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ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2006, Suppl. 1, pp. S24­S47. c Pleiades Publishing, Inc., 2006. Original Russian Text c A.V. Borisov, V.V. Kozlov, I.S. Mamaev, 2006, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2006, Vol. 12, No. 1.

On the Fall of a Heavy Rigid Body in an Ideal Fluid
A. V. Borisov1 , V. V. Kozlov2 , and I. S. Mamaev
Received Novemb er 25, 2005 3

Abstract--We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.

DOI: 10.1134/S008154380605004X

1. EQUATIONS OF MOTION AND SPECIAL CASES Let us consider the motion of a rigid b o dy in a homogeneous gravity field in an infinite volume of irrotational incompressible fluid resting at infinity. First let us give general equations of motion of a b o dy in a fluid under the action of an external force field: H H M =Mâ +pâ + K, M p p=pâ H + F, M (1.1)

where F and K are the total force and moment applied to the b o dy. These equations go back to Kirchhoff. If the external forces have the p otential nature, then Eqs. (1.1) supplemented by the equations for directional cosines and for co ordinates of a fixed p oint in the b o dy, can b e written as follows: H H H H H M =Mâ +pâ +â +â +â , M p H H H H p = pâ - - - , M x1 x2 x3 (1.2) H = â H , = â H , =â , M M M H H H x1 = , , x2 = , , x3 = , , p p p
1 2 3

Institute of Computer Science, ul. Universitetskaya 1, Izhevsk, 426034 Russia The Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia Institute of Computer Science, ul. Universitetskaya 1, Izhevsk, 426034 Russia

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where vectors p, M , , , are the pro jections of linear momentum, angular momentum, and unit vectors along axes in the fixed frame of reference on the axes connected with the b o dy; and x1 , x2 , x3 are the pro jections of p osition vector of the origin of moving co ordinate system on the fixed axes. The Hamiltonian of system (1.2) has the form 1 1 (AM , M ) + (BM , p) + (Cp, p) + U, 2 2 µb r b - µ f r f U = µ x3 + (r , ) , µ = µb - µf , r = ; µb - µ f H=

(1.3)

here, A, B, C are symmetric matrices determined by geometry of the b o dy and by its inertial prop erties, µb , µf are the weight of the b o dy and the weight of the displaced fluid, and r b , r f are the p osition vectors of the center of gravity and the center of pressure in moving axes. The case µb = µf (a susp ended b o dy) will b e also studied b elow. By straightforward calculations we can verify that there are three integrals of motion (one of which explicitly contains the time): (p, ) = P1 , (p, ) = P2 , (p, ) + µt = P3 .

This means that linear momentum of the b o dy + fluid system can b e represented in the form p = P1 + P2 + (P3 - µt) , (1.4)

i.e., vector P = (P1 , P2 , P3 ) is the initial impulse (impact, according to Chaplygin) in the fixed frame of reference. By the choice of zero time p oint (for µ b = µf ) and by rotation of the fixed axes we can obtain P2 = P3 = 0. In what follows, we consider this to b e fulfilled. Substituting (1.4) into equations of motion (1.2), we obtain a self-contained system with resp ect to M , , , , which can b e written in the Hamiltonian form: ¯ ¯ ¯ ¯ H H H H M =Mâ +â +â +â , M ¯ ¯ ¯ H H H =â , =â , =â , M M M with the Hamiltonian explicitly dep ending on time: 1 1 ¯ H = (AM , M ) + (BM , P1 - µt ) + C(P1 - µt ), P1 - µt + µ(r , ). 2 2 (1.6)

(1.5)

Remark. Equations (1.2) in various but equivalent forms can b e found in pap ers by V.A. Steklov [17], D.N. Goryachev [5], and S.A. Chaplygin [20]. V.V. Kozlov [11], we b elieve, was the first who reduced them to an elegant nonautonomous form, using the representation (1.4) (in the form of Poincar´ equations). e Let us p oint out some sp ecial cases when Eqs. (1.5) can b e simplified. They are indicated in [11, 19].
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1.1. Motion without an initial impulse [11]. Let the initial impulse b e equal to zero: P1 = 0. The equations of motion for M , in a closed form represent a (nonautonomous) Hamiltonian system on e(3) (see b elow) with the Hamiltonian 1 1 ¯ H = (AM , M ) - µt(BM , ) + µ2 t2 (C , ) + µ(r , ). 2 2 If, in addition, the b o dy has three planes of symmetry intersecting in the center of gravity, then the Hamiltonian can b e simplified further: B = 0, r = 0. 1.2. Susp ended b o dy [19]. In [19], Chaplygin also indicated the case when the gravitation is balanced by the Archimedean force (an average density of a b o dy is equal to the density of fluid, and, hence, µb = µf ), however, the center of gravity of the b o dy do es not coincide with the center of gravity of the displaced fluid volume. Thus, the b o dy is under the action of a pair of forces, and its total linear momentum in the fixed frame of reference is conserved, i.e., p = P1 + P2 + P3 , where P = (P1 , P2 , P3 ) = const. As ab ove, by the choice of fixed axes we can obtain the equality P2 = 0. Thus, in this case the evolution of vectors M , , , can b e describ ed by an autonomous Hamiltonian system with the Hamiltonian function 1 1 ¯ H = (AM , M ) + (BM , P1 + P3 ) + C(P1 + P3 ), P1 + P3 + µb (r , ), 2 2 where r is the vector connecting the center of gravity of the b o dy with the center of pressure. If the initial impulse is directed along the vertical axis, p = P , then the evolution of vectors M , ( is directed along the field of gravity) is describ ed by a system with Poisson's bracket which is determined by the algebra e(3) (i.e., {M i , Mj } = ij k Mk , {Mi , j } = ij k k , {i , j } = 0), and by the Hamiltonian function 1 1 ¯ H = (M , AM ) + P (BM , ) + P 2 (C , ) + µb (r , ). 2 2 We have equations of motion in the explicit form H H M= âM + â , M H . M

(1.7)

=â

In [19] Chaplygin indicated the case when Eqs. (1.7) are integrable with an additional integral of fourth degree in comp onents of an angular momentum. The form of the integral is similar to the Kovalevskaya integral.
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Fig. 1.

