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ISSN 1560-3547, Regular and Chaotic Dynamics, 2016, Vol. 21, No. 2, pp. 232­248. c Pleiades Publishing, Ltd., 2016.

Adiabatic Invariants, Diffusion and Acceleration in Rigid Bo dy Dynamics
Alexey V. Borisov* and Ivan S. Mamaev**
Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received Decemb er 12, 2015; accepted January 29, 2016

Abstract--The onset of adiabatic chaos in rigid bo dy dynamics is considered. A comparison of the analytically calculated diffusion co efficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincar´ ­ Zhukovsky problem. e The application of Hamiltonian metho ds to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi's acceleration). MSC2010 numbers: 70F15, 37J30, 37M25 DOI: 10.1134/S1560354716020064 Keywords: adiabatic invariants, Liouville system, transition through resonance, adiabatic chaos

INTRODUCTION We consider a numb er of problems of rigid b ody dynamics which are describ ed by Hamiltonian differential equations with one and a half and two degrees of freedom. These systems were considered by N. E. Zhukovsky and H. Poincar´ in relation to analysis of the motion of a rigid b ody having e cavities completely filled with an ideal incompressible fluid executing a uniform vortex motion. N. E. Zhukovsky found the simplest case of integrability of these equations, however, in the general case the b ehavior of such systems is chaotic. A sp ecial feature of the problems under consideration is that under additional physical assumptions regarding the parameters, the motion can b e divided into fast and slow comp onents (systems with slowly varying parameters are one of the examples of such systems). The onset and the nature of stochasticity in such systems have their p eculiarities. Random jumps in the adiabatic invariant in the course of multiple crossings of the separatrix lead to adiabatic chaos and diffusion of the adiabatic invariant. We have found uncertainty curves on the plane of slow motion, constructed a picture of split separatrices and carried out an analysis of the diffusion equation. 1. HAMILTONIAN SYSTEMS WITH ONE AND A HALF DEGREES OF FREEDOM. JUMPS IN THE ADIABATIC INVARIANT AND ADIABATIC CHAOS Consider a nonautonomous Hamiltonian system with one degree of freedom and slowly varying time, i.e., explicitly dep ending on = t, 1. We note that nevertheless all of the arguments presented b elow hold for autonomous Hamiltonian systems with two degrees of freedom, the evolution of which can b e divided into slow and fast comp onents. Some of such systems are considered in [1, 2]. For the system with one and a half degrees of freedom q=
* **

H , p

p=-

H q

(1.1)

