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ISSN 1560-3547, Regular and Chaotic Dynamics, 2010, Vol. 15, Nos. 4­5, pp. 440­461. c Pleiades Publishing, Ltd., 2010.

Special Issue: Valery Vasilievich Kozlov­60

Coupled Motion of a Rigid Bo dy and Point Vortices on a Two-Dimensional Spherical Surface
A. V. Borisov* , I. S. Mamaev** , and S. M. Ramodanov***
Institute of Computer Science Universitetskaya 1, Izhevsk 426034 Russia
Received June 6, 2009; accepted Septemb er 4, 2010

Abstract--The paper is concerned with a class of problems which involves the dynamical interaction of a rigid bo dy with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydro dynamics is further developed. The problem of motion of a dynamically symmetric circular bo dy interacting with a single vortex is shown to be integrable. Mass vortices on S 2 are intro duced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author's previous research on interaction of rigid bo dies and point vortices in a plane. MSC2000 numbers: 76B47, 70Exx, 70Hxx DOI: 10.1134/S1560354710040040 Key words: hydro dynamics on a sphere, coupled bo dy-vortex system, mass vortex, equations of motion, integrability

Contents
INTRODUCTION 1 442

FLUID DYNAMICS ON TWO-DIMENSIONAL SURFACES 443 1.1 The Dynamics of Ideal Fluid on Two-Dimensional Surfaces . . . . . . . . . . . . 443 1.2 Point Vortices on Two-Dimensional Surfaces . . . . . . . . . . . . . . . . . . . . 446 THE MOTION OF A CIRCULAR RIGID BODY WITH A CIRCULATION ON S 2 449 THE COUPLED MOTION OF A RIGID BODY WITH A CIRCULATION AND POINT VORTICES ON S 2 451 MASS VORTICES APPENDIX UNRESOLVED PROBLEMS ACKNOWLEDGEMENTS REFERENCES 454 455 459 459 459