1.3. Plane-parallel motion [5, 9, 17, 20]. The plane-parallel motion of a rigid b o dy is determined by the invariant relations M 1 = M2 = 0, 3 = 3 = 0. It is easy to show that the dynamical symmetry of the b o dy with resp ect to the considered (invariant) plane is a necessary condition for the existence of such motions. This leads to the relations b
11

=b

22

=b

33

=b

12

= 0,

c

13

=c

23

= 0.

In addition, it can b e shown that in this case, by cho osing the axes connected with the b o dy, one can obtain the equality B = 0 and the diagonal matrix C. Let an angle of rotation of the moving axes relative to the fixed axes b e clo ckwise countered, as it is shown in Fig. 1, then for the unit vectors of fixed axes we have 1 = sin , 2 = - cos , 1 = cos , 2 = sin .

For an angle of rotation we obtain the nonautonomous second-order equation
2 a3 = (c1 - c3 ) µ2 t2 sin cos + P1 µt cos 2 - P1 sin cos + µ(x sin - y cos ), ¨

(1.8)

where c1 , c3 , a3 are corresp onding elements of diagonal matrices, and r = (x, y , 0). For a balanced b o dy (x = y = 0) without an initial impulse (P 1 = 0) we obtain the remarkably simple equation µ2 (c1 - c3 ) = k t2 sin cos , k = ¨ . (1.9) a3 Remark. In [9, 10, 15] this equation is called the Chaplygin equation. In 1890, Chaplygin, b eing a student, obtained it together with other interesting results. However, he refrained from publishing it. The p ossible reason was that he could not integrate this equation explicitly. Nevertheless, this work was included in the collected works by Chaplygin, first published in his lifetime (1933, [20]). Equation (1.9) was also obtained by Goryachev (1893) [5] and Steklov (1894) [16, 17] indep endently. The latter also observed the simplest prop erties of solutions of the equation. In particular, Steklov showed that while a b o dy falls down, the amplitude of its oscillations with resp ect to the horizontal axis decreases, and the oscillation frequency grows. Steklov drew this conclusion in the supplement to his b o ok [17]. In [17], analyzing the asymptotic b ehavior of a b o dy, he made a series of inaccuracies. The Steklov problem [16, 17] ab out the asymptotic description of
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b ehavior of solutions of the equation was solved by Kozlov [9], who showed that under almost all initial conditions, the motion of a b o dy approaches the uniformly accelerated fall by the wide side upward and it oscillates around the horizontal axis with the increasing frequency of order t and 1 the decreasing amplitude of order . A numerical analysis of asymptotic motions with a different t numb er of half-turns can b e found in [23]. Analytic expressions for the asymptotics of a fall were obtained in various forms in [15, 23]. A phenomenon of emerging is describ ed and studied in [7]. Under conditions of a vortex-free flow around a b o dy it is assumed that at the initial moment the wide side of the b o dy is horizontal and the b o dy acquires a horizontal velo city. At subsequent moments the b o dy b egins to submerge. However, if its apparent additional mass in the transversal direction is sufficiently large, then, further, the b o dy abruptly emerges by the narrow side upward, rising to a greater height than at the initial moment. 1.4. The motion of an axially symmetric b o dy (circular disk). There is an imp ortant sp ecial case when system (1.6) has the additional (autonomous) linear Lagrange integral M3 = const, which exists under the condition of the axial symmetry of the b o dy. We can cho ose the moving axes so that A = diag(a1 , a1 , a3 ), B = diag(b1 , b1 , b3 ), C = diag(c1 , c1 , c3 ), r = (0, 0, z ).

In this case, the evolution of pro jections of the angular momentum on the fixed axes N = (M , ), (M , ), (M , ) and of the vector n = (3 , 3 , 3 ) of the axis of symmetry is describ ed by the Hamiltonian system on e(3) ¯ ¯ H H N= âN + â n, N n n= ¯ H â n. N (1.10)

After removing of nonessential terms, the Hamiltonian (1.6) can b e written in the form 1 1 ¯ H = a1 N 2 + b1 (P1 N1 - µtN3 ) + M3 (b3 - b1 )(P1 n1 - µtn3 ) + (c3 - c1 )(P1 n1 - µtn3 )2 + µz n3 . (1.11) 2 2 The tra jectory of the origin of the moving co ordinates C can b e obtained from the equations x1 = b1 N1 + (b3 - b1 )M3 n1 + P1 c1 + (c3 - c1 )n
2 1

- µt(c3 - c1 )n1 n3 , (1.12)

x3 = b1 N3 + (b3 - b1 )M3 n3 + P1 (c3 - c1 )n1 n3 - µt c1 + (c3 - c1 )n2 . 3 If the gravitation is balanced by the Archimedean force, then the Hamiltonian is indep endent of time: 1 1 2 ¯ H = a1 N 2 + b1 P1 N1 + M3 (b3 - b1 )P1 n1 + (c3 - c1 )P1 n2 + µb z n3 . (1.13) 1 2 2 In the case of the zero initial (horizontal) impulse, i.e., if P 1 = 0, there is one more additional integral N3 = (M , ) = const.
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x2 = b1 N2 + (b3 - b1 )M3 n2 + P1 (c3 - c1 )n1 n2 - µt(c3 - c1 )n2 n3 ,


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Moreover, for the nutation angle ( 3 = cos ) we obtain (by analogy with the plane-parallel motion) the nonautonomous second-order equation [20] ¨ (M3 cos - N3 )(M3 - N3 cos ) + (c3 - c1 )µ2 t2 sin cos - M3 (b3 - b1 )µt sin + µz sin . = sin3 (1.14) If the b o dy b egins motion from a state of rest, then M 3 = N3 = 0, and we obtain a
-1 1

a

-1 1

¨ = (c3 - c1 )µ2 t2 sin cos + µz sin .

1.5. An analog of the Hess case. Except for the cases considered ab ove there is another situation when system (1.5), (1.6) admits an invariant relation similar to the Hess case in Euler­ Poisson equations. For its existence it is necessary that the surface of the b o dy b e axially symmetric, and the axis of symmetry b e p erp endicular to the circular cross section of a gyration ellipsoid (i.e., a surface determined by the equation (x, Ax) = 1). Let one of axes connected with the b o dy b e directed along the axis of symmetry of the b o dy surface, and the other two axes b e directed so that a23 = 0, then in the Hamiltonian (1.6) a1 0 a13 A = 0 a1 0 , B = diag(b1 , b1 , b3 ), a13 0 a3 M3 = 0. The equation describing the nutation angle coincides with (1.14) if M 3 = 0, i.e., a
-1 1 2 2 ¨ N cos + (c3 - c1 )µ2 t2 sin cos + µz sin . = 3 3 sin



C = diag(c1 , c1 , c3 ),

r = (0, 0, z ).