E-mail: borisov@rcd.ru E-mail: mamaev@rcd.ru

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we assume that the Hamiltonian H = H (p, q , ) is an analytical (smooth) function of the variables 1 is q , p, and and p eriodically dep ends on the phase q and slow time = 0 + (0 < a small parameter, 0 0 < 1). The p eriodic time dep endence of the Hamiltonian allows us to define the "slow" Poincar´ map as a map for the p eriod mod 2 t mod 2 . It is also assumed e that for each fixed value of the parameter = 0 , the "frozen" integrable system with Hamiltonian H = H (p, q , 0 ) has separatrices of the same (homo- or heteroclinic) typ e. Equations of typ e (1.1) describ e, for example, a p endulum with slowly varying length and some physical problems arising in plasma theory [3, 4]. The evolution of the system far away from the separatrices is describ ed using an adiabatic approximation and an improved adiabatic approximation, which corresp ond to the first two steps of standard p erturbation theory [5]. If we pass to action-angle variables (I, ) in (1.1) for each fixed 0 ("frozen" system), then the equations of motion b ecome H1 , I = - and the Hamiltonian can b e represented as H = H0 (I, )+ H1 (I, , ). According to the theorem of averaging in one-frequency systems [3], the action variable (I ) undergoes far away from the separatrices a change of order on times 1 , i.e., it is an adiabatic invariant (AI) (according to KAM theory this is valid on arbitrary times if the nonlinearity conditions are satisfied). A more accurate description of the b ehavior of the system (1.2) can b e obtained by p erforming the following step of p erturbation theory. One can define the improved adiabatic invariant (IAI) J as 1 J = I + u(p, q , ), u = - (H1 (p, q , ) - H1 ), 0 where the brackets ... denote averaging over . The equations of motion for the improved adiabatic invariant are H2 . (1.3) J = -2 J Thus, the improved adiabatic invariant far away from the separatrices undergoes oscillations of order 2 . Remark 1. Near the separatrix the adiabatic description of the system is no longer correct and needs to b e made more accurate. Dep ending on the initial conditions, two typ es of b ehavior of the adiabatic invariant are p ossible in the region near the separatrix. In the first case corresp onding to the passage through a resonance, the tra jectory passes it without getting stuck, and the improved adiabatic invariant undergoes a jump of order [5, 6], while in regions far away from the separatrix the change in the improved adiabatic invariant has a value of order 2 . As noted in [7], if the change in the improved adiabatic invariant has b een matched to the phase change, p eriodic stable tra jectories may app ear in this case. The measure of the islands of stability near them, as 0, has order 1, even though it is very small ( 2%). In the second situation [3], which corresp onds to resonance capture, the p oint that has entered the neighb orhood of the separatrix b egins to move so that the arising commensurability is approximately preserved, which leads to a change in the improved adiabatic invariant by a value of order 1 for times of order 1 . However, as shown in [3], as 0, the measure of such tra jectories tends to zero as . Here we consider tra jectories which are not captured into resonance and do not lie in the neighb orhood of p eriodic tra jectories near the separatrix. The measure of such tra jectories is close to complete, so that a probabilistic description can b e applied to them. In [5], formulae for the value of a jump in the improved adiabatic invariant J due to separatrix crossing were obtained. The value of the jump is the function of a random quantity with a given distribution. In the course of multiple crossings of the separatrix, the changes in the adiabatic
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invariant have the character of wanderings with a random step, and a nonzero probability arises of detecting the value of the improved adiabatic invariant, which differs from the initial one by e a magnitude of order 1 after 1 steps of the Poincar´ map. Such stochastic processes in a dynamical system are usually called adiabatic chaos. However, we note that the onset of such chaos is also accompanied by usual effects, namely, splitting of separatrices and app earance of quasirandom oscillations [4, 8]. Adiabatic chaos is characterized by the fact that the region of chaos does not decrease as 0, in contrast to the standard situation [3, 9]. The general formulae for adiabatic invariant jumps in Hamiltonian systems, and their application to various mechanics and physics problems are presented in [5, 6, 10, 11]. In Section 3, we consider a system whose separatrices with fixed = 0 are in the regions shaded in Fig. 1. If the phase p ortrait of the "frozen" system remains symmetric ab out the axis Op (Fig. 3), then for tra jectories crossing the separatrix the value of a jump in the improved adiabatic invariant is calculated from the formula J = - where ( ) =
2

1 ( )a ln 2 sin + O 3/2 | ln | + 2 1-

,

(1.4)

and is the instant of crossing the separatrix, which is defined from the equation I ( ) =
H . a = - det pq conditions by the value should b e regarded as asymmetric regions the -1/2

dS dt

is the rate of change in the area of the region b ounded by one of the separatrices
S ( ) 2

,

The value (0, 1) changes under a small change ( ) in the initial 1, therefore, can b e regarded as a random quantity whose distribution uniform [5] by virtue of the Hamiltonian prop erty of the system. For formula (1.4) b ecomes more complex [5].

Fig. 1. Separatrices of the system (3.4) for = min =

1 4

and = max = 3 . 4

2. ADIABATIC CHAOS IN SYSTEMS WITH TWO DEGREES OF FREEDOM The equations of motion for a Hamiltonian system with two degrees of freedom one of which, (p, q ), is fast and the other, (y, x), is slow can b e written as [10] H , p H , x= y q= H = H (p, q , y , x) is the Hamiltonian function, Since the characteristic time of change of slow time of change of fast variables, the motion on approximation by the Hamiltonian function H = H , q H y = - , x p=-

(2.1)

1. variables is much larger than the characteristic small time intervals can b e describ ed in some H (p, q , y0 ,x0 ) in which the slow variables have
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fixed values. It is assumed that on the plane of fast motion p, q for all x0 , y0 there are separatrices dividing regions with different (regular) b ehavior of tra jectories. Consider the evolution of the initial system (2.1) at a fixed value of energy H (p, q , y , x) = h. As in Section 1, we can apply adiabatic approximation on the plane of fast variables far away from the separatrices to describ e the b ehavior of the system with an accuracy of order , while to achieve the accuracy 2 we have to use an improved adiabatic approximation. With this accuracy the tra jectories in the phase space are orbits of invariant manifolds (tori in the compact case). On the plane of fast variables with fixed x0 , y0 one can pass to action-angle variables, and then in adiabatic approximation I = const. While crossing the separatrix, the adiabatic invariant undergoes a jump (if the regions inside the separatrices are symmetric) J = - 1 ( )a ln 2 sin + O 3/2 - ln + 2 1- . (2.2)