2

3

4

* ** ***

E-mail: borisov@ics.org.ru E-mail: mamaev@rcd.ru E-mail: ramodanov@mail.ru

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In this pap er we consider the problem of motion of a circular b ody interacting dynamically with p oint vortices on the surface of a two-dimensional sphere. In its spirit this problem traces back to the classical texts of Beltrami, Bogomolov, Gromeka, Lamb, and Zermelo, which will b e discussed in detail in the main text. Since this journal issue is dedicated to the 60th birthday of our teacher Valery Vasil'evich Kozlov, we would like to give him credit for his constant encouragement and emphasize his role in directing our interest to this field. First of all let us take a quick look at V. V. Kozlov's achievements in the vortex hydrodynamics. On the one hand we should mention his works concerned with the general vortex theory [49] in which Valery Vasil'evich (V. V. ) attempts to look through the prism of the general concepts of this theory at various problems from optic, thermodynamics, quantum mechanics and other fields. In particular, V. V. has elab orated a new approach to integration of Hamiltonian system (an extension of the Hamilton­Jacobi method). On the other hand many of his recent publications address the development of the "weak limit" theory and its application to the study of asymptotic b ehavior and statistical prop erties of large clusters of p oint vortices. In addition to these two lines of investigation, V.V. Kozlov has a long-standing interest in the classical problem of motion of a rigid b ody in an ideal fluid. Among Kozlov's pap ers dealing with this topic we mention the following: study of integrability of the equations of motion [51], the falling motion of a heavy plate in an ideal fluid [45], motion of a heavy 2D rigid b ody with circulation around it [46], motion of a b ody in a resisting medium [47]. Starting with the "most trivial" model of inviscid ideal fluid and then introducing new effects (such as circulation, dissipation, and vorticity), V. V. obtained more and more realistic models for analytical treatment of motion of b odies in real medium. This naturally led him to the problem of dynamical interaction b etween p oint vortices and rigid b odies. It is known from the classical hydrodynamics that vortices shed from sharp edges of a b ody and then form rather curious vortical structures (e.g., Karman's wakes). Nevertheless, in spite of the ample literature on the sub ject (dealing for the most part with numerical analysis of various empirical models), until quiet recently there have b een no rigorous analytical expressions for the hydrodynamic reaction that p oint vortices exert on the b ody. It is also well known that the formation of new vortices (the very process of shedding) cannot b e explained within the realm of the idealfluid model. At the same time, this model provides a fairly good description of interaction b etween the b ody and the already shed vortices. So we hop e that author's strict analytical results in this direction are b oth of theoretical interest and are useful for verification of more complicated models. The use of p oint vortices in obtaining a more realistic model of interaction b etween a b ody and the ambient medium was illustrated by V.V. by the following p opular example. Consider a moving car with a small flag on the hood. The flag is made of a rough, almost inflexible material and can b e thought of as a stiff rectangular piece of cardb oard which can rotate ab out the vertical axis. When the car moves at a low sp eed, the flag is in equilibrium: its plane is aligned with car's velocity. As soon as the sp eed exceeds a certain critical value, this equilibrium loses stability: the flag starts to oscillate ab out the vertical. This phenomenon, the birth of a limit cycle from an unstable equilibrium, is referred to as the Hopf bifurcation. Kozlov tried to obtain this limit cycle analytically by introducing circulation and dissipative effects (using Rayleigh's dissipation function), but all in vain: the b ehavior of the flag changed (new equilibriums emerged), but oscillations never occurred. In this connection, V. V. suggested that, despite everything, the existence of the oscillation regime can b e proved analytically if the shedding of vortices from the flag's rear edge is taken into consideration. Thus, the oscillation is due to the interaction b etween the flag and the p oint vortices in the incident flow. From the preceding it is now clear why V. V. b ecame interested in the problem of falling motion of a rigid b ody (e.g. a plate) in an ideal fluid in the presence of p oint vortices. This is precisely the problem V. V. p osed ab out 14 years ago to one of the authors (S. M. Ramodanov). In this general statement, this problem remains unsolved. (In this connection, a mention should b e made of a recent intriguing pap er [48], which addresses this problem: the shedding of vortices in a prescrib ed manner from b ody's edges is p ostulated, the effect on the falling b ody is calculated numerically and in particular it is shown that due to the vortices the broadside-on fall can b ecome unstable.) Adopting some simplifications (according to which there is just one vortex, the b ody is a circular cylinder and there is no gravity), S. M. Ramodanov derived the equations of
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motion for the "b ody+vortices" system and partly investigated its b ehavior. Up on insp ection of numerically plotted orbits, Ramodanov came to the conclusion that the system of a circular cylinder interacting with a single p oint vortex is integrable, yet he failed to prove that. In its turn Kozlov asked for assistance from A. V. Borisov and I. S. Mamaev. It should b e noted that exploration of integrability in vortex dynamics is rather sp ecific and requires a sound knowledge of the theory of Poisson structures, Lie algebras, and modern methods of reduction of dynamical system. With this technique the determination of an additional first integral (of course, for integrable systems) is almost straightforward. Here is how A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, owing to V. V. Kozlov, b ecame a team and soon proved the predicted result. Since then we have worked together and solved a numb er of interesting problems b eing inspired by V. V.: motion of a b ody with circulation [20], interaction of several rigid b odies in an ideal fluid [22, 50], etc. In the limit of infinitely small b odies new concepts of massive vortices [22, 38] and dynamical advection [6] were introduced. The analysis of motion of two spheres in an ideal fluid [50] stimulated further investigation on non-linear reduction (the Poisson bracket of phase variables is not their linear combination). This pap er addresses a class of problems which involves the dynamical interaction of a rigid b ody with p oint vortices on the surface of a two-dimensional sphere. Our analysis is strongly based on XIX and XX century fundamental works on hydrodynamics in curved spaces and, at the same time, this pap er is a natural progression of the author's previous research on interaction of rigid b odies and p oint vortices in a plane, initiated by V. V. Kozlov. INTRODUCTION The investigation of flows of an ideal incompressible fluid with the velocities of the fluid particles parallel to a certain plane (so-called plane-parallel flows) is a classical, well-elab orated area of fluid dynamics. In contrast, the problem of fluid motion over an arbitrary two-dimensional surface app ears to b e p oorly studied. The earliest investigations of the motion of a curved fluid layer trace back to the second half of the 19th century; i.e., to electrodynamical studies by Beltrami, Boltzmann, Kirchhoff, Umov, etc. (parallels b etween electrodynamics and fluid mechanics were discussed in detail, e.g., even by Poincar´ [36]); however, these studies are restricted to the case e where a single-valued p otential exists, i.e., the case of irrotational flows. The first systematic study of vortical fluid motion on two-dimensional surfaces was done by the Russian mechanician I. S. Gromeka [7]. He p ostulated the very concept of p oint vortices (invoking hydrodynamic considerations), derived the equations of their motion (restricting himself, however, to the cases of a sphere and a circular cylinder), and analyzed, in particular, the problem of motion of a p oint vortex in b ounded regions on a sphere [7]. Some his results were indep endently obtained by modern researchers (see, e.g., Kidambi and Newton [30] and Crowdy [27]). The famous German scientist E. Zermelo (widely known for his fundamental achievements in the set theory and works on statistical mechanics) undertook the most complete and systematic investigation of two-dimensional fluid mechanics. Studies in vortical fluid mechanics were the sub ject of his dissertation, which consists of two parts [43, 44] (only the first of which has b een published). A detailed discussion of Zermelo's dissertation in a scientific­historical context was given by Borisov et al. [3]. Gromeka's studies were principally aimed at obtaining the equations of motion of p oint vortices over a spherical surface. As for Zermelo, his investigation is a fundamental construction of twodimensional fluid mechanics; he considers the flow of an ideal fluid over an arbitrary surface and proves analogs of the basic theorems of the classical fluid mechanics on a plane, (the Bernoulli integral, the Helmholtz theorem, an the conservation of energy). Only up on a comprehensive development of general theory [43, § 1], Zermelo concentrates on the problem of motion of p oint vortices [43, § 2], [44]. His study [44] (which is scientifically highly valuable but exists only as a hand-written manuscript; see [3]) presents a detailed investigation of the problem of motion of three p oint vortices on a sphere; this problem is reduced to quadratures, and a general view of motion is formed using the theory of elliptic functions (which was evolved in detail at Zermelo's time but has b een largely forgotten by now). Zermelo also obtained for the first time a Hamiltonian form of the equations of motion of p oint vortices. Similar equations were derived in the 1970s by V.A. Bogomolov [1]. In the modern
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literature, Bogomolov's studies [1, 2] are cited as historically the first systematic investigations of the dynamics of p oint vortices on a sphere; however (see [3]), as we can see, these problems were rigorously analyzed ab out a century earlier by Zermelo and Gromeka. We also note that many other results presented in Zermelo's dissertation were indep endently obtained and develop ed in modern studies (e.g., [16, 24, 31]). An alternative hydrodynamic model of the motion of p oint vortices on a sphere is prop osed in [15]. An attempt of finding the equations of motion of p oint vortices over a surface of revolution was made by Hally [29], while Boatto and Koiller consider in their preprint [14] the general case of motion on surfaces. The results obtained in [14, 29] are largely tentative. Here, we briefly describ e the ideas of and methods used by Gromeka and Zermelo [7, 43, 44], whose approaches to the problem of motion of ideal fluid over a surface differ in their ideological backgrounds. We will also develop here these approaches considering two model problems: (1) the motion of a circular rigid b ody (spherical segment) over a spherical surface with the presence of circulation and (2) the motion of a circular rigid b ody interacting with p oint vortices. For a plane, various forms of the equations of motion that describ e a planar interaction b etween a rigid circular cylinder and vortices in an ideal fluid were recently (nearly simultaneously) obtained [38­40]. A Hamiltonian form of these equations of motion with a nontrivial Poissonian structure was found and the integrability of the equations of motion of a circular cylinder interacting with a p oint vortex was noted [21, 23]. A different Hamiltonian structure of the equations of motion was also found [40]. The relationship b etween these two Poissonian structures is investigated in a recent review article [42]. 1. FLUID DYNAMICS ON TWO-DIMENSIONAL SURFACES 1.1. The Dynamics of Ideal Fluid on Two-Dimensional Surfaces Equations of motion in an Eulerian form. Let S b e a two-dimensional surface with a metric ds2 = Ed 2 + Gd 2 . The motion of a p oint with a mass m over S under the action of force with a potential mV is governed by the regular Lagrangian equations d dt where L =
m 2

L

=

L ,

d dt

L

=

L ,

(1.1)

(E 2 + G 2 ) - mV .

Remark 1. As a rule, we assume here that S is a two-dimensional surface immersed in threedimensional Eucledian space, although this constraint is not necessary with our approach. It is known (and also noted by Zermelo in his study) that a transition from the equations of motion of discrete masses (1.1) to the equations of motion of ideal fluid requires ­ replacing the mass m with the surface density (, ) and ­ adding the pressure gradient p(, ) to the right-hand side of the equations. Thus, we finally obtain (1/ E ) (1/ G) 1 p V d 2 2 u E +u E +v G =- - , dt (1/ E ) (1/ G) 1 p V d 2 2 v G +u E +v G =- - , (1.2) dt where u = E and v = G are the comp onents of the particle velocity in the directions of the coordinate lines and , and the total derivative with resp ect to time has the form d u v dt = t + E + G , V b eing the volumetric p otential of the external forces.
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To obtain a closed system, we have to complement equations (1.2) with the continuity equation (u G) (v E ) + + = 0. (1.3) EG t From here on, we will everywhere assume that the barotropy condition is satisfied, viz., the pressure p and density are related by the conditions that dp is again the differential of a certain function P , i.e., P 1 p = , 1 p P = .

