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A. V. BORISOV
Institute of Computer Science Universitetskaya, 1 426034, Izhevsk, Russia E-mail: borisov@ics.org.ru

I. S. MAMAEV
Institute of Computer Science Universitetskaya, 1 426034, Izhevsk, Russia E-mail: mamaev@ics.org.ru

S. M. RAMODANOV
Moscow, Russia E-mail: ramo danov@mail.ru

MOTION OF A CIRCULAR CYLINDER AND n POINT VORTICES IN A PERFECT FLUID
Received October 29, 2003

DOI: 10.1070/RD2003v008n04ABEH000257

The pap er studies the system of a rigid b ody interacting dynamically with p oint vortices in a p erfect fluid. For arbitrary value of vortex strengths and circulation around the cylinder the system is shown to b e Hamiltonian (the corresp onding Poisson bracket structure is rather complicated). We also reduced the numb er of degrees of freedom of the system by two using the reduction by symmetry technique and p erformed a thorough qualitative analysis of the integrable system of a cylinder interacting with one vortex.

In this pap er we consider the system of a rigid cylinder interacting with p oint vortices. We start by indicating some known results on the sub ject from the classical hydrodynamics. For the most part, these results are presented in [18, 6, 9]. As far as we can see, Kirchhgoff [5] was the first who studied the dynamics of p oint vortices on a systematic basis. In particular, he obtained the equations of motion in Hamiltonian form and indicated integrals of motion. It was enough to show that the equations of motion governing the system of two or three vortices are integrable (the problem of three vortices was first analytically solved by Grobli). ¨ The classics also considered the system of p oint vortices moving externally to rigid, stationary b oundaries (the imp ermeable condition on the b oundaries was assumed). Greenhill [12] studied in detail the motion of two vortices in a circular region. Havelock [13] investigated stability of n-gon stationary configurations in the region exterior to a circle. We should also mention F¨ pl's analysis op of interaction of a rigid b ody with an ambient flow at low Reynolds numb ers (more exactly, he investigated stability of the system of two vortices interacting with a circular cylinder emb edded in a uniform flow). At the same time, the study of the dynamics of interacting vortices-b ody systems has b een also a sub ject of interest in the classical hydrodynamics. It is known from the phenomenological theory develop ed by Prandtl [7] and Jukowski that when a b ody moves in a fluid, the thin b oundary layer p eels off the b ody thereby generating b ound vortices. In their turn, these vortices exert a lifting force on the b ody which can b e observed in aero- and hydrodynamical exp eriments. The problem of interaction of a rigid b ody and p oint vortices in a p erfect fluid (the imp ermeable condition on the b ody's surface is assumed) can b e studied within the framework of the Hamiltonian
Mathematics Sub ject Classification 76B47, 37J35, 70E40

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mechanics. Various forms of the equations of motion for a rigid circular cylinder interacting with n p oint vortices have b een recently (and practically simultaneously) obtained in [14, 15, 16]. The integrability of the equations in the case of one vortex (n = 1) was established in [10]. In this pap er we will study in greater detail this case of integrability. We will also consider the simplest chaotic system of a cylinder and two vortices to which a reduction procedure will b e applied resulting in reduction of degrees of freedom by one. Hereinafter, as in [10], the term rigid body will refer to a two-dimensional circular region. It should b e noted that even in the case of an elliptic region the equations of motion b ecome much more complicated and cannot b e written in such a compact form.

1. Hamiltonian form of the equations of motion
The equations of motion for a cylinder and vortices with resp ect to a fixed coordinate frame Oxy can b e written as [14] ri = -v +grad i |r =ri , rc = v
n n

av1 = v2 -
i=1

i (y i -yi ), av2 = -v1 +
i=1

i (xi -xi ),

(1.1)

where r

is the radius-vector from O to the center of mass of the cylinder, v is the velocity of the 2 cylinder, ri is the vector from the center of the cylinder to the i-th vortex and r i = R ri is the
c

r

i

vector from the center of the cylinder to the i-th inverse p oint (Fig. 1). Here R denotes the radius of the cylinder, the constant coefficient a involves the added mass of the cylinder; and the constants and i are connected with the circulation around the cylinder and the vortex strengths by the formulae
= , i = i . The density of the fluid is 2 . 2 2

Fig. 1

The function i (r ) represents that p ortion of the velocity p otential ( r ) which does not have a singularity at the p oint r = ri . The velocity p otential in the region exterior to the cylinder reads
2 y (r ) = - R2 (r , v ) - arctg x + r n

i arctg
i=1

y - yi y - yi - arctg x - xi x - xi

.