The formulae for calculation of and a are presented in [10]. Adiabatic invariant jumps lead to adiabatic chaos in the system (2.1), which is studied by means of the uncertainty curves introduced in [2, 12, 13]. The evolution of the slow subsystem is governed by the equations x= H , y y = - H , x (2.3)

where H dep ends on the variables I , x and y . In adiabatic approximation I = const, and the tra jectories of the slow subsystem on the energy level h are defined by I (y, x, h) = const. (2.4) Note that each p oint on the plane of slow variables with fixed h corresp onds to a tra jectory on the plane of fast variables, which is defined by the equation H (p, q , y , x) = h. Among these tra jectories there may b e separatrices of the "frozen" system. The equation for defining the p oints (x, y ) to which the separatrices corresp ond has the form H (p (y, x),q (y, x),y ,x) = h, (2.5) where p , q is an arbitrary p oint on the fixed separatrix (the dep endence on x, y arises from the fact that the location of the separatrix itself dep ends on x, y ). The curve on the plane x, y which is defined by (2.5) is called the uncertainty curve. One example of the uncertainty curve is given in [2], where the equations of the restricted three-b ody problem near a 3 : 1 resonance are analyzed in relation to the study of formation of the Kirkwood gaps. When additional restrictions are imp osed on the system parameters in the Poincar´ Zhukovsky e­ equations, one can distinguish b etween fast and slow comp onents and construct uncertainty curves (see Section 4). The result of an adiabatic invariant jump due to separatrix crossing on the plane of fast motions is that on the plane of slow variables the p oint crosses the uncertainty curve and passes from one tra jectory to another tra jectory defined by the equation I (y, x, h) = I0 +I , where the value of the jump is given by the expression (2.2). Irregular jumps in the adiabatic invariant lead to adiabatic chaos whose regions on the plane of slow variables x, y are b ounded by the phase curves I (y, x, h) = const. In addition, the curves I (y, x, h) are tangent to the uncertainty curve. A typical example of such a region is shown in Fig. 2. 3. THE DYNAMICS OF A RIGID BODY WITH SLOWLY VARYING PARAMETERS The Liouville equations [1] are 1 H , H = (M , AM ) - (M , K ), (3.1) M =Mâ M 2 where A = diag(a1 ,a2 ,a3 ) is a matrix with the elements ai = 1/Ii (Ii are the principal moments of inertia), and the vector K = (K1 ,K2 ,K3 ) describ es the constant gyrostatic momentum in the body.
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Fig. 2. The phase p ortrait of the slow subsystem of the system (5.7) for h = 0.4, b = 0.5,
c 2 -c 1 c 3 -c 1

G2 c 3 -c 1

= 0.1,

= 0.5, = 0.05.

Remark 2. Recall that if A and K do not dep end on time, one obtains the Zhukovsky ­ Volterra system [14]. In the case of explicit time dep endence of A(t) and K (t), Eqs. (3.1) were studied by J. Liouville, who also noticed a numb er of the simplest integrable cases. A more complete integrability analysis and the proof of nonintegrability of the system in the case of p eriodic time dep endence of A(t) and K (t) by the method of splitting of separatrices is presented in [15]. In [15], the time dep endence was not slow and the system did not differ much from the integrable one.
Here we consider a situation in which A and K slowly vary in time with p eriod 2 . Eqs. (3.1) 2 = G2 if we use the change of can b e written in canonical form on the surface of the integral M variables

M1 = G 1 - p2 sin q,

M2 = G 1 - p2 cos q,

M3 = Gp.