Under such a choice of the moving frame of reference, the invariant relation has the simplest form (1.15)

(1.16)

The difference from the Lagrange case analyzed ab ove app ears in the equations defining the evolution of angles of precession and prop er rotation. In [3], an analog of the Hess case for Eqs. (1.5), (1.6) was first p ointed out. 2. THE MOTION OF AN ISOTROPIC BODY Let us consider the simplest particular case when the equations can b e solved by quadratures. The case was p ointed out by Steklov [17, 18]. Here A = diag(a1 , a2 , a3 ), B = bE, C = cE, r = 0,

i.e., the added mass tensor is spherical, however, the b o dy do es not have three planes of symmetry, since B = 0. (If B = 0, then the motion is trivial: the center of gravity describ es a parab ola, and the motion of ap exes , , is the same as in the Euler­Poinsot case.) We isolate equations describing the evolution of angular momentum in the moving frame of reference. They are identical to those in the Euler­Poinsot case: M = M â AM .
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Nevertheless, in order to determine the tra jectory of the center of gravity, it is more convenient to rewrite the equations of motion in the fixed frame of reference: N1 = bµtN2 , N2 = -bµtN1 - bP1 N3 , x2 = bN2 , N3 = bP1 N2 , (2.1)

x1 = bN1 + cP1 ,

x3 = bN3 - cµt,

where N = (, M ), ( , M ), ( , M ) is the angular momentum in the fixed frame of reference. It is obvious that the squared angular momentum gives the integral of motion: M 2 = N 2 = const. If the initial impulse is equal to zero: P 1 = 0, then the first three equations in (2.1) are integrable in terms of the elementary functions: N1 = A sin(bµt2 /2 + 0 ), N2 = A cos(bµt2 /2 + 0 ), N3 = const,

where A, 0 are arbitrary constants. The b o dy moves along the vertical axis with a constant cµt2 acceleration: x3 = - , and the pro jection of the tra jectory on the plane x 1 , x2 is a spiral which 2 is describ ed by the Fresnel integrals and converges to some fixed p oint on the plane. For large times, there holds the asymptotic representation of the form x1 = x 0 - 1 A cos(bµt2 /2 + 0 ) + O (t-3 ), µ t A sin(bµt2 /2 + 0 ) x2 = x 0 + + O (t-3 ). 2 µ t

If P1 = 0, then the equations for N are nonintegrable in terms of the elementary functions. Moreover, on the plane x1 , x2 a drift app ears along the axis O x1 with the sp eed cP1 . 3. QUALITATIVE ANALYSIS OF THE PLANE-PARALLEL MOTION It was shown ab ove that for a sp ecial choice of the moving axes when the kinetic energy is diagonal the angle b etween the vertical axis and the axis connected with the b o dy (Fig. 1) is describ ed by Eq. (1.8), and the motion of the origin of the moving system C (Fig. 1) is describ ed by the equations X = (, Cp) = P1 (c1 sin2 + c2 cos2 ) - µt(c1 - c2 ) sin cos , Y = ( , Cp) = P1 (c1 - c2 ) sin cos - µt(c1 cos2 + c2 sin2 ). (3.1)

Remark. Equation (1.8) corresp onds to a nonautonomous Hamiltonian system with one degree of freedom. Such systems are studied in more detail in the case when the Hamiltonian is a p erio dic function of time. In the general case, they demonstrate a chaotic b ehavior. At the same time, as will b e shown b elow, the dep endence of angle on time t for system (1.8) is of asymptotic character. First consider the "simplest" case when a balanced b o dy (x = y = 0) falls without an initial impulse (P1 = 0). Then, after the change 2 = , Eq. (1.8) takes the form ¨ = k t2 sin , k= µ2 (c1 - c2 ) . a3 < 2 .
Suppl. 1 2006

(3.2)

In what follows we assume that c1 > c2 , i.e., k > 0, and 0

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3.1. Stationary (equilibrium) solutions. Small oscillations. Biasymptotic solutions. Equation (3.2) has the simplest "equilibrium" solutions of the form (t) = const: (1) = 0, (2) = . (3.3) µc1 t2 ), 2

µc2 t2 and the second one corresp onds to that by the wide side (X = X 0 , Y = Y0 - ). Indeed, since 2 c-1 < c-1 , the angle = n if the axis O x is vertical, and = + n if the axis O y is vertical. 1 2 2 Linearizing Eq. (3.2) near the fixed p oints (3.3), we obtain (1) (2) ¨ = k t2 , ¨ = -k t ,
2

The first solution corresp onds to the fall by the narrow side downward (X = X 0 , Y = Y0 -

= , = - . 2 k t /2 2 k t /2

The general solution of these equations is expressed in terms of Bessel function (1) (2) where of the one is Bessel I (x), K (x) are first kind. Thus, (asymptotically) functions J , Y (t) = (t) = t C1 I
1/4 1/4

2 k t /2 + C2 K 2 k t /2 + C2 Y

1/4

, (3.4) ,

t C1 J

1/4

Bessel functions of the second kind, and J (x), Y (x) are Bessel functions in the linear approximation the first solution is unstable, and the second stable with resp ect to , but not to . Indeed, using the asymptotics of for large values of the argument, we find A sin k t2 /2 + t
0

(t) =

+O t

-5/2

,

A = const.

Consequently, the amplitude of oscillations decreases similarly to t -1/2 , and their frequency increases infinitely similarly to t. As mentioned ab ove, this fact was noted in [9]. As shown in [9], using variational metho ds, one can prove that there exist two solutions (t), (t0 ) = 0 asymptotic to the unstable p osition of equilibrium ( = 0), which approach it from different sides. In addition, b ecause of the invariance of Eq. (3.2) with resp ect to the change t -t, there exists a solution (t) with the initial data (0) = for which [9] (t) + (-t) = 2 ,
t-

lim (t) = 0,

t+

lim (t) = 2 .