(1.2)

Equations (1.1) were derived using a balance of linear momentum for the fluid within a circular b oundary that encloses the b ody and the vortices. The fluid is assumed to b e at rest at infinity [14, 16]. Thus the analysis of the system of a cylinder and vortices in a p erfect fluid can b e reduced to the analysis of a finite set of ordinary differential equations. It is easy to check that equations (1.1) preserve invariant measure, i.e., the divergence of (1.1) is zero.
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As for the system of n p oint vortices [5], equations (1.1) can b e shown to b e Hamiltonian. Prop osition 1. Equations (1.1) can be represented in the form i = {i , H } =
k

{i , k } H k

(1.3)

where i are the components of the phase vector = (xc , yc , v1 , v2 , x1 , y1 , ... , xn , yn ) and H is the Hamiltonian. For the components Jij ( ) = {i , j } of the structural tensor of the Poisson bracket structure the Jacobi identity holds: Jil
l

Jjk
l

+ Jkl

Jij
l

+ Jjl

Jki l

=0

i, j, k

Proof. Equations (1.1) has an integral of motion: H = 1 av 2 + 1 2 2 2 i
i

ln(r - R ) - i ln r

2 i

2

2 i

+1 2

i j ln
i
R4 - 2R2 (ri , rj )+ ri2 r |ri - rj |2

2 j

.

(1.4)

This integral resembles the Hamiltonian of the system of n p oint vortices [5]. Let H b e our Hamiltonian. Now we have to find a skew-symmetric tensor Jij such that the equations of motion (1.3) coincide with (1.1). The non-zero comp onents of this tensor are
4 22 2 1 r -R (xi -yi ) , {v1 , xi } = a i 4 ri 2 1 2R xi yi , {v2 , xi } = - a 4 ri 2 1 2R xi yi , {v1 , yi } = - a 4 ri

4 22 2 1 r + R (xi - yi ) , {v2 , yi } = a i 4 ri

(1.5)

{v1 , v2 } = - a2

{xc , v1 } = {yc , v2 } = a-1 . It is easy to verify that for the Poisson bracket (1.5) the Jacobi identity holds. The Poisson bracket structure (1.5) is non-degenerate, therefore, by the Darb oux theorem, it can b e reduced to a canonical form ({qi , pj } = ij ). However, in our further analysis canonical coordinates will not b e used. The Lie-Poisson bracket structure for (1.1), (1.2) under the condition = - i was studied in [16]. In this work stability of an equilibrium configuration of the system of two vortices b ehind a steadily moving cylinder is discussed. The Poissom bracket (1.5) was first obtained in [10]. The fact that the general equations are Hamiltonian is not a priori obvious and does not seem to follow from the Lagrangian formalism. The Hamiltonian form of the equations allows us to apply the highly develop ed p erturbation theory techniques (e.g. the KAM theory) and other sp ecific methods of qualitative analysis. The Liouville theorem on integrability and its geometrical extension suggested by Arnold [1] can b e applied to the analysis of these equations.

i

4 i ri - R4 , 4 a2 ri

{xi , yi } = - 1 , i

2. Problem of advection
The problem of finding pathlines of the fluid for a given motion of the cylinder rc = rc (t) and the vortices ri = ri (t), i = 1, ... , n is known as the problem of advection. As mentioned ab ove, the
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ambient flow is p otential with p otential (1.2), therefore the equation of motion for a passive particle with resp ect to the cylinder-fixed frame of reference looks like r = grad (r )
ri =ri (t)

,

r = (x, y ).

(2.1)

Obviously, equations (2.1) are Hamiltonianian with resp ect to the standard Poisson bracket structure {x; y } = 1; the Hamiltonian is time-dep endent and coincides with the stream function for the p otential (1.2), that is, Ha (r , t) = R2 - 1 (v (t)y - v (t)x)+ 1 ln r 2 + 1 2 2 r2 +1 2
N i=1

(2.2) i ln |r - ri (t)|2 - ln |r - ri (t)|2 .