(3.2)

If we also normalize the time and parameters by the formulae dt = (a3 - a1 )dt, = (a2 - al )/(a3 - a1 ), vi = ai Ki (a3 - a1 ), then we obtain a canonical system with the Hamiltonian 12 1 p + (1 - p2 )cos2 q - v1 1 - p2 sin q - v2 1 - p2 cos q - v3 p. (3.3) 2 2 In the case of absence of gyrostatic momentum K = 0, the Hamiltonian (3.3) simplifies to H= H= 12 1 p + (1 - p2 )cos2 q, 2 2 1 2 = ( ) = (0 + t). (3.4)

Assume that the parameter in (3.4) varies according to the law = 1+ 1 cos 2 . 2 (3.5)

For every = 0 the phase p ortrait of the "frozen" system contains symmetric separatrices (Fig. 3). The dep endence of the adiabatic invariant on the slow time is shown in Fig. 4 (by virtue of symmetry the action variables coincide in the regions G1 and G2 , and also in G3 and G4 ). The heavy line indicates the curve of adiabatic invariant change for the separatrix of the "frozen" system. The motion far away from the separatrix is regular, and, to order , I = const, and the region of chaos corresp onds to the segment [Imin ,Imax ] (in this case Imin = 1/3, Imax = 2/3). For the Poincar´ map T constructed by sampling every seconds, in the region b ounded by the curves e I = Imin and I = Imax the system exhibits stochastic b ehavior, and the b ehavior of the adiabatic invariant is characterized by jumps with a random step of order . The value of a jump in the improved adiabatic invariant in the case of a single separatrix crossing is given by the expression 1 (3.6) J = - a( )ln 2 sin + O 3/2 | ln | +(1 - )-1 , 2
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Fig. 3. The phase p ortrait of the "frozen" system (3.4) for = 0.5.

Fig. 4. Dep endence of action on time for individual tra jectories from various regions. (The p eriodic curve corresp onds to the action for the separatrix of the "frozen" system.)

where (t) = tonian (3.4), a

2 is (1-) = 1 (1-)

the velocity of increase in the area under the separatrix for Hamil.

Since the tra jectory crosses the separatrix twice in one p eriod in , so that the second time the quantity ( ) has the same value with opp osite sign, the total jump in the improved adiabatic invariant is, to order , J = - sin 1 sin 1 ( ) a( ) ln ln = - . 2 sin 2 2 (1 - ) sin 2 J = 0, (J )
2

(3.7)

Note that the averages over the random variables 1 and 2 are equal to

a2 2 ( ) = 2 2
2 0

1

ln2 (2 sin ) d =

a2 2 ( ) . 24

(3.8)

Below we present the results of numerical analysis of the Poincar´ map and jumps in the improved e adiabatic invariant under the condition (A = A(t) and K = 0). The phase p ortraits for 0 = 0.5 and various are shown in Figs. 5a­5c. It is seen that as increases, the region of chaos decreases, but remains b ounded by the curves I (p, q , 0 ) = Imin . The onset of chaos is due to a random change in the improved adiabatic invariant at the instant of crossing the separatrix. The change in the improved adiabatic invariant over the time interval 1/ dep ending on the initial conditions is shown in Fig. 6. In this figure, the initial value of the improved adiabatic invariant J0 is plotted on 2 the axis x, and the jump J is plotted on the axis y . On crossing the separatrix, the improved adiabatic invariant changes by a value of order , unless a resonance capture occurs. These
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J 2 J 2

instants corresp ond to low p eaks on the dep endence

(J0 ). If the tra jectory was captured into

resonance, then J 1 and the dep endence (J0 ) exhibits sharp maxima. The width and height of the maximum are determined by the parameters and initial conditions of the system (3.4). In some cases, for the tra jectories that are almost the same at the initial instant, the value of the improved adiabatic invariant may b e significantly different after crossing the separatrix (see Fig. 7).

Fig. 5. The phase p ortrait for the "slow" Poincar´ map of the system (3.4) at = 0.01 (a), = 0.05 (b) and e = 0.001 (c).

Fig. 6. A change in the improved adiabatic invariant over the time interval J0 . The digit 1 indicates the analytical dep endence (3.8).

1

dep ending on the initial value

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Fig. 7. Demonstration of how the system (3.4) with almost the same initial conditions can b e significantly different at the instant t 1 after passage through the separatrix.