Thus, the solution (t) is biasymptotic (there is also a similar biasymptotic solution passing around the circle [0, 2 ] contrariwise). Here the b o dy makes one half-turn. Its tra jectory is describ ed by Eqs. (3.1) and shown in Fig. 2a. Note that the upp er p oint of the tra jectory is a cusp p oint: the equation of the curve near this p oint has the form Y = X 2/3 , = const. In Fig. 2b the change of angle for this biasymptotic solution is shown. The existence of biasymptotic tra jectories with an arbitrary numb er of half-turns was proved in [22].
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(a)

(b)

Fig. 2. The tra jectory of the body and the value of angle depending on the coordinate X for the a3 biasymptotic solution with k = 1, = 0.1 (P1 = 0); the upper point of the tra jectory is singular (see µ text).

3.2. The asymptotic b ehavior of solutions of the Chaplygin equation. As shown in [9] (the idea of the pro of in a more general case is presented b elow), for all solutions of the equation, either 0 or as t ± (i.e., the asymptotic motion of the b o dy is the fall by the wide or narrow side forward). There is a hyp othesis stated by Kozlov [9] that the measure of tra jectories which tend to unstable equilibrium state = 0(mo d2 ) as t ±, is equal to zero, and thus almost all tra jectories tend to lim (t) = (mo d2 ) (i.e., to the fall by the wide side forward).
t

(a)

(b)

Fig. 3. Domains of the phase plane corresponding to the initial conditions with t0 = 0, when the body makes the same number of half-turns while t changes from 0 to + in the case (a), and while t changes from - to 0 in the case (b) (k = 1, the white colour corresponds to an even number of half-turns, the black one corresponds to an odd number).

3.3. Numerical analysis. On the basis of the statement ab out the asymptotic b ehavior, a numerical analysis can b e applied to Eq. (3.2) [23]. For this, on the phase plane ( , ) (more
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ing unstable equilibrium = fill two-dimensional surfaces. In addition, there is also a countable set of biasymptotic solutions that differ by the numb er of half-turns made when t varies from - to +. In Fig. 4, the tra jectories of a b o dy for the biasymptotic motions with one and three half-turns is shown.

precisely, on the cylinder â [0, 2 ) â (-, +)) at the initial moment of time t = t 0 domains are constructed where the b o dy makes the same numb er of half-turns as t + (or as t -) b efore it will b e "attracted" to the solution = . As seen in Fig. 3a, these domains are lo cated regularly, moreover, their width decreases while | | increases so that for large initial | | only the probability of the fall of the b o dy by the "upp er" or "under" side as t + makes sense. Boundaries of the domains are filled by initial conditions corresp onding to motions asymptotically approaching the unstable p ositions of equilibrium = 0, 2 . Analogously, one can construct domains corresp onding to the same numb er of half-turns as t - (Fig. 3b), moreover, the domains for t + turn out to b e the mirror images ab out the line = of the domains for t -. If we overlay these domains, their b oundaries are intersected at p oints from the line = . Biasymptotic solutions of Eq. (3.2) with different numb er of half-turns corresp ond to them. Remark. On the cylinder â [0, 2 ) â (-, +), all b oundaries of the domains are glued together in one smo oth curve similar to a screw line whose step decreases when | | increases. Domains with an even numb er of half-turns lie on one side of this line, domains with an o dd numb er lie on the other. Thus, the numerical computations confirm the conjecture that for almost all solutions (t) - - , moreover, in the three-dimensional space t, , , solutions asymptotically approach--
t±

Fig. 4. The tra jectory of a body in the case of biasymptotic solutions with one (the dotted line) and three (the solid line) half-turns for k > 0, a3 /µ = 0.1. There is a singularity in the upper point of the tra jectory.

3.4. The tra jectory of the b o dy. Substituting the asymptotic decomp osition for small oscillations (3.4) into Eq. (3.1), after integration we obtain the asymptotic representation for the tra jectory of the motion in the form cos X (t) = A 2 k t /2 + t
0

+O t

-3/2

,

Y (t) = -µc2 t2 + O t

-1/2

,

where A, 0 are some constants. Therefore, the tra jectory of the motion for large times is close µc2 to the sinusoid with the constant step y = and a decreasing amplitude [9]. (Step Y is k
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calculated b etween two consecutive zeros of the function X (t).) The typical tra jectory is shown in Fig. 5.

Fig. 5. The typical form of tra jectory of a body falling without an initial impulse.

3.5. General case (P1 = 0). Now let us review the main qualitative features of the b ehavior of system (1.8), (3.1) in the general case. If P 1 = 0, there are no longer time-indep endent solutions similar to (3.3). A statement ab out the asymptotic b ehavior also holds in this case. According to this statement, for any solution (t) of Eq. (1.8) we have 1. lim (t) = n or
t+

2. lim (t) =
t

+ n, 2

n Z.

+ n 2 (i.e., the motion of the b o dy approaches the fall by the wide side downward) [9]. Numerical exp eriments confirm this. It also seems that almost al l solutions of the equation tend to a solution of the form (t) =

Fig. 6.

Similarly, we can p erform a computer analysis, considering those domains on the phase plane at the initial moment t = t0 to which there corresp onds the same numb er of half-turns b efore the
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tra jectory will b e attracted to the solution =

as t + (Fig. 6). The b oundaries of these 2 domains are filled with asymptotic solutions. As in the case P 1 = 0, domains corresp onding to a different numb er of half-turns for t 0 = 0 and t - turn out to b e the mirror image of the domains for t0 = 0 and t + ab out the line = . The intersection p oints of b oundaries of 2 the domains as t + and t - corresp ond to biasymptotic solutions. The typical form of the tra jectory of a b o dy thrown at an angle to the horizon is shown in Fig. 7. In Fig. 8 the tra jectories are given in the case of biasymptotic motions with one and three half-turns of the b o dy. As shown in [9], in the general case, the asymptotic tra jectory of the b o dy is a parab ola: X (t) = -P1 t + o(t), Y (t) = - µt2 + o(t2 ). 2c3

Fig. 7. The typical form of the trajectory of a rigid body thrown at an angle to the horizon.

Fig. 8. The tra jectories of a body for biasymptotic motions with one (the upper curve) and three half-turns.