3. Symmetry and integrals of motion
The equations of motion (1.1) are invariant under the action of the Euclidean group E (2), therefore, by Noether's theorem for Hamiltonian systems, there exist three integrals of motion. The integrals Q = av2 + xc - i (xi - xi ), P = av1 - yc + i (yi - yi ) (3.1)

corresp ond to translations along the coordinate axes. These integrals are a generalization to the classical linear momentum. On the other hand, the vector (Q, P ) can b e considered as a counterpart of the vector of the center of vorticity of the system of n vortices and cylinder. For the system of n vortices this notion was introduced in [5, 6]. For = 0, the origin of coordinates can b e so chosen as to make Q = P = 0, meaning that the center of vorticity can b e always shifted to the origin of coordinates. The third integral, corresp onding to the invariance under rotations ab out an axis p erp endicular to the plane of motion, is
2 I = a(v1 yc - v2 xc ) - 1 rc - 1 2 2 n i=1

i ri2 + 1 2

n i=1

R2 - 1 (r , r ). i c ri2

(3.2)

In the pap er [10], an additional integral for equations (1.1) but with the dynamics for the center of the cylinder excluded was indicated. This integral looks like F = a2 v 2 +
n i=1 2 4 i 2a 1- R2 (xi v2 -yi v1 )+(i -)ri2 +i R2 + ri ri 2 i j (ri , rj ) 1- R2 ri 2 1 - R2 . rj

(3.3)

+2
i
The integrals F ,I , P and Q satisfy the equation F = 2I + P + Q +2R
2 2 2 n i=1

2 . i

Comment. The Hamiltonian vector field that corresponds to the integral (3.2) lo oks like XI = { , I } = (yc , -xc , v2 , -v1 , y1 , -x1 , ..., yn , -xn ) 452 REGULAR AND CHAOTIC DYNAMICS, V. 8, (3.4)

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The Poisson bracket of the integrals Q, P , and I differs from the Lie-Poisson bracket for the e(2) algebra by a constant (co-cycle [1]), that is, {Q, P } = , {I, Q} = P, {I, P } = -Q. (3.5)

Thus, for = 0 the number of degrees of freedom of the system governed by (1.1) can be reduced by two and even by three if = 0 and P = Q = 0. Corollary 1. The dynamics of a cylinder and one vortex is integrable in the Liouvil le sense. Corollary 2. For = 0 and P = Q = 0, the dynamics of a cylinder and two vortices is integrable in the Liouvil le sense.

4. Complex form of equations of motion and the Dirac bracket
As mentioned ab ove, in [10] reduced equations were used, i.e., the equations we dealt with were exactly equations (1.1) but without the equation rc = v . The elimination of this equation from the general system (1.1) now can b e interpreted as the reduction by symmetry due to the integrals Q and P (3.1). For = 0 the reduction can b e carried out by restricting the dynamics to a joint level surface of the integrals (3.1). Since = 0, we assume that the origin of the fixed coordinate frame is at the center of vorticity, i.e., P = Q = 0. Then, we substitute rc for vi in (3.1). The first order equations in the p ositional variables result. It is convenient to write these equations in the complex form
n

azc = av = -izc + i
j =1 2 ¯ + zk = -v + R 2v + i z - ik 1 ¯ zk - zk k zk

j (~j - zj ) z
n

i
j =k

1 1 - zk - zj zk - zj

,

(4.1)

zk = R zk ¯

2

k = 1, ... , n,

where zc = xc + iyc and zk = xk + iyk define the p osition of the cylinder's center and the vortices, and v = v1 + iv2 is the velocity of the cylinder's center. Obviously, equations (4.1) are Hamiltonian. The Hamiltonian can b e obtained by replacing the cylinder's velocity in (1.4) with the expression in the right-hand side of the first equation of (4.1) H=1 2a +1 2
n 2

x -
i=1

i (xi - xi )
2

+1 2a
2 i

n

2

y -
i=1

i (yi - yi )

+
2 j

2 i
i

ln(r - R ) - i ln r

2 i

+1 2

i j ln
i
R4 - 2R2 (ri , rj )+ ri2 r |ri - rj |2

(4.2) .

The Poisson bracket for (4.1) is {xc , yc } = 1 , {xi , yi } = - 1 , i i = 1, ... , n. (4.3)

This bracket can b e obtained via the Dirac reduction procedure [11], which consists in restricting the bracket (1.5) to the manifold Q = P = 0. In other words, for = 0, the cylinder can b e considered as an (n + 1)-th "comp ound vortex", and equations (4.1) govern the system of n + 1 vortices.
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A. V. BORISOV, I. S. MAMAEV, S. M. RAMODANOV Comment. The Dirac bracket {·, ·} i = 1, ..., k is given by the formula {g, h} where c = { fi , fj }
-1

D

on a manifold Nc which is a level surface fi ( ) = ci , ci = const, = {g, h} +
ij

D

{g, fi }cij {h, fj },

ij

and {·, ·} is the original Poisson bracket. In our case we have {g, h}
D

= {g, h}- 1 ({g, Q}{h, P }-{g, P }{h, Q}) ,

where {·, ·} is the Poisson bracket (1.5)

Equations (4.1) are invariant under rotations ab out the origin (the center of vorticity) and therefore have an additional integral of motion: I = r -
i=1 2 c n

i ri2 .