Figure 6 also shows the analytical function (3.8), which maps a change in the improved adiabatic invariant averaged over (see (1.2)) [5, 16]. As is seen, it is smooth and does not reflect completely the actual b ehavior of the system. At the same time, the use of numerical methods allows one to explore in detail the evolution of the system dep ending on the entire set of initial conditions with fixed parameters. Figure 8 shows a typical view of the surface describing the change in the improved adiabatic invariant over the time interval 1/ dep ending on the initial value J0 and the initial phase 0 . As is seen, this surface has a very complex pattern and cannot b e describ ed analytically. It may also b e noted that in most real physical systems even small values of are sufficiently large to correctly use the formulae (3.8). The ab ove example of investigation of jumps in the improved adiabatic invariant is one of p ossible numerical descriptions of systems with slowly varying parameters. An alternative description is based on the analysis of conditions for splitting of separatrices [1].

Fig. 8

4. SPLITTING OF SEPARATRICES AND THE CONDITIONS FOR ADIABATIC CHAOS We now consider the case where K = 0. For simplicity we consider the case where K1 = K3 = 0, K2 = K ( ) = K (t). The phase p ortrait of the "frozen" system (the integrable Zhukovsky ­ Volterra case) is shown in Fig. 9. For more details on the dynamics of an integrable system in the case considered, see [17, 18]. The conditions for the onset of adiabatic chaos which result from adiabatic invariant jumps and from conditions obtained by the method of splitting of separatrices (Section 4)
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Fig. 9. The phase p ortrait of the "frozen" system (4.2) for = 0,1, 0 = 0,5.

Fig. 10. The phase p ortrait of a "slow" mapping of the Poincar´ system (4.2) for = 0,1, e = 0,5(1 + 0,3cos 2 ), = 0,05, 0 = 0.

hold for a system with the Hamiltonian 1 1 (4.1) H = p2 + 1 - p2 cos2 q - 1 - p2 cos q. 2 2 Figure 10 shows a typical phase p ortrait of the system (4.1). As in the previous case, the regions of chaos b ounded by the curves I (p, q , 0 ) = Imin and I (p, q , 0 ) = Imax are well distinguishable. As shown in [4, 8, 19], necessary conditions for splitting of separatrices of the Poincar´ map of e the system (4.1) can b e obtained to first order in from the analysis of the "adiabatic" Poincar´ e­ Melnikov integral. If the equations of motion of the system (4.1) are written as H (p, q , (z ),k(z )), q= p H (4.2) (p, q , (z ),k(z )), p=- q z = , where (z ) and (z ) are p eriodic functions with p eriod 2 , then in the extended phase space the Poincar´ map is given by the section formed by the intersection of the tra jectories with the planes e z = z0 +2n, n Z. According to the results of [8] to first order in , the value of splitting of the separatrices on the plane of the Poincar´ section is the same along almost the entire separatrix of e e the "frozen" system and dep ends on the parameter z0 defining the Poincar´ section. This value is prop ortional to the adiabatic Poincar´ ­ Melnikov function e


MA (z0 ) =
-

H H (p0 (t, z0 ),q0 (t, z0 ),z0 ) - (P (z0 ),Q(z0 ),z0 ) dt, z z

(4.3)

where (p0 ,q0 ,z0 ) is the solution for the separatrix of the "frozen" system. The geometrical meaning of the function (4.3) [8, 19] is that dA (z ), (4.4) MA (z ) = dz where A(z ) is the area under the separatrix of the "frozen" system. To ensure that the separatrices do not split to first order, it is necessary that A(z ) = const for any z . Consider a system with Hamiltonian (4.1). In the case K = 0 (1 = 2 = 3 = 0) the separatrices of the "frozen" system have the form shown in Fig. 1 and the value of the area A(z ) is easily calculated as follows A(z ) = p(H (P (z ),Q(z ),z )) dq = arcsin (z ). (4.5)

The requirement A(z ) = const leads to the condition (z ) = const, which defines the usual Euler ­ Poinsot case. Calculating the integral (4.5) yields
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Fig. 11. Separatrices of the system (4.2) for 0 = - ( = 0.05).

1 4

Fig. 12. Separatrices of the system (4.2) for 0 = 0 ( = 0.05).

Fig. 13. Separatrices of the system (4.2) for 0 = ( = 0.05).

1 4

Fig. 14. Separatrices of the system (4.2) for 0 = 1 in 4 the case where the condition A(z ) = const ( = 0.05) is satisfied.