4. A BODY WITH THREE PLANES OF SYMMETRY As ab ove (for the plane-parallel motion), b efore studying the general system (1.6), we consider in detail the sp ecial case of motion without an initial impulse (P 1 = 0) under the additional constraints B = 0, r = 0. (4.1) Here we obtain a nonautonomous Hamiltonian system (on e(3)) for M , with the Hamiltonian 1 1 ¯ H = (AM , M ) + µ2 t2 (C , ). 2 2 (In the general case, we can assume that A is diagonal and C is arbitrary symmetric.) 4.1. Time-indep endent (equilibrium) solutions and "normal oscillations." equations of motion for system (4.2) have the form M = M â AM + µ2 t2 â C , = â AM
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(4.2)

The

(4.3)

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and admit the simplest solutions of the form M = 0, = ± i , i = 1, 2, 3, (4.4)

where i are eigenvectors of the matrix C (for C degenerate, there are infinitely many eigenvectors i ). Linearizing system (4.3) near solution (4.4), by linear transformations of co ordinates one can reduce the equations of motion to the form of "normal oscillations" xk + t2 k xk = 0, ¨ k = 1, 2, (4.5)

where xk are corresp onding lo cal co ordinates near the fixed p oints = i . Solutions of system (4.5) can b e expressed in Bessel functions (see (3.4)). If all eigenvalues of C are different, then to 1 the lo cal minimum of the function V ( ) = ( , C ) there corresp onds an (asymptotically) stable 2 solution of system (4.5) whose asymptotic form for large t is (3.4). The lo cal minimum is defined by some eigenvector of the system. Unstable (now in the linear approximation) solutions corresp ond to other two eigenvectors. 4.2. The asymptotic b ehavior of solutions. It turns out that similarly to the planeparallel case, under arbitrary initial conditions the vector tends to one of eigenvectors of the matrix C. Indeed, in [11] it is shown that for any solution (t) of Eqs. (4.3)
t

lim V (t) = Ec ,

1 ( , C ), and Ec is a critical value of function V ( ). 2 The pro of of this statement is based on a representation of Eqs. (4.3) in terms of the new time and new variables. Let us change time and variables by the formulas where V ( ) = 12 t = , 2 here the equations of motion have the form tM = m;

dm 1 d = - m + m â Am + µ2 â C , = â Am. (4.6) d 2 d 1 It is easy to show that div v = - , i.e., in fact, system (4.6) describ es the Kirchhoff equations 2 with dissipation decreasing with time. Consider the energy of "unp erturb ed" system 1 1 E = (m, Am) + µ2 ( , C ). 2 2 Calculating the derivative E along solutions (4.6), we find dE (m, Am) =- . d 2 From this equality it easily follows that (1) E - - E = const; -
t

(4.7)

(2) the integral I =
0

m( ), Am( ) d converges. 2
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The pro of is reduced to showing that E = Ec is the critical value of (4.7), and hence, of V It turns out that the assumption E = Ec contradicts the convergence of the integral I . For the fall of an arbitrary b o dy with three planes of symmetry, there also exists min for almost al l solutions of Eqs. (4.3) . Thus, as t , conjecture [11] that E = Ec b o dy almost always tends to o ccupy a p osition in space such that the axis corresp onding to maximal added mass b ecomes vertical.

( ). the the the

4.3. Computer analysis. The statement ab out the asymptotic b ehavior formulated ab ove leads to the natural question ab out the structure of domains (basins of attraction) corresp onding to different asymptotic mo des as t ± in the space of initial conditions. Let us cho ose t 0 = 0 and parametrize the joint four-dimensional level of the integrals (M , ) = c = const, 2 = 1 (4.8)

by Andoyer variables (L, G, l, g ) and fix the surface of initial conditions for t 0 = 0 by the equations g = g0 , 1 E = (M , AM ) = const. 2 (4.9)

Dep ending on the side by which the b o dy falls as t , we shall paint the p oint on this surface with a corresp onding color. The typical picture is given in Figs. 9, 10. We see that the b o dy falls so that the axis corresp onding to the maximal apparent additional mass is vertical, i.e., it falls either by one wide side downward, or by the other one. This confirms the conjecture formulated ab ove. In addition, generally, the b oundary of these domains is fractal, i.e., the surface pattern rep eats in smaller parts. Thus, by analogy with the integrable and nonintegrable (regular and chaotic) systems, the plane-parallel case can b e called integrable, and the general case of system (4.2), (4.3) can b e called nonintegrable. Indeed, in the plane-parallel case the b oundaries of domains corresp onding to the different orientations of the b o dy are regular, but in system (4.2), (4.3) they are fractal. We shall show b elow that if system (4.3) has one more additional integral (the Lagrange integral), the b oundaries of domains also b ecome regular. The fractal structure of b oundaries separating the different typ es of b ehavior as t is closely connected with probabilistic effects arising when descripting asymptotic motions. Indeed, for complex distribution of initial conditions corresp onding to different typ es of asymptotic b ehavior, under sp ecific (given) initial conditions, the asymptotic b ehavior b ecomes unpredictable and only probabilistic description makes sense. This is a kind of probabilistic chaos generated by the structure of initial conditions. The probabilistic description was also prop osed by A. I. Neishtadt when he studied the motion around a fixed p oint of a rigid b o dy under the action of constant and linear (with resp ect to the angular velo city) dissipative moments [14]. It turned out that for small values of these moments, the dynamics of the system has probabilistic nature. In [14] the explicit formulas for probabilities realizing the evolution of the system to a uniform rotation are obtained. The straight generalization of analytic results [14] to system (4.3), (4.6) is connected with substantial difficulties by virtue of the larger dimension of this system and the dep endence of 1 the "dissipation parameter" on time: . Remark. The b ehavior of a heavy b o dy in a fluid substantially differs from its inertial motion, which is describ ed by Kirchhoff equations. In general case, the last system is nonintegrable [1, 12] and shows the typical chaotic b ehavior (the Hamiltonian chaos) [2, 21].
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Fig. 9. The typical picture of domains corresponding to two different limit orientations of the body (where the eigenvector corresponding to the greatest added mass is vertical, two colors correspond to its two possible directions) as t +. At a fourdimensional level of the first integrals, the given two-dimensional surface is defined by Eqs. (4.8), (4.9). The values of the system parameters are A = diag(1.8, 1.5, 2), C = diag(0.5, 2.9, 1.4), µ = 1, (M , ) = 1, E0 = 7.