(4.4)

This integral can b e obtained up on substitution of the expression for rc from (4.1) for v in the integral (3.3). Now we can formulate some prop erties of the motion of the system of a cylinder and n vortices. These prop erties are analogous to those formulated by Synge [17] for the system of n vortices. Prop osition 2. Suppose that the vortex strengths are of the same sign and i < 0; then the trajectory of the cylinder's center and the vortices belong to a bounded region. Prop osition 3. Suppose that = 0 and the vortex motions relative to center of the cylinder are bounded (i.e., i |ri (t)| is a bounded function), then the absolute motion of the cylinder is also bounded. Here and in the sequel, a motion with resp ect to the Fig. 2. Motion of a cylinder (solid line) fixed frame will b e referred to as an absolute motion and with and a vortex couple (1 = -2 ); a = resp ect to the center of the cylinder as a relative motion.
= 20, = 8, 1 = 10, 2 = -10, R = F = 2; the initial conditions (relative the center of the cylinder) are v1 (0) = v2 (0) = 0, x1 (0) = -3, y1 (0) = 0, x2 (0) = 3, y2 (0) = 0. 1, Comment. It should be noted that for n = 1 and = 0amore to general statement is valid: the absolute motion of the cylinder is 0, = bounded if and only if the relative motion of the vortex is bounded. Simulations have shown that already for n = 2 this is not true: the tra jectories of the cylinder and two vortices for the case 1 = -2 , || < |1 | are shown in Fig. 2. As might be expected, eventually the vortices start drifting to infinity while the cylinder moves along a curve which gradually takes the shape of a circle.

5. Motion of a cylinder and one vortex
Integrability of the equations of motion. Reduction to a system with one degree of freedom. Consider in greater detail the system of a cylinder interacting dynamically with only one vortex, i.e., n = 1. Let r1 = r = (x, y ). With the assistance of the first integrals (3.1) and (3.2), the solution to the equations of motion follows in terms of quadratures. Using the integrals, our system can b e reduced to a system with one degree of freedom. The algebraic reduction that we will now use is analogous to the Routh reduction.
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As the variables of the reduced system, it is reasonable to choose integrals of the field of symmetry vI (3.4) (see, for example, [2]). We put p1 = a(xv1 + yv2 ), p2 = a(xv2 - yv1 ), = x2 + y 2 . (5.1)

The Poisson brackets for these variables are {p1 ,p2 } = ( - 1 ) + 1 (p2 +(p2 - 1 R2 )2 ), 1 1 p1 {p1 ,} = 2 + 2 (p2 - 1 R2 ), {p2 ,} = -2 . 1 1 In terms of (5.1) the integrals (1.4) and (3.3) read p2 + p2 1 2 1 2 + 1 ln - R2 - 1 1 ln , 2 2 2a 2 + p2 2 R R4 p 2 +21 1 - p2 + 2 + - 1 . F= 1 1 H= (5.3) (5.4)

(5.2)

The rank of the Poisson bracket structure (5.2) is two, i.e., the structure is degenerate. Hence, the reduced system has one degree of freedom. One variable of the set (p1 , p2 , ) can b e eliminated using the integral (5.4), which is a Casimir function for the structure (5.2). Traditionally [1], on a two-dimensional level surface of the Casimir function (a symplectic leaf ), canonical coordinates ({q, p} = 1) are introduced. The reduced equations are q = H , p p = - H . q

In contrast to this traditional approach, the local coordinates that we will use account well for the leaf 's geometry but are not canonical. The phase p ortraits in terms of these coordinates are vivid and illustrative. Our attempts to find a simple and natural set of canonical coordinates have not b een successful. General prop erties of motion. The reduced system (5.2) describ es the motion of the vortices relative to the center of the cylinder. Before proceeding to a qualitative analysis of the reduced system, we will indicate some general prop erties of the absolute motion. Prop osition 4. The absolute motion of the cylinder is bounded except maybe for the two cases: 1. = 1 ; 2. = 0;
Comment. Condition (1) corresponds to the case where the circulation around the cylinder is zero, and condition (2) represents the case where the circulation around the cylinder and the vortex is zero.