, (4.6) 2 (1 - ) which shows that when varies slowly p eriodically, the separatrices of the p erturb ed problem are always split and transversally intersect each other. This leads to analytical nonintegrability of the p erturb ed problem and to the onset of quasirandom oscillations and a stochastic layer. The picture of split separatrices for various values of the parameter z0 = 0 is shown in Figs. 11, 12 and 13, in which it is seen that almost along the entire length of the separatrices the distance b etween them is constant and dep ends on z0 , as predicted by the formula (4.3). The phase p ortrait of the "frozen" system for 1 = 3 = 0, 2 = (0 < < < 1) is shown in Fig. 9 (phase p ortrait of the Zhukovsky ­ Volterra system). We note that in this situation the conditions for splitting of different pairs of separatrices do not coincide, and in any case one pair of separatrices will split. The calculation of the area A(z ) for one pair of separatrices leads to the expression a + 1b b2 1 - ln , (4.7) A1 (z ) = 4 arcsin b - b 1 - 2a a - 1-b2 where b = - , a = 1- . (The area for the other pair A2 (z ) is obtained from A1 (z ) by the substitution - ). Figure 14 shows the separatrices of the p erturb ed problem for A1 (z ) = const; it is seen that one pair of separatrices "does not split": for this pair the distance b etween the separatrices is not constant along the length and is prop ortional to 2 , and the other pair remains split prop ortionally to . The picture of chaos when the conditions of "non-splitting" are satisfied is shown in Fig. 15, where it can b e well seen that the stochastic layer near the "unsplit" separatrices has thickness of order . We note that for "unsplittable" separatrices 0, therefore, the ab ove formulae (1.4) and (3.6) for the jump in the improved adiabatic invariant do not hold. We compare the conditions for the onset of chaos obtained from the analysis of splitting of the separatrices with the scenario of development of adiabatic chaos due to irregular jumps in the adiabatic invariant. Figure 16 shows the b ehavior of the adiabatic invariant in the variables I and when the separatrices split, and Fig. 17 shows its b ehavior for A1 (z ) const. These figures MA (z ) =
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Fig. 15. The phase p ortrait of the "slow" Poincar´ map of the system (4.2) for A(z ) = const (0 = 1 , = 0.05) e 4

show that the condition A(z ) const prevents the onset of adiabatic chaos in regions lying outside the -neighb orhood of the separatrix. The chaos in this neighb orhood has characteristic scales of order 2 .

Fig. 16. Dep endence of the action variable for various tra jectories of the system (4.2) for = 0.1, = 0.5(1 + 0.3cos 2 ). (The heavy line indicates the action for separatrices of the "frozen" system.)

Fig. 17. Dep endence of the action variable for various tra jectories of the system (4.2) for A1 (z ) = 0.8.

´ 5. UNCERTAINTY CURVES IN THE POINCARE ­ ZHUKOVSKY SYSTEM As an illustration of the system with two degrees of freedom, we consider the Poincar´ e­ Zhukovsky equations governing the motion of a rigid b ody with an ellipsoidal cavity completely filled with a homogeneous incompressible vortex fluid. These equations also arise in the analysis of free motion of a rigid b ody in the space of constant curvature (in an ideal fluid) and inertial rotation of a 4-dimensional rigid b ody ab out a fixed p oint. In pro jections onto the b ody-fixed axes of the angular momentum vector M and the vector of vorticity describing a "quasi-rigid motion" (in the sense of Helmholtz) of fluid in a cavity), these equations are a Hamiltonian system on the Lie algebra so(4) H , M =Mâ M H , =â

(5.1)

where the Hamiltonian H = 1 (AM , M )+ (BM, )+ 1 (C , ) is a homogeneous quadratic function 2 2 of the variables (M, ). For our further analysis we restrict ourselves to the case where all matrices A, B and C are diagonal.
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In order to divide the motion of the system (5.1) into fast and slow comp onents, we introduce a small parameter by the substitutions M M , , A -1 A, B B, C C. (5.2)

In the new variables the equations of motion can b e written as M = M â AM + M â B, = { â C + â BM }. Note that the variable M is "fast" and is "slow". Hamiltonian form if we redefine = -1 H , M =Mâ M H , = â 1 H = (M, AM )+ ( ,B M )+ 2 1 1 = (M, AM )+ (, BM )+ 2 2 (5.3)

The system (5.3) can b e written in

(5.4) 12 ( ,C ) 2 (, C ).

