Dynamic interaction of point vortices and a two ...

JOURNAL OF MATHEMATICAL PHYSICS 48, 1 共2007兲

Dynamic interaction of point vortices and a two-dimensional cylinder Alexey V. Borisov,a兲 Ivan S. Mamaev,b兲 and Sergey M. Ramodanovc兲 Institute of Computer Science, Udmurt State University, 1, Universitetskaya Str., Izhevsk 426034, Russia 共Received 29 June 2006; accepted 30 November 2006兲

In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. We present the equations of motion and discuss their integrability. Simulations show that the system of an elliptic cylinder 共with nonzero eccentricity兲 and a single point vortex already exhibits chaotic features and the equations of motion are nonintegrable. We suggest a Hamiltonian form of the equations. The problem we study here, namely, the equations of motion, the Hamiltonian structure for the interacting system of a cylinder of arbitrary cross-section shape, with zero circulation around it, and N vortices, has been addressed by Shashikanth 关Regular Chaotic Dyn. 10, 1 共2005兲兴. We slightly generalize the work by Shashikanth by allowing for nonzero circulation around the cylinder and offer a different approach than that by Shashikanth by using classical complex variable theory. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2425100兴

I. INTRODUCTION

In this paper we consider the system of an arbitrary two-dimensional cylinder interacting with point vortices in a perfect fluid. The case of a circular cylinder was examined in Refs. 1–5. The paper3 covers the case of a cylinder of arbitrary cross-section shape. It is known from the phenomenological theory developed by Prandtl and Jukowski that when a body moves in a fluid, a thin boundary layer peels off the body thereby generating bound vortices. In their turn, these vortices exert a lifting force on the body which can be observed in aero- and hydrodynamical experiments. A good example is the swimming of fishes, a phenomenon in which bound vortices occur. In this connection, the problem we explore seems to be of considerable theoretical interest. First, we obtain a general form of the equations of motion for a rigid two-dimensional body interacting with point vortices. The circulation around the body is arbitrary. Then, we study the case of an elliptic body interacting with a single vortex in greater detail. The case of circular cylinder was shown to be integrable.5 If the ellipse’s eccentricity is different from zero, the system acquires chaotic features and the equations cease to be integrable. To illustrate the chaos we solve the equations numerically and plot Poincaré sections. We also present a Hamiltonian form of the equations of motion which facilitates the application of the highly developed perturbation theory. II. FLOW AROUND A MOVING CONTOUR

In this paper only two-dimensional hydrodynamic problems are discussed. Consider a twodimensional neutrally buoyant rigid body whose boundary, C, is not necessarily a circle. We assume that the region enclosed by C is simply connected and C itself is a Jordan curve. This guarantees that the exterior of C can be conformally mapped onto the exterior of a unit disk;

a兲

Electronic mail: [email protected] Electronic mail: [email protected] Electronic mail: [email protected]

b兲 c兲

0022-2488/2007/48共6兲/1/9/$23.00

48, 1-1

© 2007 American Institute of Physics

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FIG. 1. Definition of the coordinates of body and vortices.

moreover, C is mapped onto the disk’s boundary. Such a map is unique provided some conditions at infinity 共for details, see, for example, Ref. 6兲. The body is interacting dynamically with n point vortices in an unbounded volume of perfect fluid. The fluid is assumed to be at rest at infinity and on the boundary the impermeable condition is satisfied. Among the general works on the motion of point vortices in a fluid bounded by absolutely smooth walls, the study performed by Lin7 should be mentioned. He proved that the equations of motion for vortices in the absence of any walls and inside or outside a circular cylinder are Hamiltonian with the same bracket. Let Oxy be fixed in space laboratory frame of reference. We fix another orthogonal frame Oc␰␩ to the body 共Fig. 1兲. The position of a point relative to the laboratory frame and the body-fixed frame will be represented as z = x + iy and ␨ = ␰ + i␩, respectively. The location of the contour C relative to the laboratory frame is well defined by the coordinates z0 = x0 + iy 0 of the origin Oc and the angle of rotation ␪. The complex numbers z␣, ␣ = 1 , . . . , n, describe the positions of the point vortices. It is well known that the velocity of the fluid is given by a multivalued potential ␸共z兲 or a stream function ␸共z兲. The velocity at a point z reads v f 共z兲 = vxf + ivyf =