Therefore, in the case of b orotropy, the differential of the function = P + V app ears in the right-hand side of equation (1.2). Remark 2. The barotropy condition is necessarily satisfied for incompressible fluid, = const. Equations of motion in the Gromeka­Lamb form. The Helmholtz theorem. We rewrite equations (1.2) in a somewhat different form to obtain an analog of the Cauchy­Lagrange integral, from which, in turn, we will b e able to express the pressure of the fluid. We introduce the vorticity of the fluid motion as 1 vG- uE (1.4) = EG and use (1.2) to obtain equations analogous to those known in classical fluid mechanics as the Gromeka­Lamb equations, 1 u2 + v 2 u = v - + , t 2 E 1 u2 + v 2 v = - u - + . t 2 G

(1.5)

Equations (1.5) and the continuity equation (1.3) can b e used to easily deduce a statement referred to by Zermelo as the Helmholtz theorem. Theorem 1. For a barotropic flow of an eal, incompressible fluid in a potential field of mass id forces, the circulation C = u Ed + v Gd along a closed curve C consisting of the same
C

fluid particles remains invariable in time. This theorem (for a planar case) is more widely known in the literature [34] as the Thomson theorem. The generally known Helmholtz theorem on the vorticity transfer by the flow follows from the former as a simple consequence. Incompressible fluid. The stream function. We now dwell on the case of a incompressible, p homogeneous fluid ( = const). Assume that there are no mass forces V = 0 and, therefore, = . By analogy with the planar case, we introduce a stream function (, ). We fix a p oint O on S ; then we can set at p oint A equal to the fluid flux through the curve connecting O and A: (A) =
OA

vn ds,

where vn is the velocity comp onent normal to the curve OA. In view of the incompressibility of the fluid, this flux has the same value for all homotopic curves connecting O and A. As on the plane, the velocity of fluid particles is equal to the skewed gradient of the stream function: (u, v ) = J, where J is the op erator of rotation by 90 degrees; alternatively, in a coordinate notation, 1 , u= G 1 v = - . E
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Therefore, the vorticity (1.3) and stream function are related by the equation = -, where = 1 EG G + E E G (1.8) (1.7)

is the Laplace­Beltrami op erator on S . We can use (1.3), (1.5), and (1.7) to show [43] that the vorticity satisfies the partial differential equation 1 = t EG - , (1.9)

an analog of the Helmholtz equation for the case of a plane. We also note that, in view of (1.7), equation (1.9) is a third-order partial differential equation that must be satisfied by the stream function of ideal incompressible fluid, (, , t). Potential flows. If there is no vorticity ( = 0), the flow has a velocity p otential , 1 , u= E 1 v= , G (1.10)

and the stream function and the p otential are harmonic conjugate functions on S , i.e., = = 0, = E , G =- G . E (1.11)

The Cauchy­Lagrange integral. Under the assumption of constant vorticity, an analog of the Cauchy­Lagrange integral can easily b e obtained from (1.5) (to the p oint, it was not noted by Zermelo). We find (at = 1) from (1.5) and (1.7) that u2 + v p+ 2
2

=
2

+

E , G t G . E t
t

u2 + v p+ 2

(1.12)

= -

Since we have = const for the vorticity, it can b e found from (1.7) that statement is valid:

= 0, i.e., the following

Prop osition 1. Let (, , t) be the stream function of barotropic flows on an arbitrary twodimensional surface of an incompressible fluid with = const; then the function is harmonic. t Consider a function (, , t) such that
(,,t) t t (,,t) t

is a function harmonic on S and conjugate to

(the fact that the function is harmonic in the case of constant vorticity was noted rep eatedly [34]). In view of the last two relationships (1.11), the right-hand sides of equations (1.12) are derivatives of the same function; therefore, the following theorem is valid: Theorem 2 (The Cauchy­Lagrange integral). For barotropic flows of an ideal, incompressible fluid with a constant vorticity = const, the relationship p+ holds. Note that the arbitrary function of time, f (t), can vanish if an appropriate gauge transformation + f (t) dt, which does not affect the velocity field, is applied.
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(1.13)


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1.2. Point Vortices on Two-Dimensional Surfaces Let us survey once again certain currently known results concerning surfacial motion. As already noted, the studies [7] and [43, 44] present ideologically different approaches to defining a p oint vortex on the sphere. The motion of p oint vortices over a sphere is discussed in detail in [16, 24, 25, 35]. A model of vortical motion over the sphere differing from the classical one was recently introduced in [15]. As we noted, the equations of motion of p oint vortices over surfaces of revolution were obtained by Hally [29] under the assumption that the total intensity of the vortices is zero and the solution of the Poisson equation (1.7) is known. A generalization of these particular results to the case of an arbitrary compact two-dimensional surface, interesting from the standp oint of differential geometry, is given in the recent study [14]. Note also [32], where the case of Lobachevsky's plane H 2 is considered along with the case of the sphere S 2 . In [14], the hyp othesis that the vortical dip ole moves over a two-dimensional surface in a geodesic [32] is substantiated. However, little is currently known even on the motion of a single p oint vortex (self-advance) on an arbitrary two-dimensional surface (p ossibly, a solution for a triaxial ellipsoid can b e obtained in elliptic quadratures). We consider various p ossible situations in greater detail. Point vortices on compact surfaces. A vortical flow on a compact surface S has the following particular prop erty distinguishing it from those on a plane: the total vorticity over the whole surface is zero. Indeed, let D b e a region in S . According to the Stokes theorem, the circulation of velocity along the b oundary D is equal to the double integral of the vorticity over D; on the other hand, it is equal to the double integral of over M/D , with the sign reversed; thus, dS = 0.
S

Therefore, vorticity cannot b e concentrated at one p oint of a compact surface. A p oint vortex can exist on S only with the presence of 1. either another vortex with the same but sign-reversed intensity [1, 7], 2. or a background vorticity [14, 43, 44]. Equation of motion of p oint vortices on a plane. Thus, prior to explicitly writing the equations of motion for a given surface S , it is necessary to solve the Poisson equation = , (1.14) where is a function constant on S (except for the p oints where the vortices are located) and is the Beltrami­Laplace op erator on S . Remark 3. More precisely, is the sum of Dirac functions and a certain constant. The motion of a p oint vortex s0 = (0 ,0 ) can b e describ ed in terms of a "desingularized" stream function known in Riemann geometry as the Robin function [26], 1 R (s0 ) , 0 = G 0 0 = - 1 R (s0 ) , E 0 R (s0 ) = lim
ss
0