(5.5)

On the level set of the first integrals M 2 = 1 and 2 = G2 one can introduce the coordinates p, q, z, M1 = 1 = G 1 - p2 sin q, 1 - z 2 sin , M2 = 2 = G 1 - p2 cos q, 1 - z 2 cos , M3 = p, 3 = Gz ,

and Eqs. (5.3) can b e written as H , p H , = z q= with the "Hamiltonian" H= 12 1 G2 p + (1 - p2 )cos2 q + 2 2 c3 - c1 -b
1

H , q H z = - p=- c2 - c1 (1 - z 2 )cos2 c3 - c1 1 - p2 cos q

(5.6)

z2 +

1 - p2 sin q

1 - z 2 sin + b2

1 - z 2 cos + b3 pz .

In the expression for H the new constants are uniquely expressed in terms of the original constants from (5.3). We note that the substitutions (5.2) can b e justified from physical considerations in the study of restricted formulations of the Poincar´ ­ Zhukovsky problem which are characterized e by a small value of the angular momentum of the rigid b ody M as compared with vorticity . We choose b1 = b3 = 0, b2 = b. On the fixed level of energy H = h, the phase p ortrait of the slow subsystem (in adiabatic approximation) whose tra jectories are defined by the equation I (z, , h) const (5.7) has for various h a view presented in Figs. 18­20. When the tra jectory of the slow subsystem is far away from uncertainty lines, it is in the neighb orhood of the curve defined by Eq. (5.7). On crossing an uncertainty line, the adiabatic invariant undergoes a random jump of order and the tra jectory of the system oscillates near the curve I (z, , h) = I0 +I .
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Fig. 18. The phase p ortrait of the slow subsystem 2 of the system (5.7) for h = 0.24, b = 0.5, c G c = 0.1, -
c 2 -c 1 c 3 -c 1
3 1

Fig. 19. The phase p ortrait of the slow subsystem 2 of the system (5.7) for h = 0.4, b = 0.5, c G c = 0.1, -
c 2 -c 1 c 3 -c 1
3 1

= 0.5, = 0.05.

= 0.5, = 0.05.

Fig. 20. The phase p ortrait of the slow subsystem of the system (5.7) for h = 0.6, b = 0.5, = 0.05

G2 c 3 -c 1

= 0.1,

c c

2 3

- -

c1 c1

= 0.5,

Multiple jumps due to the crossing of the uncertainty curve lead to a diffusion of the adiabatic invariant in the region where the tra jectories of the slow subsystem cross the uncertainty curves. In the region where a tra jectory crosses no uncertainty curves, the motion is regular, the improved adiabatic invariant is conserved up to order 2 and almost all tra jectories of the system are the orbits of invariant tori. 6. APPLICATIONS TO NONHOLONOMIC SYSTEMS The theory of changes in adiabatic invariants, which was set forth ab ove and used in the Liouville equations, can also b e applied to investigate the dynamics and control of nonholonomic systems. One of the model problems of nonholonomic mechanics is the problem of a Chaplygin ball controlled by means of two or three noncoplanar rotors [20­22]. The control of this system can b e p erformed not only by means of variable gyrostatic momentum, but also by changing the moments of inertia. Such a system can b e easily thought of as a balanced ball inside which orthogonal rotors are fixed and individual masses move symmetrically relative to the center along straight lines like sliders. As shown in [23­25], if the system parameters are "frozen", i.e., do not change with time, then such a system is conformally Hamiltonian: xi = (x) {xi ,H (x)}, i = 1 ... n,