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(a) E0 = 20, t0 = 0.6

(b) E0 = 70, t0 = 0.3

Fig. 10. The typical pattern on the surface of initial conditions according to the behavior of the system as t when the initial energy and the initial moment t0 increase. The values of the system parameters are A = diag(1.8, 1.5, 2), C = diag(0.5, 2.9, 1.4), µ = 1, (M , ) = 1.

5. THE FALL OF A BODY WITH SCREW SYMMETRY: STEKLOV SOLUTIONS AND THEIR STABILITY

For the general case P1 = 0, B = 0 of system (1.6), after the changes

12 t = , 2 we obtain the equations of motion in the form

M = tm

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Fig. 11. An analog of the Lagrange case, i.e., the case of existence of the integral M 3 = const. The structure of the basin of attraction is regular (b1 = 0.3, b3 = 1.7, c1 = 2.9, c3 = 1.4, µ = 1).

dm 1 H H H =- m+mâ +â +â , d 2 m d H d H =â , =â , d m d m 1 1 H = H 0 + H1 + H2 , 2 2 1 µ2 H0 = (m, Am) - µ(Bm, ) + (C , ), 2 2 P1 H1 = P1 (Bm, ) - P1 µ(C, ), H2 = (, C) + µ(r , ). 2 Now, the differentiation of energy along the system gives dH (m, Am) 1 =- + d 2 2 2P1 W1 W2 Bm, µ - + + , 3/2 (2 )2 (2 ) 2 W2 = P (C, ) + 2µ(r , ).
2 1

(5.1)

(5.2)

W1 = -P1 µ(C, ),

For this system, the asymptotic principles of motion formulated in the previous sections are no longer valid. Moreover, complex attractive regimes of motion different from translational motions
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The initial point g 3.04, L/G 0.62

The initial point g 3.04, L/G 0.62

The initial point g 3.04, L/G 0.62 Fig. 12. The typical form of limiting sets in the case of fall of a body with screw symmetry (A A22 = 1.2, A33 = 2, C11 = 1.6, C22 = 0.1, C33 = 0, P1 = 0, µ = 1, x = y = z = 0, E0 = 3, g0 = ). 2
11

= 1,

exist as t . First of all, we consider stability conditions (with B = 0) for the partial solutions of Eqs. (5.1) corresp onding to uniformly accelerated rotations and find domains of values of parameters for which all these solutions lose their stability (and more complex mo des b ecome stable). Further, we also consider the case of the zero initial impulse P 1 = 0. 5.1. The linear stability of Steklov solutions. For P 1 = 0 we separate the equations for m, , and the area integral can b e represented in the form (m, ) = , = const, (5.3) 2 i.e., (M , ) = . If, in addition, r = 0 and A, B, C are simultaneously diagonalizable, then Eqs. (5.1) have partial solutions similar to the time-indep endent solutions of (4.4). In the basis of eigenvectors of
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matrices A, B, C we have k = ±1, i = j = 0, mk = ± , 2 mi = mj = 0, i = j = k = i; (5.4)

then there exist six partial solutions in total. Here the b o dy falls so that its axis O e k remains vertical, and the angular velo city of rotation around it is determined by the relation
(k )

= -µbk t + ak ,

i.e., the rotation of the b o dy is uniformly accelerated. The velo city of the origin of the moving H co ordinate system in the moving axes is defined by the expression v = = ( B - µtC) , whence, p using (1.2), we find xi = const, xj = const, xk = -µc t2 + bk t + const, 2

k

i.e., the motion of the origin along the vertical axis is uniformly accelerated similarly to the free fall of the b o dy. These uniformly accelerated motions were found out by Steklov [18] (1895) and Chaplygin [20] (1900). In what follows we call them the Steklov solutions. Similarly to solutions (4.4), solutions (5.4) are always unstable in the whole phase space (with resp ect to variables M , ). This instability was p ointed out by Steklov [18]. At the same time, the stability with resp ect to p ositional variables dep ends on parameters of the system and requires the sp ecial consideration. To investigate the stability of solutions of (5.4) we cho ose the new variables vi = di , d vj = dj , d i = j = k = i, (5.5)

adding the area integral (5.3) to these equations, we express variables m i , mj , mk in terms of vi , vj , 12 2 . Using the relation k = ±1 ( + j ) near solutions of (5.4), we obtain linearized equations 2i for new variables in the form di = vi , d dj = vj , d

dvi (k ) = -a-1 aj i i + a-1 aj µ bi - bk + ai (bj - bk ) vj i i d + a-1 µaj ak (bi - bk )i + (ai ak + aj ak - ai aj )v 2 i 1 vi - 2 a-1 aj ak (ak - ai )j + µ(bj - bk )j , - i 2 dvj = ..., d where the expression for
(k ) i

j

(5.6)

= µ2 ai (ci - ck ) - (bi - bk )2 ,

dvj is obtained by the change of indices i j . d
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We use theorems from [4] ab out the b ehavior of the solutions for linear systems of the form dx = (A + V( )) x, where d |V ( )|d < and V ( ) 0 as t . Applying them, we conclude
0

that eigenvalues of linear system (5.6) are expanded in p ower series of variable k ( ) = and the inequalities Re
(0) k (0) k

-1/2

k + + k + ..., are eigenvalues of system (5.6) for = , are the
(0)

(1)

(2)

0, where

(0) k

necessary conditions for stability of the system (similarly, Re k > 0 are the sufficient conditions for instability). To determine them, we can obtain the (biquadratic) characteristic p olynomial 4 - 2 (
(k ) i

+

(k ) j

-

(k ) k

)+

(k ) (k ) i k

= 0.

(5.7)

Thus, the necessary condition for stability of solutions of (5.4) is that p olynomial (5.7) has pure imaginary ro ots (more precisely, it is the condition of absence of exp onential instability with resp ect to ). Hence, we find the corresp onding constraints on the parameters D = (
(k ) 2 i) (k ) i

·

(k ) j

> 0,
(k ) 2 k)



(k ) i

+

(k ) j

-

(k ) k

< 0, - 2
(k ) (k ) j k

+ (

(k ) 2 j)

+ (

- 2

(k ) (k ) i j

- 2

(k ) (k ) i k

(5.8) > 0.