(s,s0 ) -

ln d(s, s0 ) 2

,

(1.15)

where the stream function is a solution of the Poisson equation (1.14) with a singularity at the point s0 and d(s, s0 ) is the geodesic distance from s to s0 . It is noted in [14] that, in contrast to the case of a plane or sphere, a p oint vortex on a varying-curvature surface S can move under the action of the flow produced by this vortex itself. Note in this context that the explicit solution of equation (1.14) is known only for very few surfaces. In particular, the equations of motion of vortices on a triaxial ellipsoid have not yet b een found in an explicit form. Let us also mention the study [26] in this connection, which considers the motion of vortices over the surface of an ellipsoid of revolution obtained by slightly p erturbing a sphere, x2 + y 2 + x2 /(1 + ) = R2 . Approximate, to terms of order O(2 ), equations of motion of vortices are derived in [26]; the cases of two and (numerically) three vortices are also investigated.
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The Zermelo­Bogomolov p oint vortex on the sphere. On the unit sphere x2 + y 2 + z 2 = 1, we introduce spherical coordinates x = sin cos , y = sin sin , z = cos and seek a solution of equation (1.14), which now b ecomes 1 sin sin in the form = ( ) setting = const. 1 + = - , sin Then, we immediately find (1.16)

= -2 ln sin . 2

(1.17)

The circulation of the fluid along the parallel sp ecified by the angle is -2 (1 + cos ). Definition ([43]). We wil l say that a point vortex of intensity 2 is located at the point = 0 if the flow of the fluid is described by the stream function (1.17). Note that the condition of zero total vorticity is satisfied in this case. Remark. If this "naive" definition of the p oint vortex is adopted, it is not easy to prove that the vortices move together with the flow. The proof suggested by Zermelo [44] can hardly b e called transparent. Unfortunately, for unknown reasons, even some classical textb ooks assume that this statement is obvious and does not require any proof. It is more correct to define p oint vortices using a limiting process, i.e., first considering the dynamics of vortical sp ots of a small radius r and a constant vorticity and then requiring that r 0 and with the product r 2 remaining constant (see [7, 11, 14]). For the system of several vortices with spherical coordinates (i ,i ) and intensities i = 2i , the stream function is the sum of functions of the form (1.17). We assume that each vortex moves with a velocity induced by the other p oint vortices, thus representing the equations of motion in the form H H di di =- = , i sin i , i = 1,... ,n. (1.18) i sin i dt i dt i Here, the Hamiltonian function H is H=
i
i j ln sin

rij , 2

(1.19)

where rij = 2R2 (1 - cos i cos j - sin i sin j cos(i - j )) is the chord distance b etween the ith and j th vortices. As already noted, Zermelo's equations are identical in their form with Bogomolov's equations, which read as [1]: k = {H, k }, k = {H, k }, {k , cos k } = ik ; R 2 i

here, H differs from (1.19) only by an unimp ortant factor. Antip odal p oint vortices on the sphere. Except the solution (1.17), equation (1.16) also admits a solution of the form , = const, = · ln tg 2 which corresp onds to the presence of vortices of opp osite intensities at the p oles, with = 0 at all other p oints of the sphere. It turns out that such a vortex pair can b e considered a whole hydrodynamic ob ject, so-called antipodal vortex. The motion of antip odal p oint vortices is investigated in the recent study [15]; there, it is particularly shown that, if two vortices of opp osite intensities are initially located at the ends of the same diameter, they will preserve this arrangement in the course of motion. In addition, a classification of motions is given in [15] for the integrable problem of three antip odal vortices.
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Studies by I. S. Gromeka. In contrast to Zermelo, I. S. Gromeka comes to considering vortical motions on two-dimensional surfaces from studying flows of ideal fluid in three-dimensional space, which are parallel to a certain family of surfaces. We choose an orthogonal system of curvilinear coordinates 1 ,2 ,3 , in which the considered family of surfaces is sp ecified by the equation 3 = const and whose metric has the form
2 2 2 ds2 = h2 d1 + h2 d2 + h2 d3 , 1 2 3

e where hi are functions of the coordinates (the Lam´ coefficients). We also assume, following I. S. Gromeka, that h3 = const. (1.20) Remark. It is well known [28] that any two-dimensional surface can b e emb edded in a triorthogonal family; i.e., orthogonal curvilinear coordinates can b e sp ecified in the neighb orhood of any surface. At the same time, not any one-parameter family of surfaces can b e completed to forming a triorthogonal family (the corresp onding families are termed the Lam´ families). e Let u1 ,u2 ,u3 b e the comp onents of the fluid-particle velocity in the chosen coordinate system; then the condition of the flow b eing parallel to the family 3 = const can b e represented as u3 = 0, (h1 u1 ) = (h2 u2 ) = 0. 3 3 (1.21)

Under the ab ove assumption (1.20) on the family of surfaces, it can b e easily shown that, if the flow is initially parallel to the surfaces 3 = const, it will remain parallel to 3 = const at any later time. By substituting (1.21) into the equations sp ecifying the vorticity = rot v of the fluid, i =
j,k

ij

k

1 hk uk , hj hk j

(where ij k is the antisymmetric Levi-Civita tensor), we find 1 = 2 = 0, 3 = 1 h1 h2 (h2 u2 ) (h1 u1 ) - 1 2 h1 h2 h3 ui hi = (1 ,2 ,3 ). (1.22)

We use the continuity equation for incompressible fluid, = 0,
i

i

and determine the stream function for the flows parallel to the surfaces 3 = const according to the formulas 1 1 , u2 = . u1 = - h2 h3 2 h1 h3 1 Thus, to find the flows of an incompressible fluid parallel to the surface 3 = const with a given vorticity (1.22), it is necessary to solve the equation 1 h1 h2 h2 h1 1 + h1 h2 2 = = -h3 (1 ,2 ,3 ),

1

2

where is the Laplace­Beltrami op erator on S ; the variable 3 here app ears as a parameter. The simplest surfaces satisfying the ab ove-formulated conditions are ­ parallel planes; ­ coaxial circular cylinders; ­ concentric spheres. The curl vector of the velocity (1.22) is normal to these surfaces.
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Let us dwell on the case of the sphere. The Lam´ coefficients for the regular spherical coordinates e 3 are h = 1, h = r, h = r sin , i.e., the conditions (1.20) is satisfied. (r, , ) in R r Remark. I. S. Gromeka himself uses either a stereographic pro jection or the coordinates of a Mercator pro jection, for which h1 = h2 . However, this has obviously no effect on the result. Later, I. S. Gromeka assumes that = 0 everywhere but at the points where the vortic4es are located. This assumption certainly eliminates the background vorticity. In view of this, I. S. Gromeka concentrates on the problem of the motion of n point vortices of arbitrary intensity in a subregion of the sphere bounded by motionless, impermeable wal ls. In this case, the total vorticity over the whole sphere remains zero due to mirror images of the vortices with the opp osite vorticity outside the subregion considered. Thus, the limiting assumption that the total vorticity is zero, introduced by Gromeka himself, prevented him from obtaining the equations of motion of the vortices on the entire sphere (1.18) by shrinking the b oundary of the region to a p oint. 2. THE MOTION OF A CIRCULAR RIGID BODY WITH A CIRCULATION ON S 2 It is well known [33] that, in regular Eucledian space E 3 and on the plane R2 , the motion of a rigid b ody in an infinite volume of ideal incompressible fluid that flows irrotationally and rests at infinity can b e describ ed by a finite-dimensional Hamiltonian system of equations known as the Kirchhoff equations. Chaplygin [12] has shown that, in the case of a plane-parallel motion of a rigid b ody with the presence of a constant circulation around the b ody, terms linear in velocities app ear in the right-hand sides of the Kirchhoff equations. (Chaplygin has also proved the integrability of this system.) The study [20] contains a proof of the nonintegrability of the Chaplygin system in the case of a gravitational field present and a brief survey of the known results. Let us note equations similar to the Chaplygin equations for the particular case where the moving b ody is b ounded by a circular contour and circulation is present on the surface of a two-dimensional sphere. Equation of motion. Let the sphere b e sp ecified in a motionless Cartesian coordinate system by the equation x2 + y 2 + z 2 = R
2