Proof. Assume the contrary: = 1 , = 0 and the motion of the cylinder is not b ounded. Then, according to Prep ositions 2 and 3, is an unb ounded function of time and 1 > 0. Dividing (5.4) by yields p2 R 1 + - 1 Obviously,
2 2

+

2 1 2

p

-
2

1 -
2

22 R2 - F 1

=0

(5.5)

lim

p2 1 +

+

p1

= 1 .

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2 Therefore, there exist functions () = o(1) and () such that 1 + = = 1 + sin . Up on substitution of these expressions into (5.3), we get

p



1 1 + cos , =

p

2H =

( - R2 )1 k + 1 ln 2a

The second term in the right-hand s e is of O(ln ), the factor k is separated from zero, that is, id k = (1 - 1 )2 + +21 ( 1 - 1 + cos ) > k1 = const > 0. This is a contradiction with the fact that H is constatnt.
Comment. One can easily note that for = 1 and = 0 an unbounded motion of the cylinder always exists. Therefore the above preposition serves as a criterion of boundedness of the absolute motion. Most likely this criterion remains valid for the case of an arbitrary (finite) number of vortices, but this is not proved yet.

The statement given b elow provides a nice "geometro-dynamical" insight into the structure of absolute motions of the cylinder. Theorem. Suppose that = 0; then for each periodic solution of the reduced system (5.1) there exists a rotating coordinate frame with the origin at the center of vorticity such that with respect to this frame the vortices and the cylinder move along closed curves. Proof. It follows from (4.1) that yx1 - xy1 = p1 , y y1 + xx1 = - p2 1 2 + (R - ). (5.6)

By the assumption of the theorem, the functions in the right-hand side of these equations are p eriodic functions with p eriod T . Therefore x1 y1 =· x y , = 1 -
2



2 1

(5.7)

where 1 and 2 are p eriodic functions with p eriod T . Substituting these expressions into (4.1), we get (5.8) ax = y - yG2 + xG1 , ay = -x + yG1 + xG2 , where G1 and G2 are also T -p eriodic functions. Let A b e the matrix for the system of linear differential equations (5.8). It can b e easily verified that A ·
t é

t

t

A dt =
0 0

A dt · A, hence (see, for example, [4])

the fundamental matrix, X, of equations (5.8) can b e written as X=e Let us represent A as a sum A = B + C, 1 B= a 0 - + G2 - G2 0 , 1 C= a G1 G2 - G2 -G2 + G2 G1 .
0

A dt

.

Note that G1 = 0, otherwise equations (5.8) have unb ounded with Prep osition 3. Since the matrices B and C commute, we h - G2 - G2 t sin cos a a X= - G2 - G2 t cos - sin a a
456

solutions which is in a contradiction ave t · G. t

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Here G is a T -p eriodic matrix. Thus, with resp ect to a coordinate frame (with origin at the center of vorticity) rotating at a rate ( - G2 )/a the cylinder moves along a closed curve. Let us prove now that the vortex also moves along a closed curve. Let x ,y b e the coordinates of the vortex in the fixed coordinate frame. Then x = xc + x, y = yc + y . It follows from (5.7) that x y cos = ( + E) · sin x y = x(0) y (0)

- G2 t a = - G2 t - sin a



- G2 t a · ( + E) · G · - G2 cos t a

Here E is the identity matrix, and the matrix ( + E)G is T -p eriodic.

Fig. 3. a) Tra jectories of the rationally related). a = 6, the cylinder) are v1 (0) = 0.24 frame of reference rotating at

cylinder (solid line) and the = - 3 , 1 = 1 , R = 1 , F = , v2 (0) = 0, x(0) = 0, y (0) = a rate 0.029 (the origin, O+ ,

vortex (here the two frequencies of the motion are 2, ; the initial conditions (relative to the center of 1.31; b) The tra jectories shown in diagram a in the is at the center of vorticity).

An absolu In Fig. 3 b this vorticity. The F = 2 and the

te motion of the cylin motion is shown in a value of the physical initial conditions are

der (solid line) and the vortex (dashed line) is shown in Fig. 3 a. rotating frame of reference whose origin, O+ , is at the center of parameters for this motion are a = 6, = -3, 1 = 1, R = 1, v1 (0) = 0.24, v2 (0) = 0, x(0) = 0, y (0) = 1.31.

