding solutions of the equations using the technique of the canonical perturbation theory. In the perturbation theory, the original system is approximated by a close integrable system subject to a small perturbation, and the solution is sought as an expansion in powers of e. The presence of resonances disrupts the convergence of such series, because this approach implicitly assumes that the original equations are integrable. This is not the case in most situations, however. Even very simple systems may not be integrable, and their dynamics may be very complex at certain initial conditions. For example, signatures of dynamical chaos are manifested in the behavior of a nonlinear pendulum [56, 57]. The perturbation theory cannot describe such a complex behavior, which is formally reflected by the divergence of the series. If the initial conditions of the system correspond to regular trajectories (quasi-periodic motion), such trajectories undergo qualitative rearrangements under the action of perturbations in the neighborhood of the resonances. The theory of nonlinear resonance is remarkable for its ability to obtain an analytic criterion for the onset of irregular motion in a Hamiltonian system. This criterion was originally introduced in [58, 59]. We consider the theory of nonlinear resonance from a more general standpoint, following Refs [60, 61] (see also Refs [54, 62]). A more complete exposition of the theory of resonances is given in [63].

4.1 Small denominators We first consider the internal resonance. We decompose the f u n c t i o n H1 a Y J [ s e e E q n ( 8 ) ] i n t o a F o u r i e r s e r i e s , k H1 a Y J k H1 J exp ika , and substitute this decomposition in Hamilton equations (9). This yields k Jj Аie kj H1 exp ika Y k 10 qH k 1 aj oj J e exp ika Y j 1Y 2Y F F F Y n Y qJj k
k where k is a vector with real integer components, H1 are the Fourier coefficients, and oj J qH0 JaqJj . We seek a solution of perturbed system (10) in the form of a series in powers of the small parameter e: I 0 s J j Jj e s Jj Y s1 11 I 0 s e s aj Y j 1Y 2Y F F F Y n X a j aj s1

where e 5 1. If e 0, system (9) is completely integrable, and its solutions cover n-dimensional tori. We now assume that e T 0. How strong are then the changes in the character of the integrable system?

4. Nonlinear resonance
The answer to the question posed in Section 3.2 essentially depends on the relation between the per turbing force frequency and the proper frequency of the system. If the proper frequency is close to the frequency of the external force, this results in an increase in amplitude, leading to a resonance. But for nonlinear systems with perturbation (8), the frequency depends on the amplitude, and hence the system falls out of resonance some time later. This reduces the amplitude, which, in turn, changes the frequency. There-

Next, using expansion (11) and Eqns (10), we select the terms containing equal powers of e. In the zeroth approximation, we 0 0 then find Jj 0, aj oj J0 , j 1Y 2Y F F F Y n. Therefore, J
0 j

const Y

a

0 j

oj J0 t const X

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Because qoj J0 А Б АБ J oj J0 eJ1 % oj J0 e 0 qJm m
1 m

Y

we can easily obtain the first-approximation equations А Б ~k 1 kj H1 J0 exp ikxJ0 t Y Jj А i
k

km ~ km where H1 are the values of H1 with the inclusion of the c o n st an t t h at ap p e a rs i n t h e z e ro t h o rd e r. T h u s, if t he resonance relation kx А mO % 0 holds, perturbation-theory series (11) diverge; this is the essence of the small-denominator problem. To overcome this difficulty, it was suggested that a canonical transformation be used to pass to special (resonant) variables.

a

1 j



qoj J0
m

qJ

0 m

J

1 m

13 qH k J0 ~ А Б 1 exp ikxJ0 t Y qJ j k

k ~k where H1 are the values of the coefficients H1 with the constant taken into account. Equations (13) can be straightforwardly integrated:

4.2 Universal Hamiltonian Let the perturbation H1 be a periodic function of time with a period T 2pan and let the motion be described by the Hamiltonian
H H0 J eH1 aY JY t Y 16

J a

1 j 1 j

А i

qoj J0 km H k J0 ~ 1
m

kj H k J0 ~ 1 exp ikxt const Y kx k qJm
0 k

14

kx2

exp ikxt

where e 5 1. We decompose the function H1 into a Fourier series: В Г km H1 J exp ika А mnt X 17 H1 aY JY t
kY m

Аi

qH k J0 exp ikxt ~ 1 X qJj kx k

It can easily be seen that if the condition kx k1 o1 k2 o2 F F F kn on % 0 15

Then Hamilton equations (9) become В Г km kH1 J exp ika А mnt Y J Аie
kY m

dH km J В Г 1 a oJ e exp ika А mnt Y dJ kY m
А where H1 k condition Y Аm

18

(called the resonance relation) holds, terms with zero or nearzero denominators appear in Eqns (14). This leads to a s s substantial increase in the corrections aj and Jj , which obviously disrupts the convergence of series (11). If the resonance relation is not satisfied in first-approximation equations (13), it can hold for equations of higher approximations, s b 1. The resonance that manifests itself in the sth order of the perturbation theory is called the order-s resonance. We now let the perturbation H1 depend on time periodically (with a period T 2paO), i.e., H1 a Y JY t H1 a Y JY t T . A treatment similar to that used in the preceding case yields В Г km H1 a Y JY t