and let the b ody b e a circular sp ot (spherical segment) of radius R1 (see Fig. 1); we assume that, outside the b ody, the sphere is covered with a homogeneous, incompressible fluid (and its density is equal to unity). Let the circulation of the fluid around the b ody b e = const; according to the Helmholtz theorem (see Section 1), it remains conserved in the course of motion. Let r0 b e a vector connecting the center of the sphere with the center of the b ody.
z r0 x
1

z

1

R1 O x y
Fig. 1.
1

y

We will obtain the equations of motion of the b ody in view of the fact that only the normal reaction of the constraint (on the side of the sphere) and the hydrodynamic pressure act on the b ody. We fix the moving coordinate system Ox1 y1 z1 to the b ody (see Fig. 1) assuming that the Oz1
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axis runs through its center. It can b e seen from the figure that the motion of the two-dimensional b ody over the sphere is equivalent to the motion of a normal (three-dimensional) b ody with a fixed p oint coinciding with the center of the sphere, O. We denote the moments of inertia of the equivalent three-dimensional rigid b ody as A, B , C and write the theorem on angular-momentum changes in the pro jections onto the moving axes (the Euler equations): A1 +(C - B )2 3 = M1 , B 2 +(A - C )3 1 = M2 , C 3 +(B - A)1 2 = M3 . (2.1) Here, i and Mi are the pro jections of the (three-dimensional) b ody's angular velocity and of the external-forces moment onto the moving axes Ox1 y1 z1 . We assume the mass distribution in the b ody to b e arbitrary, so that, generally, A = B . Since the direction of the normal reaction acting on the b ody runs through the center of the sphere, its moment is zero; the moment of the pressure forces exerted by the fluid, M L , is calculated in App endix (Section 5); it is equal to M L = -ar0 â a0 - a= where is due of the We
2 R1

R+d v0 , 2
2 R1

(2.2)

,

d=

R-
2

,

v0 and a0 are the velocity and acceleration of the b ody's center, resp ectively. The first term to the effect of associated masses (as in the planar case, it is prop ortional to the acceleration b ody), and the second term is an analog of Zhukowski's lifting force. write the velocity v0 and acceleration a0 in terms of the angular velocity of the b ody as v0 = â r0 , a0 = â r0 + â ( â r0 ) R+d â r0 , 2 (2.3)

to obtain the equations of motion in the form I + ar0 â ( â r0 ) = I + ar0 â ( â r0 ) â - where I = diag(A, B , C ). In view of the fact that r0 = (0, 0,R) in the chosen moving coordinate system, we represent equation (2.4) in the form ~ = (~ + k) â , I I where ~ = diag(A + aR2 ,B + aR2 ,C ), k = 0, 0, I
R(R+d) 2

(2.4)

(2.5) .

A comparison of (2.5) with the equations of the equilibrated gyrostat without an external field [4] leads us to the conclusion that the following theorem is valid: Theorem. If an axisymmetric (two-dimensional) rigid body (with an arbitrary mass distribution) on a sphere is submerged in an ideal incompressible fluid with a nonzero circulation, the dynamics of this body is equivalent to the dynamics of a top in the Zhukowski­Volterra case provided that the gyrostatic moment is directed along the principal axis. Remark. As is known, the Zhukowski­Volterra system describ es free motion of a rigid b ody inside which there is a balanced, rotating rotor with a constant gyrostatic-moment vector k. Zhukowski [8] noted an analogy b etween this problem and the dynamics of a rigid b ody with cavities that are not singly connected (of a torus typ e) and are filled with a p otentially moving nonideal fluid. Our theorem reveals another p ossible analogy, with a Zhukowski­Volterra system, which is also hydrodynamic. The following natural generalization of this result app ears to b e valid; Conjecture. The equations describing the motion of an arbitrary body on a sphere with a nonzero circulation are equivalent to the Zhukowski­Volterra system with an arbitrary gyrostatic-moment vector.
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The first integrals. The system (2.5), as is known [4], admits two first integrals -- energy and squared momentum: 1 I F = (~ + k, ~ + k). I I H = ( , ~), 2 Therefore, this system is integrable; it is qualitatively analyzed in the b ook [4], where a detailed bibliography concerning various asp ects of the dynamics of the Zhukowski­Volterra system is also presented. Let us consider one very simple case in greater detail. The case of dynamical symmetry. In this case, A = B ; in particular, if the mass is uniformly distributed in the b ody, µ = (C - A)(, r0 )/R2 , A=m 4R3 - d 3R2 + d2 , 6(R - d) C = m(d +2R)(R - d).

The equations of motion (2.5) in this case also admit the linear integral 3 = const. (2.6)

To find the tra jectory of the motion, we utilize the fact that the momentum vector ~ + k is I constant in the motionless axes Oxy z ; we use the integral (2.6) to represent it in the form ~ + k = + µr0 , I (2.7) C - A - aR2 ¯ 3 . µ = d + R Since and µ are scalar constants, this form remains unchanged in the motionless axes. We take the cross product of (2.7) by r0 and, taking into account the relationship r0 = â r0 , find = A + aR2 , r0 = K â r0 , where K is a constant vector. Therefore, the b ody moves at a constant velocity v0 ab out the vector K in a circle of radius Rv0 R. 2 R4 + 2 v 2 µ 0 Thus, we note that the presence of a circulation prevents the body from moving in a geodesic: the b ody moves in a great circle only in the limit of v0 . 3. THE COUPLED MOTION OF A RIGID BODY WITH A CIRCULATION AND POINT VORTICES ON S 2 Equations of motion. Assume that, under the conditions of the ab ove-considered problem, not only the b ody but also a vortex of intensity moves over the surface of the sphere; as previously, the contour of the b ody is circular and the mass distribution is arbitrary. The p osition of the vortex is sp ecified by the p osition vector issuing from the center of the sphere, r1 . In this case, an additional term app ears in the expression (2.2) for the moment of the pressure forces of the fluid (see App endix for the derivation); it is prop ortional to the absolute velocity of the vortex itself, v1 , ~ and to the velocity of a certain p oint, v1 , so that M L = -ar0 â a0 - ~ r1 = C0 r0 + C1 r1 , R+d ~ v0 +R(v1 - v1 ), 2 Rd - (r0 , r1 ) R(R - d) , C1 = 2 . C0 = 2 R - (r0 , r1 ) R - (r0 , r1 )


(3.1)

i The geometrical meaning of this vector is clarified in Fig. 2, where r1 is the p osition vector of the p oint inversely symmetric to the vortex, r1 ; thus, in a stereographic pro jection, their images are inversely symmetric with resp ect to the pro jection of the b ody's b oundary.