Qualitative analysis of the reduced system. The geometry of the symplectic leaf (5.4) of the Poisson bracket structure (5.2) (the phase space of the reduced system) is governed by 1 . The symplectic leaf is compact if 1 < 0 and non-compact if either 1 > 0 or = 0. Let us consider these three cases. 1. Compact case (1 < 0). From (5.4) it follows that 1 + p2 - 1 R
22

+ p2 + 1

-1 + K

2

= K 2,

K=

22 R2 - F 1 2 -1

(5.9)

The symplectic leaf is diffeomorphic to a two-dimensional sphere. For real motions F 21 R2 (1 - ). Since the phase space is compact, is a b ounded function, meaning that the distance b etween the vortex and the cylinder cannot grow infinitely. Moreover, in view of Prep osition 3, the absolute motion of the cylinder also is b ounded.
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On the leaf we intoduce coordinates and by the formulae 1 + p2 - 1 R2 = K cos cos , -1 + K = K sin , p1 = K sin cos , , [-, ] .
K (sin - 1) > R2 . -1
2

-, 22

(5.10)

Comment. For a fixed value of K , the co ordinate satisfies the relation =

To find stationary solutions of the reduced system, consider the differential equations in p1 ,p and : p1 = {p1 ,H } = + 1 R4 a - 2R2 1 a2 - 2 a3 + 2 R2 a2 + 1 a3 1 1 2 a(- + R2 )
2

(-R4 1 + a2 + R6 1 - 1 R2 2 - R2 a + R2 2 - 1 a2 + 1 3 - 3 )p 2 a(- + R2 ) + (2 - R4 )p2 +(2 - 2R2 + R4 )p 2 2 a(- + R2 )
2 1

, (5.11)
1

p2 ={p2 ,H }=

(3 -1 3 +R2 a-R6 1 +1 a2 -a2 -R2 2 +1 R2 2 +R4 1 )p 2 a(- + R2 ) + (-2R2 +2R4 )p2 p1 , 2 a(- + R2 ) (-2 +2R2 )p a
1

+

= {, H } =

The last equation implies p1 = 0 and the right-hand side of the second equation b ecomes zero. Therefore, on the phase p ortrait, all stationary solutions b elong to the lines = ± , = 0 and = ± .
2

To determine p2 and we should use (5.9) and equate to zero the right-hand side of the first equation. The numb er of stationary solutions dep ends on F . Equations (5.11) remain unaltered under the transformation (p1 , p2 , , t, , 1 ) (-p1 , - p2 , , -t, - , -1 ). In view of (5.10), this transformation implies the following change of variables and physical parameters (, , , 1 ) ( + , , -, -1 ). Therefore, a change in sign of the circulation and the vortex strength results in a shift of the phase curves along the -axis by . Thus, with no loss of generality, we can assume that < 0 and 1 > 0. Phase p ortraits for various values of F ( the value of the other parameters are fixed a = 20, = -1, 1 = 0.5 and R = 1) are shown in Figs. 4 a, 4 b and 4 c. The "non-physical" area ( R2 ) is shown in grey. The solution curves of the reduced equations are the level curves of (5.3) on the surface (5.4). The level curves are closed, hence the solutions of the reduced equations are p eriodic. The fixed p oints on the phase p ortraits (stationary solutions) represent a motion in which the cylinder and the vortex move along concentric circles whose centers are at the center of vorticity. For example, the tra jectories of the vortex and the cylinder that corresp ond to the elliptic p oint in Fig. 4 are shown in Fig. 5. With an increase in F two more fixed p oints app ear, the motion corresp onding to these p oints is qualitatively shown in Fig. 6. 2. Non-compact case (1 > 0). It follows from (5.5) that 1 + p2 - 1 R
458
22

+ p2 - 1

1 - K

2

= -K 2 ,

K=

22 R2 - F 1 . 2 1

(5.12)

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Fig. 4. Phase portraits of the reduced system in the compact case (1 < 0): a = 20, = -1, 1 = 0, 5, R = 1.

Fig. 5

Fig. 6

Symplectic leaves are diffeomorphic to a hyp erb oloid of two sheets. For real motions F>21 R2 (1 -). On symplectic leaves we introduce local coordinates: 1 + p2 - 1 R2 = K cosh sinh , p1 = K sinh , 1 - K = -K cosh cosh , (5.13)

Arguing as in the compact case, we can assume that > 0 and 1 > 0. In view of (5.11) and (5.13), on the phase p ortrait, all fixed p oints lie on the axis = 0. It is interesting to note that, unlike the compact case, the top ology of the phase p ortrait is determined by the sign of the difference - 1 and does not change qualitatively as the constant F varies. Typical phase p ortraits are given in Fig. 7 a ( = 1 ), 7 b ( < 1 ) and 7 c ( > 1 ). The phase curves are closed, hence the solutions of the reduced equations are p eriodic functions of time. The only exception is the case = 1 (Fig. 7). As shown ab ove, only in this case the cylinder may have an unb ounded motion (Prep osition 4).
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Fig. 7. Phase portraits of the reduced system in the non-compact case (1 > 0): F = 25, a = 20, 1 = 10, R = 1.