⳵␸ ⳵␸ ⳵␺ ⳵␺ +i = −i . ⳵y ⳵y ⳵x ⳵x

Obviously, by virtue of the Cauchy-Riemann equations one can express ␸ in terms of quadratures of ␺ 共and vice versa兲. Using the complex potential w共z兲 = ␸共z兲 + i␺共z兲, we can find the velocity as ¯v f = ⳵w / ⳵z 共the bar denotes the complex conjugate兲. Due to the point vortices the stream function satisfies the equation n

⌬␺共z兲 = 兺 ⌫␣␦共z − z␣兲, ␣

where z␣ and ⌫␣ are the coordinates and intensities of the vortices; ␦共z − z␣兲 = ␦共x − x␣兲␦共y − y ␣兲 is the Dirac delta function. The impermeable boundary conditions on C and the zero velocity at infinity 共see Refs. 6 or 8兲 imply that 兩␺兩C = vxy − v yx +

␻ 共共x − x0兲2 + 共y − y 0兲2兲, 2

␺共⬁兲 = 0.

Here vx = x˙0, vy = y˙ 0, and ␻ = ␪˙ are the linear and angular velocities of the body. The circulation around the body can be written as

冖

C

共v f ,dl兲 =

冖 冖 d␸ =

C

⳵␺ dl = ⌫c = const, C ⳵n

where dl is an element of the contour’s arc length. It is well known 共see, for example, Refs. 6 or 8兲 that the stream function can be represented in the form

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Point vortices

␺共z兲 = vx␺x共z兲 + vy␺y共z兲 + ␻␺␻共z兲 + ⌫*␺0共z兲 + 兺 ⌫␣␺␣v 共z兲.

共1兲

␣

Here all the terms 共except for ␺␣v , the stream function due to the ␣th vortex兲 are harmonic functions in the domain exterior to C, decay at infinity, and satisfy the following conditions: 兩␺x兩C = − y,

兩␺y兩C = x,

兩 ␺ ␻兩 C =

共x − x0兲2 + 共y − y 0兲2 , 2

兩␺0兩C = const.

For the circulation we can write

冖

d␸x =

C

冖

d␸ y =

C

冖

d␸␻ = 0,

C

冖

C

d␸0 = ⌫* = ⌫c + 兺 ⌫␣ .

Remark: The circulation ⌫* is introduced for convenience to represent the total circulation about the body minus the circulation induced by the vortices. Hereafter, the constant ⌫* will be referred to as pure circulation. The pure circulatory flow around the cylinder (the flow which remains when the cylinder’s velocity and the point vortices vanish) is governed by the stream function *␺0共z兲. All the functions ␺␣v can be expressed in terms of single Green’s function which satisfies the equation ⌬G共z,z0兲 = ␦共z − z0兲,兩

G共z,z0兲兩z苸C = const,

共2兲

therefore ␺␣共z兲 = G共z , z␣兲. We will show now that the other functions from Eq. 共1兲 can also be found in terms of quadratures of G共z , z0兲. First, let us note that the functions possess the following properties. v

1.

The function ␺␣v 共z兲 allows the following representation:

␺␣v 共z兲 = −

2.

1 log兩z − z␣兩 + ␺␣r 共z兲, 2␲

where ␺␣r 共z兲 is that part of ␺␣v which is regular at the point z = z␣, in other words ␺␣r 共z兲 is harmonic outside C and 兩␺␣r 兩C = 共1 / 2␲兲log兩z − z␣兩, ␺␣r 共⬁兲 = 0. Therefore, as z␣ → ⬁ we have ␺␣r 共z兲 → ␺0共z兲. The solution to the boundary value problem with Dirichlet conditions, that is, ⌬⌽共z兲 = 0, 兩⌽兩C = f共z兲, can be represented using Green’s function 共2兲 as follows:9 ⌽共z兲 =

冖

˜兲 f共z

C

⳵G ˜,z兲dl˜. 共z ⳵n

Therefore, we get

␺x共z␣兲 + i␺y共z␣兲 = i

冖

z

⳵␺␣v dl, ⳵n

␺␻ =

冖

C

兩z − z0兩2 ⳵␺␣v dl. 2 ⳵n

共3兲

Thus, to determine the flow around the body we need to know only the function G共z , z0兲. Given a conformal mapping of the region exterior to C onto the exterior of the unit disk z⬘ = F共z兲, one readily finds Green’s function to be G共z,z0兲 = Im

冉

冉

1 1 1 log共F共z兲 − F共z0兲兲 − log F共z兲 − 2␲i 2␲i F共z0兲

冊冊

Expanding the inverse mapping z = F−1共z⬘兲 into a Laurent series, one gets6

.