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Fig. 2.
i ~ In the limit of R , the vector r1 changes into r1 and expression (3.1) b ecomes completely analogous to the expression for the force acting on the circular cylinder in the planar case [38]. In the motionless coordinate system, the velocity of the vortex is determined by the desingularized stream function (the Kirchhoff­Routh function) according to relationships (1.15). Therefore, in the moving system Ox1 y1 z1 fixed to the b ody, the evolution of the vortex p osition is describ ed by the equation

r1 = r1 â - where (see App endix) R =

1 R r1 â , R r1

(3.2)

d-R (r1 â v0 , r0 )+ ln (r0 , r1 ) - Rd R - (r0 , r1 ) 4 ¯ - ln (r0 , r1 ) - R2 , 4
2

(3.3)

and R is the vector whose comp onents are the derivatives of the function R with resp ect to the r1 corresp onding coordinates of the vector r1 . We also note that the circulation around the b ody ¯ can b e found if we place a p oint vortex in its center, with an intensity sp ecified by the formula ¯ =


2R . d+R

(3.4)

Equations (2.1) (in which it is necessary to set Mi = ML , where ML are defined according to i i (2.7)) in combination with (3.1) form the complete system of equations describing the motion of the b ody­vortex system in the moving axes Ox1 y1 z1 . It can easily b e shown that these equations can b e represented in a nearly Lagrangian (but not Lagrangian, as is typical of the vortical-dynamics problems) form. Prop osition. Equations of motion of a rigid body bounded by a circular contour and of a vortex on the surface of a two-dimensional sphere have the form L 1 L â , r1 = - r1 â , R r1 1 ~ I L = (, ~ )+(k, )+ R , 2 ¯ ~ ~ ln Rd - (r0 , r1 ) - ln R2 - (r0 , r1 ) , R = R - R( , r1 )+ d(, r0 )= - R( , r1 - r1 )+ 4 4 (3.5) 2 ,B + aR2 ,C ), k = (0, 0, R(R+d) ). where, as before, ~ = diag(A + aR I 2 d dt =
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453

Proof. According to (2.5) and (3.1), the equations of motion of a rigid b ody in the moving axes are ~ = (~ + k) â +R(v1 - v1 ), ~ I I L L ~ â = (~ + k) â - R(r1 - r1 ) â ; I (3.6) ~ ~ where v1 and v1 are the absolute velocities of the vortex and the p oint r1 . On the other hand, ~ = ~ - R(r1 - r1 ), I

we note that the velocities in the ~ relationships v1 = r1 + â r1 , v1 = (3.2) for the vortex can b e obtained. Furthermore, the first term in the induced only by the displacement of R =
(0) 2

moving axes are linked to the absolute velocities via the ~ ~ r1 + â r1 and arrive at equations (3.6). Similarly, equation stream function (3.2), which describ es the motion of the fluid the b ody, can b e represented as

d-R ~ (r1 â v0 , r0 ) = R(, r1 ) - d(, r0 ) R - (r0 , r1 )

and, as in the planar case [22], proves to b e in a remarkable relationship with the image of the ~ vortex, r1 , in terms of which the moment of the forces (3.1) exerted by the fluid on the b ody can b e expressed. The Hamiltonian form of the equations of motion and first integrals. We carry out the Legendre transformation of system (3.5) with resp ect to the angular velocities : M= L ~ ~ = I + k, ~ ~ k = k - R(r1 - r1 ),

(3.7) 2 ¯ 1 ~ , ~-1 (M - k)) - ln(Rd - (r0 , r1 )) + ln(R2 - (r0 , r1 )). ~ H = (M , ) - L = (M - k I 2 4 4 A direct verification demonstrates the validity of the following Theorem. The equations of motion of a body bounded by a circular contour and of a point vortex on the sphere in the moving axes Ox1 y1 z1 can be represented in the Hamiltonian form Mi = {Mi ,H }, x1 = {x1 ,H }, y1 = {y1 ,H }, z1 = {z1 ,H },

where the Hamiltonian is given by relationship (3.7), and the Poissonian structure has the form {Mi ,Mj } = -ij k Mk , {x1 ,y1 } = 1 z1 , R {y1 ,z1 } = 1 x1 , R {z1 ,x1 } = 1 y1 . R (3.8)

The Lie­Poisson bracket (3.8) corresp onds to the Lie algebra so(3) so(3), is degenerate, and has two Casimir functions (which are obviously first integrals of the equations of motion)
2 2 2 1 = M1 + M2 + M3 , 2 2 2 = x2 + y1 + z1 = R2 . 1

(3.9)

We restrict the system to the symplectic sheet sp ecified by relationships (3.9) to obtain a Hamiltonian system with two degrees of freedom. According to the Liouville theorem, its integrability requires another, additional (except the Hamiltonian) first integral. Dynamically symmetric case. Assume that A = B ; then the system is integrable due to the presence of the additional, Lagrangian-typ e integral [4] 3 = H 1 = M3 C M3 -


R(R + d) +R(~1 - z1 ) z 2

= const.

Corollary. The problem of the motion of a circular, dynamical ly symmetric body and one point vortex on a spherical surface is Liouvil le-integrable. A similar result for the plane was obtained in [5] and analyzed in detail in [22, 23, 38, 39]. Note that, as numerical exp eriments show, the integrability of the system fails at A = B , and chaotic motions originate.
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Remark. The statement on the integrability can also b e obtained in a simpler way. Rewriting the equation of motion of the rigid b ody (3.6) in the motionless coordinate system and integrating it yields ~ = R(r1 - r1 ) - µr0 + K , (3.10) where K is a vector fixed in absolute space with K 2 = 1 (see (3.9)) and the coefficients and µ are sp ecified in (2.7). We take the cross product of equation (3.10) by r0 , thus obtaining ~ r0 = Rr0 â (r1 - r1 ) - r0 â K . 1 R . r1 = - r1 â R r1 The system of six equations (3.11) and (3.12) for r0 and r1 has four first integrals (a) the Hamiltonian function (3.7) in which should b e expressed from (3.10);
2 (b) r0 = R2 ; 2 (c) r1 = R2 ;

(3.11)

The equation of motion of the vortex (3.2) in the motionless coordinate system has the form (3.12)

~ (d) (R(r1 - r1 ) - µr0 + K , r0 ) = const (the pro jection of the angular velocity onto the symmetry axis of the b ody is constant). The system also preserves a standard measure, b eing thus integrable, according to the theory of the last Jacobi multiplier. 4. MASS VORTICES Equations of motion. Assume that the b ody is dynamically symmetric and let the radius of the b ody approach zero, leaving its mass unchanged; we then obtain an ob ject that, as in the planar case [22], we will term a mass vortex. The same reasoning as in [22] shows that the dynamics of two mass vortices in the motionless axes Oxy z is describ ed by the equations 1 2 r2 â r1 4 R2 - (r1 , r2 1 2 r1 â r2 ¨ m2 R2 r2 â r2 = -2 Rr2 + 4 R2 - (r1 , r2 ¨ m1 R2 r1 â r1 = -1 Rr1 + ) ) , (4.1) .