3. Non-compact case ( = 0). In the system of equations (1.1), the equations governing the motion of the vortex are uncoupled from the first equation. Indeed, it follows from (3.1) that av1 = -1 y1 f (x1 ,y1 )+ c1 , av2 = 1 x1 f (x1 ,y1 )+ c2 ,

2 where f (x1 ,y1 ) = -1+ R2 /(x2 + y1 ) and c1 , c2 are arbitrary constants. Substituting these expressions 1 into the first equation (1.1), we obtain a closed system of equations in the unknowns x1 and y1 . The solution curves of this system coincide with level curves of the integral (5.3). It can b e shown that the level curves are closed, hence x1 and y1 are p eriodic functions of time. The evolution of the center of the cylinder is governed by the equations

1 x = a (-1 y1 f (x1 ,y1 ) + c1 ) t + g1 (t), 1 y = a (1 x1 f (x1 ,y1 ) + c2 ) t + g2 (t), where g1 (t), g2 (t) are p eriodic functions. Thus, there exists a uniformly moving coordinate system in which the orbits of the cylinder and the vortex are closed.
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6. The case of two vortices
Supp ose now that n = 2, i.e., we are going to consider the system of a cylinder interacting dynamically with two vortices. By analogy with the case n = 1, we use the integral (3.3) to reduce the numb er of degrees of freedom by one. As b efore, we take some quantities invariant with resp ect to rotations ab out the cylinder's center as the new variables, namely, p1 = a(x1 v1 + y1 v2 ), p2 = a(x1 v2 - y1 v1 ), r1 = x2 1 +y ,
2 1

p3 = x1 x2 + y1 y2 , x2 2 +y .
2 2

p4 = x1 y2 - x2 y1 ,

r2 =

The Poisson brackets for these variables are as follows: {p1 ,p2 } = {p1 ,p3 } = {p1 ,p4 } = {p1 ,r1 } = {p2 ,p3 } = {p2 ,p4 } =
2 2 2 2 2 2 2 2 (p2 + p2 )r2 - 21 R2 p2 r2 +(R4 - r1 )2 r2 + r1 1 r2 + r1 1 2 (R4 - r2 ) 2 1 1 2 1 r1 r2 2 2 2 22 r2 (p3 p2 - p4 p1 )+ 1 (-(R2 - r1 )r2 p3 +2R2 p2 r1 - R2 r1 r2 + r1 r2 ) 4 2 1 r1 r2 2 2 r2 (p1 p3 + p2 p4 ) - 1 p4 (2R2 p3 r1 +(R2 - r1 )r2 ) 2 1 r1 r2

,

,

,

2p2 - 2(R2 - r1 )1 2p3 (R2 - r2 ) , {p1 ,r2 } = - , r2 1 2 2 - r2 (p1 p3 + p2 p4 ) - 1 p4 (2R2 p3 r1 - (R2 + r1 )r2 )
2 1 r1 r2

, ,

2 2 2 22 r2 (p3 p2 - p4 p1 )+ 1 (-(R2 + r1 )r2 p3 - 2R2 p2 r1 + R2 r1 r2 + r1 r2 ) 4 2 1 r1 r2

{p2 ,r1 } = -

2p4 (R2 - r2 ) 2p1 r - 1 r1 , {p2 ,r2 } = - , {p3 ,p4 } = 2 2 , r2 1 2 1 2p 2p {p3 ,r1 } = 4 , {p3 ,r2 } = - 4 , 1 2 2p 2p {p4 ,r1 } = - 3 , {p4 ,r2 } = 3 , {r1 ,r2 } = 0. 1 2

This Poisson bracket structure is degenerate, and its rank is four. The integral (3.3) and the obvious relation p2 + p2 = r1 r2 are Casimir functions for this structure. For the reduced system to b e 3 4 integrable, one more first integral is needed. To explore the reduced system numerically, we have used the Poincar´ surface-of-section technique. e The variables r2 , p3 , p4 can b e considered local coordinates on a three-dimensional manifold on which the integrals (1.4) and (3.3) are constant and the equation p2 + p2 = r1 r2 is fulfilled; p4 is the cross 3 4 variable. Two Poincar´ surface-of-section plots are given in Fig. ??. The chaotic b ehavior of solutions e proves that in the general case an additional first integral does not exist.