Let us now study in detail the stability of each solution dep ending on the parameters. Without loss of generality, we put k = 3 and µ = 1, c 3 = 0, b3 = 0 (the use of integrals 2 = 1 and (M , ) = const provides the fulfilment of the last two conditions). We fix a 1 , a2 , a3 , c1 , c2 and construct on the plane of parameters b 1 , b2 the domains where inequalities (5.8) hold. In this case, relations (5.8) take the form (a1 c1 - b2 )(a2 c2 - b2 ) > 0, 1 2 = a1 c2 + a2 c1 + 2b1 b2 > 0, (5.9)

D = (a1 c2 - a2 c1 )2 + 4(a1 b2 + a2 b1 )(c2 b1 + c1 b2 ) > 0. It is easy to show that there are three qualitatively different cases:

1. c3 = 0 > c1 > c2 (i.e., c1 < 0 and c2 < 0); in this case, on the plane b1 , b2 there are no domains where inequalities (5.8) hold. We can show that there will b e either two pairs of real solutions of Eq. (5.7) or four complex solutions. 2. c1 > c3 = 0 > c2 (i.e., c1 > 0 and c2 < 0); in this case, the domains given by relations (5.9) are lo cated b etween the lines b 1 = ± a1 c1 and the branches of hyp erb ola given by the relation D = 0 (see Fig. 13a). 3. c1 > c2 > c3 = 0 (i.e., c1 > 0 and c2 > 0); in this case, the domains given by relations (5.9) are lo cated b etween the lines b1 = ± a1 c1 , b2 = ± a2 c2 and the branches of hyp erb ola D = 0 (see Fig. 13b). Remark. It can b e shown that the curves = 0 and D = 0 cross each other at the same p oints where they cross any line bi = ± ai ci . If b1 = b2 = 0, then conditions (5.9) lead to the conditions mentioned ab ove [11]. Namely, only in case 3, where the axis corresp onding to the maximal apparent additional mass is vertical,
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the solution turns out to b e stable. Thus, the adding of the matrix B allows one to stabilize the motion (at least in the linear sense) for which the "middle" axis is vertical, and do es not allow one to stabilize the motion for which the "small" axis is vertical.

(a) c1 = 1.1, c2 = -0.1

(b) c1 = 1.6, c2 = 0.1

Fig. 13. The typical pattern of domains on the plane of parameters b1 , b2 (indicated by the gray color) for which the necessary conditions of stability (5.9) of Steklov solutions hold under different relations between parameters of matrix C. Here, A = diag(1, 1.2, 2).

Fig. 14. The typical pattern of domains of stability on the plane of parameters b1 , b2 of Steklov solutions corresponding to the fall by the wide (i.e., the eigenvector in the direction of the maximal added mass is vertical) and the middle side downward. A = diag(1, 1.2, 2), C = diag(1.6, 0.1, 0).

Now we put c1 > c2 > c3 domains of (linear) stability middle side; see Fig. 14 (the figure that there are domains white).

= 0 for definiteness, and plot on the plane of of the Steklov solutions corresp onding to the fall by the narrow side is always unstable). It where all three Steklov solutions are unstable

parameters b 1 , b2 the fall by the wide and is well seen from the (they are indicated in

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5.2. Lyapunov stability. For one of the Steklov solutions (5.4), namely, for the case when the b o dy falls by the "wide" side downward, one can prove the asymptotic Lyapunov stability. As mentioned ab ove, without loss of generality, we can put i = 1, j = 2, k = 3 and b 3 = 0, c1 > c2 > c3 = 0 in (5.4). We construct the Lyapunov function in the form 1 V = H2 + W, where H2 is the quadratic part of the Hamiltonian near this solution in variables 1 , 2 , v1 , v2 : 1 H2 = (a 2
-1 2 2 v1

+a 1 2a

-1 2 1 v2

)+

1 2a

1

2 2 a1 c1 - b2 - a3 b1 + 2 a3 (a1 - a3 ) 1 4 2

2 1

+

2

2 2 2 a2 c2 - b2 - a3 b2 + 2 a3 (a2 - a3 ) 2 , 2 4 2

(5.10)

and we lo ok for the function W in the form of a homogeneous quadratic form in 1 , 2 , v1 , v2 with constant co efficients. It is easy to see that for large , the function H 2 , and hence, V , is p ositive definite near the origin under the conditions a1 c1 - b2 > 0, a2 c2 - b2 > 0. (5.11) 1 2 As shown ab ove, these inequalities give us one of the domains of stability for the solution under consideration in the linear approximation (see Fig. 13b). Thus, we can show the asymptotic stability in the domain b ounded by inequalities (5.11) for those values of parameters for which we will b e able to cho ose a function V whose derivative along the solutions of a linear system is strictly negative (for sufficiently large ). The derivative of function V along solutions of system (5.6) has the form 1 1 1 dV = - G1 + 3/2 G2 + 2 G3 , d G2 , G3 are homogeneous quadratic forms in variables 1 , 2 , v1 , v2 . Thus, for large dV of the derivative is determined by the quadratic form G 1 which must b e p ositive d the case of the asymptotic stability. straightforward calculations it can b e shown that it is necessary to cho ose W in the W = k 1 v1 1 + k 2 v2 2 ; then G2 and G3 are indep endent of v1 , v2 and G1 = 2k1 a
-1 1 a2 2 (a1 c1 - b2 )v1 + 2k2 a 1 -1 2 a1 2 (a2 c2 - b2 )v2 + a 2 -1 2 2 (1 - 2a2 k1 )1 + a -1 1

where G1 , , the sign definite in By the form

1 1 + a-1 b1 - 2k1 (a1 b2 + a2 b1 ) 1 v2 - a-1 b2 - 2k2 (a1 b2 + a2 b1 ) 2 v1 . 21 22 It is easy to obtain the conditions for p ositive definiteness of the form G 1 : 0 < k1 < 1 , 2a2 0 < k2 < 1 , 2a1

(1 - 2a1 k2 )

2 2

(5.12)

2 -4(a1 b2 + a2 b1 )2 k1 - 16a1 a2 (c1 a1 - b2 )k1 k2 - b2 + 4(2a1 a2 c1 - a2 b2 + a1 b1 b2 )k1 > 0, 1 1 1 2 -4(a1 b2 + a2 b1 )2 k2 - 16a1 a2 (c2 a2 - b2 )k1 k2 - b2 + 4(2a1 a2 c2 - a1 b2 + a2 b1 b2 )k2 > 0. 2 2 2