The problem of the motion of two mass vortices is, in the general case, not integrable even on the plane [9, 14]. Therefore, system (4.1) also does not app ear to b e integrable. Nevertheless, it would b e interesting to qualitatively analyze equations (4.1). Hamiltonian form and first integrals. We define the new variables M = m r â r + and rewrite equation (4.1) as follows: U , m r = -r â M , M = -r â r (4.3) 1 2 2 ln R - (r1 , r2 ) . U =- 4R2 It can easily b e verified that these equations are Hamiltonian with the Poisson bracket (which is a Lie­Poisson bracket sp ecified by the algebra e(3) e(3)) {Mi ,Mj } = -ij k Mk , {Mi ,xj } = -ij k xk , = 1, 2, i,j,k = 1, 2, 3,
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r , R

= 1, 2,

(4.2)

(4.4)
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(here, the zero brackets are omitted), and with the Hamiltonian H= 1 11 (M1 , M1 )+ (M2 , M2 ) + U. 2 m1 m2
(2) K = (M , r ) = R -1

(4.5)

For the Poisson bracket (4.4), the constants of the Casimir functions should b e fixed as follows:
(1) K = (r , r ) = R2 ,

.

It is remarkable that the equation of motion (4.3), the Poissonian structure (4.4), and the Hamiltonian (4.5) coincide with those in the case of a conventional two-b ody problem on the (2) sphere [17]. The difference is in that, for interacting b odies, all the constants K = 0, while they are prop ortional to the circulation in the case of mass vortices. System (4.3) has a vector integral of the total momentum M = M1 + M2 = const. We also note that, in [17], results of numerical exp eriments are presented, which testify to the nonintegrability of the two-b ody problem on the sphere with a Newtonian and a Hookian p otential. The problem of n mass vortices. These equations can b e generalized without difficulties to the case of an arbitrary numb er of mass vortices, the Hamiltonian assuming the form H= 1 2
n =1

1 (M , M ) - m

n <

ln R2 - (r , r ) , 4R2

where r is the p osition vector of the corresp onding vortex and M is the momentum defimed according to formula (4.2). The Poisson structure is determined by the algebra e(3) and has the form
=1 n

{Mi ,Mj } = -ij k Mk ,

{Mi ,xj } = -ij k xk , (M , r ) = R
n -1

ij k = 1, 2, 3, = 1 ... n.

= 1 ... n,

the corresp onding Kirchhoff functions b eing fixed as follows: (r , r ) = R2 , The integral of the total momentum is M=
=1

,

M .

Remark 4. The equations of the mass vortices on the plane were indep endently obtained in [9, 22, 37], where physical and hydrodynamic problems were also noted in which this model can b e used. The mass vortices on the sphere can b e considered a real alternative to various known models of vortical motions in the Earth's atmosphere and oceans (cyclones, tornadoes, oceanic vortical flows, etc.). This model is physically more realistic due to the consideration of the vortical-column mass, whose presence is natural, since p owerful vortical features, such as tornadoes, suck in large foreign b odies in the course of their motion. Anyway, the development of applications and the exp erimental verification of a theoretical model are decisive for its applicability. As some other vortical models, we also note the motion of vorticity sources and vortical sp ots [18, 19]. APPENDIX Stream function. The fluid flow on the sphere in a region D outside the b ody b ounded by the contour C (Fig. 3) is completely determined by the stream function (, , t), which, as shown ab ove, must satisfy ­ the system of differential equations 1 = t EG
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-
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,

= -;

(A.1)


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Fig. 3.

­ the unp ermeability condition at the b oundary of the b ody, C : , (A.2) C where n and are the vectors normal and tangent to the b oundary C and v is the velocity vector of the corresp onding p oint of the b ody's b oundary; (v , n) C = ­ the condition that, in the absence of a vortex ( = 0), the circulation around the contour encircling the b ody is (v , ) dl = .
C

(A.3)

According to the sup erp osition principle, the sought-for stream function can b e represented in the form
(0) (0) (0) = vox x + voy y + voz z + c + v ,

where v0 = (vox ,voy ,voz ) is the velocity of the b ody's center and the terms c and v are prop ortional to and , resp ectively. The fluid flow induced only by the motion of the b ody with the velocity v0 can b e describ ed by the stream function d-R (0) (0) (0) (r â v0 , r0 ). (A.4) (0) (r ) = vox x + voy y + voz z = 2 R - (r , r0 ) The validity of the unp ermeability condition (A.2) can b e directly checked. The function (0) is harmonic (outside the b ody, (0) = 0) and, therefore, satisfies the condition (1.9). The corresp onding flow has zero circulation along the b ody's b oundary C . Remark 5. Formula (A.4) can b e empirically obtained as follows. It is known that the velocity field produced by a circular cylinder moving over the plane is the flow induced by a dip ole. Assume that the same result holds for the sphere. First consider a flow on the plane such that the velocities of all particles are the same. As is known, such a flow is induced by an infinitely distant dip ole. If we map this flow onto the sphere via the stereographic pro jection, simple rearrangements will lead us to formula (A.4). As in the planar case, the flow induced by the p oint vortex v can b e obtained by adding an inversely symmetric (in terms of the stereographic pro jection) vortex of the opp osite vorticity, so that i ln R2 - (r1 , r ) , v (r ) = - ln R2 - (r1 , r ) + 4 4
i r1 = R2 + d2 -

2d (r0 , r1 ) R

-1

(R2 - d2 )r1 +2(Rd - (r0 , r1 ))r0 ,
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i where r1 is the p osition vector of the p oint inversely symmetric to the vortex. The function v is harmonic outside the b ody everywhere (except for the p oint r1 ) and, therefore, satisfies equation (A.1). Let us discuss the function c , which describ es a purely circulative flow around the rigid b ody. In contrast to the case of the plane, the circulations along homotopic contours encircling the b ody on the sphere are generally different. As shown in Section 1.2, if a vortex of intensity is placed at the p oint = 0 on the sphere, the circulation along the parallel = 0 will b e (1 + cos 0 )/2. Therefore, to satisfy the condition (A.3), a p oint vortex should b e placed at the center of the b ody, ¯ and its intensity and the circulation should b e linked by the relationship (3.4), so that

c = -

2R ln R2 - (r0 , r ) . 4 R + d
¯

The function c thus defined satisfies the equation = 4R2 in the region D and, therefore, it satisfied equation (A.1). Thus, the stream function is completely sp ecified. We note in view of the following that, in the region D, it satisfies the equation ¯ - (r - r1 ), (A.5) = 4R2 where is the Laplace­Beltrami op erator (1.8), r1 is the p osition vector of the vortex, and (r - r1 ) is the Dirac delta. The Kirchhoff­Routh function. The absolute velocity of the vortex, v1 , is given by relationship (1.15), which can b e written in Cartesian coordinates as

1 (r )+ ln R2 - (r1 , r ) v1 = - r1 â R r 4 Up on simple rearrangements, we find that

r=r1

.

