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M athematical N otes, vol. 75, no. 1, 2004, pp. 19­22. Translated from Matematicheskie Zametki, vol. 75, no. 1, 2004, pp. 20­23. Original Russian Text Copyright c 2004 by A. V. Borisov, I. S. Mamaev.

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid
A. V. Borisov and I. S. Mamaev
Received July 4, 2002

Abstract--In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n = 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.
Key words: ideal fluid, motion of a circular cylinder in an ideal fluid, point vortices, Poisson structure, Poisson bracket, Casimir function.

1. THE EQUATIONS OF MOTION Consider the problem of the plane motion of a cylindrical rigid body and n point vortices with circulations i in an infinite volume of an ideal incompressible fluid, which is at rest at infinity. We assume that there are no external force fields, the surface of the cylinder is ideally smooth, and the flow around it is circulatory, i.e., the circulation along the closed contour enclosing the cylinder is equal to = 0. The equations of motion of such a system were obtained by Ramodanov [1] and also in [2]. In the latter paper, it was assumed that = 0. In what follows, we shall adhere to [1]. Consider two coordinate systems: a fixed one OX Y and the system Cxy connected with the center of the cylinder and in plane-parallel motion relative to the system Cxy (see Fig. 1). Let (v1 ,v2 ) = v be the pro jections of the velocity of the center of the cylinder on the axis Cxy , let (xi ,yi ) = ri be the pro jections of the radius vector joining the center of the cylinder to the ith vortex on the same axes, and let µ be the mass of the cylinder and R its radius. As was shown in [1], the equations for v and ri in inertial motion can be separated from the other equations and integrated independently. These equations are of the form ri = -v +grad i |
n r=r
i

,
n

µv1 = v2 -
i=1

i (y i - yi ) , ~

µv2 = -v1 +
i=1

i (xi - xi ) , ~

(1)

~~ where ~i = (xi , yi ) is the inverse image of the ith vortex conjugate to the contour of the cylinder r by the rule R2 R2 yi = 2 yi , ~ xi = 2 xi , ~ r r and the function i corresponds to the part of the flow potential without singularity at the ~ point r = ri . The flow potential outside the cylinder can be expressed as (r) = - R2 y (r , v) - arctan + 2 r x
0001-4346/2004/7512-0019
n


i=1

i

arctan

y - yi ~ x - xi ~

- arctan

y - yi x - xi

,
19

c 2004 Plenum Publishing Corporation


20

A. V. BORISOV, I. S. MAMAEV

Fig. 1

where = /(2 ) and i = i /(2 ). As is easy to verify, Eqs. (1) have an energy-type integral of the form H= 1 1 2 2 µ(v1 + v2 )+ 2 2 [2 ln(r2 - R2 ) - i ln r2 ]+ i i i
i

1 2

i j ln
i
R4 - 2R2 (ri , rj )+ r2 r2 ij (2) 2 |ri - rj |

(if we use the analogy with classical mechanics), although the origin of this integral is not quite clear, since Eqs. (1) are not deduced, as is customary, from the Lagrangian formalism, but inherit certain conservative properties of both the dynamics of a rigid body and the motion of an ideal fluid. Note that Eqs. (1) preserve the standard measure. 2. THE HAMILTONIAN STRUCTURE OF THE EQUATIONS OF MOTION The question of whether n the case = 0, i=1 i Eqs. (1) in the general case. nondegenerate. Its nonzero {v1 ,v2 } = system (1) is Hamiltonian was considered in [2] and certain results in = 0 were obtained there. Here we present the Poisson structure for It turns out that it depends nonlinearly on the phase variables and is basis Poisson brackets are as follows:
2 1 r4 - R2 (x2 - yi ) i i , µ r4 i 1 2R2 xi yi {v2 ,xi } = - , µ r4 i 1 {xi ,yi } = - . i

i r4 - R4 i - , µ2 µ2 r4 i 1 2R2 xi yi {v1 ,yi } = - , µ r4 i 2 1 r4 + R2 (x2 - yi ) i i {v2 ,yi } = , µ r4 i

{v1 ,xi } =

(3)

By Darboux's theorem, in the nondegenerate case there always exists an analytic transformation reducing (3) to the canonical form {pi ,qi } = ij {pi ,pj } = 0, {qi ,qj } = 0. However, the explicit form of this transformation is not needed in what follows. 3. INTEGRABILITY OF THE SYSTEM IN THE CASE OF ONE VORTEX Consider the case n = 1 , i.e., the joint motion of a vortex and a cylinder. We denote r1 = r = (x, y ) . It is easy to see that such a system is invariant with respect to rotation about the center of the cylinder. Hence there exists a certain integral F = µ v2 +2µ1 1 - R2 (xv2 - yv1 )+ r2
1