(5.13)

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There are two cases: 1. b1 · b2 > 0, then, cho osing k1 = 1 b1 (a1 b2 + a2 b1 )-1 , k2 = 1 b2 (a1 b2 + a2 b1 )-1 , we obtain the 2 2 diagonal quadratic form (5.12) which is obviously p ositive definite; 2. b1 ·b2 < 0, in this case the sufficient solvability conditions for inequalities (5.13) are determined by solutions of a quartic equation (and have a rather inconvenient form). At the same time, since only one term is p ositive in the last two relations (5.13), we can obtain the necessary conditions for the solvability of (5.13) in the form 1 = 2a1 a2 c1 - a2 b2 + a1 b1 b2 > 0, 1 2 = 2a1 a2 c2 - a1 b2 + a2 b1 b2 > 0, 2

(5.14)

b1 b2 > max(-a1 c2 , -a2 c1 ). In Fig. 15 by the gray color the domain is shown where the necessary conditions (5.14) hold. As can b e seen from the figure, for b1 b2 < 0 the domain of asymptotic stability do es not coincide with the whole domain of fixed sign of quadratic form (5.10) .

Fig. 15. The domain of asymptotic stability of the Steklov solution corresponding to the fall by the wide side downward with A = diag(1, 1.2, 2), C = diag(1.6, 0.1, 0).

Remark. An analysis of the (linear and nonlinear) stability for Steklov solutions was carried out in [6, 24]. In particular, conditions (5.13) were obtained in the form of general inequalities for co efficients without taking into consideration those Steklov solutions for which the relation b etween stability and instability can b e different (see ab ove). Here we carried out a geometric analysis of values of p ossible parameters for which the conditions of stability (5.13) hold, and drew the conclusion ab out the existence of a domain of values for the parameters for which all Steklov solutions are unstable. In this case, in the phase space there exists a more complex invariant attracting set of two-dimensional torus typ e (see Fig. 12) to which the tra jectories of system (5.1) tend as t +. Analytically, the existence of this invariant set remains unproven, b ecause for the present bifurcation theory and qualitative metho ds have not b een develop ed for systems of typ e (5.1) for which the linear "dissipation" decreases with time with resp ect to values of the parameter 1 . In our analysis, simpler conditions for linear stability and Lyapunov stability are also obtained due to the systematic use of the Hamiltonian form of equations of motion.
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ACKNOWLEDGMENTS The authors are grateful to K. G. Tronin for carrying out the computer calculations. This work was carried out under financial supp ort by the CRDF (pro ject RU-M1-2583-MO-04), the Russian Foundation for Basic Research (pro ject no. 05-01-01058), and the State Program for Supp ort of Leading Scientific Scho ols (pro ject no. 1312.2006.1). A. V. Borisov and I. S. Mamaev also acknowledge the supp ort by the Russian Foundation for Basic Research (pro ject 04-05-64367) and INTAS (pro ject 04-80-7297). REFERENCES
1. A. V. Borisov, Regul. Chaotic Dyn. 1 (2), 61 (1996). 2. A. V. Borisov and A. I. Kir'yanov, in Mathematical Methods in Mechanics (MGU, Moscow, 1990), pp. 16­21 [in Russian]. 3. A. V. Borisov and I. S. Mamaev, Prikl. Mat. Mekh. 67 (2), 256 (2003) [J. Appl. Math. Mech. 67 (2), 227 (2003)]. 4. A. V. Borisov, I. S. Mamaev, and A. G. Kholmskaya, Vestn. molodykh uchenykh. SPb. Prikl. Mat. Mekh., No. 4, 13 (2000). 5. D. N. Goryachev, Izv. Imper. ob-va lyubitelei estestvoznaniya pri Mosk. Imperat. Univ. 78 (2), 59 (1893). 6. M. V. Deryabin, Izv. Ross. Akad. Nauk, Ser. Mekh. Tverd. Tela, No. 5, 30 (2002). 7. M. V. Deryabin and V. V. Kozlov, Izv. Ross. Akad. Nauk, Ser. Mekh. Tverd. Tela, No. 1, 68 (2002). 8. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955; IL, Moscow, 1958). 9. V. V. Kozlov, Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela, No. 5, 10 (1989). 10. V. V. Kozlov, Mat. Zametki 45 (4), 46 (1989) [Math. Notes 45 (3­4), 296 (1989)]. 11. V. V. Kozlov, Prikl. Mat. Mekh. 55 (1), 12 (1991) [J. Appl. Math. Mech. 55 (1), 8 (1991)]. 12. V. V. Kozlov and D. A. Onishchenko, Dokl. Akad. Nauk SSSR 266 (6), 1298 (1982). 13. H. Lamb, Hydrodynamics, 6th ed. (Dover Publ., N. Y., 1945; Cambridge University Press, Cambridge, 1993; OGIZ, Gostekhizdat, Moscow, 1947). 14. A. I. Neishtadt, Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela, No. 6, 30 (1980). 15. S. M. Ramodanov, Vestn. Moskov. Univ., Ser. Mat. Mekh. No. 3, 93 (1995). 16. V. A. Steklov, The Supplements to the Work "On the Motion of a Rigid Body in a Fluid" (Khar'kov, 1895) [in Russian]. 17. V. A. Steklov, On the Motion of a Rigid Body in a Fluid (Khar'kov, 1893) [in Russian]. 18. V. A. Steklov, Trudy Otd. Fiz. Nauk Ob-va Lyubitelei Estestvoznaniya, Antropologii i Etnografii 7, 1 (1895). 19. S. A. Chaplygin, in Col lected Works (GITTL, Moscow­Leningrad, 1948), Vol. 1, pp. 337­346. 20. S. A. Chaplygin, in Complete Works (Izd. Akad Nauk SSSR, Leningrad, 1933), Vol. 1, pp. 133­150. 21. H. Aref and S. W. Jones, Phys. Fluids A 5 (12), 3026 (1993). 22. M. L. Bertolli and S. V. Bolotin, Ann. di Matem. Pura ed. Applicata CLXXIV (IV), 253 (1998). 23. M. V. Deryabin, Regul. Chaotic Dyn. 3 (1), 93 (1998). 24. M. V. Deryabin, Z. Angew. Math. Mech. 83 (3), 197 (2003).

Translated by M. Deikalova

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