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Borisov, Mamaev, and Ramodanov

F−1共z⬘兲 = kz⬘ + z0 +

k1 k2 + + ... , z ⬘ z ⬘2

共4兲

where k is some positive real number. The point z0 = x0 + iy 0, often referred to as the contour’s conformal center, is fixed in the body and can be found from the equation z0 =

1 2␲i

冖

dz⬘ . C⬘ z ⬘ z

The contour’s conformal center coincides with the point of application of the Zhukovsky lifting force.6

III. EQUATIONS OF MOTION FOR THE CONTOUR

To obtain the equations, we start with calculating the force and moment acting on the body. According to Ref. 4, the force acting on the body in the presence of point vortices looks like f = f x + if y =

d dt

冖

iz

C

⳵␺ dl − i 兺 ⌫␣v␣ , ⳵n ␣

where v␣ = z˙ ␣ is the velocity of the point vortex with coordinate z␣ and ␺ is the stream function given by Eq. 共1兲. We see that the force depends linearly on the elementary stream functions from Eq. 共1兲. Let us see how each elementary function contributes to the force. 1.

As it was shown by Kirchhoff, the elementary functions that determine the fluid’s kinetic energy for ⌫* = 0, ⌫␣ = 0 produce the well known added-mass effect, d dt

冕

iz

冉

冊

⳵T0f d ⳵T0f ⳵ 共vx␺x + vy␺y + ␻␺␻兲dl = − +i . ⳵n ⳵v y dt ⳵vx

Here T0f =

1 2

冕

R2/C

兩共v2x + v2y 兲兩⌫*=0,⌫␣=0dx dy = −

1 2

冖

C

兩␸d␺兩⌫*=⌫␣=0

1 = 共a11v2x + a22v2y + 2a12vxvy + b␻2 + 2c1vx␻ + 2c2vy␻兲. 2

2.

3.

The coefficients aij, b, ci are the body’s added masses and moments, which generally depend on the angle of rotation ␪. The body is also subject to the lifting force caused by the purely circulation flow 共⌫* ⫽ 0, ⌫␣ = 0, vx = vy = ␻ = 0兲. By Sedov’s theorem,6 in this case the body is acted on by an additional Zhukovsky lifting force i⌫*共dz0 / dt兲, which is applied to the body’s conformal center. The contribution due to the point vortices is4 f ␣ = ⌫␣

d dt

冖

iz

⳵␺␣v dl + i⌫␣v␣ . ⳵n

The contour integral can be easily calculated using property 2. We have f␣ =

d ⌫␣共␺x共z␣兲 + i␺y共z␣兲 + iz␣兲. dt

Thus, the equation of motion for the body in the fixed frame of reference looks like

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Point vortices

m

冉

冉

⳵T0f ⳵T0f d 2z 0 d ⳵␺共z␣兲 ⳵␺共z␣兲 = − + 兺 ⌫␣ + iz␣ − i +i 2 dt ⳵v y ␣ ⳵v y dt ⳵vx ⳵vx

冊冊

+ i⌫*

dz0 . dt

共5兲

The moment about the point z0 = x0 + iy 0 that the fluid exerts on the body can be written as4 MO =

d dt

冋冕 冏 C

兩z − z0兩2 ⳵␺ 2 ⳵n

冏

⌫*=0,⌫␣=0

dl + 兺 ␣

冕

C

册

兩z − z␣兩2 ⳵␺v ¯ − ¯z␣兲f ␣ . dl − 兺 Im共z 2 ⳵n ␣

By the same reasoning 关Eq. 共3兲兴, the moment on the body about the conformal center is M=

冉

冉

⳵T f d ⳵␺ 1 2 − 0 + 兺 ⌫␣ + 兩z␣兩 dt ⳵␻ l␻ 2 ␣

冊冊

=

d ⳵Tb , dt ⳵␻

共6兲

where Tb is the kinetic energy of the body only. Let us obtain the equations of motion with respect to the body-fixed frame of reference Oc␰␩. The added-mass coefficients are constant and depend only on the body’s geometry. The stream function that governs the fluid velocity relative to the moving frame reads

冉

冊

˜␺共␨兲 = u共␺ 共␨兲 − ␩兲 + v共␺ 共␨兲 + ␰兲 + ␻ ␺ 共␨兲 + 1 兩␨兩2 + ⌫*␺ 共␨兲 + 兺 ⌫ ␺v 共␨兲, u ␻ 0 ␣ ␣ v 2

共7兲

where u and v are the components of the absolute velocity of the point Oc in the body-fixed frame. Therefore,

␺u = ␺x cos ␪ + ␺y sin ␪,

␺v = − ␺x sin ␪ + ␺y cos ␪ .