1

r2 +

R4 r2

- r2 ,
Vol. 75 No. 1 2004

(4)

MATHEMATICAL NOTES


MOTION OF A CYLINDER AND A VORTEX IN AN IDEAL FLUID

21

generating (by the structure (3)) the symmetry field vF = 2(v2 , -v1 ,y , -x). (5)

The integral (4) is a generalization of the moment integral in classical mechanics and allows us to integrate the system in quadratures. Using it, we can lower the order of the system down to one degree of freedom. This reduction is similar to Routh reduction. We present it in explicit form. For the variables of the system under consideration, one should choose integrals of the symmetry field vF (5) (see, for example, [3]), such as p1 = µ(xv1 + yv2 ) , p2 = µ(xv2 - yv1 ) ,
2 2 r1 = µ2 (v1 + v2 ) ,

r2 = x2 + y 2 .

(6)

The Poisson brackets for these variables are as follows: {p1 ,p2 } = ( - 1 )r2 + {r1 ,r2 } = 4p
1

1+

R2 , r2

12 (p +(p2 - 1 R2 )2 ) , 1 1

{p1 ,r1 } = 2( - 1 )p2 - 2

p2 + p2 p2 - p2 (p2 - 1 R2 ) 1 2 +2R 1 , 2 r2 r2 2p2 - 1 R2 2 , 2 r2

2 {p1 ,r2 } = 2r2 + (p2 - 1 R2 ) , 1 {p2 ,r1 } = -2( - 1 )p1 +2R2 p {p2 ,r2 } = -2 p1 . 1
1

(7)

The system in question is degenerate, its rank is equal to two, and hence this system has one degree of freedom. To derive this system explicitly, it is necessary to eliminate two variables from the collection (r1 ,r2 ,p1 ,p2 ) by using the integral (4), which is the Casimir function of the structure (7), and using the obvious geometric relation p2 + p2 - r 1 r2 = 0 , 1 2 which is a consequence of (6). It is of interest to carry out a qualitative analysis of this system that can yield some conclusions about the motion of the complete system (1). 4. THE CASE OF n VORTICES There exists a generalization of the integral (4) in the general case. It is of the form F = µv2 + +2
i

i=1

i

2µ 1 -

R2 R4 (xi v2 - yi v1 )+(i - )r2 + i 2 i r2 ri i R2 r2 i 1- R2 . r2 j (8)

i j (ri , rj ) 1 -

Using it, we can also lower the order of the system by one degree of freedom. For the resulting system to be integrable, it is necessary to know n - 1 involutory integrals. Apparently, they do not exist in the general case, and the system of two or more vortices interacting with the cylinder
MATHEMATICAL NOTES Vol. 75 No. 1 2004


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A. V. BORISOV, I. S. MAMAEV

is no longer integrable. However, this has special cases of integrability arising under the constants of the integrals (2) and (8). It is also of interest to generalize the the problem of the interaction in a fluid naturally, in the plane setting.

not been proved so far, especially since there may exist additional constraints on the system parameters and on Poisson structure (3) and the integrals (2) and (8) to of two (or several) rigid bodies with given circulations,

ACKNOWLEDGMENTS The authors wish to express their thanks to V. V. Kozlov and S. M. Ramodanov for useful remarks. REFERENCES
1. S. M. Ramodanov, "Motion of a Circular Dyn., 6 (2001), no. 1, 33­38. 2. B. N. Shashikanth, J. E. Marsden, and J. of a 2D rigid circular cylinder interacting (2002), 1214­1227. 3. A. V. Borisov and I. S. Mamaev, Poisson Russian], RKhD, Izhevsk, 1999. Cylinder and a Vortex in an Ideal Fluid," Reg. & Chaot. W. Burdick, and S. D. Kelly, "The Hamiltonian structure dynamically with N point vortices," Phys. of Fluids, 14 Structures and Lie Algebras in Hamiltonian Mechanics [in

(A. V. Borisov) Institute of Computer Studies, Izhevsk E-mail : borisov@rcd.ru (I. S. Mamaev) Udmurt State University E-mail : mamaev@rcd.ru

MATHEMATICAL NOTES

Vol. 75

No. 1

2004