where x = (x1 ,... ,xn ) are the variables parameterizing the phase space and (x) is a p ositive definite function. This system can b e reduced to a Hamiltonian system by rescaling time as (x)dt = dt . For changing system parameters such a representation is also p ossible. Although in this case the measure and the Hamiltonian (the energy) will not b e invariants, but change with time, the formulas for adiabatic invariants will b e analogous. In particular, let Ik (), k = 1,... ,m be the
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integrals of the "frozen" system, with fixed parameter values = (1 ,... ,s ), and supp ose that the parameters dep end on "slow" time ( ), = t, 1. Then {Ik ,H } = 0 and hence Ik Ik = i ( ) , i i.e., Ik changes slowly with time. We note that there has b een a lot of discussion recently ab out the problem of acceleration in Hamiltonian systems and, in particular, as to whether it is p ossible to accelerate a system by b ounded p eriodic oscillations of the internal mechanism, i.e., to ensure its acceleration on the average. This question is closely related to Fermi's acceleration (see, e.g., [26]) and has b een actively discussed recently using billiards problems as examples [27­30]. As is well known, such an acceleration is imp ossible in analytical systems with one degree of freedom [31]. In the case of nonautonomous systems with two degrees of freedom it b ecomes p ossible to achieve acceleration. For this reason, this issue has received particularly much attention in the physical literature recently in connection with the problem of flat billiards whose b oundaries oscillate p eriodically [27­30, 32]. To date, only nonrigorous theoretical results using, in one way or another, additional assumptions have b een obtained on this problem. For example, it is assumed that if the "frozen" system has several ergodic comp onents which can b e stochastic, then acceleration is p ossible. Particularly rigorous, but unfortunately nonconstructive results were obtained in [26, 33]. The p ossibility of acceleration in a "breathing" elliptic billiard was discussed as early as 1996 in [30]. A sp ecial feature of this problem is that a frozen system is integrable. From a modern p oint of view, this system can exhibit an acceleration that is not exp onential 4 (which is a problem of its own), but this acceleration is suppressed by small dissipative terms, i.e., it is not structurally stable 6. We note that numerical calculations in [29] and more recent publications [34­37] are not completely convincing and, undoubtedly, more detailed and high-precision investigations are necessary. Arguments on additional hyp otheses in which it is assumed, for example, that the phases may b e regarded as random variables do not look convincing either. This does not hold in real systems [38, 39]. We note that the dynamics of a b ody with changing parameters is one of the classical areas of mechanics. A classical problem in this area is that of the motion of a rigid b ody on an oscillating base [40], which generalizes the well-known problem of a p endulum with a vibrating p oint of susp ension. Another dynamics problem in which the parameters are changed in a given (usually p eriodic) manner is to study the dynamics of a system, for example, a rigid b ody, inside which there are material p oints undergoing given motions. In the case of fast oscillations of these p oints, under additional constraints, the averaging procedure applies and the problem reduces to investigating the dynamics in some effective (vibrogeneous, using the expression introduced by Yudovich [41]) p otential. Approximate equations governing the dynamics of a rigid b ody in such an effective p otential were obtained by Markeev [42]. However, in the case of slow parameter changes the averaging procedure holds not for all motions, and jumps and drastic increases in adiabatic invariants are p ossible. These jumps b ecome more pronounced as the numb er of degrees of freedom increases (Arnold's diffusion is a common phenomenon in such systems). An interesting new problem, which is of practical imp ortance as well, is the study of the dynamics of a heavy rigid b ody colliding with a vibrating surface (in the absence of vibration this problem has b een studied in detail from the viewp oint of stability [43]). As is well known, Fermi's acceleration can b e observed in the case of the dynamics of a heavy material p oint (the so-called "gravitational machine" describ ed by Zaslavsky [44]). As mentioned ab ove, problems of a rigid b ody (for example, a spherical rob ot [45­48]) rolling on a plane are of great interest for modern mobile rob otics. Inside the rob ot, p eriodic vibrations of the plane or mass displacements are p ossible. The problem of creating acceleration (and deceleration as well) is of great interest, since this allows one, by minimizing energy exp enditures (i.e., by admitting only b ounded oscillations of internal masses), to accelerate or decelerate the motion. As is well known [49­54], nonholonomic systems are close to weakly dissipative systems (they have alternating divergence and admit no invariant measure in the general case), and the problem
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of acceleration is in many resp ects the main problem of dynamical control of such systems. We note that in the ab ove-mentioned problem of a balanced ball controlled by p oint masses that do not violate the balance, acceleration is imp ossible due to the existence of a vector integral of angular momentum relative to the p oint of contact. Due to the existence of integrals, acceleration is imp ossible in an ideal fluid as well [55, 56], although in the presence of circulation such an acceleration is in principle p ossible [57]. The conception of the hierarchy of dynamics of nonholonomic systems develop ed in [58­60] and elab orated up on in [61­65] shows that in the case of displacement of the center of mass one can observe weakly dissipative b ehavior leading, generally sp eaking, to strange attractors (what is essential here is the presence of involutions or reversibilities defined by the geometry and dynamics of the system). Weakly dissipative billiards are discussed, for example, in [32] and references therein. ACKNOWLEDGMENTS This work was supp orted by the Russian Science Foundation (pro ject 14-50-00005). REFERENCES
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