The fluid’s velocity becomes ˜v f = 共⳵˜␺ / ⳵␩ , −⳵˜␺ / ⳵␰兲. Let ˜␺ 共␨兲 = ˜␺共␨兲 + ⌫␣ log兩␨ − ␨ 兩, ␣ ␣ 2␲

␣ = 1, . . . ,n.

共8兲

Each ˜␺␣共␨兲 is analytic at the point coinciding with the ␣th vortex. Therefore, the velocity of the vortices in the body-fixed frame can be written as ␨˙ ␣ = ⳵˜␺␣ / ⳵␩ − 兩i共⳵˜␺␣ / ⳵␰兲兩␨=␨␣. Shifting the origin of the body-fixed frame to the conformal center and using vector notation, we arrive at the following theorem. Theorem: The equations of motion of the body and vortices in the body-fixed frame can be written in Kirchhoff’s form as follows:

冊

冉

冉

冊

d ⳵T ⳵˜␺␣共␨␣兲 ⳵˜␺␣共␨␣兲 ⳵T − 兺 ⌫␣ − 兺 ⌫␣ +⍀⫻ = ⌫*k ⫻ V, ⳵V ⳵V dt ⳵V ⳵V

冉

冊

冉

冊

d ⳵T ⳵˜␺␣共␨␣兲 ⳵˜␺␣共␨␣兲 ⳵T − 兺 ⌫␣ − 兺 ⌫␣ k+V⫻ = 0, ⳵␻ ⳵V dt ⳵␻ ⳵V

冏 冏

˜ ¯␨˙ = i ⳵␺␣共␨兲 ␣ ⳵␨

x˙ 0 = u cos ␪ − v sin ␪,

, ␨=␨␣

␣ = 1, . . . ,n,

y˙ 0 = u sin ␪ + v cos ␪,

␪˙ = ␻ ,

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V = 共u, v,0兲,

k = 共0,0,1兲,

⍀ = k␻ .

共9兲

Here x0 and y 0 are the coordinates of the conformal center in the fixed frame of reference, ␪ is the angle of rotation that describes the position of the body-fixed frame relative to the laboratory frame, and T is the kinetic energy of the body+ fluid system in the absence of the vortices and circulation around the body 共⌫* = ⌫␣ = 0兲. Hereafter, we adopt the following convention: by the derivation of a real-valued function ˜␺␣共␨兲 with respect to a complex variable ␨ = ␰ + i␩ we mean ⳵˜␺␣共␨兲 / ⳵␨ = ⳵˜␺␣共␨兲 / ⳵␰ + i⳵˜␺␣共␨兲 / ⳵␩. Remark: In the general case of asymmetric body, the kinetic energy is a quadratic form whose matrix is not diagonal. If Oc is not at the conformal center, then additional terms linear in V, ␻, and proportional to ⌫* appear in the right-hand side of the first three equations [Eqs. (9)]. Remark: Particular cases of Eqs. (9) have been obtained in Refs. 1–4. The case of a circular cylinder has been considered in greater detail in Refs. 1, 2, and 4. In Ref. 5, it has been shown that the equations of motion for the system of a circular cylinder and a single vortex are integrable.

IV. POISSON STRUCTURE AND INTEGRALS OF MOTION

Equations 共9兲 can be represented in a “nearly” Lagrangian form. As a counterpart of the Lagrangian function, we take n

L=T−

兺 ⌫␣ ␣=1

冉

冉

n

−

冊冊

n

1 共␺u共␨␣兲 − ␩␣兲u + 共␺v共␨␣兲 + ␰␣兲v + ␺␻共␨␣兲 − 兩␨␣兩2 ␻ − ⌫* 兺 ⌫␣␺0共␨␣兲 2 ␣=1 n

兩F⬘共␨␣兲兩 兩F共␨␣兲 − F共␨␤兲兩 1 1 , ⌫␣2 log ⌫␣⌫␤ log 兺 兺 2 − 4␲ ␣=1 兩1 − 兩F共␨␣兲兩 兩 4␲ ␣⫽␤ 兩1 − F共␨␣兲F共␨␤兲兩

共10兲

where F共␨兲 maps conformally the domain exterior to C onto the exterior of the unit disk. Then the equations of motion take the form

冉 冊

d ⳵L ⳵L = ⌫*k ⫻ V, +⍀⫻ dt ⳵V ⳵V

⳵L ˙ , ⌫␣¯␨␣ = 2i ⳵␨␣

冉 冊

d ⳵L ⳵L = 0, k+V⫻ dt ⳵␻ ⳵V

␣ = 1, . . . ,n.