7. Conclusion
Very often equations of motion that have not b een derived within the framework of Lagrangian formalism (i.e., using the calculus of variations) are not Hamiltonian (even though the energy may b e a conserved quantity for these equations). For example, for equations of the nonholonomic mechanics there are dynamical obstacles preventing the existence of a Poisson bracket structure [3]. Since the equations of motion (1.1) are Hamiltonian, they have the generic features of Hamiltonian dynamical systems: for any value of parameters there are no attractors (e.g., strange attractors) in the phase space; at the same time, there are invariant KAM tori separated with stochastic layers. In this pap er,
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Fig. 8. Poincar´ surface-of-section plots for the system of a cylinder and two vortices: a) H = 7, F = 5, a = 9, e = 1, 1 = 2 = 8, R = 1, p4 = -5; b) H = 10, F = 60, a = 4, = 2, 1 = -1, 2 = 10, R = 1, p4 = 0.

the chaotic system of a cylinder and two vortices has b een touched briefly. It seems that it would b e interesting to explore some particular motions of this system (b oth regular and chaotic) in greater detail. Mention should b e made of the pap er [8] where a modification of the famous Bjerknes problem, the system of two dynamically interacting 2D cylinders, is considered. The equations of motion for this system are not integrable. This work was supp orted in part by Leading Scientific School of Russia Supp ort grant ¹½¿ º¾¼¼¿º½. no.

References
[1] V. I. Arnold. Mathematical methods in classical mechanics. Moscow: Nauka. 1991. (In Russian) [2] A. V. Borisov, I. S. Mamaev. Poisson structures and Lie algebras in the Hamiltonian mechanics. Izhevsk, RCD. 1999. (In Russian) [3] A. V. Borisov, I. S. Mamaev. (Strange attractors in rattleback dynamics) UFN. V. 173. 4. P. 407­418 . [4] B. P. Demidovich. Lectures on the mathematical theory of stability. Moscow: Nauka. 1967. (In Russian) [5] G. Kirchhoff. Vorlesungen ub er mathematische ¨ Physik. Mechanik, Leipzig. 1874. [6] H. Lamb. Hydrodynamics, Ed. 6-th. N. Y. Dover publ. 1945. [7] L. Prandtl. Foundamentals of Hydro and Aeromechanics. Dover Publ., Inc. New York. 1957. [8] S. M. Ramodanov. Motion of two circular cylinders in a p erfect fluid. In: Fundamental and applied problems in the theory of vortices, eds. A. V. Borisov, I. S. Mamaev and M. A. Sokolovskij. Moscow-Izhevsk: Institute of Computer Science. 2003. P. 327­335 . [9] P. G. Saffman. Vortex Dynamics. Camb. Univ. Press. 1992. [10] A. V. Borisov, I. S. Mamaev. An integrability of the problem on motion of cylinder and vortex in the ideal fluid. Reg. & Chaot. Dyn. 2003. V. 8. P. 163­166 . [11] P. A. M. Dirac. Generalizated Hamiltonian Dynamics. Canadian Journal of Math. 1950. V. 2. 2. P. 129­148 . (See also [Borisov A. V., Mamaev I. S. Dirac bracket in geometry and mechanics, in Dirac P. A. M. Lectures on theoretical physics. Moscow-Izhevsk: RCD, 2001, in Russian) [12] A. G. Greenhil l. Plane vortex motion. Quart. J. Pure 58. P. 10­27. Appl. Math. 1877/78. V. 15. [13] T. H. Havelock. The stability of motion of rectilinear vortices in ring formation. Phil. Mc. 1931, Ser. 7. 70. P. 617­633 . V. 11. [14] S. M. Ramodanov. Motion of a Circular Cylinder and a Vortex in an Ideal Fluid. Reg. & Chaot. Dyn. 2001. 1. P. 33­38. V. 6. [15] S. M. Ramodanov. Motion of a Circular Cylinder and N Point Vortices in a Perfect Fluid. Reg. & Chaot. 3. P. 291­298 . Dyn. 2002. V. 7. [16] B. N. Shashikanth, J. E. Marsden, J. W. Burdick, S. D. Kel ly. The Hamiltonian structure of a 2D rigid circular cylinder interacting dynamically with N point vortices. Phys. of Fluids. 2002. V. 14. P. 1214­1227 . [17] J. Synge. On the motion of three vortices. Can. J. Math. 1949. V. 1. P. 257­270 . [18] H. Vil lat. Lecons sur la theorie des tourbillions. Gauthier-Villars. 1930.

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