1 R (r1 ) , v1 = - r1 â R r1 d-R (r1 â v0 , r0 )+ ln(Rd - (r0 , r1 )) R (r1 ) = 2 R - (r0 , r1 ) 4 2R ln R2 - (r0 , r1 ) . - 4 R + d The planar analog of the function R (r1 ) is traditionally referred to in vortical dynamics as the Kirchhoff­Routh function (see, e.g., [10]). The moment of the fluid-pressure forces. To calculate the moment of the fluid-pressure forces M L exerted on the b oundary of the b ody C , we use the theorem of the kinetic-momentum changes. The variation in the kinetic momentum of the fluid, K L , that fills the region D (the exterior of the contour C ) is M K L = -M L , (A.6) where ML is the sought-for moment of the fluid-pressure forces exerted on the contour C . We set the fluid density equal to unity and represent the kinetic momentum of the fluid, K L , in the region D as KL =
D

r â v ds = R
D

ds,

where all the vectors are considered to b e expressed in local orthogonal coordinates , on the sphere according to the formulas 1 1 e - e , v= G E
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1 1 e + e , = E G
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L

In the Cartesian coordinates Oxy z , the comp onents of the vector K form
L Kx = R D

can b e represented in the (z, )ds.
D

(x, )ds,

L Ky = R D

(y, )ds,

L Kz = R

Note that these integrals diverge at the point r1 ; therefore, they should be understood in terms of the theory of generalized functions (as principal values). We use the Stokes formula and employ a vector notation to obtain KC = R
C

r

E d - r G

G d . E

(A.7)

Thus, in view of (A.6), the moment of the fluid-pressure forces acting on the boundary of the body is specified by the relationship ML = -KC + R
D

r ds

.

(A.8)

Remark. The moment ML could b e found by directly integrating the moment of the pressure forces along the b oundary of the b ody, ML =
C

r â pndl = R
C

p dl = R
C

pdr ,

where p is the fluid pressure expressed from the generalized Cauchy­Lagrange integral (1.13) (we use here the relationship r â n = R , which is valid for the vectors n and normal and tangent to the contour on the sphere). However, the expression for ML thus obtained proves to b e highly cumb ersome and to have no clear physical meaning. We calculate the double integral in the right-hand side of (A.8) using equation (A.5) and the r ds = - (R2 - d2 )r0 ; finally, we obtain relationship
D

R
D

r ds = -Rr1 +

d2 - R2 ¯ r0 . 4R

(A.9)

Now we calculate the curvilinear integral KC given by formula (A.7). In this formula, we sequentially substitute the function with (0) , v , and c to directly obtain KC
=
(0)

= ar0 â r0 , KC
=v

2 where a = R1 ,

~ = -Rr1 , ¯ d(R + d) r0 . 2R

KC

=c

=

~ The coefficient a is the added mass of the circular b ody of radius R1 . The vector r1 (see Fig. 2) is ~ defined in (3.1) (the heads of the vectors r1 , r1 , and r0 are in the same straight line). We substitute these expressions together with (A.9) into (A.8) to obtain the following formula for the moment: ¨ ~ M L = -ar0 â r0 +R(r1 - r1 ) -


R+d v0 . 2
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UNRESOLVED PROBLEMS To conclude, we formulate certain issues that would b e worth further investigating based on the results of this study. · Studying various cases of the motion of a rigid b ody and p oint vortices on the sphere. In particular, analyzing the problem of the motion of an arbitrary two-dimensional b ody and several vortices. In the general case, the equations of motion of such a system will likely b e not integrable. Nevertheless, it would b e interesting to obtain a general equation in a Hamiltonian form and analyze various particular solutions, e.g., stationary configurations [21, 41]. The most interesting problem is the description of the motion of N rigid b odies on the sphere with allowances for circulation around each b ody. A particular case of this problem is the interaction of several rigid b odies and p oint vortices. Regrettably, a general formalism of such a problem has not yet b een develop ed even for the plane. Thus, the hydrodynamics of several rigid b odies and p oint vortices still remains a challenging unresolved problem. · A more systematic study of various particular cases of the dynamics of mass vortices. The equations of motion of mass vortices on the sphere were presented here; they have already b een partially analyzed for planar cases [9, 22, 41]. This form of equations makes it p ossible to consider a numb er of interesting problems. For example, a very simple application of these equations to the description of dynamical advection, i.e., the motion of an insoluble, fine-grained admixture in the fluid flow, is considered in [6]. In our opinion, the equations obtained in [6] app ear to have a more applied character compared to the classical model of chaotic advection. In addition, some issues in the dynamics of mass vortices, whose analogs are constantly under consideration in celestial mechanics, remain unexplored, viz., the existence of stationary and static configurations, their classification and stability. Note that static configurations of mass vortices, even with the same intensity, are also p ossible on the sphere. Various problems related to the classification of symmetric static configurations and search for nonsymmetric ones arise in this context. These problems have not yet completely explored even for the dynamics of regular vortices. Results concerning the static and stationary configurations of classical vortices are systematically discussed in the survey [13]. · Investigation of vortical motions on surfaces of other typ es. In view of p ossible applications, of considerable interest is the investigation of vortical dynamics not only on the sphere but also on other compact surfaces that model, in one approximation or another, the Earth's surface. From a general theoretical standp oint, it would b e interesting to consider the dynamics of rigid b odies and vortices on a Lobachevsky plane; this problem is simpler than that of motion on compact surfaces, since it does not involve difficulties due to the ambiguity of the definition of the p oint vortex. ACKNOWLEDGEMENTS The work is partly supp orted by the CRDF (grant RUM1-2943-RO-09), the RFBR (grants No. 09-01-92504-IK, 09-01-12151, 09-01-00791), the grants RFBR (pro jects 09-01-12151, 09-0100791), Federal target program "Scientific and scientific-p edagogical p ersonnel of innovative Russia" (pro jects 2009-1.5-503-004-019). The work of Ramodanov was supp orted by the State Maintenance Programs for the Leading Scientific Schools of the Russian Federation (NSh-8784.2010.1). REFERENCES
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2010