共11兲

Performing a Legendre transformation of the system 共11兲, we get

P=

⳵˜␺共␨␣兲 ⳵L ⳵T , = − 兺 ⌫␣ ⳵V ⳵V ⳵V ␣

M=

⳵˜␺共␨␣兲 ⳵L ⳵T = − 兺 ⌫␣ , ⳵␻ ⳵␻ ⳵␻ ␣

H = 共P,V兲 + M ␻ − 兩L兩V,␻→P,M ,

P = 共Pu, Pv,0兲.

共12兲

Next, we introduce the following notation: W = 共u , v , ␻兲 and ␲ = 共Pu , Pv , M兲. Assume that in the absence of circulation the kinetic energy is a homogeneous function of u, v, and ␻, that is, 1 T = 共AW,W兲. 2 The components of the matrix A involve the added masses and moments. The Hamiltonian 共12兲 takes the form

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n

兩F⬘共␨␣兲兩 1 1 H = 共A−1共␲ + b兲, ␲ + b兲 + ⌫* 兺 ⌫␣␺0共␨␣兲 + ⌫␣2 log 兺 2 4␲ ␣=1 兩1 − 兩F共␨␣兲兩2兩 ␣=1 n

+

兩F共␨␣兲 − F共␨␤兲兩 1 , ⌫␣⌫␤ log 兺 4 ␲ ␣⫽␤ 兩1 − F共␨␣兲F共␨␤兲兩

共13兲

where b = 共兺⌫␣共␺u共␨␣兲 − ␩␣兲 , 兺⌫␣共␺v共␨␣兲 + ␨␣兲 , 兺⌫␣共␺␻共␨␣兲 − 21 兩␨␣兩2兲兲. The equations of motion can now be written in Hamiltonian form. Theorem 1: The equations of motion of a two-dimensional body and point vortices are Hamiltonian and look like P˙ v = 兵Pv,H其,

P˙ u = 兵Pu,H其,

␰˙ i = 兵␰i,H其, x˙ 0 = 兵x0,H其,

˙ = 兵M,H其, M

␩˙ i = 兵␩i,H其, ␪˙ = 兵␪,H其,

y˙ 0 = 兵y 0,H其,

共14兲

where the Hamiltonian function is given by Eq. 共13兲 and the Poisson brackets are as follows: 兵Pu, Pv其 = − ⌫*, 兵␰i, ␩i其 = ⌫−1 i ,

兵M, Pu其 = − Pv,

兵x0, Pu其 = cos ␪,

兵y 0, Pu其 = sin ␪,

兵M, Pv其 = Pu ,

兵x0, Pv其 = − sin ␪ ,

兵y 0, Pv其 = cos ␪,

兵␪,M其 = 1.

共15兲

The rank of the Poisson structure 共15兲 is n + 3, meaning that the structure is nondegenerate. Since the equations of motion are invariant under rotations and translations in the plane, there are three linear in velocity integrals of motion, Px = Pu cos ␪ − Pv sin ␪ + ⌫*y 0,

Py = Pu sin ␪ + Pv cos ␪ − ⌫*x0 ,

M ⍀ = Pxy 0 − P yx0 − M .

共16兲

The Poisson brackets of these integrals are 兵Px, Py其 = ⌫*,

兵Px,M ⍀其 = Py,

兵Py,M ⍀其 = − Px .

共17兲

Using the integrals 共16兲, one can construct an integral F which is independent of x0, y 0, and ␪, for example, 1 F = 共P2u + P2v兲 − M = const. 2

共18兲

Remark: Historically, the following special cases of the system (14) and (15) have been considered: 1. ⌫i = 0 for all i but ⌫* ⫽ 0. This case was investigated already by Chaplygin.10 2. The case of a circular cylinder with 兺⌫i = 0, ⌫* = 0, and arbitrary i has been treated in Ref. 2. 3. The equations of motion of a circular cylinder for arbitrary 兺⌫i and ⌫* were obtained in Ref. 4. 4. The case of arbitrary 兺⌫i, ⌫* = 兺⌫i and the body of arbitrary shape has been covered in Ref. 3. V. INTERACTION OF AN ELLIPTIC CYLINDER WITH A SINGLE POINT VORTEX

To find the stream function ˜␺共␨兲, we map conformally 共␨⬘ = F共␨兲兲 the exterior of the ellipse onto the exterior of the circle of unit radius on the complex plane ␨⬘ = ␰⬘ + i␩⬘,

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FIG. 2. Poincaré section of the system in the 3D-space of Pu , Pv , M.

␨ = F−1共␨⬘兲 =

a+b a−b 1 ␨⬘ + , 2 2 ␨⬘

where a and b are the semiaxes of the ellipse. Using the classical circle theorem, we find the stream function 共7兲 to be

冉

冏

冊

* ˜␺共␨兲 = u共␺ 共␨兲 − ␩兲 + v共␺ + ␰兲 + ␻ ␺ 共␨兲 − 1 兩␨兩2 − ⌫ log兩F共␨兲兩 + ⌫1 log F共␨兲 − 1 u ␻ v 2 2␲ 2␲ F共␨1兲

−

⌫1 log兩F共␨兲 − F共␨1兲兩. 2␲

冏 共19兲

Here ␨1 = ␰1 + i␩1 denotes the position of the vortex and

␺u = − b

␩⬘ , ␰ ⬘ + ␩ ⬘2 2

␺v = − a

␰⬘ , ␰ ⬘ + ␩ ⬘2 2

␺␻ =

b 2 − ␻ 2 ␰ ⬘2 − ␩ ⬘2 . 4 共 ␰ ⬘2 + ␩ ⬘2兲 2

Substituting these expressions into Eqs. 共13兲 and 共14兲 yields a closed system of equations in the components of the generalized momentum P = 共Pu , Pv , M兲 and the coordinates of the vortex 共␰1 , ␩1兲. The system admits two integrals of motion: the energy and F 关Eq. 共18兲兴. On a level surface of the integral 共18兲, one gets a Hamiltonian system with two degrees of freedom. A three-dimensional 共3D兲 view of the Poincaré section in the space of Pu, Pv, and M and its two-dimensional 共2D兲 projections are shown in Figs. 2 and 3.

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Point vortices

J. Math. Phys. 48, 1 共2007兲

FIG. 3. Projections of two different branches of the Poincaré map on the plane of Pu , Pv.

One can see that for a / b = 21 there are regions of chaos which indicates the nonintegrability of the system 共in contrast to the circular case a = b, which is integrable兲.5 It seems to be interesting to explore how the integrability and chaotic behavior of the system vary with the value of a / b.

ACKNOWLEDGMENTS

The authors 共A.V.B., I.S.M., and S.M.R.兲 acknowledge support from RFBR 共Grant No. 0501-01058兲, ERG “Regular and chaotic hydrodynamics,” and the program “State Support for Leading Scientific Schools” 共NSh-1312.2006.1兲. Two of the autors 共A.V.B. and I.S.M.兲 also acknowledge the support from INTAS project 共04-80-7297兲 and RFBR 共Grant No. 04-05-64367兲. A. V. Borisov, I. S. Mamaev, and S. M. Ramodanov, Discrete Contin. Dyn. Syst., Ser. B 5, 35 共2005兲. B. N. Shashikanth, J. E. Marsden, J. W. Burdick, and S. D. Kelly, Phys. Fluids 14, 1214 共2002兲. 3 B. N. Shashikanth, Regular Chaotic Dyn. 10, 1 共2005兲. 4 S. M. Ramodanov, Regular Chaotic Dyn. 7, 291 共2002兲. 5 A. V. Borisov and I. S. Mamaev, Regular Chaotic Dyn. 8, 163 共2003兲. 6 N. E. Cochin, I. L. Kibel, and N. V. Roze, Theoretical hydrodynamics 共Fizmatgiz, Moscow, 1955兲, 共in Russian兲 p. 394. 7 C. C. Lin, Proc. Natl. Acad. Sci. U.S.A. 27, 570 共1941兲. 8 L. M. Milne-Thomson, Theoretical Hydrodynamics, 5th ed. 共Dover, New York, 1996兲. 9 R. Kurant and E. Gilbert, Methods of Mathematical Physics 共GTTI, Moscow, 1945兲 共in Russian兲. 10 S. A. Chaplygin, Izv. Akad. Nauk SSSR, Otd. Khim Nauk 3, 133 共1933兲 共in Russian兲. 1 2